A fixed point theorem for nonautonomous type superposition operators and integrable solutions of a general nonlinear functional integral equation

Fuli Wang

doi:10.1186/1029-242x-2014-487

Abstract

We first establish a new fixed point theorem for nonautonomous type superposition operators. After that, we prove the existence of integrable solutions for a general nonlinear functional integral equation in an space on an unbounded interval by using our theorem. Our main tool is the measure of weak noncompactness. MSC:47H30, 47H08.

Highlights

1 Introduction It is well known that the class of nonlinear operator equations of various types has many useful applications in describing numerous problems of the real world
The present paper is concerned with the solvability of the following quite general nonlinear functional integral equation: t x(t) = f t, x(t), k(t, s)u s, x(s) ds, t ∈ R+ := [, ∞), ( . )
U : R+ × R → R and f : R+ × R → R are two given functions, while k is a given real function defined on R+ × R+

Summary

Introduction

It is well known that the class of nonlinear operator equations of various types has many useful applications in describing numerous problems of the real world. ) is closely related to the fixed points of the nonautonomous type superposition operator, which is asked to prove that there exists x ∈ D satisfying the following operator equation: x = F(x, Ax) for two given operators F : X × Y → X and A : D ⊂ X → X, where X and Y are two Banach spaces.

Objectives

Results

Conclusion