Abstract
The purpose of this paper is to develop a qualitative stability analysis of a class of nonlinear integro-differential equation within the framework of Lyapunov-Krasovskii. We show that the existence of a Lyapunov-Krasovskii functional is a necessary and sufficient condition for the uniform asymptotic stability of the nonlinear Volterra integro-differential equations.
Highlights
Volterra integro-differential equations have wide applications in biology, ecology, medicine, physics and other scientific areas and has been extensively studied
The work of Volterra on the problem of competing species was vitally important for the development of the work in this area, since the theory and application of Volterra integro-differential equations have emerged as an important area of research
A lot of interesting results related to the qualitative behaviors of solutions; stability, boundedness etc. of Volterra integro-differential equations haven obtained by many researchers, see the papers of ([1], [[2], [5], [6], [7], [9], [11], [13], [21], [22], [23], [24], [25] ) and references therein
Summary
Volterra integro-differential equations have wide applications in biology, ecology, medicine, physics and other scientific areas and has been extensively studied. An important tool used in the discussion of the qualitative properties of solution of ordinary, functional and integro-differential equations is the Lyapunov’s second method. In [24], they established system’s stability of a class of Volterra integro-differential equation They used a known form of Lyapunov functional to establish the stability condition for the system. [23] studied certain nonlinear Volterra integro differential equations with delay He established stability and boundedness condition of the solution by defining a suitable Lyapunov functional used to prove the result. In 2007, [19] established the stability of the solutions of a class of integro-differential equations of Volterra type whose nonlinear term is assumed to be holomorphic function of variables and possible some integral form in a small neighborhood of zero. Stability in Lyapunov’s sense of single zero root and of pair of pure imaginary roots for the unperturbed equation is analyzed by relying on functional in integral form represented by Frechet series
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