Solvability of a Class of Integral Inclusions

and Applied Analysis 3 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 kset 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 C ≤ D or D ≥ C if ∀p ∈ D, ∃q ∈ C such that q ≤ p. 2.1 Amultivalued operatorA is called upper semicontinuous u.s.c. on E if for each x ∈ E the setA x is a nonempty closed subset of E, and if for each open set B of E containingA 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 A : U → CK C with x / ∈ A 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. 4 Abstract and Applied Analysis Lemma 2.1. Let E be an ordered Banach space and P a cone in E, and letΩ1 andΩ2 be bounded open sets in E such that θ ∈ Ω1 andΩ1 ⊂ Ω2. Assume thatA : Ω2 → CK P is a u.s.c., k-set contractive (here 0 ≤ k < 1) map and assume one of the following conditions hold: x / ∈ λAx, ∀λ ∈ 0, 1 , x ∈ ∂Ω2 ∩ P, 2.2 there exists a v ∈ P with x / ∈ Ax δv for x ∈ ∂Ω1 ∩ P, δ ≥ 0. 2.3