On uniform asymptotic stability for nonlinear integro-differential equations of Volterra type

Abstract

The specifics of the application of Razumikhin technique to the stability analysis of Volterra type integrodifferential equations are considered. The equation can be nonlinear and nonautonomous. We propose new sufficient conditions for uniform asymptotic stability of the zero solution using the phase space of a special construction and constraints on the right side of the equation. In the presented constraints we can analyze stability, relying not only on the behavior of the auxiliary function along the solutions, but also on the properties of the so called limiting equations.

On the Fundamental Analyses of Solutions to Nonlinear Integro-Differential Equations of the Second Order

“…Defining and then using these two new LKFs, the main results of this article are proved. In special cases of ( 6) and (10), three examples are given as numerical applications to illustrate and verify our results. We aim to provide some new contributions to qualitative theory of FDEs and some known results in the relevant literature.…”

“…In the relevant literature, the global asymptotic stability, boundedness, integrability, etc., of linear and nonlinear integro-differential equations (IDEs) of the first order without delay, scalar nonlinear delay integro-differential equations (DIDEs), nonlinear delay systems of IDEs of the first order, functional differential equations (FDEs), etc., have attracted a lot of attention from researchers. For a comprehensive treatment of the subject on the stability, boundedness, integrability, etc., of solutions of first-order IDEs without delay, see, Alahmadi et al [1], Burton [2], Furumochi and Matsuoka [3], Grimmer and Seifert [4], Jordan [5], Lakshmikantham and Rama Mohana Rao [6], Mohana Rao and Srinivas [7], Murakami [8], Rama Mohana Rao and Raghavendra [9], Sedova [10], and the bibliographies therein.…”

“…In [10], sufficient conditions for uniform asymptotic stability of the zero solution of this system are obtained using the Razumikhin method. Similar qualitative results can be found in the other sources mentioned above.…”

“…In this article, we construct new sufficient qualitative conditions on the global asymptotic stability, boundedness, and integrability of solutions for both the scalar nonlinear DIDE (6) of second order and the non-linear system of DIDEs (10) with infinite delays. Defining and then using these two new LKFs, the main results of this article are proved.…”

Optimal interface controller design for haptic system

“…Moreover, an adaptive algorithm for restoring mixed noise has been proposed by Thang et al [Thang et al, 2019]. In [Sedova, 2019] author has discussed stability for nonlinear system. Observer-based boundary control for sino-gordan energy system is posed by Dolgopolik et al [Dolgopolik et al, 2019].…”