Volterra–Stieltjes integral equations and impulsive Volterra–Stieltjes integral equations

Edgardo Alvarez,Jaqueline Mesquita,Carlos Lizama,Rogelio Grau

doi:10.14232/ejqtde.2021.1.5

Abstract

In this paper, we prove existence and uniqueness of solutions of Volterra– Stieltjes integral equations using the Henstock–Kurzweil integral. Also, we prove that these equations encompass impulsive Volterra–Stieltjes integral equations and prove the existence and uniqueness for these equations. Finally, we present some examples to illustrate our results.

Highlights

In this paper, we are interested in the study of integral equations that can be modeled in the form t x(t) = x0 + a(t, s) f (x(s), s) dg(s), t ∈ [t0, t0 + σ], (1.1)t0 where the integral on the right-hand side is in the sense of Henstock–Kurzweil–Stieltjes [22]
The subject of Volterra integral equations has been attracting the attention by several researchers, since they represent a powerful tool for applications
We investigate the Volterra–Stieltjes integral equations and we prove a result concerning the existence and uniqueness of solutions of these equations

Summary

Introduction

If a(t, s) = k(t − s) for all (t, s) ∈ [t0, t0 + σ] × [t0, t0 + σ], the integral equation (1.1) reduces to a Volterra integral equation which have many applications to the study of heat flow in the materials of fading memory type (see [7, 20, 21]), among others. Our goal is to prove existence and uniqueness results for the integral equation (1.1) under very weak conditions for the functions f , a and g. We investigate the Volterra–Stieltjes integral equations and we prove a result concerning the existence and uniqueness of solutions of these equations. The last section is devoted to present a correspondence between Volterra–Stieltjes integral equations and impulsive Volterra–Stieltjes equations and to prove a result concerning existence and uniqueness of solutions for these last equations

Henstock–Kurzweil–Stieltjes integral

Impulsive Volterra–Stieltjes integral equations