Local asymptotic stability for nonlinear quadratic functional integral equations

Bapurao Dhage,S S Bellale

doi:10.14232/ejqtde.2008.1.10

Abstract

Local asymptotic stability for nonlinear quadratic functional integral equations

Highlights

We are going to prove a theorem on the existence and uniform global attractivity of solutions for a quadratic functional integral equation
The main tool used in our considerations is the technique of measures of noncompactness and the fixed point theorem of Darbo [1, page 17]
The measure of noncompactness used in this paper allows us to obtain the existence of solutions of the mentioned functional integral equation and to characterize the solutions in terms of uniform global asymptotic attractivity. This assertion means that all possible solutions of the functional integral equation in question are globally uniformly attractive in the sense of notion defined

Summary

Introduction

We are going to prove a theorem on the existence and uniform global attractivity of solutions for a quadratic functional integral equation. We use a handy formula for ball or Hausdorff measure of noncompactness in BC(R+, R) discussed in Banas [2] To derive this formula, let us fix a nonempty and bounded subset X of the space BC(R+, R) and a positive number T. The kernel ker μ of this measure consists nonempty and bounded subsets X of BC(R+, R) such that the functions from X are locally equicontinuous on R+ and the thickness of the bundle formed by functions from X tends to zero at infinity. This particular characteristic of ker μ has been utilized in establishing the local attractivity of the solutions for quadratic integral equation. Let us mention that the concept of attractivity of solutions was introduced in Hu and Yan [9] and Banas and Rzepka [3] while the concept of asymptotic attractivity is introduced in Dhage [7]

The Integral Equation and Stability Result

An Example