1. Introduction

Let be a Banach space. A nonempty convex closed set is called a cone of if the following conditions hold:

where stands for the zero element of . A cone is said to be normal if there exists a positive constant , which is called the normal constant of , such that () implies that . Here, the partially order “≤” in is introduced as follows: if and only if for any , , if and only if and .

Given a cone of , denote that . For , denote that

For notational purposes for , let

This paper is concerned with the existence of solutions for the following multivalued integral inclusion: where is a single-valued map, is a multivalued map, and . Here, , , and the set of -selections of the multivalued map is defined by

Some problems considered in the vehicular traffic theory, biology, and queuing theory lead to the following nonlinear functional-integral equation:

(cf. [1]). The Volterra counterpart of the above equation on unbounded interval was studied by [2]. Namely, in [2], the existence of solutions of the following integral equation:

was proved by using the technique associated with measures of noncompactness, and the functions were assumed continuous and bounded on . The sufficient conditions for the existence of solutions to this equation, under the assumption of being a multivalued map, was presented by [3] via a fixed-point theorem due to Martelli [4] on ordered Banach spaces, [5] via expansion and compression fixed point theorems for multivalued mapping due to Agarwal and O’Regan [6]. When , also, [7] established the existence of solutions to the multivalued problem (1.4) in Fréchet spaces. In this paper, we give existence results of positive solutions for system (1.4).

The fundamental tool used in the proof of our main results is essentially the fixed point theorem (see Theorem 2.3) based on expansion and compression fixed point theorems for multivalued mappings. However, the hypotheses imposed on functions on the right-hand side of (1.4) and methods of the proof in this paper are different from the above-cited works.

Cone compression and expansion fixed point theorems are frequently used tools for studying the existence of positive solutions for boundary value problems of integral and differential equations. For instance, in [8–10], authors considered the existence of positive solutions for singular second-order -point boundary value problem, in [11] Leggett and Williams discussed the nonlinear equation modelling certain infectious diseases. In [12] Zima discussed a three-point boundary value problem for second-order ordinary differential equations. In [13, 14] the authors proved multiplicity of positive radial solutions for an elliptic system on an annulus and so on. The original result of Krasnoselskii fixed point theorem concerning cone compression and expansion was obtained by Krasnoselskii [15]. Afterward, a lot of generalization of this theorem has appeared (see, e.g., [8, 11, 12, 16, 17]). For instance, in [16] Guo and Lashmikantham gave the result of the norm type, and in [17] Anderson and Avery obtained a generalization of the norm type by applying conditions formulated in the terms of two functionals replacing the norm type assumptions. In [8] Zhang and Sun obtained an extension, in which the norm is replayed with some uniformly continuous convex function (see [8], Corollary 2.1). On the other hand, in [11], Leggett and Williams obtained another generalization of Krasnoselskiis original result. In [18] one can find some refinements of [11]. In [12] Zima proved another result via replacing Leggett and Williams type-ordering conditions by the conditions of the norm type (see [12], Theorem 2.1). In addition, Agarwal and O’Regan [6] extended Krasnoselskii's fixed point theorem of norm type to multivalued operator problems and obtained fixed point theorems for -set contractive multivalued operators (see [6], Theorems 2.4 and 2.8). In general, while the expansion may be easily verified for a large class of nonlinear integral operators, the compression is a rather stringent condition and is usually not easily verified. By improving the compression of the cone theorem via replacing the cone with the set , the result of Leggett and Williams [11] has the advantage which consists in its usually being easier to apply even when the compression of the cone theorem is also applicable to a large class of operators. In this paper we will extend Leggett and Williams fixed point theorem to multivalued operator problems and obtain a fixed point theorem for -set-contractive multivalued operators, in which the norm of [11] will be replayed with some nonnegative function. Our result is not only the fundamental tool to prove our main theorem, but also a generalization of corresponding results in [6, 8, 11, 12].

2. Preliminaries

We begin this section with gathering together some definitions and known facts. For two subsets , of , we write (or ) if

A multivalued operator 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 called a -set contraction if for all bounded sets of and is bounded, where denotes the Kuratowskii measure of noncompactness.

Throughout this paper, we denote by the family of nonempty, compact, and convex subsets of set and denote by the set of all u.s.c., -set-contractive maps with for .

The nonzero fixed point theorems of multivalued operators (see [6], Theorems 2.3 and 2.7) will play an important role in this section. It is not hard to extend these results on open sets, so we have the following.

Lemma 2.1. Let be an ordered Banach space and a cone in , and let and be bounded open sets in such that and . Assume that is a u.s.c., -set contractive (here ) map and assume one of the following conditions hold: Or Then has at least one fixed point with .

Lemma 2.2 (see [19]). Let be a Banach space, a closed convex subset of , and an open subset of with . Suppose that is u.s.c, -set-contractive (here ). Then either (h1) there exists with , or (h2) there exists and with .

The proof of the following theorem is not complicated but it is essential to prove our main results.

Theorem 2.3. Assume that and are bounded open sets in such that and . Let be a u.s.c, -set-contractive (here ) operator, , and a nondecreasing function with and for . Moreover, (h) , for all and . If one of the following two conditions holds: (H1) (i) for all and ,    (ii) for all and ; (H2) (i) for all and ,    (ii) for all and , then has a positive fixed point in the set .

Proof. We seek to apply Lemma 2.1. It is sufficient to check that satisfies the conditions (2.2) and (2.3) in and in , respectively, provided that the condition (H1) holds. First, (H1)(ii) with implies that (2.2) is true. To see this suppose that there exist and with . Then there exists with . Therefore, by the condition (h), we have a contradiction. Next, we will prove that for any and any , Suppose, on the contrary, that there exist and such that , that is, there exists such that . Hence, Clearly, (otherwise, this proof is completed). Noting that , we conclude that for all . Then, combining (2.8), we get that . Since , we have In virtue of the monotonicity of , we have Since , (2.11) contradicts (H1)(i). Hence, (2.7) is true. This implies that (2.3) is true. The result of Theorem 2.3 now follows from Lemma 2.1.
Similarly, we can prove that the result of Theorem 2.3 follows if (H2) holds. This proof is completed.

Corollary 2.4. Assume that , and the multivalued mapping are given as in Theorem 2.3, , and a function satisfies the condition (h), and for . Moreover, there exists a constant such that (h′) with , implies that . If either (H′1) (i) for all and ,    (ii) for all and , or (H′2) (i) for all and ,    (ii) for all and is satisfied, then has a positive fixed point in .

Proof. We seek to apply Lemma 2.1. The hypothesis (2.2) is true, the proof of which is the same as Theorem 2.3. Next, we will prove that (2.7) is satisfied for any and any . Suppose, on the contrary, that there exist and such that , that is, (2.8) holds. Similarly, we have and with . In virtue of the condition (h′), we have Since , (2.12) contradicts (H′1)(i). Hence, (2.7) is true. This shows that the conditions of Lemma 2.1 are satisfied.
Similarly, we can prove that the result of Corollary 2.4 follows if (H′2 ) holds. This proof is completed.

Remark 2.5. If the function is convex on , namely, for all , and , then the condition (h) holds provided that . From this point of view, we extend the corresponding result of [8]. Let . Then is a convex function with , for , and the condition (h) is satisfied. Obviously, is nondecreasing if be increasing with respect to . This shows that Theorem 2.3 contains the corresponding result of [6]. In addition, the condition (h′) holds if is a normal cone. Hence, Corollary 2.4 extends and improves the corresponding result of [11].

Remark 2.6. Let with the norm and the cone Define if and only if for every . Then the function defined by is nondecreasing convex and , for , and yields the condition (h).
In what follows, we combine Lemma 2.2 and Theorem 2.3 to establish existence of multiple fixed points.

Theorem 2.7. Assume that the conditions of Theorem 2.3 hold and Then has at least two fixed points and with and .

Proof. Theorem 2.3 guarantees that has at least one fixed point with . In addition, we obtain in the proof of Theorem 2.3 that for all and . Hence, we combine (2.15) and Lemma 2.2 to conclude that has a fixed point . This completes the proof of Theorem 2.7.

For constants , , with , let us suppose that (H3) for all ; (H4) (i) for all and ,   (ii) for all and ,   (iii) for all and ; (H5) (i) for all and ,   (ii) for all and ,   (iii) for all and .

Theorem 2.8. Let be a u.s.c., -set-contractive (here ) operator and the function be given as in Theorem 2.3. If either the conditions (H3) and (H4) or the conditions (H3) and (H5) hold, then has at least two positive fixed points and with and .

Proof. Theorem 2.3 implies that has a fixed point . (H3) shows that . Hence, . Again, Theorem 2.3 guarantees the existence of . This proof is completed.

3. Main Results

In this section, we shall discuss the existence of solutions of integral inclusion (1.4) by using fixed point theorems involved in Section 2. Let us start by defining that a function is said to be a solution of (1.4) if it satisfies (1.4).

By , we mean the Banach algebra consisting of all functions defined, bounded, and continuous on with the norm

For any , , define that if and only if for each , , if and only if and there exists some such that .

In following Theorem 3.1, we need impose the following hypotheses on the single valued map and the multivalued map . (S1) is a continuous function. (S2)There exists a bounded continuous function , such that(S3)There exist positive constants , , with and such that for , . (S4) is -Carathéodory, that is, is measurable for every ; is u.s.c. for a.e. .In addition, the set is nonempty for each fixed . (S5)There exist a bounded, continuous, and nondecreasing function , a function with , and a continuous function such that (S6)There exists a positive number such that and where , .

Theorem 3.1. If the conditions (S1)–(S6) hold, then (1.4) has at least one (positive) solution with on and with for given .

Proof. Let us define the multivalued map on the space by the following: We will show that has a fixed point recurring to Theorem 2.3. Define the function by and the set by It is easy to see that is a cone of and given in (3.6) is a nondecreasing convex function with and for .
First we point out that for each fixed with . In fact, for any , there exists such that for . (S1) and (S4) imply that for and , where is defined by . Applying our assumptions we have the following estimate:
and so In addition (S2) and (S5), together with (3.9), guarantee that This implies that, by the arbitrariness of , is bounded and for each . Similar to [20] we can infer that is convex for each . In the light of our assumptions and the Lebesgue dominated convergence theorem, we can see that is compact for each . Hence, maps into .
Next, we prove that has closed graph. Take , and as . We shall prove that . means that there exists such that for each , Let . From the continuity of , it follows that . From (S4) it follows that as . From (S5) and the Lebesgue dominated convergence theorem it follows that . It is easy to see that , that is, for each ,
This implies that . We want to point out that u.s.c. is equivalent to the condition of being a closed graph multivalued map when the map has nonempty compact values; that is, we have shown that is u.s.c. It is clear that is a -set-contractive multivalued map with .
It remains to prove (in virtue of Theorem 2.3) that the condition (H1) holds to conclude that has a fixed point in , that is, that (1.4) has a positive solution. Given with satisfying the condition (S6), for any , there exists such that , . Hence, In addition (S6) shows that and this together with (3.13) gives the following: Thus, satisfies condition (H1)(ii).
Take . (S5) shows that there exists a positive number small enough such that Let . To prove that (H1)(i) is true, let with and . Then there exists with . In virtue of the definition of , there exists such that Note that there exists such that . Now our assumptions imply that So for all . This shows that (H1)(i) is satisfied. Conclusively, Theorem 2.3 guarantees that has a fixed point with . This proof is completed.