Uniform Stability In Nonlinear Infinite Delay Volterra Integro-differential Equations Using Lyapunov Functionals

Abstract

Abstract:In [10] the rst author used Lyapunov functionals and studied the exponential stability of the zero solution of nite delay Volterra Integro-di erential equation. In this paper, we use modi ed version of the Lyapunov functional that were used in [10] to obtain criterion for the stability of the zero solution of the in nite delay nonlinear Volterra integro-di erential equation C(t, s)g(x(s))ds.

On the qualitative analyses solutions of new mathematical models of integro‐differential equations with infinite delay

“…Raffoul and Rai [16] used LKFs and discussed exponential stability (ES) of the IDE with finite delay:…”

“…By using spectral properties of Metzler matrices and the comparison principles, Ngoc and Anh [15], established some new explicit criteria for UAS and ES of nonlinear Volterra IDE such as $$$$ {x}^{\prime }=h\left(t,x\right)+\int_0^tq\left(t,s,x(s)\right) ds. $$$$ Raffoul and Rai [16] used LKFs and discussed exponential stability (ES) of the IDE with finite delay: $$$$ {x}^{\prime }= Px+\int_{-\infty}^tC\left(t,s\right)g\left(x(s)\right) ds. $$$$ Tunç and Tunç [17,18] take into account the scalar IDE $$$$ {x}^{\prime }=-A(t){G}_1(x)+\int_{-\infty}^tD\left(t,s\right){G}_2\left(s,x(s)\right) ds+{P}_1\left(t,x\right) $$$$ and the nonlinear system of IDEs with the constant time retardation …”

Qualitative Analyses of Integro-Fractional Differential Equations with Caputo Derivatives and Retardations via the Lyapunov–Razumikhin Method

“…In the relevant literature three methods, which are called the second Lyapunov method, Lyapunov-Krasovskiȋ method and Lyapunov-Razumikhin method, come to the forefront to investigate qualitative properties of solutions of linear and nonlinear integro-differential equations both without and with retardation. Among these methods, the second Lyapunov method and Lyapunov-Krasovskiȋ method are extensively used to study various qualitative behaviors of solutions of integro-differential equations of integer order (see, [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20]). To the best of our knowledge, the Lyapunov-Razumikhin method is less used during that kind of investigation [23,25,26].…”

New Parameter-Uniform Discretisations of Singularly Perturbed Volterra Integro-Differential Equations

“…A general overview of several techniques to integrate Volterra/Fredholm integral or integro-differential equations can be found in [1,11,12,14,17]. In 2006 Bijura [5] demonstrated the existence of the initial layers whose thickness is not of order of magnitude O(ε), ε −→ 0, and developed approximate solutions using the initial layer theory In [16], Ş evgin studied the convergence properties of a difference scheme for singularly perturbed Volterra integro-differential equations on a graded mesh.…”