New Results on Behaviors of Functional Voltera Integro-Differential Equations with Multiple Time-Lags

Abstract

The paper deals with a non-linear Volterra integro-differential equation (NVIDE) with multiple time-lags. Conditions are obtained which are sufficient for stability (S), boundedness (B), globally asymptotically stability (GAS) of solutions, and for every solution x of the given (NVIDE) to be belong to the solutions classes, such as L¹[0,∞) and L²[0,∞). We prove some results on stability, boundedness, global asymptotic stability, integrability and square integrability properties of solutions of the considered (NVIDE). The technique of the proofs involves to construct some suitable Lyapunov functionals (LFs). The given conditions involve nonlinear generalizations and extensions of those conditions found in the literature. The obtained results are new and complement that found in the literature.

Key words and phrases. Volterra integro-differential equation, first order, stability, boundedness, global asymptotic stability, integrability, square integrability, Lyapunov functional.

2000 Mathematics Subject Classification. 34K20, 45D05, 45M05