Stability criteria for certain delay integral equations of Volterra type

Abstract

In this paper we study the asymptotic stability of the solution of the following delay integral equation of Volterra type: α ∫ o x (a 0 + a 1 (x − s))y(s)ds + (1 − alpha; ∫ o x (a 0 + a 1 (x − s))y(s − τ) ds , y( x) = ψ( x), − τ ⩽ x < 0, where τ > 0 is constant and 0 ⩽ α ⩽ 1. Stability criteria are provided for certain α's and the parameters a 0, a 1 and τ. The aim of this study is to understand the effect of the delay on the asymptotic stability of the solution of Volterra integral equations. As such the parameters α and 1 − α appear with the same kernel in both integrals of the equation. We also provide four algorithmic stability tests and include several examples and stability regions for certain values of the parameters α, a 0, a 1 and τ.