Necessary and sufficient conditions for uniform stability of Volterra integro-dynamic equations using new resolvent equation

Abstract

We consider the system of Volterra integro-dynamic equations B(t, s)x(s)∆sand obtain necessary and sufficient conditions for the uniform stability of the zero solution employing the resolvent equation coupled with the variation of parameters formula. The resolvent equation that we use for the study of stability will have to be developed since it is unknown for time scales. At the end of the paper, we furnish an example in which we deploy an appropriate Lyapunov functional. In addition to generalization, the results of this paper provides improvements for its counterparts in integro-differential and integro-difference equations which are the most important particular cases of our equation.

Necessary and sufficient conditions for stability of Volterra integro-dynamic equation on time scales

“…Also, in the paper of [1] the author used the notion of fixed point theory and obtain functional bounds on solutions of Volterra integro-dynamic equations of convolution type. In addition, we shall compare our results to both [5], [12] and [15]. With respect to oscillation, the authors of [13] initiated the study of oscillation theory for integro-dynamic equations on time-scales.…”

“…In the paper of [5], Adivar and Raffoul, developed the resolvent equations for system (1) and modified them to serve as Lyapunov functionals and obtained necessary and sufficient conditions for the uniform stability of the zero solution of (1). However, this paper offers more relaxed conditions for the stability only.…”

Almost periodic solutions of Volterra difference systems

“…Qualitative properties of Volterra equations on continuous, discrete and hybrid domains became topic of many studies in the literature. By a quick literature review, we refer to [1][2][3][4][5][6][7][8][9] and references therein. Investigation of periodic solutions of dynamic equations and systems with periodic structures is of special importance for studies on population dynamics and control theory.…”

Boundedness and Asymptotic Stability for the Solution of Homogeneous Volterra Discrete Equations

“…It is well known that one of the most used tools in the stability analysis of VDEs is the Lyapunov approach [22][23][24][25][26][27][28]. As already mentioned in the introduction, among the results that can be obtained by Lyapunov techniques, the most popular are based on the hypothesis that the coefficients are summable (e.g., the result in [28,Th.…”

“…This together with (22) and (v bis ) leads to 2 +1 ≤ − 2 * ( ). Since ( ) → +∞, as increases, this is absurd.…”