On Monotonic and Nonnegative Solutions of a Nonlinear Volterra-Stieltjes Integral Equation

Tomasz Zając

doi:10.1155/2014/601824

Abstract

We study the existence of monotonic and nonnegative solutions of a nonlinear quadratic Volterra-Stieltjes integral equation in the space of real functions being continuous on a bounded interval. The main tools used in our considerations are the technique of measures of noncompactness in connection with the theory of functions of bounded variation and the theory of Riemann-Stieltjes integral. The obtained results can be easily applied to the class of fractional integral equations and Volterra-Chandrasekhar integral equations, among others.

Highlights

The aim of this paper is to study of monotonic and nonnegative solutions of the nonlinear quadratic Volterra-Stieltjes integral equation having the form t x (t) = (F1x) (t) + (F2x) (t) ∫ u (t, τ, (Tx) (τ)) dτg (t, τ), (1)where t ∈ [a, b] and F1, F2 are superposition operators defined on the function space C[a, b]
We study the existence of monotonic and nonnegative solutions of a nonlinear quadratic Volterra-Stieltjes integral equation in the space of real functions being continuous on a bounded interval
It is worth pointing out that differential and integral equations of fractional order create an important branch of nonlinear analysis and the theory of integral equations

Summary

Introduction

The aim of this paper is to study of monotonic and nonnegative solutions of the nonlinear quadratic Volterra-Stieltjes integral equation having the form t x (t) = (F1x) (t) + (F2x) (t) ∫ u (t, τ, (Tx) (τ)) dτg (t, τ) ,. As particular cases, the classical Volterra integral equation, the integral equation of fractional order, and the Volterra counterpart of the famous integral equation of Chandrasekhar type. It is worth pointing out that differential and integral equations of fractional order create an important branch of nonlinear analysis and the theory of integral equations. These equations have found a lot of applications connected with real world problems. This paper can be considered as a continuation of [1, 2] (cf. [3,4,5])

Preliminaries

Main Result

Findings

Applications and an Example