Krasnoselskii-type fixed point theorem in ordered Banach spaces and application to integral equations

Abstract

In this paper, we present a variant of Krasnoselskii's fixed point theorem in the case of ordered Banach spaces, where the order is generated by a normal and minihedral cone. In such a structure, there is a possibility to give a new sence to the concept of contraction.

Uniqueness of fixed points for sums of operators in ordered Banach spaces and applications

“…Proposition 2.8. ( [5])Let K be a minihedral cone in E, then the following assertions are equivalents.…”

“…Recently, in [6] and [5] a new version of hypothesis of contraction is introduced. In these works, authors consider the case where X is an ordered Banach space and the order is induced by a normal and minihédral cone.…”

“…It is well known that in such a framework an absolute value function |•| is defined. Hence the authors of [6] and [5] replaced the condition of contraction on A by…”

“…where r(L) is the spectral radius of L. Then they proved that any operator T mapping continuously the closed and convex set Ω into itself and satisfying Hypothesis (1.2) under hypothesis if the contraction mapping T maps continuously the closed convex set Ω into itself, has a unique fixed point. In [5], the author considered the case of Equation (1.1) in such an ordered Banach space X. Assuming that the mappings A and B satisfy Hypotheses (1.2), (b) and (c), he proved that the sum A + B admits at least one fixed point in Ω.…”

“…Motivated by the work in [5], we consider in this paper Equation (1.1) in the same framework of [6] and [5]. In this contribution, we introduce a new version of expansivity, we say that…”