Open Access

First-Order Singular and Discontinuous Differential Equations

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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq1_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq2_HTML.gif be fixed and let https://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
https://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 https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq7_HTML.gif -bound for https://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 https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq12_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq13_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq14_HTML.gif for almost all (a.a.) https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq17_HTML.gif is the least one if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq18_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq19_HTML.gif for any other solution https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq21_HTML.gif is https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq22_HTML.gif -bounded for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq23_HTML.gif is measurable, and for a.a. https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq24_HTML.gif is quasi-semicontinuous, namely, for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq25_HTML.gif we have
https://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 https://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 https://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
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_Equ4_HTML.gif
(2.3)

The mappings https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq28_HTML.gif and https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq31_HTML.gif such that for a.a. https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq32_HTML.gif one has either https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq33_HTML.gif , or
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_Equ5_HTML.gif
(2.4)
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq34_HTML.gif and
https://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 https://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 https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq39_HTML.gif and https://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) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq41_HTML.gif for a.a. https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq42_HTML.gif and all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq43_HTML.gif ,

(2) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq44_HTML.gif for a.a. https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq45_HTML.gif and all https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq47_HTML.gif in several ways. First, if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq48_HTML.gif is a solution of https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq51_HTML.gif around the graph of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq52_HTML.gif . On the other hand, if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq53_HTML.gif holds then solutions of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq54_HTML.gif can cross https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq56_HTML.gif holds then solutions can cross https://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 https://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 https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq62_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq63_HTML.gif , such that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_Equ8_HTML.gif
(2.7)
Then the mapping
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_Equ9_HTML.gif
(2.8)
is absolutely continuous on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq64_HTML.gif and satisfies https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq65_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq66_HTML.gif for a.a. https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq67_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq68_HTML.gif and for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq69_HTML.gif the set
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_Equ11_HTML.gif
(2.10)
is absolutely continuous on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq70_HTML.gif and satisfies https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq71_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq72_HTML.gif for a.a. https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq73_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq74_HTML.gif and for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq75_HTML.gif the set
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq76_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq77_HTML.gif are measurable then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq78_HTML.gif and https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq81_HTML.gif the mapping https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq82_HTML.gif satisfies the following condition.

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

Then the mappings https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq88_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq89_HTML.gif are measurable for each pair https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq90_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq91_HTML.gif for all https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq94_HTML.gif such that https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq97_HTML.gif and https://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 https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq102_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq103_HTML.gif is measurable;

(ii)for every https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq104_HTML.gif and all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq105_HTML.gif one has either (2.1) or
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq106_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq107_HTML.gif , such that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_Equ15_HTML.gif
(2.14)

Then the mapping https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq110_HTML.gif in [9] actually asserts that https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq114_HTML.gif and let
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_Equ16_HTML.gif
(2.15)

Theorem https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq115_HTML.gif in [9] implies that also https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq117_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq118_HTML.gif . Hence https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq119_HTML.gif .

Analogously we can prove that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq120_HTML.gif can be replaced by https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq121_HTML.gif in the statement of [21, Theorem https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq125_HTML.gif and an upper solution https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq126_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq127_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq128_HTML.gif and let https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq132_HTML.gif , the mapping https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq133_HTML.gif with domain https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq134_HTML.gif is measurable;

() for every https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq136_HTML.gif , https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq138_HTML.gif ;

() there exists an integrable function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq140_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq141_HTML.gif , such that
https://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
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq142_HTML.gif is given by
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq143_HTML.gif is given by
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq144_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq145_HTML.gif exist and satisfy https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq146_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq147_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq148_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq149_HTML.gif on https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq151_HTML.gif , say https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq152_HTML.gif , satisfies for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq153_HTML.gif either https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq155_HTML.gif we define
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_Equ22_HTML.gif
(3.6)

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

To show that condition https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq164_HTML.gif replaced by https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq165_HTML.gif , let https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq167_HTML.gif at https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq168_HTML.gif is trivial in the following cases: https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq169_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq170_HTML.gif satisfies (2.1) at https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq171_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq172_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq173_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq175_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq176_HTML.gif satisfies (2.1) at https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq177_HTML.gif , for which we have https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq178_HTML.gif and
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq179_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq181_HTML.gif satisfies (2.1) at every https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq183_HTML.gif for https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq187_HTML.gif . As long as the graph of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq188_HTML.gif remains in the interior of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq189_HTML.gif we have nothing to prove because https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq190_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq192_HTML.gif on a positive measure set https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq193_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq194_HTML.gif . Since https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq195_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq197_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq198_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq199_HTML.gif , thus for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq200_HTML.gif we have
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq201_HTML.gif replaced by https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq202_HTML.gif is satisfied on https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq204_HTML.gif for those https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq205_HTML.gif at which we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_Equ25_HTML.gif
(3.9)
We distinguish two cases: https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq206_HTML.gif and https://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
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_Equ26_HTML.gif
(3.10)

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

Analogous arguments show that either https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq213_HTML.gif at almost every point where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq214_HTML.gif coincides with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq215_HTML.gif , so we conclude that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq216_HTML.gif is an admissible nqsc curve for https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq218_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq219_HTML.gif defined as
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_Equ27_HTML.gif
(3.11)
are absolutely continuous on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq220_HTML.gif and satisfy https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq221_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq222_HTML.gif for a.a. https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq223_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq224_HTML.gif and for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq225_HTML.gif the set
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq226_HTML.gif for a.a. https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq227_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq228_HTML.gif and for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq229_HTML.gif the set
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq230_HTML.gif we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_Equ30_HTML.gif
(3.14)
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_Equ31_HTML.gif
(3.15)
Let https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq232_HTML.gif for all https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq234_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq235_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq236_HTML.gif and
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_Equ32_HTML.gif
(3.16)
For a.a. https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq237_HTML.gif we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_Equ33_HTML.gif
(3.17)

which together with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq238_HTML.gif imply https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq239_HTML.gif on https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq241_HTML.gif , and, on the other hand, https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq243_HTML.gif a.e., thus https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq245_HTML.gif satisfies (3.15).

Claim 3.

https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq247_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq249_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq251_HTML.gif and https://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 https://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
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_Equ34_HTML.gif
(3.18)

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

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

Therefore https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq266_HTML.gif and thus https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq268_HTML.gif can be expressed as https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq269_HTML.gif , where for each https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq270_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_Equ37_HTML.gif
(3.21)
For https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq271_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq272_HTML.gif , we have https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq273_HTML.gif , so the definition of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq274_HTML.gif implies that
https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq277_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq278_HTML.gif , and since https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq279_HTML.gif increases with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq280_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq281_HTML.gif , we conclude that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq282_HTML.gif is a null measure set. Finally https://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 https://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.

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq285_HTML.gif satisfies (3.3) and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq286_HTML.gif satisfies (3.4). Let https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq288_HTML.gif , and for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq289_HTML.gif let
https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq291_HTML.gif , thus https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq292_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq293_HTML.gif . Hence https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq297_HTML.gif and https://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 https://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 https://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
https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq302_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq303_HTML.gif , and
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq304_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq305_HTML.gif , https://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
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_Equ43_HTML.gif
(3.27)

so condition https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq308_HTML.gif and https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq312_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq313_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq314_HTML.gif defined as
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq315_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq316_HTML.gif for all https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq319_HTML.gif in the interior of https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq321_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq322_HTML.gif are absolutely continuous functions on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq323_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq324_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq325_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq326_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_Equ45_HTML.gif
(3.29)

and let https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq328_HTML.gif such that conditions https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq329_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq330_HTML.gif hold and, moreover,

() for every https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq332_HTML.gif , https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq335_HTML.gif by
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq336_HTML.gif replaced by https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq338_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq339_HTML.gif thanks to the definition of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq340_HTML.gif .

Remarks

(i)The function https://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 https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq345_HTML.gif such that

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

and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq352_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq354_HTML.gif and an upper solution https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq355_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_Equ49_HTML.gif
(4.3)

(ii)if https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq357_HTML.gif then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq358_HTML.gif , and if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq359_HTML.gif is a lower solution with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq360_HTML.gif then https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq362_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq363_HTML.gif lies between https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq364_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq366_HTML.gif because the proofs for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq367_HTML.gif are analogous.

To construct https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq369_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq370_HTML.gif with the graph of https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq372_HTML.gif a.e., https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq373_HTML.gif being the function given in https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq376_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq377_HTML.gif .

Moreover, if https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq379_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq380_HTML.gif then we have
https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq383_HTML.gif we denote the set of functions https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq384_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq385_HTML.gif for all https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq387_HTML.gif .

Definition 5.1.

We say that https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq389_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq390_HTML.gif for a.a. https://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
https://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:
https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq394_HTML.gif be such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq395_HTML.gif and let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq396_HTML.gif be continuous on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq397_HTML.gif and locally absolutely continuous on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq398_HTML.gif .

A necessary and sufficient condition for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq399_HTML.gif to be absolutely continuous on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq400_HTML.gif is that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq401_HTML.gif be of bounded variation on https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq403_HTML.gif ]. Let https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq406_HTML.gif be such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq407_HTML.gif . Since https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq408_HTML.gif is absolutely continuous on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq409_HTML.gif the set https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq411_HTML.gif . Therefore https://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
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq413_HTML.gif and a weak upper solution https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq414_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq415_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq416_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq418_HTML.gif such that conditions https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq419_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq420_HTML.gif in Theorem 3.1 hold for https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq423_HTML.gif such that for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq424_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq425_HTML.gif , one has https://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
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq427_HTML.gif is given by
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq428_HTML.gif is given by
https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq430_HTML.gif is a weak lower solution between https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq431_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq432_HTML.gif , so https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq433_HTML.gif is well defined.

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq434_HTML.gif be a decreasing sequence in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq435_HTML.gif such that https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq437_HTML.gif the problem
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq438_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq439_HTML.gif . Let https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq441_HTML.gif and https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq444_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq446_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq447_HTML.gif for each https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq448_HTML.gif .

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

The restriction to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq452_HTML.gif of each weak lower solution between https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq453_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq455_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq456_HTML.gif , thus https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq457_HTML.gif is, on the interval https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq459_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq460_HTML.gif . The definition of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq461_HTML.gif implies then that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq462_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq463_HTML.gif .

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

First, since https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq467_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq468_HTML.gif we have https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq470_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq471_HTML.gif . Then there is some https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq472_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq473_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq474_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq475_HTML.gif , but then the mapping
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq476_HTML.gif replaced by https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq477_HTML.gif ) between https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq478_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq479_HTML.gif which is greater than https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq480_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq481_HTML.gif , a contradiction.

The above properties of https://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:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_Equ59_HTML.gif
(5.9)

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

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

and therefore https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq491_HTML.gif . Since https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq493_HTML.gif .

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

Step 4 ( https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq501_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq502_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq503_HTML.gif we have (5.10) with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq504_HTML.gif , hence https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq505_HTML.gif , and for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq506_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq507_HTML.gif , Fatou's lemma yields
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_Equ63_HTML.gif
(5.13)
Hence for a.a. https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq508_HTML.gif we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_Equ64_HTML.gif
(5.14)

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

For https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq512_HTML.gif and a.a. https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq513_HTML.gif we have https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq514_HTML.gif , thus https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq515_HTML.gif for a.a. https://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. https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq518_HTML.gif yield
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_Equ65_HTML.gif
(5.15)

Let https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq520_HTML.gif at https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq522_HTML.gif , or https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq523_HTML.gif , where https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq527_HTML.gif and then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq528_HTML.gif , or https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq529_HTML.gif and then (5.15), with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq530_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq532_HTML.gif .

The above arguments prove that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq533_HTML.gif a.e. on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq534_HTML.gif , and since https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq536_HTML.gif and Step 1 imply that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq537_HTML.gif and the definition of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq538_HTML.gif and Step 4 imply that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq539_HTML.gif . Therefore for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq540_HTML.gif we have https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq541_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq542_HTML.gif and then https://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, https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq545_HTML.gif .

The assumption https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq547_HTML.gif has a weak lower solution https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq548_HTML.gif and a weak upper solution https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq549_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq550_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq551_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq552_HTML.gif on https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq554_HTML.gif such that conditions https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq555_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq556_HTML.gif in Theorem 3.1 hold for https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq559_HTML.gif there exists https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq560_HTML.gif such that for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq561_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq562_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq563_HTML.gif one has https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq565_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq566_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq567_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq568_HTML.gif . If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq569_HTML.gif then it suffices to take https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq570_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq571_HTML.gif . If https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq573_HTML.gif : let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq574_HTML.gif be a decreasing sequence in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq575_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq576_HTML.gif and for every https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq577_HTML.gif let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq578_HTML.gif be the greatest solution between https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq579_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq580_HTML.gif of
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_Equ66_HTML.gif
(5.16)

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

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq582_HTML.gif be so small that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq583_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq584_HTML.gif . Condition https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq585_HTML.gif implies that there exists https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq586_HTML.gif such that for a.a. https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq587_HTML.gif and all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq588_HTML.gif we have https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq589_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq590_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq591_HTML.gif and let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq592_HTML.gif be such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq593_HTML.gif on https://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
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq595_HTML.gif and the upper solution https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq596_HTML.gif , so there exists https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq597_HTML.gif the greatest solution between https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq598_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq599_HTML.gif of (5.17). Notice that if https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq601_HTML.gif , so https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq602_HTML.gif is also the greatest solution between https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq603_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq604_HTML.gif of (5.17).

Now condition https://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
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq606_HTML.gif and the upper solution https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq607_HTML.gif (both functions restricted to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq608_HTML.gif ). Hence there exists https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq610_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq611_HTML.gif .

Obviously we have
https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq613_HTML.gif .

Finally we show that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq614_HTML.gif holds with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq615_HTML.gif replaced by https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq616_HTML.gif . We consider a decreasing sequence https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq617_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq618_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq619_HTML.gif . As https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq620_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq621_HTML.gif are positive on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq622_HTML.gif , we can find https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq623_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq624_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq625_HTML.gif . We deduce then from https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq626_HTML.gif the existence of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq627_HTML.gif so that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq628_HTML.gif for a.e. https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq629_HTML.gif and all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq630_HTML.gif . The function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq631_HTML.gif defined by https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq632_HTML.gif for https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq635_HTML.gif replaced by https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq636_HTML.gif . Furthermore if https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq638_HTML.gif then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq639_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq640_HTML.gif . Assume, on the contrary, that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq641_HTML.gif for some https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq642_HTML.gif , then there would exist https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq643_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq644_HTML.gif and then https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq646_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq647_HTML.gif which is strictly greater than https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq650_HTML.gif ordinary differential equations:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_Equ70_HTML.gif
(6.1)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq651_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq652_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq653_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq654_HTML.gif , and https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq657_HTML.gif denote the set of functions https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq658_HTML.gif such that for each https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq659_HTML.gif the component https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq660_HTML.gif is absolutely continuous on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq661_HTML.gif for each https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq662_HTML.gif . Also, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq663_HTML.gif stands for the class of https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq666_HTML.gif such that for each https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq667_HTML.gif we have https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq668_HTML.gif and for a.a. https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq669_HTML.gif we have https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq672_HTML.gif , we write https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq673_HTML.gif if every component of https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq675_HTML.gif on the whole of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq676_HTML.gif . If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq677_HTML.gif are such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq678_HTML.gif then we define
https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq680_HTML.gif the https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq683_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq684_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq685_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq686_HTML.gif , and let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq687_HTML.gif .

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

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

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

() for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq699_HTML.gif and a.a. https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq700_HTML.gif one has either
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_Equ72_HTML.gif
(6.3)
or
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq701_HTML.gif contained in the sector https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq702_HTML.gif ;

() there exists https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq704_HTML.gif such that for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq705_HTML.gif and a.a. https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq706_HTML.gif one has https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq708_HTML.gif . Moreover the least weak solution https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq709_HTML.gif is given by
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq710_HTML.gif is given by
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_Equ75_HTML.gif
(6.6)

Proof.

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

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

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

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

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

Claim 2.

https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq741_HTML.gif . For each weak lower solution https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq742_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq743_HTML.gif a.e., the quasimonotonicity of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq744_HTML.gif yields
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_Equ79_HTML.gif
(6.10)
Hence https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq745_HTML.gif is a weak lower solution between https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq746_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq747_HTML.gif of the scalar problem
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq748_HTML.gif , where https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq750_HTML.gif . Then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq751_HTML.gif .

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

hence https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq753_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq754_HTML.gif a.e. on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq755_HTML.gif , thus https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq756_HTML.gif . Therefore https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq758_HTML.gif , it is the greatest one in https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq762_HTML.gif , https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq764_HTML.gif was fixed arbitrarily, so https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq766_HTML.gif . On the other hand, https://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 https://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:
https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq771_HTML.gif we have https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq772_HTML.gif and therefore https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq773_HTML.gif , https://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
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq775_HTML.gif we have https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq776_HTML.gif and then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq777_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq778_HTML.gif , is a lower solution.

The function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq779_HTML.gif is continuous between the graphs of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq780_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq781_HTML.gif except over the lines https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq782_HTML.gif , https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq784_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq785_HTML.gif (note that https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq787_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq788_HTML.gif ).

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

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq790_HTML.gif is such that https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq792_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq793_HTML.gif . Moreover, this weak solution between https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq794_HTML.gif and https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq797_HTML.gif to some https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq800_HTML.gif , as (7.1) with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq801_HTML.gif replaced by https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq804_HTML.gif such that
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_Equ86_HTML.gif
(7.5)
Since
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq805_HTML.gif is defined at least on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq806_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq807_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq808_HTML.gif .

Example 7.2.

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq809_HTML.gif be measurable and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq810_HTML.gif for a.a. https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq811_HTML.gif . We will prove that for each https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq812_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq813_HTML.gif , the problem
https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq815_HTML.gif on https://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, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq817_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq818_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq819_HTML.gif .

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

This shows that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq842_HTML.gif is positive between https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq843_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq845_HTML.gif it suffices to take https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq846_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq847_HTML.gif to have https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq848_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq849_HTML.gif between the graphs of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq850_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq851_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq853_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq854_HTML.gif . Moreover, since https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq855_HTML.gif is positive between https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq856_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq858_HTML.gif be fixed and consider the problem
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq859_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq860_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq861_HTML.gif . Since https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq862_HTML.gif is nonnegative between https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq863_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq864_HTML.gif the lines https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq865_HTML.gif , https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq867_HTML.gif if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq868_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq869_HTML.gif , thus one can construct https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq870_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq871_HTML.gif for a.a. https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq872_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq874_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq876_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F507671/MediaObjects/13661_2009_Article_850_IEq877_HTML.gif as https://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.