International Journal of Engineering Inventions e-ISSN: 2278-7461, p-ISSN: 2319-6491 Volume 2, Issue 2 (January 2013) PP: 57-64
Periodic Solutions for Nonlinear Systems of IntegroDifferential Equations of Operators with Impulsive Action Dr. Raad N. Butris Assistant Prof. Department of Mathematics Faculty of Science University of Zakho Iraq-Duhok-Kurdistan Region
Abstract:- In this paper we investigate the existence and approximation of the construction of periodic solutions for nonlinear systems of integro-differential equation of operators with impulsive action, by using the numerical-analytic method for periodic solutions which is given by Samoilenko. This investigation leads us to the improving and extending to the above method and expands the results gained by Butris. Keyword and Phrases:- Numerical-analytic methods existence of periodic solutions, nonlinear integrodifferential equations, subjected to impulsive action of operators.
I.
INTRODUCTION
They are many subjects in mechanics, physics, biology, and other subjects in science and engineering requires the investigation of periodic solution of integro-differential equations with impulsive action of various types and their systems in particular, they are many works devoted to problems of the existence of periodic solutions of integro-differential equations with impulsive action . samoilenko[5,6] assumed a numerical analytic method to study to periodic solution for ordinary differential equation and this method include uniformly sequences of periodic functions’ as Intel studies [2,3,4,7,8]. The present work continues the investigations in this direction. These investigations lead us to the improving and extending the above method and expand the results gained by Butris [1]. đ?‘‘đ?‘Ľ = đ?‘“(đ?‘Ą, đ?‘Ľ, đ??´đ?‘Ľ, đ??ľđ?‘Ľ, đ?‘‘đ?‘Ą Δđ?‘Ľ
đ?‘Ą=đ?‘Ą đ?‘–
đ?‘Ą
đ?‘”(đ?‘Ą, đ?‘Ľ đ?‘ , đ??´đ?‘Ľ đ?‘ , đ??ľđ?‘Ľ(đ?‘ ))đ?‘‘đ?‘ ), đ?‘Ą ≠đ?‘Ąđ?‘– đ?‘Ąâˆ’đ?‘‡
đ?‘Ą
= đ??źđ?‘– (đ?‘Ľ, đ??´đ?‘Ľ, đ??ľđ?‘Ľ,
đ?‘” đ?‘ , đ?‘Ľ đ?‘ , đ??´đ?‘Ľ đ?‘ , đ??ľđ?‘Ľ đ?‘ đ?‘‘đ?‘ ) đ?‘Ąâˆ’đ?‘‡
â‹Ż (1.1) Where đ?‘Ľ ∈ đ??ˇ ⊂ đ?‘…đ?‘› , đ??ˇ is closed bounded domain. The vector functions đ?‘“(đ?‘Ą, đ?‘Ľ, đ?‘Ś, đ?‘§, đ?‘¤) and đ?‘”(đ?‘Ą, đ?‘Ľ, đ?‘Ś, đ?‘§) are defined on the domain đ?‘Ą, đ?‘Ľ, đ?‘Ś, đ?‘§, đ?‘¤ ∈ đ?‘…1 Ă— đ??ˇ Ă— đ??ˇ1 Ă— đ??ˇ2 Ă— đ??ˇ3 = −∞, ∞ Ă— đ??ˇ Ă— đ??ˇ1 Ă— đ??ˇ2 Ă— đ??ˇ3 â‹Ż 1.2 Which are piecewise continuous functions in đ?‘Ą, đ?‘Ľ, đ?‘Ś, đ?‘§, đ?‘¤ and periodic in đ?‘Ą of period đ?‘‡, where đ??ˇ1 , đ??ˇ2 and đ??ˇ3 are bounded domain subset of Euclidean spaces đ?‘…đ?‘š . Let đ??źđ?‘– đ?‘Ľ, đ?‘Ś, đ?‘§, đ?‘¤ be continuous vector functions, defined on the domain (1.2) and đ??źđ?‘–+đ?‘? đ?‘Ľ, đ?‘Ś, đ?‘§, đ?‘¤ = đ??źđ?‘– đ?‘Ľ, đ?‘Ś, đ?‘§, đ?‘¤ , đ?‘Ąđ?‘–+đ?‘? = đ?‘Ąđ?‘– + đ?‘‡ â‹Ż 1.3 For all đ?‘– ∈ đ?‘§ + , đ?‘Ľ ∈ đ??ˇ, đ?‘Ś ∈ đ??ˇ1 , đ?‘§ ∈ đ??ˇ2 , đ?‘¤ ∈ đ??ˇ3 and for some number đ?‘? . Let the operators A and B transform any piecewise continuous functions from the domain đ??ˇ to the piecewise contiuous function in the domain đ??ˇ1 and đ??ˇ2 respectively. Moreover đ??´đ?‘Ľ đ?‘Ą + đ?‘‡ = đ??´đ?‘Ľ(đ?‘Ą) and đ??ľđ?‘Ľ đ?‘Ą + đ?‘‡ = đ??ľđ?‘Ľ đ?‘Ą . Suppose that đ?‘“ đ?‘Ą, đ?‘Ľ, đ?‘Ś, đ?‘§, đ?‘¤ , đ?‘” đ?‘Ą, đ?‘Ľ, đ?‘Ś, đ?‘§ , đ??źđ?‘– (đ?‘Ą, đ?‘Ľ, đ?‘Ś, đ?‘§, đ?‘¤) and the operators A, B satisfy the following inequalities: đ?‘“ đ?‘Ą, đ?‘Ľ, đ?‘Ś, đ?‘§, đ?‘¤ ≤ đ?‘€, đ??źđ?‘– (đ?‘Ą, đ?‘Ľ, đ?‘Ś, đ?‘§, đ?‘¤) ≤ đ?‘ , â‹Ż 1.4 đ?‘“ đ?‘Ą, đ?‘Ľ1 , đ?‘Ś1 , đ?‘§1 , đ?‘¤1 − đ?‘“ đ?‘Ą, đ?‘Ľ2 , đ?‘Ś2 , đ?‘§2 , đ?‘¤2 ≤ đ??ž( đ?‘Ľ1 − đ?‘Ľ2 + đ?‘Ś1 − đ?‘Ś2 + đ?‘§1 − đ?‘§2 + đ?‘¤1 − đ?‘¤2 ) đ?‘” đ?‘Ą, đ?‘Ľ1 , đ?‘Ś1 , đ?‘§1 − đ?‘” đ?‘Ą, đ?‘Ľ2 , đ?‘Ś2 , đ?‘§2 ≤ đ?‘„( đ?‘Ľ1 − đ?‘Ľ2 + đ?‘Ś1 − đ?‘Ś2 + đ?‘§1 − đ?‘§2 ) â‹Ż 1.5 And đ??źđ?‘– đ?‘Ą, đ?‘Ľ1 , đ?‘Ś1 , đ?‘§1 , đ?‘¤1 − đ??źđ?‘– đ?‘Ą, đ?‘Ľ2 , đ?‘Ś2 , đ?‘§2 , đ?‘¤2 ≤ đ??ż( đ?‘Ľ1 − đ?‘Ľ2 + đ?‘Ś1 − đ?‘Ś2 + đ?‘§1 − đ?‘§2 + đ?‘¤1 − đ?‘¤2 ) â‹Ż 1.6
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