A03080116

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International Journal of Engineering Inventions e-ISSN: 2278-7461, p-ISSN: 2319-6491 Volume 3, Issue 8 (April 2014) PP: 01-16

Periodic Solutions for Non-Linear Systems of Integral Equations Prof. Dr. Raad. N. Butris1, Baybeen S. Fars2 1, 2

Department of Mathematics, Faculty of Science, University of Zakho

Abstract: The aim of this paper is to study the existence and approximation of periodic solutions for non-linear systems of integral equations, by using the numerical-analytic method which were introduced by Samoilenko[ 10, 11]. The study of such nonlinear integral equations is more general and leads us to improve and extend the results of Butris [2]. Keyword And Phrases: Numerical-analytic methods, existence and approximation of periodic solutions, nonlinear system, integral equations.

I. Introduction Integral equation has been arisen in many mathematical and engineering field, so that solving this kind of problems are more efficient and useful in many research branches. Analytical solution of this kind of equation is not accessible in general form of equation and we can only get an exact solution only in special cases. But in industrial problems we have not spatial cases so that we try to solve this kind of equations numerically in general format. Many numerical schemes are employed to give an approximate solution with sufficient accuracy [3,4,6, ,12,13,14,15]. Many author create and develop numerical-analytic methods [1, 2,5, 7,8,9] and schemes to investigate periodic solution of integral equations describing many applications in mathematical and engineering field. Consider the following system of non-linear integral equations which has the form: ฤ‘?‘ฤ„

ฤ‘?‘ฤฝ ฤ‘?‘ฤ„, ฤ‘?‘ฤฝ0 , ฤ‘?‘ลš0 = ฤ‘??ลก0 ฤ‘?‘ฤ„ +

ฤ‘?‘“1 (ฤ‘?‘ , ฤ‘?‘ฤฝ ฤ‘?‘ , ฤ‘?‘ฤฝ0 , ฤ‘?‘ลš0 , ฤ‘?‘ลš ฤ‘?‘ , ฤ‘?‘ฤฝ0 , ฤ‘?‘ลš0 , 0

ฤ‘?‘

, ,

ฤ‘?‘? ฤ‘?‘ ฤ‘?‘Ž ฤ‘?‘

ฤ‘??ลŸ1 ฤ‘?‘ , ฤ‘?œ? ฤ‘?‘”1 (ฤ‘?œ?, ฤ‘?‘ฤฝ ฤ‘?œ?, ฤ‘?‘ฤฝ0 , ฤ‘?‘ลš0 , ฤ‘?‘ลš ฤ‘?œ?, ฤ‘?‘ฤฝ0 , ฤ‘?‘ลš0 ) ฤ‘?‘‘ฤ‘?œ?, รขˆ’รขˆž

ฤ‘?‘”1 (ฤ‘?œ?, ฤ‘?‘ฤฝ ฤ‘?œ?, ฤ‘?‘ฤฝ0 , ฤ‘?‘ลš0 , ฤ‘?‘ลš ฤ‘?œ?, ฤ‘?‘ฤฝ0 , ฤ‘?‘ลš0 )ฤ‘?‘‘ฤ‘?œ?)ฤ‘?‘‘ฤ‘?‘

รข‹ลป( I1 )

ฤ‘?‘ฤ„

ฤ‘?‘ลš ฤ‘?‘ฤ„, ฤ‘?‘ฤฝ0 , ฤ‘?‘ลš0 = ฤ‘??ลŸ0 ฤ‘?‘ฤ„ +

ฤ‘?‘“2 (ฤ‘?‘ , ฤ‘?‘ฤฝ ฤ‘?‘ , ฤ‘?‘ฤฝ0 , ฤ‘?‘ลš0 , ฤ‘?‘ลš ฤ‘?‘ , ฤ‘?‘ฤฝ0 , ฤ‘?‘ลš0 , 0

ฤ‘?‘

, ฤ‘?‘? ฤ‘?‘

,

ฤ‘??ลŸ2 ฤ‘?‘ , ฤ‘?œ? ฤ‘?‘”2 (ฤ‘?œ?, ฤ‘?‘ฤฝ ฤ‘?œ?, ฤ‘?‘ฤฝ0 , ฤ‘?‘ลš0 , ฤ‘?‘ลš ฤ‘?œ?, ฤ‘?‘ฤฝ0 , ฤ‘?‘ลš0 ) ฤ‘?‘‘ฤ‘?œ?, รขˆ’รขˆž

ฤ‘?‘”2 (ฤ‘?œ?, ฤ‘?‘ฤฝ ฤ‘?œ?, ฤ‘?‘ฤฝ0 , ฤ‘?‘ลš0 , ฤ‘?‘ลš ฤ‘?œ?, ฤ‘?‘ฤฝ0 , ฤ‘?‘ลš0 )ฤ‘?‘‘ฤ‘?œ?)ฤ‘?‘‘ฤ‘?‘

รข‹ลป ( I2 )

ฤ‘?‘Ž ฤ‘?‘

where ฤ‘?‘ฤฝ รขˆˆ ฤ‘??ห‡ รขŠ‚ ฤ‘?‘…ฤ‘?‘› , ฤ‘??ห‡ is closed and bounded domain subset of Euclidean space ฤ‘?‘…ฤ‘?‘› . Let the vector functions ฤ‘?‘“1 ฤ‘?‘ฤ„, ฤ‘?‘ฤฝ, ฤ‘?‘ลš, ฤ‘?‘ยง, ฤ‘?‘ยค = ฤ‘?‘“11 ฤ‘?‘ฤ„, ฤ‘?‘ฤฝ, ฤ‘?‘ลš, ฤ‘?‘ยง, ฤ‘?‘ยค , ฤ‘?‘“12 ฤ‘?‘ฤ„, ฤ‘?‘ฤฝ, ฤ‘?‘ลš, ฤ‘?‘ยง, ฤ‘?‘ยค , รข€ลš , ฤ‘?‘“1ฤ‘?‘› ฤ‘?‘ฤ„, ฤ‘?‘ฤฝ, ฤ‘?‘ลš, ฤ‘?‘ยง, ฤ‘?‘ยค , ฤ‘?‘“2 ฤ‘?‘ฤ„, ฤ‘?‘ฤฝ, ฤ‘?‘ลš, ฤ‘?‘ห˜, ฤ‘?‘ล = ฤ‘?‘“21 ฤ‘?‘ฤ„, ฤ‘?‘ฤฝ, ฤ‘?‘ลš, ฤ‘?‘ห˜, ฤ‘?‘ล , ฤ‘?‘“22 ฤ‘?‘ฤ„, ฤ‘?‘ฤฝ, ฤ‘?‘ลš, ฤ‘?‘ห˜, ฤ‘?‘ล , รข€ลš , ฤ‘?‘“2ฤ‘?‘› ฤ‘?‘ฤ„, ฤ‘?‘ฤฝ, ฤ‘?‘ลš, ฤ‘?‘ห˜, ฤ‘?‘ล , ฤ‘?‘”1 ฤ‘?‘ฤ„, ฤ‘?‘ฤฝ, ฤ‘?‘ลš = ฤ‘?‘”11 ฤ‘?‘ฤ„, ฤ‘?‘ฤฝ, ฤ‘?‘ลš , ฤ‘?‘”12 ฤ‘?‘ฤ„, ฤ‘?‘ฤฝ, ฤ‘?‘ลš , รข€ลš , ฤ‘?‘”1ฤ‘?‘› ฤ‘?‘ฤ„, ฤ‘?‘ฤฝ, ฤ‘?‘ลš , ฤ‘?‘”2 ฤ‘?‘ฤ„, ฤ‘?‘ฤฝ, ฤ‘?‘ลš = ฤ‘?‘”21 ฤ‘?‘ฤ„, ฤ‘?‘ฤฝ, ฤ‘?‘ลš , ฤ‘?‘”22 ฤ‘?‘ฤ„, ฤ‘?‘ฤฝ, ฤ‘?‘ลš , รข€ลš , ฤ‘?‘”2ฤ‘?‘› ฤ‘?‘ฤ„, ฤ‘?‘ฤฝ, ฤ‘?‘ลš , ฤ‘??ลก0 ฤ‘?‘ฤ„ = (ฤ‘??ลก01 ฤ‘?‘ฤ„ , ฤ‘??ลก02 ฤ‘?‘ฤ„ , รข€ลš , ฤ‘??ลก0ฤ‘?‘› ฤ‘?‘ฤ„ ), and ฤ‘??ลŸ0 ฤ‘?‘ฤ„ = (ฤ‘??ลŸ01 ฤ‘?‘ฤ„ , ฤ‘??ลŸ02 ฤ‘?‘ฤ„ , รข‹ลป , ฤ‘??ลŸ0ฤ‘?‘› ฤ‘?‘ฤ„ ) are defined and continuous on the domains: ฤ‘?‘ฤ„, ฤ‘?‘ฤฝ, ฤ‘?‘ลš, ฤ‘?‘ยง, ฤ‘?‘ยค รขˆˆ ฤ‘?‘…1 ฤ‚— ฤ‘??ห‡ ฤ‚— ฤ‘??ห‡1 ฤ‚— ฤ‘??ห‡2 = รขˆ’รขˆž, รขˆž ฤ‚— ฤ‘??ห‡ ฤ‚— ฤ‘??ห‡1 ฤ‚— ฤ‘??ห‡2 ฤ‚— ฤ‘??ห‡3 รข‹ลป 1.1 ฤ‘?‘ฤ„, ฤ‘?‘ฤฝ, ฤ‘?‘ลš, ฤ‘?‘ห˜, ฤ‘?‘ล รขˆˆ ฤ‘?‘…1 ฤ‚— ฤ‘??ห‡ ฤ‚— ฤ‘??ห‡1 ฤ‚— ฤ‘??ห‡2 = รขˆ’รขˆž, รขˆž ฤ‚— ฤ‘??ห‡ ฤ‚— ฤ‘??ห‡1 ฤ‚— ฤ‘??ห‡2 ฤ‚— ฤ‘??ห‡3 and periodic in t of period T, Also a(t) and b(t) are continuous and periodic in t of period ฤ‘?‘‡. Suppose that the functions ฤ‘?‘“1 ฤ‘?‘ฤ„, ฤ‘?‘ฤฝ, ฤ‘?‘ลš, ฤ‘?‘ยง, ฤ‘?‘ยค , ฤ‘?‘“2 ฤ‘?‘ฤ„, ฤ‘?‘ฤฝ, ฤ‘?‘ลš, ฤ‘?‘ห˜, ฤ‘?‘ล , ฤ‘?‘”1 ฤ‘?‘ฤ„, ฤ‘?‘ฤฝ, ฤ‘?‘ลš and ฤ‘?‘”2 ฤ‘?‘ฤ„, ฤ‘?‘ฤฝ, ฤ‘?‘ลš satisfies the following inequalities: www.ijeijournal.com

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Periodic Solutions for Non-Linear Systems of Integral Equations ฤ‘?‘“1 ฤ‘?‘ฤ„, ฤ‘?‘ฤฝ, ฤ‘?‘ลš, ฤ‘?‘ยง, ฤ‘?‘ยค รข‰ยค ฤ‘?‘€1 , ฤ‘?‘“2 ฤ‘?‘ฤ„, ฤ‘?‘ฤฝ, ฤ‘?‘ลš, ฤ‘?‘ห˜, ฤ‘?‘ล รข‰ยค ฤ‘?‘€2 , ฤ‘?‘€1 , ฤ‘?‘€2 > 0 รข‹ลป 1.2 ฤ‘?‘”1 ฤ‘?‘ฤ„, ฤ‘?‘ฤฝ, ฤ‘?‘ลš รข‰ยค ฤ‘?‘ 1 , ฤ‘?‘”1 ฤ‘?‘ฤ„, ฤ‘?‘ฤฝ, ฤ‘?‘ลš รข‰ยค ฤ‘?‘ 1 , ฤ‘?‘ 1 , ฤ‘?‘ 2 > 0 ฤ‘?‘“1 ฤ‘?‘ฤ„, ฤ‘?‘ฤฝ1 , ฤ‘?‘ลš1 , ฤ‘?‘ยง1 , ฤ‘?‘ยค1 รขˆ’ ฤ‘?‘“1 ฤ‘?‘ฤ„, ฤ‘?‘ฤฝ2 , ฤ‘?‘ลš2 , ฤ‘?‘ยง2 , ฤ‘?‘ยค2 รข‰ยค ฤ‘??ลพ1 ฤ‘?‘ฤฝ1 รขˆ’ ฤ‘?‘ฤฝ2 + ฤ‘??ลพ2 ฤ‘?‘ลš1 รขˆ’ฤ‘?‘ลš2 + +ฤ‘??ลพ3 ฤ‘?‘ยง1 รขˆ’ฤ‘?‘ยง2 + ฤ‘??ลพ4 ฤ‘?‘ยค1 รขˆ’ฤ‘?‘ยค2 รข‹ลป 1.3 ฤ‘?‘“2 ฤ‘?‘ฤ„, ฤ‘?‘ฤฝ1 , ฤ‘?‘ลš1 , ฤ‘?‘ห˜1 , ฤ‘?‘ล1 รขˆ’ ฤ‘?‘“2 ฤ‘?‘ฤ„, ฤ‘?‘ฤฝ2 , ฤ‘?‘ลš2 , ฤ‘?‘ห˜2 , ฤ‘?‘ล2 รข‰ยค ฤ‘??ลผ1 ฤ‘?‘ฤฝ1 รขˆ’ ฤ‘?‘ฤฝ2 + ฤ‘??ลผ2 ฤ‘?‘ลš1 รขˆ’ฤ‘?‘ลš2 + +ฤ‘??ลผ3 ฤ‘?‘ห˜1 รขˆ’ฤ‘?‘ห˜2 + ฤ‘??ลผ4 ฤ‘?‘ล1 รขˆ’ฤ‘?‘ล2 , รข‹ลป 1.4 ฤ‘?‘”1 ฤ‘?‘ฤ„, ฤ‘?‘ฤฝ1 , ฤ‘?‘ลš1 รขˆ’ ฤ‘?‘”1 ฤ‘?‘ฤ„, ฤ‘?‘ฤฝ2 , ฤ‘?‘ลš2 รข‰ยค ฤ‘?‘…1 ฤ‘?‘ฤฝ1 รขˆ’ ฤ‘?‘ฤฝ2 + ฤ‘?‘…2 ฤ‘?‘ลš1 รขˆ’ ฤ‘?‘ลš2 , รข‹ลป 1.5 ฤ‘?‘”2 ฤ‘?‘ฤ„, ฤ‘?‘ฤฝ1 , ฤ‘?‘ลš1 รขˆ’ ฤ‘?‘”2 ฤ‘?‘ฤ„, ฤ‘?‘ฤฝ2 , ฤ‘?‘ลš2 รข‰ยค ฤ‘??ลฅ1 ฤ‘?‘ฤฝ1 รขˆ’ ฤ‘?‘ฤฝ2 + ฤ‘??ลฅ2 ฤ‘?‘ลš1 รขˆ’ ฤ‘?‘ลš2 , รข‹ลป 1.6 for all ฤ‘?‘ฤ„ รขˆˆ ฤ‘?‘…1 , ฤ‘?‘ฤฝ, ฤ‘?‘ฤฝ1 , ฤ‘?‘ฤฝ2 รขˆˆ ฤ‘??ห‡ , ฤ‘?‘ลš , ฤ‘?‘ลš1 , ฤ‘?‘ลš2 รขˆˆ ฤ‘??ห‡1 , ฤ‘?‘ยง, ฤ‘?‘ยง1 , ฤ‘?‘ยง2 , ฤ‘?‘ห˜, ฤ‘?‘ห˜1 , ฤ‘?‘ห˜2 รขˆˆ ฤ‘??ห‡2 , and ฤ‘?‘ยค, ฤ‘?‘ยค1, ฤ‘?‘ยค2 , ฤ‘?‘ล, ฤ‘?‘ล1 , ฤ‘?‘ล2 รขˆˆ ฤ‘??ห‡3 , where ฤ‘?‘€1 = ฤ‘?‘€11 , ฤ‘?‘€12 , รข€ลš , ฤ‘?‘€1ฤ‘?‘› , ฤ‘?‘€2 = ฤ‘?‘€21 , ฤ‘?‘€22 , รข€ลš , ฤ‘?‘€2ฤ‘?‘› , ฤ‘?‘ 1 = (ฤ‘?‘ 11 , ฤ‘?‘ 12 , รข€ลš , ฤ‘?‘ 1ฤ‘?‘› ) and ฤ‘?‘ 2 = ฤ‘?‘ 21 , ฤ‘?‘ 22 , รข€ลš , ฤ‘?‘ 2ฤ‘?‘› are positive constant vectors and ฤ‘??ลพ1 = (ฤ‘??ลพ1 ฤ‘?‘–ฤ‘?‘— ), ฤ‘??ลพ2 = (ฤ‘??ลพ2 ฤ‘?‘–ฤ‘?‘— ), ฤ‘??ลพ3 = (ฤ‘??ลพ3 ฤ‘?‘–ฤ‘?‘— ), ฤ‘??ลพ4 = (ฤ‘??ลพ4 ฤ‘?‘–ฤ‘?‘— ), ฤ‘??ลผ1 = (ฤ‘??ลผ1 ฤ‘?‘–ฤ‘?‘— ), ฤ‘??ลผ2 = (ฤ‘??ลผ2 ฤ‘?‘–ฤ‘?‘— ), ฤ‘??ลผ3 = ( ฤ‘??ลผ3 ฤ‘?‘–ฤ‘?‘— ), ฤ‘??ลผ4 = (ฤ‘??ลผ4 ฤ‘?‘–ฤ‘?‘— ), ฤ‘?‘…1 = (ฤ‘?‘…1 ฤ‘?‘–ฤ‘?‘— ), ฤ‘?‘…2 = (ฤ‘?‘…2 ฤ‘?‘–ฤ‘?‘— ), ฤ‘??ลฅ1 = (ฤ‘??ลฅ1 ฤ‘?‘–ฤ‘?‘— ), and ฤ‘??ลฅ2 = (ฤ‘??ลฅ2 ฤ‘?‘–ฤ‘?‘— ) are positive constant matrices. Also ฤ‘??ลŸ1 ฤ‘?‘ฤ„, ฤ‘?‘ and ฤ‘??ลŸ2 ฤ‘?‘ฤ„, ฤ‘?‘ are (nฤ‚—n) continuous positive matrix and periodic in t, s of period T in the domain ฤ‘?‘…1 ฤ‚— ฤ‘?‘…1 and satisfy the following conditions: ฤ‘??ลŸ1 ฤ‘?‘ฤ„, ฤ‘?‘ รข‰ยค ฤ‘?›ลพ ฤ‘?‘’ รขˆ’ฤ‘?œ† 1 ฤ‘?‘ฤ„รขˆ’ฤ‘?‘ , ฤ‘?›ลพ, ฤ‘?œ†1 > 0 รข‹ลป 1.7 and ฤ‘??ลŸ2 (ฤ‘?‘ฤ„, ฤ‘?‘ ) รข‰ยค ฤ‘?›ลผ ฤ‘?‘’ รขˆ’ฤ‘?œ† 2 ฤ‘?‘ฤ„รขˆ’ฤ‘?‘ , ฤ‘?›ลผ, ฤ‘?œ†2 > 0 รข‹ลป 1.8 where รขˆ’รขˆž < 0 รข‰ยค ฤ‘?‘ รข‰ยค ฤ‘?‘ฤ„ รข‰ยค ฤ‘?‘‡ < รขˆž , ฤ‘?‘–, ฤ‘?‘— = 1,2, รข‹ลป , ฤ‘?‘›, and ฤ‘?‘• = ฤ‘?‘šฤ‘?‘Žฤ‘?‘ฤฝฤ‘?‘ฤ„รขˆˆ 0,ฤ‘?‘‡ ฤ‘?‘? ฤ‘?‘ฤ„ รขˆ’ ฤ‘?‘Ž ฤ‘?‘ฤ„ , . = ฤ‘?‘šฤ‘?‘Žฤ‘?‘ฤฝฤ‘?‘ฤ„รขˆˆ 0,ฤ‘?‘‡ .

Now define a non-empty sets as follows: ฤ‘?‘‡ ฤ‘??ห‡ฤ‘?‘“1 = ฤ‘??ห‡ รขˆ’ ฤ‘?‘€ 1 , 2 ฤ‘?‘‡ ฤ‘??ห‡1ฤ‘?‘“2 = ฤ‘??ห‡1 รขˆ’ ฤ‘?‘€ 2 , 2 ฤ‘?‘‡ ฤ‘?›ลพ ฤ‘??ห‡2ฤ‘?‘“1 = ฤ‘??ห‡2 รขˆ’ ฤ‘?‘…1 ฤ‘?‘€1 + ฤ‘?‘…2 ฤ‘?‘€2 , 2 ฤ‘?œ†1 รข‹ลป 1.9 ฤ‘?‘‡ ฤ‘??ห‡3ฤ‘?‘“1 = ฤ‘??ห‡3 รขˆ’ ฤ‘?‘• ฤ‘?‘…1 ฤ‘?‘€1 + ฤ‘?‘…2 ฤ‘?‘€2 , 2 ฤ‘?‘‡ ฤ‘?›ลผ ฤ‘??ห‡2ฤ‘?‘“2 = ฤ‘??ห‡2 รขˆ’ ฤ‘??ลฅ1 ฤ‘?‘€1 + ฤ‘??ลฅ2 ฤ‘?‘€2 , 2 ฤ‘?œ†2 ฤ‘?‘‡ ฤ‘??ห‡3ฤ‘?‘“2 = ฤ‘??ห‡3 รขˆ’ ฤ‘?‘• ฤ‘??ลฅ1 ฤ‘?‘€1 + ฤ‘??ลฅ2 ฤ‘?‘€2 . 2 Forever, we suppose that the greatest eigen-value ฤ‘?‘žฤ‘?‘šฤ‘?‘Žฤ‘?‘ฤฝ of the following matrix: ฤ‘?‘‡ ฤ‘?›ลพ ฤ‘?‘‡ ฤ‘?›ลพ [ฤ‘??ลพ + ฤ‘?‘…1 ( ฤ‘??ลพ3 + ฤ‘?‘•ฤ‘??ลพ4 )] [ฤ‘??ลพ + ฤ‘?‘…2 ( ฤ‘??ลพ3 + ฤ‘?‘•ฤ‘??ลพ4 )] 2 1 ฤ‘?œ†1 2 2 ฤ‘?œ†1 Q0 = ฤ‘?‘‡ ฤ‘?›ลผ ฤ‘?‘‡ ฤ‘?›ลผ [ฤ‘??ลผ1 + ฤ‘??ลฅ1 (ฤ‘??ลผ3 + ฤ‘?‘•ฤ‘??ลผ4 )] [ฤ‘??ลผ2 + ฤ‘??ลฅ2 ( ฤ‘??ลผ3 + ฤ‘?‘•ฤ‘??ลผ4 )] 2 ฤ‘?œ†2 2 ฤ‘?œ†2 is less than unity, i.e. ฤŽ†1 + ฤ‘?œ‘1 2 + 4(ฤ‘?œ‘2 รขˆ’ ฤ‘?œ‘3 ) ฤ‘?‘žฤ‘?‘šฤ‘?‘Žฤ‘?‘ฤฝ ฤ‘?‘„0 = <1 , 2 ฤ‘?‘‡ ฤ‘?›ลพ ฤ‘?‘‡ ฤ‘?›ลผ where ฤŽ†1 = [ฤ‘??ลพ1 + ฤ‘?‘…1 ( ฤ‘??ลพ3 + ฤ‘?‘•ฤ‘??ลพ4 )] + [ฤ‘??ลผ2 + ฤ‘??ลฅ2 ( ฤ‘??ลผ3 + ฤ‘?‘•ฤ‘??ลผ4 )], 2

ฤ‘?‘‡

ฤ‘?›ลพ

2 ฤ‘?‘‡ 2

ฤ‘?œ†1

2

ฤ‘?‘‡

ฤ‘?›ลผ

ฤ‘?œ†1 ฤ‘?›ลพ

2 ฤ‘?‘‡

ฤ‘?œ†2 ฤ‘?›ลผ

ฤ‘?œ†1

2

ฤ‘?œ†2

รข‹ลป 1.10

รข‹ลป 1.11

ฤ‘?œ†2

ฤŽ†2 = ( [ฤ‘??ลพ2 + ฤ‘?‘…2 ( ฤ‘??ลพ3 + ฤ‘?‘•ฤ‘??ลพ4 )])( [ฤ‘??ลผ1 + ฤ‘??ลฅ1 ( ฤ‘??ลผ3 + ฤ‘?‘•ฤ‘??ลผ4 )]) and ฤ‘?œ‘3 = ( [ฤ‘??ลพ1 + ฤ‘?‘…1 ( ฤ‘??ลพ3 + ฤ‘?‘•ฤ‘??ลพ4 )])( [ฤ‘??ลผ2 + ฤ‘??ลฅ2 ( ฤ‘??ลผ3 + ฤ‘?‘•ฤ‘??ลผ4 )]). By using lemma 3.1[10],we can state and prove the following lemma: Lemma 1.1. Let ฤ‘?‘“1 (ฤ‘?‘ฤ„, ฤ‘?‘ฤฝ, ฤ‘?‘ลš, ฤ‘?‘ยง, ฤ‘?‘ยค) and ฤ‘?‘“2 (ฤ‘?‘ฤ„, ฤ‘?‘ฤฝ, ฤ‘?‘ลš, ฤ‘?‘ห˜, ฤ‘?‘ล) be a vector functions which are defined and continuous in the interval 0, ฤ‘?‘‡ , then the inequality: www.ijeijournal.com

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Periodic Solutions for Non-Linear Systems of Integral Equations ฤ‘??ลผ1 ฤ‘?‘ฤ„, ฤ‘?‘ฤฝ0 , ฤ‘?‘ลš0 ฤ‘?‘€1 ฤ‘?›ลบ ฤ‘?‘ฤ„ รข‰ยค ฤ‘??ลผ2 (ฤ‘?‘ฤ„, ฤ‘?‘ฤฝ0 , ฤ‘?‘ลš0 ) ฤ‘?‘€2 ฤ‘?›ลบ ฤ‘?‘ฤ„ ฤ‘?‘‡ satisfies for 0 รข‰ยค ฤ‘?‘ฤ„ รข‰ยค ฤ‘?‘‡ and ฤ‘?›ลบ ฤ‘?‘ฤ„ รข‰ยค ,

รข‹ลป 1.12

2

ฤ‘?‘ฤ„

where ฤ‘?›ลบ ฤ‘?‘ฤ„ = 2ฤ‘?‘ฤ„(1 รขˆ’ ) for all ฤ‘?‘ฤ„ รขˆˆ 0, ฤ‘?‘‡ , ฤ‘?‘ฤ„

ฤ‘??ลผ1 ฤ‘?‘ฤ„, ฤ‘?‘ฤฝ0 , ฤ‘?‘ลš0 =

ฤ‘?‘‡

ฤ‘?‘

[ฤ‘?‘“1 (ฤ‘?‘ , ฤ‘?‘ฤฝ0 , ฤ‘?‘ลš0 , 0

ฤ‘??ลŸ1 ฤ‘?‘ , ฤ‘?œ? ฤ‘?‘”1 (ฤ‘?œ?, ฤ‘?‘ฤฝ0 , ฤ‘?‘ลš0 ) ฤ‘?‘‘ฤ‘?œ?, รขˆ’รขˆž ฤ‘?‘? ฤ‘?‘

, ฤ‘?‘Ž ฤ‘?‘ ฤ‘?‘

,

ฤ‘?‘‡

ฤ‘?‘“1 (ฤ‘?‘ , ฤ‘?‘ฤฝ0 , ฤ‘?‘ลš0 , 0 ฤ‘?‘? ฤ‘?‘

ฤ‘??ลŸ1 ฤ‘?‘ , ฤ‘?œ? ฤ‘?‘”1 (ฤ‘?œ?, ฤ‘?‘ฤฝ0 , ฤ‘?‘ลš0 ) ฤ‘?‘‘ฤ‘?œ?, รขˆ’รขˆž ฤ‘?‘

ฤ‘?‘ฤ„

ฤ‘??ลผ2 ฤ‘?‘ฤ„, ฤ‘?‘ฤฝ0 , ฤ‘?‘ลš0 =

1 ฤ‘?‘”1 (ฤ‘?œ?, ฤ‘?‘ฤฝ0 , ฤ‘?‘ลš0 )ฤ‘?‘‘ฤ‘?œ?) รขˆ’ ฤ‘?‘‡

[ฤ‘?‘“2 (ฤ‘?‘ , ฤ‘?‘ฤฝ0 , ฤ‘?‘ลš0 , 0

ฤ‘?‘”1 (ฤ‘?œ?, ฤ‘?‘ฤฝ0 , ฤ‘?‘ลš0 )ฤ‘?‘‘ฤ‘?œ? )ฤ‘?‘‘ฤ‘?‘ ]ฤ‘?‘‘ฤ‘?‘ ฤ‘?‘Ž ฤ‘?‘

ฤ‘??ลŸ2 ฤ‘?‘ , ฤ‘?œ? ฤ‘?‘”2 (ฤ‘?œ?, ฤ‘?‘ฤฝ0 , ฤ‘?‘ลš0 ) ฤ‘?‘‘ฤ‘?œ?, รขˆ’รขˆž ฤ‘?‘? ฤ‘?‘

, ฤ‘?‘Ž ฤ‘?‘ ฤ‘?‘

,

1 ฤ‘?‘”2 (ฤ‘?œ?, ฤ‘?‘ฤฝ0 , ฤ‘?‘ลš0 )ฤ‘?‘‘ฤ‘?œ?) รขˆ’ ฤ‘?‘‡

ฤ‘?‘‡

ฤ‘?‘“2 (ฤ‘?‘ , ฤ‘?‘ฤฝ0 , ฤ‘?‘ลš0 , 0 ฤ‘?‘? ฤ‘?‘

ฤ‘??ลŸ2 ฤ‘?‘ , ฤ‘?œ? ฤ‘?‘”2 (ฤ‘?œ?, ฤ‘?‘ฤฝ0 , ฤ‘?‘ลš0 ) ฤ‘?‘‘ฤ‘?œ?, รขˆ’รขˆž

ฤ‘?‘”2 (ฤ‘?œ?, ฤ‘?‘ฤฝ0 , ฤ‘?‘ลš0 )ฤ‘?‘‘ฤ‘?œ?)ฤ‘?‘‘ฤ‘?‘ ]ฤ‘?‘‘ฤ‘?‘ ฤ‘?‘Ž ฤ‘?‘

Proof. ฤ‘?‘ฤ„ รข‰ยค (1 รขˆ’ ) ฤ‘?‘‡

ฤ‘??ลผ1 ฤ‘?‘ฤ„, ฤ‘?‘ฤฝ0 , ฤ‘?‘ลš0

ฤ‘?‘ฤ„

ฤ‘?‘

ฤ‘?‘“1 (ฤ‘?‘ , ฤ‘?‘ฤฝ0 , ฤ‘?‘ลš0 , 0

ฤ‘??ลŸ1 ฤ‘?‘ , ฤ‘?œ? ฤ‘?‘”1 ฤ‘?œ?, ฤ‘?‘ฤฝ0 , ฤ‘?‘ลš0 ฤ‘?‘‘ฤ‘?œ?, รขˆ’รขˆž

ฤ‘?‘?(ฤ‘?‘ )

,

ฤ‘?‘”1 ฤ‘?œ?, ฤ‘?‘ฤฝ0 , ฤ‘?‘ลš0 ฤ‘?‘‘ฤ‘?œ?) ฤ‘?‘‘ฤ‘?‘ + ฤ‘?‘Ž(ฤ‘?‘ )

ฤ‘?‘ฤ„ + ฤ‘?‘‡ ฤ‘?‘ฤ„ รข‰ยค (1 รขˆ’ ) ฤ‘?‘‡

ฤ‘?‘“1 (ฤ‘?‘ , ฤ‘?‘ฤฝ0 , ฤ‘?‘ลš0 , ฤ‘?‘ฤ„

ฤ‘?‘ฤ„

ฤ‘?‘?(ฤ‘?‘ )

ฤ‘?‘

ฤ‘?‘‡

ฤ‘??ลŸ1 ฤ‘?‘ , ฤ‘?œ? ฤ‘?‘”1 ฤ‘?œ?, ฤ‘?‘ฤฝ0 , ฤ‘?‘ลš0 ฤ‘?‘‘ฤ‘?œ?, รขˆ’รขˆž

ฤ‘?‘”1 ฤ‘?œ?, ฤ‘?‘ฤฝ0 , ฤ‘?‘ลš0 ฤ‘?‘‘ฤ‘?œ?) ฤ‘?‘‘ฤ‘?‘ ฤ‘?‘Ž(ฤ‘?‘ )

ฤ‘?‘‡

ฤ‘?‘ฤ„ ฤ‘?‘€1 ฤ‘?‘‘ฤ‘?‘ + ฤ‘?‘€1 ฤ‘?‘‘ฤ‘?‘ ฤ‘?‘‡ 0 ฤ‘?‘ฤ„ ฤ‘?‘ฤ„ ฤ‘?‘ฤ„ รข‰ยค ฤ‘?‘€1 [(1 รขˆ’ )ฤ‘?‘ฤ„ + (ฤ‘?‘‡ รขˆ’ ฤ‘?‘ฤ„)] ฤ‘?‘‡ ฤ‘?‘‡ so that: ฤ‘??ลผ1 ฤ‘?‘ฤ„, ฤ‘?‘ฤฝ0 , ฤ‘?‘ลš0 รข‰ยค ฤ‘?‘€1 ฤ‘?›ลบ ฤ‘?‘ฤ„ And similarly, we get also

รข‹ลป 1.13

ฤ‘??ลผ2 ฤ‘?‘ฤ„, ฤ‘?‘ฤฝ0 , ฤ‘?‘ลš0 รข‰ยค ฤ‘?‘€2 ฤ‘?›ลบ ฤ‘?‘ฤ„ รข‹ลป 1.14 From ( 1.13) and ( 1.14), then the inequality ( 1.12) satisfies for all ฤ‘?‘‡ 0 รข‰ยค ฤ‘?‘ฤ„ รข‰ยค ฤ‘?‘‡ and ฤ‘?›ลบ ฤ‘?‘ฤ„ รข‰ยค . รขˆŽ 2

II. Approximation Solution Of (I1) And (I2) In this section, we study the approximate periodic solution of (I1) and (I2) by proving the following theorem: Theorem 2.1. If the system (I1) and (I2) satisfies the inequalities ( 1.2), ( 1.3), 1.4), (1.5),(1.6) and conditions(1.7),(1.8) has periodic solutions ฤ‘?‘ฤฝ = ฤ‘?‘ฤฝ ฤ‘?‘ฤ„, ฤ‘?‘ฤฝ0 , ฤ‘?‘ลš0 and ฤ‘?‘ลš = ฤ‘?‘ลš(ฤ‘?‘ฤ„, ฤ‘?‘ฤฝ0 , ฤ‘?‘ลš0 ), then the sequence of functions: ฤ‘?‘ฤ„

ฤ‘?‘ฤฝฤ‘?‘š +1 ฤ‘?‘ฤ„, ฤ‘?‘ฤฝ0 , ฤ‘?‘ลš0 = ฤ‘??ลก0 ฤ‘?‘ฤ„ +

[ฤ‘?‘“1 (ฤ‘?‘ , ฤ‘?‘ฤฝฤ‘?‘š ฤ‘?‘ , ฤ‘?‘ฤฝ0 , ฤ‘?‘ลš0 , ฤ‘?‘ลšฤ‘?‘š ฤ‘?‘ , ฤ‘?‘ฤฝ0 , ฤ‘?‘ลš0 , 0

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Periodic Solutions for Non-Linear Systems of Integral Equations ๐‘ 

,

๐บ1 ๐‘ , ๐œ ๐‘”1 (๐œ, ๐‘ฅ๐‘š ๐œ, ๐‘ฅ0 , ๐‘ฆ0 , ๐‘ฆ๐‘š ๐œ, ๐‘ฅ0 , ๐‘ฆ0 ) ๐‘‘๐œ, โˆ’โˆž

๐‘ ๐‘ 

, ๐‘ 

, ๐‘ฆ๐‘š ๐‘ , ๐‘ฅ0 , ๐‘ฆ0 ,

๐‘”1 (๐œ, ๐‘ฅ๐‘š ๐œ, ๐‘ฅ0 , ๐‘ฆ0 , ๐‘ฆ๐‘š ๐‘Ž ๐‘ 

1 ๐œ, ๐‘ฅ0 , ๐‘ฆ0 )๐‘‘๐œ) โˆ’ ๐‘‡

๐‘‡

๐‘“1 (๐‘ , ๐‘ฅ๐‘š ๐‘ , ๐‘ฅ0 , ๐‘ฆ0 , 0

๐บ1 ๐‘ , ๐œ ๐‘”1 (๐œ, ๐‘ฅ๐‘š ๐œ, ๐‘ฅ0 , ๐‘ฆ0 , ๐‘ฆ๐‘š ๐œ, ๐‘ฅ0 , ๐‘ฆ0 ) ๐‘‘๐œ, โˆ’โˆž

๐‘ ๐‘ 

,

๐‘”1 (๐œ, ๐‘ฅ๐‘š ๐œ, ๐‘ฅ0 , ๐‘ฆ0 , ๐‘ฆ๐‘š ๐œ, ๐‘ฅ0 , ๐‘ฆ0 )๐‘‘๐œ)๐‘‘๐‘ ]๐‘‘๐‘ 

โ‹ฏ 2.1

๐‘Ž ๐‘ 

with ๐‘ฅ0 ๐‘ก, ๐‘ฅ0 , ๐‘ฆ0 = ๐น0 ๐‘ก = ๐‘ฅ0

,

for all m = 0, 1,2, โ‹ฏ

๐‘ก

๐‘ฆ๐‘š +1 ๐‘ก, ๐‘ฅ0 , ๐‘ฆ0 = ๐บ0 ๐‘ก +

[๐‘“2 (๐‘ , ๐‘ฅ๐‘š ๐‘ , ๐‘ฅ0 , ๐‘ฆ0 , ๐‘ฆ๐‘š ๐‘ , ๐‘ฅ0 , ๐‘ฆ0 , 0

๐‘ 

,

๐บ2 ๐‘ , ๐œ ๐‘”2 (๐œ, ๐‘ฅ๐‘š ๐œ, ๐‘ฅ0 , ๐‘ฆ0 , ๐‘ฆ๐‘š ๐œ, ๐‘ฅ0 , ๐‘ฆ0 ) ๐‘‘๐œ, โˆ’โˆž

๐‘ ๐‘ 

, ๐‘ 

๐‘”2 (๐œ, ๐‘ฅ๐‘š ๐œ, ๐‘ฅ0 , ๐‘ฆ0 , ๐‘ฆ๐‘š ๐‘Ž ๐‘ 

, ๐‘ฆ๐‘š ๐‘ , ๐‘ฅ0 , ๐‘ฆ0 ,

1 ๐œ, ๐‘ฅ0 , ๐‘ฆ0 )๐‘‘๐œ) โˆ’ ๐‘‡

๐‘‡

๐‘“2 (๐‘ , ๐‘ฅ๐‘š ๐‘ , ๐‘ฅ0 , ๐‘ฆ0 , 0

๐บ2 ๐‘ , ๐œ ๐‘”2 (๐œ, ๐‘ฅ๐‘š ๐œ, ๐‘ฅ0 , ๐‘ฆ0 , ๐‘ฆ๐‘š ๐œ, ๐‘ฅ0 , ๐‘ฆ0 ) ๐‘‘๐œ, โˆ’โˆž

๐‘ ๐‘ 

,

๐‘”2 (๐œ, ๐‘ฅ๐‘š ๐œ, ๐‘ฅ0 , ๐‘ฆ0 , ๐‘ฆ๐‘š ๐œ, ๐‘ฅ0 , ๐‘ฆ0 )๐‘‘๐œ)๐‘‘๐‘ ]๐‘‘๐‘ 

โ‹ฏ 2.2

๐‘Ž ๐‘ 

with ๐‘ฆ0 ๐‘ก, ๐‘ฅ0 , ๐‘ฆ0 = ๐บ0 ๐‘ก = ๐‘ฆ0 , for all m = 0, 1,2, โ‹ฏ periodic in t of period T, and uniformly converges as ๐‘š โ†’ โˆž in the domain: ๐‘ก, ๐‘ฅ0 , ๐‘ฆ0 โˆˆ 0, ๐‘‡ ร— ๐ท๐‘“1 ร— ๐ท1๐‘“2 โ‹ฏ 2.3 to the limit functions ๐‘ฅ 0 ๐‘ก, ๐‘ฅ0 , ๐‘ฆ0 , ๐‘ฆ 0 ๐‘ก, ๐‘ฅ0 , ๐‘ฆ0 defined in the domain ( 2.3) which are periodic in t of period T and satisfying the system of integral equations: ๐‘ก

๐‘ฅ ๐‘ก, ๐‘ฅ0 , ๐‘ฆ0 = ๐น0 ๐‘ก +

[๐‘“1 (๐‘ , ๐‘ฅ ๐‘ , ๐‘ฅ0 , ๐‘ฆ0 , ๐‘ฆ ๐‘ , ๐‘ฅ0 , ๐‘ฆ0 , 0

๐‘ 

,

๐บ1 ๐‘ , ๐œ ๐‘”1 (๐œ, ๐‘ฅ ๐œ, ๐‘ฅ0 , ๐‘ฆ0 , ๐‘ฆ ๐œ, ๐‘ฅ0 , ๐‘ฆ0 ) ๐‘‘๐œ, โˆ’โˆž

๐‘ ๐‘ 

, ๐‘ 

, ๐‘ฆ ๐‘ , ๐‘ฅ0 , ๐‘ฆ0 ,

๐‘Ž ๐‘ 

1 ๐‘”1 (๐œ, ๐‘ฅ ๐œ, ๐‘ฅ0 , ๐‘ฆ0 , ๐‘ฆ ๐œ, ๐‘ฅ0 , ๐‘ฆ0 )๐‘‘๐œ) โˆ’ ๐‘‡

๐‘‡

๐‘“1 (๐‘ , ๐‘ฅ ๐‘ , ๐‘ฅ0 , ๐‘ฆ0 , 0

๐บ1 ๐‘ , ๐œ ๐‘”1 (๐œ, ๐‘ฅ ๐œ, ๐‘ฅ0 , ๐‘ฆ0 , ๐‘ฆ ๐œ, ๐‘ฅ0 , ๐‘ฆ0 ) ๐‘‘๐œ, โˆ’โˆž

๐‘ ๐‘ 

,

๐‘”1 (๐œ, ๐‘ฅ ๐œ, ๐‘ฅ0 , ๐‘ฆ0 , ๐‘ฆ ๐œ, ๐‘ฅ0 , ๐‘ฆ0 )๐‘‘๐œ)๐‘‘๐‘ ]๐‘‘๐‘ 

โ‹ฏ 2.4

๐‘Ž ๐‘ 

and ๐‘ก

๐‘ฆ ๐‘ก, ๐‘ฅ0 , ๐‘ฆ0 = ๐บ0 ๐‘ก +

[๐‘“2 (๐‘ , ๐‘ฅ ๐‘ , ๐‘ฅ0 , ๐‘ฆ0 , ๐‘ฆ ๐‘ , ๐‘ฅ0 , ๐‘ฆ0 , 0

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Periodic Solutions for Non-Linear Systems of Integral Equations ๐‘ 

,

๐บ2 ๐‘ , ๐œ ๐‘”2 (๐œ, ๐‘ฅ ๐œ, ๐‘ฅ0 , ๐‘ฆ0 , ๐‘ฆ ๐œ, ๐‘ฅ0 , ๐‘ฆ0 ) ๐‘‘๐œ, โˆ’โˆž

๐‘ ๐‘ 

1 ๐‘”2 (๐œ, ๐‘ฅ ๐œ, ๐‘ฅ0 , ๐‘ฆ0 , ๐‘ฆ ๐œ, ๐‘ฅ0 , ๐‘ฆ0 )๐‘‘๐œ) โˆ’ ๐‘‡

, ๐‘ 

, ๐‘ฆ ๐‘ , ๐‘ฅ0 , ๐‘ฆ0 , ,

๐‘‡

๐ถ0 =

2 ๐‘‡ 2

๐‘€1 ๐‘€2

๐‘“2 (๐‘ , ๐‘ฅ ๐‘ , ๐‘ฅ0 , ๐‘ฆ0 , 0

๐บ2 ๐‘ , ๐œ ๐‘”2 (๐œ, ๐‘ฅ ๐œ, ๐‘ฅ0 , ๐‘ฆ0 , ๐‘ฆ ๐œ, ๐‘ฅ0 , ๐‘ฆ0 ) ๐‘‘๐œ, โˆ’โˆž

provided that: ๐‘ฅ 0 ๐‘ก, ๐‘ฅ0 , ๐‘ฆ0 โˆ’ ๐‘ฅ๐‘š ๐‘ก, ๐‘ฅ0 , ๐‘ฆ0 ๐‘ฆ 0 ๐‘ก, ๐‘ฅ0 , ๐‘ฆ0 โˆ’ ๐‘ฆ๐‘š ๐‘ก, ๐‘ฅ0 , ๐‘ฆ0 for all ๐‘š โ‰ฅ 0 , where

๐‘Ž ๐‘ 

๐‘‡

๐‘ ๐‘  ๐‘Ž ๐‘ 

๐‘”2 (๐œ, ๐‘ฅ ๐œ, ๐‘ฅ0 , ๐‘ฆ0 , ๐‘ฆ ๐œ, ๐‘ฅ0 , ๐‘ฆ0 )๐‘‘๐œ)๐‘‘๐‘ ]๐‘‘๐‘ 

โ‰ค ๐‘„0 ๐‘š (๐ธ โˆ’ ๐‘„0 )โˆ’1 ๐ถ0

โ‹ฏ 2.5

โ‹ฏ 2.6

, ๐ธ is identity matrix.

Proof. Consider the sequence of functions ๐‘ฅ1 ๐‘ก, ๐‘ฅ0 , ๐‘ฆ0 , ๐‘ฅ2 ๐‘ก, ๐‘ฅ0 , ๐‘ฆ0 , โ‹ฏ, ๐‘ฅ๐‘š ๐‘ก, ๐‘ฅ0 , ๐‘ฆ0 , โ‹ฏ, and ๐‘ฆ1 ๐‘ก, ๐‘ฅ0 , ๐‘ฆ0 , ๐‘ฆ2 ๐‘ก, ๐‘ฅ0 , ๐‘ฆ0 , โ‹ฏ , ๐‘ฆ๐‘š ๐‘ก, ๐‘ฅ0 , ๐‘ฆ0 , โ‹ฏ, defined by the recurrences relations ( 2.1) and ( 2.2). Each of these functions of sequence defined and continuous in the domain ( 1.1) and periodic in ๐‘ก of period ๐‘‡. Now, by the lemma 1.1, and using the sequence of functions (2.1), when ๐‘š = 0, we get: By mathematical induction and lemma1.1 , we have the following inequality: ๐‘‡ ๐‘ฅ๐‘š ๐‘ก, ๐‘ฅ0 , ๐‘ฆ0 โˆ’ ๐‘ฅ0 โ‰ค ๐‘€ โ‹ฏ 2.13 2 1 i.e. ๐‘ฅ๐‘š ๐‘ก, ๐‘ฅ0 , ๐‘ฆ0 โˆˆ ๐ท, for all ๐‘ก โˆˆ ๐‘…1 , ๐‘ฅ0 โˆˆ ๐ท๐‘“1 , ๐‘ฆ0 โˆˆ ๐ท1๐‘“2 , ๐‘š = 0,1,2, โ‹ฏ, and ๐‘‡ ๐‘ฆ๐‘š ๐‘ก, ๐‘ฅ0 , ๐‘ฆ0 โˆ’ ๐‘ฆ0 โ‰ค ๐‘€ โ‹ฏ 2.14 2 2 1 i.e. ๐‘ฆ๐‘š ๐‘ก, ๐‘ฅ0 , ๐‘ฆ0 โˆˆ ๐ท, for all ๐‘ก โˆˆ ๐‘… , ๐‘ฅ0 โˆˆ ๐ท๐‘“1 , ๐‘ฆ0 โˆˆ ๐ท1๐‘“2 , ๐‘š = 0,1,2, โ‹ฏ, Now, from ( 2.13), we get: ๐‘‡ ๐›พ ๐‘ง๐‘š ๐‘ก, ๐‘ฅ0 , ๐‘ฆ0 โˆ’ ๐‘ง0 (๐‘ก, ๐‘ฅ0 , ๐‘ฆ0 ) โ‰ค ( ) ๐‘…1 ๐‘€1 + ๐‘…2 ๐‘€2 โ‹ฏ 2.15 2 ๐œ†1 and ๐‘‡ ๐‘ค๐‘š ๐‘ก, ๐‘ฅ0 , ๐‘ฆ0 โˆ’ ๐‘ค0 (๐‘ก, ๐‘ฅ0 , ๐‘ฆ0 ) โ‰ค ๐‘• ๐‘…1 ๐‘€1 + ๐‘…2 ๐‘€2 โ‹ฏ 2.16 2 1 for all ๐‘ก โˆˆ ๐‘… , ๐‘ฅ0 โˆˆ ๐ท๐‘“1 , ๐‘ฆ0 โˆˆ ๐ท1๐‘“2 , ๐‘ง0 ๐‘ก, ๐‘ฅ0 , ๐‘ฆ0 โˆˆ ๐ท2๐‘“1 and ๐‘ค0 ๐‘ก, ๐‘ฅ0 , ๐‘ฆ0 โˆˆ ๐ท3๐‘“1 , i.e. ๐‘ง๐‘š ๐‘ก, ๐‘ฅ0 , ๐‘ฆ0 โˆˆ ๐ท2 and ๐‘ค๐‘š ๐‘ก, ๐‘ฅ0 , ๐‘ฆ0 โˆˆ ๐ท3 , for all ๐‘ฅ0 โˆˆ ๐ท๐‘“1 , ๐‘ฆ0 โˆˆ ๐ท1๐‘“2 , where ๐‘ก

๐‘ง๐‘š ๐‘ก, ๐‘ฅ0 , ๐‘ฆ0 =

๐บ1 ๐‘ก, ๐‘  ๐‘”1 ๐‘ , ๐‘ฅ๐‘š ๐‘ , ๐‘ฅ0 , ๐‘ฆ0 , ๐‘ฆ๐‘š ๐‘ , ๐‘ฅ0 , ๐‘ฆ0 ๐‘‘๐‘  โˆ’โˆž

and ๐‘(๐‘ก)

๐‘ค๐‘š ๐‘ก, ๐‘ฅ0 , ๐‘ฆ0 =

๐‘”1 ๐‘ , ๐‘ฅ๐‘š ๐‘ , ๐‘ฅ0 , ๐‘ฆ0 , ๐‘ฆ๐‘š ๐‘ , ๐‘ฅ0 , ๐‘ฆ0 ๐‘‘๐‘  ,

๐‘š = 0,1,2, โ‹ฏ ,

๐‘Ž(๐‘ก)

Also from (4.2.14), we get: ๐‘ข๐‘š ๐‘ก, ๐‘ฅ0 , ๐‘ฆ0 โˆ’ ๐‘ข0 (๐‘ก, ๐‘ฅ0 , ๐‘ฆ0 ) โ‰ค

๐‘‡ ๐›ฟ ( ) ๐ป1 ๐‘€1 + ๐ป2 ๐‘€2 2 ๐œ†2

โ‹ฏ 2.17

๐‘‰๐‘š ๐‘ก, ๐‘ฅ0 , ๐‘ฆ0 โˆ’ ๐‘‰0 (๐‘ก, ๐‘ฅ0 , ๐‘ฆ0 ) โ‰ค

๐‘‡ ๐‘• ๐ป1 ๐‘€1 + ๐ป2 ๐‘€2 2

โ‹ฏ 2.18

and

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Periodic Solutions for Non-Linear Systems of Integral Equations for all ๐‘ก โˆˆ ๐‘…1 , ๐‘ฅ0 โˆˆ ๐ท๐‘“1 , ๐‘ฆ0 โˆˆ ๐ท1๐‘“2 , ๐‘ข0 ๐‘ก, ๐‘ฅ0 , ๐‘ฆ0 โˆˆ ๐ท2๐‘“2 and ๐‘ฃ0 ๐‘ก, ๐‘ฅ0 , ๐‘ฆ0 โˆˆ ๐ท3๐‘“2 , i.e. ๐‘ข๐‘š ๐‘ก, ๐‘ฅ0 , ๐‘ฆ0 โˆˆ ๐ท2 and ๐‘ฃ๐‘š ๐‘ก, ๐‘ฅ0 , ๐‘ฆ0 โˆˆ ๐ท3 , for all ๐‘ฅ0 โˆˆ ๐ท๐‘“1 , ๐‘ฆ0 โˆˆ ๐ท1๐‘“2 , where ๐‘ก

๐‘ข๐‘š ๐‘ก, ๐‘ฅ0 , ๐‘ฆ0 =

๐บ2 ๐‘ก, ๐‘  ๐‘”2 ๐‘ , ๐‘ฅ๐‘š ๐‘ , ๐‘ฅ0 , ๐‘ฆ0 , ๐‘ฆ๐‘š ๐‘ , ๐‘ฅ0 , ๐‘ฆ0 ๐‘‘๐‘  โˆ’โˆž

and ๐‘(๐‘ก)

๐‘ฃ๐‘š ๐‘ก, ๐‘ฅ0 , ๐‘ฆ0 =

๐‘”2 ๐‘ , ๐‘ฅ๐‘š ๐‘ , ๐‘ฅ0 , ๐‘ฆ0 , ๐‘ฆ๐‘š ๐‘ , ๐‘ฅ0 , ๐‘ฆ0 ๐‘‘๐‘  ,

๐‘š = 0,1,2, โ‹ฏ ,

๐‘Ž(๐‘ก) โˆž Next, we claim that two sequences ๐‘ฅ๐‘š (๐‘ก, ๐‘ฅ0 , ๐‘ฆ0 )}โˆž ๐‘š =0 , ๐‘ฆ๐‘š (๐‘ก, ๐‘ฅ0 , ๐‘ฆ0 )}๐‘š =0 are convergent uniformly to the limit functions ๐‘ฅ, ๐‘ฆ on the domain ( 2.3). By mathematical induction and lemma 1.1, we find that:

๐‘ฅ๐‘š +1 ๐‘ก, ๐‘ฅ0 , ๐‘ฆ0 โˆ’ ๐‘ฅ๐‘š ๐‘ก, ๐‘ฅ0 , ๐‘ฆ0

โ‰ค

๐›พ ๐พ + ๐‘•๐พ4 )] ๐‘ฅ๐‘š ๐‘ก, ๐‘ฅ0 , ๐‘ฆ0 โˆ’ ๐‘ฅ๐‘š โˆ’1 ๐‘ก, ๐‘ฅ0 , ๐‘ฆ0 ๐œ†1 3 ๐›พ +๐›ผ(๐‘ก)[๐พ2 + ๐‘…2 ( ๐พ3 + ๐‘•๐พ4 )] ๐‘ฆ๐‘š ๐‘ก, ๐‘ฅ0 , ๐‘ฆ0 โˆ’ ๐‘ฆ๐‘š โˆ’1 ๐‘ก, ๐‘ฅ0 , ๐‘ฆ0 ๐œ†1

โ‰ค ๐›ผ(๐‘ก)[๐พ1 + ๐‘…1 (

โ‹ฏ 2.19 and ๐‘ฆ๐‘š +1 ๐‘ก, ๐‘ฅ0 , ๐‘ฆ0 โˆ’ ๐‘ฆ๐‘š ๐‘ก, ๐‘ฅ0 , ๐‘ฆ0

โ‰ค

๐›ฟ ๐ฟ + ๐‘•๐ฟ4 )] ๐‘ฅ๐‘š ๐‘ก, ๐‘ฅ0 , ๐‘ฆ0 โˆ’ ๐‘ฅ๐‘š โˆ’1 ๐‘ก, ๐‘ฅ0 , ๐‘ฆ0 + ๐œ†2 3 ๐›ฟ +๐›ผ(๐‘ก)[๐ฟ2 + ๐ป2 ( ๐ฟ3 + ๐‘•๐ฟ4 )] ๐‘ฆ๐‘š ๐‘ก, ๐‘ฅ0 , ๐‘ฆ0 โˆ’ ๐‘ฆ๐‘š โˆ’1 ๐‘ก, ๐‘ฅ0 , ๐‘ฆ0 ๐œ†2

โ‰ค ๐›ผ(๐‘ก)[๐ฟ1 + ๐ป1 (

โ‹ฏ 2.20 We can write the inequalities ( 2.19) and ( 2.20) in a vector form: ๐ถ๐‘š +1 ๐‘ก, ๐‘ฅ0 , ๐‘ฆ0 โ‰ค ๐‘„ ๐‘ก ๐ถ๐‘š ๐‘ก, ๐‘ฅ0 , ๐‘ฆ0 where ๐‘ฅ๐‘š +1 ๐‘ก, ๐‘ฅ0 , ๐‘ฆ0 โˆ’ ๐‘ฅ๐‘š ๐‘ก, ๐‘ฅ0 , ๐‘ฆ0 ๐ถ๐‘š +1 ๐‘ก, ๐‘ฅ0 , ๐‘ฆ0 = , ๐‘ฆ๐‘š +1 ๐‘ก, ๐‘ฅ0 , ๐‘ฆ0 โˆ’ ๐‘ฆ๐‘š ๐‘ก, ๐‘ฅ0 , ๐‘ฆ0 and ๐›พ ๐›พ ๐›ผ(๐‘ก)[๐พ1 + ๐‘…1 ( ๐พ3 + ๐‘•๐พ4 )] ๐›ผ(๐‘ก)[๐พ2 + ๐‘…2 ( ๐พ3 + ๐‘•๐พ4 )] ๐œ†1 ๐œ†1 ๐‘„ ๐‘ก = , ๐›ฟ ๐›ฟ ๐›ผ(๐‘ก)[๐ฟ1 + ๐ป1 ( ๐ฟ3 + ๐‘•๐ฟ4 )] ๐›ผ(๐‘ก)[๐ฟ2 + ๐ป2 ( ๐ฟ3 + ๐‘•๐ฟ4 )] ๐œ†2 ๐œ†2 ๐‘ฅ๐‘š ๐‘ก, ๐‘ฅ0 , ๐‘ฆ0 โˆ’ ๐‘ฅ๐‘š โˆ’1 ๐‘ก, ๐‘ฅ0 , ๐‘ฆ0 ๐ถ๐‘š ๐‘ก, ๐‘ฅ0 , ๐‘ฆ0 = ๐‘ฆ๐‘š ๐‘ก, ๐‘ฅ0 , ๐‘ฆ0 โˆ’ ๐‘ฆ๐‘š โˆ’1 ๐‘ก, ๐‘ฅ0 , ๐‘ฆ0 Now, we take the maximum value to both sides of the inequality ( 2.21), ๐‘‡ for all 0 โ‰ค ๐‘ก โ‰ค ๐‘‡ and ๐›ผ ๐‘ก โ‰ค , we get: 2 ๐ถ๐‘š +1 โ‰ค ๐‘„0 ๐ถ๐‘š , where ๐‘„0 = ๐‘š๐‘Ž๐‘ฅ๐‘กโˆˆ 0,๐‘‡ ๐‘„(๐‘ก) . ๐‘‡ ๐›พ ๐‘‡ ๐›พ [๐พ + ๐‘…1 ( ๐พ3 + ๐‘•๐พ4 )] [๐พ + ๐‘…2 ( ๐พ3 + ๐‘•๐พ4 )] 2 1 ๐œ†1 2 2 ๐œ†1 Q0 = ๐‘‡ ๐›ฟ ๐‘‡ ๐›ฟ [๐ฟ1 + ๐ป1 ( ๐ฟ3 + ๐‘•๐ฟ4 )] [๐ฟ2 + ๐ป2 ( ๐ฟ3 + ๐‘•๐ฟ4 )] 2 ๐œ†2 2 ๐œ†2 By iterating the inequality (4.2.22), we find that: ๐ถ๐‘š +1 โ‰ค ๐‘„0 ๐‘š ๐ถ0 , which leads to the estimate: ๐‘š

โ‹ฏ 2.22

โ‹ฏ 2.23

๐‘š

๐‘„0 ๐‘–โˆ’1 ๐ถ0

๐ถ๐‘– โ‰ค ๐‘–=1

โ‹ฏ 2.21

โ‹ฏ 2.24

๐‘–=1

Since the matrix ๐‘„0 has maximum eigen-values of (4.1.13) and the series (4.2.24) is uniformly convergent, i. e. www.ijeijournal.com

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Periodic Solutions for Non-Linear Systems of Integral Equations ๐‘š

lim

๐‘š โ†’โˆž

โˆž

๐‘„0

๐‘–โˆ’1

๐‘–=1

๐‘„0 ๐‘–โˆ’1 ๐ถ0 = (๐ธ โˆ’ ๐‘„0 )โˆ’1 ๐ถ0

๐ถ0 =

โ‹ฏ ( 2.25)

๐‘–=1

The limiting relation (4.2.25) signifies a uniform convergence of the sequence ๐‘ฅ๐‘š (๐‘ก, ๐‘ฅ0 , ๐‘ฆ0 ), ๐‘ฆ๐‘š (๐‘ก, ๐‘ฅ0 , ๐‘ฆ0 ) โˆž ๐‘š =0 in the domain (4.2.3) as ๐‘š โ†’ โˆž. Let lim ๐‘ฅ๐‘š ๐‘ก, ๐‘ฅ0 , ๐‘ฆ0 = ๐‘ฅ 0 ๐‘ก, ๐‘ฅ0 , ๐‘ฆ0 , ๐‘š โ†’โˆž

lim ๐‘ฆ๐‘š ๐‘ก, ๐‘ฅ0 , ๐‘ฆ0 = ๐‘ฆ 0 ๐‘ก, ๐‘ฅ0 , ๐‘ฆ0

๐‘š โ†’โˆž

.

โ‹ฏ ( 2.26)

Finally, we show that ๐‘ฅ ๐‘ก, ๐‘ฅ0 , ๐‘ฆ0 โ‰ก ๐‘ฅ 0 ๐‘ก, ๐‘ฅ0 , ๐‘ฆ0 โˆˆ ๐ท and ๐‘ฆ ๐‘ก, ๐‘ฅ0 , ๐‘ฆ0 โ‰ก ๐‘ฆ 0 (๐‘ก, ๐‘ฅ0 , ๐‘ฆ0 ) โˆˆ ๐ท1 , for all ๐‘ฅ0 โˆˆ ๐ท๐‘“1 and ๐‘ฆ0 โˆˆ ๐ท1๐‘“2 . By using inequalities ( 2.1) and ( 2.4) and lemma 1.1, such that: ๐‘ก

[๐‘“1 (๐‘ , ๐‘ฅ๐‘š (๐‘ , ๐‘ฅ0 , ๐‘ฆ0 ), ๐‘ฆ๐‘š (๐‘ , ๐‘ฅ0 , ๐‘ฆ0 ), ๐‘ง๐‘š ๐‘ , ๐‘ฅ0 , ๐‘ฆ0 , ๐‘ค๐‘š (๐‘ , ๐‘ฅ0 , ๐‘ฆ0 )) โˆ’ 0

1 โˆ’ ๐‘‡ ๐‘ก

๐‘‡

๐‘“1 (๐‘ , ๐‘ฅ๐‘š (๐‘ , ๐‘ฅ0 , ๐‘ฆ0 ), ๐‘ฆ๐‘š (๐‘ , ๐‘ฅ0 , ๐‘ฆ0 ), ๐‘ง๐‘š ๐‘ , ๐‘ฅ0 , ๐‘ฆ0 , ๐‘ค๐‘š (๐‘ , ๐‘ฅ0 , ๐‘ฆ0 ))๐‘‘๐‘ ]๐‘‘๐‘  โˆ’ 0

โˆ’ [๐‘“1 (๐‘ , ๐‘ฅ(๐‘ , ๐‘ฅ0 , ๐‘ฆ0 ), ๐‘ฆ(๐‘ , ๐‘ฅ0 , ๐‘ฆ0 ), ๐‘ง ๐‘ , ๐‘ฅ0 , ๐‘ฆ0 , ๐‘ค(๐‘ , ๐‘ฅ0 , ๐‘ฆ0 )) โˆ’ 0

1 โˆ’ ๐‘‡

๐‘‡

๐‘“1 (๐‘ , ๐‘ฅ(๐‘ , ๐‘ฅ0 , ๐‘ฆ0 ), ๐‘ฆ(๐‘ , ๐‘ฅ0 , ๐‘ฆ0 ), ๐‘ง ๐‘ , ๐‘ฅ0 , ๐‘ฆ0 , ๐‘ค(๐‘ , ๐‘ฅ0 , ๐‘ฆ0 ))๐‘‘๐‘ ]๐‘‘๐‘  0

๐›พ โ‰ค ๐›ผ(๐‘ก) [๐พ1 + ๐‘…1 ( ๐พ3 + ๐‘•๐พ4 )] ๐‘ฅ๐‘š ๐‘ , ๐‘ฅ0 , ๐‘ฆ0 โˆ’ ๐‘ฅ ๐‘ , ๐‘ฅ0 , ๐‘ฆ0 ๐œ†1 ๐›พ +[๐พ2 + ๐‘…2 ( ๐พ3 + ๐‘•๐พ4 )] ๐‘ฆ๐‘š ๐‘ , ๐‘ฅ0 , ๐‘ฆ0 โˆ’ ๐‘ฆ ๐‘ , ๐‘ฅ0 , ๐‘ฆ0 โ‰ค ๐œ†1 ๐‘‡ ๐›พ โ‰ค [๐พ1 + ๐‘…1 ( ๐พ3 + ๐‘•๐พ4 )] ๐‘ฅ๐‘š ๐‘ก, ๐‘ฅ0 , ๐‘ฆ0 โˆ’ ๐‘ฅ ๐‘ก, ๐‘ฅ0 , ๐‘ฆ0 + 2 ๐œ†1 ๐›พ +[๐พ2 + ๐‘…2 ( ๐พ3 + ๐‘•๐พ4 )] ๐‘ฆ๐‘š ๐‘ก, ๐‘ฅ0 , ๐‘ฆ0 โˆ’ ๐‘ฆ ๐‘ก, ๐‘ฅ0 , ๐‘ฆ0 .

+

๐œ†1

From inequality ( 2.26) and on the other hand suppose that:. ๐‘ฅ๐‘š ๐‘ก, ๐‘ฅ0 , ๐‘ฆ0 โˆ’ ๐‘ฅ ๐‘ก, ๐‘ฅ0 , ๐‘ฆ0 โ‰ค ๐œ–1 , ๐‘ฆ๐‘š ๐‘ก, ๐‘ฅ0 , ๐‘ฆ0 โˆ’ ๐‘ฆ ๐‘ก, ๐‘ฅ0 , ๐‘ฆ0 โ‰ค ๐œ–2 . Thus ๐‘ก

[๐‘“1 (๐‘ , ๐‘ฅ๐‘š (๐‘ , ๐‘ฅ0 , ๐‘ฆ0 ), ๐‘ฆ๐‘š (๐‘ , ๐‘ฅ0 , ๐‘ฆ0 ), ๐‘ง๐‘š ๐‘ , ๐‘ฅ0 , ๐‘ฆ0 , ๐‘ค๐‘š (๐‘ , ๐‘ฅ0 , ๐‘ฆ0 )) โˆ’ 0

1 โˆ’ ๐‘‡ ๐‘ก

๐‘‡

๐‘“1 (๐‘ , ๐‘ฅ๐‘š (๐‘ , ๐‘ฅ0 , ๐‘ฆ0 ), ๐‘ฆ๐‘š (๐‘ , ๐‘ฅ0 , ๐‘ฆ0 ), ๐‘ง๐‘š ๐‘ , ๐‘ฅ0 , ๐‘ฆ0 , ๐‘ค๐‘š (๐‘ , ๐‘ฅ0 , ๐‘ฆ0 ))๐‘‘๐‘ ]๐‘‘๐‘  โˆ’ 0

โˆ’ [๐‘“1 (๐‘ , ๐‘ฅ(๐‘ , ๐‘ฅ0 , ๐‘ฆ0 ), ๐‘ฆ(๐‘ , ๐‘ฅ0 , ๐‘ฆ0 ), ๐‘ง ๐‘ , ๐‘ฅ0 , ๐‘ฆ0 , ๐‘ค(๐‘ , ๐‘ฅ0 , ๐‘ฆ0 )) โˆ’ 0

1 โˆ’ ๐‘‡

๐‘‡

๐‘“1 (๐‘ , ๐‘ฅ(๐‘ , ๐‘ฅ0 , ๐‘ฆ0 ), ๐‘ฆ(๐‘ , ๐‘ฅ0 , ๐‘ฆ0 ), ๐‘ง ๐‘ , ๐‘ฅ0 , ๐‘ฆ0 , ๐‘ค(๐‘ , ๐‘ฅ0 , ๐‘ฆ0 ))๐‘‘๐‘ ]๐‘‘๐‘  0

๐‘‡ ๐›พ ๐‘‡ ๐›พ [๐พ + ๐‘…1 ( ๐พ3 + ๐‘•๐พ4 )]๐œ–1 + [๐พ2 + ๐‘…2 ( ๐พ3 + ๐‘•๐พ4 )]๐œ–2 2 1 ๐œ†1 2 ๐œ†1 ๐œ–3 ๐œ–4 Putting ๐œ–1 = ๐‘‡ and ๐œ–2 = ๐‘‡ and substituting in the last equation, we have: ๐›พ ๐›พ โ‰ค

2

๐‘ก

[๐พ1 +๐‘…1 (

๐พ +๐‘• ๐พ4 )] ๐œ†1 3

2

[๐พ2 +๐‘…2 (

๐พ +๐‘•๐พ4 )] ๐œ†1 3

[๐‘“1 (๐‘ , ๐‘ฅ๐‘š (๐‘ , ๐‘ฅ0 , ๐‘ฆ0 ), ๐‘ฆ๐‘š (๐‘ , ๐‘ฅ0 , ๐‘ฆ0 ), ๐‘ง๐‘š ๐‘ , ๐‘ฅ0 , ๐‘ฆ0 , ๐‘ค๐‘š (๐‘ , ๐‘ฅ0 , ๐‘ฆ0 )) โˆ’ 0

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Periodic Solutions for Non-Linear Systems of Integral Equations ๐‘‡

1 โˆ’ ๐‘‡

๐‘“1 (๐‘ , ๐‘ฅ๐‘š (๐‘ , ๐‘ฅ0 , ๐‘ฆ0 ), ๐‘ฆ๐‘š (๐‘ , ๐‘ฅ0 , ๐‘ฆ0 ), ๐‘ง๐‘š ๐‘ , ๐‘ฅ0 , ๐‘ฆ0 , ๐‘ค๐‘š (๐‘ , ๐‘ฅ0 , ๐‘ฆ0 ))๐‘‘๐‘ ]๐‘‘๐‘  โˆ’

๐‘ก

0

โˆ’ [๐‘“1 (๐‘ , ๐‘ฅ(๐‘ , ๐‘ฅ0 , ๐‘ฆ0 ), ๐‘ฆ(๐‘ , ๐‘ฅ0 , ๐‘ฆ0 ), ๐‘ง ๐‘ , ๐‘ฅ0 , ๐‘ฆ0 , ๐‘ค(๐‘ , ๐‘ฅ0 , ๐‘ฆ0 )) โˆ’ 0

1 โˆ’ ๐‘‡

๐‘‡

๐‘“1 (๐‘ , ๐‘ฅ(๐‘ , ๐‘ฅ0 , ๐‘ฆ0 ), ๐‘ฆ(๐‘ , ๐‘ฅ0 , ๐‘ฆ0 ), ๐‘ง ๐‘ , ๐‘ฅ0 , ๐‘ฆ0 , ๐‘ค(๐‘ , ๐‘ฅ0 , ๐‘ฆ0 ))๐‘‘๐‘ ]๐‘‘๐‘  0

๐‘‡ ๐›พ ๐œ–3 โ‰ค [๐พ1 + ๐‘…1 ( ๐พ3 + ๐‘•๐พ4 )] + ๐‘‡ ๐›พ 2 ๐œ†1 [๐พ1 + ๐‘…1 ( ๐พ3 + ๐‘•๐พ4 )] 2 ๐œ†1 ๐‘‡ ๐›พ ๐œ–4 + [๐พ2 + ๐‘…2 ( ๐พ3 + ๐‘•๐พ4 )] ๐‘‡ ๐›พ 2 ๐œ†1 [๐พ + ๐‘…2 ( ๐พ3 + ๐‘•๐พ4 )] 2 2 ๐œ†1 โ‰ค ๐œ–3 + ๐œ–4, and choosing ๐œ–3 + ๐œ–4 = ๐œ– , we get: ๐‘ก

[๐‘“1 (๐‘ , ๐‘ฅ๐‘š (๐‘ , ๐‘ฅ0 , ๐‘ฆ0 ), ๐‘ฆ๐‘š (๐‘ , ๐‘ฅ0 , ๐‘ฆ0 ), ๐‘ง๐‘š ๐‘ , ๐‘ฅ0 , ๐‘ฆ0 , ๐‘ค๐‘š (๐‘ , ๐‘ฅ0 , ๐‘ฆ0 )) โˆ’ 0 ๐‘‡

1 โˆ’ ๐‘‡

๐‘“1 (๐‘ , ๐‘ฅ๐‘š (๐‘ , ๐‘ฅ0 , ๐‘ฆ0 ), ๐‘ฆ๐‘š (๐‘ , ๐‘ฅ0 , ๐‘ฆ0 ), ๐‘ง๐‘š ๐‘ , ๐‘ฅ0 , ๐‘ฆ0 , ๐‘ค๐‘š (๐‘ , ๐‘ฅ0 , ๐‘ฆ0 ))๐‘‘๐‘ ]๐‘‘๐‘  โˆ’

๐‘ก

0

โˆ’ [๐‘“1 (๐‘ , ๐‘ฅ(๐‘ , ๐‘ฅ0 , ๐‘ฆ0 ), ๐‘ฆ(๐‘ , ๐‘ฅ0 , ๐‘ฆ0 ), ๐‘ง ๐‘ , ๐‘ฅ0 , ๐‘ฆ0 , ๐‘ค(๐‘ , ๐‘ฅ0 , ๐‘ฆ0 )) โˆ’ 0

1 โˆ’ ๐‘‡

๐‘‡

๐‘“1 (๐‘ , ๐‘ฅ(๐‘ , ๐‘ฅ0 , ๐‘ฆ0 ), ๐‘ฆ(๐‘ , ๐‘ฅ0 , ๐‘ฆ0 ), ๐‘ง ๐‘ , ๐‘ฅ0 , ๐‘ฆ0 , ๐‘ค(๐‘ , ๐‘ฅ0 , ๐‘ฆ0 ))๐‘‘๐‘ ]๐‘‘๐‘  โ‰ค ๐œ– 0

for all ๐‘š โ‰ฅ 0, ๐‘ก

i. e. lim

[๐‘“1 (๐‘ , ๐‘ฅ๐‘š (๐‘ , ๐‘ฅ0 , ๐‘ฆ0 ), ๐‘ฆ๐‘š (๐‘ , ๐‘ฅ0 , ๐‘ฆ0 ), ๐‘ง๐‘š ๐‘ , ๐‘ฅ0 , ๐‘ฆ0 , ๐‘ค๐‘š (๐‘ , ๐‘ฅ0 , ๐‘ฆ0 )) โˆ’

๐‘š โ†’โˆž

1 โˆ’ ๐‘‡ ๐‘ก

๐‘‡

0

๐‘“1 (๐‘ , ๐‘ฅ๐‘š (๐‘ , ๐‘ฅ0 , ๐‘ฆ0 ), ๐‘ฆ๐‘š (๐‘ , ๐‘ฅ0 , ๐‘ฆ0 ), ๐‘ง๐‘š ๐‘ , ๐‘ฅ0 , ๐‘ฆ0 , ๐‘ค๐‘š (๐‘ , ๐‘ฅ0 , ๐‘ฆ0 ))๐‘‘๐‘ ]๐‘‘๐‘  = 0

โˆ’ [๐‘“1 (๐‘ , ๐‘ฅ(๐‘ , ๐‘ฅ0 , ๐‘ฆ0 ), ๐‘ฆ(๐‘ , ๐‘ฅ0 , ๐‘ฆ0 ), ๐‘ง ๐‘ , ๐‘ฅ0 , ๐‘ฆ0 , ๐‘ค(๐‘ , ๐‘ฅ0 , ๐‘ฆ0 )) โˆ’ 0

1 โˆ’ ๐‘‡

๐‘‡

๐‘“1 (๐‘ , ๐‘ฅ(๐‘ , ๐‘ฅ0 , ๐‘ฆ0 ), ๐‘ฆ(๐‘ , ๐‘ฅ0 , ๐‘ฆ0 ), ๐‘ง ๐‘ , ๐‘ฅ0 , ๐‘ฆ0 , ๐‘ค(๐‘ , ๐‘ฅ0 , ๐‘ฆ0 ))๐‘‘๐‘ ]๐‘‘๐‘ . 0

So ๐‘ฅ ๐‘ก, ๐‘ฅ0 , ๐‘ฆ0 โˆˆ ๐ท, and ๐‘ฅ ๐‘ก, ๐‘ฅ0 , ๐‘ฆ0 โ‰ก ๐‘ฅ 0 (๐‘ก, ๐‘ฅ0 , ๐‘ฆ0 ) is a periodic solution of ( I1), Also by using the same method above we can prove ๐‘ฆ ๐‘ก, ๐‘ฅ0 , ๐‘ฆ0 โˆˆ ๐ท1 , and ๐‘ฆ ๐‘ก, ๐‘ฅ0 , ๐‘ฆ0 โ‰ก ๐‘ฆ 0 ๐‘ก, ๐‘ฅ0 , ๐‘ฆ0 also periodic solution of (๐ผ2 ).

is

Theorem 2.2. With the hypotheses and all conditions of the theorem 2.1, the periodic solution of integral equations ( ๐ผ1 ) and ( ๐ผ2 ) are a unique on the domain (1.3). Proof. Suppose that ๐‘ฅ ๐‘ก, ๐‘ฅ0 , ๐‘ฆ0 and ๐‘ฆ ๐‘ก, ๐‘ฅ0 , ๐‘ฆ0 be another periodic solutions for the systems (๐ผ1 ) and ( ๐ผ2 ) defined and continuous and periodic in ๐‘ก of period ๐‘‡, this means that: ๐‘ก

๐‘ฅ ๐‘ก, ๐‘ฅ0 , ๐‘ฆ0 = ๐น0 ๐‘ก +

[๐‘“1 (๐‘ , ๐‘ฅ ๐‘ , ๐‘ฅ0 , ๐‘ฆ0 , ๐‘ฆ ๐‘ , ๐‘ฅ0 , ๐‘ฆ0 , 0

๐‘ 

,

๐บ1 ๐‘ , ๐œ ๐‘”1 (๐œ, ๐‘ฅ ๐œ, ๐‘ฅ0 , ๐‘ฆ0 , ๐‘ฆ ๐œ, ๐‘ฅ0 , ๐‘ฆ0 ) ๐‘‘๐œ, โˆ’โˆž

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Periodic Solutions for Non-Linear Systems of Integral Equations ๐‘ ๐‘ 

1 ๐‘”1 (๐œ, ๐‘ฅ ๐œ, ๐‘ฅ0 , ๐‘ฆ0 , ๐‘ฆ ๐œ, ๐‘ฅ0 , ๐‘ฆ0 )๐‘‘๐œ) โˆ’ ๐‘‡

, ๐‘Ž ๐‘ 

๐‘ 

, ๐‘ฆ ๐‘ , ๐‘ฅ0 , ๐‘ฆ0 ,

๐‘‡

๐‘“1 (๐‘ , ๐‘ฅ ๐‘ , ๐‘ฅ0 , ๐‘ฆ0 , 0

๐บ1 ๐‘ , ๐œ ๐‘”1 (๐œ, ๐‘ฅ ๐œ, ๐‘ฅ0 , ๐‘ฆ0 , ๐‘ฆ ๐œ, ๐‘ฅ0 , ๐‘ฆ0 ) ๐‘‘๐œ, โˆ’โˆž

๐‘ ๐‘ 

,

๐‘”1 (๐œ, ๐‘ฅ ๐œ, ๐‘ฅ0 , ๐‘ฆ0 , ๐‘ฆ ๐œ, ๐‘ฅ0 , ๐‘ฆ0 )๐‘‘๐œ)๐‘‘๐‘ ]๐‘‘๐‘ 

โ‹ฏ 2.27

๐‘Ž ๐‘ 

and ๐‘ก

๐‘ฆ ๐‘ก, ๐‘ฅ0 , ๐‘ฆ0 = ๐บ0 ๐‘ก +

[๐‘“2 (๐‘ , ๐‘ฅ ๐‘ , ๐‘ฅ0 , ๐‘ฆ0 , ๐‘ฆ ๐‘ , ๐‘ฅ0 , ๐‘ฆ0 , 0

๐‘ 

,

๐บ2 ๐‘ , ๐œ ๐‘”2 (๐œ, ๐‘ฅ ๐œ, ๐‘ฅ0 , ๐‘ฆ0 , ๐‘ฆ ๐œ, ๐‘ฅ0 , ๐‘ฆ0 ) ๐‘‘๐œ, โˆ’โˆž

๐‘ ๐‘ 

, ๐‘ 

, ๐‘ฆ ๐‘ , ๐‘ฅ0 , ๐‘ฆ0 ,

๐‘Ž ๐‘ 

1 ๐‘”2 (๐œ, ๐‘ฅ ๐œ, ๐‘ฅ0 , ๐‘ฆ0 , ๐‘ฆ ๐œ, ๐‘ฅ0 , ๐‘ฆ0 )๐‘‘๐œ) โˆ’ ๐‘‡

๐‘‡

๐‘“2 (๐‘ , ๐‘ฅ ๐‘ , ๐‘ฅ0 , ๐‘ฆ0 , 0

๐บ2 ๐‘ , ๐œ ๐‘”2 (๐œ, ๐‘ฅ ๐œ, ๐‘ฅ0 , ๐‘ฆ0 , ๐‘ฆ ๐œ, ๐‘ฅ0 , ๐‘ฆ0 ) ๐‘‘๐œ, โˆ’โˆž

๐‘ ๐‘ 

,

๐‘”2 (๐œ, ๐‘ฅ ๐œ, ๐‘ฅ0 , ๐‘ฆ0 , ๐‘ฆ ๐œ, ๐‘ฅ0 , ๐‘ฆ0 )๐‘‘๐œ)๐‘‘๐‘ ]๐‘‘๐‘ 

โ‹ฏ 2.28

๐‘Ž ๐‘ 

For their difference, we should obtain the inequality: ๐‘ฅ ๐‘ก, ๐‘ฅ0 , ๐‘ฆ0 โˆ’ ๐‘ฅ ๐‘ก, ๐‘ฅ0 , ๐‘ฆ0

โ‰ค ๐‘‡ ๐›พ โ‰ค [๐พ1 + ๐‘…1 ( ๐พ3 + ๐‘•๐พ4 )] ๐‘ฅ ๐‘ก, ๐‘ฅ0 , ๐‘ฆ0 โˆ’ ๐‘ฅ ๐‘ก, ๐‘ฅ0 , ๐‘ฆ0 + 2 ๐œ†1 ๐‘‡ ๐›พ + [๐พ2 + ๐‘…2 ( ๐พ3 + ๐‘•๐พ4 )] ๐‘ฆ ๐‘ก, ๐‘ฅ0 , ๐‘ฆ0 โˆ’ ๐‘ฆ ๐‘ก, ๐‘ฅ0 , ๐‘ฆ0 2 ๐œ†1

And also ๐‘ฆ ๐‘ , ๐‘ฅ0 , ๐‘ฆ0 โˆ’ ๐‘ฆ ๐‘ , ๐‘ฅ0 , ๐‘ฆ0

โ‹ฏ 2.29

โ‰ค ๐‘ก

๐‘ก โ‰ค (1 โˆ’ ) [๐ฟ1 ๐‘ฅ ๐‘ , ๐‘ฅ0 , ๐‘ฆ0 โˆ’ ๐‘ฅ ๐‘ , ๐‘ฅ0 , ๐‘ฆ0 ๐‘‡

+ ๐ฟ2 ๐‘ฆ ๐‘ , ๐‘ฅ0 , ๐‘ฆ0 โˆ’ ๐‘ฆ ๐‘ , ๐‘ฅ0 , ๐‘ฆ0

0

๐›ฟ ๐ฟ ๐ป ๐‘ฅ ๐‘ , ๐‘ฅ0 , ๐‘ฆ0 โˆ’ ๐‘ฅ ๐‘ , ๐‘ฅ0 , ๐‘ฆ0 + ๐ป2 ๐‘ฆ ๐‘ , ๐‘ฅ0 , ๐‘ฆ0 โˆ’ ๐‘ฆ ๐‘ , ๐‘ฅ0 , ๐‘ฆ0 + ๐œ†2 3 1 +๐‘•๐ฟ4 (๐ป1 ๐‘ฅ ๐‘ , ๐‘ฅ0 , ๐‘ฆ0 โˆ’ ๐‘ฅ ๐‘ , ๐‘ฅ0 , ๐‘ฆ0 + ๐ป2 ๐‘ฆ ๐‘ , ๐‘ฅ0 , ๐‘ฆ0 โˆ’ ๐‘ฆ ๐‘ , ๐‘ฅ0 , ๐‘ฆ0 )]๐‘‘๐‘  +

๐‘ก + ๐‘‡

๐‘‡

๐ฟ1 ๐‘ฅ ๐‘ , ๐‘ฅ0 , ๐‘ฆ0 โˆ’ ๐‘ฅ ๐‘ , ๐‘ฅ0 , ๐‘ฆ0

+ ๐ฟ2 ๐‘ฆ ๐‘ , ๐‘ฅ0 , ๐‘ฆ0 โˆ’ ๐‘ฆ ๐‘ , ๐‘ฅ0 , ๐‘ฆ0

+

๐‘ก

๐›ฟ ๐ฟ ๐ป ๐‘ฅ ๐‘ , ๐‘ฅ0 , ๐‘ฆ0 โˆ’ ๐‘ฅ ๐‘ , ๐‘ฅ0 , ๐‘ฆ0 + ๐ป2 ๐‘ฆ ๐‘ , ๐‘ฅ0 , ๐‘ฆ0 โˆ’ ๐‘ฆ ๐‘ , ๐‘ฅ0 , ๐‘ฆ0 + ๐œ†2 3 1 +๐‘•๐ฟ4 (๐ป1 ๐‘ฅ ๐‘ , ๐‘ฅ0 , ๐‘ฆ0 โˆ’ ๐‘ฅ ๐‘ , ๐‘ฅ0 , ๐‘ฆ0 + ๐ป2 ๐‘ฆ ๐‘ , ๐‘ฅ0 , ๐‘ฆ0 โˆ’ ๐‘ฆ ๐‘ , ๐‘ฅ0 , ๐‘ฆ0 ๐›ฟ โ‰ค ๐›ผ(๐‘ก)[๐ฟ1 + ๐ป1 ( ๐ฟ3 + ๐‘•๐ฟ4 )] ๐‘ฅ ๐‘ก, ๐‘ฅ0 , ๐‘ฆ0 โˆ’ ๐‘ฅ ๐‘ก, ๐‘ฅ0 , ๐‘ฆ0 + ๐œ†2 ๐›ฟ +๐›ผ(๐‘ก)[๐ฟ2 + ๐ป2 ( ๐ฟ3 + ๐‘•๐ฟ4 )] ๐‘ฆ ๐‘ก, ๐‘ฅ0 , ๐‘ฆ0 โˆ’ ๐‘ฆ ๐‘ก, ๐‘ฅ0 , ๐‘ฆ0 +

)]๐‘‘๐‘ 

๐œ†2

so that: ๐‘ฆ ๐‘ก, ๐‘ฅ0 , ๐‘ฆ0 โˆ’ ๐‘ฆ ๐‘ก, ๐‘ฅ0 , ๐‘ฆ0

โ‰ค ๐‘‡ ๐›ฟ โ‰ค [๐ฟ1 + ๐ป1 ( ๐ฟ3 + ๐‘•๐ฟ4 )] ๐‘ฅ ๐‘ก, ๐‘ฅ0 , ๐‘ฆ0 โˆ’ ๐‘ฅ ๐‘ก, ๐‘ฅ0 , ๐‘ฆ0 + 2 ๐œ†2 ๐‘‡ ๐›ฟ + [๐ฟ2 + ๐ป2 ( ๐ฟ3 + ๐‘•๐ฟ4 )] ๐‘ฆ ๐‘ก, ๐‘ฅ0 , ๐‘ฆ0 โˆ’ ๐‘ฆ ๐‘ก, ๐‘ฅ0 , ๐‘ฆ0 2

๐œ†2

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โ‹ฏ 2.30 Page | 9

Periodic Solutions for Non-Linear Systems of Integral Equations and the inequalities (4.2.29) and (4.2.30) would lead to the estimate: ฤ‘?‘ฤฝ ฤ‘?‘ฤ„, ฤ‘?‘ฤฝ0 , ฤ‘?‘ลš0 รขˆ’ ฤ‘?‘ฤฝ ฤ‘?‘ฤ„, ฤ‘?‘ฤฝ0 , ฤ‘?‘ลš0 ฤ‘?‘ฤฝ ฤ‘?‘ฤ„, ฤ‘?‘ฤฝ0 , ฤ‘?‘ลš0 รขˆ’ ฤ‘?‘ฤฝ ฤ‘?‘ฤ„, ฤ‘?‘ฤฝ0 , ฤ‘?‘ลš0 รข‰ยค ฤ‘?‘„0 ฤ‘?‘ลš ฤ‘?‘ฤ„, ฤ‘?‘ฤฝ0 , ฤ‘?‘ลš0 รขˆ’ ฤ‘?‘ลš ฤ‘?‘ฤ„, ฤ‘?‘ฤฝ0 , ฤ‘?‘ลš0 ฤ‘?‘ลš ฤ‘?‘ฤ„, ฤ‘?‘ฤฝ0 , ฤ‘?‘ลš0 รขˆ’ ฤ‘?‘ลš ฤ‘?‘ฤ„, ฤ‘?‘ฤฝ0 , ฤ‘?‘ลš0 รข‹ลป 2.31 By iterating the inequality (4.2.27), which should find: ฤ‘?‘ฤฝ ฤ‘?‘ฤ„, ฤ‘?‘ฤฝ0 , ฤ‘?‘ลš0 รขˆ’ ฤ‘?‘ฤฝ ฤ‘?‘ฤ„, ฤ‘?‘ฤฝ0 , ฤ‘?‘ลš0 ฤ‘?‘ฤฝ ฤ‘?‘ฤ„, ฤ‘?‘ฤฝ0 , ฤ‘?‘ลš0 รขˆ’ ฤ‘?‘ฤฝ ฤ‘?‘ฤ„, ฤ‘?‘ฤฝ0 , ฤ‘?‘ลš0 รข‰ยค ฤ‘?‘„0 ฤ‘?‘š . ฤ‘?‘ลš ฤ‘?‘ฤ„, ฤ‘?‘ฤฝ0 , ฤ‘?‘ลš0 รขˆ’ ฤ‘?‘ลš ฤ‘?‘ฤ„, ฤ‘?‘ฤฝ0 , ฤ‘?‘ลš0 ฤ‘?‘ลš ฤ‘?‘ฤ„, ฤ‘?‘ฤฝ0 , ฤ‘?‘ลš0 รขˆ’ ฤ‘?‘ลš ฤ‘?‘ฤ„, ฤ‘?‘ฤฝ0 , ฤ‘?‘ลš0 But ฤ‘?‘„0 ฤ‘?‘š รข†’ 0 as ฤ‘?‘š รข†’ รขˆž, so that proceeding in the last inequality which is contradict the supposition It follows immediately ฤ‘?‘ฤฝ ฤ‘?‘ฤ„, ฤ‘?‘ฤฝ0 , ฤ‘?‘ลš0 = ฤ‘?‘ฤฝ ฤ‘?‘ฤ„, ฤ‘?‘ฤฝ0 , ฤ‘?‘ลš0 and ฤ‘?‘ลš ฤ‘?‘ฤ„, ฤ‘?‘ฤฝ0 , ฤ‘?‘ลš0 = ฤ‘?‘ลš ฤ‘?‘ฤ„, ฤ‘?‘ฤฝ0 , ฤ‘?‘ลš0 .

III. Existence of Solution of (ฤ‘?‘ยฐฤ‘?&#x;? ) and (ฤ‘?‘ยฐฤ‘?&#x;? ) The problem of existence of periodic solution of period T of the system ( ฤ‘??ลบ1 ) and ( ฤ‘??ลบ2 ) are uniquely connected with the existence of zero of the functions รขˆ†1 0, ฤ‘?‘ฤฝ0 , ฤ‘?‘ลš0 = รขˆ†1 and รขˆ†2 0, ฤ‘?‘ฤฝ0 , ฤ‘?‘ลš0 = รขˆ†2 which has the form: รขˆ†1 : ฤ‘??ห‡ฤ‘?‘“1 ฤ‚— ฤ‘??ห‡1ฤ‘?‘“2 รข†’ ฤ‘?‘…ฤ‘?‘› รขˆ†1 0, ฤ‘?‘ฤฝ0 , ฤ‘?‘ลš0

1 = ฤ‘?‘‡

ฤ‘?‘‡

ฤ‘?‘“1 (ฤ‘?‘ฤ„, ฤ‘?‘ฤฝ 0 ฤ‘?‘ฤ„, ฤ‘?‘ฤฝ0 , ฤ‘?‘ลš0 , ฤ‘?‘ลš 0 ฤ‘?‘ฤ„, ฤ‘?‘ฤฝ0 , ฤ‘?‘ลš0 , 0

ฤ‘?‘ฤ„

ฤ‘??ลŸ1 ฤ‘?‘ฤ„, ฤ‘?‘ ฤ‘?‘”1 ฤ‘?‘ , ฤ‘?‘ฤฝ 0 ฤ‘?‘ , ฤ‘?‘ฤฝ0 , ฤ‘?‘ลš0 , ฤ‘?‘ลš 0 ฤ‘?‘ , ฤ‘?‘ฤฝ0 , ฤ‘?‘ลš0

, รขˆ’รขˆž

ฤ‘?‘? ฤ‘?‘ฤ„

ฤ‘?‘”1 ฤ‘?‘ , ฤ‘?‘ฤฝ 0 ฤ‘?‘ , ฤ‘?‘ฤฝ0 , ฤ‘?‘ลš0 , ฤ‘?‘ลš 0 ฤ‘?‘ , ฤ‘?‘ฤฝ0 , ฤ‘?‘ลš0 ฤ‘?‘‘ฤ‘?‘ )ฤ‘?‘‘ฤ‘?‘ฤ„

,

รขˆ†2 0, ฤ‘?‘ฤฝ0 , ฤ‘?‘ลš0

รข‹ลป 3.1

ฤ‘?‘Ž ฤ‘?‘ฤ„

รขˆ†2 : ฤ‘??ห‡ฤ‘?‘“1 ฤ‚— ฤ‘??ห‡1ฤ‘?‘“2 รข†’ ฤ‘?‘…ฤ‘?‘› 1 = ฤ‘?‘‡

ฤ‘?‘‘ฤ‘?‘

ฤ‘?‘‡

ฤ‘?‘“2 (ฤ‘?‘ฤ„, ฤ‘?‘ฤฝ 0 ฤ‘?‘ฤ„, ฤ‘?‘ฤฝ0 , ฤ‘?‘ลš0 , ฤ‘?‘ลš 0 ฤ‘?‘ฤ„, ฤ‘?‘ฤฝ0 , ฤ‘?‘ลš0 , 0

ฤ‘?‘ฤ„

ฤ‘??ลŸ2 ฤ‘?‘ฤ„, ฤ‘?‘ ฤ‘?‘”2 ฤ‘?‘ , ฤ‘?‘ฤฝ 0 ฤ‘?‘ , ฤ‘?‘ฤฝ0 , ฤ‘?‘ลš0 , ฤ‘?‘ลš 0 ฤ‘?‘ , ฤ‘?‘ฤฝ0 , ฤ‘?‘ลš0

, รขˆ’รขˆž

ฤ‘?‘‘ฤ‘?‘

ฤ‘?‘? ฤ‘?‘ฤ„

ฤ‘?‘”2 ฤ‘?‘ , ฤ‘?‘ฤฝ 0 ฤ‘?‘ , ฤ‘?‘ฤฝ0 , ฤ‘?‘ลš0 , ฤ‘?‘ลš 0 ฤ‘?‘ , ฤ‘?‘ฤฝ0 , ฤ‘?‘ลš0 ฤ‘?‘‘ฤ‘?‘ )ฤ‘?‘‘ฤ‘?‘ฤ„

,

รข‹ลป 3.2

ฤ‘?‘Ž ฤ‘?‘ฤ„

where the function ฤ‘?‘ฤฝ 0 ฤ‘?‘ฤ„, ฤ‘?‘ฤฝ0 , ฤ‘?‘ลš0 is the limit of the sequence of the functions ฤ‘?‘ฤฝฤ‘?‘š (ฤ‘?‘ฤ„, ฤ‘?‘ฤฝ0 , ฤ‘?‘ลš0 ) and the function ฤ‘?‘ลš 0 ฤ‘?‘ฤ„, ฤ‘?‘ฤฝ0 , ฤ‘?‘ลš0 is the limit of the sequence of the functions ฤ‘?‘ลšฤ‘?‘š ฤ‘?‘ฤ„, ฤ‘?‘ฤฝ0 , ฤ‘?‘ลš0 . Since this two functions are approximately determined from the sequences of functions: รขˆ†1ฤ‘?‘š : ฤ‘??ห‡ฤ‘?‘“1 ฤ‚— ฤ‘??ห‡1ฤ‘?‘“2 รข†’ ฤ‘?‘…ฤ‘?‘› รขˆ†1ฤ‘?‘š 0, ฤ‘?‘ฤฝ0 , ฤ‘?‘ลš0

1 = ฤ‘?‘‡

ฤ‘?‘‡

ฤ‘?‘“1 (ฤ‘?‘ฤ„, ฤ‘?‘ฤฝฤ‘?‘š ฤ‘?‘ฤ„, ฤ‘?‘ฤฝ0 , ฤ‘?‘ลš0 , ฤ‘?‘ลšฤ‘?‘š ฤ‘?‘ฤ„, ฤ‘?‘ฤฝ0 , ฤ‘?‘ลš0 , 0

ฤ‘?‘ฤ„

,

ฤ‘??ลŸ1 ฤ‘?‘ฤ„, ฤ‘?‘ ฤ‘?‘”1 ฤ‘?‘ , ฤ‘?‘ฤฝฤ‘?‘š ฤ‘?‘ , ฤ‘?‘ฤฝ0 , ฤ‘?‘ลš0 , ฤ‘?‘ลšฤ‘?‘š ฤ‘?‘ , ฤ‘?‘ฤฝ0 , ฤ‘?‘ลš0 รขˆ’รขˆž

ฤ‘?‘? ฤ‘?‘ฤ„

,

รขˆ†2ฤ‘?‘š 0, ฤ‘?‘ฤฝ0 , ฤ‘?‘ลš0

ฤ‘?‘”1 ฤ‘?‘ , ฤ‘?‘ฤฝฤ‘?‘š ฤ‘?‘ , ฤ‘?‘ฤฝ0 , ฤ‘?‘ลš0 , ฤ‘?‘ลšฤ‘?‘š ฤ‘?‘ , ฤ‘?‘ฤฝ0 , ฤ‘?‘ลš0 ฤ‘?‘‘ฤ‘?‘ )ฤ‘?‘‘ฤ‘?‘ฤ„

รข‹ลป 3.3

ฤ‘?‘Ž ฤ‘?‘ฤ„

รขˆ†2ฤ‘?‘š : ฤ‘??ห‡ฤ‘?‘“1 ฤ‚— ฤ‘??ห‡1ฤ‘?‘“2 รข†’ ฤ‘?‘…ฤ‘?‘› 1 = ฤ‘?‘‡

ฤ‘?‘‘ฤ‘?‘

ฤ‘?‘‡

ฤ‘?‘“2 (ฤ‘?‘ฤ„, ฤ‘?‘ฤฝฤ‘?‘š ฤ‘?‘ฤ„, ฤ‘?‘ฤฝ0 , ฤ‘?‘ลš0 , ฤ‘?‘ลšฤ‘?‘š ฤ‘?‘ฤ„, ฤ‘?‘ฤฝ0 , ฤ‘?‘ลš0 , 0

ฤ‘?‘ฤ„

,

ฤ‘??ลŸ2 ฤ‘?‘ฤ„, ฤ‘?‘ ฤ‘?‘”2 ฤ‘?‘ , ฤ‘?‘ฤฝฤ‘?‘š ฤ‘?‘ , ฤ‘?‘ฤฝ0 , ฤ‘?‘ลš0 , ฤ‘?‘ลšฤ‘?‘š ฤ‘?‘ , ฤ‘?‘ฤฝ0 , ฤ‘?‘ลš0

ฤ‘?‘‘ฤ‘?‘

รขˆ’รขˆž

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Periodic Solutions for Non-Linear Systems of Integral Equations ฤ‘?‘? ฤ‘?‘ฤ„

,

ฤ‘?‘”2 ฤ‘?‘ , ฤ‘?‘ฤฝฤ‘?‘š ฤ‘?‘ , ฤ‘?‘ฤฝ0 , ฤ‘?‘ลš0 , ฤ‘?‘ลšฤ‘?‘š ฤ‘?‘ , ฤ‘?‘ฤฝ0 , ฤ‘?‘ลš0 ฤ‘?‘‘ฤ‘?‘ )ฤ‘?‘‘ฤ‘?‘ฤ„

รข‹ลป 3.4

ฤ‘?‘Ž ฤ‘?‘ฤ„

for all ฤ‘?‘š = 0,1,2, รข‹ลป Theorem 3.1. If the hypotheses and all conditions of the theorem 2.1 and 2.2 are satisfied, then the following inequality satisfied: รขˆ†1 0, ฤ‘?‘ฤฝ0 , ฤ‘?‘ลš0 รขˆ’ รขˆ†1ฤ‘?‘š 0, ฤ‘?‘ฤฝ0 , ฤ‘?‘ลš0 รข‰ยค ฤ‘?‘‘ฤ‘?‘š รข‹ลป 3.5 รขˆ†2 0, ฤ‘?‘ฤฝ0 , ฤ‘?‘ลš0 รขˆ’ รขˆ†2ฤ‘?‘š 0, ฤ‘?‘ฤฝ0 , ฤ‘?‘ลš0 รข‰ยค ฤ‘?œ‚ฤ‘?‘š รข‹ลป 3.6 satisfied for all ฤ‘?‘š รข‰ฤฝ 0 , ฤ‘?‘ฤฝ0 รขˆˆ ฤ‘??ห‡ฤ‘?‘“1 and ฤ‘?‘ลš0 รขˆˆ ฤ‘??ห‡1ฤ‘?‘“2 , where ฤ‘?›ลพ ฤ‘?›ลพ ฤ‘?‘‘ฤ‘?‘š = [ฤ‘??ลพ1 + ฤ‘?‘…1 ( ฤ‘??ลพ3 + ฤ‘?‘•ฤ‘??ลพ4 )] [ฤ‘??ลพ2 + ฤ‘?‘…2 ( ฤ‘??ลพ3 + ฤ‘?‘•ฤ‘??ลพ4 )] , ฤ‘?œ†1 ฤ‘?œ†1 , ฤ‘?‘„0 ฤ‘?‘š +1 ฤ‘??ยธ รขˆ’ ฤ‘?‘„0 รขˆ’1 ฤ‘??ล›0 and ฤ‘?œ‚ฤ‘?‘š =

ฤ‘?›ลผ

[ฤ‘??ลผ1 + ฤ‘??ลฅ1 ( ฤ‘??ลผ3 + ฤ‘?‘•ฤ‘??ลผ4 )] ฤ‘?œ†2

ฤ‘?›ลผ

[ฤ‘??ลผ2 + ฤ‘??ลฅ2 ( ฤ‘??ลผ3 + ฤ‘?‘•ฤ‘??ลผ4 )] , ฤ‘?œ†2

, ฤ‘?‘„0 ฤ‘?‘š +1 ฤ‘??ยธ รขˆ’ ฤ‘?‘„0 รขˆ’1 ฤ‘??ล›0 . Proof. By using the relation ( 3.1) and ( 3.3), we have: รขˆ†1 0, ฤ‘?‘ฤฝ0 , ฤ‘?‘ลš0 รขˆ’ รขˆ†1ฤ‘?‘š 0, ฤ‘?‘ฤฝ0 , ฤ‘?‘ลš0 ฤ‘?›ลพ รข‰ยค [ฤ‘??ลพ1 + ฤ‘?‘…1 ( ฤ‘??ลพ3 + ฤ‘?‘•ฤ‘??ลพ4 )] ฤ‘?‘ฤฝ 0 ฤ‘?‘ฤ„, ฤ‘?‘ฤฝ0 , ฤ‘?‘ลš0 รขˆ’ ฤ‘?‘ฤฝฤ‘?‘š ฤ‘?‘ฤ„, ฤ‘?‘ฤฝ0 , ฤ‘?‘ลš0 + ฤ‘?œ†1 ฤ‘?›ลพ +[ฤ‘??ลพ2 + ฤ‘?‘…2 ( ฤ‘??ลพ3 + ฤ‘?‘•ฤ‘??ลพ4 )] ฤ‘?‘ลš 0 ฤ‘?‘ฤ„, ฤ‘?‘ฤฝ0 , ฤ‘?‘ลš0 รขˆ’ ฤ‘?‘ลšฤ‘?‘š ฤ‘?‘ฤ„, ฤ‘?‘ฤฝ0 , ฤ‘?‘ลš0 ฤ‘?œ†1 ฤ‘?›ลพ ฤ‘?›ลพ รข‰ยค [ฤ‘??ลพ1 + ฤ‘?‘…1 ( ฤ‘??ลพ3 + ฤ‘?‘•ฤ‘??ลพ4 )] [ฤ‘??ลพ2 + ฤ‘?‘…2 ( ฤ‘??ลพ3 + ฤ‘?‘•ฤ‘??ลพ4 )] , ฤ‘?œ†1 ฤ‘?œ†1 , ฤ‘?‘„0 ฤ‘?‘š +1 ฤ‘??ยธ รขˆ’ ฤ‘?‘„0 รขˆ’1 ฤ‘??ล›0 = ฤ‘?‘‘ฤ‘?‘š And also by using the relation ( 3.2) and ( 3.4), we get: รขˆ†2 0, ฤ‘?‘ฤฝ0 , ฤ‘?‘ลš0 รขˆ’ รขˆ†2ฤ‘?‘š 0, ฤ‘?‘ฤฝ0 , ฤ‘?‘ลš0 รข‰ยค ฤ‘?›ลผ ฤ‘?›ลผ รข‰ยค [ฤ‘??ลผ1 + ฤ‘??ลฅ1 ( ฤ‘??ลผ3 + ฤ‘?‘•ฤ‘??ลผ4 )] [ฤ‘??ลผ2 + ฤ‘??ลฅ2 ( ฤ‘??ลผ3 + ฤ‘?‘•ฤ‘??ลผ4 )] , ฤ‘?œ†2 ฤ‘?œ†2 , ฤ‘?‘„0 ฤ‘?‘š +1 ฤ‘??ยธ รขˆ’ ฤ‘?‘„0 รขˆ’1 ฤ‘??ล›0 = ฤ‘?œ‚ฤ‘?‘š where . denotes the ordinary scalar product in the space ฤ‘?‘…ฤ‘?‘› . รขˆŽ Theorem 3.2. Let the vector functions ฤ‘?‘“1 ฤ‘?‘ฤ„, ฤ‘?‘ฤฝ, ฤ‘?‘ลš, ฤ‘?‘ยง, ฤ‘?‘ยค , ฤ‘?‘“2 ฤ‘?‘ฤ„, ฤ‘?‘ฤฝ, ฤ‘?‘ลš, ฤ‘?‘ห˜, ฤ‘?‘ล , ฤ‘?‘”1 ฤ‘?‘ฤ„, ฤ‘?‘ฤฝ, ฤ‘?‘ลš and ฤ‘?‘”2 ฤ‘?‘ฤ„, ฤ‘?‘ฤฝ, ฤ‘?‘ลš be defined on the domain: ฤ‘??ลŸ = 0 รข‰ยค ฤ‘?‘ รข‰ยค ฤ‘?‘ฤ„ รข‰ยค ฤ‘?‘‡, ฤ‘?‘Ž รข‰ยค ฤ‘?‘ฤฝ, ฤ‘?‘ลš รข‰ยค ฤ‘?‘?, ฤ‘?‘? รข‰ยค ฤ‘?‘ยง, ฤ‘?‘ห˜ รข‰ยค ฤ‘?‘‘, ฤ‘?‘’ รข‰ยค ฤ‘?‘ยค, ฤ‘?‘ล รข‰ยค ฤ‘?‘“} รขŠ† ฤ‘?‘…1 , and periodic in t of period T. Assume that the sequence of functions (4.3.3) and (4.3.4) satisfies the inequalities: min ฤ‘?‘‡ รขˆ†1m 0, x0 , y0 รข‰ยค รขˆ’ ฤ‘?‘‘m , ฤ‘?‘‡ a+ ฤ‘?‘€1 รข‰ยคx 0 รข‰ยคbรขˆ’ ฤ‘?‘€1 2 2 ฤ‘?‘‡ ฤ‘?‘‡ c+ ฤ‘?‘€2 รข‰ยคy 0 รข‰ยคdรขˆ’ ฤ‘?‘€2 2 2

max

ฤ‘?‘‡ ฤ‘?‘‡ a+ ฤ‘?‘€1 รข‰ยคx 0 รข‰ยคbรขˆ’ ฤ‘?‘€1 2 2 ฤ‘?‘‡ ฤ‘?‘‡ c+ ฤ‘?‘€2 รข‰ยคy 0 รข‰ยคdรขˆ’ ฤ‘?‘€2 2 2

รขˆ†1m 0, x0 , y0 รข‰ฤฝ ฤ‘?‘‘m min

รขˆ†2m 0, x0 , y0 รข‰ยค รขˆ’ รŽห‡m ,

max

รขˆ†2m 0, x0 , y0 รข‰ฤฝ รŽห‡m

ฤ‘?‘‡ ฤ‘?‘‡ a+ ฤ‘?‘€1 รข‰ยคx 0 รข‰ยคbรขˆ’ ฤ‘?‘€1 2 2 ฤ‘?‘‡ ฤ‘?‘‡ c+ ฤ‘?‘€2 รข‰ยคy 0 รข‰ยคdรขˆ’ ฤ‘?‘€2 2 2

for all ฤ‘?‘š รข‰ฤฝ 0, where ฤ‘?›ลพ ฤ‘?‘‘ฤ‘?‘š = [ฤ‘??ลพ1 + ฤ‘?‘…1 ( ฤ‘??ลพ3 + ฤ‘?‘•ฤ‘??ลพ4 )] ฤ‘?œ†1

รข‹ลป 3.7

,

ฤ‘?‘‡ ฤ‘?‘‡ a+ ฤ‘?‘€1 รข‰ยคx 0 รข‰ยคbรขˆ’ ฤ‘?‘€1 2 2 ฤ‘?‘‡ ฤ‘?‘‡ c+ ฤ‘?‘€2 รข‰ยคy 0 รข‰ยคdรขˆ’ ฤ‘?‘€2 2 2

,

รข‹ลป 3.8

ฤ‘?›ลพ [ฤ‘??ลพ2 + ฤ‘?‘…2 ( ฤ‘??ลพ3 + ฤ‘?‘•ฤ‘??ลพ4 )] , ฤ‘?œ†1 , ฤ‘?‘„0 ฤ‘?‘š +1 ฤ‘??ยธ รขˆ’ ฤ‘?‘„0 รขˆ’1 ฤ‘??ล›0

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Periodic Solutions for Non-Linear Systems of Integral Equations ฤ‘?›ลผ ฤ‘??ลผ + ฤ‘?‘•ฤ‘??ลผ4 )] , ฤ‘?œ†2 3 ฤ‘?‘š +1 , ฤ‘?‘„0 ฤ‘??ยธ รขˆ’ ฤ‘?‘„0 รขˆ’1 ฤ‘??ล›0 Then the system (4.1.1) and (4.1.2) has periodic solution of period T ฤ‘?‘‡ ฤ‘?‘‡ ฤ‘?‘ฤฝ = ฤ‘?‘ฤฝ(ฤ‘?‘ฤ„, ฤ‘?‘ฤฝ0 , ฤ‘?‘ลš0 ) and ฤ‘?‘ลš = ฤ‘?‘ลš(ฤ‘?‘ฤ„, ฤ‘?‘ฤฝ0 , ฤ‘?‘ลš0 ) for which ฤ‘?‘ฤฝ0 รขˆˆ [ฤ‘?‘Ž + ฤ‘?‘€1 , ฤ‘?‘? รขˆ’ ฤ‘?‘€1 ] and 2 2 ฤ‘?‘‡ ฤ‘?‘‡ ฤ‘?‘ลš0 รขˆˆ [ฤ‘?‘? + ฤ‘?‘€2 , ฤ‘?‘‘ รขˆ’ ฤ‘?‘€2 ]. 2 2 ฤ‘?‘‡ ฤ‘?‘‡ Proof. Let ฤ‘?‘ฤฝ1 , ฤ‘?‘ฤฝ2 be any two points in the interval [ฤ‘?‘Ž + ฤ‘?‘€1 , ฤ‘?‘? รขˆ’ ฤ‘?‘€1 ] and ฤ‘?‘ลš1 , ฤ‘?‘ลš2 be any two points in the ฤ‘?œ‚ฤ‘?‘š =

ฤ‘?›ลผ [ฤ‘??ลผ1 + ฤ‘??ลฅ1 ( ฤ‘??ลผ3 + ฤ‘?‘•ฤ‘??ลผ4 )] ฤ‘?œ†2

ฤ‘?‘‡

[ฤ‘??ลผ2 + ฤ‘??ลฅ2 (

2

ฤ‘?‘‡

2

interval [ฤ‘?‘? + ฤ‘?‘€2 , ฤ‘?‘‘ รขˆ’ ฤ‘?‘€2 ], 2 2 such that: รŽ”1m 0, ฤ‘?‘ฤฝ1 , ฤ‘?‘ลš1 =

รŽ”1m 0, ฤ‘?‘ฤฝ2 , ฤ‘?‘ลš2 = รŽ”2m 0, ฤ‘?‘ฤฝ1 , ฤ‘?‘ลš1 =

รŽ”2m 0, ฤ‘?‘ฤฝ2 , ฤ‘?‘ลš2 =

min

รขˆ†2m 0, x0 , y0

,

max

รขˆ†2m 0, x0 , y0

,

ฤ‘?‘‡ ฤ‘?‘‡ a+ ฤ‘?‘€1 รข‰ยคx 0 รข‰ยคbรขˆ’ ฤ‘?‘€1 2 2 ฤ‘?‘‡ ฤ‘?‘‡ c+ ฤ‘?‘€2 รข‰ยคy 0 รข‰ยคdรขˆ’ ฤ‘?‘€2 2 2 ฤ‘?‘‡ ฤ‘?‘‡ a+ ฤ‘?‘€1 รข‰ยคx 0 รข‰ยคbรขˆ’ ฤ‘?‘€1 2 2 ฤ‘?‘‡ ฤ‘?‘‡ c+ ฤ‘?‘€2 รข‰ยคy 0 รข‰ยคdรขˆ’ ฤ‘?‘€2 2 2

min

รขˆ†1m 0, ฤ‘?‘ฤฝ0 , ฤ‘?‘ลš0

max

รขˆ†1ฤ‘?‘š 0, ฤ‘?‘ฤฝ0 , ฤ‘?‘ลš0

ฤ‘?‘‡ ฤ‘?‘‡ a+ ฤ‘?‘€1 รข‰ยคx 0 รข‰ยคbรขˆ’ ฤ‘?‘€1 2 2 ฤ‘?‘‡ ฤ‘?‘‡ c+ ฤ‘?‘€2 รข‰ยคy 0 รข‰ยคdรขˆ’ ฤ‘?‘€2 2 2 ฤ‘?‘‡ ฤ‘?‘‡ a+ ฤ‘?‘€1 รข‰ยคx 0 รข‰ยคbรขˆ’ ฤ‘?‘€1 2 2 ฤ‘?‘‡ ฤ‘?‘‡ c+ ฤ‘?‘€2 รข‰ยคy 0 รข‰ยคdรขˆ’ ฤ‘?‘€2 2 2

,

,

รข‹ลป 3.9

รข‹ลป 3.10

By using the inequalities ( 3.5), ( 3.6), ( 3.7) and ( 3.8), we have: รขˆ†1 0, ฤ‘?‘ฤฝ1 , ฤ‘?‘ลš1 = รขˆ†1ฤ‘?‘š 0, ฤ‘?‘ฤฝ1 , ฤ‘?‘ลš1 + รขˆ†1 0, ฤ‘?‘ฤฝ1 , ฤ‘?‘ลš1 รขˆ’ รขˆ†1ฤ‘?‘š (0, ฤ‘?‘ฤฝ1 , ฤ‘?‘ลš1 ) รข‰ยค 0 , รขˆ†1 0, ฤ‘?‘ฤฝ2 , ฤ‘?‘ลš2 = รขˆ†1ฤ‘?‘š 0, ฤ‘?‘ฤฝ2 , ฤ‘?‘ลš2 + รขˆ†1 0, ฤ‘?‘ฤฝ2 , ฤ‘?‘ลš2 รขˆ’ รขˆ†1ฤ‘?‘š (0, ฤ‘?‘ฤฝ2 , ฤ‘?‘ลš2 ) รข‰ยค 0 . รข‹ลป 3.11 รขˆ†2 0, ฤ‘?‘ฤฝ1 , ฤ‘?‘ลš1 = รขˆ†2ฤ‘?‘š 0, ฤ‘?‘ฤฝ1 , ฤ‘?‘ลš1 + รขˆ†2 0, ฤ‘?‘ฤฝ1 , ฤ‘?‘ลš1 รขˆ’ รขˆ†2ฤ‘?‘š (0, ฤ‘?‘ฤฝ1 , ฤ‘?‘ลš1 ) รข‰ยค 0 , รขˆ†2 0, ฤ‘?‘ฤฝ2 , ฤ‘?‘ลš2 = รขˆ†2ฤ‘?‘š 0, ฤ‘?‘ฤฝ2 , ฤ‘?‘ลš2 + รขˆ†2 0, ฤ‘?‘ฤฝ2 , ฤ‘?‘ลš2 รขˆ’ รขˆ†2ฤ‘?‘š (0, ฤ‘?‘ฤฝ2 , ฤ‘?‘ลš2 ) รข‰ยค 0 . รข‹ลป 3.12 It follows from the inequalities ( 3.11) and ( 3.12) in virtue of the continuity of the functions รขˆ†1 (0, ฤ‘?‘ฤฝ0 , ฤ‘?‘ลš0 ) and รขˆ†1 (0, ฤ‘?‘ฤฝ0 , ฤ‘?‘ลš0 ) that there exists an isolated singular point (ฤ‘?‘ฤฝ 0 , ฤ‘?‘ลš 0 ) = (ฤ‘?‘ฤฝ0 , ฤ‘?‘ลš0 ) , ฤ‘?‘ฤฝ 0 รขˆˆ ฤ‘?‘ฤฝ1 , ฤ‘?‘ฤฝ2 and ฤ‘?‘ลš 0 รขˆˆ ฤ‘?‘ลš1 , ฤ‘?‘ลš2 , so that รขˆ†1 (0, ฤ‘?‘ฤฝ 0 , ฤ‘?‘ลš 0 ) = 0 and รขˆ†2 (0, ฤ‘?‘ฤฝ 0 , ฤ‘?‘ลš 0 ) = 0. This means that the system ( 3.1) and (4.3.2) has a periodic solutions ฤ‘?‘ฤฝ ฤ‘?‘ฤ„, ฤ‘?‘ฤฝ0 , ฤ‘?‘ลš0 , ฤ‘?‘ลš ฤ‘?‘ฤ„, ฤ‘?‘ฤฝ0 , ฤ‘?‘ลš0 for which ฤ‘?‘‡ ฤ‘?‘‡ ฤ‘?‘‡ ฤ‘?‘‡ ฤ‘?‘ฤฝ0 รขˆˆ [ฤ‘?‘Ž + ฤ‘?‘€1 , ฤ‘?‘? รขˆ’ ฤ‘?‘€1 ] and ฤ‘?‘ลš0 รขˆˆ [ฤ‘?‘? + ฤ‘?‘€2 , ฤ‘?‘‘ รขˆ’ ฤ‘?‘€2 ]. รขˆŽ 2

2

2

2

Remark 3.1. Theorem 3.2 is proved when ฤ‘?‘…ฤ‘?‘› = ฤ‘?‘…1 , on the other hand as ฤ‘?‘ฤฝ0 , ฤ‘?‘ลš0 are a scalar singular point which should be isolated (For this remark, see [5]).

IV. Stability Theorem Of Solution (ฤ‘?‘ยฐฤ‘?&#x;? ) And (ฤ‘?‘ยฐฤ‘?&#x;? )

In this section, we study theorem on stability of a periodic solution for the integral equations ( ฤ‘??ลบ1 ) and ( ฤ‘??ลบ2 ). Theorem 4.1. If the function รขˆ†1 0, ฤ‘?‘ฤฝ0 , ฤ‘?‘ลš0 , รŽ”2 (0, ฤ‘?‘ฤฝ0 , ฤ‘?‘ลš0 ) are defined by equations ( 3.1) and ( 3.2), where the function ฤ‘?‘ฤฝ 0 (ฤ‘?‘ฤ„, ฤ‘?‘ฤฝ0 , ฤ‘?‘ลš0 ) is a limit of the sequence of the functions ( 2.1) , the function ฤ‘?‘ลš 0 (ฤ‘?‘ฤ„, ฤ‘?‘ฤฝ0 , ฤ‘?‘ลš0 ) is the limit of the sequence of the functions ( 2.2) , Then the following inequalities yields: รขˆ†1 0, ฤ‘?‘ฤฝ0 , ฤ‘?‘ลš0 รข‰ยค ฤ‘?‘€1 รข‹ลป 4.1 รขˆ†2 0, ฤ‘?‘ฤฝ0 , ฤ‘?‘ลš0 รข‰ยค ฤ‘?‘€2 รข‹ลป 4.2 and รขˆ†2 0, ฤ‘?‘ฤฝ01 , ฤ‘?‘ลš0 1 รขˆ’ รขˆ†2 0, ฤ‘?‘ฤฝ02 , ฤ‘?‘ลš0 2

รข‰ยค ฤ‘??ลก1 ฤ‘??ลก2 ฤ‘??ยธ3 ฤ‘??ลก01 ฤ‘?‘ฤ„ รขˆ’ ฤ‘??ลก02 ฤ‘?‘ฤ„ + ฤ‘?‘‡ ฤ‘?‘‡ + ฤ‘??ลก1 ฤ‘??ลก2 ฤ‘??ยธ2 ( ฤ‘??ยธ3 + ฤ‘??ยธ4 + ฤ‘??ลก1 ฤ‘??ยธ4 (1 รขˆ’ ฤ‘??ยธ1 )) ฤ‘??ลŸ0 1 ฤ‘?‘ฤ„ รขˆ’ ฤ‘??ลŸ0 2 (ฤ‘?‘ฤ„) 2 2 รข‹ลป 4.4 ฤ‘?‘ฤ„ ฤ‘?‘‡ for all ฤ‘?‘ฤฝ 0 , ฤ‘?‘ฤฝ01 , ฤ‘?‘ฤฝ02 รขˆˆ ฤ‘??ห‡ฤ‘?‘“1 , ฤ‘?‘ลš 0 , ฤ‘?‘ลš01 , ฤ‘?‘ลš02 รขˆˆ ฤ‘??ห‡1ฤ‘?‘“2 , and ฤ‘?›ลบ ฤ‘?‘ฤ„ = 2ฤ‘?‘ฤ„(1 รขˆ’ ) รข‰ยค , ฤ‘?›ลพ

ฤ‘?›ลพ

ฤ‘?œ†1

ฤ‘?œ†1

ฤ‘?‘‡

2

where ฤ‘??ยธ1 = [ฤ‘??ลพ1 + ฤ‘?‘…1 ( ฤ‘??ลพ3 + ฤ‘?‘•ฤ‘??ลพ4 )], ฤ‘??ยธ2 = [ฤ‘??ลพ2 + ฤ‘?‘…2 ( ฤ‘??ลพ3 + ฤ‘?‘•ฤ‘??ลพ4 )],

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Periodic Solutions for Non-Linear Systems of Integral Equations ๐›ฟ

๐›ฟ

๐œ†2

๐œ†2 2

๐ธ3 = [๐ฟ1 + ๐ป1 ( ๐ฟ3 + ๐‘•๐ฟ4 )], ๐ธ4 = [๐ฟ2 + ๐ป2 ( ๐ฟ3 + ๐‘•๐ฟ4 )], ๐‘‡ ๐‘‡ ๐‘‡ ๐ธ )(1 โˆ’ ๐ธ4 )]โˆ’1 and ๐น2 = (1 โˆ’ ๐‘2 ๐‘3 ๐น1 )โˆ’1 2 1 2 4 Proof. From the properties of the functions ๐‘ฅ 0 ๐‘ก, ๐‘ฅ0 , ๐‘ฆ0 and ๐‘ฆ 0 ๐‘ก, ๐‘ฅ0 , ๐‘ฆ0 as in the theorem 4.2.1, the functions ฮ”1 = โˆ†1 ๐‘ฅ0 , ๐‘ฆ0 , ฮ”1 = โˆ†1 ๐‘ฅ0 , ๐‘ฆ0 , ๐‘ฅ0 โˆˆ ๐ท๐‘“1 , ๐‘ฆ0 โˆˆ ๐ท1๐‘“2 are continuous and bounded by ๐‘€1 , ๐‘€2 in the domain ( 1.3). ๐น1 = [(1 โˆ’

From relation ( 3.1), we find: โˆ†1 (0, ๐‘ฅ0 , ๐‘ฆ0

1 โ‰ค ๐‘‡

๐‘‡

๐‘“1 (๐‘ก, ๐‘ฅ 0 ๐‘ก, ๐‘ฅ0 , ๐‘ฆ0 , ๐‘ฆ 0 ๐‘ก, ๐‘ฅ0 , ๐‘ฆ0 , 0

๐‘ก

๐บ1 ๐‘ก, ๐‘  ๐‘”1 ๐‘ , ๐‘ฅ 0 ๐‘ , ๐‘ฅ0 , ๐‘ฆ0 , ๐‘ฆ 0 ๐‘ , ๐‘ฅ0 , ๐‘ฆ0 ๐‘‘๐‘ ,

,

โˆ’โˆž ๐‘(๐‘ก)

๐‘”1 ๐‘ , ๐‘ฅ 0 ๐‘ , ๐‘ฅ0 , ๐‘ฆ0 , ๐‘ฆ 0 ๐‘ , ๐‘ฅ0 , ๐‘ฆ0 ๐‘‘ ๐‘ ) ๐‘‘๐‘ก

, ๐‘Ž(๐‘ก)

by using the Lemma 4.1.1, gives: โˆ†1 (0, ๐‘ฅ0 , ๐‘ฆ0 ) โ‰ค ๐‘€1 . And from relation ( 3.2), we get: โˆ†2 (0, ๐‘ฅ0 , ๐‘ฆ0

1 โ‰ค ๐‘‡

๐‘‡

๐‘“2 (๐‘ก, ๐‘ฅ 0 ๐‘ก, ๐‘ฅ0 , ๐‘ฆ0 , ๐‘ฆ 0 ๐‘ก, ๐‘ฅ0 , ๐‘ฆ0 , 0

๐‘ก

๐บ2 ๐‘ก, ๐‘  ๐‘”2 ๐‘ , ๐‘ฅ 0 ๐‘ , ๐‘ฅ0 , ๐‘ฆ0 , ๐‘ฆ 0 ๐‘ , ๐‘ฅ0 , ๐‘ฆ0 ๐‘‘๐‘ ,

,

โˆ’โˆž ๐‘(๐‘ก)

๐‘”2 ๐‘ , ๐‘ฅ 0 ๐‘ , ๐‘ฅ0 , ๐‘ฆ0 , ๐‘ฆ 0 ๐‘ , ๐‘ฅ0 , ๐‘ฆ0 ๐‘‘๐‘ ) ๐‘‘๐‘ก

, ๐‘Ž(๐‘ก)

and using Lemma 4.1.1, we have: โˆ†2 (0, ๐‘ฅ0 , ๐‘ฆ0 ) โ‰ค ๐‘€2 By using equation ( 3.1) and lemma 1.1, we get: โˆ†1 0, ๐‘ฅ01 , ๐‘ฆ0 1 โˆ’ โˆ†1 0, ๐‘ฅ02 , ๐‘ฆ0 2 1 โ‰ค ๐‘‡

๐‘‡

[๐พ1 ๐‘ฅ 0 ๐‘ก, ๐‘ฅ01 , ๐‘ฆ0 1 โˆ’ ๐‘ฅ 0 ๐‘ก, ๐‘ฅ02 , ๐‘ฆ0 2

+

0

+๐พ2 ๐‘ฆ 0 ๐‘ก, ๐‘ฅ01 , ๐‘ฆ0 1 โˆ’ ๐‘ฆ 0 ๐‘ก, ๐‘ฅ02 , ๐‘ฆ0 2 + ๐›พ + ๐พ3 (๐‘…1 ๐‘ฅ 0 ๐‘ก, ๐‘ฅ01 , ๐‘ฆ0 1 โˆ’ ๐‘ฅ 0 ๐‘ก, ๐‘ฅ02 , ๐‘ฆ0 2 + ๐œ†1 +๐‘…2 ๐‘ฆ 0 ๐‘ก, ๐‘ฅ01 , ๐‘ฆ0 1 โˆ’ ๐‘ฆ 0 ๐‘ก, ๐‘ฅ02 , ๐‘ฆ0 2 ) + + ๐‘•๐พ4 (๐‘…1 ๐‘ฅ 0 ๐‘ก, ๐‘ฅ01 , ๐‘ฆ0 1 โˆ’ ๐‘ฅ 0 ๐‘ก, ๐‘ฅ02 , ๐‘ฆ0 2 + +๐‘…2 ๐‘ฆ 0 ๐‘ก, ๐‘ฅ01 , ๐‘ฆ0 1 โˆ’ ๐‘ฆ 0 ๐‘ก, ๐‘ฅ02 , ๐‘ฆ0 2 )]๐‘‘๐‘ก ๐›พ โ‰ค [๐พ1 + ๐‘…1 ( ๐พ3 + ๐‘•๐พ4 )] ๐‘ฅ 0 ๐‘ก, ๐‘ฅ01 , ๐‘ฆ0 1 โˆ’ ๐‘ฅ 0 ๐‘ก, ๐‘ฅ02 , ๐‘ฆ0 2 + ๐œ†1 ๐›พ +[๐พ2 + ๐‘…2 ( ๐พ3 + ๐‘•๐พ4 )] ๐‘ฆ 0 ๐‘ก, ๐‘ฅ01 , ๐‘ฆ0 1 โˆ’ ๐‘ฆ 0 ๐‘ก, ๐‘ฅ02 , ๐‘ฆ0 2 ๐œ†1 so, โˆ†1 0, ๐‘ฅ01 , ๐‘ฆ0 1 โˆ’ โˆ†1 0, ๐‘ฅ02 , ๐‘ฆ0 2 โ‰ค ๐ธ1 ๐‘ฅ 0 ๐‘ก, ๐‘ฅ01 , ๐‘ฆ0 1 โˆ’ ๐‘ฅ 0 ๐‘ก, ๐‘ฅ02 , ๐‘ฆ0 2 + +๐ธ2 ๐‘ฆ 0 ๐‘ก, ๐‘ฅ01 , ๐‘ฆ0 1 โˆ’ ๐‘ฆ 0 ๐‘ก, ๐‘ฅ02 , ๐‘ฆ0 2 โ‹ฏ 4.5 And also by using equation ( 3.2) and lemma 1.1, gives: โˆ†2 0, ๐‘ฅ01 , ๐‘ฆ0 1 โˆ’ โˆ†2 0, ๐‘ฅ02 , ๐‘ฆ0 2 1 โ‰ค ๐‘‡

๐‘‡

[๐ฟ1 ๐‘ฅ 0 ๐‘ก, ๐‘ฅ01 , ๐‘ฆ0 1 โˆ’ ๐‘ฅ 0 ๐‘ก, ๐‘ฅ02 , ๐‘ฆ0 2

+

0

+๐ฟ2 ๐‘ฆ 0 ๐‘ก, ๐‘ฅ01 , ๐‘ฆ0 1 โˆ’ ๐‘ฆ 0 ๐‘ก, ๐‘ฅ02 , ๐‘ฆ0 2

+ www.ijeijournal.com

Page | 13

Periodic Solutions for Non-Linear Systems of Integral Equations ๐›ฟ ๐ฟ (๐ป ๐‘ฅ 0 ๐‘ก, ๐‘ฅ01 , ๐‘ฆ0 1 โˆ’ ๐‘ฅ 0 ๐‘ก, ๐‘ฅ02 , ๐‘ฆ0 2 + ๐œ†2 3 1 +๐ป2 ๐‘ฆ 0 ๐‘ก, ๐‘ฅ01 , ๐‘ฆ0 1 โˆ’ ๐‘ฆ 0 ๐‘ก, ๐‘ฅ02 , ๐‘ฆ0 2 ) + + ๐‘•๐ฟ4 (๐ป1 ๐‘ฅ 0 ๐‘ก, ๐‘ฅ01 , ๐‘ฆ0 1 โˆ’ ๐‘ฅ 0 ๐‘ก, ๐‘ฅ02 , ๐‘ฆ0 2 + +๐ป2 ๐‘ฆ 0 ๐‘ก, ๐‘ฅ01 , ๐‘ฆ0 1 โˆ’ ๐‘ฆ 0 ๐‘ก, ๐‘ฅ02 , ๐‘ฆ0 2 )]๐‘‘๐‘ก ๐›ฟ โ‰ค [๐ฟ1 + ๐ป1 ( ๐ฟ3 + ๐‘•๐ฟ4 )] ๐‘ฅ 0 ๐‘ก, ๐‘ฅ01 , ๐‘ฆ0 1 โˆ’ ๐‘ฅ 0 ๐‘ก, ๐‘ฅ02 , ๐‘ฆ0 2 ๐œ†2 ๐›ฟ +[๐ฟ2 + ๐ป2 ( ๐ฟ3 + ๐‘•๐ฟ4 )] ๐‘ฆ 0 ๐‘ก, ๐‘ฅ01 , ๐‘ฆ0 1 โˆ’ ๐‘ฆ 0 ๐‘ก, ๐‘ฅ02 , ๐‘ฆ0 2 +

๐œ†2

+ .

Therefore, โˆ†2 0, ๐‘ฅ01 , ๐‘ฆ0 1 โˆ’ โˆ†2 0, ๐‘ฅ02 , ๐‘ฆ0 2

โ‰ค ๐ธ3 ๐‘ฅ 0 ๐‘ก, ๐‘ฅ01 , ๐‘ฆ0 1 โˆ’ ๐‘ฅ 0 ๐‘ก, ๐‘ฅ02 , ๐‘ฆ0 2 + +๐ธ4 ๐‘ฆ 0 ๐‘ก, ๐‘ฅ01 , ๐‘ฆ0 1 โˆ’ ๐‘ฆ 0 ๐‘ก, ๐‘ฅ02 , ๐‘ฆ0 2 โ‹ฏ 4.6 where the functions ๐‘ฅ 0 ๐‘ก, ๐‘ฅ01 , ๐‘ฆ0 1 , ๐‘ฆ 0 ๐‘ก, ๐‘ฅ01 , ๐‘ฆ0 1 , ๐‘ฅ 0 ๐‘ก, ๐‘ฅ02 , ๐‘ฆ0 2 and ๐‘ฆ 0 ๐‘ก, ๐‘ฅ02 , ๐‘ฆ0 2 are solutions of the equation: ๐‘ก

๐‘ฅ

๐‘ก, ๐‘ฅ0๐‘˜ , ๐‘ฆ0๐‘˜

=

๐น0๐‘˜

[๐‘“1 (๐‘ , ๐‘ฅ(๐‘ , ๐‘ฅ0๐‘˜ , ๐‘ฆ0๐‘˜ ), ๐‘ฆ(๐‘ , ๐‘ฅ0๐‘˜ , ๐‘ฆ0๐‘˜ ),

๐‘ก + 0

๐‘ 

๐บ1 ๐‘ , ๐œ ๐‘”1 (๐œ, ๐‘ฅ(๐œ, ๐‘ฅ0๐‘˜ , ๐‘ฆ0๐‘˜ ), ๐‘ฆ(๐œ, ๐‘ฅ0๐‘˜ , ๐‘ฆ0๐‘˜ )) ๐‘‘๐œ,

,

โˆ’โˆž ๐‘ ๐‘ 

๐‘”1 (๐œ, ๐‘ฅ(๐œ, ๐‘ฅ0๐‘˜ , ๐‘ฆ0๐‘˜ ), ๐‘ฆ(๐œ, ๐‘ฅ0๐‘˜ , ๐‘ฆ0๐‘˜ ))๐‘‘๐œ ) โˆ’

, ๐‘Ž ๐‘ 

1 โˆ’ ๐‘‡ ๐‘ 

๐‘‡

๐‘“1 (๐‘ , ๐‘ฅ(๐‘ , ๐‘ฅ0๐‘˜ , ๐‘ฆ0๐‘˜ ), ๐‘ฆ(๐‘ , ๐‘ฅ0๐‘˜ , ๐‘ฆ0๐‘˜ ), 0

๐บ1 ๐‘ , ๐œ ๐‘”1 (๐œ, ๐‘ฅ(๐œ, ๐‘ฅ0๐‘˜ , ๐‘ฆ0๐‘˜ ), ๐‘ฆ(๐œ, ๐‘ฅ0๐‘˜ , ๐‘ฆ0๐‘˜ )) ๐‘‘๐œ,

, โˆ’โˆž

๐‘ ๐‘ 

๐‘”1 (๐œ, ๐‘ฅ(๐œ, ๐‘ฅ0๐‘˜ , ๐‘ฆ0๐‘˜ ), ๐‘ฆ(๐œ, ๐‘ฅ0๐‘˜ , ๐‘ฆ0๐‘˜ ))๐‘‘๐œ)๐‘‘๐‘ ]๐‘‘๐‘ 

,

โ‹ฏ ( 4.7)

๐‘Ž ๐‘ 

and ๐‘ก

๐‘ฆ

๐‘ก, ๐‘ฅ0๐‘˜ , ๐‘ฆ0๐‘˜

=

๐บ0๐‘˜

[๐‘“2 (๐‘ , ๐‘ฅ(๐‘ , ๐‘ฅ0๐‘˜ , ๐‘ฆ0๐‘˜ ), ๐‘ฆ(๐‘ , ๐‘ฅ0๐‘˜ , ๐‘ฆ0๐‘˜ ),

๐‘ก + 0

๐‘ 

๐บ2 ๐‘ , ๐œ ๐‘”2 (๐œ, ๐‘ฅ(๐œ, ๐‘ฅ0๐‘˜ , ๐‘ฆ0๐‘˜ ), ๐‘ฆ(๐œ, ๐‘ฅ0๐‘˜ , ๐‘ฆ0๐‘˜ )) ๐‘‘๐œ,

, โˆ’โˆž ๐‘ ๐‘ 

๐‘”2 (๐œ, ๐‘ฅ(๐œ, ๐‘ฅ0๐‘˜ , ๐‘ฆ0๐‘˜ ), ๐‘ฆ(๐œ, ๐‘ฅ0๐‘˜ , ๐‘ฆ0๐‘˜ ))๐‘‘๐œ) โˆ’

, ๐‘Ž ๐‘ 

1 โˆ’ ๐‘‡ ๐‘ 

๐‘‡

๐‘“2 (๐‘ , ๐‘ฅ(๐‘ , ๐‘ฅ0๐‘˜ , ๐‘ฆ0๐‘˜ ), ๐‘ฆ(๐‘ , ๐‘ฅ0๐‘˜ , ๐‘ฆ0๐‘˜ ), 0

๐บ2 ๐‘ , ๐œ ๐‘”2 (๐œ, ๐‘ฅ(๐œ, ๐‘ฅ0๐‘˜ , ๐‘ฆ0๐‘˜ ), ๐‘ฆ(๐œ, ๐‘ฅ0๐‘˜ , ๐‘ฆ0๐‘˜ )) ๐‘‘๐œ,

, โˆ’โˆž

๐‘ ๐‘ 

๐‘”2 (๐œ, ๐‘ฅ(๐œ, ๐‘ฅ0๐‘˜ , ๐‘ฆ0๐‘˜ ), ๐‘ฆ(๐œ, ๐‘ฅ0๐‘˜ , ๐‘ฆ0๐‘˜ ))๐‘‘๐œ)๐‘‘๐‘ ]๐‘‘๐‘ 

,

โ‹ฏ ( 4.8)

๐‘Ž ๐‘ 

where ๐‘˜ = 1, 2. From (4.4.7), we get: โ‰ค

๐น01

๐‘ก โˆ’

๐น02

๐‘ก

๐‘ฅ 0 ๐‘ก, ๐‘ฅ01 , ๐‘ฆ0 1 โˆ’๐‘ฅ 0 ๐‘ก, ๐‘ฅ02 , ๐‘ฆ0 2 + www.ijeijournal.com

Page | 14

Periodic Solutions for Non-Linear Systems of Integral Equations ๐›พ +๐›ผ(๐‘ก)[๐พ1 + ๐‘…1 ( ๐พ3 + ๐‘•๐พ4 )] ๐‘ฅ 0 ๐‘ก, ๐‘ฅ01 , ๐‘ฆ0 1 โˆ’ ๐‘ฅ 0 ๐‘ก, ๐‘ฅ02 , ๐‘ฆ0 2 + ๐œ†1 ๐›พ +๐›ผ ๐‘ก [๐พ2 + ๐‘…2 ( ๐พ3 + ๐‘•๐พ4 )] ๐‘ฆ 0 ๐‘ก, ๐‘ฅ01 , ๐‘ฆ0 1 โˆ’ ๐‘ฆ 0 ๐‘ก, ๐‘ฅ02 , ๐‘ฆ0 2 ๐œ†1 โ‰ค ๐น01 ๐‘ก โˆ’ ๐น02 ๐‘ก + ๐‘‡ ๐›พ + [๐พ1 + ๐‘…1 ( ๐พ3 + ๐‘•๐พ4 )] ๐‘ฅ 0 ๐‘ก, ๐‘ฅ01 , ๐‘ฆ0 1 โˆ’ ๐‘ฅ 0 ๐‘ก, ๐‘ฅ02 , ๐‘ฆ0 2 + 2 ๐œ†1 ๐‘‡ ๐›พ + [๐พ2 + ๐‘…2 ( ๐พ3 + ๐‘•๐พ4 )] ๐‘ฆ 0 ๐‘ก, ๐‘ฅ01 , ๐‘ฆ0 1 โˆ’ ๐‘ฆ 0 ๐‘ก, ๐‘ฅ02 , ๐‘ฆ0 2 2 ๐œ†1 such that: ๐‘ฅ 0 ๐‘ก, ๐‘ฅ01 , ๐‘ฆ0 1 โˆ’๐‘ฅ 0 ๐‘ก, ๐‘ฅ02 , ๐‘ฆ0 2

โ‰ค ๐น01 ๐‘ก โˆ’ ๐น02 ๐‘ก + ๐‘‡ + ๐ธ1 ๐‘ฅ 0 ๐‘ก, ๐‘ฅ01 , ๐‘ฆ0 1 โˆ’ ๐‘ฅ 0 ๐‘ก, ๐‘ฅ02 , ๐‘ฆ0 2 + 2 ๐‘‡ + ๐ธ2 ๐‘ฆ 0 ๐‘ก, ๐‘ฅ01 , ๐‘ฆ0 1 โˆ’ ๐‘ฆ 0 ๐‘ก, ๐‘ฅ02 , ๐‘ฆ0 2 2

therefore:

๐‘‡ ๐ธ )โˆ’1 ๐น01 ๐‘ก โˆ’ ๐น02 ๐‘ก + 2 1 ๐‘‡ ๐‘‡ + ๐ธ2 (1 โˆ’ ๐ธ1 )โˆ’1 ๐‘ฆ 0 ๐‘ก, ๐‘ฅ01 , ๐‘ฆ0 1 โˆ’ ๐‘ฆ 0 ๐‘ก, ๐‘ฅ02 , ๐‘ฆ0 2 2 2 โ‹ฏ 4.9 And also from relation ( 4.8), we have: ๐‘ฆ 0 ๐‘ก, ๐‘ฅ01 , ๐‘ฆ0 1 โˆ’๐‘ฆ 0 ๐‘ก, ๐‘ฅ02 , ๐‘ฆ0 2 โ‰ค 1 2 โ‰ค ๐บ0 ๐‘ก โˆ’ ๐บ0 ๐‘ก + ๐›ฟ +๐›ผ(๐‘ก)[๐ฟ1 + ๐ป1 ( ๐ฟ3 + ๐‘•๐ฟ4 )] ๐‘ฅ 0 ๐‘ก, ๐‘ฅ01 , ๐‘ฆ0 1 โˆ’ ๐‘ฅ 0 ๐‘ก, ๐‘ฅ02 , ๐‘ฆ0 2 + ๐œ†2 ๐›ฟ +๐›ผ(๐‘ก)[๐ฟ2 + ๐ป2 ( ๐ฟ3 + ๐‘•๐ฟ4 )] ๐‘ฆ 0 ๐‘ก, ๐‘ฅ01 , ๐‘ฆ0 1 โˆ’ ๐‘ฆ 0 ๐‘ก, ๐‘ฅ02 , ๐‘ฆ0 2 โ‰ค ๐œ†2 โ‰ค ๐บ01 ๐‘ก โˆ’ ๐บ02 ๐‘ก + ๐‘‡ ๐›ฟ + [๐ฟ1 + ๐ป1 ( ๐ฟ3 + ๐‘•๐ฟ4 )] ๐‘ฅ 0 ๐‘ก, ๐‘ฅ01 , ๐‘ฆ0 1 โˆ’ ๐‘ฅ 0 ๐‘ก, ๐‘ฅ02 , ๐‘ฆ0 2 + 2 ๐œ†2 ๐‘‡ ๐›ฟ + [๐ฟ2 + ๐ป2 ( ๐ฟ3 + ๐‘•๐ฟ4 )] ๐‘ฆ 0 ๐‘ก, ๐‘ฅ01 , ๐‘ฆ0 1 โˆ’ ๐‘ฆ 0 ๐‘ก, ๐‘ฅ02 , ๐‘ฆ0 2 2 ๐œ†2 so that: ๐‘ฆ 0 ๐‘ก, ๐‘ฅ01 , ๐‘ฆ0 1 โˆ’๐‘ฆ 0 ๐‘ก, ๐‘ฅ02 , ๐‘ฆ0 2 โ‰ค ๐บ01 ๐‘ก โˆ’ ๐บ02 ๐‘ก + ๐‘‡ + ๐ธ3 ๐‘ฅ 0 ๐‘ก, ๐‘ฅ01 , ๐‘ฆ0 1 โˆ’ ๐‘ฅ 0 ๐‘ก, ๐‘ฅ02 , ๐‘ฆ0 2 + 2 ๐‘‡ + ๐ธ4 ๐‘ฆ 0 ๐‘ก, ๐‘ฅ01 , ๐‘ฆ0 1 โˆ’ ๐‘ฆ 0 ๐‘ก, ๐‘ฅ02 , ๐‘ฆ0 2 2 and hence: ๐‘‡ ๐‘ฆ 0 ๐‘ก, ๐‘ฅ01 , ๐‘ฆ0 1 โˆ’๐‘ฆ 0 ๐‘ก, ๐‘ฅ02 , ๐‘ฆ0 2 โ‰ค (1 โˆ’ ๐ธ4 )โˆ’1 ๐บ01 ๐‘ก โˆ’ ๐บ02 ๐‘ก + 2 ๐‘‡ ๐‘‡ + ๐ธ3 (1 โˆ’ ๐ธ4 )โˆ’1 ๐‘ฅ 0 ๐‘ก, ๐‘ฅ01 , ๐‘ฆ0 1 โˆ’ ๐‘ฅ 0 ๐‘ก, ๐‘ฅ02 , ๐‘ฆ0 2 2 2 โ‹ฏ 4.10 Now, by substituting inequality ( 4.10) in ( 4.9), we get: ๐‘‡ ๐‘ฅ 0 ๐‘ก, ๐‘ฅ01 , ๐‘ฆ0 1 โˆ’ ๐‘ฅ 0 ๐‘ก, ๐‘ฅ02 , ๐‘ฆ0 2 โ‰ค (1 โˆ’ ๐ธ1 )โˆ’1 ๐น01 ๐‘ก โˆ’ ๐น02 ๐‘ก + 2 ๐‘‡ ๐‘‡ ๐‘‡ + ๐ธ2 (1 โˆ’ ๐ธ1 )โˆ’1 [(1 โˆ’ ๐ธ4 )โˆ’1 ๐บ01 ๐‘ก โˆ’ ๐บ02 ๐‘ก + 2 2 2 ๐‘‡ ๐‘‡ โˆ’1 0 + ๐ธ3 (1 โˆ’ ๐ธ4 ) ๐‘ฅ ๐‘ก, ๐‘ฅ01 , ๐‘ฆ0 1 โˆ’ ๐‘ฅ 0 ๐‘ก, ๐‘ฅ02 , ๐‘ฆ0 2 ] 2 2 ๐‘‡ โ‰ค (1 โˆ’ ๐ธ1 )โˆ’1 ๐น01 ๐‘ก โˆ’ ๐น02 ๐‘ก + 2 ๐‘‡ ๐‘‡ ๐‘‡ + ๐ธ2 [(1 โˆ’ ๐ธ1 )(1 โˆ’ ๐ธ4 )]โˆ’1 ๐บ01 ๐‘ก โˆ’ ๐บ02 ๐‘ก + 2 2 2 ๐‘‡2 ๐‘‡ ๐‘‡ + ๐ธ2 ๐ธ3 [(1 โˆ’ ๐ธ1 )(1 โˆ’ ๐ธ4 )]โˆ’1 ๐‘ฅ 0 ๐‘ก, ๐‘ฅ01 , ๐‘ฆ0 1 โˆ’ ๐‘ฅ 0 ๐‘ก, ๐‘ฅ02 , ๐‘ฆ0 2 ๐‘ฅ 0 ๐‘ก, ๐‘ฅ01 , ๐‘ฆ0 1 โˆ’๐‘ฅ 0 ๐‘ก, ๐‘ฅ02 , ๐‘ฆ0 2

๐‘‡

โ‰ค (1 โˆ’

๐‘‡

4

2

2

]

putting ๐น1 = [(1 โˆ’ ๐ธ1 )(1 โˆ’ ๐ธ4 )]โˆ’1 , 2 2 and substituting in the last inequality, we obtain: www.ijeijournal.com

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Periodic Solutions for Non-Linear Systems of Integral Equations ฤ‘?‘‡ ฤ‘??ยธ )รขˆ’1 ฤ‘??ลก01 ฤ‘?‘ฤ„ รขˆ’ ฤ‘??ลก02 ฤ‘?‘ฤ„ + 2 1 ฤ‘?‘‡ ฤ‘?‘‡2 + ฤ‘??ยธ2 ฤ‘??ลก1 ฤ‘??ลŸ01 ฤ‘?‘ฤ„ รขˆ’ ฤ‘??ลŸ02 ฤ‘?‘ฤ„ + + ฤ‘??ยธ2 ฤ‘??ยธ3 ฤ‘??ลก1 ฤ‘?‘ฤฝ 0 ฤ‘?‘ฤ„, ฤ‘?‘ฤฝ01 , ฤ‘?‘ลš0 1 รขˆ’ ฤ‘?‘ฤฝ 0 ฤ‘?‘ฤ„, ฤ‘?‘ฤฝ02 , ฤ‘?‘ลš0 2

ฤ‘?‘ฤฝ 0 ฤ‘?‘ฤ„, ฤ‘?‘ฤฝ01 , ฤ‘?‘ลš0 1 รขˆ’ ฤ‘?‘ฤฝ 0 ฤ‘?‘ฤ„, ฤ‘?‘ฤฝ02 , ฤ‘?‘ลš0 2

as the ฤ‘?‘ฤฝ

0

ฤ‘?‘‡

รข‰ยค (1 รขˆ’ ฤ‘?‘‡

ฤ‘??ลก1 (1 รขˆ’ ฤ‘??ยธ4 ) = (1 รขˆ’ ฤ‘??ยธ1 ) 2

ฤ‘?‘ฤ„, ฤ‘?‘ฤฝ01 , ฤ‘?‘ลš0 1

รขˆ’ฤ‘?‘ฤฝ

2

0

ฤ‘?‘ฤ„, ฤ‘?‘ฤฝ02 , ฤ‘?‘ลš0 2

2 รขˆ’1

4

ฤ‘?‘‡ ฤ‘??ยธ ) ฤ‘??ลก01 ฤ‘?‘ฤ„ รขˆ’ ฤ‘??ลก02 ฤ‘?‘ฤ„ + 2 4 ฤ‘?‘‡ ฤ‘?‘‡2 + ฤ‘??ยธ2 ฤ‘??ลก1 ฤ‘??ลŸ01 ฤ‘?‘ฤ„ รขˆ’ ฤ‘??ลŸ02 ฤ‘?‘ฤ„ + ฤ‘??ยธ2 ฤ‘??ยธ3 ฤ‘??ลก1 ฤ‘?‘ฤฝ 0 ฤ‘?‘ฤ„, ฤ‘?‘ฤฝ01 , ฤ‘?‘ลš0 1 รขˆ’ ฤ‘?‘ฤฝ 0 ฤ‘?‘ฤ„, ฤ‘?‘ฤฝ02 , ฤ‘?‘ลš0 2

]

รข‰ยค ฤ‘??ลก1 (1 รขˆ’ 2

4

]

which implies that: ฤ‘?‘‡ ฤ‘?‘‡2 ฤ‘??ยธ4 )(1 รขˆ’ ฤ‘??ยธ2 ฤ‘??ยธ3 ฤ‘??ลก1 )รขˆ’1 ฤ‘??ลก01 ฤ‘?‘ฤ„ รขˆ’ ฤ‘??ลก02 ฤ‘?‘ฤ„ + 2 4 ฤ‘?‘‡ ฤ‘?‘‡2 + ฤ‘??ยธ2 ฤ‘??ลก1 (1 รขˆ’ ฤ‘??ยธ2 ฤ‘??ยธ3 ฤ‘??ลก1 )รขˆ’1 ฤ‘??ลŸ01 ฤ‘?‘ฤ„ รขˆ’ ฤ‘??ลŸ02 ฤ‘?‘ฤ„ 2 4 2 ฤ‘?‘‡ รขˆ’1 putting ฤ‘??ลก2 = (1 รขˆ’ ฤ‘??ยธ2 ฤ‘??ยธ3 ฤ‘??ลก1 ) 4 and substituting in the last inequality, we obtain: ฤ‘?‘‡ ฤ‘?‘ฤฝ 0 ฤ‘?‘ฤ„, ฤ‘?‘ฤฝ01 , ฤ‘?‘ลš0 1 รขˆ’ ฤ‘?‘ฤฝ 0 ฤ‘?‘ฤ„, ฤ‘?‘ฤฝ02 , ฤ‘?‘ลš0 2 รข‰ยค ฤ‘??ลก1 ฤ‘??ลก2 (1 รขˆ’ ฤ‘??ยธ4 ) ฤ‘??ลก01 ฤ‘?‘ฤ„ รขˆ’ ฤ‘??ลก02 ฤ‘?‘ฤ„ + 2 ฤ‘?‘‡ + ฤ‘??ยธ2 ฤ‘??ลก1 ฤ‘??ลก2 ฤ‘??ลŸ01 ฤ‘?‘ฤ„ รขˆ’ ฤ‘??ลŸ02 ฤ‘?‘ฤ„ รข‹ลป 4.11 2 Also, substituting the inequalities ( 4.11) in ( 4.10), we find that: ฤ‘?‘‡ 0 ฤ‘?‘ลš ฤ‘?‘ฤ„, ฤ‘?‘ฤฝ01 , ฤ‘?‘ลš0 1 รขˆ’ฤ‘?‘ลš 0 ฤ‘?‘ฤ„, ฤ‘?‘ฤฝ02 , ฤ‘?‘ลš0 2 รข‰ยค (1 รขˆ’ ฤ‘??ยธ4 )รขˆ’1 ฤ‘??ลŸ01 ฤ‘?‘ฤ„ รขˆ’ ฤ‘??ลŸ02 ฤ‘?‘ฤ„ + 2 ฤ‘?‘‡ ฤ‘?‘‡ ฤ‘?‘‡ + ฤ‘??ยธ3 (1 รขˆ’ ฤ‘??ยธ4 )รขˆ’1 [ฤ‘??ลก1 ฤ‘??ลก2 (1 รขˆ’ ฤ‘??ยธ4 ) ฤ‘??ลก01 ฤ‘?‘ฤ„ รขˆ’ ฤ‘??ลก02 ฤ‘?‘ฤ„ + 2 2 2 ฤ‘?‘‡ 1 2 + ฤ‘??ยธ2 ฤ‘??ลก1 ฤ‘??ลก2 ฤ‘??ลŸ0 ฤ‘?‘ฤ„ รขˆ’ ฤ‘??ลŸ0 ฤ‘?‘ฤ„ ] 2 and hence ฤ‘?‘‡ ฤ‘?‘ลš 0 ฤ‘?‘ฤ„, ฤ‘?‘ฤฝ01 , ฤ‘?‘ลš0 1 รขˆ’ฤ‘?‘ลš 0 ฤ‘?‘ฤ„, ฤ‘?‘ฤฝ02 , ฤ‘?‘ลš0 2 รข‰ยค ฤ‘??ยธ3 ฤ‘??ลก1 ฤ‘??ลก2 ฤ‘??ลก01 ฤ‘?‘ฤ„ รขˆ’ ฤ‘??ลก02 ฤ‘?‘ฤ„ + 2 ฤ‘?‘‡ ฤ‘?‘‡ +[ฤ‘??ลก1 (1 รขˆ’ ฤ‘??ยธ1 ) + ฤ‘??ลก1 ฤ‘??ลก2 ฤ‘??ยธ2 ] ฤ‘??ลŸ01 ฤ‘?‘ฤ„ รขˆ’ ฤ‘??ลŸ02 ฤ‘?‘ฤ„ 2 2 รข‹ลป 4.12 so, substituting inequalities ( 4.11) and ( 4.12) in inequality ( 4.5), we get the inequality ( 4.3). and the same, substituting inequalities ( 4.11) and ( 4.12) in inequality ( 4.6), gives the inequality ( 4.4). รขˆŽ ฤ‘?‘ฤฝ 0 ฤ‘?‘ฤ„, ฤ‘?‘ฤฝ01 , ฤ‘?‘ลš0 1 รขˆ’ ฤ‘?‘ฤฝ 0 ฤ‘?‘ฤ„, ฤ‘?‘ฤฝ02 , ฤ‘?‘ลš0 2

รข‰ยครข‰ยค ฤ‘??ลก1 (1 รขˆ’

REFERENCES [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]

Aziz, M.A., (2006), Periodic solutions for some systems of non-linear ordinary differential equations, M. Sc. Thesis, college of Education, University of Mosul. Butris, R. N. and Rafeq, A. Sh., (2011), Existence and Uniqueness, Solution for Non-linear Volterra Integral Equation, J. Duhok Univ. Vol.No. 1, (Pure and Eng. Sciences). Jaswon, M. A. and Symm, G. T., (1977), Integral Equations Methods in Potential Theory and Elastostatics, A subsidiary of Hart court brace Jovanovich Publishers, Academic press, London. Korol, I. I., (2005), On periodic solutions of one class of systems of differential equations, Ukraine, Math. J. Vol. 57, No. 4. Mitropolsky, Yu. A. and Martynyuk, D. I., (1979), For Periodic Solutions for the Oscillations System with Retarded Argument, Kiev, Ukraine. Rama, M. M., (1981), Ordinary Differential Equations Theory and Applications, Britain. Rafeq, A. Sh., (2009), Periodic solutions for some classes of non-linear systems of integro-differential equations, M. Sc. Thesis, college of Education, University of Duhok. Perestyuk, N. A., (1971), The periodic solutions for non-linear systems of differential equations, Math. and Meca. J., Univ. of Kiev, Kiev, Ukraine,5, 136-146. Perestyuk, N. A. and Martynyuk, D. I., (1976), Periodic solutions of a certain class systems of differential equations, Math. J., Univ. of Kiev , Ukraine,No 3. Samoilenko, A. M. and Ronto, N. I., (1976), A Numerical รข€“ Analytic Methods for Investigating of Periodic Solutions, Kiev, Ukraine. Samoilenko, A. M., (1966), A numerical รข€“ analytic methods for investigations of periodic systems of ordinary differential equations II, Kiev, Ukraine, Math. J.,No 5. Shestopalov, Y. V. and Smirnov, Y. G., (2002), Integral Equations, Karlstad University. Struble, R. A., (1962), Non-Linear Differential Equations, Mc Graw-Hall Book Company Inc., New York. Tarang, M., (2004), Stability of the spline collocation method for Volterra integro-differential equations, Thesis, University of Tartu. Tricomi, F. G., (1965), Integral Equations, Turin University, Turin, Italy.

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