On the stability and uniform stability of retarded integro-differential equations

Abstract

In this paper, the authors obtain new sufficient conditions for stability (S) and uniform stability (US) of solutions of the first order retarded Volterra integro-differential equations (VIDEs) in the formx′=A(t)x+∫t-τtC(t,s)ϕ(s,x(s))ds+f(t,x,x(t-τ)).The analysis of the obtained (S) and (US) results mainly depend on the definition of an appropriate Lyapunov functional (LF). An example is provided to illustrate the effectiveness of the proposed results. MATLAB-Simulink is applied to show the behaviors of the paths of solutions of the considered (VIDEs) for a particular case. Keywords: Non-linear, (VIDE), First order, Stability, Uniformly stability, Lyapunov functional, 2010 AMS subject classifications: 34D05, 34K20, 45J05