The Method of Coupled Fixed Points and Coupled Quasisolutions When Working with ODE’s with Arguments of Bounded Variation

Abstract

The aim of this paper is to show the use of the coupled quasisolutions method as a useful technique when dealing with ordinary differential equations with functional arguments of bounded variation. We will do this by looking for solutions for a first-order ordinary differential equation with an advanced argument of bounded variation. The main trick is to use the Jordan decomposition of this argument in a nondecreasing part and a nonincreasing one. As a necessary step, we will also talk about coupled fixed points of multivalued operators.

1. Introduction

In the paper [1], we proved a new result on the existence of coupled fixed points for multivalued operators, and then we used it to guarantee the existence of coupled quasisolutions and solutions to a certain first-order ordinary differential equation with state-dependent delay. In that paper, the nonlinearity was allowed to have both nondecreasing and nonincreasing arguments and the existence of solutions was obtained under strong Lipschitz conditions. We pointed out there that this tool could be useful when working with arguments of bounded variation, but no literature about this was written since then. So, the main goal in the present paper is to develop the application of the coupled quasisolutions technique in the framework of arguments of bounded variation, and we do it in an appropriate way, in order to take advantage of the Jordan decomposition and avoid the use of strong assumptions, as Lipschitz-continuity.

To show the application of this technique, we will study throughout this paper the existence of solutions for the following first-order problem: 𝑥^′(𝑡)((𝑡)(𝜏(𝑡[],𝑥(𝑡)(𝑡)[],=𝑓𝑡,𝑥,𝑥))),fora.a.𝑡∈𝐼=𝑎,𝑏=𝜙,∀𝑡∈𝑏,𝑏+𝑟 (1) where 𝑟≥0, 𝜏 is a measurable function such that 𝜏(𝑡)≥𝑡 for a.a. 𝑡; that is, 𝜏 is an advanced argument, and 𝜙 is a bounded function which represents the final state of the solution. By a solution of (1), we mean a function 𝑥∈𝒞[𝑎,𝑏+𝑟] such that 𝑥_|𝐼∈𝐴𝐶(𝐼) and 𝑥 satisfies both the differential equation (a.e. on 𝐼) and the final condition. We refer the readers to papers [2–4] to see more results on the existence of solutions and some applications of first-order problems with advance.

This paper is organized as follows. In Section 2, we gather some preliminary concepts and results involving functions of bounded variation and coupled fixed points of multivalued operators. These preliminaries are used later, in Section 3, to prove the existence of quasisolutions and solutions for problem (1). In Section 4, we show how our results can be adapted to deal with delay problems. Finally, in Section 5, some examples of application are available.

2. Preliminaries on Bounded Variation and Coupled Fixed Points of Multivalued Operators

In this section, we introduce some preliminaries that we will use throughout this work. First, we remember some concepts about functions of bounded variation. The reader can see more about this in the monographs [5, 6].

Definition 1. Given a function 𝑓∶𝐼=[𝑎,𝑏]⊂ℝ→ℝ and a partition 𝑃={𝑥₀,…,𝑥_𝑛} of 𝐼, one defines the variation of 𝑓 relative to the partition 𝑃 as the number 𝑉𝑓,𝑃=^𝑛_𝑖=1𝑓𝑥_𝑖−𝑓𝑥_𝑖−1, (2) and one defines the total variation of 𝑓 on 𝐼 as 𝑉^𝑏_𝑎𝑓=sup_𝑃∈𝒫𝑉𝑓,𝑃, (3) where 𝒫={𝑃∶𝑃isapartitionof𝐼}.
One says that 𝑓 is a function of bounded variation on 𝐼 if 𝑉^𝑏_𝑎(𝑓)<+∞. In that case, one writes 𝑓∈BV(𝐼).

Functions of bounded variation satisfy the following well-known result, which becomes essential now for our purposes.

Proposition 2 (Jordan decomposition). A function 𝑓 is of bounded variation on 𝐼 if and only if there exist a nondecreasing function, 𝑔, and a nonincreasing one, ℎ, such that 𝑓(𝑡)=𝑔(𝑡)+ℎ(𝑡),∀𝑡∈𝐼. (4)

The proof of Proposition 2 uses the fact that the function 𝑡∈𝐼→𝑉^𝑡_𝑎(𝑓) is nondecreasing and 𝑡→𝑓(𝑡)−𝑉^𝑡_𝑎(𝑓) is nonincreasing, and thus the desired decomposition is 𝑓(𝑡)=𝑉^𝑡_𝑎𝑓+𝑓(𝑡)−𝑉^𝑡_𝑎𝑓. (5) We remark that this decomposition is not unique. Finally, notice that, as a consequence of this result, every function of bounded variation is a.e. differentiable.

The set BV(𝐼) is an algebra which is included neither in the set of continuous functions nor in its complementary. Indeed, if 𝑓 is monotone on [𝑎,𝑏], then 𝑉^𝑏_𝑎(𝑓)=|𝑓(𝑏)−𝑓(𝑎)|, and thus 𝑓∈BV(𝐼). Then, there exist discontinuous functions which are of bounded variation (e.g., step function). In fact, it is also a well-known fact that if 𝑓∈BV(𝐼), then 𝑓 has only “jump” discontinuities. On the other hand, there exist continuous functions which are not of bounded variation, as (see [5, Example 6.3.1]) 𝑓(𝑡)=𝜋𝑡cos2𝑡if0<𝑡≤1,0if𝑡=0. (6)

To obtain our main result, we will use a generalized monotone method in presence of lower and upper solutions. This is a very well-known tool which is extensively used in the literature of ordinary differential equations. The classical version of this technique uses a pair of monotone sequences which will converge to the extremal solutions of the problem. The generalized version of this technique was developed in [7], and it is used when the nonlinearity has discontinuous arguments and therefore the pair of monotone sequences is replaced by a monotone operator. As a novelty which respect to the method developed in [7] and related references, we will use here a multivalued operator (i.e., a set-valued mapping) defined in a product space and then we will look for coupled fixed points. We concrete this idea in the following lines.

Definition 3. A metric space 𝑋 equipped with a partial ordering ≤ is an ordered metric space if the intervals [𝑥)={𝑦∈𝑋∶𝑥≤𝑦} and (𝑥]={𝑦∈𝑋∶𝑦≤𝑥} are closed for every 𝑥∈𝑋. Let 𝑃 be a subset of an ordered metric space. An operator 𝐴∶𝑃×𝑃→𝑃 is said to be mixed monotone if 𝐴(⋅,𝑥) is nondecreasing and 𝐴(𝑥,⋅) is nonincreasing for each 𝑥∈𝑃. One says that 𝐴 satisfies the mixed monotone convergence property (m.m.c.p.) if (𝐴(𝑣_𝑗,𝑤_𝑗))^∞_𝑗=1 converges in 𝑋 whenever (𝑣_𝑗)^∞_𝑗=1 and (𝑤_𝑗)^∞_𝑗=1 are sequences in 𝑃, one being nondecreasing and the other nonincreasing.

Definition 4. Let 𝑋 be a subset of an ordered metric space 𝑋. One defines a multivalued operator in the product 𝑋×𝑋 as a mapping 𝒜∶𝑋×𝑋⟶2^𝑋⧵∅. (7) We say that 𝑣, 𝑤∈𝑋 are coupled fixed points of 𝒜 if 𝑣∈𝒜(𝑣,𝑤) and 𝑤∈𝒜(𝑤,𝑣). We say that 𝑣_∗, 𝑤^∗∈𝑋 are the extremal coupled fixed points of 𝒜 in 𝑋 if 𝑣_∗, 𝑤^∗ are coupled fixed points of 𝒜 and if 𝑣, 𝑤∈𝑋 are another pair of coupled fixed points of 𝒜; then 𝑣_∗≤𝑣 and 𝑤≤𝑤^∗.

Theorem 5 (see [1, Theorem 2.1]). Let 𝑌 be a subset of an ordered metric space 𝑋, [𝛼,𝛽] be a nonempty closed interval in 𝑌, and 𝒜∶[𝛼,𝛽]×[𝛼,𝛽]→2^[𝛼,𝛽]⧵∅ be a multivalued operator.
If for all 𝑣, 𝑤∈[𝛼,𝛽], there exist 𝐴_∗()()∈,𝐴𝑣,𝑤=min𝒜𝑣,𝑤𝛼,𝛽_∗(𝑣,𝑤)=max𝒜(𝑣,𝑤)∈𝛼,𝛽, (8) and the (single-valued) operators 𝐴_∗ and 𝐴^∗ are mixed monotone and satisfy the m.m.c.p., then 𝒜 has the extremal coupled fixed points in [𝛼,𝛽], 𝑣_∗, 𝑣^∗. Moreover, they satisfy the following characterization: 𝑣_∗,𝑤^∗=min_⪯(𝑣,𝑤)∶𝐴_∗(𝑣,𝑤),𝐴^∗(𝑤,𝑣)⪯(𝑣,𝑤), (9) where (𝑣,𝑤)⪯(𝑣,𝑤)⟺𝑣≤𝑣,𝑤≥𝑤. (10)

3. Main Result

Now, we develop our generalized monotone method applied to problem (1). To do this, throughout this section, we will assume the following.(𝐻₁) There exists a closed interval 𝐽⊂ℝ, such that for a.a. 𝑡∈𝐼 and all 𝑥∈ℝ the function 𝑓(𝑡,𝑥,⋅) is of bounded variation on 𝐽.

Assumption (𝐻₁) implies that there exists a nondecreasing function, 𝑔, and a nonincreasing one, ℎ, such that 𝑓(𝑡,𝑥,⋅)=𝑔(𝑡,𝑥,⋅)+ℎ(𝑡,𝑥,⋅), (11) for all (𝑡,𝑥)∈𝐼×ℝ.

Now, we define what we mean by lower and upper solutions for problem (1).

Definition 6. One says that 𝛼, 𝛽∈𝒞[𝑎,𝑏+𝑟] are, respectively, a lower and upper solutions for problem (1), and if 𝛼_|𝐼, 𝛽_|𝐼∈𝐴𝐶(𝐼), min_{[]𝑡∈𝑎,𝑏+𝑟}𝛼(𝑡),max_{[]𝑡∈𝑎,𝑏+𝑟}𝛽(𝑡)⊂𝐽, (12) the compositions 𝑡⟼𝑓𝑡,𝛼(𝑡),𝑦,𝑡⟼𝑓𝑡,𝛽(𝑡),𝑦 (13) are measurable for all 𝑦∈𝐽 and the following inequalities hold: 𝛼^′(𝑡)(𝑡)(𝜏(𝑡((𝑡)(𝜏(𝑡,𝛼(𝑡)(𝑡)[],𝛽≥𝑔𝑡,𝛼,𝛽))+ℎ𝑡,𝛼,𝛼)))fora.a.𝑡∈𝐼,≤𝜙,∀𝑡∈𝑏,𝑏+𝑟_′(𝑡)(𝑡)(𝜏(𝑡(𝑡)(𝜏(𝑡,𝛽(𝑡)(𝑡)[].≤𝑔𝑡,𝛽,𝛼))+ℎ𝑡,𝛽,𝛽))fora.a.𝑡∈𝐼,≥𝜙,∀𝑡∈𝑏,𝑏+𝑟 (14)

Remark 7. Notice that, under the previous definition, the lower and the upper solutions appear “coupled.” On the other hand, it is assumed that min_{[]𝑡∈𝑎,𝑏+𝑟}𝛼(𝑡)≤max_{[]𝑡∈𝑎,𝑏+𝑟}𝛽(𝑡). (15) This is not a strong assumption, taking into account that, as usual, we will ask the lower and the upper solutions to be well ordered in the whole interval [𝑎,𝑏+𝑟].
On the other hand, the fact that 𝑡↦𝑓(𝑡,𝛼(𝑡),𝑦) and 𝑡∈𝐼↦𝑓(𝑡,𝛽(𝑡),𝑦) are being measurable for all 𝑦∈𝐽 implies that the compositions (𝑡)(𝜏(𝑡((𝑡)(𝜏(𝑡,(𝑡)(𝜏(𝑡(𝑡)(𝜏(t𝑡∈𝐼⟼𝑔𝑡,𝛼,𝛽))+ℎ𝑡,𝛼,𝛼)))𝑡∈𝐼⟼𝑔𝑡,𝛽,𝛼))+ℎ𝑡,𝛽,𝛽)) (16) are measurable too, because 𝑔 and ℎ are being monotone with respect to their last variables.

As we said in the Introduction, an essential tool in our work is the use of coupled quasisolutions. So, we introduce now this concept.

Definition 8. One says that two functions 𝑥_∗, 𝑥^∗∈𝒞[𝑎,𝑏+𝑟] are coupled quasisolutions of problem (1), and if 𝑥_∗|𝐼, 𝑥^∗_|𝐼∈𝐴𝐶(𝐼), 𝑥_∗(𝑡)=𝑥^∗(𝑡)=𝜙(𝑡) for all 𝑡∈[𝑏,𝑏+𝑟] and for a.a. 𝑡∈𝐼, they satisfy 𝑥^′_∗(𝑡)=𝑔𝑡,𝑥_∗(𝑡),𝑥^∗(𝜏(𝑡))+ℎ𝑡,𝑥_∗(𝑡),𝑥_∗(𝜏(𝑡,𝑥))_∗′(𝑡)=𝑔𝑡,𝑥^∗(𝑡),𝑥_∗(𝜏(𝑡))+ℎ𝑡,𝑥^∗(𝑡),𝑥^∗(𝜏(𝑡)). (17) We say that these coupled quasisolutions are extremal in a subset 𝑋⊂𝒞[𝑎,𝑏+𝑟] if 𝑥_∗, 𝑥^∗∈𝑋 and 𝑥_∗(𝑡)≤𝑥₁(𝑡), 𝑥₂(𝑡)≤𝑥^∗(𝑡) whenever 𝑥₁, 𝑥₂∈𝑋 is another pair of quasisolutions.

We need the following maximum principle related to problems with advance, as an auxiliar tool, for proving our main result. Compare it with [3, Lemma 3.2], [4, Lemma 1].

Lemma 9. Let 𝜏∶𝐼→[𝑎,𝑏+𝑟] be a measurable function such that 𝜏(𝑡)≥𝑡 for a.a. 𝑡∈𝐼 and assume that 𝑝∈𝒞[𝑎,𝑏+𝑟] such that 𝑝_|𝐼∈𝐴𝐶(𝐼) and satisfies 𝑝^′(𝑡)(𝑡)𝑝(𝑡)(𝑡)𝑝(𝜏(𝑡𝑝(𝑡)[],≥𝐾−𝐿))𝑓𝑜𝑟𝑎.𝑎.𝑡∈𝐼,=0,∀𝑡∈𝑏,𝑏+𝑟 (18) where 𝐾, 𝐿∈𝐿¹(𝐼) and 𝐿≥0 a.e.
If ^𝑏_𝑎𝐾₋(𝑡)+𝐿(𝑡)𝑑𝑡<1, (19) where 𝐾₋=max{−𝐾,0}, then 𝑝(𝑡)≤0 for all 𝑡∈[𝑎,𝑏+𝑟].

Proof. Let 𝑡₁∈[𝑎,𝑏+𝑟] such that 𝑝𝑡₁=max_{[]𝑡∈𝑎,𝑏+𝑟}𝑝(𝑡) (20) and assume by contradiction that 𝑝(𝑡₁)>0. Then, 𝑡₁∈[𝑎,𝑏). Now, let 𝑡₂∈(𝑡₁,𝑏] such that 𝑝(𝑡₂)=0 and 𝑝(𝑡)≥0 for all 𝑡∈[𝑡₁,𝑡₂]. Now, integrating 𝑡₁ and 𝑡₂, we obtain 𝑝𝑡₁=−^𝑡2_𝑡1𝑝^′(𝑡)𝑑𝑡≤−^𝑡2_𝑡1𝐾(𝑡)𝑝(𝑡)𝑑𝑡+^𝑡2_𝑡1𝐿(𝑡)𝑝(𝜏(𝑡𝑡))𝑑𝑡≤𝑝₁^𝑡2_𝑡1𝐾₋(𝑡)+𝐿(𝑡)𝑑𝑡, (21) and then condition (49) provides the contradiction 𝑝(𝑡₁)<𝑝(𝑡₁).

The main result on this paper concerns the existence of extremal quasisolutions and solutions for problem (1). It is as follows.

Theorem 10. Assume (𝐻₁) and that there exist 𝛼, 𝛽∈𝒞[𝑎,𝑏+𝑟] which are, respectively, lower and upper solutions for problem (1) such that 𝛼(𝑡)≤𝛽(𝑡) for all 𝑡∈[𝑎,𝑏+𝑟] and 𝐸=min_{[]𝑡∈𝑎,𝑏+𝑟}𝛼(𝑡),max_{[]𝑡∈𝑎,𝑏+𝑟}𝛽(𝑡)⊂𝐽. (22) Assume moreover that the following conditions hold: (𝐻₂) for each 𝛾₁, 𝛾₂∈[𝛼,𝛽]={𝛾∈𝒞[𝑎,𝑏+𝑟]∶𝛼(𝑡)≤𝛾(𝑡)≤𝛽(𝑡) for all 𝑡∈[𝑎,𝑏+𝑟]}, the final value problem 𝑃_𝛾1,𝛾2𝑥^′(𝑡)=𝐹_𝛾2,𝛾1((𝑡(𝑡)𝑡,𝑥))∶=𝑔𝑡,𝑥,𝛾₂(𝜏(𝑡(𝑡)))+ℎ𝑡,𝑥,𝛾₁(𝜏(𝑡𝑥(𝑏)(𝑏))),𝑓𝑜𝑟𝑎.𝑎.𝑡∈𝐼,=𝜙 (23) has the extremal solutions in [𝛼,𝛽]; (𝐻₃) there exists 𝜓∈𝐿¹(𝐼,[0,+∞)) such that for a.a. 𝑡∈𝐼, all 𝑥∈[𝛼(𝑡),𝛽(𝑡)], and all 𝑦₁, 𝑦₂∈[𝛼(𝜏(𝑡)),𝛽(𝜏(𝑡))], one has 𝑔𝑡,𝑥,𝑦₁+ℎ𝑡,𝑥,𝑦₂≤𝜓(𝑡); (24) (𝐻₄) there exists 𝐾₁, 𝐾₂, 𝐿₁, 𝐿₂∈𝐿¹(𝐼) such that 𝐿₁, 𝐿₂≥0 a.e. and 𝑔𝑡,𝑥,𝑦−𝑔𝑡,𝑥,𝑦≥𝐾₁(𝑡)(𝑥−𝑥)−𝐿₁(𝑡),ℎ𝑦−𝑦𝑡,𝑥,𝑦−ℎ𝑡,𝑥,𝑦≥𝐾₂(𝑡)(𝑥−𝑥)−𝐿₂(𝑡)𝑦−𝑦 (25) whenever 𝛼(𝑡)≤𝑥≤𝑥≤𝛽(𝑡) and min_{[]𝑠∈𝑏,𝑏+𝑟}𝜙(𝑠)−^𝑏_𝑡𝜓(𝑠)𝑑𝑠≤𝑦≤𝑦≤max_{[]𝑠∈𝑏,𝑏+𝑟}𝜙(𝑠)+^𝑏_𝑡𝜓(𝑠)𝑑𝑠. (26) Moreover, ^𝑏_𝑎𝐾₋(𝑡)+𝐿(𝑡)𝑑𝑡<1, (27) where 𝐾=𝐾₁+𝐾₂, 𝐿=𝐿₁+𝐿₂ and 𝐾₋(𝑡)=max{−𝐾(𝑡),0}.
In these conditions, problem (1) has a unique solution in [𝛼,𝛽].

Proof. We consider the space 𝑋=𝒞[𝑎,𝑏+𝑟] endowed with the ordering 𝛾₁≤𝛾₂⟺𝛾₁(𝑡)≤𝛾₂(𝑡),∀𝑡∈[𝑎,𝑏+𝑟], (28) and we define a multivalued operator 𝒜∶𝛼,𝛽×𝛼,𝛽⊂𝑋×𝑋⟶2^[𝛼,𝛽]⧵∅ (29) as follows: for each 𝛾₁, 𝛾₂∈[𝛼,𝛽], we have 𝑥∈𝒜(𝛾₁,𝛾₂) if and only if 𝑥∈[𝛼,𝛽], 𝑥_𝐼 is a solution of (𝑃_𝛾1,𝛾2) and 𝑥_{|[𝑏,𝑏+𝑟]}=𝜙.
Step 1. Operator 𝒜 has the extremal coupled fixed points in [𝛼,𝛽]. By virtue of condition (𝐻₂), operator 𝒜 is well defined and there exist 𝐴_∗=min𝒜𝛾₁,𝛾₂,𝐴^∗=max𝒜𝛾₁,𝛾₂. (30) We will show now that 𝐴_∗, 𝐴^∗ are mixed monotone and satisfy m.m.c.p. So, let 𝛾₁,𝛾₁,𝛾₂,𝛾₂∈𝛼,𝛽 (31) such that 𝛾₁≤𝛾₁, 𝛾₂≤𝛾₂ and put 𝑥₁=𝐴_∗𝛾₁,𝛾₂,𝑥₁=𝐴_∗𝛾₁,𝛾₂,𝑥₂=𝐴_∗𝛾₁,𝛾₂. (32) Then, for all 𝑡∈[𝑏,𝑏+𝑟], we have that 𝑥₁(𝑡)=𝑥₁(𝑡)=𝑥₂(𝑡), and for a.a. 𝑡∈𝐼, we have 𝑥^′₁(𝑡)=𝑔𝑡,𝑥₁(𝑡),𝛾₂(𝜏(𝑡))+ℎ𝑡,𝑥₁,𝛾₁(𝜏(𝑡))≤𝑔𝑡,𝑥₁(𝑡),𝛾₂(𝜏(𝑡))+ℎ𝑡,𝑥₁,𝛾₁(𝜏(𝑡)). (33) And so, 𝑥₁ is an upper solution for problem (𝑃_𝛾1,𝛾2). The fact that 𝑥₁ is being the least solution of this problem in [𝛼,𝛽] implies that 𝑥₁≥𝑥₁ and then 𝐴_∗(⋅,𝛾₂) is nondecreasing. On the other hand, 𝑥^′₁(𝑡)=𝑔𝑡,𝑥₁(𝑡),𝛾₂(𝜏(𝑡))+ℎ𝑡,𝑥₁(𝑡),𝛾₁(𝜏(𝑡))≤𝑔𝑡,𝑥₁(𝑡),𝛾₂(𝜏(𝑡))+ℎ𝑡,𝑥₁(𝑡),𝛾₁(𝜏(𝑡)), (34) and therefore 𝑥₁ is an upper solution for problem (𝑃_𝛾1,𝛾2). Then, 𝑥₁≥𝑥₂, and so the mapping 𝐴_∗(𝛾₁,⋅) is nonincreasing. In the same way, we show that 𝐴^∗ is mixed monotone.
To see that 𝐴_∗, 𝐴^∗ satisfy the m.m.c.p., let (𝑣_𝑗)^∞_𝑗=1, (𝑤_𝑗)^∞_𝑗=1 be sequences in [𝛼,𝛽], one being nondecreasing and the other being nonincreasing. As 𝐴_∗, 𝐴^∗ are mixed monotone and bounded, we obtain that the sequences (𝐴_∗(𝑣_𝑗,𝑤_𝑗))^∞_𝑗=1, (𝐴^∗(𝑣_𝑗,𝑤_𝑗))^∞_𝑗=1 have their pointwise limit; say 𝑧_∗, 𝑧^∗. As (𝐴_∗(𝑣_𝑗,𝑤_𝑗))^∞_𝑗=1, (𝐴^∗(𝑣_𝑗,𝑤_𝑗))^∞_𝑗=1 are constant in [𝑏,𝑏+𝑟], the convergence is uniform in this interval. On the other hand, for 𝑡, 𝑠∈𝐼, 𝑠<𝑡, and 𝑗∈ℕ, we have 𝑧^∗_𝑗(𝑡)−𝑧^∗_𝑗(𝑠≤_𝑡𝑠)𝑔𝑟,𝑧^∗_𝑗(𝑟),𝑤_𝑗(𝜏(𝑟))+ℎ𝑡,𝑧^∗_𝑗(𝑟),𝑣_𝑗(𝜏(𝑟≤))𝑑𝑟_𝑡𝑠𝜓(𝑟)𝑑𝑟, (35) and thus (𝑧^∗_𝑗)^∞_𝑗=1 converges to 𝑧^∗ uniformly on 𝐼. The same argument is valid for 𝑧_∗.
By application of Theorem 5, operator 𝒜 has the extremal coupled fixed points in [𝛼,𝛽]; say 𝑥_∗, 𝑥^∗.
Step 2. Problem (1) has the extremal quasisolutions in [𝛼,𝛽]. Indeed, we will show that the extremal coupled fixed points of operators 𝒜, 𝑥_∗, and 𝑥^∗ correspond with these extremal quasisolutions. First, it is clear that if 𝑥, 𝑥∈[𝛼,𝛽] are coupled fixed points of 𝒜, then they are coupled quasisolutions of problem (1). On the other hand, if 𝑥, 𝑥 are quasisolutions of problem (1), then 𝐴_∗(𝑥,𝑥)≤𝑥 and 𝐴^∗(𝑥,𝑥)≥𝑥, and then characterization (9) implies that 𝑥_∗≤𝑥 and 𝑥≤𝑥^∗. This shows that 𝑥_∗, 𝑥^∗ are the extremal quasisolutions of problem (1) in [𝛼,𝛽].
Step 3. Problem (1) has a unique solution in [𝛼,𝛽]. We will prove this by showing that the extremal quasisolutions 𝑥_∗, 𝑥^∗ are, in fact, the same functions, and thus defining a solution of the problem. This solution must be unique in [𝛼,𝛽] because if 𝑥∈[𝛼,𝛽] is a solution of (1), then the pair 𝑥, 𝑥 is also a quasisolution, and then 𝑥_∗≤𝑥≤𝑥^∗.
To see that 𝑥_∗=𝑥^∗, first notice that as (𝑥_∗,𝑥^∗) is a pair of quasisolutions; then, the reversed pair, (𝑥^∗,𝑥_∗), is quasisolutions too, and then, extremality implies 𝑥_∗≤𝑥^∗. Moreover, condition (𝐻₃) implies that for a.a. 𝑡∈𝐼𝑥_∗(𝑡),𝑥^∗(𝑡)∈𝜙(𝑏)−^𝑏_𝑡𝜓(𝑠)𝑑𝑠,𝜙(𝑏)+^𝑏_𝑡𝜓(𝑠)𝑑𝑠. (36)
Now, define the function 𝑝(𝑡)=𝑥^∗(𝑡)−𝑥_∗(𝑡)≥0. On the one hand, 𝑝(𝑡)=0 for all 𝑡∈[𝑏,𝑏+𝑟]. On the other hand, condition (𝐻₄) implies for a.a. 𝑡∈𝐼 that 𝑝^′(𝑡)=𝑔𝑡,𝑥^∗(𝑡),𝑥_∗(𝜏(𝑡))−𝑔𝑡,𝑥_∗(𝑡),𝑥^∗(𝜏(𝑡))+ℎ𝑡,𝑥^∗(𝑡),𝑥^∗(𝜏(𝑡))−ℎ𝑡,𝑥_∗(𝑡),𝑥_∗(𝜏(𝑡(𝑡)𝑥))≥𝐾^∗(𝑡)−𝑥_∗(𝑡)−𝐿(𝑡)𝑥^∗(𝜏(𝑡))−𝑥_∗(𝜏(𝑡)), (37) and then by virtue of Lemma 9, we obtain that 𝑝(𝑡)≤0 on 𝐼. We conclude that 𝑝(𝑡)=0 for all 𝑡∈[𝑎,𝑏+𝑟]; that is, 𝑥_∗=𝑥^∗. This ends the proof.

Remark 11. Now, we point out some remarks related to Theorem 10.(1) Condition (𝐻₂) could be replaced by any result on the existence of extremal solutions between lower and upper solutions for problem (𝑃_𝛾1,𝛾2). For example, as it is well known, if 𝐹_𝛾1,𝛾2 is a Carathéodory function, then (𝐻₃) implies that (𝑃_𝛾1,𝛾2) has the extremal solutions between 𝛼 and 𝛽. Moreover, there exists a very extensive literature about the existence of extremal solutions for problem (𝑃_𝛾1,𝛾2) for discontinuous 𝐹_𝛾1,𝛾2. The reader is referred to [1, 8–10] and references therein for some results of this type. Notice that although most of these references deal with initial value problems, these results can easily be adapted for final value problems. Finally, notice that (𝐻₂) implies, in particular, measurability of the composition 𝑡∈𝐼↦𝐹_𝛾1,𝛾2(𝑡,𝑥(𝑡)) for all 𝑥∈[𝛼,𝛽].(2) As we said in Section 2, a function of bounded variation has only “jump” discontinuities. Although condition (𝐻₄) implies that for a.a. 𝑡∈𝐼 the function 𝑓 is continuous with respect to its third variable in the interval min_{[]𝑠∈𝑏,𝑏+𝑟}𝜙(𝑠)−^𝑏_𝑡𝜓(𝑠)𝑑𝑠,max_{[]𝑠∈𝑏,𝑏+𝑟}𝜙(𝑠)+^𝑏_𝑡𝜓(s)𝑑𝑠, (38) a countable number of discontinuities are allowed to exist outside this interval. Moreover, notice that this interval can be improved if we find another function 𝜓 satisfying (𝐻₃) and such that 𝜓(𝑡)≤𝜓(𝑡) for a.a. 𝑡.(3) For almost all 𝑡∈𝐼 and all 𝑥∈[𝛼(𝑡),𝛽(𝑡)] the function 𝑓_𝑡,𝑥(⋅)=𝑓(𝑡,𝑥,⋅) is of bounded variation in [𝛼(𝜏(𝑡)),𝛽(𝜏(𝑡))], and thus there exists in this interval a decomposition 𝑓_𝑡,𝑥(⋅)=𝑔_𝑡,𝑥(⋅)+ℎ_𝑡,𝑥(⋅), with 𝑔 nondecreasing and ℎ nonincreasing. Although all conditions in Theorem 10 are stated for an arbitrary Jordan decomposition of this type, all of them can be rewritten with 𝑔_𝑡,𝑥𝑦=𝑉^𝑦_𝐴𝑓,ℎ_𝑡,𝑥𝑦=𝑓_𝑡,𝑥𝑦−𝑉^𝑦_𝐴𝑓, (39) for any choice of 𝐴≤min{𝛼(𝑡)∶𝑡∈[𝑎,𝑏+𝑟]}, 𝐴≥min𝐽.

Theorem 10 provides, in particular, a new result on the existence of extremal solutions for problem (1) in the case that function 𝑓 is nonincreasing with respect to its third variable. In this case, the nondecreasing part of the Jordan decomposition of 𝑓 does not exist, and therefore the lower and upper solutions introduced in Definition 6 appear uncoupled. Moreover, a pair of quasisolutions in the sense of Definition 8 becomes, in fact, a pair of solutions, and then extremal quasisolutions provided by Theorem 10 reduce to extremal solutions. We specify these ideas in the following corollary.

Corollary 12. Assume that there exist 𝛼, 𝛽∈𝒞[𝑎,𝑏+𝑟] such that 𝛼_𝐼, 𝛽_𝐼∈𝐴𝐶(𝐼), 𝛼≤𝛽 on [𝑎,𝑏+𝑟] and the following inequalities hold: 𝛼^′(𝑡)((𝑡)(𝜏(𝑡(𝑡)(𝑡)[],𝛽≥𝑓𝑡,𝛼,𝛼))),𝑓𝑜𝑟𝑎.𝑎.𝑡∈𝐼,𝛼≤𝜙∀𝑡∈𝑏,𝑏+𝑟_′(𝑡)(𝑡)(𝜏(𝑡(𝑡)(𝑡)[].≤𝑓𝑡,𝛽,𝛽)),𝑓𝑜𝑟𝑎.𝑎.𝑡∈𝐼,𝛽≥𝜙𝑡∈𝑏,𝑏+𝑟 (40) Assume moreover that the following conditions hold: (𝐻₂)^′ for all 𝛾∈[𝛼,𝛽], the final value problem 𝑥^′(𝑡)(𝑡)𝑥(𝑏)(𝑏)=𝑓𝑡,𝑥,𝛾,𝑓𝑜𝑟𝑎.𝑎.𝑡∈𝐼,=𝜙 (41) has the extremal solutions in [𝛼,𝛽]; (𝐻₃)^′ there exists 𝜓∈𝐿¹(𝐼,[0,+∞)) such that for a.a. 𝑡∈𝐼, all 𝑥∈[𝛼(𝑡),𝛽(𝑡)], and all 𝑦∈[𝛼(𝜏(𝑡)),𝛽(𝜏(𝑡))], one has 𝑓𝑡,𝑥,𝑦≤𝜓(𝑡); (42) (𝐻₄)^′ for a.a. 𝑡∈𝐼 and all 𝑥∈[𝛼(𝑡),𝛽(𝑡)], the function 𝑓(𝑡,𝑥,⋅) is nonincreasing.
In these conditions problem (1) has the extremal solutions in [𝛼,𝛽].

4. Delay Problems

The results obtained in the previous section can be easily reformulated in order to deal with problems with delay. We concrete this idea in the following lines.

Consider the following problem: 𝑥^′(𝑡)((𝑡)(𝜏(𝑡[],𝑥(𝑡)(𝑡)[],=𝑓𝑡,𝑥,𝑥))),fora.a.𝑡∈𝐼=𝑎,𝑏=𝜙,∀𝑡∈𝑎−𝑟,𝑎 (43) where 𝑟≥0, 𝜏 is a measurable function such that 𝜏(𝑡)≤𝑡 for a.a. 𝑡; that is, 𝜏 is a delayed argument and 𝜙 is a bounded function which represents the initial state of the solution. Now, by a solution of (43), we mean a function 𝑥∈𝒞[𝑎−𝑟,𝑏] such that 𝑥_|𝐼∈𝐴𝐶(𝐼) and 𝑥 satisfies both the differential equation (a.e. on 𝐼) and the initial condition.

As we said, we will show now that we can use our technique to obtain a new result on the existence of solutions for problem (43) in the case that function 𝑓(𝑡,𝑥,⋅) is of bounded variation. We begin by reformulating the concept of lower and upper solutions and coupled quasisolutions in order to adapt them to our new problem. As in previous section, we assume (𝐻₁).

Definition 13. One says that 𝛼, 𝛽∈𝒞[𝑎−𝑟,𝑏] are, respectively, lower and upper solutions for problem (43), and if 𝛼_|𝐼, 𝛽_|𝐼∈𝐴𝐶(𝐼), min_{[]𝑡∈𝑎−𝑟,𝑏}𝛼(𝑡),max_{[]𝑡∈𝑎−𝑟,𝑏}𝛽(𝑡)⊂𝐽, (44) the compositions 𝑡⟼𝑓𝑡,𝛼(𝑡),𝑦,𝑡⟼𝑓𝑡,𝛽(𝑡),𝑦 (45) are measurable for all 𝑦∈𝐽 and the following inequalities hold: 𝛼^′(𝑡)((𝑡)(𝜏(𝑡(𝑡)(𝜏(𝑡,𝛼(𝑡)(𝑡)[],𝛽≤𝑔𝑡,𝛼,𝛼)))+ℎ𝑡,𝛼,𝛽))fora.a.𝑡∈𝐼,≤𝜙,∀𝑡∈𝑎−𝑟,𝑎_′(𝑡)(𝑡)(𝜏(𝑡(𝑡)(𝜏(𝑡,𝛽(𝑡)(𝑡)[].≥𝑔𝑡,𝛽,𝛽))+ℎ𝑡,𝛽,𝛼))fora.a.𝑡∈𝐼,≥𝜙,∀𝑡∈𝑎−𝑟,𝑎 (46)

Definition 14. One says that two functions 𝑥_∗, 𝑥^∗∈𝒞[𝑎−𝑟,𝑏] are coupled quasisolutions of problem (43) if 𝑥_∗|𝐼, 𝑥^∗_|𝐼∈𝐴𝐶(𝐼), 𝑥_∗(𝑡)=𝑥^∗(𝑡)=𝜙(𝑡) for all 𝑡∈[𝑎−𝑟,𝑎] and for a.a. 𝑡∈𝐼, they satisfy 𝑥^′_∗(𝑡)=𝑔𝑡,𝑥_∗(𝑡),𝑥_∗(𝜏(𝑡))+ℎ𝑡,𝑥_∗(𝑡),𝑥^∗(𝜏(𝑡,𝑥))_∗′(𝑡)=𝑔𝑡,𝑥^∗(𝑡),𝑥^∗(𝜏(𝑡))+ℎ𝑡,𝑥^∗(𝑡),𝑥_∗(𝜏(𝑡)). (47) We say that these coupled quasisolutions are extremal in a subset 𝑋⊂𝒞[𝑎−𝑟,𝑏]; if 𝑥_∗, 𝑥^∗∈𝑋 and 𝑥_∗(𝑡)≤𝑥₁(𝑡), 𝑥₂(𝑡)≤𝑥^∗(𝑡) whenever 𝑥₁, 𝑥₂∈𝑋 is another pair of quasisolutions.

Before introducing our main result for problem (43), we need a maximum principle for problems with delay, which is as follows. Its proof is analogous to that done in Lemma 9, so we omit it.

Lemma 15. Let 𝜏∶𝐼→[𝑎−𝑟,𝑏] be a measurable function such that 𝜏(𝑡)≤𝑡 for a.a. 𝑡∈𝐼 and assume that 𝑝∈𝒞[𝑎−𝑟,𝑏] is such that 𝑝_|𝐼∈𝐴𝐶(𝐼) and satisfies 𝑝^′(𝑡)(𝑡)𝑝(𝑡)(𝑡)𝑝(𝜏(𝑡𝑝(𝑡)[],≤𝐾+𝐿)),𝑓𝑜𝑟𝑎.𝑎.𝑡∈𝐼,=0,∀𝑡∈𝑎−𝑟,𝑎 (48) where 𝐾, 𝐿∈𝐿¹(𝐼) and 𝐿≥0 a.e.
If ^𝑏_𝑎𝐾₊(𝑡)+𝐿(𝑡)𝑑𝑡<1, (49) where 𝐾₊=max{𝐾,0}, then 𝑝(𝑡)≤0 for all 𝑡∈[𝑎−𝑟,𝑏].

Now, we state our main result in this Section.

Theorem 16. Assume (𝐻₁) and that there exist 𝛼, 𝛽∈𝒞[𝑎−𝑟,𝑏] which are, respectively, lower and upper solutions for problem (43) such that 𝛼(𝑡)≤𝛽(𝑡) for all 𝑡∈[𝑎−𝑟,𝑏] and 𝐸=min_{[]𝑡∈𝑎−𝑟,𝑏}𝛼(𝑡),max_{[]𝑡∈𝑎−𝑟,𝑏}𝛽(𝑡)⊂𝐽. (50) Assume moreover that the following conditions hold: (𝐻₂) for each 𝛾₁, 𝛾₂∈[𝛼,𝛽]={𝛾∈𝒞[𝑎−𝑟,𝑏]∶𝛼(𝑡)≤𝛾(𝑡)≤𝛽(𝑡) for all 𝑡∈[𝑎−𝑟,𝑏]}, the initial value problem 𝑃_𝛾1,𝛾2𝑥^′(𝑡)=𝐹_𝛾1,𝛾2((𝑡(𝑡)𝑡,𝑥))∶=𝑔𝑡,𝑥,𝛾₁(𝜏(𝑡(𝑡)))+ℎ𝑡,𝑥,𝛾₂(𝜏(𝑡𝑥(a)(𝑎))),𝑓𝑜𝑟𝑎.𝑎.𝑡∈𝐼,=𝜙 (51) has the extremal solutions in [𝛼,𝛽]; (𝐻₃) there exists 𝜓∈𝐿¹(𝐼,[0,+∞)) such that for a.a. 𝑡∈𝐼, all 𝑥∈[𝛼(𝑡),𝛽(𝑡)], and all 𝑦₁, 𝑦₂∈[𝛼(𝜏(𝑡)),𝛽(𝜏(𝑡))], one has 𝑔𝑡,𝑥,𝑦₁+ℎ𝑡,𝑥,𝑦₂≤𝜓(𝑡); (52) (𝐻₄) there exists 𝐾₁, 𝐾₂, 𝐿₁, 𝐿₂∈𝐿¹(𝐼) such that 𝐿₁, 𝐿₂≥0 a.e. and 𝑔𝑡,𝑥,𝑦−𝑔𝑡,𝑥,𝑦≤𝐾₁(𝑡)(𝑥−𝑥)+𝐿₁(𝑡),ℎ𝑦−𝑦𝑡,𝑥,𝑦−ℎ𝑡,𝑥,𝑦≤𝐾₂(𝑡)(𝑥−𝑥)+𝐿₂(𝑡)𝑦−𝑦 (53) whenever 𝛼(𝑡)≤𝑥≤𝑥≤𝛽(𝑡) and min_{[]𝑠∈𝑎−𝑟,𝑎}𝜙(𝑠)−^𝑡_𝑎𝜓(𝑠)𝑑𝑠≤𝑦≤𝑦≤max_{[]𝑠∈𝑎−𝑟,𝑎}𝜙(𝑠)+^𝑡_𝑎𝜓(𝑠)𝑑𝑠. (54) Moreover, ^𝑏_𝑎𝐾₊(𝑡)+𝐿(𝑡)𝑑𝑡<1, (55) where 𝐾=𝐾₁+𝐾₂, 𝐿=𝐿₁+𝐿₂ and 𝐾₊(𝑡)=max{𝐾(𝑡),0}.
In these conditions, problem (43) has a unique solution in [𝛼,𝛽].

Proof. The proof is analogous to that done in Theorem 10, but now, redefining operator 𝒜 in this way, first, we consider the space 𝑋=𝒞[𝑎−𝑟,𝑏] endowed with the ordering 𝛾₁≤𝛾₂⟺𝛾₁(𝑡)≤𝛾₂(𝑡),∀𝑡∈[𝑎−𝑟,𝑏]. (56) Then, we consider the operator 𝒜∶𝛼,𝛽×𝛼,𝛽⊂𝑋×𝑋⟶2^[𝛼,𝛽]⧵∅ (57) as follows: for each 𝛾₁, 𝛾₂∈[𝛼,𝛽], we have 𝑥∈𝒜(𝛾₁,𝛾₂) if and only if 𝑥∈[𝛼,𝛽], 𝑥_𝐼 is a solution of (𝑃_𝛾1,𝛾2) and 𝑥_{|[𝑎−𝑟,𝑎]}=𝜙.
The rest of the proof is analogous, with obvious changes.

Now, Theorem 16 provides, in particular, a new result on the existence of extremal solutions in the case that function 𝑓 is nondecreasing with respect to its third variable. For the sake of completeness, we concrete this idea in the following Corollary, which is the analogous to Corollary 12.

Corollary 17. Assume that there exist 𝛼, 𝛽∈𝒞[𝑎−𝑟,𝑏] such that 𝛼_𝐼, 𝛽_𝐼∈𝐴𝐶(𝐼), 𝛼≤𝛽 on [𝑎−𝑟,𝑏] and the following inequalities hold: 𝛼^′(𝑡)((𝑡)(𝜏(𝑡(𝑡)(𝑡)[],𝛽≤𝑓𝑡,𝛼,𝛼))),𝑓𝑜𝑟𝑎.𝑎.𝑡∈𝐼,𝛼≤𝜙∀𝑡∈𝑎−𝑟,𝑎_′(𝑡)(𝑡)(𝜏(𝑡(𝑡)(𝑡)[].≥𝑓𝑡,𝛽,𝛽)),𝑓𝑜𝑟𝑎.𝑎.𝑡∈𝐼,𝛽≥𝜙∀𝑡∈𝑎−𝑟,𝑎 (58) Assume moreover that the following conditions hold: (𝐻₂)^′ for all 𝛾∈[𝛼,𝛽], the initial value problem 𝑥^′(𝑡)(𝑡)𝑥(𝑎)(𝑎)=𝑓𝑡,𝑥,𝛾𝑓𝑜𝑟𝑎.𝑎.𝑡∈𝐼,=𝜙 (59) has the extremal solutions in [𝛼,𝛽]; (𝐻₃)^′ there exists 𝜓∈𝐿¹(𝐼,[0,+∞)) such that for a.a. 𝑡∈𝐼, all 𝑥∈[𝛼(𝑡),𝛽(𝑡)], and all 𝑦∈[𝛼(𝜏(𝑡)),𝛽(𝜏(𝑡))], one has 𝑓𝑡,𝑥,𝑦≤𝜓(𝑡); (60) (𝐻₄)^′ for a.a. 𝑡∈𝐼 and all 𝑥∈[𝛼(𝑡),𝛽(𝑡)], the function 𝑓(𝑡,𝑥,⋅) is nondecreasing.
In these conditions, problem (43) has the extremal solutions in [𝛼,𝛽].

5. Examples of Application

We finish this work with two applications of our main results.

Example 1. Consider the following problem with advance: 𝑥^′(𝑡)(𝑥(𝜋=𝑓4𝑡)),fora.a.𝑡∈𝐼=0,8,𝑥(𝑡)(𝑡)=1=𝜙2𝜋𝑥−21sin𝜋𝑥−𝜋/2,∀𝑡∈8,𝜋2, (61) where 𝑓 is defined as follows: for each 𝑛∈{1,2,…}, we have 𝑓𝑦=1(],10𝑦,if𝑦∈2𝑛−2,2𝑛−14𝑛−1−110(],10𝑦,if𝑦∈2𝑛−1,2𝑛 (62) and for 𝑦≤0, we define 𝑓(𝑦)=−𝑓(−𝑦).
Defined that way, 𝑓 is a function of bounded variation in any bounded interval of ℝ. Moreover, 𝑓 has a countable number of both downwards and upwards discontinuities. We will construct later a pair (𝛼,𝛽) of coupled lower and upper solutions for problem (1) such that for all 𝑡∈[0,𝜋/2], we have −𝜋2(𝑡)(𝑡)≤𝜋≤𝛼≤𝛽2. (63) And then, it suffices to consider a Jordan decomposition of 𝑓 in the interval [−2,2]. So, we put 𝑓=𝑔+ℎ, with 𝑔𝑦=𝑉^𝑦₋₂𝑓=1210𝑦+[),110,if𝑦∈−2,−1310𝑦+[],110,if𝑦∈−1,1410𝑦+(],ℎ𝑦𝑦10,if𝑦∈1,2=𝑓−𝑉^𝑦₋₂𝑓=−2510𝑦−[),−310,if𝑦∈−2,−1[],−210,if𝑦∈−1,1110𝑦−(].10,if𝑦∈1,2 (64)
We will show now that the functions 𝛼(𝑡)=𝑡−𝜋/2=−𝛽(𝑡) are coupled lower and upper solutions for problem (1). First, we have that 𝛼(𝑡)≤𝜙(𝑡)≤𝛽(𝑡) for all 𝑡∈[𝜋/8,𝜋/2]. On the other hand, for a.a. 𝑡∈𝐼 we have the following.(i) If (𝜋/2)−4𝑡∈(1,2], then 4𝑡−(𝜋/2)∈[−2,−1) and then 𝑉^𝛽(4𝑡)₋₂𝑓+𝛼(4𝑡)−𝑉^𝛼(4𝑡)₋₂𝑓=8𝜋102−4𝑡=−8𝜋102+2−4𝑡10≤1=𝛼^′(𝑡),𝑉^𝛼(4𝑡)₋₂𝑓+𝛽(4𝑡)−𝑉^𝛽(4𝑡)₋₂𝑓=8𝜋102−2−4𝑡10≥−1=𝛽^′(𝑡). (65)(ii) If (𝜋/2)−4𝑡∈[0,1], then 4𝑡−(𝜋/2)∈[−1,0] and then 𝑉^𝛽(4𝑡)₋₂𝑓+𝛼(4𝑡)−𝑉^𝛼(4𝑡)₋₂𝑓=8𝜋102−4𝑡=−8𝜋102−4𝑡≤1=𝛼^′(𝑡),𝑉^𝛼(4𝑡)₋₂𝑓+𝛽(4𝑡)−𝑉^𝛽(4𝑡)₋₂𝑓=8𝜋102−4𝑡≥−1=𝛽^′(𝑡). (66)
Then, 𝛼 and 𝛽 are coupled lower and upper solutions for problem (61), satisfying 𝛼≤𝛽 on [0,𝜋/2].
Now, we check condition (𝐻₃). We have for a.a. 𝑡∈𝐼, all 𝑥∈[𝛼(𝑡),𝛽(𝑡)], and all 𝑦₁, 𝑦₂∈[𝛼(4𝑡),𝛽(4𝑡)], 𝑔𝑡,𝑥,𝑦₁+ℎ𝑡,𝑥,𝑦₂≤6+510,10 (67) and thus condition (𝐻₂) is satisfied with 𝜓≡11/10.
Finally, notice that for a.a. 𝑡∈𝐼, it is min_{[]𝑠∈𝜋/8,𝜋/2}𝜙(𝑠)−_𝑡^𝜋/8𝜓(𝑠)𝑑𝑠,max_{[]𝑠∈𝜋/8,𝜋/2}𝜙(𝑠)+_𝑡^𝜋/8𝜓(𝑠)𝑑𝑠⊂[−1,1], (68) and thus condition (𝐻₄) is satisfied with 𝐿₁≡110,𝐿₂≡0. (69)
By application of Theorem 10, we conclude that problem (61) has exactly one solution in the functional interval 𝜋4𝑡−2,𝜋2−4𝑡. (70)

In the following example, we consider a practical application of Corollary 17. It involves a modified logistic-type model with delay.

Example 2. Consider a bacterial culture governed by a logistic-type equation of the form 𝑥^′(𝑡)=𝑟₁𝑥(𝑡)(𝐾−𝑥(𝑡)), (71) where 𝑥 represents the population in thousands.
To counteract the effects of saturation term, we introduce an electronic mechanism which acts as follows. It counts the number of individuals and then it provides some food that makes the population grow. The amount of food supplied by the machine is proportional to the number of individuals. Moreover, the machine can distinguish only thousands of individuals and it supplies the food with a delay 𝜏 which also depends on time; as time goes by, this delay increases. Therefore, we can model this process with a differential equation of the form 𝑥^′(𝑡)=𝑟₁𝑥(𝑡)(𝐾−𝑥(𝑡))+𝑟₂[𝑥(𝑡−𝜏(𝑡))],𝑡∈[0,1], (72) where 𝑟₁, 𝑟₂≥0, 𝐾>1, [⋅] means integer part and 𝜏∶𝑡∈[0,1]→𝜏(𝑡)∈[0,𝑡] is measurable. We consider the normalized time interval [0,1] for simplicity.
Finally, we consider an initial population of one thousand individuals. Therefore, we deal with the following initial value problem with delay: 𝑥^′(𝑡)=𝑟₁𝑥(𝑡)(𝐾−𝑥(𝑡))+𝑟₂[𝑥((𝑡,[](0)𝑡−𝜏))]fora.a.𝑡∈0,1,𝑥=1. (73)
We will show now that problem (73) has extremal solutions between suitable lower and upper solutions.
First, notice that 𝛼≡1 and 𝛽≡𝐿, for large enough 𝐿, are, respectively, lower and upper solutions for the problem. Indeed, for all 𝑡∈[0,1], we have 0=𝛼^′(𝑡)≤𝑟₁(𝐾−1)+𝑟₂,0=𝛽^′(𝑡)≥𝑟₁𝐿(𝐾−𝐿)+𝑟₂𝐿,forlargeenough𝐿. (74)
Then, 𝛼 and 𝛽 are, respectively, lower and upper solutions for problem (73), which moreover satisfy 𝛼(𝑡)≤𝛽(𝑡) for all 𝑡∈[0,1].
To check conditions (𝐻₂)^′, (𝐻₃)^′, and (𝐻₄)^′, notice that the differential equation in (73) is defined by the function 𝑓𝑡,𝑥,𝑦=𝑟₁𝑥(𝐾−𝑥)+𝑟₂𝑦. (75) First, for each continuous 𝛾 such that 1≤𝛾(𝑡)≤𝐿 for all 𝑡∈[0,1], the function 𝑓_𝛾(𝑡,𝑥)=𝑓𝑡,𝑥,𝛾 (76) is the classical logistic function and then the initial value problem 𝑥^′(𝑡)=𝑓_𝛾(𝑡,𝑥(𝑡)),fora.a.𝑡∈[0,1],𝑥(0)=1 (77) has extremal solutions (in fact, a unique solution) between 𝛼 and 𝛽. Therefore, condition (𝐻₂)^′ is satisfied.
Finally, as for 𝑥, 𝑦∈[1,𝐿], the function 𝑓 is bounded and, moreover, 𝑓 is nondecreasing with respect to its third variable; we conclude that (𝐻₃)^′ and (𝐻₄)^′ hold.
Therefore, we can apply Corollary 17 to ensure that problem (73) has the extremal solutions between 𝛼 and 𝛽.

Acknowledgments

The author wants to thank the reviewers for their pertinent comments, which contribute to improve the quality of this paper. The paper is partially supported by FEDER and Ministerio de Educación y Ciencia, Spain, Project MTM2010-15314.