Abstract

We discuss the existence of solutions for a boundary value problem of second-order differential inclusions with three-point integral boundary conditions involving convex and nonconvex multivalued maps. Our results are based on the nonlinear alternative of Leray-Schauder type and some suitable theorems of fixed point theory.

1. Introduction

Boundary value problems for nonlinear differential equations arise in a variety of areas of applied mathematics, physics, and variational problems of control theory. A point of central importance in the study of nonlinear boundary value problems is to understand how the properties of nonlinearity in a problem influence the nature of the solutions to the boundary value problems. The topic of multipoint nonlocal boundary conditions, initiated by Bicadze and Samarski [1], has been addressed by many authors, for instance, [2–13]. The multi-point boundary conditions appear in certain problems of thermodynamics, elasticity, and wave propagation; see [5] and the references therein. The multi-point boundary conditions may be understood in the sense that the controllers at the end points dissipate or add energy according to censors located at intermediate positions. However, much of the literature dealing with three-point boundary value problems involves the three-point boundary condition restrictions on the solution or gradient of the solution of the problem.

In this paper, we consider the following second-order differential inclusion with three-point integral boundary conditions: where is such that  ,     is a multivalued map, and is the family of all subsets of . We emphasize that the present work is motivated by [14], where the authors discussed the existence of positive solutions for the problem (1.1) with as a single-valued map (i.e., . Note that the three-point boundary condition in (1.1) corresponds to the area under the curve of solutions from to .

Differential inclusions arise in the mathematical modelling of certain problems in economics, optimal control, stochastic analysis, and so forth and are widely studied by many authors, see [15–21] and the references therein.

The aim of our paper is to present existence results for the problem (1.1), when the right-hand side is convex as well as nonconvex valued. The first result relies on the nonlinear alternative of Leray-Schauder type. In the second result, we will combine the nonlinear alternative of Leray-Schauder type for single-valued maps with a selection theorem due to Bressan and Colombo for lower semicontinuous multivalued maps with nonempty closed and decomposable values, while, in the third result, we will use the fixed point theorem for contraction multivalued maps due to Covitz and Nadler. The methods used are standard; however, their exposition in the framework of problem (1.1) is new.

The paper is organized as follows. In Section 2 we recall some preliminary facts that we need in the sequel, and in Section 3 we prove our main results.

2. Preliminaries

Let us recall some basic definitions on multivalued maps [22, 23].

For a normed space , let , , , and . A multi-valued map is convex (closed) valued if is convex (closed) for all . The map is bounded on bounded sets if is bounded in for all (i.e., is called upper semicontinuous (u.s.c.) on if for each , the set is a nonempty closed subset of and if, for each open set of containing , there exists an open neighborhood of such that .   is said to be completely continuous if is relatively compact for every . If the multi-valued map is completely continuous with nonempty compact values, then is u.s.c. if and only if has a closed graph, that is, ,   imply that has a fixed point if there is such that . The fixed point set of the multivalued operator will be denoted by . A multivalued map is said to be measurable if, for every , the function is measurable.

Let denote a Banach space of continuous functions from into with the norm . Let be the Banach space of measurable functions which are Lebesgue integrable and normed by .

Definition 2.1. A multivalued map is said to be Carathéodory if (i) is measurable for each ,(ii) is upper semicontinuous for almost all , and further a Carathéodory function is called Carathéodory if(iii)for each , there exists such that for all and for a.e. .
For each , define the set of selections of by
Let be a nonempty closed subset of a Banach space and a multivalued operator with nonempty closed values. is lower semi-continuous (l.s.c.) if the set is open for any open set in . Let be a subset of . is measurable if belongs to the algebra generated by all sets of the form , where is Lebesgue measurable in and is Borel measurable in . A subset of is decomposable if, for all and measurable , the function , where stands for the characteristic function of .

Definition 2.2. Let be a separable metric space and let be a multivalued operator. One says that has a property (BC) if is lower semi-continuous (l.s.c.) and has nonempty closed and decomposable values.
Let be a multivalued map with nonempty compact values.
Define a multivalued operator associated with as which is called the Nemytskii operator associated with

Definition 2.3. Let be a multivalued function with nonempty compact values. One says that is of lower semi-continuous type (l.s.c. type) if its associated Nemytskii operator is lower semi-continuous and has nonempty closed and decomposable values.
Let be a metric space induced from the normed space . Consider given by where and . Then is a metric space and is a generalized metric space (see [24]).

Definition 2.4. A multivalued operator is called (a)Lipschitz if and only if there exists such that (b)a contraction if and only if it is Lipschitz with .

The following lemmas will be used in the sequel.

Lemma 2.5 (see [25]). Let be a Banach space. Let be an Carathéodory multivalued map and let be a linear continuous mapping from to . Then the operator is a closed graph operator in .

Lemma 2.6 (see [26]). Let be a separable metric space, and let be a multivalued operator satisfying the property (BC). Then has a continuous selection, that is, there exists a continuous function (single valued) such that for every .

Lemma 2.7 (see [27]). Let be a complete metric space. If is a contraction, then .
In order to define the solution of (1.1), we consider the following lemma whose proof is given in [14].

Lemma 2.8. Assume that . For a given , the unique solution of the boundary value problem is given by

Definition 2.9. A function is a solution of the problem (1.1) if there exists a function such that a.e. on and

3. Main Results

Theorem 3.1. Assume that is Carathéodory and has convex values, there exists a continuous nondecreasing function     and a function such that there exists a number such that Then the boundary value problem (1.1) has at least one solution on .

Proof. Define the operator by for . We will show that satisfies the assumptions of the nonlinear alternative of Leray-Schauder type. The proof consists of several steps. As a first step, we show that is convex for each . For that, let . Then there exist such that, for each , we have Let . Then, for each , we have Since is convex ( has convex values), it follows that .
Next, we show that maps bounded sets (balls) into bounded sets in . For a positive number , let be a bounded ball in . Then, for each , there exists such that Thus, Now we show that maps bounded sets into equicontinuous sets of . Let with and , where is a bounded set of . For each , we obtain Obviously the right-hand side of the above inequality tends to zero independently of as . As satisfies the above three assumptions, it follows by the Ascoli-Arzelá theorem that is completely continuous.
In our next step, we show that has a closed graph. Let and . Then we need to show that . Associated with , there exists such that, for each , Thus we have to show that there exists such that, for each , Let us consider the continuous linear operator given by Observe that Thus, it follows by Lemma 2.5 that is a closed graph operator. Further, we have . Since , therefore, we have for some .
Finally, we discuss a priori bounds on solutions. Let be a solution of (1.1). Then there exists with such that, for , we have In view of , for each , we obtain Consequently, we have
In view of , there exists such that . Let us set
Note that the operator is upper semicontinuous and completely continuous. From the choice of , there is no such that for some . Consequently, by the nonlinear alternative of Leray-Schauder type [28], we deduce that has a fixed point which is a solution of the problem (1.1). This completes the proof.

As a next result, we study the case when is not necessarily convex valued. Our strategy to deal with this problem is based on the nonlinear alternative of Leray-Schauder type together with the selection theorem of Bressan and Colombo [26] for lower semi-continuous maps with decomposable values.

Theorem 3.2. Assume that , and the following conditions hold: is a nonempty compact-valued multivalued map such that (a) is measurable, (b) is lower semicontinuous for each , for each , there exists such that Then the boundary value problem (1.1) has at least one solution on .

Proof. It follows from and that is of l.s.c. type. Then, from Lemma 2.6, there exists a continuous function such that for all .
Consider the problem
Observe that, if is a solution of (3.19), then is a solution to the problem (1.1). In order to transform the problem (3.19) into a fixed point problem, we define the operator as It can easily be shown that is continuous and completely continuous. The remaining part of the proof is similar to that of Theorem 3.1. So we omit it. This completes the proof.

Now we prove the existence of solutions for the problem (1.1) with a nonconvex-valued right-hand side by applying a fixed point theorem for multivalued map due to Covitz and Nadler [27].

Theorem 3.3. Assume that the following conditions hold: () is such that is measurable for each .() for almost all and with and for almost all . Then the boundary value problem (1.1) has at least one solution on if

Proof. Observe that the set is nonempty for each by the assumption , so has a measurable selection (see Theorem III.6 [29]). Now we show that the operator satisfies the assumptions of Lemma 2.7. To show that for each , let be such that in . Then and there exists such that, for each , As has compact values, we pass onto a subsequence to obtain that converges to in . Thus, and, for each , Hence, .
Next we show that there exists such that Let and . Then there exists such that, for each , By , we have So, there exists such that Define by Since the multivalued operator is measurable (Proposition III.4 [29]), there exists a function which is a measurable selection for . So , and, for each , we have .
For each , let us define Thus, Hence, Analogously, interchanging the roles of and , we obtain
Since is a contraction, it follows by Lemma 2.7 that has a fixed point which is a solution of (1.1). This completes the proof.

Remark 3.4. By fixing the functions and parameters involved in the problem (1.1), we obtain some interesting results: (i)if we take , where is a continuous function, then our results correspond to the ones for a single-valued problem, which are new results in the present configuration;(ii)in the limit , our results correspond to an inclusion problem with integral boundary condition (see [30]) of the type In this case, the solution defined by (2.10) takes the form (iii)By fixing in (1.1), our results reduce to the ones for a second-order inclusion problem with Dirichlet boundary conditions ().