Abstract

This paper presents sufficient conditions for the existence of positive solutions for a class of integral inclusions. Our results are obtained via a new fixed point theorem for multivalued operators developed in the paper, in which some nonnegative function is used to describe the cone expansion and compression instead of the classical norm-type, and lead to new existence principles.

1. Introduction

Let (๐ธ,โ€–โ‹…โ€–) be a Banach space. A nonempty convex closed set ๐‘ƒโŠ‚๐ธ is called a cone of ๐ธ if the following conditions hold:๐‘ฅโˆˆ๐‘ƒ,๐œ†โ‰ฅ0,implies๐œ†๐‘ฅโˆˆ๐‘ƒ,๐‘ฅโˆˆ๐‘ƒ,โˆ’๐‘ฅโˆˆ๐‘ƒ,implies๐‘ฅ=๐œƒ,(1.1)

where ๐œƒ stands for the zero element of ๐ธ. A cone ๐‘ƒ is said to be normal if there exists a positive constant ๐‘, which is called the normal constant of ๐‘ƒ, such that ๐œƒโ‰ค๐‘ฅโ‰ค๐‘ฆ (๐‘ฅ,๐‘ฆโˆˆ๐ธ) implies that โ€–๐‘ฅโ€–โ‰ค๐‘โ€–๐‘ฆโ€–. Here, the partially order โ€œโ‰คโ€ in ๐ธ is introduced as follows: ๐‘ฅโ‰ค๐‘ฆ if and only if ๐‘ฆโˆ’๐‘ฅโˆˆ๐‘ƒ for any ๐‘ฅ, ๐‘ฆโˆˆ๐ธ, ๐‘ฅ<๐‘ฆ if and only if ๐‘ฅโ‰ค๐‘ฆ and ๐‘ฅโ‰ ๐‘ฆ.

Given a cone ๐‘ƒ of ๐ธ, denote that ๐‘ƒ+=๐‘ƒโงต{๐œƒ}. For ๐‘ข0โˆˆ๐‘ƒ+, denote that๐‘ƒ๎€ท๐‘ข0๎€ธ=๎€ฝ๐‘ฅโˆˆ๐‘ƒโˆถ๐œ†๐‘ข0๎€พโ‰ค๐‘ฅ,forsome๐œ†>0.(1.2)

For notational purposes for ๐œ‚>0, let ฮฉ๐œ‚={๐‘ฆโˆˆ๐ธโˆถโ€–๐‘ฆโ€–<๐œ‚},๐œ•ฮฉ๐œ‚={๐‘ฆโˆˆ๐ธโˆถโ€–๐‘ฆโ€–=๐œ‚},ฮฉ๐œ‚={๐‘ฆโˆˆ๐ธโˆถโ€–๐‘ฆโ€–โ‰ค๐œ‚},๐œ•ฮฉdenotetheboundaryofsetฮฉ.(1.3)

This paper is concerned with the existence of solutions for the following multivalued integral inclusion:๐‘ฅ๎€œ(๐‘ก)=๐‘“(๐‘ก,๐‘ฅ)0+โˆž๐‘ข๐‘ฅ(๐‘ก,๐‘ )๐‘‘๐‘ ,(1.4) where ๐‘“โˆถโ„+ร—โ„โ†’โ„ is a single-valued map, ๐‘ˆโˆถ๐ปร—โ„โŠธ2โ„ is a multivalued map, and ๐‘ข๐‘ฅโˆˆ๐‘†๐‘ˆ,๐‘ฅ. Here, โ„+=[0,+โˆž), ๐ป=โ„+ร—โ„+, and the set of ๐ฟ1-selections ๐‘†๐‘ˆ,๐‘ฅ of the multivalued map ๐‘ˆ is defined by๐‘†๐‘ˆ,๐‘ฅ๎€ฝ๐‘“โˆถ=๐‘ฅโˆˆ๐ฟ1๎€ทโ„+๎€ธ,โ„โˆถ๐‘“๐‘ฅ๎€พ.(๐‘ก,๐‘ )โˆˆ๐‘ˆ(๐‘ก,๐‘ ,๐‘ฅ(๐‘ ))a.e.for๐‘กโ‰ฅ0(1.5)

Some problems considered in the vehicular traffic theory, biology, and queuing theory lead to the following nonlinear functional-integral equation:๎€œ๐‘ฅ(๐‘ก)=๐‘“(๐‘ก,๐‘ฅ(๐‘ก))10[].๐‘ข(๐‘ก,๐‘ ,๐‘ฅ(๐‘ ))๐‘‘๐‘ ,๐‘กโˆˆ0,1(1.6)

(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:๎€œ๐‘ฅ(๐‘ก)=๐‘“(๐‘ก,๐‘ฅ(๐‘ก))๐‘ก0๐‘ข(๐‘ก,๐‘ ,๐‘ฅ(๐‘ ))๐‘‘๐‘ ,๐‘กโ‰ฅ0(1.7)

was proved by using the technique associated with measures of noncompactness, and the functions were assumed continuous and bounded on โ„+. The sufficient conditions for the existence of solutions to this equation, under the assumption of ๐‘ข 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 Oโ€™Regan [6]. When ๐‘“(๐‘ก,๐‘ฅ)=1, also, [7] established the existence of solutions to the multivalued problem (1.4) in Frรฉchet 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 [8โ€“10], authors considered the existence of positive solutions for singular second-order ๐‘š-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 Oโ€™Regan [6] extended Krasnoselskii's fixed point theorem of norm type to multivalued operator problems and obtained fixed point theorems for ๐‘˜-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 ๐‘ƒ with the set ๐‘ƒ(๐‘ข0), 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 ๐‘˜-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 ๐ถ, ๐ท of ๐ธ, we write ๐ถโ‰ค๐ท (or ๐ทโ‰ฅ๐ถ) ifโˆ€๐‘โˆˆ๐ท,โˆƒ๐‘žโˆˆ๐ถsuchthat๐‘žโ‰ค๐‘.(2.1)

A multivalued operator ๐ด is called upper semicontinuous (u.s.c.) on ๐ธ if for each ๐‘ฅโˆˆ๐ธ the set ๐ด(๐‘ฅ) is a nonempty closed subset of ๐ธ, and if for each open set ๐ต of ๐ธ containing ๐ด(๐‘ฅ), there exists an open neighborhood ๐‘‰ of ๐‘ฅ such that ๐ด(๐‘‰)โŠ†๐ต.

๐ด is called a ๐‘˜-set contraction if ๐›พ(๐ด(๐ท))โ‰ค๐‘˜๐›พ(๐ท) for all bounded sets ๐ท of ๐ธ and ๐ด(๐ท) is bounded, where ๐›พ denotes the Kuratowskii measure of noncompactness.

Throughout this paper, we denote by ๐ถ๐พ(๐ถ) the family of nonempty, compact, and convex subsets of set ๐ถ and denote by ๐พ๐œ•๐‘ˆ(๐‘ˆ,๐ถ) the set of all u.s.c., ๐‘˜-set-contractive maps ๐ดโˆถ๐‘ˆโ†’๐ถ๐พ(๐ถ) with ๐‘ฅโˆ‰๐ด(๐‘ฅ) for ๐‘ฅโˆˆ๐œ•๐‘ˆ.

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 ๐ธ be an ordered Banach space and ๐‘ƒ a cone in ๐ธ, and let ฮฉ1 and ฮฉ2 be bounded open sets in ๐ธ such that ๐œƒโˆˆฮฉ1 and ฮฉ1โŠ‚ฮฉ2. Assume that ๐ดโˆถฮฉ2โ†’๐ถ๐พ(๐‘ƒ) is a u.s.c., ๐‘˜-set contractive (here 0โ‰ค๐‘˜<1) map and assume one of the following conditions hold: [๐‘ฅโˆ‰๐œ†๐ด๐‘ฅ,โˆ€๐œ†โˆˆ0,1),๐‘ฅโˆˆ๐œ•ฮฉ2โˆฉ๐‘ƒ,(2.2)thereexistsa๐‘ฃโˆˆ๐‘ƒ+with๐‘ฅโˆ‰๐ด๐‘ฅ+๐›ฟ๐‘ฃfor๐‘ฅโˆˆ๐œ•ฮฉ1โˆฉ๐‘ƒ,๐›ฟโ‰ฅ0.(2.3) Or [๐‘ฅโˆ‰๐œ†๐ด๐‘ฅ,โˆ€๐œ†โˆˆ0,1),๐‘ฅโˆˆ๐œ•ฮฉ1โˆฉ๐‘ƒ,(2.4)thereexistsa๐‘ฃโˆˆ๐‘ƒ+with๐‘ฅโˆ‰๐ด๐‘ฅ+๐›ฟ๐‘ฃfor๐‘ฅโˆˆ๐œ•ฮฉ2โˆฉ๐‘ƒ,๐›ฟโ‰ฅ0.(2.5) Then ๐ด has at least one fixed point ๐‘ฆ with ๐‘ฆโˆˆ(ฮฉ2โงตฮฉ1)โˆฉ๐‘ƒ.

Lemma 2.2 (see [19]). Let ๐ธ be a Banach space, ๐ท a closed convex subset of ๐ธ, and ๐‘ˆ an open subset of ๐ท with ๐œƒโˆˆ๐‘ˆ. Suppose that ๐ดโˆถ๐‘ˆโŠธ๐ถ๐พ(๐ท) is u.s.c, ๐‘˜-set-contractive (here 0โ‰ค๐‘˜<1). Then either (h1) there exists ๐‘ฅโˆˆ๐‘ˆ with ๐‘ฅโˆˆ๐ด๐‘ฅ, or (h2) there exists ๐‘ขโˆˆ๐œ•๐‘ˆ and ๐œ†โˆˆ(0,1) with ๐‘ขโˆˆ๐œ†๐ด๐‘ฅ.

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

Theorem 2.3. Assume that ฮฉ1 and ฮฉ2 are bounded open sets in ๐ธ such that ๐œƒโˆˆฮฉ1 and ฮฉ1โŠ‚ฮฉ2. Let ๐ดโˆถ๐‘ƒโˆฉฮฉ2โ†’๐ถ๐พ(๐‘ƒ) be a u.s.c, ๐‘˜-set-contractive (here 0โ‰ค๐‘˜<1) operator, ๐‘ข0โˆˆ๐‘ƒ+, and ๐œŒโˆถ๐‘ƒโ†’[0,+โˆž) a nondecreasing function with ๐œŒ(๐œƒ)=0 and ๐œŒ(๐‘ฅ)>0 for ๐‘ฅโˆˆ๐‘ƒ+. Moreover, (h) ๐œŒ(๐œ†๐‘ฅ)โ‰ค๐œ†๐œŒ(๐‘ฅ), for all ๐‘ฅโˆˆ๐‘ƒ and ๐œ†โˆˆ[0,1]. If one of the following two conditions holds: (H1) (i) ๐œŒ(๐‘ฆ)>๐œŒ(๐‘ฅ) for all ๐‘ฆโˆˆ๐ด(๐‘ฅ) and ๐‘ฅโˆˆ๐‘ƒ(๐‘ข0)โˆฉ๐œ•ฮฉ1, โ€ƒโ€ƒ (ii) ๐œŒ(๐‘ฆ)โ‰ค๐œŒ(๐‘ฅ) for all ๐‘ฆโˆˆ๐ด(๐‘ฅ) and ๐‘ฅโˆˆ๐‘ƒโˆฉ๐œ•ฮฉ2; (H2) (i) ๐œŒ(๐‘ฆ)โ‰ค๐œŒ(๐‘ฅ) for all ๐‘ฆโˆˆ๐ด(๐‘ฅ) and ๐‘ฅโˆˆ๐‘ƒโˆฉ๐œ•ฮฉ1, โ€ƒโ€ƒ (ii) ๐œŒ(๐‘ฆ)>๐œŒ(๐‘ฅ) for all ๐‘ฆโˆˆ๐ด(๐‘ฅ) and ๐‘ฅโˆˆ๐‘ƒ(๐‘ข0)โˆฉ๐œ•ฮฉ2, then ๐ด has a positive fixed point in the set ๐‘ƒโˆฉ(ฮฉ2โงตฮฉ1).

Proof. We seek to apply Lemma 2.1. It is sufficient to check that ๐ด satisfies the conditions (2.2) and (2.3) in ฮฉ1 and in ฮฉ2, respectively, provided that the condition (H1) holds. First, (H1)(ii) with ๐‘ฅโˆˆ๐‘ƒโˆฉ๐œ•ฮฉ2 implies that (2.2) is true. To see this suppose that there exist ๐‘ฅโˆˆ๐‘ƒโˆฉ๐œ•ฮฉ2 and ๐œ†โˆˆ[0,1) with ๐‘ฅโˆˆ๐œ†๐ด(๐‘ฅ). Then there exists ๐‘ฆโˆˆ๐ด(๐‘ฅ) with ๐‘ฅ=๐œ†๐‘ฆ. Therefore, by the condition (h), we have 0<๐œŒ(๐‘ฅ)=๐œŒ(๐œ†๐‘ฆ)โ‰ค๐œ†๐œŒ(๐‘ฆ)<๐œŒ(๐‘ฆ)โ‰ค๐œŒ(๐‘ฅ),(2.6) a contradiction. Next, we will prove that for any ๐‘ฅโˆˆ๐‘ƒโˆฉ๐œ•ฮฉ1 and any ๐›ฟโ‰ฅ0, ๐‘ฅโˆ‰๐ด(๐‘ฅ)+๐›ฟ๐‘ข0.(2.7) Suppose, on the contrary, that there exist ๐‘ฅ0โˆˆ๐‘ƒโˆฉ๐œ•ฮฉ1 and ๐‘กโ‰ฅ0 such that ๐‘ฅ0โˆˆ๐ด(๐‘ฅ0)+๐‘ก๐‘ข0, that is, there exists ๐‘ฆ0โˆˆ๐ด(๐‘ฅ0) such that ๐‘ฅ0=๐‘ฆ0+๐‘ก๐‘ข0. Hence, ๐‘ฅ0โˆ’๐‘ก๐‘ข0=๐‘ฆ0๎€ท๐‘ฅโˆˆ๐ด0๎€ธ.(2.8) Clearly, ๐‘กโ‰ 0 (otherwise, this proof is completed). Noting that ๐ด(๐‘ฅ0)โŠ‚๐‘ƒ, we conclude that ๐‘ก๐‘ข0โ‰ค๐‘ก๐‘ข0+๐‘ฆ(2.9) for all ๐‘ฆโˆˆ๐ด(๐‘ฅ0). Then, combining (2.8), we get that ๐‘ฅ0โˆˆ๐‘ƒ(๐‘ข0). Since ๐‘ฅ0โˆ’๐‘ก๐‘ข0=๐‘ฆ0, we have ๐œƒโ‰ค๐‘ฆ0โ‰ค๐‘ฅ0,๐‘ฅ0โ‰ ๐‘ฆ0.(2.10) In virtue of the monotonicity of ๐œŒ, we have ๐œŒ๎€ท๐‘ฆ0๎€ธ๎€ท๐‘ฅโ‰ค๐œŒ0๎€ธ.(2.11) Since ๐‘ฅ0โˆˆ๐‘ƒ(๐‘ข0)โˆฉ๐œ•ฮฉ1, (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 ฮฉ1, ฮฉ2 and the multivalued mapping ๐ด are given as in Theorem 2.3, ๐‘ข0โˆˆ๐‘ƒ+, and a function ๐œŒโˆถ๐‘ƒโ†’[0,+โˆž) satisfies the condition (h), ๐œŒ(๐œƒ)=0 and ๐œŒ(๐‘ฅ)>0 for ๐‘ฅโˆˆ๐‘ƒ+. Moreover, there exists a constant ๐‘>0 such that (hโ€ฒ) ๐œƒโ‰ค๐‘ฅโ‰ค๐‘ฆ with ๐‘ฅ, ๐‘ฆโˆˆ๐ธ implies that ๐œŒ(๐‘ฅ)โ‰ค๐‘๐œŒ(๐‘ฆ). If either (Hโ€ฒ1) (i) ๐œŒ(๐‘ฆ)>๐‘๐œŒ(๐‘ฅ) for all ๐‘ฆโˆˆ๐ด(๐‘ฅ) and ๐‘ฅโˆˆ๐‘ƒ(๐‘ข0)โˆฉ๐œ•ฮฉ1, โ€ƒโ€ƒ (ii) ๐œŒ(๐‘ฆ)โ‰ค๐œŒ(๐‘ฅ) for all ๐‘ฆโˆˆ๐ด(๐‘ฅ) and ๐‘ฅโˆˆ๐‘ƒโˆฉ๐œ•ฮฉ2, or (Hโ€ฒ2) (i) ๐œŒ(๐‘ฆ)โ‰ค๐œŒ(๐‘ฅ) for all ๐‘ฆโˆˆ๐ด(๐‘ฅ) and ๐‘ฅโˆˆ๐‘ƒโˆฉ๐œ•ฮฉ1, โ€ƒโ€ƒ (ii) ๐œŒ(๐‘ฆ)>๐‘๐œŒ(๐‘ฅ) for all ๐‘ฆโˆˆ๐ด(๐‘ฅ) and ๐‘ฅโˆˆ๐‘ƒ(๐‘ข0)โˆฉ๐œ•ฮฉ2 is satisfied, then ๐ด has a positive fixed point in ๐‘ƒโˆฉ(ฮฉ2โงตฮฉ1).

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 ๐‘ฅโˆˆ๐‘ƒโˆฉ๐œ•ฮฉ1 and any ๐›ฟโ‰ฅ0. Suppose, on the contrary, that there exist ๐‘ฅ0โˆˆ๐‘ƒโˆฉ๐œ•ฮฉ1 and ๐‘กโ‰ฅ0 such that ๐‘ฅ0โˆˆ๐ด(๐‘ฅ0)+๐‘ก๐‘ข0, that is, (2.8) holds. Similarly, we have ๐‘ฅ0โˆˆ๐‘ƒ(๐‘ข0) and ๐œƒโ‰ค๐‘ฆ0โ‰ค๐‘ฅ0 with ๐‘ฅ0โˆ’๐‘ก๐‘ข0=๐‘ฆ0โˆˆ๐ด(๐‘ฅ0). In virtue of the condition (hโ€ฒ), we have ๐œŒ๎€ท๐‘ฆ0๎€ธ๎€ท๐‘ฅโ‰ค๐‘๐œŒ0๎€ธ.(2.12) Since ๐‘ฅ0โˆˆ๐‘ƒโˆฉ๐œ•ฮฉ1, (2.12) contradicts (Hโ€ฒ1)(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 (Hโ€ฒ2 ) holds. This proof is completed.

Remark 2.5. If the function ๐œŒ is convex on ๐‘ƒ, namely, ๐œŒ(๐‘ก๐‘ฅ+(1โˆ’๐‘ก)๐‘ฆ)โ‰ค๐‘ก๐œŒ(๐‘ฅ)+(1โˆ’๐‘ก)๐œŒ(๐‘ฆ) for all ๐‘ฅ, ๐‘ฆโˆˆ๐‘ƒ and ๐‘กโˆˆ[0,1], then the condition (h) holds provided that ๐œŒ(๐œƒ)=0. From this point of view, we extend the corresponding result of [8]. Let ๐œŒ(๐‘ฅ)=โ€–๐‘ฅโ€–. Then ๐œŒ(๐‘ฅ) is a convex function with ๐œŒ(๐œƒ)=0, ๐œŒ(๐‘ฅ)>0 for ๐‘ฅโ‰ 0, and the condition (h) is satisfied. Obviously, ๐œŒ is nondecreasing if โ€–โ‹…โ€– be increasing with respect to ๐‘ƒ. This shows that Theorem 2.3 contains the corresponding result of [6]. In addition, the condition (hโ€ฒ) holds if ๐‘ƒ is a normal cone. Hence, Corollary 2.4 extends and improves the corresponding result of [11].

Remark 2.6. Let ๐ธ=๐ถ[0,1] with the norm โ€–๐œ‘โ€–=max0โ‰ค๐‘กโ‰ค1|๐œ‘(๐‘ก)| and the cone ๐‘ƒ0[][]={๐œ‘โˆˆ๐ถ0,1โˆถ๐œ‘(๐‘ก)โ‰ฅ0,โˆ€๐‘กโˆˆ0,1}.(2.13) Define ๐œ‘1โ‰ค๐œ‘2 if and only if ๐œ‘1(๐‘ก)โ‰ค๐œ‘2(๐‘ก) for every ๐‘กโˆˆ[0,1]. Then the function ๐œŒโˆถ๐‘ƒ0โ†’[0,+โˆž) defined by ๎‚ต๎€œ๐œŒ(๐œ‘)=10๐œ‘๐‘๎‚ถ(๐‘ก)๐‘‘๐‘ก1/๐‘,(๐‘โ‰ฅ1)(2.14) is nondecreasing convex and ๐œŒ(๐œƒ)=0, ๐œŒ(๐œ‘)>0 for ๐œ‘โ‰ 0, and ๐œŒ 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 ๐‘ฅโˆ‰๐ด(๐‘ฅ),โˆ€๐‘ฅโˆˆ๐œ•ฮฉ1โˆฉ๐‘ƒ.(2.15) Then ๐ด has at least two fixed points ๐‘ฅ1 and ๐‘ฅ2 with ๐‘ฅ1โˆˆฮฉ1โˆฉ๐‘ƒ and ๐‘ฅ2โˆˆ๐‘ƒโˆฉ(ฮฉ2โงตฮฉ1).

Proof. Theorem 2.3 guarantees that ๐ด has at least one fixed point ๐‘ฅ2 with ๐‘ฅ2โˆˆ๐‘ƒโˆฉ(ฮฉ2โงตฮฉ1). In addition, we obtain in the proof of Theorem 2.3 that ๐‘ฅโˆ‰๐œ†๐ด(๐‘ฅ) for all ๐œ†โˆˆ[0,1) and ๐‘ฅโˆˆ๐œ•ฮฉ1โˆฉ๐‘ƒ. Hence, we combine (2.15) and Lemma 2.2 to conclude that ๐ด has a fixed point ๐‘ฅ1โˆˆฮฉ1โˆฉ๐‘ƒ. This completes the proof of Theorem 2.7.

For constants ๐ฟ, ๐‘Ÿ, ๐‘… with 0<๐‘Ÿ<๐ฟ<๐‘…, let us suppose that (H3) ๐‘ฅโˆ‰๐ด(๐‘ฅ) for all ๐‘ฅโˆˆ๐œ•ฮฉ๐ฟโˆฉ๐‘ƒ; (H4) (i) ๐œŒ(๐‘ฅ)<๐œŒ(๐‘ฆ) for all ๐‘ฆโˆˆ๐ด(๐‘ฅ) and ๐‘ฅโˆˆ๐‘ƒ(๐‘ข0)โˆฉ๐œ•ฮฉ๐‘Ÿ, โ€ƒโ€ƒ(ii) ๐œŒ(๐‘ฆ)โ‰ค๐œŒ(๐‘ฅ) for all ๐‘ฆโˆˆ๐ด(๐‘ฅ) and ๐‘ฅโˆˆ๐‘ƒโˆฉ๐œ•ฮฉ๐ฟ, โ€ƒโ€ƒ(iii) ๐œŒ(๐‘ฅ)<๐œŒ(๐‘ฆ) for all ๐‘ฆโˆˆ๐ด(๐‘ฅ) and ๐‘ฅโˆˆ๐‘ƒ(๐‘ข0)โˆฉ๐œ•ฮฉ๐‘…; (H5) (i) ๐œŒ(๐‘ฆ)โ‰ค๐œŒ(๐‘ฅ) for all ๐‘ฆโˆˆ๐ด(๐‘ฅ) and ๐‘ฅโˆˆ๐‘ƒโˆฉ๐œ•ฮฉ๐‘Ÿ, โ€ƒโ€ƒ(ii) ๐œŒ(๐‘ฅ)<๐œŒ(๐‘ฆ) for all ๐‘ฆโˆˆ๐ด(๐‘ฅ) and ๐‘ฅโˆˆ๐‘ƒ(๐‘ข0)โˆฉ๐œ•ฮฉ๐ฟ, โ€ƒโ€ƒ(iii) ๐œŒ(๐‘ฆ)โ‰ค๐œŒ(๐‘ฅ) for all ๐‘ฆโˆˆ๐ด(๐‘ฅ) and ๐‘ฅโˆˆ๐‘ƒโˆฉ๐œ•ฮฉ๐‘….

Theorem 2.8. Let ๐ดโˆถฮฉ๐‘…โˆฉ๐‘ƒโ†’๐ถ๐พ(๐‘ƒ) be a u.s.c., ๐‘˜-set-contractive (here 0โ‰ค๐‘˜<1) operator and the function ๐œŒ be given as in Theorem 2.3. If either the conditions (H3) and (H4) or the conditions (H3) and (H5) hold, then ๐ด has at least two positive fixed points ๐‘ฅ1 and ๐‘ฅ2 with ๐‘ฅ1โˆˆ๐‘ƒโˆฉ(ฮฉ๐ฟโงตฮฉ๐‘Ÿ) and ๐‘ฅ2โˆˆ๐‘ƒโˆฉ(ฮฉ๐‘…โงตฮฉ๐ฟ).

Proof. Theorem 2.3 implies that ๐ด has a fixed point ๐‘ฅ1โˆˆ๐‘ƒโˆฉ(ฮฉ๐ฟโงตฮฉ๐‘Ÿ). (H3) shows that ๐‘ฅ1โˆ‰๐œ•ฮฉ๐ฟ. Hence, ๐‘ฅ1โˆˆ๐‘ƒโˆฉ(ฮฉ๐ฟโงตฮฉ๐‘Ÿ). Again, Theorem 2.3 guarantees the existence of ๐‘ฅ2. 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 ๐‘ฅโˆˆ๐ถ(โ„+,โ„) is said to be a solution of (1.4) if it satisfies (1.4).

By ๐ต๐ถโˆถ=๐ต๐ถ(โ„+,โ„), we mean the Banach algebra consisting of all functions defined, bounded, and continuous on โ„+ with the norm๎€ฝ||๐‘ฅ||๎€พโ€–๐‘ฅโ€–=sup(๐‘ก)โˆถ๐‘กโ‰ฅ0.(3.1)

For any ๐‘ฅ, ๐‘ฆโˆˆ๐ต๐ถ, define that ๐‘ฅโ‰ค๐‘ฆ if and only if ๐‘ฅ(๐‘ก)โ‰ค๐‘ฆ(๐‘ก) for each ๐‘กโ‰ฅ0, ๐‘ฅ<๐‘ฆ, if and only if ๐‘ฅโ‰ค๐‘ฆ and there exists some ๐‘กโ‰ฅ0 such that ๐‘ฅ(๐‘ก)โ‰ ๐‘ฆ(๐‘ก).

In following Theorem 3.1, we need impose the following hypotheses on the single valued map ๐‘“ and the multivalued map ๐‘ˆ. (S1)๐‘“โˆถโ„+ร—โ„+โ†’โ„+ is a continuous function. (S2)There exists a bounded continuous function ๐‘”โˆถโ„+โ†’โ„+, such that||||๐‘“(๐‘ก,๐‘ฅ)โ‰ค๐‘”(๐‘ก),forany๐‘กโ‰ฅ0,๐‘ฅโ‰ฅ0.(3.2)(S3)There exist positive constants ๐’ž, ๐›ฟ, ๐œ‰ with ๐’ž>0 and 0<๐›ฟ<๐œ‰<+โˆž such that |๐‘“(๐‘ก,๐‘ฅ)|โ‰ฅ๐’ž for ๐‘กโˆˆ[๐›ฟ,๐œ‰], ๐‘ฅโ‰ฅ0. (S4)๐‘ˆโˆถ๐ปร—โ„+โ†’๐ถ๐พ(โ„+) is ๐ฟ1-Carathรฉodory, that is,(๐‘ก,๐‘ )โ†’๐‘ˆ(๐‘ก,๐‘ ,๐‘ฅ) is measurable for every ๐‘ฅโˆˆโ„+;๐‘ฅโ†’๐‘ˆ(๐‘ก,๐‘ ,๐‘ฅ) is u.s.c. for a.e. (๐‘ก,๐‘ )โˆˆ๐ป.In addition, the set ๐‘†๐‘ˆ,๐‘ฅ is nonempty for each fixed ๐‘ฅโˆˆ๐ต๐ถ. (S5)There exist a bounded, continuous, and nondecreasing function ๐›ฝโˆถโ„+โ†’โ„+, a function ๐›ผโˆˆ๐ฟ1(โ„+,โ„+) with โˆซ๐œ‰๐›ฟ๐›ผ(๐‘ก)๐‘‘๐‘ก>0, and a continuous function ๐›พโˆถ[๐›ฟ,๐œ‰]โ†’(0,+โˆž) such that ||๐‘ข๐‘ฅ||(๐‘ก,๐‘ )โ‰ค๐›ผ(๐‘ )๐›ฝ(๐‘ฅ(๐‘ )),fora.e.(๐‘ก,๐‘ )โˆˆ๐ป,๐‘ฅโ‰ฅ0,๐‘ข๐‘ฅโˆˆ๐‘†๐‘ˆ,๐‘ฅ;||๐‘ข๐‘ฅ||[]ร—[(๐‘ก,๐‘ )โ‰ฅ๐›พ(๐‘ก)๐›ผ(๐‘ )๐›ฝ(๐‘ฅ(๐‘ )),fora.e.(๐‘ก,๐‘ )โˆˆ๐›ฟ,๐œ‰0,+โˆž),๐‘ฅโ‰ฅ0,๐‘ข๐‘ฅโˆˆ๐‘†๐‘ˆ,๐‘ฅ;lim๐œ€โ†’0+๐›ฝ(๐œ€)๐œ€=+โˆž.(3.3)(S6)There exists a positive number ๐‘…>0 such that ๐›ฝ(๐‘…)>0 and ๐‘…โˆซ๐›ฝ(๐‘…)๐œ‰๐›ฟโˆซ๐‘”(๐‘ )๐‘‘๐‘ 0+โˆžโ‰ฅ๐‘€๐›ผ(๐œ)๐‘‘๐œ,๐’žโ„ณ(3.4)where ๐‘€=sup๐‘กโ‰ฅ0๐‘”(๐‘ก), โ„ณ=min๐›ฟโ‰ค๐‘กโ‰ค๐œ‰๐›พ(๐‘ก).

Theorem 3.1. If the conditions (S1)โ€“(S6) hold, then (1.4) has at least one (positive) solution ๐‘ฅโˆˆ๐ต๐ถ(โ„+,โ„) with ๐‘ฅโ‰ฅ0 on โ„+ and with ๐‘Ÿ<โ€–๐‘ฅโ€–โ‰ค๐‘… for given 0<๐‘Ÿ<๐‘….

Proof. Let us define the multivalued map ๐ด on the space ๐ต๐ถ by the following: ๎€œ(๐ด(๐‘ฅ))(๐‘ก)={๐‘“(๐‘ก,๐‘ฅ(๐‘ก))}0+โˆž๐‘ข๐‘ฅ(๐‘ก,๐‘ )๐‘‘๐‘ ,๐‘ข๐‘ฅโˆˆ๐‘†๐‘ˆ,๐‘ฅ,๐‘กโ‰ฅ0.(3.5)We will show that ๐ด has a fixed point recurring to Theorem 2.3. Define the function ๐œŒโˆถ๐ต๐ถ(โ„+,โ„)โ†’[0,+โˆž) by ๐œŒ(๐‘ฅ)=max๐›ฟโ‰ค๐‘กโ‰ค๐œ‰||||๐‘ฅ(๐‘ก)(3.6) and the set ๐‘‹+ by ๐‘‹+=๎‚†๐‘ฅโˆˆ๐ต๐ถโˆถ๐‘ฅ(๐‘ก)โ‰ฅ0,for๐‘กโ‰ฅ0,๐‘ฅ(๐‘ก)โ‰ฅ๐’žโ„ณ๐‘€[]๎‚‡.โ€–๐‘ฅโ€–,for๐‘กโˆˆ๐›ฟ,๐œ‰(3.7) It is easy to see that ๐‘‹+ is a cone of ๐ต๐ถ and ๐œŒโˆถ๐‘‹+โ†’[0,+โˆž) given in (3.6) is a nondecreasing convex function with ๐œŒ(๐œƒ)=0 and ๐œŒ(๐œ‘)>0 for ๐œ‘โ‰ ๐œƒ.
First we point out that ๐ด(๐‘ฅ)โˆˆ๐ถ๐พ(๐‘‹+) for each fixed ๐‘ฅโˆˆฮฉ๐‘โˆฉ๐‘‹+ with ๐‘>0. In fact, for any ๐‘ฆโˆˆ๐ด(๐‘ฅ), there exists ๐‘ข๐‘ฅโˆˆ๐‘†๐‘ˆ,๐‘ฅ such that โˆซ๐‘ฆ(๐‘ก)=๐‘“(๐‘ก,๐‘ฅ)โˆž0๐‘ข๐‘ฅ(๐‘ก,๐‘ )๐‘‘๐‘  for ๐‘กโ‰ฅ0. (S1) and (S4) imply that ๐‘ฆ(๐‘ก)โ‰ฅ0 for ๐‘กโ‰ฅ0 and ๐›ผ๐›ฝ(๐‘ฅ)โˆˆ๐ฟ1(โ„+,โ„+), where ๐›ฝ(๐‘ฅ) is defined by (๐›ฝ(๐‘ฅ))(๐‘ก)=๐›ฝ(๐‘ฅ(๐‘ก)). Applying our assumptions we have the following estimate: ||๐‘ฆ||โ‰ค||๐‘“||๎€œ(๐‘ก)(๐‘ก,๐‘ฅ)0+โˆž||๐‘ข๐‘ฅ||๎€œ(๐‘ก,๐‘ )๐‘‘๐‘ โ‰ค๐‘”(๐‘ก)0+โˆž๐›ผ(๐‘ )๐›ฝ(๐‘ฅ(๐‘ ))๐‘‘๐‘ โ‰คsup๐‘กโ‰ฅ0๎€œ๐‘”(๐‘ก)โˆž0๐›ผ(๐‘ )๐›ฝ(๐‘ฅ(๐‘ ))๐‘‘๐‘ ,(3.8)
and so ๎€œโ€–๐‘ฆโ€–โ‰ค๐‘€0+โˆž๐›ผ(๐‘ )๐›ฝ(๐‘ฅ(๐‘ ))๐‘‘๐‘ .(3.9) In addition (S2) and (S5), together with (3.9), guarantee that ๐‘ฆ๎€œ(๐‘ก)=๐‘“(๐‘ก,๐‘ )0+โˆž๐‘ข๐‘ฅ๎€œ(๐‘ก,๐‘ )๐‘‘๐‘ โ‰ฅ๐’ž0+โˆž๐›พ(๐‘ก)๐›ผ(๐‘ )๐›ฝ(๐‘ฅ(๐‘ ))๐‘‘๐‘ โ‰ฅ๐’žmin๐›ฟโ‰ค๐‘กโ‰ค๐œ‰๎€œ๐›พ(๐‘ก)0+โˆž๐›ผ(๐‘ )๐›ฝ(๐‘ฅ(๐‘ ))๐‘‘๐‘ โ‰ฅ๐’žโ„ณ๐‘€โ€–๐‘ฆโ€–.(3.10) This implies that, by the arbitrariness of ๐‘ฆ, ๐ด(๐‘ฅ) is bounded and ๐ด(๐‘ฅ)โŠ‚๐‘‹+ for each ๐‘ฅโˆˆฮฉ๐‘โˆฉ๐‘‹+. Similar to [20] we can infer that ๐ด(๐‘ฅ) is convex for each ๐‘ฅโˆˆฮฉ๐‘โˆฉ๐‘‹+. In the light of our assumptions and the Lebesgue dominated convergence theorem, we can see that ๐ด(๐‘ฅ) is compact for each ๐‘ฅโˆˆฮฉ๐‘โˆฉ๐‘‹+. Hence, ๐ด maps ฮฉ๐‘โˆฉ๐‘‹+ into ๐ถ๐พ(๐‘‹+).
Next, we prove that ๐ด has closed graph. Take ๐‘ฅ๐‘˜โ†’๐‘ฅโˆ—, โ„Ž๐‘˜โˆˆ๐ด(๐‘ฅ๐‘˜) and โ„Ž๐‘˜โ†’โ„Žโˆ— as ๐‘˜โ†’โˆž. We shall prove that โ„Žโˆ—โˆˆ๐ด(๐‘ฅโˆ—). โ„Ž๐‘˜โˆˆ๐ด(๐‘ฅ๐‘˜) means that there exists ๐‘ข๐‘˜โˆˆ๐‘†๐‘ˆ,๐‘ฅ๐‘˜ such that for each ๐‘กโ‰ฅ0, โ„Ž๐‘˜๎€ท(๐‘ก)=๐‘“๐‘ก,๐‘ฅ๐‘˜๎€ธ๎€œ0+โˆž๐‘ข๐‘˜(๐‘ก,๐‘ )๐‘‘๐‘ ,๐‘˜=1,2,โ€ฆ.(3.11) Let โ„Žโˆ—(๐‘ก)=๐‘“(๐‘ก,๐‘ฅโˆ—)โ„Ž(๐‘ก). From the continuity of ๐‘“, it follows that ๐‘“(๐‘ก,๐‘ฅ๐‘˜)โ†’๐‘“(๐‘ก,๐‘ฅโˆ—). From (S4) it follows that ๐‘ข๐‘˜โ†’๐‘ข๐‘ฅโˆ—โˆˆ๐‘†๐‘ˆ,๐‘ฅโˆ— as ๐‘˜โ†’โˆž. From (S5) and the Lebesgue dominated convergence theorem it follows that โˆซ0+โˆž๐‘ข๐‘˜โˆซ(๐‘ก,๐‘ )๐‘‘๐‘ โ†’0+โˆž๐‘ข๐‘ฅโˆ—(๐‘ก,๐‘ )๐‘‘๐‘ . It is easy to see that โˆซ0+โˆž๐‘ข๐‘ฅโˆ—(๐‘ก,๐‘ )๐‘‘๐‘ =โ„Ž(๐‘ก), that is, for each ๐‘กโ‰ฅ0, โ„Žโˆ—๎€ท(๐‘ก)=๐‘“๐‘ก,๐‘ฅโˆ—๎€ธ๎€œ0+โˆž๐‘ข๐‘ฅโˆ—(๐‘ก,๐‘ )๐‘‘๐‘ .(3.12)
This implies that โ„Žโˆ—โˆˆ๐ด(๐‘ฅโˆ—). 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 ๐ด is u.s.c. It is clear that ๐ด is a ๐‘˜-set-contractive multivalued map with ๐‘˜=0.
It remains to prove (in virtue of Theorem 2.3) that the condition (H1) holds to conclude that ๐ด has a fixed point in ๐‘‹+, that is, that (1.4) has a positive solution. Given ๐‘ฅโˆˆ๐œ•ฮฉ๐‘…โˆฉ๐‘‹+ with ๐‘… satisfying the condition (S6), for any ๐‘ฆโˆˆ๐ด(๐‘ฅ), there exists ๐‘ข๐‘ฅโˆˆ๐‘†๐‘ˆ,๐‘ฅ such that โˆซ๐‘ฆ(๐‘ก)=๐‘“(๐‘ก,๐‘ฅ(๐‘ก))โˆž0๐‘ข๐‘ฅ(๐‘ก,๐‘ )๐‘‘๐‘ , ๐‘กโ‰ฅ0. Hence, ๐œŒ(๐‘ฆ)=max๐›ฟโ‰ค๐‘กโ‰ค๐œ‰||๐‘ฆ||(๐‘ก)=max๐›ฟโ‰ค๐‘กโ‰ค๐œ‰||||๎€œ๐‘“(๐‘ก,๐‘ฅ)0+โˆž๐‘ข๐‘ฅ||||(๐‘ก,๐‘ )๐‘‘๐‘ โ‰คmax๐›ฟโ‰ค๐‘กโ‰ค๐œ‰๎€œ๐‘”(๐‘ก)0+โˆž๐›ผ(๐‘ )๐›ฝ(๐‘ฅ(๐‘ ))๐‘‘๐‘ โ‰ค๐›ฝ(๐‘…)max๐›ฟโ‰ค๐‘กโ‰ค๐œ‰๎€œ๐‘”(๐‘ก)0+โˆž๐›ผ(๐‘ )๐‘‘๐‘ .(3.13) In addition (S6) shows that ๐›ฝ(๐‘ฅ(๐œ))โ‰ค๐›ฝ(๐‘…)โ‰ค๐’žโ„ณ๐‘…๐‘€max๐›ฟโ‰ค๐‘กโ‰ค๐œ‰โˆซ๐‘”(๐‘ก)0+โˆž๐›ผ(๐‘ )๐‘‘๐‘ ,(3.14) and this together with (3.13) gives the following: ๐œŒ(๐‘ฆ)โ‰ค๐’žโ„ณ๐‘…๐‘€max๐›ฟโ‰ค๐‘กโ‰ค๐œ‰โˆซ๐‘”(๐‘ก)0+โˆž๐›ผ(๐‘ )๐‘‘๐‘ max๐›ฟโ‰ค๐‘กโ‰ค๐œ‰๐‘”๎€œ(๐‘ก)0+โˆž๐›ผ(๐‘ )๐‘‘๐‘ =๐’žโ„ณ๐‘…๐‘€โ‰ค๐œŒ(๐‘ฅ).(3.15) Thus, ๐ด satisfies condition (H1)(ii).
Take ๐’ฆโˆถ=๐‘€/๐’ž2โ„ณmin๐›ฟโ‰ค๐‘กโ‰ค๐œ‰โˆซ๐›พ(๐‘ก)๐œ‰๐›ฟ๐›ผ(๐‘ )๐‘‘๐‘ . (S5) shows that there exists a positive number ๐‘Ÿ<๐‘… small enough such that ๐›ฝ(๐œ€)>๐’ฆ๐œ€,0<๐œ€โ‰ค๐‘Ÿ.(3.16) Let ๐‘ข0(๐‘ก)โ‰ก1. To prove that (H1)(i) is true, let ๐‘ฆโˆˆ๐ด๐‘ฅ with ๐‘ฆโ‰ ๐‘ฅ and ๐‘ฅโˆˆ๐œ•ฮฉ๐‘Ÿโˆฉ๐‘‹+(1). Then there exists ๐‘ข๐‘ฅโˆˆ๐‘†๐‘ˆ,๐‘ฅ with โˆซ๐‘ฆ(๐‘ก)=๐‘“(๐‘ก,๐‘ฅ)0+โˆž๐‘ข๐‘ฅ(๐‘ก,๐‘ )๐‘‘๐‘ . In virtue of the definition of ๐‘‹+(1), there exists 0<๐œ†<1 such that ๐œ†โ‰ค๐‘ฅ(๐‘ก)โ‰ค๐‘Ÿ,๐‘กโ‰ฅ0.(3.17) Note that there exists ๐œ‚โˆˆ[๐›ฟ,๐œ‰] such that ๐œŒ(๐‘ฆ)=๐‘ฆ(๐œ‚). Now our assumptions imply that ๐œŒ๎€œ(๐‘ฆ)=๐‘ฆ(๐œ‚)=๐‘“(๐œ‚,๐‘ฅ)0+โˆž๐‘ข๐‘ฅ๎€œ(๐œ‚,๐‘ )๐‘‘๐‘ โ‰ฅ๐’ž0+โˆž๐›พ(๐œ‚)๐›ผ(๐‘ )๐›ฝ(๐‘ฅ(๐‘ ))๐‘‘๐‘ โ‰ฅ๐’žmin๐›ฟโ‰ค๐‘กโ‰ค๐œ‰๐›พ๎€œ(๐‘ก)๐œ‰๐›ฟ๐›ผ>(๐‘ )๐›ฝ(๐‘ฅ(๐‘ ))๐‘‘๐‘ ๐’ž๐‘€min๐›ฟโ‰ค๐‘กโ‰ค๐œ‰๐›พ(๐‘ก)๐’ž2โ„ณmin๐›ฟโ‰ค๐‘กโ‰ค๐œ‰โˆซ๐›พ(๐‘ก)๐œ‰๐›ฟ๎€œ๐›ผ(๐‘ )๐‘‘๐‘ ๐œ‰๐›ฟ๐›ผ(๐‘ )๐‘ฅ(๐‘ )๐‘‘๐‘ โ‰ฅโ€–๐‘ฅโ€–โ‰ฅ๐œŒ(๐‘ฅ).(3.18) So ๐œŒ(๐‘ฆ)>๐œŒ(๐‘ฅ) for all ๐‘ฆโˆˆ๐ด(๐‘ฅ). This shows that (H1)(i) is satisfied. Conclusively, Theorem 2.3 guarantees that ๐ด has a fixed point ๐‘ฆ with ๐‘Ÿโ‰คโ€–๐‘ฆโ€–โ‰ค๐‘…. This proof is completed.

Theorem 3.2. Suppose that conditions (S1)โ€“(S6) hold. Then (1.4) has at least two positive solutions if the following conditions are satisfied: lim๐œ‚โ†’+โˆž๐›ฝ(๐œ‚)๐œ‚=+โˆž.(3.19)

Example 3.3. Let ๐›ฟ=1/4, ๐œ‰=3/4, ๐›พ(๐‘ก)=1/2 for ๐‘กโˆˆ[๐›ฟ,๐œ‰], โŽงโŽชโŽจโŽชโŽฉ[],๐‘ ๐›ผ(๐‘ )=1,๐‘ โˆˆ0,1๐‘ ,๐‘ =2,3,โ€ฆ,โˆ’2,others,(3.20)โˆš๐›ฝ(๐‘ฅ)=๐‘ฅ and โˆš๐‘ˆ(๐‘ก,๐‘ ,๐‘ฅ)=[(1/2)๐›ผ(๐‘ )โˆš๐‘ฅ,๐›ผ(๐‘ )๐‘ฅ]. Let ๐‘“(๐‘ก,๐‘ฅ)=๐‘’โˆ’๐‘ก(sin๐‘ฅ+1), ๐‘”(๐‘ก)=2๐‘’โˆ’๐‘ก, โ„’=(1/2๐‘’), ๐‘…=32๐‘’3/4. It is clear that conditions (S1)โ€“(S6) are satisfied. Hence, Theorem 3.1 guarantees the problem ๐‘ฅ(๐‘ก)=๐‘’โˆ’๐‘ก๎€œ(sin๐‘ฅ+1)โˆž0๐‘ข๐‘ฅ(๐‘ก,๐‘ )๐‘‘๐‘ (3.21) with ๐‘ข๐‘ฅโˆšโˆˆ[(1/2)๐›ผ(๐‘ )โˆš๐‘ฅ,๐›ผ(๐‘ )๐‘ฅ] having at least a positive solution ๐‘ฅ with โ€–๐‘ฅโ€–โ‰ค32๐‘’3/4.

Acknowledgment

The research is supported by Foundation of Zhejiang Education Department (Y201009938) and partially by NSFC (10901043).