Solvability of an infinite system of integral equations on the real half-axis

Józef Banaś,Weronika Woś

doi:10.1515/anona-2020-0114

Józef Banaś, Weronika Woś

Open Access

https://doi.org/10.1515/anona-2020-0114

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Abstract

Abstract The aim of the paper is to investigate the solvability of an infinite system of nonlinear integral equations on the real half-axis. The considerations will be located in the space of function sequences which are bounded at every point of the half-axis. The main tool used in the investigations is the technique associated with measures of noncompactness in the space of functions defined, continuous and bounded on the real half-axis with values in the space l ∞ consisting of real bounded sequences endowed with the standard supremum norm. The essential role in our considerations is played by the fact that we will use a measure of noncompactness constructed on the basis of a measure of noncompactness in the mentioned sequence space l ∞. An example illustrating our result will be included.

Highlights

The paper is dedicated to the study of the existence of solutions of an in nite system of nonlinear integral equations on the real half-axis R+ = [, ∞)
The main tool used in the investigations is the technique associated with measures of noncompactness in the space of functions de ned, continuous and bounded on the real half-axis with values in the space l∞ consisting of real bounded sequences endowed with the standard supremum norm
The essential role in our considerations is played by the fact that we will use a measure of noncompactness constructed on the basis of a measure of noncompactness in the mentioned sequence space l∞

Summary

Introduction

Keeping in mind the fact that the space BC∞ = BC(R+, l∞) forms a Banach algebra with respect to the coordinatewise multiplication of function sequences and taking into account the de nition of the operator Q as well as assumption (i), we infer that for an arbitrarily xed function x = x(t) ∈ BC∞ the function (Qx)(t) = ((Qn x)(t)) = an(t) + (Fn x)(t)(Vn x)(t) acts from the interval R+ into the space l∞. For each xed t ∈ R+, in virtue of assumption (vii) we get (Fy)(t) − (Fx)(t) l∞ m(r ) x − y l∞ εm(r ) This shows that the operator F is continuous at every point of the ball Br. To prove the continuity of the operator V on the ball Br let us de ne the function δ = δ(ε) by putting δ(ε) = sup |gn(t, y) − gn(t, x)| : x, y ∈ l∞, y − x l∞ ε, t ∈ R+, n ∈ N.

This yields to the estimate

An example

On the other hand we obtain t