Solvability of an infinite system of nonlinear integral equations of Volterra-Hammerstein type

Agnieszka Chlebowicz

doi:10.1515/anona-2020-0045

Abstract

AbstractThe purpose of the paper is to study the solvability of an infinite system of integral equations of Volterra-Hammerstein type on an unbounded interval. We show that such a system of integral equations has at least one solution in the space of functions defined, continuous and bounded on the real half-axis with values in the spacel1consisting of all real sequences whose series is absolutely convergent. To prove this result we construct a suitable measure of noncompactness in the mentioned function space and we use that measure together with a fixed point theorem of Darbo type.

Highlights

Integral equations create a very signi cant part of nonlinear analysis and applied mathematics ([1,2,3,4])
The purpose of the paper is to study the solvability of an in nite system of integral equations of Volterra-Hammerstein type on an unbounded interval. We show that such a system of integral equations has at least one solution in the space of functions de ned, continuous and bounded on the real half-axis with values in the space l consisting of all real sequences whose series is absolutely convergent
To prove this result we construct a suitable measure of noncompactness in the mentioned function space and we use that measure together with a xed point theorem of Darbo type

Summary

Introduction

Integral equations create a very signi cant part of nonlinear analysis and applied mathematics ([1,2,3,4]). On the one hand such systems are very interesting subject of the study for researchers specialized in the theory of integral equations but on the other hand systems of integral equations play very crucial role in applications. We show that in nite system of integral equations of VolterraHammerstein type has at least one solution in the space BC(R+, l ) consisting of all functions de ned, continuous and bounded on the interval R+ with values in the sequence space l. Each such solution belongs to the space BC(R+, c ) considered in [10]

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