Existence, Uniqueness and Stability of Periodic Solution for Nonlinear System of Integro-Differential Equations

Abstract

In this paper, we investigate the existence, uniqueness, and stability of the periodic solution for the system of nonlinear integro-differential equations by using the numerical-analytic methods for investigating the solutions and the periodic solutions of ordinary differential equations, which are given by A. Samoilenko.

Existence, Uniqueness and Stability Solutions for New Nonlinear System of Integro-Differential Equations of Volterra Type

“…1996;Bailey, P.B., Shampine, L.F. and Waltman, P.E. 1968) and the Picard iteration can be used to generate the Taylor series solution for an ordinary differential equation (Butris, R. N. 1994;Butris R.N., 2015;Butris, R. N. and Ghada, Sh. J.…”

Periodic solutions for nonlinear systems of integro-differential equations of Volterra- Friedholm type

“…In the original works of Samoilenk and Ronto [16,17 ] the approach used and described here had been referred to as the numericalanalytic based upon successive approximations. The idea of the method, originally aimed at the investigation of periodic solution only, had been later applied in studies [2,3,8,9,10,11,13,14,15]. Also, it should be noted that appropriate versions of the method considered can be applied in many situations for handling periodic in the cases of the systems of first or second order ordinary differential equations, integrodifferential equations, equations with retarded arguments, systems containing unknown parameters, and countable systems of differential equations.…”

“…A survey of the investigations on the subject can be found in the studies and researches [4,5,6,12,13,18,19,20]. Butris [1], assumes the both methods Picard approximation and Banach fixed point theorem to study the existence, uniqueness and stability solution of Volterra-Friedholm of integro-differential equations which has the following form:- In this work, we study the periodic solutions for nonlinear systems of integro-differential equations of Volterra-Friedholm (VF1) and (VF2).…”

“…Let the vector functions ( , , ), ( , , ), ( , , , ) and ( , , , ) defined on the domain ( , , ) ∈ 1 × 1 × = (−∞, ∞) × (−∞, ∞) × ( , , ) ∈ 1 × 1 × 1 = (−∞, ∞) × (−∞, ∞) × 1} (1) and periodic in of period . Assume that the vector functions ( , , ), ( , , ), ( , , , ) and ( , , , ) are satisfying the following inequalities: ‖ ( , , )‖ ≤ 1 ,‖ ( , , )‖ ≤ 2…”

Study of the Existence and Uniqueness of Global Solution for the Fractional Second Order Nonlinear Integro –Differential Equations with Boundary Conditions

“…Enormous of studies have discussed the solution of the differential and integral equation for fractional order such as [1], [2], [3], [4], [5], [6] and [7]. In this study, the focus will be on the 2nd order nonlinear integro-differential fractional equations [8], and [9].…”