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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 213270, 12 pages
http://dx.doi.org/10.1155/2012/213270
Research Article

Solvability of a Class of Integral Inclusions

Institute of Applied Mathematics and Engineering Computation, Hangzhou Dianzi University, Hangzhou 310018, China

Received 13 July 2011; Accepted 12 February 2012

Academic Editor: Shaher Momani

Copyright © 2012 Ying Chen and Shihuang Hong. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 (𝐸,) be a Banach space. A nonempty convex closed set 𝑃𝐸 is called a cone of 𝐸 if the following conditions hold:𝑥𝑃,𝜆0,implies𝜆𝑥𝑃,𝑥𝑃,𝑥𝑃,implies𝑥=𝜃,(1.1)

where 𝜃 stands for the zero element of 𝐸. A cone 𝑃 is said to be normal if there exists a positive constant 𝑁, which is called the normal constant of 𝑃, such that 𝜃𝑥𝑦 (𝑥,𝑦𝐸) implies that 𝑥𝑁𝑦. Here, the partially order “≤” in 𝐸 is introduced as follows: 𝑥𝑦 if and only if 𝑦𝑥𝑃 for any 𝑥, 𝑦𝐸, 𝑥<𝑦 if and only if 𝑥𝑦 and 𝑥𝑦.

Given a cone 𝑃 of 𝐸, denote that 𝑃+=𝑃{𝜃}. For 𝑢0𝑃+, denote that𝑃𝑢0=𝑥𝑃𝜆𝑢0𝑥,forsome𝜆>0.(1.2)

For notational purposes for 𝜂>0, let Ω𝜂={𝑦𝐸𝑦<𝜂},𝜕Ω𝜂={𝑦𝐸𝑦=𝜂},Ω𝜂={𝑦𝐸𝑦𝜂},𝜕ΩdenotetheboundaryofsetΩ.(1.3)

This paper is concerned with the existence of solutions for the following multivalued integral inclusion:𝑥(𝑡)=𝑓(𝑡,𝑥)0+𝑢𝑥(𝑡,𝑠)𝑑𝑠,(1.4) where 𝑓+× is a single-valued map, 𝑈𝐻×2 is a multivalued map, and 𝑢𝑥𝑆𝑈,𝑥. Here, +=[0,+), 𝐻=+×+, and the set of 𝐿1-selections 𝑆𝑈,𝑥 of the multivalued map 𝑈 is defined by𝑆𝑈,𝑥𝑓=𝑥𝐿1+,𝑓𝑥.(𝑡,𝑠)𝑈(𝑡,𝑠,𝑥(𝑠))a.e.for𝑡0(1.5)

Some problems considered in the vehicular traffic theory, biology, and queuing theory lead to the following nonlinear functional-integral equation:𝑥(𝑡)=𝑓(𝑡,𝑥(𝑡))10[].𝑢(𝑡,𝑠,𝑥(𝑠))𝑑𝑠,𝑡0,1(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:𝑥(𝑡)=𝑓(𝑡,𝑥(𝑡))𝑡0𝑢(𝑡,𝑠,𝑥(𝑠))𝑑𝑠,𝑡0(1.7)

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 𝑢 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 𝑓(𝑡,𝑥)=1, 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 [810], authors considered the existence of positive solutions for singular second-order 𝑚-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 𝑘-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 𝑃 with the set 𝑃(𝑢0), 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 𝑘-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 𝐶, 𝐷 of 𝐸, we write 𝐶𝐷 (or 𝐷𝐶) if𝑝𝐷,𝑞𝐶suchthat𝑞𝑝.(2.1)

A multivalued operator 𝐴 is called upper semicontinuous (u.s.c.) on 𝐸 if for each 𝑥𝐸 the set 𝐴(𝑥) is a nonempty closed subset of 𝐸, and if for each open set 𝐵 of 𝐸 containing 𝐴(𝑥), there exists an open neighborhood 𝑉 of 𝑥 such that 𝐴(𝑉)𝐵.

𝐴 is called a 𝑘-set contraction if 𝛾(𝐴(𝐷))𝑘𝛾(𝐷) for all bounded sets 𝐷 of 𝐸 and 𝐴(𝐷) is bounded, where 𝛾 denotes the Kuratowskii measure of noncompactness.

Throughout this paper, we denote by 𝐶𝐾(𝐶) the family of nonempty, compact, and convex subsets of set 𝐶 and denote by 𝐾𝜕𝑈(𝑈,𝐶) the set of all u.s.c., 𝑘-set-contractive maps 𝐴𝑈𝐶𝐾(𝐶) with 𝑥𝐴(𝑥) for 𝑥𝜕𝑈.

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 𝐸 be an ordered Banach space and 𝑃 a cone in 𝐸, and let Ω1 and Ω2 be bounded open sets in 𝐸 such that 𝜃Ω1 and Ω1Ω2. Assume that 𝐴Ω2𝐶𝐾(𝑃) is a u.s.c., 𝑘-set contractive (here 0𝑘<1) map and assume one of the following conditions hold: [𝑥𝜆𝐴𝑥,𝜆0,1),𝑥𝜕Ω2𝑃,(2.2)thereexistsa𝑣𝑃+with𝑥𝐴𝑥+𝛿𝑣for𝑥𝜕Ω1𝑃,𝛿0.(2.3) Or [𝑥𝜆𝐴𝑥,𝜆0,1),𝑥𝜕Ω1𝑃,(2.4)thereexistsa𝑣𝑃+with𝑥𝐴𝑥+𝛿𝑣for𝑥𝜕Ω2𝑃,𝛿0.(2.5) Then 𝐴 has at least one fixed point 𝑦 with 𝑦(Ω2Ω1)𝑃.

Lemma 2.2 (see [19]). Let 𝐸 be a Banach space, 𝐷 a closed convex subset of 𝐸, and 𝑈 an open subset of 𝐷 with 𝜃𝑈. Suppose that 𝐴𝑈𝐶𝐾(𝐷) is u.s.c, 𝑘-set-contractive (here 0𝑘<1). Then either (h1) there exists 𝑥𝑈 with 𝑥𝐴𝑥, or (h2) there exists 𝑢𝜕𝑈 and 𝜆(0,1) with 𝑢𝜆𝐴𝑥.

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

Theorem 2.3. Assume that Ω1 and Ω2 are bounded open sets in 𝐸 such that 𝜃Ω1 and Ω1Ω2. Let 𝐴𝑃Ω2𝐶𝐾(𝑃) be a u.s.c, 𝑘-set-contractive (here 0𝑘<1) operator, 𝑢0𝑃+, and 𝜌𝑃[0,+) a nondecreasing function with 𝜌(𝜃)=0 and 𝜌(𝑥)>0 for 𝑥𝑃+. Moreover, (h) 𝜌(𝜆𝑥)𝜆𝜌(𝑥), for all 𝑥𝑃 and 𝜆[0,1]. If one of the following two conditions holds: (H1) (i) 𝜌(𝑦)>𝜌(𝑥) for all 𝑦𝐴(𝑥) and 𝑥𝑃(𝑢0)𝜕Ω1,    (ii) 𝜌(𝑦)𝜌(𝑥) for all 𝑦𝐴(𝑥) and 𝑥𝑃𝜕Ω2; (H2) (i) 𝜌(𝑦)𝜌(𝑥) for all 𝑦𝐴(𝑥) and 𝑥𝑃𝜕Ω1,    (ii) 𝜌(𝑦)>𝜌(𝑥) for all 𝑦𝐴(𝑥) and 𝑥𝑃(𝑢0)𝜕Ω2, then 𝐴 has a positive fixed point in the set 𝑃(Ω2Ω1).

Proof. We seek to apply Lemma 2.1. It is sufficient to check that 𝐴 satisfies the conditions (2.2) and (2.3) in Ω1 and in Ω2, respectively, provided that the condition (H1) holds. First, (H1)(ii) with 𝑥𝑃𝜕Ω2 implies that (2.2) is true. To see this suppose that there exist 𝑥𝑃𝜕Ω2 and 𝜆[0,1) with 𝑥𝜆𝐴(𝑥). Then there exists 𝑦𝐴(𝑥) with 𝑥=𝜆𝑦. Therefore, by the condition (h), we have 0<𝜌(𝑥)=𝜌(𝜆𝑦)𝜆𝜌(𝑦)<𝜌(𝑦)𝜌(𝑥),(2.6) a contradiction. Next, we will prove that for any 𝑥𝑃𝜕Ω1 and any 𝛿0, 𝑥𝐴(𝑥)+𝛿𝑢0.(2.7) Suppose, on the contrary, that there exist 𝑥0𝑃𝜕Ω1 and 𝑡0 such that 𝑥0𝐴(𝑥0)+𝑡𝑢0, that is, there exists 𝑦0𝐴(𝑥0) such that 𝑥0=𝑦0+𝑡𝑢0. Hence, 𝑥0𝑡𝑢0=𝑦0𝑥𝐴0.(2.8) Clearly, 𝑡0 (otherwise, this proof is completed). Noting that 𝐴(𝑥0)𝑃, we conclude that 𝑡𝑢0𝑡𝑢0+𝑦(2.9) for all 𝑦𝐴(𝑥0). Then, combining (2.8), we get that 𝑥0𝑃(𝑢0). Since 𝑥0𝑡𝑢0=𝑦0, we have 𝜃𝑦0𝑥0,𝑥0𝑦0.(2.10) In virtue of the monotonicity of 𝜌, we have 𝜌𝑦0𝑥𝜌0.(2.11) Since 𝑥0𝑃(𝑢0)𝜕Ω1, (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 Ω1, Ω2 and the multivalued mapping 𝐴 are given as in Theorem 2.3, 𝑢0𝑃+, and a function 𝜌𝑃[0,+) satisfies the condition (h), 𝜌(𝜃)=0 and 𝜌(𝑥)>0 for 𝑥𝑃+. Moreover, there exists a constant 𝑁>0 such that (h′) 𝜃𝑥𝑦 with 𝑥, 𝑦𝐸 implies that 𝜌(𝑥)𝑁𝜌(𝑦). If either (H′1) (i) 𝜌(𝑦)>𝑁𝜌(𝑥) for all 𝑦𝐴(𝑥) and 𝑥𝑃(𝑢0)𝜕Ω1,    (ii) 𝜌(𝑦)𝜌(𝑥) for all 𝑦𝐴(𝑥) and 𝑥𝑃𝜕Ω2, or (H′2) (i) 𝜌(𝑦)𝜌(𝑥) for all 𝑦𝐴(𝑥) and 𝑥𝑃𝜕Ω1,    (ii) 𝜌(𝑦)>𝑁𝜌(𝑥) for all 𝑦𝐴(𝑥) and 𝑥𝑃(𝑢0)𝜕Ω2 is satisfied, then 𝐴 has a positive fixed point in 𝑃(Ω2Ω1).

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 𝑥𝑃𝜕Ω1 and any 𝛿0. Suppose, on the contrary, that there exist 𝑥0𝑃𝜕Ω1 and 𝑡0 such that 𝑥0𝐴(𝑥0)+𝑡𝑢0, that is, (2.8) holds. Similarly, we have 𝑥0𝑃(𝑢0) and 𝜃𝑦0𝑥0 with 𝑥0𝑡𝑢0=𝑦0𝐴(𝑥0). In virtue of the condition (h′), we have 𝜌𝑦0𝑥𝑁𝜌0.(2.12) Since 𝑥0𝑃𝜕Ω1, (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 𝜌 is convex on 𝑃, namely, 𝜌(𝑡𝑥+(1𝑡)𝑦)𝑡𝜌(𝑥)+(1𝑡)𝜌(𝑦) for all 𝑥, 𝑦𝑃 and 𝑡[0,1], then the condition (h) holds provided that 𝜌(𝜃)=0. From this point of view, we extend the corresponding result of [8]. Let 𝜌(𝑥)=𝑥. Then 𝜌(𝑥) is a convex function with 𝜌(𝜃)=0, 𝜌(𝑥)>0 for 𝑥0, and the condition (h) is satisfied. Obviously, 𝜌 is nondecreasing if be increasing with respect to 𝑃. This shows that Theorem 2.3 contains the corresponding result of [6]. In addition, the condition (h′) holds if 𝑃 is a normal cone. Hence, Corollary 2.4 extends and improves the corresponding result of [11].

Remark 2.6. Let 𝐸=𝐶[0,1] with the norm 𝜑=max0𝑡1|𝜑(𝑡)| and the cone 𝑃0[][]={𝜑𝐶0,1𝜑(𝑡)0,𝑡0,1}.(2.13) Define 𝜑1𝜑2 if and only if 𝜑1(𝑡)𝜑2(𝑡) for every 𝑡[0,1]. Then the function 𝜌𝑃0[0,+) defined by 𝜌(𝜑)=10𝜑𝑝(𝑡)𝑑𝑡1/𝑝,(𝑝1)(2.14) is nondecreasing convex and 𝜌(𝜃)=0, 𝜌(𝜑)>0 for 𝜑0, and 𝜌 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 𝑥𝐴(𝑥),𝑥𝜕Ω1𝑃.(2.15) Then 𝐴 has at least two fixed points 𝑥1 and 𝑥2 with 𝑥1Ω1𝑃 and 𝑥2𝑃(Ω2Ω1).

Proof. Theorem 2.3 guarantees that 𝐴 has at least one fixed point 𝑥2 with 𝑥2𝑃(Ω2Ω1). In addition, we obtain in the proof of Theorem 2.3 that 𝑥𝜆𝐴(𝑥) for all 𝜆[0,1) and 𝑥𝜕Ω1𝑃. Hence, we combine (2.15) and Lemma 2.2 to conclude that 𝐴 has a fixed point 𝑥1Ω1𝑃. This completes the proof of Theorem 2.7.

For constants 𝐿, 𝑟, 𝑅 with 0<𝑟<𝐿<𝑅, let us suppose that (H3) 𝑥𝐴(𝑥) for all 𝑥𝜕Ω𝐿𝑃; (H4) (i) 𝜌(𝑥)<𝜌(𝑦) for all 𝑦𝐴(𝑥) and 𝑥𝑃(𝑢0)𝜕Ω𝑟,   (ii) 𝜌(𝑦)𝜌(𝑥) for all 𝑦𝐴(𝑥) and 𝑥𝑃𝜕Ω𝐿,   (iii) 𝜌(𝑥)<𝜌(𝑦) for all 𝑦𝐴(𝑥) and 𝑥𝑃(𝑢0)𝜕Ω𝑅; (H5) (i) 𝜌(𝑦)𝜌(𝑥) for all 𝑦𝐴(𝑥) and 𝑥𝑃𝜕Ω𝑟,   (ii) 𝜌(𝑥)<𝜌(𝑦) for all 𝑦𝐴(𝑥) and 𝑥𝑃(𝑢0)𝜕Ω𝐿,   (iii) 𝜌(𝑦)𝜌(𝑥) for all 𝑦𝐴(𝑥) and 𝑥𝑃𝜕Ω𝑅.

Theorem 2.8. Let 𝐴Ω𝑅𝑃𝐶𝐾(𝑃) be a u.s.c., 𝑘-set-contractive (here 0𝑘<1) operator and the function 𝜌 be given as in Theorem 2.3. If either the conditions (H3) and (H4) or the conditions (H3) and (H5) hold, then 𝐴 has at least two positive fixed points 𝑥1 and 𝑥2 with 𝑥1𝑃(Ω𝐿Ω𝑟) and 𝑥2𝑃(Ω𝑅Ω𝐿).

Proof. Theorem 2.3 implies that 𝐴 has a fixed point 𝑥1𝑃(Ω𝐿Ω𝑟). (H3) shows that 𝑥1𝜕Ω𝐿. Hence, 𝑥1𝑃(Ω𝐿Ω𝑟). Again, Theorem 2.3 guarantees the existence of 𝑥2. 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 𝑥𝐶(+,) is said to be a solution of (1.4) if it satisfies (1.4).

By 𝐵𝐶=𝐵𝐶(+,), we mean the Banach algebra consisting of all functions defined, bounded, and continuous on + with the norm||𝑥||𝑥=sup(𝑡)𝑡0.(3.1)

For any 𝑥, 𝑦𝐵𝐶, define that 𝑥𝑦 if and only if 𝑥(𝑡)𝑦(𝑡) for each 𝑡0, 𝑥<𝑦, if and only if 𝑥𝑦 and there exists some 𝑡0 such that 𝑥(𝑡)𝑦(𝑡).

In following Theorem 3.1, we need impose the following hypotheses on the single valued map 𝑓 and the multivalued map 𝑈. (S1)𝑓+×++ is a continuous function. (S2)There exists a bounded continuous function 𝑔++, such that||||𝑓(𝑡,𝑥)𝑔(𝑡),forany𝑡0,𝑥0.(3.2)(S3)There exist positive constants 𝒞, 𝛿, 𝜉 with 𝒞>0 and 0<𝛿<𝜉<+ such that |𝑓(𝑡,𝑥)|𝒞 for 𝑡[𝛿,𝜉], 𝑥0. (S4)𝑈𝐻×+𝐶𝐾(+) is 𝐿1-Carathéodory, that is,(𝑡,𝑠)𝑈(𝑡,𝑠,𝑥) is measurable for every 𝑥+;𝑥𝑈(𝑡,𝑠,𝑥) is u.s.c. for a.e. (𝑡,𝑠)𝐻.In addition, the set 𝑆𝑈,𝑥 is nonempty for each fixed 𝑥𝐵𝐶. (S5)There exist a bounded, continuous, and nondecreasing function 𝛽++, a function 𝛼𝐿1(+,+) with 𝜉𝛿𝛼(𝑡)𝑑𝑡>0, and a continuous function 𝛾[𝛿,𝜉](0,+) such that ||𝑢𝑥||(𝑡,𝑠)𝛼(𝑠)𝛽(𝑥(𝑠)),fora.e.(𝑡,𝑠)𝐻,𝑥0,𝑢𝑥𝑆𝑈,𝑥;||𝑢𝑥||[]×[(𝑡,𝑠)𝛾(𝑡)𝛼(𝑠)𝛽(𝑥(𝑠)),fora.e.(𝑡,𝑠)𝛿,𝜉0,+),𝑥0,𝑢𝑥𝑆𝑈,𝑥;lim𝜀0+𝛽(𝜀)𝜀=+.(3.3)(S6)There exists a positive number 𝑅>0 such that 𝛽(𝑅)>0 and 𝑅𝛽(𝑅)𝜉𝛿𝑔(𝑠)𝑑𝑠0+𝑀𝛼(𝜏)𝑑𝜏,𝒞(3.4)where 𝑀=sup𝑡0𝑔(𝑡), =min𝛿𝑡𝜉𝛾(𝑡).

Theorem 3.1. If the conditions (S1)–(S6) hold, then (1.4) has at least one (positive) solution 𝑥𝐵𝐶(+,) with 𝑥0 on + and with 𝑟<𝑥𝑅 for given 0<𝑟<𝑅.

Proof. Let us define the multivalued map 𝐴 on the space 𝐵𝐶 by the following: (𝐴(𝑥))(𝑡)={𝑓(𝑡,𝑥(𝑡))}0+𝑢𝑥(𝑡,𝑠)𝑑𝑠,𝑢𝑥𝑆𝑈,𝑥,𝑡0.(3.5)We will show that 𝐴 has a fixed point recurring to Theorem 2.3. Define the function 𝜌𝐵𝐶(+,)[0,+) by 𝜌(𝑥)=max𝛿𝑡𝜉||||𝑥(𝑡)(3.6) and the set 𝑋+ by 𝑋+=𝑥𝐵𝐶𝑥(𝑡)0,for𝑡0,𝑥(𝑡)𝒞𝑀[].𝑥,for𝑡𝛿,𝜉(3.7) It is easy to see that 𝑋+ is a cone of 𝐵𝐶 and 𝜌𝑋+[0,+) given in (3.6) is a nondecreasing convex function with 𝜌(𝜃)=0 and 𝜌(𝜑)>0 for 𝜑𝜃.
First we point out that 𝐴(𝑥)𝐶𝐾(𝑋+) for each fixed 𝑥Ω𝑝𝑋+ with 𝑝>0. In fact, for any 𝑦𝐴(𝑥), there exists 𝑢𝑥𝑆𝑈,𝑥 such that 𝑦(𝑡)=𝑓(𝑡,𝑥)0𝑢𝑥(𝑡,𝑠)𝑑𝑠 for 𝑡0. (S1) and (S4) imply that 𝑦(𝑡)0 for 𝑡0 and 𝛼𝛽(𝑥)𝐿1(+,+), where 𝛽(𝑥) is defined by (𝛽(𝑥))(𝑡)=𝛽(𝑥(𝑡)). Applying our assumptions we have the following estimate: ||𝑦||||𝑓||(𝑡)(𝑡,𝑥)0+||𝑢𝑥||(𝑡,𝑠)𝑑𝑠𝑔(𝑡)0+𝛼(𝑠)𝛽(𝑥(𝑠))𝑑𝑠sup𝑡0𝑔(𝑡)0𝛼(𝑠)𝛽(𝑥(𝑠))𝑑𝑠,(3.8)
and so 𝑦𝑀0+𝛼(𝑠)𝛽(𝑥(𝑠))𝑑𝑠.(3.9) In addition (S2) and (S5), together with (3.9), guarantee that 𝑦(𝑡)=𝑓(𝑡,𝑠)0+𝑢𝑥(𝑡,𝑠)𝑑𝑠𝒞0+𝛾(𝑡)𝛼(𝑠)𝛽(𝑥(𝑠))𝑑𝑠𝒞min𝛿𝑡𝜉𝛾(𝑡)0+𝛼(𝑠)𝛽(𝑥(𝑠))𝑑𝑠𝒞𝑀𝑦.(3.10) This implies that, by the arbitrariness of 𝑦, 𝐴(𝑥) is bounded and 𝐴(𝑥)𝑋+ for each 𝑥Ω𝑝𝑋+. Similar to [20] we can infer that 𝐴(𝑥) is convex for each 𝑥Ω𝑝𝑋+. In the light of our assumptions and the Lebesgue dominated convergence theorem, we can see that 𝐴(𝑥) is compact for each 𝑥Ω𝑝𝑋+. Hence, 𝐴 maps Ω𝑝𝑋+ into 𝐶𝐾(𝑋+).
Next, we prove that 𝐴 has closed graph. Take 𝑥𝑘𝑥, 𝑘𝐴(𝑥𝑘) and 𝑘 as 𝑘. We shall prove that 𝐴(𝑥). 𝑘𝐴(𝑥𝑘) means that there exists 𝑢𝑘𝑆𝑈,𝑥𝑘 such that for each 𝑡0, 𝑘(𝑡)=𝑓𝑡,𝑥𝑘0+𝑢𝑘(𝑡,𝑠)𝑑𝑠,𝑘=1,2,.(3.11) Let (𝑡)=𝑓(𝑡,𝑥)(𝑡). From the continuity of 𝑓, it follows that 𝑓(𝑡,𝑥𝑘)𝑓(𝑡,𝑥). From (S4) it follows that 𝑢𝑘𝑢𝑥𝑆𝑈,𝑥 as 𝑘. From (S5) and the Lebesgue dominated convergence theorem it follows that 0+𝑢𝑘(𝑡,𝑠)𝑑𝑠0+𝑢𝑥(𝑡,𝑠)𝑑𝑠. It is easy to see that 0+𝑢𝑥(𝑡,𝑠)𝑑𝑠=(𝑡), that is, for each 𝑡0, (𝑡)=𝑓𝑡,𝑥0+𝑢𝑥(𝑡,𝑠)𝑑𝑠.(3.12)
This implies that 𝐴(𝑥). 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 𝐴 is u.s.c. It is clear that 𝐴 is a 𝑘-set-contractive multivalued map with 𝑘=0.
It remains to prove (in virtue of Theorem 2.3) that the condition (H1) holds to conclude that 𝐴 has a fixed point in 𝑋+, that is, that (1.4) has a positive solution. Given 𝑥𝜕Ω𝑅𝑋+ with 𝑅 satisfying the condition (S6), for any 𝑦𝐴(𝑥), there exists 𝑢𝑥𝑆𝑈,𝑥 such that 𝑦(𝑡)=𝑓(𝑡,𝑥(𝑡))0𝑢𝑥(𝑡,𝑠)𝑑𝑠, 𝑡0. Hence, 𝜌(𝑦)=max𝛿𝑡𝜉||𝑦||(𝑡)=max𝛿𝑡𝜉||||𝑓(𝑡,𝑥)0+𝑢𝑥||||(𝑡,𝑠)𝑑𝑠max𝛿𝑡𝜉𝑔(𝑡)0+𝛼(𝑠)𝛽(𝑥(𝑠))𝑑𝑠𝛽(𝑅)max𝛿𝑡𝜉𝑔(𝑡)0+𝛼(𝑠)𝑑𝑠.(3.13) In addition (S6) shows that 𝛽(𝑥(𝜏))𝛽(𝑅)𝒞𝑅𝑀max𝛿𝑡𝜉𝑔(𝑡)0+𝛼(𝑠)𝑑𝑠,(3.14) and this together with (3.13) gives the following: 𝜌(𝑦)𝒞𝑅𝑀max𝛿𝑡𝜉𝑔(𝑡)0+𝛼(𝑠)𝑑𝑠max𝛿𝑡𝜉𝑔(𝑡)0+𝛼(𝑠)𝑑𝑠=𝒞𝑅𝑀𝜌(𝑥).(3.15) Thus, 𝐴 satisfies condition (H1)(ii).
Take 𝒦=𝑀/𝒞2min𝛿𝑡𝜉𝛾(𝑡)𝜉𝛿𝛼(𝑠)𝑑𝑠. (S5) shows that there exists a positive number 𝑟<𝑅 small enough such that 𝛽(𝜀)>𝒦𝜀,0<𝜀𝑟.(3.16) Let 𝑢0(𝑡)1. To prove that (H1)(i) is true, let 𝑦𝐴𝑥 with 𝑦𝑥 and 𝑥𝜕Ω𝑟𝑋+(1). Then there exists 𝑢𝑥𝑆𝑈,𝑥 with 𝑦(𝑡)=𝑓(𝑡,𝑥)0+𝑢𝑥(𝑡,𝑠)𝑑𝑠. In virtue of the definition of 𝑋+(1), there exists 0<𝜆<1 such that 𝜆𝑥(𝑡)𝑟,𝑡0.(3.17) Note that there exists 𝜂[𝛿,𝜉] such that 𝜌(𝑦)=𝑦(𝜂). Now our assumptions imply that 𝜌(𝑦)=𝑦(𝜂)=𝑓(𝜂,𝑥)0+𝑢𝑥(𝜂,𝑠)𝑑𝑠𝒞0+𝛾(𝜂)𝛼(𝑠)𝛽(𝑥(𝑠))𝑑𝑠𝒞min𝛿𝑡𝜉𝛾(𝑡)𝜉𝛿𝛼>(𝑠)𝛽(𝑥(𝑠))𝑑𝑠𝒞𝑀min𝛿𝑡𝜉𝛾(𝑡)𝒞2min𝛿𝑡𝜉𝛾(𝑡)𝜉𝛿𝛼(𝑠)𝑑𝑠𝜉𝛿𝛼(𝑠)𝑥(𝑠)𝑑𝑠𝑥𝜌(𝑥).(3.18) So 𝜌(𝑦)>𝜌(𝑥) for all 𝑦𝐴(𝑥). This shows that (H1)(i) is satisfied. Conclusively, Theorem 2.3 guarantees that 𝐴 has a fixed point 𝑦 with 𝑟𝑦𝑅. This proof is completed.

Theorem 3.2. Suppose that conditions (S1)–(S6) hold. Then (1.4) has at least two positive solutions if the following conditions are satisfied: lim𝜂+𝛽(𝜂)𝜂=+.(3.19)

Example 3.3. Let 𝛿=1/4, 𝜉=3/4, 𝛾(𝑡)=1/2 for 𝑡[𝛿,𝜉], [],𝑠𝛼(𝑠)=1,𝑠0,1𝑠,𝑠=2,3,,2,others,(3.20)𝛽(𝑥)=𝑥 and 𝑈(𝑡,𝑠,𝑥)=[(1/2)𝛼(𝑠)𝑥,𝛼(𝑠)𝑥]. Let 𝑓(𝑡,𝑥)=𝑒𝑡(sin𝑥+1), 𝑔(𝑡)=2𝑒𝑡, =(1/2𝑒), 𝑅=32𝑒3/4. It is clear that conditions (S1)–(S6) are satisfied. Hence, Theorem 3.1 guarantees the problem 𝑥(𝑡)=𝑒𝑡(sin𝑥+1)0𝑢𝑥(𝑡,𝑠)𝑑𝑠(3.21) with 𝑢𝑥[(1/2)𝛼(𝑠)𝑥,𝛼(𝑠)𝑥] having at least a positive solution 𝑥 with 𝑥32𝑒3/4.

Acknowledgment

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

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