%0 Journal Article
%T Positive Fixed Point of Strict Set Contraction Operators on Ordered Banach Spaces and Applications
%A Meiqiang Feng
%A Xuemei Zhang
%A Weigao Ge
%J Abstract and Applied Analysis
%D 2010
%I Hindawi Publishing Corporation
%R 10.1155/2010/439137
%X The fixed point theorem of cone expansion and compression of norm type for a strict set contraction operator is generalized by replacing the norms with a convex functional satisfying certain conditions. We then show how to apply our theorem to prove the existence of a positive solution to a second-order differential equation with integral boundary conditions in an ordered Banach space. An example is worked out to demonstrate the main results. 1. Introduction The theory of integral and differential equations in Banach spaces, as two new branches of nonlinear functional analysis, has developed for no more than forty years, but it has extensive applications in such domains as the critical point theory, the theory of partial differential equations, and eigenvalue problems. For an introduction of the basic theory of integral and differential equations in Banach spaces, see Guo et al. [1], Guo and Lakshmikantham [2], Lakshmikantham and Leela [3], and Demling [4], and the references therein. In recent years, the theory of integral and differential equations in Banach spaces has become an important area of investigation in both pure and applied mathematics (see, for instance, [5¨C18] and references cited therein). On the other hand, the theory of fixed point is an important tool to study various boundary value problems of ordinary differential equations, difference differential equations, and dynamic equations on time scales. An overview of such results can be found in Guo et al. [1], in Guo and Lakshmikantham [2], and in Demling [4]. The Krasnoselskii's fixed point theorem concerning cone compression and expansion of norm type is worth mentioing here as follows (see [1, 2, 4]). Theorem 1.1. Let and be two bounded open sets in Banach space , such that and . Let be a cone in and let operator be completely continuous. Suppose that one of the following two conditions is satisfied: (a) and ; (b) and .Then, has at least one fixed point in . To generalize Theorem 1.1, one may consider the weakening of one or more of the following hypotheses: (i) the operator , (ii) the norm. In [19], Sun generalized Theorem 1.1 for completely continuous operator to strict set contraction operator and obtain the following results. Theorem 1.2. Let be a cone of Banach space and with Suppose that is a strict set contraction such that one of the following two conditions is satisfied: (a) (b) Then, has a fixed point . Recently, in [20], Anderson and Avery generalized the fixed point theorem of cone expansion and compression of norm type by replacing the norms with two functionals
%U http://www.hindawi.com/journals/aaa/2010/439137/