Existence of a periodic solutions for certain system of nonlinear integro-differential equations

R Butris

doi:10.33899/edusj.2008.51249

Abstract

In this paper we investigate the existence and approximation of the periodic solutions for certain systems of nonlinear integro-differential equations, by using the method of successive periodic approximation of ordinary differential equations which is given by A. M. Samoilenko. Also these investigation lead us to the improving the extending the above method.

Highlights

Theorem 1: If the system of integro-differential equations (1) satisfy the inequalities (3), (4) with assumptions (5) and the conditions (6), (7) has a periodic solution x = x(t, x0 ), passing through the point (0, x0 ) , x0 D f , the sequence of functions:
Consider the following system of integro-differential equation, which has the form: dx dt = f t, x, x, t g (s, x( s), x ( s) )ds t −T ... ... (1)where x D Rn, D is a closed and bounded domain.The vectors functions f (t, x, x,v) and g(t, x, x ) are defined on the domain: ... ... (2)
W m → 0 as m →, proceeding in the last inequality to the limit we obtain that x(t, x0 ) = y(t, x0 ) which proves that the solution is unique, and this completes the proof of theorem 1

Summary

Introduction

Theorem 1: If the system of integro-differential equations (1) satisfy the inequalities (3), (4) with assumptions (5) and the conditions (6), (7) has a periodic solution x = x(t, x0 ), passing through the point (0, x0 ) , x0 D f , the sequence of functions: Theorem 4: If the system of equations (33) satisfies the above assumptions and conditions has a periodic solution x = (t), passing through the point

Results

Conclusion