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International Journal of Engineering Inventions ISSN: 2278-7461, ISBN: 2319-6491 Volume 1, Issue 12 (December 2012) PP: 75-87

Periodic Solution for Nonlinear System of Differential Equations with Pulse Action of Parameters Dr. Raad. N. Butris Department of Mathematics Faculty of Science, University of Zakho

Abstract:- In this paper we study the existence of a periodic solution for nonlinear system of differential equations with pulse action of parameters. The numerical-analytic method has been used to study the periodic solutions of the nonlinear ordinary differential equations that were introduced by Somioleko And the result of this study which is the using the periodic solutions on a wide range in difference processes in industry and technology.

I.

INTRODUCTION

There are many subjects in physics and technology using mathematical methods that depends on the nonlinear differential equations, and it became clear that the existence of the periodic solutions and it's algorithm structure from more important problems in the present time. Where many of studies and researches dedicates for treatment the autonomous and non-autonomous periodic systems and specially with the integral equations and differential equations and the linear and nonlinear differential and which is dealing in general shape with the problems about periodic solutions theory and the modern methods in its quality treatment for the periodic differential equations. Somioleko [6] assumes the numerical analytic method to study the periodic solutions for the ordinary differential equations and its algorithm structure and this method include uniformly sequences of the periodic functions and the result of that study is the using of the periodic solutions on a wide range for example see [4, 5, 6]. Consider the following system of nonlinear differential equation, which has the form: đ?‘‘đ?‘Ľ = đ?œ†đ?‘Ľ + đ?‘“ đ?‘Ą, đ?‘Ľ, đ?‘Ś , đ?‘Ą ≠đ?‘Ąđ?‘– , ∆đ?‘Ľ = đ??źđ?‘– đ?‘Ľ, đ?‘Ś đ?‘Ą = đ?‘Ąđ?‘– đ?‘‘đ?‘Ą . . . (1) đ?‘‘đ?‘Ś = đ?›˝đ?‘Ľ + đ?‘” đ?‘Ą, đ?‘Ľ, đ?‘Ś , đ?‘Ą ≠ đ?‘Ąđ?‘– , ∆đ?‘Ś = đ??şđ?‘– đ?‘Ľ, đ?‘Ś đ?‘Ą = đ?‘Ąđ?‘– đ?‘‘đ?‘Ą Where đ?‘Ľ ∈ đ??ˇđ?œ† ⊆ đ?‘…đ?‘› , đ?‘Ś ∈ đ??ˇđ?›˝ ⊆ đ?‘…đ?‘› , đ??ˇđ?œ† is a closed and bounded domain. The vector functions đ?‘“ đ?‘Ą, đ?‘Ľ, đ?‘Ś đ?‘Žđ?‘›đ?‘‘ đ?‘” đ?‘Ą, đ?‘Ľ, đ?‘Ś are defined on the domain: đ?‘Ą, đ?‘Ľ, đ?‘Ś ∈ đ?‘…1 Ă— đ??ˇđ?œ† Ă— đ??ˇđ?›˝ = −∞, ∞ Ă— đ??ˇđ?œ† Ă— đ??ˇđ?›˝ . . . (2) Which are continuous inđ?‘Ą , đ?‘Ľ , đ?‘Śand periodic in t of period T, where đ??ˇđ?›˝ is bounded domains subset of Euclidean spaces đ?‘…đ?‘š , and the functions đ??źđ?‘– đ?‘Ľ, đ?‘Ś , đ??şđ?‘– đ?‘Ľ, đ?‘Ś are continuous in the domain (2), where đ??źđ?‘–+đ?‘? đ?‘Ľ, đ?‘Ś = đ??źđ?‘– đ?‘Ľ, đ?‘Ś , đ??şđ?‘–+đ?‘? đ?‘Ľ, đ?‘Ś = đ??şđ?‘– đ?‘Ľ, đ?‘Ś đ?‘Žđ?‘›đ?‘‘ đ?‘Ąđ?‘–+đ?‘? đ?‘Ľ, đ?‘Ś = đ?‘Ąđ?‘– + đ?‘‡ for p is a positive integer and đ?‘Ąđ?‘– is finite positive sequence of numbers. Suppose that the vector functions in (1)are satisfying the following inequalities: max đ?‘“(đ?‘Ą, đ?‘Ľ, đ?‘Ś) ≤ đ?‘€1 , max đ?‘”(đ?‘Ą, đ?‘Ľ, đ?‘Ś) ≤ đ?‘€2 . . . (3) đ?‘Ľ,đ?‘Ś ∈đ??ˇ đ?œ† Ă— đ??ˇ đ?›˝ đ?‘Ąâˆˆ[0,đ?‘‡]

max

đ?‘Ľ,đ?‘Ś ∈đ??ˇ đ?œ† Ă— đ??ˇ đ?›˝ 1≤đ?‘–≤đ?‘?

đ?‘Ľ ,đ?‘Ś ∈đ??ˇ đ?œ† Ă— đ??ˇ đ?›˝ đ?‘Ąâˆˆ[0,đ?‘‡]

đ??źđ?‘– (đ?‘Ľ, đ?‘Ś) ≤ đ?‘€3

,

max

đ?‘Ľ,đ?‘Ś ∈đ??ˇ đ?œ† Ă— đ??ˇ đ?›˝ 1≤đ?‘–≤đ?‘?

đ??şđ?‘– (đ?‘Ľ, đ?‘Ś) ≤ đ?‘€4

. . . (4)

đ?‘“ đ?‘Ą, đ?‘Ľ1 , đ?‘Ś1 − đ?‘“ đ?‘Ą, đ?‘Ľ2 , đ?‘Ś2 ≤ đ??ž1 đ?‘Ľ1 − đ?‘Ľ2 + đ??ž2 đ?‘Ś1 − đ?‘Ś2 . . . (5) đ?‘” đ?‘Ą, đ?‘Ľ1 , đ?‘Ś1 − đ?‘” đ?‘Ą, đ?‘Ľ2 , đ?‘Ś2 ≤ đ??ż1 đ?‘Ľ1 − đ?‘Ľ2 + đ??ż2 đ?‘Ś1 − đ?‘Ś2 . . . (6) đ??źđ?‘– đ?‘Ľ1 , đ?‘Ś1 − đ??źđ?‘– đ?‘Ľ2 , đ?‘Ś2 ≤ đ??ž3 đ?‘Ľ1 − đ?‘Ľ2 + đ??ž4 đ?‘Ś1 − đ?‘Ś2 . . . (7) đ??şđ?‘– đ?‘Ľ1 , đ?‘Ś1 − đ??şđ?‘– đ?‘Ľ2 , đ?‘Ś2 ≤ đ??ż3 đ?‘Ľ1 − đ?‘Ľ2 + đ??ż4 đ?‘Ś1 − đ?‘Ś2 . . . (8) Where đ?‘Ą ∈ đ?‘…′ , x ,đ?‘Ľ1 ,đ?‘Ľ2 ∈ đ??ˇđ?œ† , y ,đ?‘Ś1 ,đ?‘Ś2 ∈ đ??ˇđ?›˝ and đ?‘€1 , đ?‘€2 , đ?‘€3 , đ?‘€4 , đ??ž1 , đ??ž2 , đ??ž3 , đ??ž4 , đ??ż1 , đ??ż2 , đ??ż3 , đ??ż4 are a positive constant , . = max0≤đ?‘Ąâ‰¤đ?‘‡ . . Let, đ?›˝ are a positive parameter defined in (2), continuous and periodic at đ?œ? , đ?‘ , đ?‘Ą and satisfy both following inequalities:

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Periodic Solution for Nonlinear System of Differential‌ đ?‘’đ?œ†

đ?‘Ąâˆ’đ?‘

≤đ??ť . . . (9)

đ?‘’ đ?›˝ (đ?‘Ąâˆ’đ?‘ ) ≤ đ??š Where H and F are a positive constants. We define the non-empty sets as follows: đ?œ†đ?‘‡đ??ť2 đ?‘‡ đ?œ†đ?‘‡ đ??ˇđ?œ†đ?‘“ = đ??ˇđ?œ† − đ??ťđ?‘‡ − đ?‘€1 + đ?‘?đ??ť 1 + đ?œ†đ?‘‡ 1+đ?‘’ 1 + đ?‘’ đ?œ†đ?‘‡

,

. . . (10) đ?›˝đ?‘‡đ??š 2 đ?‘‡ đ?›˝đ?‘‡ đ??ˇđ?›˝đ?‘“ = đ??ˇđ?›˝ − đ??šđ?‘‡ − đ?‘€2 + đ?‘?đ??š 1 + 1 + đ?‘’ đ?›˝đ?‘‡ 1 + đ?‘’ đ?›˝đ?‘‡ Furthermore, we suppose that the larges Eigen-value for the following matrix: đ?‘Š1 đ?‘Š2 Λ0 = đ?‘Š3 đ?‘Š4 is less than one, i.e. : đ?‘Š1 + đ?‘Š4 + (đ?‘Š1 + đ?‘Š4 )2 + 4(đ?‘Š2 đ?‘Š3 − đ?‘Š1 đ?‘Š4 ) đ?‘žđ?‘šđ?‘Žđ?‘Ľ = , . . . (11) 2 where đ?œ†đ?‘‡đ??ť2 đ?‘‡ đ?œ†đ?‘‡ đ?‘Š1 = đ??ťđ?‘‡ − đ??ž1 + 2đ?‘?đ??ť 1 + đ??ž , đ?œ†đ?‘‡ 1+đ?‘’ 1 + đ?‘’ đ?œ†đ?‘‡ 3 đ?œ†đ?‘‡đ??ť2 đ?‘‡ đ?œ†đ?‘‡ đ?‘Š2 = đ??ťđ?‘‡ − đ??ž2 + 2đ?‘?đ??ť 1 + đ??ž , đ?œ†đ?‘‡ 1+đ?‘’ 1 + đ?‘’ đ?œ†đ?‘‡ 4 đ?›˝đ?‘‡đ??š 2 đ?‘‡ đ?›˝đ?‘‡ đ?‘Š3 = đ??šđ?‘‡ − đ??ż + 2đ?‘?đ??š 1 + đ??ż , 1 + đ?‘’ đ?›˝đ?‘‡ 1 1 + đ?‘’ đ?›˝đ?‘‡ 3 đ?›˝đ?‘‡đ??š 2 đ?‘‡ đ?›˝đ?‘‡ đ?‘Š4 = đ??šđ?‘‡ − đ??ż + 2đ?‘?đ??š 1 + đ??ż , 1 + đ?‘’đ?›˝ 2 1 + đ?‘’đ?›˝ 4 2 đ?œ†đ?‘‡đ??ť đ?‘‡ đ?œ†đ?‘‡ đ?‘€5 = đ??ťđ?‘‡ − đ?‘€1 + 2đ?‘?đ??ť 1 + đ?‘€3 , đ?œ†đ?‘‡ 1+đ?‘’ 1 + đ?‘’ đ?œ†đ?‘‡ đ?›˝đ?‘‡đ??š 2 đ?‘‡ đ?›˝đ?‘‡ đ?‘€6 = đ??šđ?‘‡ − đ?‘€2 + 2đ?‘?đ??š 1 + đ?‘€4 . đ?›˝đ?‘‡ 1+đ?‘’ 1 + đ?‘’ đ?›˝đ?‘‡ Approximation solution of (1) The investigation of approximation solution of the system (1) will be introduced by the following theorem: Theorem 1 If the system of nonlinear differential equations with pulse action (1) satisfy the inequalities (3) -- (8) and the conditions (9) , (10) then the sequences of functions : đ?‘Ą

đ?‘’đ?œ†

�� +1 �, �ο , �ο = �ο + �

− 0

đ?‘Ąâˆ’đ?‘

[đ?‘“ đ?‘ , đ?‘Ľđ?‘š đ?‘ , đ?‘ĽÎż , đ?‘ŚÎż , đ?‘Śđ?‘š đ?‘ , đ?‘ĽÎż , đ?‘ŚÎż

−

0

đ?œ† đ?‘’đ?œ† 1 + đ?‘’ đ?œ†đ?‘‡ đ?‘’đ?œ†

+ 0<đ?&#x2018;Ą đ?&#x2018;&#x2013; <đ?&#x2018;Ą

đ?&#x153;&#x2020; â&#x2C6;&#x2019; 1 + đ?&#x2018;&#x2019; đ?&#x153;&#x2020;đ?&#x2018;&#x2021; with đ?&#x2018;ĽÎż đ?&#x2018;Ą, đ?&#x2018;ĽÎż = đ?&#x2018;ĽÎż đ?&#x2018;&#x2019; đ?&#x153;&#x2020;đ?&#x2018;Ą , end

đ?&#x2018;&#x2021;â&#x2C6;&#x2019;đ?&#x2018;

đ?&#x2018;&#x2021;â&#x2C6;&#x2019;đ?&#x2018;

đ?&#x2018;&#x201C; đ?&#x2018; , đ?&#x2018;Ľđ?&#x2018;&#x161; đ?&#x2018; , đ?&#x2018;ĽÎż , đ?&#x2018;ŚÎż , đ?&#x2018;Śđ?&#x2018;&#x161; đ?&#x2018; , đ?&#x2018;ĽÎż , đ?&#x2018;ŚÎż đ?&#x2018;&#x2018;đ?&#x2018; ]đ?&#x2018;&#x2018;đ?&#x2018;  +

đ??źđ?&#x2018;&#x2013; đ?&#x2018;Ľđ?&#x2018;&#x161; đ?&#x2018;Ą, đ?&#x2018;ĽÎż , đ?&#x2018;ŚÎż , đ?&#x2018;Śđ?&#x2018;&#x161; đ?&#x2018;Ą, đ?&#x2018;ĽÎż , đ?&#x2018;ŚÎż

â&#x2C6;&#x2019;

đ?&#x2018;?

đ?&#x2018;&#x2019;đ?&#x153;&#x2020;

đ?&#x2018;&#x2021;â&#x2C6;&#x2019;đ?&#x2018;

đ??źđ?&#x2018;&#x2013; đ?&#x2018;Ľđ?&#x2018;&#x161; đ?&#x2018;Ą, đ?&#x2018;ĽÎż , đ?&#x2018;ŚÎż , đ?&#x2018;Śđ?&#x2018;&#x161; đ?&#x2018;Ą, đ?&#x2018;ĽÎż , đ?&#x2018;ŚÎż

. . . (12)

đ?&#x2018;&#x2013;=1

m =0 , 1 , 2 , . . . . . . , đ?&#x2018;Ą

đ?&#x2018;&#x2019;đ?&#x203A;˝

đ?&#x2018;Śđ?&#x2018;&#x161; +1 đ?&#x2018;Ą, đ?&#x2018;ĽÎż , đ?&#x2018;ŚÎż = đ?&#x2018;ŚÎż +

đ?&#x2018;Ąâ&#x2C6;&#x2019;đ?&#x2018;

[đ?&#x2018;&#x201C; đ?&#x2018; , đ?&#x2018;Ľđ?&#x2018;&#x161; đ?&#x2018; , đ?&#x2018;ĽÎż , đ?&#x2018;ŚÎż , đ?&#x2018;Śđ?&#x2018;&#x161; đ?&#x2018; , đ?&#x2018;ĽÎż , đ?&#x2018;ŚÎż

â&#x2C6;&#x2019;

0

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Periodic Solution for Nonlinear System of Differential… 𝑇

− 0

𝛽 𝑒𝛽 1 + 𝑒 𝛽𝑇 𝑒𝛽

+ 0<𝑡 𝑖 <𝑡

𝑓 𝑠, 𝑥𝑚 𝑠, 𝑥ο , 𝑦ο , 𝑦𝑚 𝑠, 𝑥ο , 𝑦ο 𝑑𝑠]𝑑𝑠 +

𝐺𝑖 𝑥𝑚 𝑡, 𝑥ο , 𝑦ο , 𝑦𝑚 𝑡, 𝑥ο , 𝑦ο

𝑝

𝛽 − 1 + 𝑒 𝛽𝑇 With 𝑦ο 𝑡, 𝑥ο = 𝑦ο 𝑒 𝛽𝑡 ,

𝑇−𝑠

𝑇−𝑠

𝑒𝛽

𝑇−𝑠

𝐺𝑖 𝑥𝑚 𝑡, 𝑥ο , 𝑦ο , 𝑦𝑚 𝑡, 𝑥ο , 𝑦ο

. . . (13)

𝑖=1

m =0 , 1 , 2 , . . . . . . ,

Are periodic in t of period T, and are uniformly convergent as 𝑚 ∞ in the domain : 𝑡, 𝑥ο , 𝑦ο ∈ 𝑅′ × 𝐷𝜆𝑓 × 𝐷𝛽𝑓 . . . (14) To the limit function 𝑥∞ 𝑡, 𝑥ο , 𝑦ο and𝑦∞ 𝑡, 𝑥ο , 𝑦ο define in the domain (14), which is periodic in t of period T and satisfying the system of integral equations : 𝑡

𝑒𝜆

𝑥 𝑡, 𝑥ο , 𝑦ο = 𝑥ο + 0 𝑇

− 0

𝑡−𝑠

[𝑓 𝑠, 𝑥 𝑠, 𝑥ο , 𝑦ο , 𝑦 𝑠, 𝑥ο , 𝑦ο

𝜆 𝑒𝜆 1 + 𝑒 𝜆𝑇 𝑒𝜆

+ 0<𝑡 𝑖 <𝑡

𝜆 − 1 + 𝑒 𝜆𝑇

𝑇−𝑠

𝑇−𝑠

𝑓 𝑠, 𝑥 𝑠, 𝑥ο , 𝑦ο , 𝑦 𝑠, 𝑥ο , 𝑦ο 𝑑𝑠]𝑑𝑠 +

𝐼𝑖 𝑥 𝑡, 𝑥ο , 𝑦ο , 𝑦 𝑡, 𝑥ο , 𝑦ο

𝑝

𝑒𝜆

𝑇−𝑠

𝐼𝑖 𝑥 𝑡, 𝑥ο , 𝑦ο , 𝑦 𝑡, 𝑥ο , 𝑦ο

,

. . . (15)

𝑖=1

and 𝑡

𝑒𝛽

𝑦 𝑡, 𝑥ο , 𝑦ο = 𝑦ο + 0 𝑇

− 0

𝑡−𝑠

[𝑓 𝑠, 𝑥 𝑠, 𝑥ο , 𝑦ο , 𝑦 𝑠, 𝑥ο , 𝑦ο

𝛽 𝑒𝛽 1 + 𝑒 𝛽𝑇 𝑒𝛽

+ 0<𝑡 𝑖 <𝑡

𝛽 − 1 + 𝑒 𝛽𝑇

𝑇−𝑠

𝑇−𝑠

𝑓 𝑠, 𝑥 𝑠, 𝑥ο , 𝑦ο , 𝑦 𝑠, 𝑥ο , 𝑦ο 𝑑𝑠]𝑑𝑠 +

𝐺𝑖 𝑥 𝑡, 𝑥ο , 𝑦ο , 𝑦 𝑡, 𝑥ο , 𝑦ο

𝑝

𝑒𝛽

𝑇−𝑠

𝐺𝑖 𝑥 𝑡, 𝑥ο , 𝑦ο , 𝑦 𝑡, 𝑥ο , 𝑦ο

,

. . . (16)

𝑖=1

Which is are unique solutions of the system (1), provided that : 𝜆𝑇𝐻2 𝑇 𝜆𝑇 𝑥 ο 𝑡, 𝑥ο , 𝑦ο − 𝑥ο ≤ 𝐻𝑇 − 𝑀1 + 2𝑝𝐻 1 + 𝑀3 . . . (17) 1 + 𝑒 𝜆𝑇 1 + 𝑒 𝜆𝑇 𝛽𝑇𝐹 2 𝑇 𝛽𝑇 𝑦 ο 𝑡, 𝑥ο , 𝑦ο − 𝑦ο ≤ 𝐹𝑇 − 𝑀2 + 2𝑝𝐹 1 + 𝑀4 . . . (18) 1 + 𝑒 𝛽𝑇 1 + 𝑒 𝛽𝑇 𝑥∞ 𝑡, 𝑥ο , 𝑦ο − 𝑥𝑚 𝑡, 𝑥ο , 𝑦ο −1 ≤ Λ𝑚 . . . (19) ο (𝐸 − Λο ) 𝑉ο , 𝑦∞ 𝑡, 𝑥ο , 𝑦ο − 𝑦𝑚 𝑡, 𝑥ο , 𝑦ο for all 𝑡 ∈ [0, 𝑇] , 𝑥ο ∈ 𝐷𝜆𝑓 , 𝑦ο ∈ 𝐷𝛽𝑓 , when: 𝑁1 (𝑡) 𝑁3 (𝑡) Λο = 𝑁2 (𝑡) 𝑁4 (𝑡) And where 𝜆𝑇𝐻2 𝑇 𝜆𝑇 𝑁1 𝑡 = 𝐻𝑇 − 𝐾 + 2𝑝𝐻 1 + 𝐾 , 𝑊1 = max 𝑁1 𝑡 𝑡∈[0,𝑇] 1 + 𝑒 𝜆𝑇 1 1 + 𝑒 𝜆𝑇 3 2 𝜆𝑇𝐻 𝑇 𝜆𝑇 𝑁2 𝑡 = 𝐻𝑇 − 𝐾 + 2𝑝𝐻 1 + 𝐾 , 𝑊2 = max 𝑁2 𝑡 𝑡∈ 0,𝑇 1 + 𝑒 𝜆𝑇 2 1 + 𝑒 𝜆𝑇 4

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Periodic Solution for Nonlinear System of Differential… 𝛽𝑇𝐹 2 𝑇 𝛽𝑇 𝐿1 + 2𝑝𝐹 1 + 𝐿 , 𝜆𝑇 1+𝑒 1 + 𝑒 𝜆𝑇 3 𝛽𝑇𝐹 2 𝑇 𝛽𝑇 𝑁4 𝑡 = 𝐹𝑇 − 𝐿2 + 2𝑝𝐹 1 + 𝐿 , 𝜆𝑇 1+𝑒 1 + 𝑒 𝜆𝑇 4 𝜆𝑇𝐻2 𝑇 𝜆𝑇 𝐻𝑇 − 𝑀1 + 2𝑝𝐻 1 + 𝑀3 1 + 𝑒 𝜆𝑇 1 + 𝑒 𝜆𝑇 𝑉ο = 𝛽𝑇𝐹 2 𝑇 𝛽𝑇 𝐹𝑇 − 𝑀2 + 2𝑝𝐹 1 + 𝑀4 1 + 𝑒 𝛽𝑇 1 + 𝑒 𝛽𝑇 Proof: Setting m = 0 and using (12) , and the condition (9), we get 𝑁3 𝑡 = 𝐹𝑇 −

𝑊3 = max 𝑁3 𝑡 𝑡∈ 0,𝑇

𝑊4 = max 𝑁4 𝑡 𝑡∈ 0,𝑇

𝑡

𝑒𝜆

𝑥1 𝑡, 𝑥ο , 𝑦ο − 𝑥ο = 0

𝜆𝑡 − 1 + 𝑒 𝜆𝑇

𝑡−𝑠

𝑓 𝑠, 𝑥ο , 𝑦ο

𝑑𝑠 −

𝑇

𝑒 2𝜆 0

𝜆𝑡 − 1 + 𝑒 𝜆𝑇

𝑇−𝑠

𝑓 𝑠, 𝑥ο , 𝑦ο

𝑇−𝑠

𝐼𝑖 𝑥ο , 𝑦ο

0<𝑡 𝑖 <𝑡

𝑝

𝑒𝜆

𝑇−𝑠

𝑖=1

𝐼𝑖 𝑥ο , 𝑦ο

𝑡

𝜆𝑡𝐻2 ≤ 𝐻− 1 + 𝑒 𝜆𝑇 +𝐻

𝑒𝜆

𝑑𝑠 +

𝑓 𝑠, 𝑥ο , 𝑦ο 0

𝐼𝑖 𝑥ο , 𝑦ο 0<𝑡 𝑖 <𝑡

𝜆𝑡𝐻2 𝑑𝑠 − 1 + 𝑒 𝜆𝑇

𝜆𝑡𝐻 − 1 + 𝑒 𝜆𝑇

𝑝

𝑇

𝑓 𝑠, 𝑥ο , 𝑦ο

𝑑𝑠 +

0

𝐼𝑖 𝑥ο , 𝑦ο 𝑖=1

Hence 𝜆𝑇𝐻2 𝑇 𝜆𝑇 𝑀1 + 2𝑝𝐻 1 + 𝑀3 . . . (19) 𝜆𝑇 1+𝑒 1 + 𝑒 𝜆𝑇 So that𝑥1 𝑡, 𝑥ο , 𝑦ο ∈ 𝐷𝜆 , for all 𝑡 ∈ 𝑅′ , 𝑥ο ∈ 𝐷𝜆𝑓 ,and by mathematic induction we get: 𝜆𝑇𝐻2 𝑇 𝜆𝑇 𝑥𝑚 𝑡, 𝑥ο , 𝑦ο − 𝑥ο ≤ 𝐻𝑇 − 𝑀1 + 2𝑝𝐻 1 + 𝑀3 𝜆𝑇 1+𝑒 1 + 𝑒 𝜆𝑇 and 𝛽𝑇𝐹 2 𝑇 𝛽𝑇 𝑦1 𝑡, 𝑥ο , 𝑦ο − 𝑦ο ≤ 𝐹𝑇 − 𝑀2 + 2𝑝𝐹 1 + 𝑀4 . . . (20) 𝛽𝑇 1+𝑒 1 + 𝑒 𝛽𝑇 Hence 𝑦1 𝑡, 𝑥ο , 𝑦ο ∈ 𝐷𝛽 , for all 𝑡 ∈ 𝑅′ , 𝑦ο ∈ 𝐷𝛽𝑓 . and by mathematic induction we get : 𝛽𝑇𝐹 2 𝑇 𝛽𝑇 𝑦𝑚 𝑡, 𝑥ο , 𝑦ο − 𝑦ο ≤ 𝐹𝑇 − 𝑀2 + 2𝑝𝐹 1 + 𝑀4 𝛽𝑇 1+𝑒 1 + 𝑒 𝛽𝑇 Then 𝑥𝑚 𝑡, 𝑥ο , 𝑦ο ∈ 𝐷𝜆 , 𝑦𝑚 𝑡, 𝑥ο , 𝑦ο ∈ 𝐷𝛽 , 𝑥ο ∈ 𝐷𝜆𝑓 , 𝑦ο ∈ 𝐷𝛽𝑓 . We claim that the sequence of functions (12) and (13) are uniformly convergent on the domain (14). By using (19), and when m = 1 , we get 𝑥1 𝑡, 𝑥ο , 𝑦ο − 𝑥ο ≤ 𝐻𝑇 −

𝑡

𝑥2 𝑡, 𝑥ο , 𝑦ο − 𝑥1 𝑡, 𝑥ο , 𝑦ο 𝜆𝑡 − 1 + 𝑒 𝜆𝑇

𝑒

𝜆 𝑇−𝑠

𝑓 𝑠, 𝑥1 , 𝑦1

0 𝑇−𝑠

𝐼𝑖 𝑥1 , 𝑦1

0<𝑡 𝑖 <𝑡 𝑡

𝑒𝜆

− 0

𝑡−𝑠

𝑡−𝑠

𝑓 𝑠, 𝑥1 , 𝑦1

𝑑𝑠 −

0

𝑡

𝑒𝜆

+

𝑒𝜆

=

𝑓 𝑠, 𝑥ο , 𝑦ο

𝜆𝑡 𝑑𝑠 + 1 + 𝑒 𝜆𝑇

𝜆𝑡 + 1 + 𝑒 𝜆𝑇 𝑑𝑠 +

𝜆𝑡 1 + 𝑒 𝜆𝑇

𝑝

𝑇

𝑒𝜆

𝑇−𝑠

𝑓 𝑠, 𝑥1 , 𝑦1

𝑑𝑠 +

0

𝑒𝜆

𝑇−𝑠

𝐼𝑖 𝑥1 , 𝑦1

𝑒𝜆

𝑇−𝑠

𝑓 𝑠, 𝑥ο , 𝑦ο

𝑖=1 𝑡

𝑑𝑠 +

0

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Periodic Solution for Nonlinear System of Differential… 𝜆𝑡 + 1 + 𝑒 𝜆𝑇 𝜆𝑡 − 1 + 𝑒 𝜆𝑇

𝑇

𝑒𝜆 0

𝑇−𝑠

𝑓 𝑠, 𝑥ο , 𝑦ο

𝑒𝜆

𝑑𝑠 −

𝑇−𝑠

𝐼𝑖 𝑥ο , 𝑦ο

0<𝑡 𝑖 <𝑡

𝑝

𝑒𝜆

𝑇−𝑠

𝐼𝑖 𝑥ο , 𝑦ο

,

𝑖=1

then 𝑥2 𝑡, 𝑥ο , 𝑦ο − 𝑥1 𝑡, 𝑥ο , 𝑦ο

𝐻𝑇 −

𝜆𝑇𝐻2 𝑇 𝜆𝑇 𝐾 + 2𝑝𝐻 1 + 𝐾 1 + 𝑒 𝜆𝑇 1 1 + 𝑒 𝜆𝑇 3

𝑥1 𝑡, 𝑥ο , 𝑦ο − 𝑥ο + +

𝐻𝑇 −

𝑦1 𝑡, 𝑥ο , 𝑦ο − 𝑦ο And by mathematic induction we get : 𝑥𝑚 +1 𝑡, 𝑥ο , 𝑦ο − 𝑥m 𝑡, 𝑥ο , 𝑦ο

𝑥𝑚 𝑡, 𝑥ο , 𝑦ο − 𝑥𝑚 −1 𝑡, 𝑥ο , 𝑦ο

+

𝐻𝑇 −

+ 𝑦𝑚 𝑡, 𝑥ο , 𝑦ο − 𝑦𝑚 −1 𝑡, 𝑥ο , 𝑦ο and 𝑦2 𝑡, 𝑥ο , 𝑦ο − 𝑦1 𝑡, 𝑥ο , 𝑦ο

𝜆𝑇𝐻2 𝑇 𝜆𝑇 𝐾 + 2𝑝𝐻 1 + 𝐾 1 + 𝑒 𝜆𝑇 2 1 + 𝑒 𝜆𝑇 4 . . . (22)

𝐹𝑇 −

𝑥1 𝑡, 𝑥ο , 𝑦ο − 𝑥ο + +

𝐹𝑇 −

𝑦1 𝑡, 𝑥ο , 𝑦ο − 𝑦ο And by mathematic induction we get : 𝑦𝑚 +1 𝑡, 𝑥ο , 𝑦ο − 𝑦m 𝑡, 𝑥ο , 𝑦ο

𝑥𝑚 𝑡, 𝑥ο , 𝑦ο − 𝑥𝑚 −1 𝑡, 𝑥ο , 𝑦ο

+

𝜆𝑇𝐻2 𝑇 𝜆𝑇 𝐾 + 2𝑝𝐻 1 + 𝐾 1 + 𝑒 𝜆𝑇 1 1 + 𝑒 𝜆𝑇 3

𝐻𝑇 −

,

𝜆𝑇𝐻2 𝑇 𝜆𝑇 𝐾2 + 2𝑝𝐻 1 + 𝐾 𝜆𝑇 1+𝑒 1 + 𝑒 𝜆𝑇 4 . . . (21)

𝛽𝑇𝐹 2 𝑇 𝛽𝑇 𝐿1 + 2𝑝𝐹 1 + 𝐿 𝜆𝑇 1+𝑒 1 + 𝑒 𝜆𝑇 3 𝛽𝑇𝐹 2 𝑇 𝛽𝑇 𝐿 + 2𝑝𝐹 1 + 𝐿 1 + 𝑒 𝜆𝑇 2 1 + 𝑒 𝜆𝑇 4 . . . (23)

𝐻𝑇 −

𝜆𝑇𝐻2 𝑇 𝜆𝑇 𝐾1 + 2𝑝𝐻 1 + 𝐾 𝜆𝑇 1+𝑒 1 + 𝑒 𝜆𝑇 3

𝜆𝑇𝐻2 𝑇 𝜆𝑇 𝐾2 + 2𝑝𝐻 1 + 𝐾 𝜆𝑇 1+𝑒 1 + 𝑒 𝜆𝑇 4 𝑦𝑚 𝑡, 𝑥ο , 𝑦ο − 𝑦𝑚 −1 𝑡, 𝑥ο , 𝑦ο , . . . (24) Rewrite inequalities (22) and (24) in vector form as : 𝑉𝑚 +1 𝑡, 𝑥ο , 𝑦ο ≤ Λ 𝑡 𝑉𝑚 𝑡, 𝑥ο , 𝑦ο . . . (25) where 𝑥𝑚 +1 𝑡, 𝑥ο , 𝑦ο − 𝑥𝑚 𝑡, 𝑥ο , 𝑦ο 𝑉𝑚 +1 𝑡, 𝑥ο , 𝑦ο = 𝑦𝑚 +1 𝑡, 𝑥ο , 𝑦ο − 𝑦𝑚 𝑡, 𝑥ο , 𝑦ο 𝑁1 (𝑡) 𝑁3 (𝑡) Λ 𝑡 = 𝑁2 (𝑡) 𝑁4 (𝑡) and 𝑥𝑚 𝑡, 𝑥ο , 𝑦ο − 𝑥𝑚 −1 𝑡, 𝑥ο , 𝑦ο 𝑉𝑚 𝑡, 𝑥ο , 𝑦ο = 𝑦𝑚 𝑡, 𝑥ο , 𝑦ο − 𝑦𝑚 −1 𝑡, 𝑥ο , 𝑦ο It follows form the inequality (25) that : 𝑉𝑚 +1 ≤ Λο 𝑡 𝑉𝑚 . . . (26) where Λο = max Λ(𝑡) +

𝐻𝑇 −

𝑡∈ 0,𝑇

By iterating the inequality (3.20) , we find that 𝑉𝑚 +1 ≤ Λm ο 𝑉ο where

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Periodic Solution for Nonlinear System of Differentialâ&#x20AC;Ś đ??ťđ?&#x2018;&#x2021; â&#x2C6;&#x2019;

đ?&#x153;&#x2020;đ?&#x2018;&#x2021;đ??ť2 đ?&#x2018;&#x2021; đ?&#x153;&#x2020;đ?&#x2018;&#x2021; đ?&#x2018;&#x20AC;1 + 2đ?&#x2018;?đ??ť 1 + đ?&#x2018;&#x20AC;3 đ?&#x153;&#x2020;đ?&#x2018;&#x2021; 1+đ?&#x2018;&#x2019; 1 + đ?&#x2018;&#x2019; đ?&#x153;&#x2020;đ?&#x2018;&#x2021;

đ?&#x2018;&#x2030;Îż =

đ?&#x203A;˝đ?&#x2018;&#x2021;đ??š 2 đ?&#x2018;&#x2021; đ?&#x203A;˝đ?&#x2018;&#x2021; đ?&#x2018;&#x20AC;2 + 2đ?&#x2018;?đ??š 1 + đ?&#x2018;&#x20AC;4 đ?&#x203A;˝đ?&#x2018;&#x2021; 1+đ?&#x2018;&#x2019; 1 + đ?&#x2018;&#x2019; đ?&#x203A;˝đ?&#x2018;&#x2021; which leads to the estimate đ??šđ?&#x2018;&#x2021; â&#x2C6;&#x2019;

đ?&#x2018;&#x161;

đ?&#x2018;&#x161;

Î&#x203A;đ?&#x2018;&#x2013;â&#x2C6;&#x2019;1 đ?&#x2018;&#x2030;Îż Îż

đ?&#x2018;&#x2030;i â&#x2030;¤ đ?&#x2018;&#x2013;=1

. . . (27)

đ?&#x2018;&#x2013;=1

Since the matrix Î&#x203A;Îż has Eigen-values like as (12), then the series (27) is uniformly convergent, i.e. â&#x2C6;&#x17E;

đ?&#x2018;&#x161;

Î&#x203A;đ?&#x2018;&#x2013;â&#x2C6;&#x2019;1 đ?&#x2018;&#x2030;Îż = Îż

lim

đ?&#x2018;&#x161; â&#x2C6;&#x17E;

đ?&#x2018;&#x2013;=1

Î&#x203A;đ?&#x2018;&#x2013;â&#x2C6;&#x2019;1 đ?&#x2018;&#x2030;Îż = (đ??¸ â&#x2C6;&#x2019; Î&#x203A;Îż )â&#x2C6;&#x2019;1 đ?&#x2018;&#x2030;Îż Îż

. . . (28)

đ?&#x2018;&#x2013;=1

where E is a unity matrix. The limiting relation (28) signifies a uniform convergent of the sequence đ?&#x2018;Ľđ?&#x2018;&#x161; đ?&#x2018;Ą, đ?&#x2018;ĽÎż , đ?&#x2018;ŚÎż andđ?&#x2018;Śđ?&#x2018;&#x161; đ?&#x2018;Ą, đ?&#x2018;ĽÎż , đ?&#x2018;ŚÎż in the domain (14). Let lim đ?&#x2018;Ľđ?&#x2018;&#x161; đ?&#x2018;Ą, đ?&#x2018;ĽÎż , đ?&#x2018;ŚÎż = đ?&#x2018;Ľâ&#x2C6;&#x17E; đ?&#x2018;Ą, đ?&#x2018;ĽÎż , đ?&#x2018;ŚÎż đ?&#x2018;&#x161; â&#x2C6;&#x17E;

. . . (29) lim đ?&#x2018;Śđ?&#x2018;&#x161; đ?&#x2018;Ą, đ?&#x2018;ĽÎż , đ?&#x2018;ŚÎż = đ?&#x2018;Śâ&#x2C6;&#x17E; đ?&#x2018;Ą, đ?&#x2018;ĽÎż , đ?&#x2018;ŚÎż đ?&#x2018;&#x161; â&#x2C6;&#x17E;

Since the sequences of functions đ?&#x2018;Ľđ?&#x2018;&#x161; đ?&#x2018;Ą, đ?&#x2018;ĽÎż , đ?&#x2018;ŚÎż andđ?&#x2018;Śđ?&#x2018;&#x161; đ?&#x2018;Ą, đ?&#x2018;ĽÎż , đ?&#x2018;ŚÎż are periodic in t with period T , then the limiting of them are also periodic in t with period T , end thus đ?&#x2018;Ľâ&#x2C6;&#x17E; đ?&#x2018;Ą, đ?&#x2018;ĽÎż , đ?&#x2018;ŚÎż = đ?&#x2018;Ľ đ?&#x2018;Ą, đ?&#x2018;ĽÎż , đ?&#x2018;ŚÎż , đ?&#x2018;Śâ&#x2C6;&#x17E; đ?&#x2018;Ą, đ?&#x2018;ĽÎż , đ?&#x2018;ŚÎż = đ?&#x2018;Ś đ?&#x2018;Ą, đ?&#x2018;ĽÎż , đ?&#x2018;ŚÎż . Also from (29) the following inequality đ?&#x2018;Ľâ&#x2C6;&#x17E; đ?&#x2018;Ą, đ?&#x2018;ĽÎż , đ?&#x2018;ŚÎż â&#x2C6;&#x2019; đ?&#x2018;Ľđ?&#x2018;&#x161; đ?&#x2018;Ą, đ?&#x2018;ĽÎż , đ?&#x2018;ŚÎż â&#x2C6;&#x2019;1 â&#x2030;¤ Î&#x203A;đ?&#x2018;&#x161; . . . (30) Îż (đ??¸ â&#x2C6;&#x2019; Î&#x203A;Îż ) đ?&#x2018;&#x2030;Îż , đ?&#x2018;Śâ&#x2C6;&#x17E; đ?&#x2018;Ą, đ?&#x2018;ĽÎż , đ?&#x2018;ŚÎż â&#x2C6;&#x2019; đ?&#x2018;Śđ?&#x2018;&#x161; đ?&#x2018;Ą, đ?&#x2018;ĽÎż , đ?&#x2018;ŚÎż is hold for đ?&#x2018;&#x161; â&#x2030;Ľ 1 , and hence đ?&#x2018;Ľâ&#x2C6;&#x17E; đ?&#x2018;Ą, đ?&#x2018;ĽÎż , đ?&#x2018;ŚÎż = đ?&#x2018;Ľ đ?&#x2018;Ą, đ?&#x2018;ĽÎż , đ?&#x2018;ŚÎż , đ?&#x2018;Śâ&#x2C6;&#x17E; đ?&#x2018;Ą, đ?&#x2018;ĽÎż , đ?&#x2018;ŚÎż = đ?&#x2018;Ś đ?&#x2018;Ą, đ?&#x2018;ĽÎż , đ?&#x2018;ŚÎż which are the solutions of the system (1).

Uniqueness of Solution of (1) Let all assumptions and conditions of theorem 1 were given, then the two functions đ?&#x2018;Ľ đ?&#x2018;Ą, đ?&#x2018;ĽÎż , đ?&#x2018;ŚÎż and đ?&#x2018;Ś đ?&#x2018;Ą, đ?&#x2018;ĽÎż , đ?&#x2018;ŚÎż are uniqueness of solution of (1) in the domain (14). Proof : Let đ?&#x2018;Ą

đ?&#x2018;&#x2019;đ?&#x153;&#x2020;

đ?&#x2018;Ľ đ?&#x2018;Ą, đ?&#x2018;ĽÎż , đ?&#x2018;ŚÎż = đ?&#x2018;ĽÎż + 0 đ?&#x2018;&#x2021;

â&#x2C6;&#x2019; 0

đ?&#x2018;Ąâ&#x2C6;&#x2019;đ?&#x2018;

[đ?&#x2018;&#x201C; đ?&#x2018; , đ?&#x2018;Ľ đ?&#x2018; , đ?&#x2018;ĽÎż , đ?&#x2018;ŚÎż , đ?&#x2018;Ś đ?&#x2018; , đ?&#x2018;ĽÎż , đ?&#x2018;ŚÎż

đ?&#x153;&#x2020; đ?&#x2018;&#x2019;đ?&#x153;&#x2020; 1 + đ?&#x2018;&#x2019; đ?&#x153;&#x2020;đ?&#x2018;&#x2021; đ?&#x2018;&#x2019;đ?&#x153;&#x2020;

+ 0<đ?&#x2018;Ą đ?&#x2018;&#x2013; <đ?&#x2018;Ą

đ?&#x153;&#x2020; â&#x2C6;&#x2019; 1 + đ?&#x2018;&#x2019; đ?&#x153;&#x2020;đ?&#x2018;&#x2021;

đ?&#x2018;&#x2021;â&#x2C6;&#x2019;đ?&#x2018;

đ?&#x2018;&#x2021;â&#x2C6;&#x2019;đ?&#x2018;

â&#x2C6;&#x2019;

đ?&#x2018;&#x201C; đ?&#x2018; , đ?&#x2018;Ľ đ?&#x2018; , đ?&#x2018;ĽÎż , đ?&#x2018;ŚÎż , đ?&#x2018;Ś đ?&#x2018; , đ?&#x2018;ĽÎż , đ?&#x2018;ŚÎż đ?&#x2018;&#x2018;đ?&#x2018; ]đ?&#x2018;&#x2018;đ?&#x2018;  +

đ??źđ?&#x2018;&#x2013; đ?&#x2018;Ľ đ?&#x2018;Ą, đ?&#x2018;ĽÎż , đ?&#x2018;ŚÎż , đ?&#x2018;Ś đ?&#x2018;Ą, đ?&#x2018;ĽÎż , đ?&#x2018;ŚÎż

â&#x2C6;&#x2019;

đ?&#x2018;?

đ?&#x2018;&#x2019;đ?&#x153;&#x2020;

đ?&#x2018;&#x2021;â&#x2C6;&#x2019;đ?&#x2018;

đ??źđ?&#x2018;&#x2013; đ?&#x2018;Ľ đ?&#x2018;Ą, đ?&#x2018;ĽÎż , đ?&#x2018;ŚÎż , đ?&#x2018;Ś đ?&#x2018;Ą, đ?&#x2018;ĽÎż , đ?&#x2018;ŚÎż

,

. . . (31)

đ?&#x2018;&#x2013;=1

and đ?&#x2018;Ą

đ?&#x2018;&#x2019;đ?&#x203A;˝

đ?&#x2018;Ś đ?&#x2018;Ą, đ?&#x2018;ĽÎż , đ?&#x2018;ŚÎż = đ?&#x2018;ŚÎż + 0 đ?&#x2018;&#x2021;

â&#x2C6;&#x2019; 0

đ?&#x2018;Ąâ&#x2C6;&#x2019;đ?&#x2018;

[đ?&#x2018;&#x201C; đ?&#x2018; , đ?&#x2018;Ľ đ?&#x2018; , đ?&#x2018;ĽÎż , đ?&#x2018;ŚÎż , đ?&#x2018;Ś đ?&#x2018; , đ?&#x2018;ĽÎż , đ?&#x2018;ŚÎż

đ?&#x203A;˝ đ?&#x2018;&#x2019;đ?&#x203A;˝ 1 + đ?&#x2018;&#x2019; đ?&#x203A;˝đ?&#x2018;&#x2021;

đ?&#x2018;&#x2021;â&#x2C6;&#x2019;đ?&#x2018;

â&#x2C6;&#x2019;

đ?&#x2018;&#x201C; đ?&#x2018; , đ?&#x2018;Ľ đ?&#x2018; , đ?&#x2018;ĽÎż , đ?&#x2018;ŚÎż , đ?&#x2018;Ś đ?&#x2018; , đ?&#x2018;ĽÎż , đ?&#x2018;ŚÎż đ?&#x2018;&#x2018;đ?&#x2018; ]đ?&#x2018;&#x2018;đ?&#x2018;  +

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Periodic Solution for Nonlinear System of Differential… 𝑒𝛽

+

𝑇−𝑠

0<𝑡 𝑖 <𝑡

𝛽 − 1 + 𝑒 𝛽𝑇

𝐺𝑖 𝑥 𝑡, 𝑥ο , 𝑦ο , 𝑦 𝑡, 𝑥ο , 𝑦ο

𝑝

𝑒𝛽

𝑇−𝑠

𝐺𝑖 𝑥 𝑡, 𝑥ο , 𝑦ο , 𝑦 𝑡, 𝑥ο , 𝑦ο

,

. . . (32)

𝑖=1

Are another solutions for the system (1), then we shall prove that 𝑥 𝑡, 𝑥ο , 𝑦ο = 𝑥 𝑡, 𝑥ο , 𝑦ο , 𝑡, 𝑥ο , 𝑦ο = 𝑦 𝑡, 𝑥ο , 𝑦ο , and to do this we need to prove the following inequality by induction, 𝑥 𝑡, 𝑥ο , 𝑦ο − 𝑥𝑚 𝑡, 𝑥ο , 𝑦ο 𝑥 𝑡, 𝑥ο , 𝑦ο − 𝑥𝑚 −1 𝑡, 𝑥ο , 𝑦ο ≤ Λ𝑚 . . . (33) ο 𝑦 𝑡, 𝑥ο , 𝑦ο − 𝑦𝑚 𝑡, 𝑥ο , 𝑦ο 𝑦 𝑡, 𝑥ο , 𝑦ο − 𝑦𝑚 −1 𝑡, 𝑥ο , 𝑦ο for 𝑚 ≥ 1 , where 𝑀1∗ = max 𝑓(𝑡, 𝑥, 𝑦) , 𝑀2∗ = max 𝑔(𝑡, 𝑥, 𝑦) 𝑥,𝑦 ∈𝐷 𝜆 × 𝐷 𝛽 𝑡∈[0,𝑇]

𝑀3∗ =

max

𝑥,𝑦 ∈𝐷 𝜆 × 𝐷 𝛽 1≤𝑖≤𝑝

𝑥 ,𝑦 ∈𝐷 𝜆 × 𝐷 𝛽 𝑡∈[0,𝑇]

𝐼𝑖 (𝑥, 𝑦)

,

𝑀4∗ =

max

𝑥,𝑦 ∈𝐷 𝜆 × 𝐷 𝛽 1≤𝑖≤𝑝

𝐺𝑖 (𝑥, 𝑦)

For m = 0 in (15) and (16), we have 𝜆𝑇𝐻2 𝑇 𝜆𝑇 𝑥 𝑡, 𝑥ο , 𝑦ο − 𝑥ο ≤ 𝐻𝑇 − 𝑀1∗ + 2𝑝𝐻 1 + 𝑀3∗ . . . (34) 1 + 𝑒 𝜆𝑇 1 + 𝑒 𝜆𝑇 and 𝛽𝑇𝐹 2 𝑇 𝛽𝑇 𝑦 𝑡, 𝑥ο , 𝑦ο − 𝑦ο ≤ 𝐹𝑇 − 𝑀2∗ + 2𝑝𝐹 1 + 𝑀4∗ . . . (35) 𝛽𝑇 1+𝑒 1 + 𝑒 𝛽𝑇 and for m = 1, we get also 𝜆𝑇𝐻2 𝑇 𝜆𝑇 𝑥 𝑡, 𝑥ο , 𝑦ο − 𝑥1 𝑡, 𝑥ο , 𝑦ο ≤ 𝐻𝑇 − 𝐾 + 2𝑝𝐻 1 + 𝐾 1 + 𝑒 𝜆𝑇 1 1 + 𝑒 𝜆𝑇 3 𝑥 𝑡, 𝑥ο , 𝑦ο − 𝑥ο + 𝜆𝑇𝐻2 𝑇 𝜆𝑇 + 𝐻𝑇 − 𝐾2 + 2𝑝𝐻 1 + 𝐾 𝜆𝑇 1+𝑒 1 + 𝑒 𝜆𝑇 4 𝑦 𝑡, 𝑥ο , 𝑦ο − 𝑦ο . . . (36) and 𝛽𝑇𝐹 2 𝑇 𝛽𝑇 𝑦 𝑡, 𝑥ο , 𝑦ο − 𝑦1 𝑡, 𝑥ο , 𝑦ο ≤ 𝐹𝑇 − 𝐿1 + 2𝑝𝐹 1 + 𝐿 𝜆𝑇 1+𝑒 1 + 𝑒 𝜆𝑇 3 𝑥 𝑡, 𝑥ο , 𝑦ο − 𝑥ο + 𝛽𝑇𝐹 2 𝑇 𝛽𝑇 + 𝐹𝑇 − 𝐿 + 2𝑝𝐹 1 + 𝐿 1 + 𝑒 𝜆𝑇 2 1 + 𝑒 𝜆𝑇 4 𝑦 𝑡, 𝑥ο , 𝑦ο − 𝑦ο . . . (37) Suppose that (33) is true for m = p -1 , i.e. 𝑥 𝑡, 𝑥ο , 𝑦ο − 𝑥𝑝−1 𝑡, 𝑥ο , 𝑦ο ≤

Λ𝑝−1 ο

𝑦 𝑡, 𝑥ο , 𝑦ο − 𝑦𝑝 −1 𝑡, 𝑥ο , 𝑦ο then 𝑥 𝑡, 𝑥ο , 𝑦ο − 𝑥p 𝑡, 𝑥ο , 𝑦ο

𝑥 𝑡, 𝑥ο , 𝑦ο − 𝑥𝑝−2 𝑡, 𝑥ο , 𝑦ο . . . (38) 𝑦 𝑡, 𝑥ο , 𝑦ο − 𝑦𝑝 −2 𝑡, 𝑥ο , 𝑦ο

𝑥 𝑡, 𝑥ο , 𝑦ο − 𝑥p−1 𝑡, 𝑥ο , 𝑦ο

𝐻𝑇 −

𝜆𝑇𝐻2 𝑇 𝜆𝑇 𝐾1 + 2𝑝𝐻 1 + 𝐾 𝜆𝑇 1+𝑒 1 + 𝑒 𝜆𝑇 3

+ +

𝐻𝑇 −

𝜆𝑇𝐻2 𝑇 𝜆𝑇 𝐾2 + 2𝑝𝐻 1 + 𝐾 𝜆𝑇 1+𝑒 1 + 𝑒 𝜆𝑇 4

𝑦 𝑡, 𝑥ο , 𝑦ο − 𝑦p−1 𝑡, 𝑥ο , 𝑦ο and 𝑦 𝑡, 𝑥ο , 𝑦ο − 𝑦p 𝑡, 𝑥ο , 𝑦ο 𝑥 𝑡, 𝑥ο , 𝑦ο − 𝑥p−1 𝑡, 𝑥ο , 𝑦ο

𝐹𝑇 −

𝛽𝑇𝐹 2 𝑇 𝛽𝑇 𝐿1 + 2𝑝𝐹 1 + 𝐿 𝜆𝑇 1+𝑒 1 + 𝑒 𝜆𝑇 3

+ +

𝐹𝑇 −

𝛽𝑇𝐹 2 𝑇 𝛽𝑇 𝐿 + 2𝑝𝐹 1 + 𝐿 1 + 𝑒 𝜆𝑇 2 1 + 𝑒 𝜆𝑇 4

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Periodic Solution for Nonlinear System of Differentialâ&#x20AC;Ś đ?&#x2018;Ś đ?&#x2018;Ą, đ?&#x2018;ĽÎż , đ?&#x2018;ŚÎż â&#x2C6;&#x2019; đ?&#x2018;Śpâ&#x2C6;&#x2019;1 đ?&#x2018;Ą, đ?&#x2018;ĽÎż , đ?&#x2018;ŚÎż i.e. đ?&#x2018;Ľ đ?&#x2018;Ą, đ?&#x2018;ĽÎż , đ?&#x2018;ŚÎż â&#x2C6;&#x2019; đ?&#x2018;Ľđ?&#x2018;? đ?&#x2018;Ą, đ?&#x2018;ĽÎż , đ?&#x2018;ŚÎż

â&#x2030;¤ Î&#x203A;đ?&#x2018;?Îż

đ?&#x2018;Ľ đ?&#x2018;Ą, đ?&#x2018;ĽÎż , đ?&#x2018;ŚÎż â&#x2C6;&#x2019; đ?&#x2018;Ľđ?&#x2018;?â&#x2C6;&#x2019;1 đ?&#x2018;Ą, đ?&#x2018;ĽÎż , đ?&#x2018;ŚÎż

đ?&#x2018;Ś đ?&#x2018;Ą, đ?&#x2018;ĽÎż , đ?&#x2018;ŚÎż â&#x2C6;&#x2019; đ?&#x2018;Śđ?&#x2018;? đ?&#x2018;Ą, đ?&#x2018;ĽÎż , đ?&#x2018;ŚÎż đ?&#x2018;Ś đ?&#x2018;Ą, đ?&#x2018;ĽÎż , đ?&#x2018;ŚÎż â&#x2C6;&#x2019; đ?&#x2018;Śđ?&#x2018;?â&#x2C6;&#x2019;1 đ?&#x2018;Ą, đ?&#x2018;ĽÎż , đ?&#x2018;ŚÎż Then the inequality (33) is true for m = 0, 1, 2, . . . By iterating the inequality (33) gives : đ?&#x2018;Ľ đ?&#x2018;Ą, đ?&#x2018;ĽÎż , đ?&#x2018;ŚÎż â&#x2C6;&#x2019; đ?&#x2018;Ľđ?&#x2018;&#x161; đ?&#x2018;Ą, đ?&#x2018;ĽÎż , đ?&#x2018;ŚÎż đ?&#x2018;Ľ đ?&#x2018;Ą, đ?&#x2018;ĽÎż , đ?&#x2018;ŚÎż â&#x2C6;&#x2019; đ?&#x2018;Ľđ?&#x2018;&#x161; â&#x2C6;&#x2019;1 đ?&#x2018;Ą, đ?&#x2018;ĽÎż , đ?&#x2018;ŚÎż đ?&#x2018;&#x161; â&#x2030;¤ Î&#x203A;Îż đ?&#x2018;Ś đ?&#x2018;Ą, đ?&#x2018;ĽÎż , đ?&#x2018;ŚÎż â&#x2C6;&#x2019; đ?&#x2018;Śđ?&#x2018;&#x161; đ?&#x2018;Ą, đ?&#x2018;ĽÎż , đ?&#x2018;ŚÎż đ?&#x2018;Ś đ?&#x2018;Ą, đ?&#x2018;ĽÎż , đ?&#x2018;ŚÎż â&#x2C6;&#x2019; đ?&#x2018;Śđ?&#x2018;&#x161; â&#x2C6;&#x2019;1 đ?&#x2018;Ą, đ?&#x2018;ĽÎż , đ?&#x2018;ŚÎż But from the condition (11) we obtainÎ&#x203A;đ?&#x2018;&#x161; 0 as đ?&#x2018;&#x161; â&#x2C6;&#x17E;, hence, proceeding in the last inequality to the limit Îż we obtain that đ?&#x2018;Ľ đ?&#x2018;Ą, đ?&#x2018;ĽÎż , đ?&#x2018;ŚÎż = đ?&#x2018;Ľ đ?&#x2018;Ą, đ?&#x2018;ĽÎż , đ?&#x2018;ŚÎż and đ?&#x2018;Ś đ?&#x2018;Ą, đ?&#x2018;ĽÎż , đ?&#x2018;ŚÎż = đ?&#x2018;Ś đ?&#x2018;Ą, đ?&#x2018;ĽÎż , đ?&#x2018;ŚÎż which proves that the two solutions đ?&#x2018;Ľ đ?&#x2018;Ą, đ?&#x2018;ĽÎż , đ?&#x2018;ŚÎż and đ?&#x2018;Ś đ?&#x2018;Ą, đ?&#x2018;ĽÎż , đ?&#x2018;ŚÎż are unique, and this completes the proof of theorem 2. Existence of solution of (1) The problem of existence solution of the system (1) is uniquely connected with the existence of zero of the function â&#x2C6;&#x2020;(0, đ?&#x2018;ĽÎż , đ?&#x2018;ŚÎż ) and â&#x2C6;&#x2020;â&#x2C6;&#x2014; (0, đ?&#x2018;ĽÎż , đ?&#x2018;ŚÎż ), which has the form: đ?&#x2018;&#x2021;

đ?&#x153;&#x2020; â&#x2C6;&#x2020;(0, đ?&#x2018;ĽÎż , đ?&#x2018;ŚÎż ) = [ 1 + đ?&#x2018;&#x2019; đ?&#x153;&#x2020;đ?&#x2018;&#x2021;

đ?&#x2018;&#x2019;đ?&#x153;&#x2020;

đ?&#x2018;&#x2019;đ?&#x153;&#x2020;

đ?&#x2018;&#x2021;â&#x2C6;&#x2019;đ?&#x2018;

đ?&#x2018;&#x2013;=1

đ?&#x203A;˝ â&#x2C6;&#x2020; (0, đ?&#x2018;ĽÎż , đ?&#x2018;ŚÎż ) = [ 1 + đ?&#x2018;&#x2019; đ?&#x203A;˝đ?&#x2018;&#x2021; đ?&#x2018;&#x2019;đ?&#x203A;˝

+

đ??źđ?&#x2018;&#x2013; đ?&#x2018;Ľâ&#x2C6;&#x17E; đ?&#x2018;Ąđ?&#x2018;&#x2013; , đ?&#x2018;ĽÎż , đ?&#x2018;ŚÎż , đ?&#x2018;Śâ&#x2C6;&#x17E; đ?&#x2018;Ąđ?&#x2018;&#x2013; , đ?&#x2018;ĽÎż , đ?&#x2018;ŚÎż ]

. . . (39)

đ?&#x2018;&#x2021;

â&#x2C6;&#x2014;

đ?&#x2018;?

đ?&#x2018;&#x201C; đ?&#x2018;Ą, đ?&#x2018;Ľâ&#x2C6;&#x17E; đ?&#x2018;Ą, đ?&#x2018;ĽÎż , đ?&#x2018;ŚÎż , đ?&#x2018;Śâ&#x2C6;&#x17E; đ?&#x2018;Ą, đ?&#x2018;ĽÎż , đ?&#x2018;ŚÎż đ?&#x2018;&#x2018;đ?&#x2018;Ą +

0

đ?&#x2018;?

+

đ?&#x2018;&#x2021;â&#x2C6;&#x2019;đ?&#x2018;

đ?&#x2018;&#x2019;đ?&#x203A;˝

đ?&#x2018;&#x2021;â&#x2C6;&#x2019;đ?&#x2018;

đ?&#x2018;&#x201C; đ?&#x2018;Ą, đ?&#x2018;Ľâ&#x2C6;&#x17E; đ?&#x2018;Ą, đ?&#x2018;ĽÎż , đ?&#x2018;ŚÎż , đ?&#x2018;Śâ&#x2C6;&#x17E; đ?&#x2018;Ą, đ?&#x2018;ĽÎż , đ?&#x2018;ŚÎż đ?&#x2018;&#x2018;đ?&#x2018;Ą +

0

đ?&#x2018;&#x2021;â&#x2C6;&#x2019;đ?&#x2018;

đ??şđ?&#x2018;&#x2013; đ?&#x2018;Ľâ&#x2C6;&#x17E; đ?&#x2018;Ąđ?&#x2018;&#x2013; , đ?&#x2018;ĽÎż , đ?&#x2018;ŚÎż , đ?&#x2018;Śâ&#x2C6;&#x17E; đ?&#x2018;Ąđ?&#x2018;&#x2013; , đ?&#x2018;ĽÎż , đ?&#x2018;ŚÎż ]

. . . (40)

đ?&#x2018;&#x2013;=1

Where đ?&#x2018;Ľâ&#x2C6;&#x17E; đ?&#x2018;Ą, đ?&#x2018;ĽÎż , đ?&#x2018;ŚÎż andđ?&#x2018;Śâ&#x2C6;&#x17E; đ?&#x2018;Ą, đ?&#x2018;ĽÎż , đ?&#x2018;ŚÎż the sequence's limiting (12) and (13) successively, and this function is approximately determined from the sequence of functions: đ?&#x153;&#x2020; â&#x2C6;&#x2020;đ?&#x2018;&#x161; (0, đ?&#x2018;ĽÎż , đ?&#x2018;ŚÎż ) = [ 1 + đ?&#x2018;&#x2019; đ?&#x153;&#x2020;đ?&#x2018;&#x2021; đ?&#x2018;?

đ?&#x2018;&#x2019;đ?&#x153;&#x2020;

+

đ?&#x203A;˝ = [ 1 + đ?&#x2018;&#x2019; đ?&#x203A;˝đ?&#x2018;&#x2021; đ?&#x2018;?

đ?&#x2018;&#x2019;đ?&#x203A;˝

+

đ?&#x2018;&#x2019;đ?&#x153;&#x2020;

đ?&#x2018;&#x2021;â&#x2C6;&#x2019;đ?&#x2018;

đ?&#x2018;&#x201C; đ?&#x2018;Ą, đ?&#x2018;Ľđ?&#x2018;&#x161; đ?&#x2018;Ą, đ?&#x2018;ĽÎż , đ?&#x2018;ŚÎż , đ?&#x2018;Śđ?&#x2018;&#x161; đ?&#x2018;Ą, đ?&#x2018;ĽÎż , đ?&#x2018;ŚÎż đ?&#x2018;&#x2018;đ?&#x2018;Ą +

0

đ?&#x2018;&#x2021;â&#x2C6;&#x2019;đ?&#x2018;

đ?&#x2018;&#x2013;=1

â&#x2C6;&#x2020;â&#x2C6;&#x2014;đ?&#x2018;&#x161; (0, đ?&#x2018;ĽÎż , đ?&#x2018;ŚÎż )

đ?&#x2018;&#x2021;

đ??źđ?&#x2018;&#x2013; đ?&#x2018;Ľđ?&#x2018;&#x161; đ?&#x2018;Ąđ?&#x2018;&#x2013; , đ?&#x2018;ĽÎż , đ?&#x2018;ŚÎż , đ?&#x2018;Śđ?&#x2018;&#x161; đ?&#x2018;Ąđ?&#x2018;&#x2013; , đ?&#x2018;ĽÎż , đ?&#x2018;ŚÎż ]

. . . (41)

đ?&#x2018;&#x2021;

đ?&#x2018;&#x2019;đ?&#x203A;˝

đ?&#x2018;&#x2021;â&#x2C6;&#x2019;đ?&#x2018;

đ?&#x2018;&#x201C; đ?&#x2018;Ą, đ?&#x2018;Ľđ?&#x2018;&#x161; đ?&#x2018;Ą, đ?&#x2018;ĽÎż , đ?&#x2018;ŚÎż , đ?&#x2018;Śđ?&#x2018;&#x161; đ?&#x2018;Ą, đ?&#x2018;ĽÎż , đ?&#x2018;ŚÎż đ?&#x2018;&#x2018;đ?&#x2018;Ą +

0

đ?&#x2018;&#x2021;â&#x2C6;&#x2019;đ?&#x2018;

đ??şđ?&#x2018;&#x2013; đ?&#x2018;Ľđ?&#x2018;&#x161; đ?&#x2018;Ąđ?&#x2018;&#x2013; , đ?&#x2018;ĽÎż , đ?&#x2018;ŚÎż , đ?&#x2018;Śđ?&#x2018;&#x161; đ?&#x2018;Ąđ?&#x2018;&#x2013; , đ?&#x2018;ĽÎż , đ?&#x2018;ŚÎż ]

. . . (42)

đ?&#x2018;&#x2013;=1

where đ?&#x2018;&#x161; = 0 , 1 , 2 , . .. Theorem 3 : Let all assumptions and conditions of theorem 1 were given, then the following inequality : â&#x2C6;&#x2020;(0, đ?&#x2018;ĽÎż , đ?&#x2018;ŚÎż ) â&#x2C6;&#x2019; â&#x2C6;&#x2020;đ?&#x2018;&#x161; (0, đ?&#x2018;ĽÎż , đ?&#x2018;ŚÎż ) â&#x2C6;&#x2019;1 â&#x2030;¤ đ?&#x2018;&#x201E; Î&#x203A;đ?&#x2018;&#x161; . . . (43) Îż (đ??¸ â&#x2C6;&#x2019; Î&#x203A;Îż ) đ?&#x2018;&#x2030;Îż â&#x2C6;&#x2014; â&#x2C6;&#x2014; â&#x2C6;&#x2020; đ?&#x2018;Ą, đ?&#x2018;ĽÎż , đ?&#x2018;ŚÎż â&#x2C6;&#x2019; â&#x2C6;&#x2020;đ?&#x2018;&#x161; (0, đ?&#x2018;ĽÎż , đ?&#x2018;ŚÎż where đ?&#x153;&#x2020;đ??ťđ?&#x2018;&#x2021; đ?&#x153;&#x2020;đ?&#x2018;?đ??ť đ?&#x153;&#x2020;đ??ťđ?&#x2018;&#x2021; đ?&#x153;&#x2020;đ?&#x2018;?đ??ť đ??ž1 + đ??ž3 đ??ž2 + đ??ž đ?&#x153;&#x2020;đ?&#x2018;&#x2021; đ?&#x153;&#x2020;đ?&#x2018;&#x2021; đ?&#x153;&#x2020;đ?&#x2018;&#x2021; 1+đ?&#x2018;&#x2019; 1+đ?&#x2018;&#x2019; 1+đ?&#x2018;&#x2019; 1 + đ?&#x2018;&#x2019; đ?&#x153;&#x2020;đ?&#x2018;&#x2021; 4 đ?&#x2018;&#x201E;= đ?&#x203A;˝đ??šđ?&#x2018;&#x2021; đ?&#x203A;˝đ?&#x2018;?đ??š đ?&#x203A;˝đ??šđ?&#x2018;&#x2021; đ?&#x203A;˝đ?&#x2018;?đ??š đ??ż1 + đ??ż3 đ??ż2 + đ??ż đ?&#x203A;˝đ?&#x2018;&#x2021; đ?&#x203A;˝đ?&#x2018;&#x2021; đ?&#x203A;˝đ?&#x2018;&#x2021; 1+đ?&#x2018;&#x2019; 1+đ?&#x2018;&#x2019; 1+đ?&#x2018;&#x2019; 1 + đ?&#x2018;&#x2019; đ?&#x203A;˝đ?&#x2018;&#x2021; 4 Is holds for đ?&#x2018;&#x161; â&#x2030;Ľ 0 , đ?&#x2018;Ą â&#x2C6;&#x2C6; 0, đ?&#x2018;&#x2021; , đ?&#x2018;ĽÎż â&#x2C6;&#x2C6; đ??ˇđ?&#x153;&#x2020;đ?&#x2018;&#x201C; , đ?&#x2018;ŚÎż â&#x2C6;&#x2C6; đ??ˇđ?&#x203A;˝đ?&#x2018;&#x201C; . proof :

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P a g e | 82


Periodic Solution for Nonlinear System of Differential… According to (39) and (41) , we have. 𝜆 ∆(0, 𝑥ο , 𝑦ο ) − ∆𝑚 (0, 𝑥ο , 𝑦ο ) ≤ 1 + 𝑒 𝜆𝑇

+

𝑇

𝑒𝜆

𝑇−𝑠

𝑓 𝑡, 𝑥∞ 𝑡, 𝑥ο , 𝑦ο , 𝑦∞ 𝑡, 𝑥ο , 𝑦ο

0

𝜆 1 + 𝑒 𝜆𝑇

𝑝

−𝑓 𝑡, 𝑥𝑚 𝑡, 𝑥ο , 𝑦ο , 𝑦𝑚 𝑡, 𝑥ο , 𝑦ο 𝑒𝜆

𝑇−𝑠

𝑑𝑡 +

𝐼𝑖 𝑥∞ 𝑡𝑖 , 𝑥ο , 𝑦ο , 𝑦∞ 𝑡𝑖 , 𝑥ο , 𝑦ο

𝑖=1

− 𝐼𝑖 𝑥𝑚 𝑡𝑖 , 𝑥ο , 𝑦ο , 𝑦𝑚 𝑡𝑖 , 𝑥ο , 𝑦ο ≤ 𝜆𝐻𝑇 ≤ 𝐾 𝑥 𝑡, 𝑥ο , 𝑦ο − 𝑥𝑚 𝑡, 𝑥ο , 𝑦ο + 𝐾2 𝑦∞ 𝑡, 𝑥ο , 𝑦ο − 𝑦𝑚 𝑡, 𝑥ο , 𝑦ο + 1 + 𝑒 𝜆𝑇 1 ∞ 𝜆𝑝𝐻 + 𝐾 𝑥 𝑡, 𝑥ο , 𝑦ο − 𝑥𝑚 𝑡, 𝑥ο , 𝑦ο + 𝐾4 𝑦∞ 𝑡, 𝑥ο , 𝑦ο − 𝑦𝑚 𝑡, 𝑥ο , 𝑦ο 1 + 𝑒 𝜆𝑇 3 ∞ So that 𝜆𝐻𝑇 𝜆𝑝𝐻 ∆ 0, 𝑥ο , 𝑦ο − ∆𝑚 0, 𝑥ο , 𝑦ο ≤ 𝐾1 + 𝐾 𝑥∞ 𝑡, 𝑥ο , 𝑦ο − 𝑥𝑚 𝑡, 𝑥ο , 𝑦ο 𝜆𝑇 1+𝑒 1 + 𝑒 𝜆𝑇 3 𝜆𝐻𝑇 𝜆𝑝𝐻 + 𝐾2 + 𝐾 𝑦∞ 𝑡, 𝑥ο , 𝑦ο − 𝑦𝑚 𝑡, 𝑥ο , 𝑦ο 𝜆𝑇 1+𝑒 1 + 𝑒 𝜆𝑇 4 . . . (44) By the same method and by (40) and (42) , we have 𝛽𝐹𝑇 𝛽𝑝𝐹 ∆∗ 𝑡, 𝑥ο , 𝑦ο − ∆∗𝑚 (0, 𝑥ο , 𝑦ο ≤ 𝐿 + 𝐿 𝑥∞ 𝑡, 𝑥ο , 𝑦ο − 𝑥𝑚 𝑡, 𝑥ο , 𝑦ο 1 + 𝑒 𝛽𝑇 1 1 + 𝑒 𝛽𝑇 3 𝛽𝐹𝑇 𝛽𝑝𝐹 + 𝐿 + 𝐿 𝑦∞ 𝑡, 𝑥ο , 𝑦ο − 𝑦𝑚 𝑡, 𝑥ο , 𝑦ο 1 + 𝑒 𝛽𝑇 2 1 + 𝑒 𝛽𝑇 4 . . . (45) And so on, rewrite the inequalities (44) and (45) in vector form as : ∆ 0, 𝑥ο , 𝑦ο − ∆𝑚 0, 𝑥ο , 𝑦ο 𝑥∞ 𝑡, 𝑥ο , 𝑦ο − 𝑥𝑚 𝑡, 𝑥ο , 𝑦ο ≤𝑄 ∆∗ 𝑡, 𝑥ο , 𝑦ο − ∆∗𝑚 (0, 𝑥ο , 𝑦ο 𝑦∞ 𝑡, 𝑥ο , 𝑦ο − 𝑦𝑚 𝑡, 𝑥ο , 𝑦ο And by (30) , we get ∆(0, 𝑥ο , 𝑦ο ) − ∆𝑚 (0, 𝑥ο , 𝑦ο ) −1 ≤ 𝑄 Λ𝑚 ο (𝐸 − Λο ) 𝑉ο ∗ ∗ ∆ 𝑡, 𝑥ο , 𝑦ο − ∆𝑚 (0, 𝑥ο , 𝑦ο Theorem 4 : Let thefunction 𝑓(𝑡, 𝑥, 𝑦) and 𝑔 𝑡, 𝑥, 𝑦 in the system (1) are defined on the intervals [a, b]and [c, d] respectively, and periodic in t with period T. Let that the sequence of functions (41) satisfying the next inequalities : min ∆𝑚 (0, 𝑥ο , 𝑦ο ) ≤ −𝛿𝑚 𝑎 + 𝑀5 ≤ 𝑥ο ≤ 𝑏 − 𝑀5 . . . (46) max ∆𝑚 0, 𝑥ο , 𝑦ο ≥ 𝛿𝑚 𝑎 + 𝑀5 ≤ 𝑥ο ≤ 𝑏 − 𝑀5 Let that the sequence of functions (42) satisfying the next inequalities : min ∆∗𝑚 (0, 𝑥ο , 𝑦ο ) ≤ −𝜀𝑚 𝑐 + 𝑀6 ≤ 𝑦ο ≤ 𝑑 − 𝑀6 . . . (47) max ∆∗𝑚 0, 𝑥ο , 𝑦ο ≥ 𝜀𝑚 , 𝑐 + 𝑀6 ≤ 𝑦ο ≤ 𝑑 − 𝑀6 for 𝑚 ≥ 0 where : 𝜆𝐻𝑇 𝜆𝑝𝐻 𝜆𝐻𝑇 𝜆𝑝𝐻 −1 𝛿𝑚 = 𝐾1 + 𝐾3 + 𝐾2 + 𝐾 Λ𝑚 ο (𝐸 − Λο ) 𝑀5 , 𝜆𝑇 𝜆𝑇 𝜆𝑇 1+𝑒 1+𝑒 1+𝑒 1 + 𝑒 𝜆𝑇 4 𝛽𝐹𝑇 𝛽𝑝𝐹 𝛽𝐹𝑇 𝛽𝑝𝐹 −1 𝜀𝑚 = 𝐿 + 𝐿 + 𝐿 + 𝐿 Λ𝑚 ο (𝐸 − Λο ) 𝑀6 1 + 𝑒 𝛽𝑇 1 1 + 𝑒 𝛽𝑇 3 1 + 𝑒 𝛽𝑇 2 1 + 𝑒 𝛽𝑇 4 Then the system (1) has periodic solution of period T, 𝑥 = 𝑥 𝑡, 𝑥ο , 𝑦ο and 𝑦 = 𝑦 𝑡, 𝑥ο , 𝑦ο for which 𝑎 + 𝑀5 ≤ 𝑥ο ≤ 𝑏 − 𝑀5 , 𝑐 + 𝑀6 ≤ 𝑦ο ≤ 𝑑 − 𝑀6 . Proof: Let 𝑥1 , 𝑥2 be any two points in the interval [𝑎 + 𝑀5 , 𝑏 − 𝑀5 ] such that :

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Periodic Solution for Nonlinear System of Differentialâ&#x20AC;Ś â&#x2C6;&#x2020;đ?&#x2018;&#x161; 0, đ?&#x2018;Ľ1 , đ?&#x2018;Ś1 = min â&#x2C6;&#x2020;đ?&#x2018;&#x161; 0, đ?&#x2018;ĽÎż , đ?&#x2018;ŚÎż đ?&#x2018;&#x17D; + đ?&#x2018;&#x20AC;5 â&#x2030;¤ đ?&#x2018;ĽÎż â&#x2030;¤ đ?&#x2018;? â&#x2C6;&#x2019; đ?&#x2018;&#x20AC;5 . . . (48) â&#x2C6;&#x2020;đ?&#x2018;&#x161; 0, đ?&#x2018;Ľ2 , đ?&#x2018;Ś2 = max â&#x2C6;&#x2020;đ?&#x2018;&#x161; 0, đ?&#x2018;ĽÎż , đ?&#x2018;ŚÎż đ?&#x2018;&#x17D; + đ?&#x2018;&#x20AC;5 â&#x2030;¤ đ?&#x2018;ĽÎż â&#x2030;¤ đ?&#x2018;? â&#x2C6;&#x2019; đ?&#x2018;&#x20AC;5 Let đ?&#x2018;Ś1 , đ?&#x2018;Ś2 be any two points in the interval [đ?&#x2018;? + đ?&#x2018;&#x20AC;6 , đ?&#x2018;&#x2018; â&#x2C6;&#x2019; đ?&#x2018;&#x20AC;6 ] such that : â&#x2C6;&#x2020;â&#x2C6;&#x2014;đ?&#x2018;&#x161; 0, đ?&#x2018;Ľ1 , đ?&#x2018;Ś1 = min â&#x2C6;&#x2020;â&#x2C6;&#x2014;đ?&#x2018;&#x161; 0, đ?&#x2018;ĽÎż , đ?&#x2018;ŚÎż đ?&#x2018;? + đ?&#x2018;&#x20AC;6 â&#x2030;¤ đ?&#x2018;ŚÎż â&#x2030;¤ đ?&#x2018;&#x2018; â&#x2C6;&#x2019; đ?&#x2018;&#x20AC;6 . . . (49) â&#x2C6;&#x2020;â&#x2C6;&#x2014;đ?&#x2018;&#x161; 0, đ?&#x2018;Ľ2 , đ?&#x2018;Ś2 = max â&#x2C6;&#x2020;â&#x2C6;&#x2014;đ?&#x2018;&#x161; 0, đ?&#x2018;ĽÎż , đ?&#x2018;ŚÎż đ?&#x2018;? + đ?&#x2018;&#x20AC;6 â&#x2030;¤ đ?&#x2018;ŚÎż â&#x2030;¤ đ?&#x2018;&#x2018; â&#x2C6;&#x2019; đ?&#x2018;&#x20AC;6 By (43) and (46), we get : â&#x2C6;&#x2020; 0, đ?&#x2018;Ľ1 , đ?&#x2018;Ś1 = â&#x2C6;&#x2020;đ?&#x2018;&#x161; 0, đ?&#x2018;Ľ1 , đ?&#x2018;Ś1 + â&#x2C6;&#x2020; 0, đ?&#x2018;Ľ1 , đ?&#x2018;Ś1 â&#x2C6;&#x2019; â&#x2C6;&#x2020;đ?&#x2018;&#x161; 0, đ?&#x2018;Ľ1 , đ?&#x2018;Ś1

<0 . . . (50)

â&#x2C6;&#x2020; 0, đ?&#x2018;Ľ2 , đ?&#x2018;Ś2 = â&#x2C6;&#x2020;đ?&#x2018;&#x161; 0, đ?&#x2018;Ľ2 , đ?&#x2018;Ś2 + â&#x2C6;&#x2020; 0, đ?&#x2018;Ľ2 , đ?&#x2018;Ś2 â&#x2C6;&#x2019; â&#x2C6;&#x2020;đ?&#x2018;&#x161; 0, đ?&#x2018;Ľ2 , đ?&#x2018;Ś2 > 0 From (43) and (47), we get : â&#x2C6;&#x2020;â&#x2C6;&#x2014; 0, đ?&#x2018;Ľ1 , đ?&#x2018;Ś1 = â&#x2C6;&#x2020;â&#x2C6;&#x2014;đ?&#x2018;&#x161; 0, đ?&#x2018;Ľ1 , đ?&#x2018;Ś1 + â&#x2C6;&#x2020;â&#x2C6;&#x2014; 0, đ?&#x2018;Ľ1 , đ?&#x2018;Ś1 â&#x2C6;&#x2019; â&#x2C6;&#x2020;â&#x2C6;&#x2014;đ?&#x2018;&#x161; 0, đ?&#x2018;Ľ1 , đ?&#x2018;Ś1

<0

. . . (51) â&#x2C6;&#x2020;â&#x2C6;&#x2014; 0, đ?&#x2018;Ľ2 , đ?&#x2018;Ś2 = â&#x2C6;&#x2020;â&#x2C6;&#x2014;đ?&#x2018;&#x161; 0, đ?&#x2018;Ľ2 , đ?&#x2018;Ś2 + â&#x2C6;&#x2020;â&#x2C6;&#x2014; 0, đ?&#x2018;Ľ2 , đ?&#x2018;Ś2 â&#x2C6;&#x2019; â&#x2C6;&#x2020;â&#x2C6;&#x2014;đ?&#x2018;&#x161; 0, đ?&#x2018;Ľ2 , đ?&#x2018;Ś2 > 0 It follows from the functions (39) , (40) and the relations (50) , (51) in virtue of the continuity of the â&#x2C6;&#x2020;constant, that there exists đ?&#x2018;Ľâ&#x2C6;&#x17E; = đ?&#x2018;ĽÎż , đ?&#x2018;Ľâ&#x2C6;&#x17E; â&#x2C6;&#x2C6; [đ?&#x2018;Ľ1 , đ?&#x2018;Ľ2 ] and đ?&#x2018;Śâ&#x2C6;&#x17E; = đ?&#x2018;ŚÎż , đ?&#x2018;Śâ&#x2C6;&#x17E; â&#x2C6;&#x2C6; [đ?&#x2018;Ś1 , đ?&#x2018;Ś2 ] such that â&#x2C6;&#x2020; 0, đ?&#x2018;Ľâ&#x2C6;&#x17E; , đ?&#x2018;Śâ&#x2C6;&#x17E; = 0 , â&#x2C6;&#x2020;â&#x2C6;&#x2014; 0, đ?&#x2018;Ľâ&#x2C6;&#x17E; , đ?&#x2018;Śâ&#x2C6;&#x17E; = 0. And this proved that the system (1) has a periodic solution đ?&#x2018;Ľ = đ?&#x2018;Ľ đ?&#x2018;Ą, đ?&#x2018;ĽÎż , đ?&#x2018;ŚÎż for đ?&#x2018;ĽÎż â&#x2C6;&#x2C6; [đ?&#x2018;&#x17D; + đ?&#x2018;&#x20AC;5 , đ?&#x2018;? â&#x2C6;&#x2019; đ?&#x2018;&#x20AC;5 ]and đ?&#x2018;Ś = đ?&#x2018;Ś đ?&#x2018;Ą, đ?&#x2018;ĽÎż , đ?&#x2018;ŚÎż for đ?&#x2018;ŚÎż â&#x2C6;&#x2C6; [đ?&#x2018;? + đ?&#x2018;&#x20AC;6 , đ?&#x2018;&#x2018; â&#x2C6;&#x2019; đ?&#x2018;&#x20AC;6 ]. Remark 1 [6] : When đ?&#x2018;&#x2026;đ?&#x2018;&#x203A; = đ?&#x2018;&#x2026;â&#x20AC;˛ , i.e. when đ?&#x2018;Ľ is a scalar theorem 4can be strengthens by giving up the requirement that the singular point shout be isolated, thus we have Theorem 5 : Let the function â&#x2C6;&#x2020; 0, đ?&#x2018;ĽÎż , đ?&#x2018;ŚÎż defined asâ&#x2C6;&#x2020; â&#x2C6;ś đ??ˇđ?&#x153;&#x2020;đ?&#x2018;&#x201C; â&#x2C6;&#x2020;(0, đ?&#x2018;ĽÎż , đ?&#x2018;ŚÎż ) =

đ?&#x153;&#x2020; [ 1 + đ?&#x2018;&#x2019; đ?&#x153;&#x2020;đ?&#x2018;&#x2021;

đ?&#x2018;&#x2019;đ?&#x153;&#x2020;

đ?&#x2018;&#x2019;đ?&#x153;&#x2020;

đ?&#x2018;&#x2021;â&#x2C6;&#x2019;đ?&#x2018;

đ?&#x2018;&#x201C; đ?&#x2018;Ą, đ?&#x2018;Ľâ&#x2C6;&#x17E; đ?&#x2018;Ą, đ?&#x2018;ĽÎż , đ?&#x2018;ŚÎż , đ?&#x2018;Śâ&#x2C6;&#x17E; đ?&#x2018;Ą, đ?&#x2018;ĽÎż , đ?&#x2018;ŚÎż đ?&#x2018;&#x2018;đ?&#x2018;Ą +

0

đ?&#x2018;?

+

đ?&#x2018;&#x2026;â&#x20AC;˛

đ?&#x2018;&#x2021;

đ?&#x2018;&#x2021;â&#x2C6;&#x2019;đ?&#x2018;

đ??źđ?&#x2018;&#x2013; đ?&#x2018;Ľâ&#x2C6;&#x17E; đ?&#x2018;Ąđ?&#x2018;&#x2013; , đ?&#x2018;ĽÎż , đ?&#x2018;ŚÎż , đ?&#x2018;Śâ&#x2C6;&#x17E; đ?&#x2018;Ąđ?&#x2018;&#x2013; , đ?&#x2018;ĽÎż , đ?&#x2018;ŚÎż ]

. . . (52)

đ?&#x2018;&#x2013;=1

Let the function â&#x2C6;&#x2020;â&#x2C6;&#x2014; 0, đ?&#x2018;ĽÎż , đ?&#x2018;ŚÎż defined asâ&#x2C6;&#x2020;â&#x2C6;&#x2014; â&#x2C6;ś đ??ˇđ?&#x203A;˝đ?&#x2018;&#x201C; â&#x2C6;&#x2020;â&#x2C6;&#x2014; (0, đ?&#x2018;ĽÎż , đ?&#x2018;ŚÎż ) =

đ?&#x203A;˝ [ 1 + đ?&#x2018;&#x2019; đ?&#x203A;˝đ?&#x2018;&#x2021; đ?&#x2018;?

đ?&#x2018;&#x2019;đ?&#x203A;˝

+

đ?&#x2018;&#x2026;â&#x20AC;˛

đ?&#x2018;&#x2021;

đ?&#x2018;&#x2019;đ?&#x203A;˝

đ?&#x2018;&#x2021;â&#x2C6;&#x2019;đ?&#x2018;

đ?&#x2018;&#x201C; đ?&#x2018;Ą, đ?&#x2018;Ľâ&#x2C6;&#x17E; đ?&#x2018;Ą, đ?&#x2018;ĽÎż , đ?&#x2018;ŚÎż , đ?&#x2018;Śâ&#x2C6;&#x17E; đ?&#x2018;Ą, đ?&#x2018;ĽÎż , đ?&#x2018;ŚÎż đ?&#x2018;&#x2018;đ?&#x2018;Ą +

0

đ?&#x2018;&#x2021;â&#x2C6;&#x2019;đ?&#x2018;

đ??şđ?&#x2018;&#x2013; đ?&#x2018;Ľâ&#x2C6;&#x17E; đ?&#x2018;Ąđ?&#x2018;&#x2013; , đ?&#x2018;ĽÎż , đ?&#x2018;ŚÎż , đ?&#x2018;Śâ&#x2C6;&#x17E; đ?&#x2018;Ąđ?&#x2018;&#x2013; , đ?&#x2018;ĽÎż , đ?&#x2018;ŚÎż ]

. . . (53)

đ?&#x2018;&#x2013;=1

Where the functions đ?&#x2018;Ľâ&#x2C6;&#x17E; đ?&#x2018;Ą, đ?&#x2018;ĽÎż , đ?&#x2018;ŚÎż andđ?&#x2018;Śâ&#x2C6;&#x17E; đ?&#x2018;Ą, đ?&#x2018;ĽÎż , đ?&#x2018;ŚÎż are the limit of asequence of periodic functions (12) , (13) respectively, then the following inequalities are holds : đ?&#x153;&#x2020;đ?&#x2018;Ąđ??ť2 đ?&#x2018;&#x2021; đ?&#x153;&#x2020;đ?&#x2018;Ą â&#x2C6;&#x2020;(0, đ?&#x2018;ĽÎż , đ?&#x2018;ŚÎż ) â&#x2030;¤ đ??ťđ?&#x2018;Ą â&#x2C6;&#x2019; đ?&#x2018;&#x20AC;1 + 2đ?&#x2018;?đ??ť 1 + đ?&#x2018;&#x20AC;3 . . . (54) 1 + đ?&#x2018;&#x2019; đ?&#x153;&#x2020;đ?&#x2018;&#x2021; 1 + đ?&#x2018;&#x2019; đ?&#x153;&#x2020;đ?&#x2018;&#x2021; 2 đ?&#x203A;˝đ?&#x2018;Ąđ??š đ?&#x2018;&#x2021; đ?&#x203A;˝đ?&#x2018;Ą â&#x2C6;&#x2020;â&#x2C6;&#x2014; (0, đ?&#x2018;ĽÎż , đ?&#x2018;ŚÎż ) â&#x2030;¤ đ??šđ?&#x2018;Ą â&#x2C6;&#x2019; đ?&#x2018;&#x20AC;2 + 2đ?&#x2018;?đ??š 1 + đ?&#x2018;&#x20AC;4 . . . (55) đ?&#x203A;˝đ?&#x2018;&#x2021; 1+đ?&#x2018;&#x2019; 1 + đ?&#x2018;&#x2019; đ?&#x203A;˝đ?&#x2018;&#x2021; â&#x2C6;&#x2020; 0, đ?&#x2018;ĽÎż1 , đ?&#x2018;ŚÎż1 â&#x2C6;&#x2019; â&#x2C6;&#x2020; 0, đ?&#x2018;ĽÎż2 , đ?&#x2018;ŚÎż2 â&#x2030;¤ đ?&#x2018;&#x160;1 (1 â&#x2C6;&#x2019; đ?&#x2018;&#x160;1 â&#x2C6;&#x2019; đ?&#x2018;&#x160;2 đ?&#x2018;&#x160;3 1 â&#x2C6;&#x2019; đ?&#x2018;&#x160;4 â&#x2C6;&#x2019;1 )â&#x2C6;&#x2019;1 [ đ?&#x2018;ĽÎż1 â&#x2C6;&#x2019; đ?&#x2018;ĽÎż2 + +đ?&#x2018;&#x160;2 1 â&#x2C6;&#x2019; đ?&#x2018;&#x160;4 â&#x2C6;&#x2019;1 đ?&#x2018;ŚÎż1 â&#x2C6;&#x2019; đ?&#x2018;ŚÎż2 ] + +đ?&#x2018;&#x160;2 (1 â&#x2C6;&#x2019; đ?&#x2018;&#x160;4 â&#x2C6;&#x2019; đ?&#x2018;&#x160;2 đ?&#x2018;&#x160;3 1 â&#x2C6;&#x2019; đ?&#x2018;&#x160;1 â&#x2C6;&#x2019;1 )â&#x2C6;&#x2019;1 [ đ?&#x2018;ŚÎż1 â&#x2C6;&#x2019; đ?&#x2018;ŚÎż2 + +đ?&#x2018;&#x160;3 1 â&#x2C6;&#x2019; đ?&#x2018;&#x160;1 â&#x2C6;&#x2019;1 đ?&#x2018;ĽÎż1 â&#x2C6;&#x2019; đ?&#x2018;ĽÎż2 ] . . . (56) â&#x2C6;&#x2014; 1 1 â&#x2C6;&#x2014; 2 2 â&#x2C6;&#x2020; 0, đ?&#x2018;ĽÎż , đ?&#x2018;ŚÎż â&#x2C6;&#x2019; â&#x2C6;&#x2020; 0, đ?&#x2018;ĽÎż , đ?&#x2018;ŚÎż â&#x2030;¤ đ?&#x2018;&#x160;3 (1 â&#x2C6;&#x2019; đ?&#x2018;&#x160;1 â&#x2C6;&#x2019; đ?&#x2018;&#x160;2 đ?&#x2018;&#x160;3 1 â&#x2C6;&#x2019; đ?&#x2018;&#x160;4 â&#x2C6;&#x2019;1 )â&#x2C6;&#x2019;1 [ đ?&#x2018;ĽÎż1 â&#x2C6;&#x2019; đ?&#x2018;ĽÎż2 + +đ?&#x2018;&#x160;2 1 â&#x2C6;&#x2019; đ?&#x2018;&#x160;4 â&#x2C6;&#x2019;1 đ?&#x2018;ŚÎż1 â&#x2C6;&#x2019; đ?&#x2018;ŚÎż2 ] + +đ?&#x2018;&#x160;4 (1 â&#x2C6;&#x2019; đ?&#x2018;&#x160;4 â&#x2C6;&#x2019; đ?&#x2018;&#x160;2 đ?&#x2018;&#x160;3 1 â&#x2C6;&#x2019; đ?&#x2018;&#x160;1 â&#x2C6;&#x2019;1 )â&#x2C6;&#x2019;1 [ đ?&#x2018;ŚÎż1 â&#x2C6;&#x2019; đ?&#x2018;ŚÎż2 + +đ?&#x2018;&#x160;3 1 â&#x2C6;&#x2019; đ?&#x2018;&#x160;1 â&#x2C6;&#x2019;1 đ?&#x2018;ĽÎż1 â&#x2C6;&#x2019; đ?&#x2018;ĽÎż2 ] . . . (57) Where

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Periodic Solution for Nonlinear System of Differential… 𝜆𝑇𝐾1 𝜆𝐾3 𝑝 + 𝐾3 + , 1 + 𝑒 𝜆𝑇 1 + 𝑒 𝜆𝑇 𝜆𝑇𝐾2 𝜆𝐾4 𝑝 𝑊2 = 𝐻𝑇 𝐾2 − + 𝐾4 + , 1 + 𝑒 𝜆𝑇 1 + 𝑒 𝜆𝑇 𝛽𝑇𝐿1 𝛽𝐿3 𝑝 𝑊3 = 𝐹𝑇 𝐿1 − + 𝐿3 + , 1 + 𝑒 𝛽𝑇 1 + 𝑒 𝛽𝑇 𝛽𝑇𝐿2 𝛽𝐿4 𝑝 𝑊4 = 𝐹𝑇 𝐿2 − + 𝐿4 + , 𝛽𝑇 1+𝑒 1 + 𝑒 𝛽𝑇 1 2 1 2 for all 𝑥ο , 𝑥ο , 𝑥ο ∈ 𝐷𝜆𝑓 and 𝑦ο , 𝑦ο , 𝑦ο ∈ 𝐷𝛽𝑓 . Proof: From the properties of the function𝑥∞ 𝑡, 𝑥ο , 𝑦ο and𝑦∞ 𝑡, 𝑥ο , 𝑦ο established by theorem 1, it follows that the functions∆= ∆ 0, 𝑥ο , 𝑦ο and ∆∗ = ∆∗ 0, 𝑥ο , 𝑦ο are continuous and bounded with the positive constants 𝜆𝑡𝐻2 𝑇 𝜆𝑡 𝐻𝑡 − 𝑀1 + 2𝑝𝐻 1 + 𝑀3 for𝑥ο ∈ 𝐷𝜆𝑓 , and 𝜆𝑇 1+𝑒 1 + 𝑒 𝜆𝑇 𝛽𝑡𝐹 2 𝑇 𝛽𝑡 𝐹𝑡 − 𝑀2 + 2𝑝𝐹 1 + 𝑀4 for𝑦ο ∈ 𝐷𝛽𝑓 , respectively. 𝛽𝑇 1+𝑒 1 + 𝑒 𝛽𝑇 By using (52) , we have 𝑊1 = 𝐻𝑇

0, 𝑥ο1 , 𝑦ο1

𝐾1 −

−∆

𝜆 ≤ 1 + 𝑒 𝜆𝑇

0, 𝑥ο2 , 𝑦ο2

+

𝑇

𝑒𝜆 0

𝜆 1 + 𝑒 𝜆𝑇

𝑝

𝑇−𝑠

𝑓 𝑡, 𝑥∞ 𝑡, 𝑥ο1 , 𝑦ο1 , 𝑦∞ 𝑡, 𝑥ο1 , 𝑦ο1

−𝑓 𝑡, 𝑥∞ 𝑡, 𝑥ο2 , 𝑦ο2 , 𝑦∞ 𝑡, 𝑥ο2 , 𝑦ο2 𝑒𝜆

𝑇−𝑠

𝑑𝑡 +

𝐼𝑖 𝑥∞ 𝑡𝑖 , 𝑥ο1 , 𝑦ο1 , 𝑦∞ 𝑡𝑖 , 𝑥ο2 , 𝑦ο2

𝑖=1

− 𝐼𝑖 𝑥∞ 𝑡𝑖 , 𝑥ο1 , 𝑦ο1 , 𝑦∞ 𝑡𝑖 , 𝑥ο2 , 𝑦ο2 ≤ 𝜆𝐻𝑇 ≤ 𝐾 𝑥 𝑡, 𝑥ο1 , 𝑦ο1 − 𝑥∞ 𝑡, 𝑥ο2 , 𝑦ο2 + 𝐾2 𝑦∞ 𝑡, 𝑥ο1 , 𝑦ο1 − 𝑦∞ 𝑡, 𝑥ο2 , 𝑦ο2 + 1 + 𝑒 𝜆𝑇 1 ∞ 𝜆𝑝𝐻 + 𝐾 𝑥 𝑡, 𝑥ο1 , 𝑦ο1 − 𝑥∞ 𝑡, 𝑥ο2 , 𝑦ο2 + 𝐾4 𝑦∞ 𝑡, 𝑥ο1 , 𝑦ο1 − 𝑦∞ 𝑡, 𝑥ο2 , 𝑦ο2 1 + 𝑒 𝜆𝑇 3 ∞ So that 𝜆𝐻 ∆ 0, 𝑥ο1 , 𝑦ο1 − ∆ 0, 𝑥ο2 , 𝑦ο2 ≤ [ 𝑇𝐾1 + 𝑝𝐾3 𝑥∞ 𝑡, 𝑥ο1 , 𝑦ο1 − 𝑥∞ 𝑡, 𝑥ο2 , 𝑦ο2 + 1 + 𝑒 𝜆𝑇 + 𝑇𝐾2 + 𝑝𝐾4 𝑦∞ 𝑡, 𝑥ο1 , 𝑦ο1 − 𝑦∞ 𝑡, 𝑥ο2 , 𝑦ο2 ] . . . (58) And we will find by the same method and by (53) , we have : 𝛽𝐹 ∆∗ 0, 𝑥ο1 , 𝑦ο1 − ∆∗ 0, 𝑥ο2 , 𝑦ο2 ≤ [ 𝑇𝐿1 + 𝑝𝐿3 𝑥∞ 𝑡, 𝑥ο1 , 𝑦ο1 − 𝑥∞ 𝑡, 𝑥ο2 , 𝑦ο2 + 1 + 𝑒 𝜆𝑇 + 𝑇𝐿2 + 𝑝𝐿4 𝑦∞ 𝑡, 𝑥ο1 , 𝑦ο1 − 𝑦∞ 𝑡, 𝑥ο2 , 𝑦ο2 ] . . . (59) 1 1 2 2 Where 𝑥∞ 𝑡, 𝑥ο , 𝑦ο , 𝑥∞ 𝑡, 𝑥ο , 𝑦ο and𝑦∞ 𝑡, 𝑥ο1 , 𝑦ο1 , 𝑦∞ 𝑡, 𝑥ο2 , 𝑦ο2 are the solutions of the following equations :

integral

𝑡

𝑥 𝑡, 𝑥ο𝑘 , 𝑦ο𝑘 = 𝑥ο𝑘 +

𝑒𝜆 0 𝑇

− 0

𝑡−𝑠

[𝑓 𝑠, 𝑥 𝑠, 𝑥ο𝑘 , 𝑦ο𝑘 , 𝑦 𝑠, 𝑥ο𝑘 , 𝑦ο𝑘

𝜆 𝑒𝜆 1 + 𝑒 𝜆𝑇 𝑒𝜆

+ 0<𝑡 𝑖 <𝑡

𝜆 − 1 + 𝑒 𝜆𝑇

𝑇−𝑠

𝑇−𝑠

𝑓 𝑠, 𝑥 𝑠, 𝑥ο𝑘 , 𝑦ο𝑘 , 𝑦 𝑠, 𝑥ο𝑘 , 𝑦ο𝑘 𝑑𝑠]𝑑𝑠 +

𝐼𝑖 𝑥 𝑡𝑖 , 𝑥ο𝑘 , 𝑦ο𝑘 , 𝑦 𝑡𝑖 , 𝑥ο𝑘 , 𝑦ο𝑘

𝑝

𝑒𝜆

𝑇−𝑠

𝐼𝑖 𝑥 𝑡𝑖 , 𝑥ο𝑘 , 𝑦ο𝑘 , 𝑦 𝑡𝑖 , 𝑥ο𝑘 , 𝑦ο𝑘

,

. . . (60)

𝑖=1

And 𝑡

𝑦

𝑡, 𝑥ο𝑘 , 𝑦ο𝑘

=

𝑦ο𝑘

𝑒𝛽

+

𝑡−𝑠

[𝑓 𝑠, 𝑥 𝑠, 𝑥ο𝑘 , 𝑦ο𝑘 , 𝑦 𝑠, 𝑥ο𝑘 , 𝑦ο𝑘

0 𝑇

− 0

𝛽 𝑒𝛽 1 + 𝑒 𝛽𝑇

𝑇−𝑠

𝑓 𝑠, 𝑥 𝑠, 𝑥ο𝑘 , 𝑦ο𝑘 , 𝑦 𝑠, 𝑥ο𝑘 , 𝑦ο𝑘 𝑑𝑠]𝑑𝑠 +

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Periodic Solution for Nonlinear System of Differential… 𝑒𝛽

+ 0<𝑡 𝑖 <𝑡

𝑇−𝑠

𝐺𝑖 𝑥 𝑡𝑖 , 𝑥ο𝑘 , 𝑦ο𝑘 , 𝑦 𝑡𝑖 , 𝑥ο𝑘 , 𝑦ο𝑘

𝑝

𝛽 − 1 + 𝑒 𝛽𝑇

𝑒𝛽

𝑇−𝑠

=

𝑥ο1

𝐺𝑖 𝑥 𝑡𝑖 , 𝑥ο𝑘 , 𝑦ο𝑘 , 𝑦 𝑡𝑖 , 𝑥ο𝑘 , 𝑦ο𝑘

,

. . . (61)

𝑖=1

Where 𝑘 = 1,2 From (60) , we have

𝑡

𝑥∞ 𝑡, 𝑥ο1 , 𝑦ο1

𝑥∞ 𝑡, 𝑥ο2 , 𝑦ο2 𝑇

− 0

𝜆 𝑒𝜆 1 + 𝑒 𝜆𝑇 𝑒𝜆

0<𝑡 𝑖 <𝑡

𝑇−𝑠

𝑇−𝑠

𝑓 𝑠, 𝑥∞ 𝑠, 𝑥ο1 , 𝑦ο1 , 𝑦∞ 𝑠, 𝑥ο1 , 𝑦ο1 𝑑𝑠]𝑑𝑠 +

𝐼𝑖 𝑥∞ 𝑡𝑖 , 𝑥ο1 , 𝑦ο1 , 𝑦∞ 𝑡𝑖 , 𝑥ο1 , 𝑦ο1

𝑒𝜆

𝑇−𝑠

𝐼𝑖 𝑥∞ 𝑡𝑖 , 𝑥ο1 , 𝑦ο1 , 𝑦∞ 𝑡𝑖 , 𝑥ο1 , 𝑦ο1

𝑖=1

𝑡

− 𝑥ο2 −

0

[𝑓 𝑠, 𝑥∞ 𝑠, 𝑥ο1 , 𝑦ο1 , 𝑦∞ 𝑠, 𝑥ο1 , 𝑦ο1

𝑝

𝜆 − 1 + 𝑒 𝜆𝑇

𝑡−𝑠

0

+

𝑇

𝑒𝜆

+

𝑒𝜆

𝑡−𝑠

[𝑓 𝑠, 𝑥∞ 𝑠, 𝑥ο2 , 𝑦ο2 , 𝑦∞ 𝑠, 𝑥ο2 , 𝑦ο2

0

𝜆 𝑒𝜆 1 + 𝑒 𝜆𝑇 𝑒𝜆

− 0<𝑡 𝑖 <𝑡

𝜆 + 1 + 𝑒 𝜆𝑇

𝑇−𝑠

𝑇−𝑠

𝑓 𝑠, 𝑥∞ 𝑠, 𝑥ο2 , 𝑦ο2 , 𝑦∞ 𝑠, 𝑥ο2 , 𝑦ο2 𝑑𝑠]𝑑𝑠 −

𝐼𝑖 𝑥∞ 𝑡𝑖 , 𝑥ο2 , 𝑦ο2 , 𝑦∞ 𝑡𝑖 , 𝑥ο2 , 𝑦ο2

+

𝑝

𝑒𝜆

𝑇−𝑠

𝐼𝑖 𝑥∞ 𝑡𝑖 , 𝑥ο2 , 𝑦ο2 , 𝑦∞ 𝑡, 𝑥ο2 , 𝑦ο2

𝑖=1

≤ 𝑥ο1 − 𝑥ο2 + 𝐻𝐾1 𝑡 𝑥∞ 𝑡, 𝑥ο1 , 𝑦ο1 − 𝑥∞ 𝑡, 𝑥ο2 , 𝑦ο2 + 𝜆𝑡𝐻𝑇𝐾1 + 𝐻𝐾2 𝑡 𝑦∞ 𝑡, 𝑥ο1 , 𝑦ο1 − 𝑦∞ 𝑡, 𝑥ο2 , 𝑦ο2 + 𝑥 𝑡, 𝑥ο1 , 𝑦ο1 − 𝑥∞ 𝑡, 𝑥ο2 , 𝑦ο2 + 1 + 𝑒 𝜆𝑇 ∞ 𝜆𝑡𝐻𝑇𝐾2 + 𝑦 𝑡, 𝑥ο1 , 𝑦ο1 − 𝑦∞ 𝑡, 𝑥ο2 , 𝑦ο2 + 𝐻𝐾3 𝑡 𝑥∞ 𝑡, 𝑥ο1 , 𝑦ο1 − 𝑥∞ 𝑡, 𝑥ο2 , 𝑦ο2 + 1 + 𝑒 𝜆𝑇 ∞ 𝜆𝑡𝐻𝑇𝐾3 + 𝐻𝐾4 𝑡 𝑦∞ 𝑡, 𝑥ο1 , 𝑦ο1 − 𝑦∞ 𝑡, 𝑥ο2 , 𝑦ο2 + 𝑝 𝑥∞ 𝑡, 𝑥ο1 , 𝑦ο1 − 𝑥∞ 𝑡, 𝑥ο2 , 𝑦ο2 + 1 + 𝑒 𝜆𝑇 𝜆𝑡𝐻𝑇𝐾4 + 𝑝 𝑦∞ 𝑡, 𝑥ο1 , 𝑦ο1 − 𝑦∞ 𝑡, 𝑥ο2 , 𝑦ο2 1 + 𝑒 𝜆𝑇 𝜆𝑇𝐾1 𝜆𝑇𝐾3 𝑝 ≤ 𝑥ο1 − 𝑥ο2 + 𝐻𝑡 𝐾1 + + 𝐾3 + 𝑥∞ 𝑡, 𝑥ο1 , 𝑦ο1 − 𝑥∞ 𝑡, 𝑥ο2 , 𝑦ο2 + 𝜆𝑇 1+𝑒 1 + 𝑒 𝜆𝑇 𝜆𝑇𝐾2 𝜆𝑇𝐾4 𝑝 + 𝐻𝑡 𝐾2 + + 𝐾4 + 𝑦∞ 𝑡, 𝑥ο1 , 𝑦ο1 − 𝑦∞ 𝑡, 𝑥ο2 , 𝑦ο2 . . . (62) 𝜆𝑇 1+𝑒 1 + 𝑒 𝜆𝑇 By the same method and by (61), we find : 𝑦∞ 𝑡, 𝑥ο1 , 𝑦ο1 − 𝑦∞ 𝑡, 𝑥ο2 , 𝑦ο2 ≤ 𝑦ο1 − 𝑦ο2 + 𝛽𝑇𝐿1 𝛽𝑇𝐿3 𝑝 + 𝐹𝑡 𝐿1 + + 𝐿3 + 𝑥∞ 𝑡, 𝑥ο1 , 𝑦ο1 − 𝑥∞ 𝑡, 𝑥ο2 , 𝑦ο2 + 1 + 𝑒 𝛽𝑇 1 + 𝑒 𝛽𝑇 𝛽𝑇𝐿2 𝛽𝑇𝐿4 𝑝 + 𝐹𝑡 𝐿2 + + 𝐿4 + 𝑦∞ 𝑡, 𝑥ο1 , 𝑦ο1 − 𝑦∞ 𝑡, 𝑥ο2 , 𝑦ο2 . . . (63) 𝛽𝑇 1+𝑒 1 + 𝑒 𝛽𝑇 From (62), we have: −1

𝜆𝑇𝐾1 𝜆𝑇𝐾3 𝑝 − ≤ 1 − 𝐻𝑡 𝐾1 + + 𝐾3 + 𝜆𝑇 1+𝑒 1 + 𝑒 𝜆𝑇 𝜆𝑇𝐾2 𝜆𝑇𝐾4 𝑝 + 𝐻𝑡 𝐾2 + + 𝐾4 + 𝑦∞ 𝑡, 𝑥ο1 , 𝑦ο1 − 𝑦∞ 𝑡, 𝑥ο2 , 𝑦ο2 𝜆𝑇 1+𝑒 1 + 𝑒 𝜆𝑇

𝑥∞ 𝑡, 𝑥ο1 , 𝑦ο1 𝑥ο1 − 𝑥ο2

𝑥∞ 𝑡, 𝑥ο2 , 𝑦ο2

. . . (64)

By (63), we have:

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Periodic Solution for Nonlinear System of Differentialâ&#x20AC;Ś â&#x2C6;&#x2019;1

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đ?&#x2018;Śâ&#x2C6;&#x17E; đ?&#x2018;Ą, đ?&#x2018;ĽÎż1 , đ?&#x2018;ŚÎż1 đ?&#x2018;ŚÎż1 â&#x2C6;&#x2019; đ?&#x2018;ŚÎż2

đ?&#x2018;Śâ&#x2C6;&#x17E; đ?&#x2018;Ą, đ?&#x2018;ĽÎż2 , đ?&#x2018;ŚÎż2

. . . (65) By substitute (65) in (64) , we obtain :đ?&#x2018;&#x160;3 1 â&#x2C6;&#x2019; đ?&#x2018;&#x160;4 â&#x2C6;&#x2019;1 đ?&#x2018;Ľâ&#x2C6;&#x17E; đ?&#x2018;Ą, đ?&#x2018;ĽÎż1 , đ?&#x2018;ŚÎż1 â&#x2C6;&#x2019; đ?&#x2018;Ľâ&#x2C6;&#x17E; đ?&#x2018;Ą, đ?&#x2018;ĽÎż2 , đ?&#x2018;ŚÎż2 â&#x2030;¤ đ?&#x2018;ĽÎż1 â&#x2C6;&#x2019; đ?&#x2018;ĽÎż2 + đ?&#x2018;&#x160;1 đ?&#x2018;Ľâ&#x2C6;&#x17E; đ?&#x2018;Ą, đ?&#x2018;ĽÎż1 , đ?&#x2018;ŚÎż1 â&#x2C6;&#x2019; đ?&#x2018;Ľâ&#x2C6;&#x17E; đ?&#x2018;Ą, đ?&#x2018;ĽÎż2 , đ?&#x2018;ŚÎż2 + + đ?&#x2018;&#x160;2 1 â&#x2C6;&#x2019; đ?&#x2018;&#x160;4 â&#x2C6;&#x2019;1 đ?&#x2018;ŚÎż1 â&#x2C6;&#x2019; đ?&#x2018;ŚÎż2 + đ?&#x2018;&#x160;3 đ?&#x2018;Ľâ&#x2C6;&#x17E; đ?&#x2018;Ą, đ?&#x2018;ĽÎż1 , đ?&#x2018;ŚÎż1 â&#x2C6;&#x2019; đ?&#x2018;Ľâ&#x2C6;&#x17E; đ?&#x2018;Ą, đ?&#x2018;ĽÎż2 , đ?&#x2018;ŚÎż2 So that đ?&#x2018;Ľâ&#x2C6;&#x17E; đ?&#x2018;Ą, đ?&#x2018;ĽÎż1 , đ?&#x2018;ŚÎż1 â&#x2C6;&#x2019; đ?&#x2018;Ľâ&#x2C6;&#x17E; đ?&#x2018;Ą, đ?&#x2018;ĽÎż2 , đ?&#x2018;ŚÎż2 â&#x2030;¤ 1 â&#x2C6;&#x2019; đ?&#x2018;&#x160;1 â&#x2C6;&#x2019; đ?&#x2018;&#x160;2 đ?&#x2018;&#x160;3 1 â&#x2C6;&#x2019; đ?&#x2018;&#x160;4 â&#x2C6;&#x2019;1 â&#x2C6;&#x2019;1 ( đ?&#x2018;ĽÎż1 â&#x2C6;&#x2019; đ?&#x2018;ĽÎż2 + +đ?&#x2018;&#x160;2 1 â&#x2C6;&#x2019; đ?&#x2018;&#x160;4 â&#x2C6;&#x2019;1 đ?&#x2018;ŚÎż1 â&#x2C6;&#x2019; đ?&#x2018;ŚÎż2 ) . . . (66) And also, we have đ?&#x2018;Śâ&#x2C6;&#x17E; đ?&#x2018;Ą, đ?&#x2018;ĽÎż1 , đ?&#x2018;ŚÎż1 â&#x2C6;&#x2019; đ?&#x2018;Śâ&#x2C6;&#x17E; đ?&#x2018;Ą, đ?&#x2018;ĽÎż2 , đ?&#x2018;ŚÎż2 â&#x2030;¤ đ?&#x2018;ŚÎż1 â&#x2C6;&#x2019; đ?&#x2018;ŚÎż2 + đ?&#x2018;&#x160;4 đ?&#x2018;Śâ&#x2C6;&#x17E; đ?&#x2018;Ą, đ?&#x2018;ĽÎż1 , đ?&#x2018;ŚÎż1 â&#x2C6;&#x2019; đ?&#x2018;Śâ&#x2C6;&#x17E; đ?&#x2018;Ą, đ?&#x2018;ĽÎż2 , đ?&#x2018;ŚÎż2 +ď&#x20AC; + đ?&#x2018;&#x160;3 1 â&#x2C6;&#x2019; đ?&#x2018;&#x160;1 â&#x2C6;&#x2019;1 đ?&#x2018;ĽÎż1 â&#x2C6;&#x2019; đ?&#x2018;ĽÎż2 + đ?&#x2018;&#x160;2 đ?&#x2018;Śâ&#x2C6;&#x17E; đ?&#x2018;Ą, đ?&#x2018;ĽÎż1 , đ?&#x2018;ŚÎż1 â&#x2C6;&#x2019; đ?&#x2018;Śâ&#x2C6;&#x17E; đ?&#x2018;Ą, đ?&#x2018;ĽÎż2 , đ?&#x2018;ŚÎż2 So that đ?&#x2018;Śâ&#x2C6;&#x17E; đ?&#x2018;Ą, đ?&#x2018;ĽÎż1 , đ?&#x2018;ŚÎż1 â&#x2C6;&#x2019; đ?&#x2018;Śâ&#x2C6;&#x17E; đ?&#x2018;Ą, đ?&#x2018;ĽÎż2 , đ?&#x2018;ŚÎż2 â&#x2030;¤ 1 â&#x2C6;&#x2019; đ?&#x2018;&#x160;4 â&#x2C6;&#x2019; đ?&#x2018;&#x160;2 đ?&#x2018;&#x160;3 1 â&#x2C6;&#x2019; đ?&#x2018;&#x160;1 â&#x2C6;&#x2019;1 â&#x2C6;&#x2019;1 ( đ?&#x2018;ŚÎż1 â&#x2C6;&#x2019; đ?&#x2018;ŚÎż2 + +đ?&#x2018;&#x160;3 1 â&#x2C6;&#x2019; đ?&#x2018;&#x160;1 â&#x2C6;&#x2019;1 đ?&#x2018;ĽÎż1 â&#x2C6;&#x2019; đ?&#x2018;ĽÎż2 ) . . . (67) Now by substitute (66) and (67) in (58) and (59), we obtain on (56) and (57) respectively. Remark 2 [1]: The theorem 5 to ensure solution's to the system (1), in view of to happen small change in the point đ?&#x2018;Ľ 0 , to requite small change on the function's behaviorâ&#x2C6;&#x2020;= â&#x2C6;&#x2020; đ?&#x2018;Ą, đ?&#x2018;ĽÎż , đ?&#x2018;ŚÎż .

REFERENCES [1]. [2]. [3]. [4]. [5]. [6]. [7].

Butris R.N. Existence of periodic for non-linear systems of differential equations of operators with pulse action, Ukraine, Kiev , Math. J. No.9,pp.1260-1264,(1991). Perestyuk N. A. Stability solutions for linear systems of pulse action, Ukraine, Kiev, Uesnki, J. , No. 19, pp 71 - 76, (1977). Rama M. M. Ordinary differential equations theory and applications, Britain, (1981). Ronto, N. I. Existence of periodic solution of differential equations with pulse action, Dok1. Nauk Ukraine, pp.54-55, (1987). Samoilenko A. M. and Perestyuk N. A. Impulsive differential equations, USA: World Scientific; Ser A Nonlinear Science, vol. 14, pp.462, (1995). Samoilenko A. M. and Ronto N. I. , A numerical â&#x20AC;&#x201C; analytic methods for investigations of periodic solutions, Ukraine, Kiev, (1976). Yan Juran, JianhuaShen, Razumikhin type stability theorems for impulsive functional differential equations, Non Linear Analysis, vol.33,pp.519-531, (1998).

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