Minimal and maximal solutions to first-order differential equations with state-dependent deviated arguments

  • Rubén Figueroa1Email author and

    Affiliated with

    • Rodrigo López Pouso1

      Affiliated with

      Boundary Value Problems20122012:7

      DOI: 10.1186/1687-2770-2012-7

      Received: 13 May 2011

      Accepted: 20 January 2012

      Published: 20 January 2012

      Abstract

      We prove some new results on existence of solutions to first-order ordinary differential equations with deviated arguments. Delay differential equations are included in our general framework, which even allows deviations to depend on the unknown solutions. Our existence results lean on new definitions of lower and upper solutions introduced in this article, and we show with an example that similar results with the classical definitions are false. We also introduce an example showing that the problems considered need not have the least (or the greatest) solution between given lower and upper solutions, but we can prove that they do have minimal and maximal solutions in the usual set-theoretic sense. Sufficient conditions for the existence of lower and upper solutions, with some examples of application, are provided too.

      1 Introduction

      Let I0 = [t0, t0 + L] be a closed interval, r ≥ 0, and put I- = [t0 - r, t0] and I = I-I0. In this article, we are concerned with the existence of solutions for the following problem with deviated arguments:
      x ( t ) = f ( t , x ( t ) , x ( τ ( t , x ) ) ) for almost all ( a . a . ) t I 0 , x ( t ) = Λ ( x ) + k ( t ) for all t I - , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_Equ1_HTML.gif
      (1)

      where f : I × ℝ2 → ℝ and τ : I 0 × C ( I ) I http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_IEq1_HTML.gif are Carathéodory functions, Λ : C ( I ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_IEq2_HTML.gif is a continuous nonlinear operator and k C ( I - ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_IEq3_HTML.gif. Here C ( J ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_IEq4_HTML.gif denotes the set of real functions which are continuous on the interval J.

      For example, our framework admits deviated arguments of the form
      τ ( t , x ) = sin 2 ( x ( t ) ) t 0 + ( 1 - sin 2 ( x ( t ) ) ) ( t 0 + L ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_Equa_HTML.gif
      or
      τ ( t , x ) = t - I x ( s ) d s 1 + I x ( s ) d s r . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_Equb_HTML.gif

      We define a solution of problem (1) to be a function x C ( I ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_IEq5_HTML.gif such that x | I 0 A C ( I 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_IEq6_HTML.gif (i.e., x | I 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_IEq7_HTML.gif is absolutely continuous on I0) and x fulfills (1).

      In the space C ( I ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_IEq8_HTML.gif we consider the usual pointwise partial ordering, i.e., for γ 1 , γ 2 C ( I ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_IEq9_HTML.gif we define γ1γ2 if and only if γ1(t) ≤ γ2(t) for all tI. A solution of (1), x*, is a minimal (respectively, maximal) solution of (1) in a certain subset Y C ( I ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_IEq10_HTML.gif if x*Y and the inequality xx*, (respectively, xx*) implies x = x*, whenever x is a solution to (1) and xY. We say that x* is the least (respectively the greatest) solution of (1) in Y if x*x (respectively x*x) for any other solution xY. Notice that the least solution in a subset Y is a minimal solution in Y, but the converse is false in general, and an analgous remark is true for maximal and greatest solutions.

      Interestingly, we will show that problem (1) may have minimal (maximal) solutions between given lower and upper solutions and not have the least (greatest) solution. This seems to be a peculiar feature of equations with deviated arguments, see [1] for an example with a second-order equation. Therefore, we are obliged to distinguish between the concepts of minimal solution and least solution (or maximal and greatest solutions), unfortunately often identified in the literature on lower and upper solutions.

      First-order differential equations with state-dependent deviated arguments have received a lot of attention in the last years. We can cite the recent articles [27] which deal with existence results for this kind of problems. For the qualitative study of this type of problems we can cite the survey of Hartung et al. [8] and references therein.

      As main improvements in this article with regard to previous works in the literature we can cite the following:
      1. (1)

        The deviating argument τ depends at each moment t on the global behavior of the solution, and not only on the values that it takes at the instant t.

         
      2. (2)

        Delay problems, which correspond to differential equations of the form x'(t) = f(t, x(t), x(t - r)) along with a functional start condition, are included in the framework of problem (1). This is not allowed in articles [36].

         
      3. (3)

        No monotonicity conditions are required for the functions f and τ, and they need not be continuous with respect to their first variable.

         

      This article is organized as follows. In Section 2, we state and prove the main results in this article, which are two existence results for problem (1) between given lower and upper solutions. The first result ensures the existence of maximal and minimal solutions, and the second one establishes the existence of the greatest and the least solutions in a particular case. The concepts of lower and upper solutions introduced in Section 2 are new, and we show with an example that our existence results are false if we consider lower and upper solutions in the usual sense. We also show with an example that our problems need not have the least or the greatest solution between given lower and upper solutions. In Section 3, we prove some results on the existence of lower and upper solutions with some examples of application.

      2 Main results

      We begin this section by introducing adequate new definitions of lower and upper solutions for problem (1).

      Notice first that τ(t, γ) ∈ I = I-I0 for all ( t , γ ) I 0 × C ( I ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_IEq11_HTML.gif, so for each tI0we can define
      τ * ( t ) = inf γ C ( I ) τ ( t , γ ) I , τ * ( t ) = sup γ C ( I ) τ ( t , γ ) I . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_Equc_HTML.gif
      Definition 1 We say that α , β C ( I ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_IEq12_HTML.gif, with αβ on I, are a lower and an upper solution for problem (1) if α | I 0 , β | I 0 A C ( I 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_IEq13_HTML.gif and the following inequalities hold:
      α ( t ) min ξ E ( t ) f ( t , α ( t ) , ξ ) f o r a . a . t I 0 , α inf γ [ α , β ] Λ ( γ ) + k o n I - , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_Equ2_HTML.gif
      (2)
      β ( t ) min ξ E ( t ) f ( t , β ( t ) , ξ ) f o r a . a . t I 0 , β sup γ [ α , β ] Λ ( γ ) + k o n I - , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_Equ3_HTML.gif
      (3)
      where
      E ( t ) = min s [ τ * ( t ) , τ * ( t ) ] α ( s ) , max s [ τ * ( t ) , τ * ( t ) ] β ( s ) ( t I 0 ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_Equd_HTML.gif

      and [ α , β ] = { γ C ( I ) : α γ β } . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_IEq14_HTML.gif.

      Remark 1 Definition 1 requires implicitly that Λ be bounded in [α, β].

      On the other hand, the values
      min ξ E ( t ) f ( t , α ( t ) , ξ ) a n d max ξ E ( t ) f ( t , β ( t ) , ξ ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_Eque_HTML.gif

      are really attained for almost every fixed tI0 thanks to the continuity of f(t, α(t), ·) and f(t, β(t), ·) on the compact set E(t).

      Now we introduce the main result of this article.

      Theorem 1 Assume that the following conditions hold:

      (H1) (Lower and upper solutions) There exist α , β C ( I ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_IEq12_HTML.gif, with αβ on I, which are a lower and an upper solution for problem (1).

      (H2) (Carathéodory conditions)

      (H2) - (a) For all x, y ∈ [mintIα(t), maxtIβ(t)] the function f(·,x,y) is measurable and for a.a. tI0, all x ∈ [α(t), β(t)] and all yE(t) (as defined in Definition 1) the functions f(t, ·, y) and f(t, x, •) are continuous.

      (H2) - (b) For all γ [ α , β ] = { ξ C ( I ) : α ξ β } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_IEq15_HTML.gif the function τ(·, γ) is measurable and for a.a. tI0 the operator τ(t, ·) is continuous in C ( I ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_IEq8_HTML.gif (equipped with it usual topology of uniform convergence).

      (H2) - (c) The nonlinear operator Λ : C ( I ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_IEq16_HTML.gif is continuous.

      (H3) (L1-bound) There exists ψL1( I0) such that for a.a. tI0, all x ∈ [α(t), β(t)] and all yE(t) we have
      f ( t , x , y ) φ ( t ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_Equf_HTML.gif

      Then problem (1) has maximal and minimal solutions in [α, β].

      Proof. As usual, we consider the function
      p ( t , x ) = α ( t ) , if x < α ( t ) , x , if α ( t ) x β ( t ) , β ( t ) , if x > β ( t ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_Equg_HTML.gif
      and the modified problem
      x ( t ) = f ( t , p ( t , x ( t ) ) , p ( τ ( t , x ) , x ( τ ( t , x ) ) ) ) for a .a . t I 0 , x ( t ) = Λ ( p ( , x ( ) ) ) + k ( t ) for all t I - . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_Equ4_HTML.gif
      (4)
      Claim 1: Problem (4) has a nonempty and compact set of solutions. Consider the operator T : C ( I ) C ( I ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_IEq17_HTML.gif which maps each γ C ( I ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_IEq18_HTML.gif to a continuous function defined for each tI- as
      T γ ( t ) = Λ ( p ( , γ ( ) ) ) + k ( t ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_Equh_HTML.gif
      and for each tI0 as
      T γ ( t ) = Λ ( p ( , γ ( ) ) ) + k ( t 0 ) + t 0 t f ( s , p ( t , γ ( s ) ) , p ( τ ( s , γ ) , γ ( τ ( s , γ ) ) ) ) d s . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_Equi_HTML.gif

      It is an elementary matter to check that T is a completely continuous opera-tor from C ( I ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_IEq8_HTML.gif into itself (one has to take Remark 1 into account). Therefore, Schauder's Theorem ensures that T has a nonempty and compact set of fixed points in C ( I ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_IEq8_HTML.gif, which are exactly the solutions of problem (4).

      Claim 2: Every solution x of (4) satisfies αxβ on I and, therefore, it is a solution of (1) in [α, β]. First, notice that if x is a solution of (4) then p(·,x(·)) ∈ [α, β]. Hence the definition of lower solution implies that for all tI- we have
      α ( t ) Λ ( p ( , x ( ) ) ) + k ( t ) = x ( t ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_Equj_HTML.gif
      Assume now, reasoning by contradiction, that x α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_IEq19_HTML.gif on I0. Then we can find t ^ 0 t 0 , t 0 + L http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_IEq20_HTML.gif and ε > 0 such that α ( t ^ 0 ) = x ( t ^ 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_IEq21_HTML.gif and
      α ( t ) > x ( t ) for all t [ t ^ 0 , t ^ 0 + ε ] . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_Equ5_HTML.gif
      (5)
      Therefore, for all t [ t ^ 0 , t ^ 0 + ε ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_IEq22_HTML.gif we have p(t, x(t)) = α(t) and
      p ( τ ( t , x ) , x ( τ ( t , x ) ) ) [ α ( τ ( t , x ) ) , β ( τ ( t , x ) ) ] E ( t ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_Equk_HTML.gif
      so for a.a. s [ t ^ 0 , t ^ 0 + ε ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_IEq23_HTML.gif we have
      α ( s ) f ( s , p ( s , x ( s ) ) , p ( τ ( s , x ) , x ( τ ( s , x ) ) ) ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_Equl_HTML.gif
      Hence for t [ t ^ 0 , t ^ 0 + ε ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_IEq22_HTML.gif we have
      α ( t ) - x ( t ) = t 0 t α ( s ) d s - t ^ 0 t f ( s , p ( s , x ( s ) ) , p ( τ ( s , x ) , x ( τ ( s , x ) ) ) ) d s 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_Equm_HTML.gif

      a contradiction with (5).

      Similar arguments prove that all solutions x of (4) obey xβ on I. Claim 3: The set of solutions of problem (1) in [α, β] has maximal and minimal elements. The set
      S = { x C ( I ) : x is a solution of ( 1 ) , x [ α , β ] } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_Equn_HTML.gif
      is nonempty and compact in C ( I ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_IEq8_HTML.gif, beacuse it coincides with the set of fixed points of the operator T. Then, the real-valued continuous mapping
      x S ( x ) = t 0 t 0 + L x ( s ) d s http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_Equo_HTML.gif
      attains its maximum and its minimum, that is, there exist x * , x * S http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_IEq24_HTML.gif such that
      ( x * ) = max { ( x ) : x S } , ( x * ) = min { ( x ) : x S } . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_Equ6_HTML.gif
      (6)

      Now, if x S http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_IEq25_HTML.gif is such that xx* on I then we have ( x ) ( x * ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_IEq26_HTML.gif and, by (6), ( x ) ( x * ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_IEq27_HTML.gif. So we conclude that ( x ) = ( x * ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_IEq28_HTML.gif which, along with xx*, implies that x = x* on I. Hence x* is a maximal element of S http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_IEq29_HTML.gif. In the same way, we can prove that x* is a minimal element.

      One might be tempted to follow the standard ideas with lower and upper solutions to define a lower solution of (1) as some function α such that
      α ( t ) f ( t , α ( t ) , α ( τ ( t , α ) ) ) for a .a . t I 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_Equ7_HTML.gif
      (7)
      and an upper solution as some function β such that
      β ( t ) f ( t , β ( t ) , β ( τ ( t , β ) ) ) for a .a . t I 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_Equ8_HTML.gif
      (8)

      These definitions are not adequate to ensure the existence of solutions of (1) between given lower and upper solutions, as we show in the following example.

      Example 1 Consider the problem with delay
      x ( t ) = - x ( t - 1 ) for a .a . t [ 0 , 1 ] , x ( t ) = k ( t ) = - t for t [ - 1 , 0 ] . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_Equ9_HTML.gif
      (9)
      Notice that functions α(t) = 0 and β(t) = 1, t ∈ [-1, 1], are lower and upper solutions in the usual sense for problem (9). However, if x is a solution for problem (9) then for a.a. t ∈ [0, 1] we have
      x ( t ) = - x ( t - 1 ) = k ( t - 1 ) = - [ - ( t - 1 ) ] = t - 1 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_Equp_HTML.gif
      so for all t ∈ [0,1] we compute
      x ( t ) = x ( 0 ) + 0 t ( s - 1 ) d s = t 2 2 - t , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_Equq_HTML.gif

      and then x(t) < α(t) for all t ∈ (0,1]. Hence (9) has no solution at all between α and β.

      Remark 2 Notice that inequalities (2) and (3) imply (7) and (8), so lower and upper solutions in the sense of Definition 1 are lower and upper solutions in the usual sense, but the converse is false in general.

      Definition 1 is probably the best possible for (1) because it reduces to some definitions that one can find in the literature in connection with particular cases of (1). Indeed, when the function τ does not depend on the second variable then for all tI0 we have E(t) = [α(τ(t)), β(τ(t))] in Definition 1. Therefore, if f is nondecreasing with respect to its third variable, then Definition 1 and the usual definition of lower and upper solutions are the same (we will use this fact in the proof of Theorem 2). If, in turn, f is nonincreasing with respect to its third variable, then Definition 1 coincides with the usual definition of coupled lower and upper solutions (see for example [5]).

      In general, in the conditions of Theorem 1 we cannot expect problem (1) to have the extremal solutions in [α, β] (that is, the greatest and the least solutions in [α, β ]). This is justified by the following example.

      Example 2 Consider the problem
      x ( t ) = f ( t , x ( t ) , x ( τ ( t ) ) ) for a .a . t I 0 = - π 2 , π , x - π 2 = 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_Equ10_HTML.gif
      (10)
      where
      f ( t , x , y ) = 1 , if y < - 1 , - y , if - 1 y 1 , - 1 , if y > 1 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_Equr_HTML.gif

      and τ ( t ) = π 2 - t http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_IEq30_HTML.gif.

      First we check that α ( t ) = - t - π 2 = - β ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_IEq31_HTML.gif, tI0, are lower and upper solutions for problem (10). The definition of f implies that for all (t, x, y) ∈ I0 × ℝ2 we have |f(t, x, y)| ≤ 1, so for all tI0 we have
      min ξ E ( t ) f ( t , α ( t ) , ξ ) - 1 = α ( t ) a n d max ξ E ( t ) f ( t , β ( t ) , ξ ) 1 = β ( t ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_Equs_HTML.gif
      where, according to Definition 1,
      E ( t ) = α π 2 - t , β π 2 - t = [ t - π , π - t ] . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_Equt_HTML.gif

      Moreover, α - π 2 = β - π 2 = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_IEq32_HTML.gif, so α and β are, respectively, a lower and an upper solution for (10), and then condition (H1) of Theorem 1 is fulfilled. As conditions (H2) and (H3) are also satisfied (take, for example, ψ ≡ 1) we deduce that problem (1) has maximal and minimal solutions in [α, β]. However we will show that this problem does not have the extremal solutions in [α, β].

      The family x x (t) = λ cos t, tI0, with λ ∈ [-1,1], defines a set of solutions of problem (10) such that αx λ β for each λ ∈ [-1,1]. Notice that the zero solution is neither the least nor the greatest solution of (10) in [α, β]. Now let x ^ [ α , β ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_IEq33_HTML.gif be an arbitrary solution of problem (10) and let us prove that x ^ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_IEq34_HTML.gif is neither the least nor the greatest solution of (10) in [α, β]. First, if x ^ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_IEq34_HTML.gif changes sign in I0 then x ^ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_IEq34_HTML.gif cannot be an extremal solution of problem (10) because it cannot be compared with the solution x ≡ 0. If, on the other hand, x ^ 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_IEq35_HTML.gif in I0 then the differential equation yields x ^ 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_IEq36_HTML.gif a.e. on I0, which implies, along with the initial condition x ^ - π 2 = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_IEq37_HTML.gif, that x ^ ( t ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_IEq38_HTML.gif for all tI0. Reasoning in the same way, we can prove that x ^ 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_IEq39_HTML.gif in I0 implies x ^ 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_IEq40_HTML.gif. Hence, problem (10) does not have extremal solutions in [α, β].

      The previous example notwithstanding, existence of extremal solutions for problem (1) between given lower and upper solutions can be proven under a few more assumptions. Specifically, we have the following extremality result.

      Theorem 2 Consider the problem
      x ( t ) = f ( t , x ( t ) , x ( τ ( t ) ) ) f o r a . a . t I 0 , x ( t ) = Λ ( x ) + k ( t ) f o r a l l t I - . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_Equ11_HTML.gif
      (11)

      If (11) satisfies all the conditions in Theorem 1 and, moreover, f is nondecreasing with respect to its third variable and Λ is nondecreasing in [α, β], then problem (11) has the extremal solutions in [α, β].

      Proof. Theorem 1 guarantees that problem (11) has a nonempty set of solutions between α and β. We will show that this set of solutions is, in fact, a directed set, and then we can conclude that it has the extremal elements by virtue of [9, Theorem 1.2].

      According to Remark 2, the lower solution α and the upper solution β satisfy, respectively, inequalities (7) and (8) and, conversely, if α and β satisfy (7) and (8) then they are lower and upper solutions in the sense of Definition 1.

      Let x1, x2 ∈ [α, β] be two solutions of problem (11). We are going to prove that there is a solution x3 ∈ [α, β] such that x i x3 (i = 1, 2), thus showing that the set of solutions in [α, β] is upwards directed. To do so, we consider the function x ^ ( t ) = max { x 1 ( t ) , x 2 ( t ) } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_IEq41_HTML.gif, tI0, which is absolutely continuous on I0. For a.a. tI0 we have either
      x ^ ( t ) = f ( t , x ^ ( t ) , x 1 ( τ ( t ) ) ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_Equu_HTML.gif
      or
      x ^ ( t ) = f ( t , x ^ ( t ) , x 2 ( τ ( t ) ) ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_Equv_HTML.gif
      and, since f is nondecreasing with respect to its third variable, we obtain
      x ^ ( t ) f ( t , x ^ ( t ) , x 2 ( τ ( t ) ) ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_Equw_HTML.gif

      We also have x ^ ( t ) Λ ( x ^ ) + k ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_IEq42_HTML.gif in I- because Λ is nondecreasing, so x ^ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_IEq34_HTML.gif is a lower solution for problem (11). Theorem 1 ensures now that (11) has at least one solution x 3 [ x ^ , β ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_IEq43_HTML.gif.

      Analogous arguments show that the set of solutions of (11) in [α, β] is downwards directed and, therefore, it is a directed set.

      Next we show the applicability of Theorem 2.

      Example 3 Let L > 0 and consider the following differential equation with reflection of argument and a singularity at x = 0:
      x ( t ) = - t x ( - t ) for a .a . t [ 0 , L ] , x ( t ) = k ( t ) = t cos t - 3 t for all t [ - L , 0 ] . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_Equ12_HTML.gif
      (12)
      In this case, the function defining the equation is f ( t , y ) = - t y http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_IEq44_HTML.gif, which is nondecreasing with respect to y. On the other hand, functions
      α ( t ) = - 2 t if t < 0 , - 1 2 t , if 0 t L , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_Equx_HTML.gif
      and
      β ( t ) = - 4 t if t < 0 , 0 , if 0 t L , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_Equy_HTML.gif
      are lower and upper solutions for problem (12). Indeed, for t ∈ [-L,0] we have -2tk(t) ≤ -4t and for a.a. tI0 we have
      f ( t , α ( - t ) ) = - 1 2 = α ( t ) , f ( t , β ( - t ) ) = - 1 4 < β ( t ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_Equz_HTML.gif

      Hence α and β are lower and upper solutions for problem (12) by virtue of Remark 2.

      Finally, for a.a. tI0 and all y ∈ [α(-t), β(-t)] we have
      f ( t , x , y ) 1 4 , 1 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_Equaa_HTML.gif
      so problem (12) has the extremal solutions in [α, β ]. Notice that f admits a Carathéodory extension to I0 × ℝ outside the set
      { ( t , y ) I 0 × : α ( - t ) y β ( - t ) } , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_Equab_HTML.gif

      so Theorem 2 can be applied.

      In fact, we can explicitly solve problem (12) because the differential equation and the initial condition yield
      x ( t ) = 1 cos t - 3 for all t [ 0 , L ] , and x ( 0 ) = 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_Equac_HTML.gif
      hence problem (12) has a unique solution (see Figure 1) which is given by
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_Fig1_HTML.jpg
      Figure 1

      Solution of (12) bracketed by the lower and the upper solution.

      x ( t ) = 0 t d r cos r - 3 , t [ 0 , L ] . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_Equad_HTML.gif

      3 Construction of lower and upper solutions

      In general, condition (H1) is the most difficult to check among all the hypotheses in Theorem 1. Because of this, we include in this section some sufficient conditions on the existence of linear lower and upper solutions for problem (1) in particular cases We begin by considering a problem of the form
      x ( t ) = f ( x ( τ ( t , x ) ) ) for a .a . t I 0 = [ t 0 , t 0 + L ] , x ( t ) = k ( t ) for all t I - = [ t 0 - r , t 0 ] , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_Equ13_HTML.gif
      (13)

      where f C ( ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_IEq45_HTML.gif and k C ( I - ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_IEq46_HTML.gif.

      Proposition 1 Assume that f is a continuous function satisfying
      lim y + f ( y ) = + ; http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_Equ14_HTML.gif
      (14)
      lim y - f ( y ) = - ; http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_Equ15_HTML.gif
      (15)
      lim y ± f ( y ) y < 1 L . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_Equ16_HTML.gif
      (16)
      Then there exist m , m ̄ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_IEq47_HTML.gif such that the functions
      α ( t ) = φ * , i f t < t 0 , m ( t 0 - t ) + φ * , i f t t 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_Equ17_HTML.gif
      (17)
      and
      β ( t ) = φ * , i f t < t 0 , m ̄ ( t - t 0 ) + φ * , i f t t 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_Equ18_HTML.gif
      (18)
      are, respectively, a lower and an upper solution for problem (13), where
      φ * = min t I - k ( t ) , φ * = max t I - k ( t ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_Equae_HTML.gif

      In particular, problem (13) has maximal and minimal solutions between α and β, and this does not depend on the choice of τ.

      Proof. Conditions (15) and (16) imply that
      lim y - y - φ * f ( y ) > L , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_Equaf_HTML.gif
      so there exists y1 < min{0, φ t } such that
      0 > f ( y ) > y - φ * L if y y 1 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_Equ19_HTML.gif
      (19)
      On the other hand, condition (14) implies that there exists y2 > 0 such that
      f ( y ) > 0 if y y 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_Equ20_HTML.gif
      (20)
      Let λ = min{f(y): y1yy2}. By condition (15) and continuity of f, there exists y3y1 such that
      f ( y 3 ) = λ and f ( y ) λ for all y [ y 3 , y 1 ] , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_Equ21_HTML.gif
      (21)
      and this choice of y3 also provides that
      f ( y 3 ) f ( y ) for all y y 3 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_Equ22_HTML.gif
      (22)
      and, by virtue of (19),
      f ( y 3 ) > y 3 - φ * L . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_Equ23_HTML.gif
      (23)
      Now, define α as in (17), with m = φ * - y 3 L http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_IEq48_HTML.gif. Notice that α(t) ≤ k(t) for all tI-, α ( t ) = y 3 - φ * L http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_IEq49_HTML.gif for all tI0 and
      min t I α ( t ) = α ( t 0 + L ) = - m L + φ * = y 3 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_Equag_HTML.gif
      so we deduce from (22) and (23) that for all tI0 we have
      α ( t ) = - m < f ( y 3 ) = min y min I α ( t ) f ( y ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_Equ24_HTML.gif
      (24)
      In the same way, we can find ȳ 3 max { 0 , φ * } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_IEq50_HTML.gif such that β defined as in (18) with m ̄ = φ * - ȳ 3 L http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_IEq51_HTML.gif satisfies that β(t) ≥ k(t) for all tI- and
      β ( t ) = m ̄ max y max I β ( t ) f ( y ) for all t I 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_Equ25_HTML.gif
      (25)

      So we deduce from (24) and (25) that α and β are lower and upper solutions for problem (13).

      Example 4 The function
      f ( y ) = sgn ( y ) log y , if y ( - , - 1 ) ( 1 , ) , sin ( π y ) , if y [ - 1 , 1 ] , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_Equah_HTML.gif

      satisfies all the conditions in Proposition 1 for every compact interval I0. So the corresponding problem (13) has at least one solution for any choice of k C ( I - ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_IEq52_HTML.gif and τ C ( I , I ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_IEq53_HTML.gif.

      We use now the ideas of Proposition 1 to construct lower and upper solutions for the general problem (1).

      Proposition 2 Let k C ( I 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_IEq54_HTML.gif and let f : I0 × ℝ2 → ℝ be a Carathéodory function. Assume that there exist F α , F β C ( ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_IEq55_HTML.gif such that for a.a. tI0 and all y ∈ ℝ we have
      f ( t , x , y ) F α ( y ) f o r a l l x φ * http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_Equ26_HTML.gif
      (26)
      and
      f ( t , x , y ) F β ( y ) f o r a l l x φ * http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_Equ27_HTML.gif
      (27)
      Moreover, assume that the next conditions involving F α and F β hold:
      lim y - F α ( y ) = - , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_Equ28_HTML.gif
      (28)
      F α i s b o u n d e d f r o m b e l o w i n 0 , + , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_Equ29_HTML.gif
      (29)
      lim y - F α ( y ) y < 1 L , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_Equ30_HTML.gif
      (30)
      lim y + F β ( y ) = + , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_Equ31_HTML.gif
      (31)
      F β i s b o u n d e d f r o m a b o v e i n - , 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_Equ32_HTML.gif
      (32)
      lim y + F β ( y ) y < 1 L . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_Equ33_HTML.gif
      (33)

      Then there exist m , m ̄ 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_IEq56_HTML.gif such that α and β defined as in (17), (18) are lower and upper solutions for problem (1) with Λ = 0, and this does not depend on the choice of τ.

      Proof. Reasoning in the same way as in the proof of Proposition 1, we obtain that there exists m ≥ 0 such that α(t) ≤ φ* for all tI- and
      α ( t ) = - m min y min I α F α ( y ) for a .a . t I 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_Equai_HTML.gif
      As α(t) ≤ φ * for all tI, we obtain by virtue of (26) that
      α ( t ) min y min I α f ( t , α ( t ) , y ) for a .a . t I 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_Equaj_HTML.gif
      In the same way there exists m ̄ 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_IEq57_HTML.gif such that β(t) ≥ φ* for all tI- and
      β ( t ) = m ̄ min y max I β f ( t , β ( t ) , y ) for a .a . t I 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_Equak_HTML.gif

      Therefore, α and β are lower and upper solutions for problem (1).

      Example 5 Let F be the function defined in Example 4 and consider the problem
      x ( t ) = - ( x + π ) x + π γ g ( t , x ) + F ( x ( τ ( t , x ) ) ) for a .a . t [ 0 , L ] , x ( t ) = - t cos t for all t [ - π , 0 ] , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_Equ34_HTML.gif
      (34)

      where γ ≥ 0, L > 0, and g is a nonnegative Carathéodory function.

      In this case, we have φ* = -π, φ* ≈ 0.5611, and the function f(t,x,y) which defines the equation satisfies
      f ( t , x , y ) F ( y ) if x - π and f ( t , x , y ) F ( y ) if x - π , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_Equal_HTML.gif

      so in particular conditions (26) and (27) hold. As conditions (28)-(33) also hold (see Example 4) we obtain that there exist m , m ̄ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-7/MediaObjects/13661_2011_Article_111_IEq58_HTML.gif such that α and β defined as in (17), (18) are lower and upper solutions for problem (34) for any choice of τ. In particular, if there exists ψL1(I 0) such that for a.a. tI0 and all x ∈ [α(t), β(t)] we have g(t, x) ≤ ψ(t), then problem (34) has maximal and minimal solutions between α and β.

      Remark 3 Notice that the lower and upper solutions obtained both in Propositions 1 and 2 satisfy a slightly stronger condition than the one required in Definition 1.

      Declarations

      Acknowledgements

      This study was partially supported by the FEDER and Ministerio de Edu-cación y Ciencia, Spain, project MTM2010-15314.

      Authors’ Affiliations

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

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      © Figueroa and Pouso; licensee Springer. 2012

      This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://​creativecommons.​org/​licenses/​by/​2.​0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.