On some fixed-point theorems for generalized contractions and their perturbations

Abstract

The aim of this paper is to prove a collection of fixed-point theorems for mappings which can be roughly called generalized contractions or their perturbations. In particular, we are going to consider operators (single-valued or multi-valued) in Banach spaces with a quasimodulus, in hyperconvex subsets of normed spaces, or finally in non-Archimedean spaces. A particular attention will be paid to Krasnoselskii-type fixed-point theorems as well as to a Schaefer-type fixed-point theorem. Some applications to nonlinear functionalintegral equations will be given. Our results extend and complement some commonly known theorems.

Solvability of Hammerstein Integral Equations with Applications to Boundary Value Problems

“…Remark 2.7. Let us also add that a Krasnosel ′ slkiȋ-Schaefer type result similar to Theorem 2.5 can be found in [3], where the mapping F 2 , defined on the whole Banach space X, is required to be completely continuous (that is, F 2 is continuous and maps bounded sets into relatively compact ones) rather than compact (see [3,Theorem 4.2]).…”

Existence and uniqueness of fixed point for ordered contraction type operator in Banach Space

“…Existence of fixed points for contraction type maps in partially ordered metric space has been considered recently in [6]- [11], where some applications to matrix equation, ordinary differential equations and integral equations are presented, see [12]- [18]. The following generalization of Banach's contraction principle is due to Geraghty [21].…”