On Behaviours of Functional Volterra Integro-Differential Equations With Multiple Time Lags

Abstract

In this paper, the authors consider a non-linear Volterra integro-differential equation (NVIDE) of first order with multiple constant time lags. They obtain new sufficient conditions on stability (S), boundedness (B), global asymptotic stability (GAS) of solutions, and in addition, every solution x of the considered NVIDE belong to L-1[0, infinity) and L-2[0, infinity). The authors prove five new theorems on S, B, GAS, L-1[0, infinity) and L-2[0, infinity) properties of solutions. The technique of the proofs involves the construction of suitable Lyapunov functionals. The obtained conditions are nonlinear generalizations and extensions of those of Becker [Uniformly continuous L-1 solutions of Volterra equations and global asymptotic stability. Cubo 11(3); 2009: 1-24], Graef et al. [Behavior of solutions of non-linear functional Voltera integro-differential equations with multiple delays. Dynam Syst Appl. 25(1-2); 2016: 39-46] and Tunc [A note on the qualitative behaviors of non-linear Volterra integro-differential equation. J Egyptian Math Soc. 24 (2); 2016: 187-192; New stability and boundedness results to Volterra integro-differential equations with delay. J Egyptian Math Soc. 24(2); 2016: 210-213] and they improve some results can found in the literature. The results of this paper are new, and they have novelty and complete some results exist in the literature.