First-Order Singular and Discontinuous Differential Equations

  • DanielC Biles1 and

    Affiliated with

    • Rodrigo López Pouso2Email author

      Affiliated with

      Boundary Value Problems20092009:507671

      DOI: 10.1155/2009/507671

      Received: 10 March 2009

      Accepted: 4 May 2009

      Published: 9 June 2009

      Abstract

      We use subfunctions and superfunctions to derive sufficient conditions for the existence of extremal solutions to initial value problems for ordinary differential equations with discontinuous and singular nonlinearities.

      1. Introduction

      Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq1_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq2_HTML.gif be fixed and let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq3_HTML.gif be a given mapping. We are concerned with the existence of solutions of the initial value problem
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_Equ1_HTML.gif
      (1.1)

      It is well-known that Peano's theorem ensures the existence of local continuously differentiable solutions of (1.1) in case http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq4_HTML.gif is continuous. Despite its fundamental importance, it is probably true that Peano's proof of his theorem is even more important than the result itself, which nowadays we know can be deduced quickly from standard fixed point theorems (see [1, Theorem http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq5_HTML.gif ] for a proof based on the Schauder's theorem). The reason for believing this is that Peano's original approach to the problem in [2] consisted in obtaining the greatest solution as the pointwise infimum of strict upper solutions. Subsequently this idea was improved by Perron in [3], who also adapted it to study the Laplace equation by means of what we call today Perron's method. For a more recent and important revisitation of the method we mention the work by Goodman [4] on (1.1) in case http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq6_HTML.gif is a Carathéodory function. For our purposes in this paper, the importance of Peano's original ideas is that they can be adapted to prove existence results for (1.1) under such weak conditions that standard functional analysis arguments are no longer valid. We refer to differential equations which depend discontinuously on the unknown and several results obtained in papers as [59], see also the monographs [10, 11].

      On the other hand, singular differential equations have been receiving a lot of attention in the last years, and we can quote [7, 1219]. The main objective in this paper is to establish an existence result for (1.1) with discontinuous and singular nonlinearities which generalizes in some aspects some of the previously mentioned works.

      This paper is organized as follows. In Section 2 we introduce the relevant definitions together with some previously published material which will serve as a basis for proving our main results. In Section 3 we prove the existence of the greatest and the smallest Carathéodory solutions for (1.1) between given lower and upper solutions and assuming the existence of a http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq7_HTML.gif -bound for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq8_HTML.gif on the sector delimited by the graphs of the lower and upper solutions (regular problems), and we give some examples. In Section 4 we show that looking for piecewise continuous lower and upper solutions is good in practice, but once we have found them we can immediately construct a pair of continuous lower and upper solutions which provide better information on the location of the solutions. In Section 5 we prove two existence results in case http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq9_HTML.gif does not have such a strong bound as in Section 3 (singular problems), which requires the addition of some assumptions over the lower and upper solutions. Finally, we prove a result for singular quasimonotone systems in Section 6 and we give some examples in Section 7. Comparison with the literature is provided throughout the paper.

      2. Preliminaries

      In the following definition http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq10_HTML.gif stands for the set of absolutely continuous functions on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq11_HTML.gif .

      Definition 2.1.

      A lower solution of (1.1) is a function http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq12_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq13_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq14_HTML.gif for almost all (a.a.) http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq15_HTML.gif ; an upper solution is defined analogously reversing the inequalities. One says that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq16_HTML.gif is a (Carathéodory) solution of (1.1) if it is both a lower and an upper solution. On the other hand, one says that a solution http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq17_HTML.gif is the least one if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq18_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq19_HTML.gif for any other solution http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq20_HTML.gif , and one defines the greatest solution in a similar way. When both the least and the greatest solutions exist, one calls them the extremal solutions.

      It is proven in [8] that (1.1) has extremal solutions if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq21_HTML.gif is http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq22_HTML.gif -bounded for all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq23_HTML.gif is measurable, and for a.a. http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq24_HTML.gif is quasi-semicontinuous, namely, for all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq25_HTML.gif we have
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_Equ2_HTML.gif
      (2.1)

      A similar result was established in [20] assuming moreover that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq26_HTML.gif is superpositionally measurable, and the systems case was considered in [5, 8]. The term "quasi-semicontinuous" in connection with (2.1) was introduced in [5] for the first time and it appears to be conveniently short and descriptive. We note however that, rigorously speaking, we should say that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq27_HTML.gif is left upper and right lower semicontinuous.

      On the other hand, the above assumptions imply that the extremal solutions of (1.1) are given as the infimum of all upper solutions and the supremum of all lower solutions, that is, the least solution of (1.1) is given by
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_Equ3_HTML.gif
      (2.2)
      and the greatest solution is
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_Equ4_HTML.gif
      (2.3)

      The mappings http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq28_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq29_HTML.gif turn out to be the extremal solutions even under more general conditions. It is proven in [9] that solutions exist even if (2.1) fails on the points of a countable family of curves in the conditions of the following definition.

      Definition 2.2.

      An admissible non-quasi-semicontinuity (nqsc) curve for the differential equation http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq30_HTML.gif is the graph of an absolutely continuous function http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq31_HTML.gif such that for a.a. http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq32_HTML.gif one has either http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq33_HTML.gif , or
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_Equ5_HTML.gif
      (2.4)
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_Equ6_HTML.gif
      (2.5)

      Remark 2.3.

      The condition (2.1) cannot fail over arbitrary curves. As an example note that (1.1) has no solution for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq34_HTML.gif and
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_Equ7_HTML.gif
      (2.6)

      In this case (2.1) only fails over the line http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq35_HTML.gif , but solutions coming from above that line collide with solutions coming from below and there is no way of continuing them to the right once they reach the level http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq36_HTML.gif . Following Binding [21] we can say that the equation "jams" at http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq37_HTML.gif .

      An easily applicable sufficient condition for an absolutely continuous function http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq38_HTML.gif to be an admissible nqsc curve is that either it is a solution or there exist http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq39_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq40_HTML.gif such that one of the following conditions hold:

      (1) http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq41_HTML.gif for a.a. http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq42_HTML.gif and all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq43_HTML.gif ,

      (2) http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq44_HTML.gif for a.a. http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq45_HTML.gif and all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq46_HTML.gif .

      These conditions prevent the differential equation from exhibiting the behavior of the previous example over the line http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq47_HTML.gif in several ways. First, if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq48_HTML.gif is a solution of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq49_HTML.gif then any other solution can be continued over http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq50_HTML.gif once they contact each other and independently of the definition of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq51_HTML.gif around the graph of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq52_HTML.gif . On the other hand, if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq53_HTML.gif holds then solutions of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq54_HTML.gif can cross http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq55_HTML.gif from above to below (hence at most once), and if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq56_HTML.gif holds then solutions can cross http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq57_HTML.gif from below to above, so in both cases the equation does not jam over the graph of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq58_HTML.gif .

      For the convenience of the reader we state the main results in [9]. The next result establishes the fact that we can have "weak" solutions in a sense just by assuming very general conditions over http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq59_HTML.gif .

      Theorem 2.4.

      Suppose that there exists a null-measure set http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq60_HTML.gif such that the following conditions hold:

      (1)condition (2.1) holds for all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq61_HTML.gif except, at most, over a countable family of admissible non-quasi-semicontinuity curves;

      (2)there exists an integrable function http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq62_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq63_HTML.gif , such that
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_Equ8_HTML.gif
      (2.7)
      Then the mapping
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_Equ9_HTML.gif
      (2.8)
      is absolutely continuous on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq64_HTML.gif and satisfies http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq65_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq66_HTML.gif for a.a. http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq67_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq68_HTML.gif and for all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq69_HTML.gif the set
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_Equ10_HTML.gif
      (2.9)

      contains no positive measure set.

      Analogously, the mapping
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_Equ11_HTML.gif
      (2.10)
      is absolutely continuous on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq70_HTML.gif and satisfies http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq71_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq72_HTML.gif for a.a. http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq73_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq74_HTML.gif and for all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq75_HTML.gif the set
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_Equ12_HTML.gif
      (2.11)

      contains no positive measure set.

      Note that if the sets http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq76_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq77_HTML.gif are measurable then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq78_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq79_HTML.gif immediately become the extremal Carathéodory solutions of (1.1). In turn, measurability of those sets can be deduced from some measurability assumptions on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq80_HTML.gif . The next lemma is a slight generalization of some results in [8] and the reader can find its proof in [9].

      Lemma 2.5.

      Assume that for a null-measure set http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq81_HTML.gif the mapping http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq82_HTML.gif satisfies the following condition.

      For each http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq84_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq85_HTML.gif is measurable, and for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq86_HTML.gif one has
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_Equ13_HTML.gif
      (2.12)

      Then the mappings http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq88_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq89_HTML.gif are measurable for each pair http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq90_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq91_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq92_HTML.gif .

      Remark 2.6.

      A revision of the proof of [9, Lemma http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq93_HTML.gif ] shows that it suffices to impose (2.12) for all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq94_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq95_HTML.gif . This fact will be taken into account in this paper.

      As a consequence of Theorem 2.4 and Lemma 2.5 we have a result about existence of extremal Carathéodory solutions for (1.1) and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq96_HTML.gif -bounded nonlinearities. Note that the assumptions in Lemma 2.5 include a restriction over the type of discontinuities that can occur over the admissible nonqsc curves, but remember that such a restriction only plays the role of implying that the sets http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq97_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq98_HTML.gif in Theorem 2.4 are measurable. Therefore, only using the axiom of choice one can find a mapping http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq99_HTML.gif in the conditions of Theorem 2.4 which does not satisfy the assumptions in Lemma 2.5 and for which the corresponding problem (1.1) lacks the greatest (or the least) Carathéodory solution.

      Theorem 2.7 ([9, Theorem http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq100_HTML.gif ]).

      Suppose that there exists a null-measure set http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq101_HTML.gif such that the following conditions hold:

      (i)for every http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq102_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq103_HTML.gif is measurable;

      (ii)for every http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq104_HTML.gif and all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq105_HTML.gif one has either (2.1) or
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_Equ14_HTML.gif
      (2.13)

      and (2.1) can fail, at most, over a countable family of admissible nonquasisemicontinuity curves;

      (iii)there exists an integrable function http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq106_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq107_HTML.gif , such that
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_Equ15_HTML.gif
      (2.14)

      Then the mapping http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq108_HTML.gif defined in (2.2) is the least Carathéodory solution of (1.1) and the mapping http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq109_HTML.gif defined in (2.3) is the greatest one.

      Remark 2.8.

      Theorem http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq110_HTML.gif in [9] actually asserts that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq111_HTML.gif , as defined in (2.8), is the least Carathéodory solution, but it is easy to prove that in that case http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq112_HTML.gif , as defined in (2.2). Indeed, let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq113_HTML.gif be an arbitrary upper solution of (1.1), let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq114_HTML.gif and let
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_Equ16_HTML.gif
      (2.15)

      Theorem http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq115_HTML.gif in [9] implies that also http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq116_HTML.gif is the least Carathéodory solution of (1.1), thus http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq117_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq118_HTML.gif . Hence http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq119_HTML.gif .

      Analogously we can prove that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq120_HTML.gif can be replaced by http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq121_HTML.gif in the statement of [21, Theorem http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq122_HTML.gif ].

      3. Existence between Lower and Upper Solutions

      Condition http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq123_HTML.gif in Theorem 2.7 is rather restrictive and can be relaxed by assuming boundedness of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq124_HTML.gif between a lower and an upper solution.

      In this section we will prove the following result.

      Theorem 3.1.

      Suppose that (1.1) has a lower solution http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq125_HTML.gif and an upper solution http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq126_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq127_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq128_HTML.gif and let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq129_HTML.gif .

      Suppose that there exists a null-measure set http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq130_HTML.gif such that the following conditions hold:

      () for every http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq132_HTML.gif , the mapping http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq133_HTML.gif with domain http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq134_HTML.gif is measurable;

      () for every http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq136_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq137_HTML.gif , one has either (2.1) or (2.13), and (2.1) can fail, at most, over a countable family of admissible non-quasisemicontinuity curves contained in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq138_HTML.gif ;

      () there exists an integrable function http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq140_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq141_HTML.gif , such that
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_Equ17_HTML.gif
      (3.1)
      Then (1.1) has extremal solutions in the set
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_Equ18_HTML.gif
      (3.2)
      Moreover the least solution of (1.1) in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq142_HTML.gif is given by
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_Equ19_HTML.gif
      (3.3)
      and the greatest solution of (1.1) in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq143_HTML.gif is given by
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_Equ20_HTML.gif
      (3.4)

      Proof.

      Without loss of generality we suppose that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq144_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq145_HTML.gif exist and satisfy http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq146_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq147_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq148_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq149_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq150_HTML.gif . We also may (and we do) assume that every admissible nqsc curve in condition http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq151_HTML.gif , say http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq152_HTML.gif , satisfies for all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq153_HTML.gif either http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq154_HTML.gif or (2.4)-(2.5).

      For each http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq155_HTML.gif we define
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_Equ21_HTML.gif
      (3.5)

      Claim 1.

      The modified problem
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_Equ22_HTML.gif
      (3.6)

      satisfies conditions http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq156_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq157_HTML.gif in Theorem 2.4 with http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq158_HTML.gif replaced by http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq159_HTML.gif . First we note that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq160_HTML.gif is an immediate consequence of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq161_HTML.gif and the definition of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq162_HTML.gif .

      To show that condition http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq163_HTML.gif in Theorem 2.4 is satisfied with http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq164_HTML.gif replaced by http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq165_HTML.gif , let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq166_HTML.gif be fixed. The verification of (2.1) for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq167_HTML.gif at http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq168_HTML.gif is trivial in the following cases: http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq169_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq170_HTML.gif satisfies (2.1) at http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq171_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq172_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq173_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq174_HTML.gif . Let us consider the remaining situations: we start with the case http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq175_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq176_HTML.gif satisfies (2.1) at http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq177_HTML.gif , for which we have http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq178_HTML.gif and
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_Equ23_HTML.gif
      (3.7)

      and an analogous argument is valid when http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq179_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq180_HTML.gif satisfies (2.1).

      The previous argument shows that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq181_HTML.gif satisfies (2.1) at every http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq182_HTML.gif except, at most, over the graphs of the countable family of admissible nonquasisemicontinuity curves in condition http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq183_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq184_HTML.gif . Therefore it remains to show that if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq185_HTML.gif is one of those admissible nqsc curves for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq186_HTML.gif then it is also an admissible nqsc curve for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq187_HTML.gif . As long as the graph of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq188_HTML.gif remains in the interior of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq189_HTML.gif we have nothing to prove because http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq190_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq191_HTML.gif are the same, so let us assume that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq192_HTML.gif on a positive measure set http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq193_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq194_HTML.gif . Since http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq195_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq196_HTML.gif are absolutely continuous there is a null measure set http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq197_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq198_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq199_HTML.gif , thus for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq200_HTML.gif we have
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_Equ24_HTML.gif
      (3.8)
      so condition (2.5) with http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq201_HTML.gif replaced by http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq202_HTML.gif is satisfied on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq203_HTML.gif . On the other hand, we have to check whether http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq204_HTML.gif for those http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq205_HTML.gif at which we have
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_Equ25_HTML.gif
      (3.9)
      We distinguish two cases: http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq206_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq207_HTML.gif . In the first case (3.9) is equivalent to
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_Equ26_HTML.gif
      (3.10)

      and therefore either http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq208_HTML.gif or condition (2.4) holds, yielding http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq209_HTML.gif . If http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq210_HTML.gif then we have http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq211_HTML.gif .

      Analogous arguments show that either http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq212_HTML.gif or (2.4)-(2.5) hold for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq213_HTML.gif at almost every point where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq214_HTML.gif coincides with http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq215_HTML.gif , so we conclude that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq216_HTML.gif is an admissible nqsc curve for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq217_HTML.gif .

      By virtue of Claim 1 and Theorem 2.4 we can ensure that the functions http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq218_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq219_HTML.gif defined as
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_Equ27_HTML.gif
      (3.11)
      are absolutely continuous on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq220_HTML.gif and satisfy http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq221_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq222_HTML.gif for a.a. http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq223_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq224_HTML.gif and for all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq225_HTML.gif the set
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_Equ28_HTML.gif
      (3.12)
      contains no positive measure set, and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq226_HTML.gif for a.a. http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq227_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq228_HTML.gif and for all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq229_HTML.gif the set
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_Equ29_HTML.gif
      (3.13)

      contains no positive measure set.

      Claim 2.

      For all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq230_HTML.gif we have
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_Equ30_HTML.gif
      (3.14)
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_Equ31_HTML.gif
      (3.15)
      Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq231_HTML.gif be an upper solution of (3.6) and let us show that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq232_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq233_HTML.gif . Reasoning by contradiction, assume that there exist http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq234_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq235_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq236_HTML.gif and
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_Equ32_HTML.gif
      (3.16)
      For a.a. http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq237_HTML.gif we have
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_Equ33_HTML.gif
      (3.17)

      which together with http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq238_HTML.gif imply http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq239_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq240_HTML.gif , a contradiction with (3.16). Therefore every upper solution of (3.6) is greater than or equal to http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq241_HTML.gif , and, on the other hand, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq242_HTML.gif is an upper solution of (3.6) with http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq243_HTML.gif a.e., thus http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq244_HTML.gif satisfies (3.14).

      One can prove by means of analogous arguments that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq245_HTML.gif satisfies (3.15).

      Claim 3.

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq246_HTML.gif is the least solution of (1.1) in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq247_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq248_HTML.gif is the greatest one. From (3.14) and (3.15) it suffices to show that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq249_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq250_HTML.gif are actually solutions of (3.6). Therefore we only have to prove that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq251_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq252_HTML.gif are null measure sets.

      Let us show that the set http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq253_HTML.gif is a null measure set. First, note that
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_Equ34_HTML.gif
      (3.18)

      and we can split http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq254_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq255_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq256_HTML.gif

      Let us show that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq257_HTML.gif is a null measure set. Since http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq258_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq259_HTML.gif are absolutely continuous the set
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_Equ35_HTML.gif
      (3.19)
      is null. If http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq260_HTML.gif then there is some http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq261_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq262_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq263_HTML.gif , but then the definitions of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq264_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq265_HTML.gif yield
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_Equ36_HTML.gif
      (3.20)

      Therefore http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq266_HTML.gif and thus http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq267_HTML.gif is a null measure set.

      The set http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq268_HTML.gif can be expressed as http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq269_HTML.gif , where for each http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq270_HTML.gif
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_Equ37_HTML.gif
      (3.21)
      For http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq271_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq272_HTML.gif , we have http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq273_HTML.gif , so the definition of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq274_HTML.gif implies that
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_Equ38_HTML.gif
      (3.22)

      which is a measurable set by virtue of Lemma 2.5 and Remark 2.6.

      Since http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq275_HTML.gif contains no positive measure subset we can ensure that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq276_HTML.gif is a null measure set for all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq277_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq278_HTML.gif , and since http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq279_HTML.gif increases with http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq280_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq281_HTML.gif , we conclude that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq282_HTML.gif is a null measure set. Finally http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq283_HTML.gif is null because it is the union of countably many null measure sets.

      Analogous arguments show that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq284_HTML.gif is a null measure set, thus the proof of Claim 3 is complete.

      Claim 4.

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq285_HTML.gif satisfies (3.3) and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq286_HTML.gif satisfies (3.4). Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq287_HTML.gif be an upper solution of (1.1), let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq288_HTML.gif , and for all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq289_HTML.gif let
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_Equ39_HTML.gif
      (3.23)

      Repeating the previous arguments we can prove that also http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq290_HTML.gif is the least Carathéodory solution of (1.1) in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq291_HTML.gif , thus http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq292_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq293_HTML.gif . Hence http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq294_HTML.gif satisfies (3.3).

      Analogous arguments show that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq295_HTML.gif satisfies (3.4).

      Remark 3.2.

      Problem (3.6) may not satisfy condition http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq296_HTML.gif in Theorem 2.7 as the compositions http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq297_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq298_HTML.gif need not be measurable. That is why we used Theorem 2.4, instead of Theorem 2.7, to establish Theorem 3.1.

      Next we show that even singular problems may fall inside the scope of Theorem 3.1 if we have adequate pairs of lower and upper solutions.

      Example 3.3.

      Let us denote by http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq299_HTML.gif the integer part of a real number http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq300_HTML.gif . We are going to show that the problem
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_Equ40_HTML.gif
      (3.24)

      has positive solutions. Note that the limit of the right hand side as http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq301_HTML.gif tends to the origin does not exist, so the equation is singular at the initial condition.

      In order to apply Theorem 3.1 we consider (1.1) with http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq302_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq303_HTML.gif , and
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_Equ41_HTML.gif
      (3.25)
      It is elementary matter to check that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq304_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq305_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq306_HTML.gif , are lower and upper solutions for the problem. Condition (2.1) only fails over the graphs of the functions
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_Equ42_HTML.gif
      (3.26)

      which are a countable family of admissible nqsc curves at which condition (2.13) holds.

      Finally note that
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_Equ43_HTML.gif
      (3.27)

      so condition http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq307_HTML.gif is satisfied.

      Theorem 3.1 ensures that our problem has extremal solutions between http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq308_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq309_HTML.gif which, obviously, are different from zero almost everywhere. Therefore (3.24) has positive solutions.

      The result of Theorem 3.1 may fail if we assume that condition http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq310_HTML.gif is satisfied only in the interior of the set http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq311_HTML.gif . This is shown in the following example.

      Example 3.4.

      Let us consider problem (1.1) with http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq312_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq313_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq314_HTML.gif defined as
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_Equ44_HTML.gif
      (3.28)

      It is easy to check that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq315_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq316_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq317_HTML.gif are lower and upper solutions for this problem and that all the assumptions of Theorem 3.1 are satisfied in the interior of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq318_HTML.gif . However this problem has no solution at all.

      In order to complete the previous information we can say that condition http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq319_HTML.gif in the interior of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq320_HTML.gif is enough if we modify the definitions of lower and upper solutions in the following sense.

      Theorem 3.5.

      Suppose that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq321_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq322_HTML.gif are absolutely continuous functions on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq323_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq324_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq325_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq326_HTML.gif ,
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_Equ45_HTML.gif
      (3.29)

      and let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq327_HTML.gif .

      Suppose that there exists a null-measure set http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq328_HTML.gif such that conditions http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq329_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq330_HTML.gif hold and, moreover,

      () for every http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq332_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq333_HTML.gif , one has either (2.1) or (2.13), and (2.1) can fail, at most, over a countable family of admissible non-quasisemicontinuity curves contained in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq334_HTML.gif .

      Then the conclusions of Theorem 3.1 hold true.

      Proof (Outline)

      It follows the same steps as the proof of Theorem 3.1 but replacing http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq335_HTML.gif by
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_Equ46_HTML.gif
      (3.30)

      Note that condition (2.1) with http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq336_HTML.gif replaced by http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq337_HTML.gif is immediately satisfied over the graphs of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq338_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq339_HTML.gif thanks to the definition of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq340_HTML.gif .

      Remarks

      (i)The function http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq341_HTML.gif in Example 3.4 does not satisfy the conditions in Theorem 3.5.
      1. (ii)

        When http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq342_HTML.gif satisfies (2.1) everywhere or almost all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq343_HTML.gif then every couple of lower and upper solutions satisfies the conditions in Theorem 3.5, so this result is not really new in that case (which includes the Carathéodory and continuous cases).

         

      4. Discontinuous Lower and Upper Solutions

      Another modification of the concepts of lower and upper solutions concerns the possibility of allowing jumps in their graphs. Since the task of finding a pair of lower and upper solutions is by no means easy in general, and bearing in mind that constant lower and upper solutions are the first reasonable attempt, looking for lower and upper solutions "piece by piece" might make it easier to find them in practical situations. Let us consider the following definition.

      Definition 4.1.

      One says that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq344_HTML.gif is a piecewise continuous lower solution of (1.1) if there exist http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq345_HTML.gif such that

      1. (a)
        for all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq346_HTML.gif , one has http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq347_HTML.gif and for a.a. http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq348_HTML.gif
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_Equ47_HTML.gif
        (4.1)
         
      (b) http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq349_HTML.gif , for all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq350_HTML.gif
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_Equ48_HTML.gif
      (4.2)

      and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq351_HTML.gif .

      A piecewise continuous upper solution of (1.1) is defined reversing the relevant inequalities.

      The existence of a pair of well-ordered piecewise continuous lower and upper solutions implies the existence of a better pair of continuous lower and upper solutions. We establish this more precisely in our next proposition. Note that the proof is constructive.

      Proposition 4.2.

      Assume that all the conditions in Theorem 3.1 hold with piecewise continuous lower and upper solutions http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq352_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq353_HTML.gif .

      Then the following statements hold:

      (i)there exist a lower solution http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq354_HTML.gif and an upper solution http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq355_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_Equ49_HTML.gif
      (4.3)

      (ii)if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq356_HTML.gif is an upper solution of (1.1) with http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq357_HTML.gif then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq358_HTML.gif , and if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq359_HTML.gif is a lower solution with http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq360_HTML.gif then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq361_HTML.gif .

      In particular, the conclusions of Theorem 3.1 remain valid and, moreover, every solution of (1.1) between http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq362_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq363_HTML.gif lies between http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq364_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq365_HTML.gif .

      Proof.

      We will only prove the assertions concerning http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq366_HTML.gif because the proofs for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq367_HTML.gif are analogous.

      To construct http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq368_HTML.gif we simply have to join the points http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq369_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq370_HTML.gif with the graph of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq371_HTML.gif by means of an absolutely continuous curve with derivative less than or equal to http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq372_HTML.gif a.e., http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq373_HTML.gif being the function given in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq374_HTML.gif . It can be easily proven that this http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq375_HTML.gif is a lower solution of (1.1) that lies between http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq376_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq377_HTML.gif .

      Moreover, if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq378_HTML.gif is an upper solution of (1.1) between http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq379_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq380_HTML.gif then we have
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_Equ50_HTML.gif
      (4.4)

      so it cannot go below http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq381_HTML.gif .

      Piecewise continuous lower and upper solutions in the sense of Definition 4.1 were already used in [15, 22]. It is possible to generalize further the concept of lower and upper solutions, as a piecewise continuous lower solution is a particular case of a bounded variation function that has a nonincreasing singular part. Bounded variation lower and upper solutions with monotone singular parts were used in [23, 24], but it is not clear whether Theorem 3.1 is valid with this general type of lower and upper solutions. Anyway, piecewise continuous lower and upper solutions are enough in practical situations, and since these can be transformed into continuous ones which provide better information we will only consider from now on continuous lower and upper solutions as defined in Definition 2.1.

      5. Singular Differential Equations

      It is the goal of the present section to establish a theorem on existence of solutions for (1.1) between a pair of well-ordered lower and upper solutions and in lack of a local http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq382_HTML.gif bound. Solutions will be weak, in the sense of the following definition. By http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq383_HTML.gif we denote the set of functions http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq384_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq385_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq386_HTML.gif , and in a similar way we define http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq387_HTML.gif .

      Definition 5.1.

      We say that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq388_HTML.gif is a weak lower solution of (1.1) if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq389_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq390_HTML.gif for a.a. http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq391_HTML.gif . A weak upper solution is defined analogously reversing inequalities. A weak solution of (1.1) is a function which is both a weak lower solution and a weak upper solution.

      We will also refer to extremal weak solutions with obvious meaning.

      Note that (lower/upper) solutions, as defined in Definition 2.1, are weak (lower/upper) solutions but the converse is false in general. For instance the singular linear problem
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_Equ51_HTML.gif
      (5.1)
      has exactly the following weak solutions:
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_Equ52_HTML.gif
      (5.2)

      and none of them is absolutely continuous on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq392_HTML.gif . Another example, which uses lower and upper solutions, can be found in [15, Remark http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq393_HTML.gif ].

      However weak (lower/upper) solutions are of Carathéodory type provided they have bounded variation. We establish this fact in the next proposition.

      Proposition 5.2.

      Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq394_HTML.gif be such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq395_HTML.gif and let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq396_HTML.gif be continuous on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq397_HTML.gif and locally absolutely continuous on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq398_HTML.gif .

      A necessary and sufficient condition for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq399_HTML.gif to be absolutely continuous on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq400_HTML.gif is that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq401_HTML.gif be of bounded variation on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq402_HTML.gif .

      Proof.

      The necessary part is trivial. To estalish the sufficiency of our condition we use Banach-Zarecki's theorem, see [18, Theorem http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq403_HTML.gif ]. Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq404_HTML.gif be a null measure set, we have to prove that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq405_HTML.gif is also a null measure set. To do this let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq406_HTML.gif be such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq407_HTML.gif . Since http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq408_HTML.gif is absolutely continuous on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq409_HTML.gif the set http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq410_HTML.gif is a null measure set for each http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq411_HTML.gif . Therefore http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq412_HTML.gif is also a null measure set because
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_Equ53_HTML.gif
      (5.3)

      Next we present our main result on existence of weak solutions for (1.1) in absence of integrable bounds.

      Theorem 5.3.

      Suppose that (1.1) has a weak lower solution http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq413_HTML.gif and a weak upper solution http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq414_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq415_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq416_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq417_HTML.gif .

      Suppose that there is a null-measure set http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq418_HTML.gif such that conditions http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq419_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq420_HTML.gif in Theorem 3.1 hold for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq421_HTML.gif and assume moreover that the following condition holds:

      () there exists http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq423_HTML.gif such that for all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq424_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq425_HTML.gif , one has http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq426_HTML.gif . Then (1.1) has extremal weak solutions in the set
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_Equ54_HTML.gif
      (5.4)
      Moreover the least weak solution of (1.1) in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq427_HTML.gif is given by
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_Equ55_HTML.gif
      (5.5)
      and the greatest weak solution of (1.1) in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq428_HTML.gif is given by
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_Equ56_HTML.gif
      (5.6)

      Proof.

      We will only prove that (5.6) defines the greatest weak solution of (1.1) in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq429_HTML.gif , as the arguments to show that (5.5) is the least one are analogous.

      First note that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq430_HTML.gif is a weak lower solution between http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq431_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq432_HTML.gif , so http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq433_HTML.gif is well defined.

      Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq434_HTML.gif be a decreasing sequence in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq435_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq436_HTML.gif . Theorem 2.7 ensures that for every http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq437_HTML.gif the problem
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_Equ57_HTML.gif
      (5.7)

      has extremal Carathéodory solutions between http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq438_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq439_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq440_HTML.gif denote the greatest solution of (5.7) between http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq441_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq442_HTML.gif . By virtue of Theorem 2.7 we also know that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq443_HTML.gif is the greatest lower solution of (5.7) between http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq444_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq445_HTML.gif .

      Next we prove in several steps that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq446_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq447_HTML.gif for each http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq448_HTML.gif .

      Step 1 ( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq449_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq450_HTML.gif for each http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq451_HTML.gif ).

      The restriction to http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq452_HTML.gif of each weak lower solution between http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq453_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq454_HTML.gif is a lower solution of (5.7) between http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq455_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq456_HTML.gif , thus http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq457_HTML.gif is, on the interval http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq458_HTML.gif , greater than or equal to any weak lower solution of (1.1) between http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq459_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq460_HTML.gif . The definition of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq461_HTML.gif implies then that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq462_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq463_HTML.gif .

      Step 2 ( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq464_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq465_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq466_HTML.gif ).

      First, since http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq467_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq468_HTML.gif we have http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq469_HTML.gif . Reasoning by contradiction, assume that there exists http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq470_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq471_HTML.gif . Then there is some http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq472_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq473_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq474_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq475_HTML.gif , but then the mapping
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_Equ58_HTML.gif
      (5.8)

      would be a solution of (5.7) (with http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq476_HTML.gif replaced by http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq477_HTML.gif ) between http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq478_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq479_HTML.gif which is greater than http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq480_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq481_HTML.gif , a contradiction.

      The above properties of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq482_HTML.gif imply that the following function is well defined:
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_Equ59_HTML.gif
      (5.9)

      Step 3 ( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq483_HTML.gif ).

      Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq484_HTML.gif be fixed. Condition http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq485_HTML.gif implies that for all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq486_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq487_HTML.gif we have
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_Equ60_HTML.gif
      (5.10)
      with http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq488_HTML.gif . Hence for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq489_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq490_HTML.gif , we have
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_Equ61_HTML.gif
      (5.11)

      and therefore http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq491_HTML.gif . Since http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq492_HTML.gif was fixed arbitrarily in the previous arguments, we conclude that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq493_HTML.gif .

      The continuity of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq494_HTML.gif at http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq495_HTML.gif follows from the continuity of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq496_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq497_HTML.gif at http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq498_HTML.gif , the assumption http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq499_HTML.gif , and the relation
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_Equ62_HTML.gif
      (5.12)

      Step 4 ( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq500_HTML.gif is a weak lower solution of (1.1)).

      For http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq501_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq502_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq503_HTML.gif we have (5.10) with http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq504_HTML.gif , hence http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq505_HTML.gif , and for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq506_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq507_HTML.gif , Fatou's lemma yields
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_Equ63_HTML.gif
      (5.13)
      Hence for a.a. http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq508_HTML.gif we have
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_Equ64_HTML.gif
      (5.14)

      Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq509_HTML.gif where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq510_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq511_HTML.gif .

      For http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq512_HTML.gif and a.a. http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq513_HTML.gif we have http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq514_HTML.gif , thus http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq515_HTML.gif for a.a. http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq516_HTML.gif .

      On the other hand, for a.a. http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq517_HTML.gif the relation (5.14) and the increasingness of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq518_HTML.gif yield
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_Equ65_HTML.gif
      (5.15)

      Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq519_HTML.gif be such that (5.15) holds. We have two possibilities: either (2.1) holds for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq520_HTML.gif at http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq521_HTML.gif and then from (5.15) we deduce http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq522_HTML.gif , or http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq523_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq524_HTML.gif is an admissible curve of non quasisemicontinuity. In the last case we have that either http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq525_HTML.gif belongs to a null-measure set or http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq526_HTML.gif , which, in turn, yields two possibilities: either http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq527_HTML.gif and then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq528_HTML.gif , or http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq529_HTML.gif and then (5.15), with http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq530_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq531_HTML.gif , and the definition of admissible curve of non quasisemicontinuity imply that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq532_HTML.gif .

      The above arguments prove that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq533_HTML.gif a.e. on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq534_HTML.gif , and since http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq535_HTML.gif was fixed arbitrarily, the proof of Step 4 is complete.

      Conclusion

      The construction of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq536_HTML.gif and Step 1 imply that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq537_HTML.gif and the definition of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq538_HTML.gif and Step 4 imply that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq539_HTML.gif . Therefore for all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq540_HTML.gif we have http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq541_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq542_HTML.gif and then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq543_HTML.gif is a weak solution of (1.1). Since every weak solution is a weak lower solution, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq544_HTML.gif is the greatest weak solution of (1.1) in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq545_HTML.gif .

      The assumption http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq546_HTML.gif in Theorem 5.3 can be replaced by other types of conditions. The next theorem generalizes the main results in [7, 1214] concerning existence of solutions of singular problems of the type of (1.1).

      Theorem 5.4.

      Suppose that (1.1) with http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq547_HTML.gif has a weak lower solution http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq548_HTML.gif and a weak upper solution http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq549_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq550_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq551_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq552_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq553_HTML.gif .

      Suppose that there is a null-measure set http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq554_HTML.gif such that conditions http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq555_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq556_HTML.gif in Theorem 3.1 hold for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq557_HTML.gif and assume moreover that the following condition holds:

      () for every http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq559_HTML.gif there exists http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq560_HTML.gif such that for all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq561_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq562_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq563_HTML.gif one has http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq564_HTML.gif .

      Then the conclusions of Theorem 5.3 hold true.

      Proof.

      We start observing that there exists a weak upper solution http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq565_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq566_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq567_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq568_HTML.gif . If http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq569_HTML.gif then it suffices to take http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq570_HTML.gif as http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq571_HTML.gif . If http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq572_HTML.gif we proceed as follows in order to construct http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq573_HTML.gif : let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq574_HTML.gif be a decreasing sequence in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq575_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq576_HTML.gif and for every http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq577_HTML.gif let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq578_HTML.gif be the greatest solution between http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq579_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq580_HTML.gif of
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_Equ66_HTML.gif
      (5.16)

      Claim [ http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq581_HTML.gif exists]

      Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq582_HTML.gif be so small that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq583_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq584_HTML.gif . Condition http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq585_HTML.gif implies that there exists http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq586_HTML.gif such that for a.a. http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq587_HTML.gif and all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq588_HTML.gif we have http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq589_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq590_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq591_HTML.gif and let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq592_HTML.gif be such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq593_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq594_HTML.gif . We can apply Theorem 2.7 to the problem
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_Equ67_HTML.gif
      (5.17)

      and with respect to the lower solution http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq595_HTML.gif and the upper solution http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq596_HTML.gif , so there exists http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq597_HTML.gif the greatest solution between http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq598_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq599_HTML.gif of (5.17). Notice that if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq600_HTML.gif is a solution of (5.17) then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq601_HTML.gif , so http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq602_HTML.gif is also the greatest solution between http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq603_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq604_HTML.gif of (5.17).

      Now condition http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq605_HTML.gif ensures that Theorem 2.7 can be applied to the problem
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_Equ68_HTML.gif
      (5.18)

      with respect to the lower solution http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq606_HTML.gif and the upper solution http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq607_HTML.gif (both functions restricted to http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq608_HTML.gif ). Hence there exists http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq609_HTML.gif the greatest solution of (5.18) between http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq610_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq611_HTML.gif .

      Obviously we have
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_Equ69_HTML.gif
      (5.19)

      Analogous arguments to those in the proof of Theorem 5.3 show that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq612_HTML.gif is a weak upper solution and it is clear that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq613_HTML.gif .

      Finally we show that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq614_HTML.gif holds with http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq615_HTML.gif replaced by http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq616_HTML.gif . We consider a decreasing sequence http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq617_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq618_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq619_HTML.gif . As http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq620_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq621_HTML.gif are positive on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq622_HTML.gif , we can find http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq623_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq624_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq625_HTML.gif . We deduce then from http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq626_HTML.gif the existence of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq627_HTML.gif so that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq628_HTML.gif for a.e. http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq629_HTML.gif and all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq630_HTML.gif . The function http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq631_HTML.gif defined by http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq632_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq633_HTML.gif works.

      Theorem 5.3 implies that (1.1) has extremal weak solutions in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq634_HTML.gif which, moreover, satisfy (5.6) and (5.5) with http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq635_HTML.gif replaced by http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq636_HTML.gif . Furthermore if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq637_HTML.gif is a weak solution of (1.1) in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq638_HTML.gif then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq639_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq640_HTML.gif . Assume, on the contrary, that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq641_HTML.gif for some http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq642_HTML.gif , then there would exist http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq643_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq644_HTML.gif and then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq645_HTML.gif would be a solution of (5.16) between http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq646_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq647_HTML.gif which is strictly greater than http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq648_HTML.gif on some subinterval, a contradiction. Hence (1.1) has extremal weak solutions in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq649_HTML.gif which, moreover, satisfy (5.6) and (5.5).

      6. Systems

      Let us consider the following system of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq650_HTML.gif ordinary differential equations:
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_Equ70_HTML.gif
      (6.1)

      where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq651_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq652_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq653_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq654_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq655_HTML.gif .

      Our goal is to extend Theorem 5.3 to this multidimensional setting, which, as usual, requires the right hand side http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq656_HTML.gif to be quasimonotone, as we will define later.

      We start extending to the vector case the definitions given before for scalar problems. To do so, let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq657_HTML.gif denote the set of functions http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq658_HTML.gif such that for each http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq659_HTML.gif the component http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq660_HTML.gif is absolutely continuous on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq661_HTML.gif for each http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq662_HTML.gif . Also, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq663_HTML.gif stands for the class of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq664_HTML.gif -valued functions which are defined and continuous on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq665_HTML.gif .

      A weak lower solution of (6.1) is a function http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq666_HTML.gif such that for each http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq667_HTML.gif we have http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq668_HTML.gif and for a.a. http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq669_HTML.gif we have http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq670_HTML.gif Weak upper solutions are defined similarly by reversing the relevant inequalities, and weak solutions of (6.1) are functions which are both weak lower and weak upper solutions.

      In the set http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq671_HTML.gif we define a partial ordering as follows: let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq672_HTML.gif , we write http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq673_HTML.gif if every component of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq674_HTML.gif is less than or equal to the corresponding component of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq675_HTML.gif on the whole of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq676_HTML.gif . If http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq677_HTML.gif are such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq678_HTML.gif then we define
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_Equ71_HTML.gif
      (6.2)

      Extremal (least and greatest) weak solutions of (6.1) in a certain subset of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq679_HTML.gif are defined in the obvious way considering the previous ordering.

      Now we are ready to extend Theorem 5.3 to the vector case. We will denote by http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq680_HTML.gif the http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq681_HTML.gif th canonical vector. The proof follows the line of that of [8, Theorem http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq682_HTML.gif ].

      Theorem 6.1.

      Suppose that (6.1) has weak lower and upper solutions http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq683_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq684_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq685_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq686_HTML.gif , and let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq687_HTML.gif .

      Suppose that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq688_HTML.gif is quasimonotone nondecreasing in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq689_HTML.gif , that is, for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq690_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq691_HTML.gif the relations http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq692_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq693_HTML.gif imply http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq694_HTML.gif .

      Suppose, moreover, that for each http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq695_HTML.gif the following conditions hold:

      () the function http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq697_HTML.gif is measurable;

      () for all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq699_HTML.gif and a.a. http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq700_HTML.gif one has either
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_Equ72_HTML.gif
      (6.3)
      or
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_Equ73_HTML.gif
      (6.4)

      and (6.3) fails, at most, over a countable family of admissible nqsc curves of the scalar differential equation http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq701_HTML.gif contained in the sector http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq702_HTML.gif ;

      () there exists http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq704_HTML.gif such that for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq705_HTML.gif and a.a. http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq706_HTML.gif one has http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq707_HTML.gif .

      Then (6.1) has extremal weak solutions in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq708_HTML.gif . Moreover the least weak solution http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq709_HTML.gif is given by
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_Equ74_HTML.gif
      (6.5)
      and the greatest weak solution http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq710_HTML.gif is given by
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_Equ75_HTML.gif
      (6.6)

      Proof.

      Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq711_HTML.gif be a weak lower solution of (6.1), and let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq712_HTML.gif be as in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq713_HTML.gif and such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq714_HTML.gif a.e. on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq715_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq716_HTML.gif . Now let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq717_HTML.gif be defined for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq718_HTML.gif as
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_Equ76_HTML.gif
      (6.7)

      In particular, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq719_HTML.gif . Further, every possible solution of (6.1) in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq720_HTML.gif is less than or equal to http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq721_HTML.gif by (6.7) and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq722_HTML.gif , independently of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq723_HTML.gif .

      Claim 1 ( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq724_HTML.gif ).

      If http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq725_HTML.gif is a weak lower solution in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq726_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq727_HTML.gif a.e. on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq728_HTML.gif then for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq729_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq730_HTML.gif , we have
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_Equ77_HTML.gif
      (6.8)
      which implies
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_Equ78_HTML.gif
      (6.9)

      and, therefore, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq731_HTML.gif . Further http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq732_HTML.gif is continuous at http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq733_HTML.gif because http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq734_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq735_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq736_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq737_HTML.gif are continuous at http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq738_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq739_HTML.gif .

      Claim 2.

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq740_HTML.gif is the greatest weak solution of (6.1) in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq741_HTML.gif . For each weak lower solution http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq742_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq743_HTML.gif a.e., the quasimonotonicity of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq744_HTML.gif yields
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_Equ79_HTML.gif
      (6.10)
      Hence http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq745_HTML.gif is a weak lower solution between http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq746_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq747_HTML.gif of the scalar problem
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_Equ80_HTML.gif
      (6.11)

      and then Theorem 5.3 implies that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq748_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq749_HTML.gif is the greatest weak solution of (6.11) in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq750_HTML.gif . Then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq751_HTML.gif .

      On the other hand, we have
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_Equ81_HTML.gif
      (6.12)

      hence http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq752_HTML.gif is a weak lower solution of (6.1) in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq753_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq754_HTML.gif a.e. on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq755_HTML.gif , thus http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq756_HTML.gif . Therefore http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq757_HTML.gif is a weak solution of (6.1), and, by (6.7) and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq758_HTML.gif , it is the greatest one in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq759_HTML.gif . In particular, the greatest weak solution of (6.1) in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq760_HTML.gif exists and it is greater than or equal to http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq761_HTML.gif .

      Claim 3.

      The greatest weak solution of (6.1) in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq762_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq763_HTML.gif , satisfies (6.6). The weak lower solution http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq764_HTML.gif was fixed arbitrarily, so http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq765_HTML.gif is greater than or equal to any weak lower solution in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq766_HTML.gif . On the other hand, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq767_HTML.gif is a weak lower solution.

      Analogously, the least weak solution of (6.1) in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq768_HTML.gif is given by (6.5).

      7. Examples

      Example 7.1.

      Let us show that the following singular and non-quasisemicontinous problem has a unique positive Carathéodory solution:
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_Equ82_HTML.gif
      (7.1)

      Here square brackets mean integer part, and by positive solution we mean a solution which is positive on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq769_HTML.gif .

      First note that (7.1) has at most one positive weak solution because the right hand side in the differential equation is nonincreasing with respect to the unknown on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq770_HTML.gif , thus at no point can solutions bifurcate.

      For all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq771_HTML.gif we have http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq772_HTML.gif and therefore http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq773_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq774_HTML.gif , is an upper solution of (7.1) as it solves the majorant problem
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_Equ83_HTML.gif
      (7.2)

      On the other hand it is easy to check that for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq775_HTML.gif we have http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq776_HTML.gif and then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq777_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq778_HTML.gif , is a lower solution.

      The function http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq779_HTML.gif is continuous between the graphs of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq780_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq781_HTML.gif except over the lines http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq782_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq783_HTML.gif , which are admissible nqsc curves for all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq784_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq785_HTML.gif (note that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq786_HTML.gif is not an admissible nqsc curve but it does not lie between http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq787_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq788_HTML.gif ).

      Finally, for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq789_HTML.gif we have
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_Equ84_HTML.gif
      (7.3)

      where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq790_HTML.gif is such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq791_HTML.gif .

      Therefore Theorem 5.4 implies the existence of a weak solution of (7.1) between http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq792_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq793_HTML.gif . Moreover, this weak solution between http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq794_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq795_HTML.gif is increasing, so Proposition 5.2 ensures that it is, in fact, a Carathéodory solution on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq796_HTML.gif .

      It is possible to extend the solution on the right of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq797_HTML.gif to some http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq798_HTML.gif where the solution will assume the value http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq799_HTML.gif . The solution cannot be extended further on the right of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq800_HTML.gif , as (7.1) with http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq801_HTML.gif replaced by http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq802_HTML.gif has no solution on the right of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq803_HTML.gif .

      We owe to the anonymous referee the following remarks. Problem (7.1) is autonomous, so it falls inside the scope of the results in [21], which ensure that if we find http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq804_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_Equ85_HTML.gif
      (7.4)
      then (7.1) has a positive absolutely continuous solution defined implicitly by
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_Equ86_HTML.gif
      (7.5)
      Since
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_Equ87_HTML.gif
      (7.6)

      we deduce that the solution http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq805_HTML.gif is defined at least on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq806_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq807_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq808_HTML.gif .

      Example 7.2.

      Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq809_HTML.gif be measurable and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq810_HTML.gif for a.a. http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq811_HTML.gif . We will prove that for each http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq812_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq813_HTML.gif , the problem
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_Equ88_HTML.gif
      (7.7)

      has a unique positive Carathéodory solution.

      Note that the equation is not separable and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq814_HTML.gif assumes positive and negative values on every neighborhood of the initial condition. Moreover the equation is singular at the initial condition with respect to both of its variables.

      Once again the right hand side in the differential equation is nonincreasing with respect to the unknown http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq815_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq816_HTML.gif , thus we have at most one positive weak solution.

      Lower and upper solutions are given by, respectively, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq817_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq818_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq819_HTML.gif .

      For each http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq820_HTML.gif the function http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq821_HTML.gif is continuous between the graphs of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq822_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq823_HTML.gif except over the lines http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq824_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq825_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq826_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq827_HTML.gif . Let us show that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq828_HTML.gif is positive between http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq829_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq830_HTML.gif , thus http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq831_HTML.gif will be an admissible nqsc curve for each http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq832_HTML.gif . For http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq833_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq834_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq835_HTML.gif , we have
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_Equ89_HTML.gif
      (7.8)
      and if, moreover, we restrict our attention to those http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq836_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq837_HTML.gif then we have http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq838_HTML.gif which implies
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_Equ90_HTML.gif
      (7.9)
      and thus for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq839_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq840_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq841_HTML.gif , we have
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_Equ91_HTML.gif
      (7.10)

      This shows that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq842_HTML.gif is positive between http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq843_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq844_HTML.gif and, moreover, we can say that for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq845_HTML.gif it suffices to take http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq846_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq847_HTML.gif to have http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq848_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq849_HTML.gif between the graphs of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq850_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq851_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq852_HTML.gif .

      Therefore Theorem 5.4 implies the existence of a weak solution of (7.7) between http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq853_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq854_HTML.gif . Moreover, since http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq855_HTML.gif is positive between http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq856_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq857_HTML.gif the solution is increasing and, therefore, it is a Carathéodory solution.

      The previous two examples fit the conditions of Theorems 5.3 and 5.4. Next we show an example where Theorem 5.3 can be used but it is not clear whether or not we can also apply Theorem 5.4.

      Example 7.3.

      Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq858_HTML.gif be fixed and consider the problem
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_Equ92_HTML.gif
      (7.11)

      Lower and upper solutions are given by http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq859_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq860_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq861_HTML.gif . Since http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq862_HTML.gif is nonnegative between http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq863_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq864_HTML.gif the lines http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq865_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq866_HTML.gif , are admissible nqsc curves for the differential equation. Finally it is easy to check that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq867_HTML.gif if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq868_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq869_HTML.gif , thus one can construct http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq870_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq871_HTML.gif for a.a. http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq872_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq873_HTML.gif .

      Theorem 5.3 ensures that (7.11) has extremal weak solutions between http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq874_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq875_HTML.gif . Moreover (7.11) has a unique solution between http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq876_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq877_HTML.gif as http://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq878_HTML.gif is nonincreasing with respect to the unknown. Further, the unique solution is monotone and therefore it is a Carathéodory solution.

      Declarations

      Acknowledgments

      The research of Rodrigo López Pouso is partially supported by Ministerio de Educación y Ciencia, Spain, Project MTM2007-61724, and by Xunta de Galicia, Spain, Project PGIDIT06PXIB207023PR.

      Authors’ Affiliations

      (1)
      Department of Mathematics, Belmont University
      (2)
      Department of Mathematical Analysis, University of Santiago de Compostela

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      Copyright

      © D. C. Biles and R. López Pouso. 2009

      This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.