Periodic solution of integro-differential equations depended on special function with singular Kernels

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In this paper, we investigate the existence, uniqueness and stability of periodic solution of new integro-differential equations depended on special function with singular kernels.

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Nội dung Text: Periodic solution of integro-differential equations depended on special function with singular Kernels

International Journal of Mechanical Engineering and Technology (IJMET) Volume 10, Issue 03, March 2019, pp. 1681–1695, Article ID: IJMET_10_03_170 Available online at http://www.iaeme.com/ijmet/issues.asp?JType=IJMET&VType=10&IType=3 ISSN Print: 0976-6340 and ISSN Online: 0976-6359 © IAEME Publication Scopus Indexed PERIODIC SOLUTION OF INTEGRO- DIFFERENTIAL EQUATIONS DEPENDED ON SPECIAL FUNCTION WITH SINGULAR KERNELS Raad N. Butris and Raveen F. Taher University of Duhok, College of Basic Education, Department of Mathematics, Kurdistan Region, Iraq ABSTRACT In this paper, we investigate the existence, uniqueness and stability of periodic solution of new integro-differential equations depended on special function with singular kernels. The numerical-analytic method has been used to study the periodic solutions for the ordinary differential equations that were introduced by Samoilenko. Also these investigation lead us to the improving and extending the results of Butris and extended Samoilenko method. Key words: Numerical-analytic methods, existence, uniqueness and stability, periodic solution, integro-differential equations, special function, singular kernels Cite this Article: Raad N. Butris and Raveen F. Taher, Periodic Solution of Integro- Differential Equations Depended on Special Function with Singular Kernels, International Journal of Mechanical Engineering and Technology 10(3), 2019, pp. 1681–1695. http://www.iaeme.com/IJMET/issues.asp?JType=IJMET&VType=10&IType=3 1. INTRODUCTION Many results about the existence and approximation of periodic solutions of integro- differential equations have been obtained by the numerical-analytic methods that were proposed by Samoilenko[18] which had been later applied in many subjects in physics and technology using mathematical methods that depends on the linear and nonlinear integro- differential equations, and it became clear that the existence of periodic solutions and it is algorithm structure from more important problems, to present time where many of studies and researches [9,14,16,17] dedicates for treatment the autonomous and no autonomous periodic systems and specially with integro-differential equations. Numerical-analytic method [11,15,18,20, 21] owing to the great possibilities of exploiting computers are becoming versatile means of the finding and approximate construction of periodic solutions of integro- differential equations. Samoilenko [1,3,4,5,6,7,8,12,19] assumes the numerical-analytic method to study the periodic solutions for ordinary differential equations and it is algorithm structure and this method include uniformly sequences of periodic functions and the results of http://www.iaeme.com/IJMET/index.asp 1681 editor@iaeme.com
Periodic Solution of Integro-Differential Equations Depended on Special Function with Singular Kernels that study is using of the periodic solutions on wide range in the difference of new processes industry and technology as in the some studies [1,2,3,12,13]. Burris[3] has been used the numerical-analytic method of periodic solution for ordinary differential equations which were introduced by Samoilenko [18] to study the periodic solution of the system, nonlinear integro-differential equation which has the form 𝑑𝑥 𝑡+𝑇 = 𝑓 (𝑡, 𝛾(𝑡, 𝛼), 𝑥, ∫𝑡 𝑔(𝑠, 𝛾(𝑠, 𝛼), 𝑥(𝑠))𝑑𝑠), 𝑑𝑡 where 𝑥 ∈ 𝐷 ⊂ 𝑅 𝑛 , 𝐷 is a closed and bounded domain. The vector functions 𝑓(𝑡, 𝛾(𝑡, 𝛼), 𝑥) and 𝑔(𝑡, 𝛾(𝑡, 𝛼), 𝑥) are defined on the domain :- (𝑡, 𝛾(𝑡, 𝛼), 𝑥) ∈ 𝑅1 × [0, 𝑇] × 𝐷 × 𝐷1 = (−∞, ∞) × [0, 𝑇] × 𝐷 × 𝐷1 where 𝐷1 is a bounded domain subset of Euclidean space R𝑛 . Our work is studying the existence, uniqueness and stability of periodic solution of integro- differential equations which has the form :- 𝑡 𝑑𝑥 = 𝑓(𝑡, 𝛾(𝑡, 𝛼), 𝑥(𝑡), 𝜇, ∫ 𝑅(𝑡, 𝜏)(𝑥(𝜏) − 𝑦(𝜏))𝑑𝜏 𝑑𝑡 −∞ 𝑡 … (1.1) 𝑑𝑦 = 𝑔(𝑡, 𝛾(𝑡, 𝛼), 𝑦(𝑡), 𝜔, ∫ 𝐺(𝑡, 𝜏)(𝑥(𝜏) − 𝑦(𝜏))𝑑𝜏 𝑑𝑡 −∞ } where 𝑏 𝑑 𝜇 = ∫𝑎 𝛾(𝜏, 𝛼)𝑥(𝜏)𝑑𝜏 and 𝜔 = ∫𝑐 𝛾(𝜏, 𝛼)𝑦(𝜏)𝑑𝜏 , 𝑥 ∈ 𝐷1 ⊂ 𝑅 𝑛 , 𝑦 ∈ 𝐷2 ⊂ 𝑅 𝑛 , where 𝐷1 and 𝐷2 are compact domains. Let the vector functions 𝑓(𝑡, 𝛾(𝑡, 𝛼), 𝑥(𝑡), 𝜇, 𝑢) and 𝑔(𝑡, 𝛾(𝑡, 𝛼), 𝑦(𝑡), 𝜔, 𝑣) are defined and continuous on the domain :- (𝑡, 𝛾(𝑡, 𝛼), 𝑥, 𝜇, 𝑢) ∈ 𝑅1 × 𝐺1 = (−∞, ∞) × 𝐷 × 𝐷1 × 𝐷𝜇 × 𝐷𝑢 } … (1.2) (𝑡, 𝛾(𝑡, 𝛼), 𝑦, 𝜔, 𝑣) ∈ 𝑅1 × 𝐺2 = (−∞, ∞) × 𝐷 × 𝐷2 × 𝐷𝜔 × 𝐷𝑣 where 𝐷𝜇 , 𝐷𝜔 , 𝐷𝑢 and 𝐷𝑣 are bounded domains subset of Euclidean space 𝑅 𝑚 . Also 𝑡 𝑡 𝑢 = ∫−∞ 𝑅(𝑡, 𝜏)(𝑥(𝜏) − 𝑦(𝜏))𝑑𝜏, 𝑣 = ∫−∞ 𝐺(𝑡, 𝜏) (𝑥(𝜏) − 𝑦(𝜏))𝑑𝜏, 𝐷 = [𝜏, 𝜏 + 𝑇] × (0,1]. and periodic in 𝑡 of period 𝑇 and satisfy the following inequalities : ∥ 𝑓(𝑡, 𝛾(𝑡, 𝛼), 𝑥 , 𝜇, 𝑢) ∥ ≤ ‖𝛾(𝑡, 𝛼)‖‖ 𝑓(𝑡, 𝑥 , 𝜇, 𝑢)‖ ≤ 𝑀𝛾 𝑀 } … (1.3) ∥ 𝑔(𝑡, 𝛾(𝑡, 𝛼), 𝑦 , 𝜔, 𝑣) ∥ ≤ ‖𝛾(𝑡, 𝛼)‖ ∥ 𝑔(𝑡, 𝑦 , 𝜔, 𝑣) ∥≤ 𝑁𝛾 𝑁 ∥ 𝑓(𝑡, 𝛾(𝑡, 𝛼), 𝑥1 , 𝜇1 , 𝑢1 ) − 𝑓(𝑡, 𝛾(𝑡, 𝜏), 𝑥2 , 𝜇2 , 𝑢2 ) ∥ ≤ 𝑀𝛾 (𝐾1 ∥ 𝑥1 − 𝑥2 ∥ + 𝐾2 ∥ 𝜇1 − 𝜇2 ∥ +𝐾3 ∥ 𝑢1 − 𝑢2 ∥) … (1.4) ∥ 𝑔(𝑡, 𝛾(𝑡, 𝛼), 𝑦1 , 𝜔1 , 𝑣1 ) − 𝑔(𝑡, 𝛾(𝑡, 𝜏), 𝑦2 , 𝜔2 , 𝑣2 ) ∥ ≤ 𝑁𝛾 (𝐿1 ∥ 𝑦1 − 𝑦2 ∥ + 𝐿2 ∥ 𝜔1 − 𝜔2 ∥ +𝐿3 ∥ 𝑣1 − 𝑣2 ∥) … (1.5) For all 𝑡 ∈ 𝑅1 , 𝑥, 𝑥1 , 𝑥2 ∈ 𝐷1 , 𝑦, 𝑦1 , 𝑦2 ∈ 𝐷2 , 𝜇, 𝜔 and 𝑣 are belong to 𝐷𝜇 , 𝐷𝜔 , 𝐷𝑢 and 𝐷𝑣 respectively, where 𝑀, 𝑁, 𝐾1 , 𝐾2 , 𝐾3 , 𝐿1 , 𝐿2 and 𝐿3 are positive constants. Also 𝛾(𝑡, 𝛼) is said to be special function provided that 𝛾(𝑡 + 𝑇, 𝛼) = 𝛾(𝑡, 𝛼). The singular kernels 𝑅(𝑡, 𝜏) and 𝐺(𝑡, 𝜏) satisfying the following conditions :- ∥ 𝑅(𝑡, 𝜏) ∥≤ ℎ𝑒 −𝛼(𝑡−𝜏) } … (1.6) ∥ 𝐺(𝑡, 𝜏) ∥≤ 𝜎𝑒 −𝛽(𝑡−𝜏) http://www.iaeme.com/IJMET/index.asp 1682 editor@iaeme.com
Raad N. Butris and Raveen F. Taher where −∞ < 0 ≤ 𝜏 ≤ 𝑡 ≤ 𝜏 + 𝑇, 𝛼 and 𝛽 are positive constants. We defined the non-empty sets as follows:- 𝑇 𝐷𝑓 = 𝐺1 − 𝑀𝛾 𝑀 2 } … (1.7) 𝑇 𝐷𝑔 = 𝐺2 − 𝑁𝛾 𝑁 2 Furthermore, we suppose that the largest eigen-value of the matrix 𝑇 𝑇 𝑀𝛾 𝐶1 𝑀𝛾 𝐶2 2 2 Λ=( ) is less than one, i.e. 𝑇 𝑇 𝑁𝛾 𝐶3 𝑁 𝐶 2 2 𝛾 4 𝜑1 + √𝜑12 + 4(𝜑2 − 𝜑3 ) 𝜆𝑚𝑎𝑥 (Λ) =
Periodic Solution of Integro-Differential Equations Depended on Special Function with Singular Kernels 𝑇 satisfies for 𝜏 ≤ 𝑡 ≤ 𝜏 + 𝑇 and 𝛼(𝑡) ≤ 2, 𝑡−𝜏 where 𝛼(𝑡 )= 2(𝑡 − 𝜏) (1 − ) for all 𝑡 ∈ [𝜏, 𝜏 + 𝑇], 𝑇 𝑡 𝐹1 (𝑡, 𝑥0 , 𝑦0 ) = 𝑥0 + ∫ [𝑓( 𝜏, 𝛾(𝜏, 𝛼), 𝑥(𝜏, 𝑥0 , 𝑦0 ), 𝜇, 𝑢) 𝜏 1 𝜏+𝑇 − ∫ 𝑓(𝜏, 𝛾(𝜏, 𝛼), 𝑥(𝜏, 𝑥0 , 𝑦0 ), 𝜇, 𝑢)𝑑𝜏] 𝑑𝜏 𝑇 𝜏 𝑡 𝐹2 (𝑡, 𝑥0 , 𝑦0 ) = 𝑦0 + ∫ [𝑔( 𝜏, 𝛾(𝜏, 𝛼), 𝑦(𝜏, 𝑥0 , 𝑦0 ), 𝜔, 𝑣) 𝜏 1 𝜏+𝑇 − ∫ 𝑔(𝜏, 𝛾(𝜏, 𝛼), 𝑦(𝜏, 𝑥0 , 𝑦0 ), 𝜔, 𝑣)𝑑𝜏] 𝑑𝜏 𝑇 𝜏 Proof. 𝑡 𝑡−𝜏 ∥ 𝐹1 (𝑡, 𝑥0 , 𝑦0 ) ∥≤ (1 − ) ∫ ‖𝛾(𝜏, 𝛼)‖ ‖𝑓(𝜏, 𝑥(𝜏, 𝑥0 , 𝑦0 ), 𝜇, 𝑢)‖ 𝑑𝜏 𝑇 𝜏 𝑡 − 𝜏 𝜏+𝑇 + ∫ ‖𝛾(𝜏, 𝛼)‖ ‖𝑓(𝜏, 𝑥(𝜏, 𝑥0 , 𝑦0 ), 𝜇, 𝑢)‖ 𝑑𝜏 𝑇 𝑡 ≤ 𝛼(𝑡)𝑀𝛾 𝑀 So that ∥ 𝐹1 (𝑡, 𝑥0 , 𝑦0 ) ∥ ≤ 𝛼(𝑡)𝑀𝛾 𝑀 … (1.12) and ∥ 𝐹2 (𝑡, 𝑥0 , 𝑦0 ) ∥ ≤ 𝛼(𝑡)𝑁𝛾 𝑁 … (1.13) From (1.12) and (1.13) we conclude that the inequality (1.11) holds. Approximation solution of (𝟏. 𝟏). The investigation of approximation periodic solution of (1.1) will be introduced by the following theorem . Theorem 2.1. Let the vector functions 𝑓(𝑡, 𝛾(𝑡, 𝛼), 𝑥, 𝜇, 𝑢) and 𝑔(𝑡, 𝛾(𝑡, 𝛼), 𝑦, 𝜔, 𝑣) are defined, continuous on the domain (1.2) and periodic in 𝑡 of period 𝑇. Suppose the above functions satisfy the inequalities (1.3) to (1.6) and the conditions (1.7), (1.8). Then there exist a sequence of functions (1.9) and (1.10) converges uniformly on the domain :- ( 𝑡, 𝑥0 , 𝑦0 ) ∈ [𝜏, 𝜏 + 𝑇] × 𝐷𝑓 × 𝐷𝑔 . .. (2.1) to the limit functions 𝑥(𝑡, 𝑥0 , 𝑦0 ) and 𝑦(𝑡, 𝑥0 , 𝑦0 ) defined in the domain (2.1) which is periodic in 𝑡 of period 𝑇 and satisfies the following vector form :- 𝑡 𝑥(𝑡, 𝑥0 , 𝑦0 ) = 𝑥0 + ∫ [𝑓( 𝜏, 𝛾(𝜏, 𝛼), 𝑥(𝑡, 𝑥0 , 𝑦0 ), 𝜇, 𝑢) 𝜏 1 𝜏+𝑇 − ∫ 𝑓(𝜏, 𝛾(𝜏, 𝛼), 𝑥(𝜏, 𝑥0 , 𝑦0 ), 𝜇, 𝑢)𝑑𝜏] 𝑑𝜏 … (2.2) 𝑇 𝜏 http://www.iaeme.com/IJMET/index.asp 1684 editor@iaeme.com
Raad N. Butris and Raveen F. Taher 𝑡 𝑦(𝑡, 𝑥0 , 𝑦0 ) = 𝑦0 + ∫ [𝑔( 𝜏, 𝛾(𝜏, 𝛼), 𝑦(𝜏, 𝑥0 , 𝑦0 ), 𝜔, 𝑣) 𝜏 1 𝜏+𝑇 − ∫ 𝑔(𝜏, 𝛾(𝜏, 𝛼), 𝑦(𝜏, 𝑥0 , 𝑦0 ), 𝜔, 𝑣)𝑑𝜏] 𝑑𝜏 … (2.3) 𝑇 𝜏 and it's a unique solution of (1.1) Provided that :- 𝑇 ∥ 𝑥(𝑡, 𝑥0 , 𝑦0 ) − 𝑥0 ) ∥ 𝑀𝛾 𝑀 2 ( )≤( ) … (2.4) ∥ 𝑦(𝑡, 𝑥0 , 𝑦0 ) − 𝑦0 ) ∥ 𝑇 𝐻𝛾 𝐻 2 and ∥ 𝑥(𝑡, 𝑥0 , 𝑦0 ) − 𝑥𝑚 (𝑡, 𝑥0 , 𝑦0 ) ∥ ( ) ≤ Λ𝑚 (𝐸 − Λ)−1 Ω1 … (2.5) ∥ 𝑧(𝑡, 𝑥0 , 𝑦0 ) − 𝑧𝑚 (𝑡, 𝑥0 , 𝑦0 ) ∥ Proof. Setting 𝑚 = 0 in (1.9) and by using Lemma 1.1, we have 𝑡 𝑡−𝜏 ∥ 𝑥1 (𝑡, 𝑥0 , 𝑦0 ) − 𝑥0 ∥ ≤ (1 − ) ∫ ‖𝛾(𝜏, 𝛼)‖ ‖𝑓(𝜏, 𝑥(𝜏, 𝑥0 , 𝑦0 ), 𝜇, 𝑢)‖ 𝑑𝜏 𝑇 𝜏 𝑡 − 𝜏 𝜏+𝑇 + ∫ ‖𝛾(𝜏, 𝛼)‖ ‖𝑓(𝜏, 𝑥(𝜏, 𝑥0 , 𝑦0 ), 𝜇, 𝑢)‖ 𝑑𝜏 𝑇 𝑡 ≤ 𝛼(𝑡)𝑀𝛾 𝑀 and hence 𝑇 ∥ 𝑥1 (𝑡, 𝑥0 , 𝑦0 ) − 𝑥0 ∥ ≤ 𝛼(𝑡)𝑀𝛾 𝑀 ≤ 2 𝑀𝛾 𝑀 … (2.6) so that 𝑥1 (𝑡, 𝑥0 , 𝑦0 ) ∈ 𝐺1 , for all 𝑡 ∈ 𝑅1 , 𝑥0 ∈ 𝐷𝑓 , 𝑦0 ∈ 𝐷ℎ . Also from (1.10), and by using Lemma 1.1, when 𝑚 = 0, we have 𝑇 ∥ 𝑦1 (𝑡, 𝑥0 , 𝑦0 ) − 𝑦0 ∥≤ 𝛼(𝑡)𝑁𝛾 𝑁 ≤ 2 𝑁𝛾 𝑁 … (2.7) i.e. 𝑦1 (𝑡, 𝑥0 , 𝑦0 ) ∈ 𝐺2 , for all 𝑡 ∈ 𝑅1 , 𝑥0 ∈ 𝐷𝑓 , 𝑦0 ∈ 𝐷ℎ . Then by mathematical induction we can prove that 𝑇 ∥ 𝑥𝑚 (𝑡, 𝑥0 , 𝑦0 ) − 𝑥0 ∥ ≤ 2 𝑀𝛾 𝑀 } … (2.8) 𝑇 ∥ 𝑦𝑚 (𝑡, 𝑥0 , 𝑦0 ) − 𝑦0 ∥ ≤ 2 𝑁𝛾 𝑁 i.e. 𝑥𝑚 (𝑡, 𝑥0 , 𝑦0 ) ∈ 𝐷1 , 𝑦𝑚 (𝑡, 𝑥0 , 𝑦0 ) ∈ 𝐷2 , 𝑥0 ∈ 𝐷𝑓 , 𝑦0 ∈ 𝐷𝑔 , for all 𝑡 ∈ [𝜏, 𝜏 + 𝑇], 𝑚 = 0,1,2, … . Rewrite (2.8) by the vector from , we get (2.4). Now, we shall prove that the sequence of functions (1.9) and (1.10) convergent uniformly on the domain (2.1). By using Lemma 1.1 and from (1.4) when 𝑚 = 1 in (1.9) , we find that 𝑡 𝑡−𝜏 ∥ 𝑥2 (𝑡, 𝑥0 , 𝑦0 ) − 𝑥1 (𝑡, 𝑥0 , 𝑦0 ) ∥≤ (1 − ) ∫ ‖𝛾(𝜏, 𝛼)‖ 𝑇 𝜏 http://www.iaeme.com/IJMET/index.asp 1685 editor@iaeme.com
Periodic Solution of Integro-Differential Equations Depended on Special Function with Singular Kernels [𝐾1 ‖𝑥1 (𝜏, 𝑥0 , 𝑦0 )−𝑥0 ‖+𝐾2 𝑀𝛾 (𝑏 − 𝑎)‖𝑥1 (𝜏, 𝑥0 , 𝑦0 )−𝑥0 ‖ ℎ + 𝐾3 [‖𝑥1 (𝜏, 𝑥0 , 𝑦0 )−𝑥0 ‖ + ‖𝑦1 (𝜏, 𝑥0 , 𝑦0 )−𝑦0 ‖]] 𝑑𝜏 𝛼 𝑡 − 𝜏 𝜏+𝑇 + ∫ ‖𝛾(𝜏, 𝛼)‖ [𝐾1 ‖𝑥1 (𝜏, 𝑥0 , 𝑦0 )−𝑥0 ‖ 𝑇 𝑡 ℎ +𝐾2 𝑀𝛾 (𝑏 − 𝑎)‖𝑥1 (𝜏, 𝑥0 , 𝑦0 )−𝑥0 ‖ + 𝐾 [‖𝑥 (𝜏, 𝑥0 , 𝑦0 )−𝑥0 ‖ 𝛼 3 1 +‖𝑦1 (𝜏, 𝑥0 , 𝑦0 )−𝑦0 ‖]] 𝑑𝜏 therefore ∥ 𝑥2 (𝑡, 𝑥0 , 𝑦0 ) − 𝑥1 (𝑡, 𝑥0 , 𝑦0 ) ∥≤ 𝛼(𝑡)𝑀𝛾 𝐶1 ‖𝑥1 (𝑡, 𝑥0 , 𝑦0 )−𝑥0 ‖ +𝛼(𝑡)𝑀𝛾 𝐶2 ‖𝑦1 (𝑡, 𝑥0 , 𝑦0 )−𝑦0 ‖ ℎ ℎ where 𝐶1 = 𝐾1 + 𝐾2 𝑀𝛾 (𝑏 − 𝑎) + 𝛼 𝐾3 and 𝐶2 = 𝛼 𝐾3 . And by lemma 1.1 and from (1.5) ,we have ∥ 𝑦2 (𝑡, 𝑥0 , 𝑦0 ) − 𝑦1 (𝑡, 𝑥0 , 𝑦0 ) ∥≤ 𝛼(𝑡)𝑁𝛾 𝐶3 ‖𝑥1 (𝑡, 𝑥0 , 𝑦0 )−𝑥0 ‖ +𝛼(𝑡)𝑁𝛾 𝐶4 ‖𝑦1 (𝑡, 𝑥0 , 𝑦0 )−𝑦0 ‖ 𝜎 𝜎 where 𝐶3 = 𝐿1 + 𝐿2 𝑁𝛾 (𝑑 − 𝑐) + 𝛽 𝐿3 and 𝐶4 = 𝛽 𝐿3 . Then by mathematical induction we can prove that ∥ 𝑥𝑚+1 (𝑡, 𝑥0 , 𝑦0 ) − 𝑥𝑚 (𝑡, 𝑥0 , 𝑦0 ) ∥≤ 𝛼(𝑡)𝑀𝛾 𝐶1 ‖𝑥𝑚 (𝑡, 𝑥0 , 𝑦0 )−𝑥𝑚−1 (𝑡, 𝑥0 , 𝑦0 )‖ +𝛼(𝑡)𝑀𝛾 𝐶2 ‖𝑦𝑚 (𝑡, 𝑥0 , 𝑦0 )−𝑦𝑚−1 (𝑡, 𝑥0 , 𝑦0 )‖ … (2.9) and ∥ 𝑦𝑚+1 (𝑡, 𝑥0 , 𝑦0 ) − 𝑦𝑚 (𝑡, 𝑥0 , 𝑦0 ) ∥≤ 𝛼(𝑡)𝑁𝛾 𝐶3 ‖𝑥𝑚 (𝑡, 𝑥0 , 𝑦0 )−𝑥𝑚−1 (𝑡, 𝑥0 , 𝑦0 )‖ +𝛼(𝑡)𝑁𝛾 𝐶4 ‖𝑦𝑚 (𝑡, 𝑥0 , 𝑦0 )−𝑦𝑚−1 (𝑡, 𝑥0 , 𝑦0 )‖ … (2.10) Rewrite inequalities (2.9) and (2.10) by vector form, i.e. 𝛺𝑚+1 (𝑡) ≤ Λ(𝑡)Ω1 … (2.11) where ∥ 𝑥𝑚+1 (𝑡, 𝑥0 , 𝑦0 ) − 𝑥𝑚 (𝑡, 𝑥0 , 𝑦0 ) ∥ 𝛺𝑚+1 = ( ) ∥ 𝑦𝑚+1 (𝑡, 𝑥0 , 𝑦0 ) − 𝑦𝑚 (𝑡, 𝑥0 , 𝑦0 ) ∥ ∥ 𝑥𝑚 (𝑡, 𝑥0 , 𝑦0 ) − 𝑥𝑚−1 (𝑡, 𝑥0 , 𝑦0 ) ∥ Ω𝑚 = ( ) ∥ 𝑦𝑚 (𝑡, 𝑥0 , 𝑦0 ) − 𝑦𝑚−1 (𝑡, 𝑥0 , 𝑦0 ) ∥ and 𝛼(𝑡)𝑀𝛾 𝐶1 𝛼(𝑡)𝑀𝛾 𝐶2 Λ(𝑡) = ( ) 𝛼(𝑡)𝐻𝑁𝛾 𝐶3 𝛼(𝑡)𝑁𝛾 𝐶4 http://www.iaeme.com/IJMET/index.asp 1686 editor@iaeme.com
Raad N. Butris and Raveen F. Taher Now, we take the maximum value of the both sides of the (2.11), we get 𝛺𝑚+1 ≤ ΛΩ𝑚 . . . (2.12) where Λ = 𝑚𝑎𝑥 Λ(𝑡), we obtain 𝑡∈[𝜏,𝜏+𝑇] 𝑇 𝑇 𝑀 𝐶 𝑀 𝐶 2 𝛾 1 2 𝛾 2 Λ= 𝑇 𝑇 𝑁𝛾 𝐶3 𝑁𝐶 ( 2 2 𝛾 4 ) and by repetition of (2.12), we find that 𝛺𝑚+1 ≤ Λ𝑚 Ω1 … (2.13) where 𝑇 𝑀𝛾 𝑀 2 Ω1 = ( ) and also we get 𝑇 𝑁𝛾 𝑁 2 𝑚 𝑚 ∑ Ω𝑖 ≤ ∑ Λ𝑖−1 Ω1 … (2.14) 𝑖=1 𝑖=1 By using (1.8) then the series (2.14) is uniformly convergent that is 𝑙𝑖𝑚 ∑𝑚𝑖=1 Λ 𝑖−1 Ω1 ≤ ∑∞𝑖=1 Λ 𝑖−1 Ω1 = (𝐸 − Λ)−1 Ω1 … (2.15) 𝑚→∞ Let 𝑥𝑚 (𝑡, 𝑥0 , 𝑦0 ) 𝑥 (𝑡, 𝑥0 , 𝑦0 ) 𝑙𝑖𝑚 ( )=( ) … (2.16) 𝑚→∞ 𝑦𝑚 (𝑡, 𝑥0 , 𝑦0 ) 𝑦(𝑡, 𝑥0 , 𝑦0 ) Since the sequence of functions (1.9) and (1.10) are defined and continuous in the 𝑥 (𝑡, 𝑥0 , 𝑦0 ) domain (1.2 ) then the limiting vector function ( ) is also defined , continuous 𝑦(𝑡, 𝑥0 , 𝑦0 ) 𝑥 (𝑡, 𝑥0 , 𝑦0 ) and periodic on the same domain and hence the vector function ( ) is a solution 𝑦(𝑡, 𝑥0 , 𝑦0 ) of (1.1). 𝑥 (𝑡, 𝑥0 , 𝑦0 ) Finally, we prove that ( ) is a unique solution of (1.1). 𝑦(𝑡, 𝑥0 , 𝑦0 ) 𝑥 ∗ (𝑡, 𝑥0 , 𝑦0 ) Let ( ) be another solution of (1.1). ∗ (𝑡, 𝑦 𝑥0 , 𝑦0 ) where 𝑡 𝑥 ∗ (𝑡, 𝑥0 , 𝑦0 ) = 𝑥0 + ∫ [𝑓( 𝜏, 𝛾(𝜏, 𝛼), 𝑥 ∗ (𝜏, 𝑥0 , 𝑦0 ), 𝜇 ∗ , 𝑢∗ ) 𝜏 http://www.iaeme.com/IJMET/index.asp 1687 editor@iaeme.com
Periodic Solution of Integro-Differential Equations Depended on Special Function with Singular Kernels 1 𝜏+𝑇 − ∫ 𝑓(𝜏, 𝛾(𝜏, 𝛼), 𝑥 ∗ (𝜏, 𝑥0 , 𝑦0 ), 𝜇 ∗ , 𝑢∗ )𝑑𝜏] 𝑑𝜏 … (2.17) 𝑇 𝜏 𝑡 ∗ (𝑡, 𝑦 𝑥0 , 𝑦0 ) = 𝑦0 + ∫ [𝑔( 𝜏, 𝛾(𝜏, 𝛼), 𝑦 ∗ (𝜏, 𝑥0 , 𝑦0 ), 𝜔∗ , 𝑣 ∗ ) 𝜏 1 𝜏+𝑇 − ∫ 𝑔(𝜏, 𝛾(𝜏, 𝛼), 𝑦 ∗ (𝜏, 𝑥0 , 𝑦0 ), 𝜔∗ , 𝑣 ∗ )𝑑𝜏] 𝑑𝜏 … (2.18) 𝑇 𝜏 where 𝑏 𝑑 𝜇 = ∫ 𝛾(𝜏, 𝛼)𝑥 (𝜏, 𝑥0 , 𝑦0 )𝑑𝜏, 𝜔 = ∫ 𝛾(𝜏, 𝛼)𝑦 ∗ (𝜏, 𝑥0 , 𝑦0 )𝑑𝜏 ∗ ∗ ∗ 𝑎 𝑐 𝑡 𝑢∗ = ∫ 𝑅(𝑡, 𝜏)(𝑥 ∗ (𝜏, 𝑥0 , 𝑦0 ) − 𝑦 ∗ (𝜏, 𝑥0 , 𝑦0 ))𝑑𝜏 −∞ 𝑡 𝑣 ∗ = ∫ 𝐺(𝑡, 𝜏)(𝑥 ∗ (𝜏, 𝑥0 , 𝑦0 ) − 𝑦 ∗ (𝜏, 𝑥0 , 𝑦0 ))𝑑𝜏 −∞ Now 𝑇 ∥ 𝑥(𝑡, 𝑥0 , 𝑦0 ) − 𝑥 ∗ (𝑡, 𝑥0 , 𝑦0 ) ∥≤ 𝑀 𝐶 ‖𝑥(𝑡, 𝑥0 , 𝑦0 )−𝑥 ∗ (𝑡, 𝑥0 , 𝑦0 )‖ 2 𝛾 1 𝑇 + 𝑀𝛾 𝐶2 ‖𝑦𝑚 (𝑡, 𝑥0 , 𝑦0 )−𝑦𝑚−1 (𝑡, 𝑥0 , 𝑦0 )‖ … (2.19) 2 and also 𝑇 ∥ 𝑦(𝑡, 𝑥0 , 𝑦0 ) − 𝑦 ∗ (𝑡, 𝑥0 , 𝑦0 ) ∥≤ 𝑁 𝐶 ‖𝑥(𝑡, 𝑥0 , 𝑦0 )−𝑥 ∗ (𝑡, 𝑥0 , 𝑦0 )‖ 2 𝛾 3 𝑇 + 𝑁𝛾 𝐶3 ‖𝑦𝑚 (𝑡, 𝑥0 , 𝑦0 )−𝑦𝑚−1 (𝑡, 𝑥0 , 𝑦0 )‖ … (2.20) 2 Rewrite the inequalities (2.19) and (2.20) in a vector form :- ∥ 𝑥(𝑡, 𝑥0 , 𝑦0 ) − 𝑥 ∗ (𝑡, 𝑥0 , 𝑦0 ) ∥ ∥ 𝑥(𝑡, 𝑥0 , 𝑦0 ) − 𝑥 ∗ (𝑡, 𝑥0 , 𝑦0 ) ∥ ( ) ≤ Λ𝑚 ( ) … (2.21) ∗ ∗ ∥ 𝑦(𝑡, 𝑥0 , 𝑦0 ) − 𝑦 (𝑡, 𝑥0 , 𝑦0 ) ∥ ∥ 𝑧(𝑡, 𝑥0 , 𝑦0 ) − 𝑧 (𝑡, 𝑥0 , 𝑦0 ) ∥ Then by the condition (1.8), we have ∥ 𝑥(𝑡, 𝑥0 , 𝑦0 ) − 𝑥 ∗ (𝑡, 𝑥0 , 𝑦0 ) ∥ 0 ( ) → (0 ) ∥ 𝑦(𝑡, 𝑥0 , 𝑦0 ) − 𝑦 ∗ (𝑡, 𝑥0 , 𝑦0 ) ∥ 0 That is 𝑥 (𝑡, 𝑥0 , 𝑦0 ) 𝑥 ∗ (𝑡, 𝑥0 , 𝑦0 ) ( )=( ) ∗ 𝑦(𝑡, 𝑥0 , 𝑦0 ) 𝑦 (𝑡, 𝑥0 , 𝑦0 ) 𝑥 (𝑡, 𝑥0 , 𝑦0 ) and hence ( )is a unique solution of (1.1). 𝑦(𝑡, 𝑥0 , 𝑦0 ) http://www.iaeme.com/IJMET/index.asp 1688 editor@iaeme.com
Raad N. Butris and Raveen F. Taher Existence of periodic solution of (𝟏. 𝟏). The problem of existence of periodic solution of (1.1) is uniquely connected with the existence of zero of the vector functions 1 𝜏+𝑇 ∫ 𝑓(𝜏, 𝛾(𝜏, 𝛼), 𝑥(𝜏, 𝑥0 , 𝑦0 ), 𝜇, 𝑢)𝑑𝜏 𝛥1 (0, 𝑥0 , 𝑦0 ) 𝑇 𝜏 ( )= … (3.1) 𝛥2 (0, 𝑥0 , 𝑦0 ) 1 𝜏+𝑇 ∫ 𝑔(𝜏, 𝛾(𝜏, 𝛼), 𝑦(𝜏, 𝑥0 , 𝑦0 ), 𝜔, 𝑣)𝑑𝜏 (𝑇 𝜏 ) 𝛥1 (0, 𝑥0 , 𝑦0 ) the vector function ( ) is approximately determined by the following vector 𝛥2 (0, 𝑥0 , 𝑦0 ) sequence :- 1 𝜏+𝑇 ∫ 𝑓(𝜏, 𝛾(𝜏, 𝛼), 𝑥𝑚 (𝜏, 𝑥0 , 𝑦0 ), 𝜇𝑚 , 𝑢𝑚 )𝑑𝜏 𝛥1𝑚 (0, 𝑥0 , 𝑦0 ) 𝑇 𝜏 ( )= . . . ( 3.2) 𝛥2𝑚 (0, 𝑥0 , 𝑦0 ) 1 𝜏+𝑇 ∫ 𝑔(𝜏, 𝛾(𝜏, 𝛼), 𝑦𝑚 (𝜏, 𝑥0 , 𝑦0 ), 𝜔𝑚 , 𝑣𝑚 )𝑑𝜏 (𝑇 𝜏 ) Theorem 3.1. Under the conditions and hypothesis of theorem (1.1) then the following inequality: ∥ 𝛥1 (0, 𝑥0 , 𝑦0 ) − 𝛥1𝑚 (, 𝑥0 , 𝑦0 ) ∥ 𝑏1𝑚 ( )≤( ) … (3.3) ∥ 𝛥2 (0, 𝑥0 , 𝑦0 ) − 𝛥2𝑚 (0, 𝑥0 , 𝑦0 ) ∥ 𝑏 2𝑚 𝑀𝛾 𝐶1 is satisfied where 𝑏1𝑚 = 〈( ) , Λ𝑚 (𝐸 − Λ)−1 Ω1 〉 and 𝑀𝛾 𝐶2 𝑁𝛾 𝐶3 𝑏2𝑚 = 〈( ) , Λ𝑚 (𝐸 − Λ)−1 Ω1 〉 for all m ≥ 0. 𝑁𝛾 𝐶4 Proof . From the equations (3.1) and (3.2) we have 𝑀𝛾 𝐶1 ∥ 𝛥1 (0, 𝑥0 , 𝑦0 ) − 𝛥1𝑚 (0, 𝑥0 , 𝑦0 ) ∥ ≤ 〈( ) , Λ𝑚 (𝐸 − Λ)−1 Ω1 〉 = 𝑏1𝑚 … (3.4) 𝑀𝛾 𝐶2 and similarly 𝑁𝛾 𝐶3 ∥ 𝛥2 (0, 𝑥0 , 𝑦0 ) − 𝛥2𝑚 (0, 𝑥0 , 𝑦0 ) ∥ ≤ 〈( ) , Λ𝑚 (𝐸 − Λ)−1 Ω1 〉 = 𝑏2𝑚 … (3.5) 𝑁𝛾 𝐶4 From (3.4) and (3.5), we get (3.3). Now, we prove the following theorem taking into account that the inequality (3.3) will be satisfied for all 𝑚 ≥ 0. Theorem 3.2. Let the vector functions 𝑓(𝑡, 𝛾(𝑡, 𝛼), 𝑥, 𝜇, 𝑢) and 𝑔(𝑡, 𝛾(𝑡, 𝛼), 𝑦, 𝜔, 𝑣) be defined on the intervals [𝑎, 𝑏] and [𝑐, 𝑑] on 𝑅1 and periodic in 𝑡 of period 𝑇, suppose that for all 𝑚 ≥ 0, the sequences of functions 𝛥1𝑚 (0, 𝑥0 , 𝑦0 ) and 𝛥2𝑚 (0, 𝑥0 , 𝑦0 ) which are defined in (3.2) and satisfy the inequalities :- http://www.iaeme.com/IJMET/index.asp 1689 editor@iaeme.com
Periodic Solution of Integro-Differential Equations Depended on Special Function with Singular Kernels 𝑚𝑖𝑛 𝛥1𝑚 (0, 𝑥0 , 𝑦0 ) ≤ −𝑏1𝑚 𝑥0 ∈𝐼1 ,𝑦0 ∈𝐼2 } … (3.6) 𝑚𝑎𝑥 𝛥1𝑚 (0, 𝑥0 , 𝑦0 ) ≥ 𝑏1𝑚 𝑥0 ∈𝐼1 ,𝑦0 ∈𝐼2 𝑚𝑖𝑛 𝛥2𝑚 (0, 𝑥0 , 𝑦0 ) ≤ −𝑏2𝑚 𝑥0 ∈𝐼1 ,𝑦0 ∈𝐼2 } . . . (3.7) 𝑚𝑎𝑥 𝛥2𝑚 (0, 𝑥0 , 𝑦0 ) ≥ 𝑏2𝑚 𝑥0 ∈𝐼1 ,𝑦0 ∈𝐼2 𝑥 (𝑡, 𝑥0 , 𝑦0 ) Then (1.1) has a periodic solution ( ) such that : 𝑦(𝑡, 𝑥0 , 𝑦0 ) 𝑇 𝑇 𝑇 𝑇 𝑥0 ∈ 𝐼1 = [𝑎 + 2 𝑀𝛾 𝑀 , 𝑏 − 2 𝑀𝛾 𝑀 ] and 𝑦0 ∈ 𝐼2 = [𝑐 + 2 𝑁𝛾 𝑁, 𝑑 − 2 𝑁𝛾 𝑁 ]. Proof. Let 𝑥1 , 𝑥2 and 𝑦1 , 𝑦2 be any points belonging on the intervals 𝐼1 and 𝐼2 respectively, such that 𝛥1𝑚 (0, 𝑥1 , 𝑦1 ) = 𝑚𝑖𝑛 𝛥1𝑚 (0, 𝑥0 , 𝑦0 ) 𝑥0 ∈𝐼1 ,𝑦0 ∈𝐼2 } … (3.8) 𝛥1𝑚 (0, 𝑥2 , 𝑦2 ) = 𝑚𝑎𝑥 𝛥1𝑚 (0, 𝑥0 , 𝑦0 ) 𝑥0 ∈𝐼1 ,𝑦0 ∈𝐼2 𝛥2𝑚 (0, 𝑥1 , 𝑦1 ) = 𝑚𝑖𝑛 𝛥2𝑚 (0, 𝑥0 , 𝑦0 ) 𝑥0 ∈𝐼1 ,𝑦0 ∈𝐼2 } … (3.9) 𝛥2𝑚 (0, 𝑥2 , 𝑦2 ) = 𝑚𝑎𝑥 𝛥2𝑚 (0, 𝑥0 , 𝑦0 ) 𝑥0 ∈𝐼1 ,𝑦0 ∈𝐼2 By using the inequalities ( 3.4), (3.5), (3.6), (3.7) and (3.8), we obtains 𝛥1 (0, 𝑥1 , 𝑦1 ) = 𝛥1𝑚 (0, 𝑥1 , 𝑦1 ) + (𝛥1 (0, 𝑥1 , 𝑦1 ) − 𝛥1𝑚 (0, 𝑥1 , 𝑦1 )) < 0 } … (3.10) 𝛥1 (0, 𝑥2 , 𝑦2 ) = 𝛥1𝑚 (0, 𝑥2 , 𝑦2 ) + (𝛥1 (0, 𝑥2 , 𝑦2 ) − 𝛥1𝑚 (0, 𝑥2 , 𝑦2 )) > 0 𝛥2 (0, 𝑥1 , 𝑦1 ) = 𝛥2𝑚 (0, 𝑥1 , 𝑦1 ) + (𝛥2 (0, 𝑥1 , 𝑦1 ) − 𝛥2𝑚 (0, 𝑥1 , 𝑦1 )) < 0 } … (3.11) 𝛥2 (0, 𝑥1 , 𝑦1 ) = 𝛥2𝑚 (0, 𝑥2 , 𝑦2 ) + (𝛥2 (0, 𝑥2 , 𝑦2 ) − 𝛥2𝑚 (0, 𝑥2 , 𝑦2 )) > 0 and from the continuity of the functions 𝛥1 (0, 𝑥0 , 𝑦0 ) and 𝛥2 (0 , 𝑥0 , 𝑦0 ) and the inequalities(3.10) and (3.11 ), then there exist and isolated points 𝑥 0 ∈ [𝑥1 , 𝑥2 ] and 𝑦 0 ∈ [𝑦1 , 𝑦2 ] such that 𝛥1 (0, 𝑥0 , 𝑦0 ) =0 and 𝛥1 (0, 𝑥0 , 𝑦0 ) = 0. This means that (1.1) has a periodic solution 𝑥 = 𝑥(𝑡, 𝑥0 , 𝑦0 ) and 𝑦 = 𝑦(𝑡 , 𝑥0 , 𝑦0 ). Stability of solution of (𝟏. 𝟏). In this section, we study the stability periodic solution of (1.1). Theorem 4.1. Let the vector functions 𝛥1 (0, 𝑥0 , 𝑦0 ) and 𝛥2 (0, 𝑥0 , 𝑦0 ) are defined by equations (3.1), where 𝑥(𝑡, 𝑥0 , 𝑦0 ) is a limit of the sequence of the functions (1.9), the function 𝑦(𝑡, 𝑥0 , 𝑦0 ) is the limit of the sequence of the functions (1.10), then the following inequalities yields :- ∥ 𝛥1 (0, 𝑥0 , 𝑦0 ) ∥ 𝑀𝛾 𝑀 ( )≤( ) … (4.1) ∥ 𝛥2 (0, 𝑥0 , 𝑦0 ) ∥ 𝑁𝛾 𝑁 And http://www.iaeme.com/IJMET/index.asp 1690 editor@iaeme.com
Raad N. Butris and Raveen F. Taher ∥ 𝛥1 (0, 𝑥01 , 𝑦01 ) − 𝛥1 (0, 𝑥02 , 𝑦02 ) ∥ 𝑀𝛾 𝐺1 𝑀𝛾 𝐺2 ∥ 𝑥01 − 𝑥02 ∥ ( )≤( )( ) … (4.2) 1 1 2 2 𝑁𝛾 𝐺3 𝑁𝛾 𝐺4 1 2 ∥ 𝛥2 (0, 𝑥0 , 𝑥0 ) − 𝛥2 (0, 𝑥0 , 𝑦0 ) ∥ ∥ 𝑦0 − 𝑦0 ∥ 𝑇 𝑇 𝑇 where 𝐻1 = 𝐹1 𝐹2 (1 − 2 𝑁𝛾 𝐶4 ) , 𝐻2 = 2 𝑀𝛾 𝐶2 𝐹1 𝐹2 , 𝐻3 = 2 𝑁𝛾 𝐶3 𝐹1 𝐹2 𝑇 𝑇2 𝐻4 = 𝐹1 (1 − 2 𝑀𝛾 𝐶1 ) [1 + 𝑁𝛾 𝑀𝛾 𝐶2 𝐶3 𝐹1 𝐹2 ] , 𝐺1 = 𝐶1 𝐻1 + 𝐶2 𝐻3 , 4 𝐺2 = 𝐶1 𝐻2 + 𝐶2 𝐻4 , 𝐺3 = 𝐶3 𝐻1 + 𝐶4 𝐻3 , 𝐺4 = 𝐶3 𝐻2 + 𝐶4 𝐻4 Proof. From the properties of the functions 𝑥(𝑡, 𝑥0 , 𝑦0 ) and 𝑦 0 (𝑡, 𝑥0 , 𝑦0 ) as in the theorem 1.1, the functions 𝛥1 = 𝛥1 (0, 𝑥0 , 𝑦0 )and 𝛥2 = 𝛥2 (0, 𝑥0 , 𝑦0 ), 𝑥0 ∈ 𝐷𝑓 , 𝑦0 ∈ 𝐷𝑔 are continuous and bounded by 𝑀𝛾 𝑀, 𝑁𝛾 𝑁 in the domain (1.2). By using (3.1) and (1.3), we get ∥ 𝛥1 (0, 𝑥0 , 𝑦0 ) ∥≤ 𝑀𝛾 𝑀 … (4.3) In similar way, we have ∥ 𝛥2 (0, 𝑥0 , 𝑦0 ) ∥≤ 𝑁𝛾 𝑁 … (4.4) then we rewrite (4.4) and (4.5) by the vector from, we get (4.2). By using (3.1) and also from inequality (1.4) and (1.5) we get ∥ 𝛥1 (0, 𝑥01 , 𝑦01 ) − 𝛥1𝑚 (0, 𝑥02 , 𝑦02 ) ∥ ≤ 𝑀𝛾 𝐶1 ‖𝑥(𝑡, 𝑥01 , 𝑦01 ) − 𝑥(𝑡, 𝑥02 , 𝑦02 ) ∥ +𝑀𝛾 𝐶2 ‖𝑦(𝑡, 𝑥01 , 𝑦01 ) − 𝑦(𝑡, 𝑥02 , 𝑦02 ) ∥ … (4.5) Similarly ∥ 𝛥2 (0, 𝑥01 , 𝑦01 ) − 𝛥2𝑚 (0, 𝑥02 , 𝑦02 ) ∥ ≤ 𝑁𝛾 𝐶3 ‖𝑥(𝑡, 𝑥01 , 𝑦01 ) − 𝑥(𝑡, 𝑥02 , 𝑦02 ) ∥ +𝑁𝛾 𝐶4 ‖𝑦(𝑡, 𝑥01 , 𝑦01 ) − 𝑦(𝑡, 𝑥02 , 𝑦02 ) ∥ … (4.6) where the functions 𝑥(𝑡, 𝑥01 , 𝑦01 ), 𝑦(𝑡, 𝑥01 , 𝑦01 ), 𝑥(𝑡, 𝑥02 , 𝑦02 ) and 𝑦(𝑡, 𝑥02 , 𝑦02 ) are solutions of the equation :- 𝑡 𝑥(𝜏, 𝑥0𝑘 , 𝑦0𝑘 ) = 𝑥0𝑘 + ∫ [𝑓( 𝜏, 𝛾(𝜏, 𝛼), 𝑥(𝜏, 𝑥0𝑘 , 𝑦0𝑘 ), 𝜇, 𝑢) 𝜏 1 𝜏+𝑇 − ∫ 𝑓(𝜏, 𝛾(𝜏, 𝛼), 𝑥(𝜏, 𝑥0𝑘 , 𝑦0𝑘 ), 𝜇, 𝑢)𝑑𝜏] 𝑑𝜏 … (4.7) 𝑇 𝜏 𝑡 𝑦(𝑡, 𝑥0𝑘 , 𝑦0𝑘 ) = 𝑦0𝑘 + ∫ [𝑔( 𝜏, 𝛾(𝜏, 𝛼), 𝑦(𝜏, 𝑥0𝑘 , 𝑦0𝑘 ), 𝜔, 𝑣) 𝜏 1 𝜏+𝑇 − ∫ 𝑔(𝜏, 𝛾(𝜏, 𝛼), 𝑦(𝜏, 𝑥0𝑘 , 𝑦0𝑘 ), 𝜔, 𝑣)𝑑𝜏] 𝑑𝜏 … (4.8) 𝑇 𝜏 where 𝑏 𝑑 𝜇 = ∫ 𝛾(𝜏, 𝛼)𝑥(𝜏, 𝑥0𝑘 , 𝑦0𝑘 )𝑑𝜏, 𝜔∗ = ∫ 𝛾(𝜏, 𝛼)𝑦(𝜏, 𝑥0𝑘 , 𝑦0𝑘 )𝑑𝜏 𝑎 𝑐 𝑡 𝑢∗ = ∫ 𝑅(𝑡, 𝜏)(𝑥(𝜏, 𝑥0𝑘 , 𝑦0𝑘 ) − 𝑦(𝜏, 𝑥0𝑘 , 𝑦0𝑘 ))𝑑𝜏 −∞ http://www.iaeme.com/IJMET/index.asp 1691 editor@iaeme.com
Periodic Solution of Integro-Differential Equations Depended on Special Function with Singular Kernels 𝑡 𝑣 = ∫ 𝐺(𝑡, 𝜏)(𝑥(𝜏, 𝑥0𝑘 , 𝑦0𝑘 ) − 𝑦(𝜏, 𝑥0𝑘 , 𝑦0𝑘 ))𝑑𝜏 ∗ −∞ where 𝑘 = 1,2. From (4.7) we get 𝑇 ∥ 𝑥(𝑡, 𝑥01 , 𝑦01 ) − 𝑥(𝑡, 𝑥02 , 𝑦02 ) ∥≤ ‖𝑥01 − 𝑥02 ‖ + 𝑀𝛾 𝐶1 ‖𝑥(𝑡, 𝑥01 , 𝑦01 ) 2 𝑇 −𝑥(𝑡, 𝑥02 , 𝑦02 ) ∥ + 𝑀𝛾 𝐶2 ‖𝑦(𝑡, 𝑥01 , 𝑦01 ) − 𝑦(𝑡, 𝑥02 , 𝑦02 ) ∥ . 2 Therefore 𝑇 ∥ 𝑥(𝑡, 𝑥01 , 𝑦01 ) − 𝑥(𝑡, 𝑥02 , 𝑦02 ) ∥≤ (1 − 𝑀𝛾 𝐶1 )−1 ‖𝑥01 − 𝑥02 ‖ 2 𝑇 𝑇 +(1 − 𝑀𝛾 𝐶1 )−1 𝑀𝛾 𝐶2 ‖𝑦(𝑡, 𝑥01 , 𝑦01 ) − 𝑦(𝑡, 𝑥02 , 𝑦02 ) ∥ … (4.9) 2 2 Also from (4.8), we have 𝑇 ∥ 𝑦(𝑡, 𝑥01 , 𝑦01 ) − 𝑦(𝑡, 𝑥02 , 𝑦02 ) ∥≤ ‖𝑦01 − 𝑦02 ‖ + 𝑁𝛾 𝐶3 ‖𝑥(𝑡, 𝑥01 , 𝑦01 ) 2 𝑇 −𝑥(𝑡, 𝑥02 , 𝑦02 ) ∥ + 𝑁𝛾 𝐶4 ‖𝑦(𝑡, 𝑥01 , 𝑦01 ) − 𝑦(𝑡, 𝑥02 , 𝑦02 ) ∥ 2 and hence 𝑇 ∥ 𝑦(𝑡, 𝑥01 , 𝑦01 ) − 𝑦(𝑡, 𝑥02 , 𝑦02 ) ∥≤ (1 − 𝑁𝛾 𝐶4 )−1 ‖𝑦01 − 𝑦02 ‖ 2 𝑇 𝑇 +(1 − 𝑁𝛾 𝐶4 )−1 𝑁𝛾 𝐶3 ‖𝑥(𝑡, 𝑥01 , 𝑦01 ) − 𝑥(𝑡, 𝑥02 , 𝑦02 ) ∥ … (4.10) 2 2 Now, by substituting (4.9) in (4.10), we get 𝑇 𝑇 ∥ 𝑥(𝑡, 𝑥01 , 𝑦01 ) − 𝑥(𝑡, 𝑥02 , 𝑦02 ) ∥ ≤ (1 − 𝑀𝛾 𝐶1 )−1 ‖𝑥01 − 𝑥02 ‖ + 𝑀𝛾 𝐶2 2 2 𝑇 𝑇 𝑇2 [(1 − 𝑀𝛾 𝐶1 ) (1 − 𝑁𝛾 𝐶4 )]−1 ‖𝑦01 − 𝑦02 ‖ + 𝑀𝛾 𝐶2 𝑁𝛾 𝐶3 2 2 4 𝑇 𝑇 [(1 − 𝑀𝛾 𝐶1 ) (1 − 𝑁𝛾 𝐶4 )]−1 ‖𝑥(𝑡, 𝑥01 , 𝑦01 ) − 𝑥(𝑡, 𝑥02 , 𝑦02 ) ∥ 2 2 𝑇 𝑇 Putting 𝐹1 = [(1 − 2 𝑀𝛾 𝐶1 ) (1 − 2 𝑁𝛾 𝐶4 )]−1 and substituting in the last inequality, we obtain 𝑇 𝑇 ∥ 𝑥(𝑡, 𝑥01 , 𝑦01 ) − 𝑥(𝑡, 𝑥02 , 𝑦02 ) ∥ ≤ (1 − 𝑀𝛾 𝐶1 )−1 ‖𝑥01 − 𝑥02 ‖ + 𝑀𝛾 𝐶2 𝐹1 2 2 2 𝑇 ‖𝑦01 − 𝑦02 ‖ + 𝑀𝛾 𝐶2 𝑁𝛾 𝐶3 𝐹1 ‖𝑥(𝑡, 𝑥01 , 𝑦01 ) − 𝑥(𝑡, 𝑥02 , 𝑦02 ) ∥ 4 𝑇 𝑇 −1 as the 𝐹1 (1 − 2 𝑁𝛾 𝐶4 ) = (1 − 2 𝑀𝛾 𝐶1 ) 𝑇 𝑇 ∥ 𝑥(𝑡, 𝑥01 , 𝑦01 ) − 𝑥(𝑡, 𝑥02 , 𝑦02 ) ∥ ≤ 𝐹1 (1 − 𝑁𝛾 𝐶4 ) ‖𝑥01 − 𝑥02 ‖ + 𝑀𝛾 𝐶2 𝐹1 2 2 2 𝑇 ‖𝑦01 − 𝑦02 ‖ + 𝑀𝛾 𝐶2 𝑁𝛾 𝐶3 𝐹1 ‖𝑥(𝑡, 𝑥01 , 𝑦01 ) − 𝑥(𝑡, 𝑥02 , 𝑦02 ) ∥ 4 This implies that http://www.iaeme.com/IJMET/index.asp 1692 editor@iaeme.com
Raad N. Butris and Raveen F. Taher −1 𝑇 𝑇2 ∥ 𝑥(𝑡, 𝑥01 , 𝑦01 ) − 𝑥(𝑡, 𝑥02 , 𝑦02 ) ∥≤ 𝐹1 (1 − 𝑁𝛾 𝐶4 ) (1 − 𝑀𝛾 𝐶2 𝑁𝛾 𝐶3 𝐹1 ) 2 4 −1 𝑇2 𝑇 ‖𝑥01 − 𝑥02 ‖ + (1 − 𝑀 𝐶𝑁𝐶𝐹) 𝑀𝛾 𝐶2 𝐹1 ‖𝑦01 − 𝑦02 ‖ 4 𝛾 2 𝛾 3 1 2 −1 𝑇2 Putting 𝐹2 = (1 − 𝑀𝛾 𝐶2 𝑁𝛾 𝐶3 𝐹1 ) 4 and substituting in the last inequality, we obtain 𝑇 ∥ 𝑥(𝑡, 𝑥01 , 𝑦01 ) − 𝑥(𝑡, 𝑥02 , 𝑦02 ) ∥≤ 𝐹1 𝐹2 (1 − 𝑁𝛾 𝐶4 ) ‖𝑥01 − 𝑥02 ‖ 2 𝑇 + 𝑀𝛾 𝐶2 𝐹1 𝐹2 ‖𝑦01 − 𝑦02 ‖ … (4.11) 2 Also, substituting the inequalities (4.11) in (4.10), we find that 𝑇 ∥ 𝑦(𝑡, 𝑥01 , 𝑦01 ) − 𝑦(𝑡, 𝑥02 , 𝑦02 ) ∥≤ (1 − 𝑁𝛾 𝐶4 )−1 ‖𝑦01 − 𝑦02 ‖ 2 𝑇 𝑇 𝑇 +(1 − 𝑁𝛾 𝐶4 )−1 𝑁𝛾 𝐶3 [𝐹1 𝐹2 (1 − 𝑁𝛾 𝐶4 ) ‖𝑥01 − 𝑥02 ‖ 2 2 2 𝑇 + 𝑀𝛾 𝐶2 𝐹1 𝐹2 ‖𝑦01 − 𝑦02 ‖] 2 and hence 𝑇 ∥ 𝑦(𝑡, 𝑥01 , 𝑦01 ) − 𝑦(𝑡, 𝑥02 , 𝑦02 ) ∥≤ 𝑁𝛾 𝐶3 𝐹1 𝐹2 ‖𝑥01 − 𝑥02 ‖ 2 𝑇 𝑇2 +𝐹1 (1 − 2 𝑀𝛾 𝐶1 ) [1 + 𝑁𝛾 𝑀𝛾 𝐶2 𝐶3 𝐹1 𝐹2 ] ‖𝑦01 − 𝑦02 ‖ … (4.12) 4 So, substituting inequalities (4.11) and (4.12) in inequality (4.5), we get ∥ 𝛥1 (0, 𝑥01 , 𝑦01 ) − 𝛥1𝑚 (0, 𝑥02 , 𝑦02 ) ∥ ≤ 𝑀𝛾 𝐶1 [𝐻1 ‖𝑥01 − 𝑥02 ‖ +𝐻2 ‖𝑦01 − 𝑦02 ‖] + 𝑀𝛾 𝐶2 [𝐻3 ‖𝑥01 − 𝑥02 ‖ + 𝐻4 ‖𝑦01 − 𝑦02 ‖ . Therefore ∥ 𝛥1 (0, 𝑥01 , 𝑦01 ) − 𝛥1𝑚 (0, 𝑥02 , 𝑦02 ) ∥ ≤ 𝑀𝛾 [𝐺1 ‖𝑥01 − 𝑥02 ‖ + 𝐺2 ‖𝑦01 − 𝑦02 ‖ … (4.13) And by the same techniques, substituting inequalities (4.11) and (4.12) in inequality (4.6), we get ∥ 𝛥2 (0, 𝑥01 , 𝑦01 ) − 𝛥2𝑚 (0, 𝑥02 , 𝑦02 ) ∥ ≤ 𝑁𝛾 𝐶3 [𝐻1 ‖𝑥01 − 𝑥02 ‖ +𝐻2 ‖𝑦01 − 𝑦02 ‖] + 𝑁𝛾 𝐶4 [𝐻3 ‖𝑥01 − 𝑥02 ‖ + 𝐻4 ‖𝑦01 − 𝑦02 ‖ So ∥ 𝛥2 (0, 𝑥01 , 𝑦01 ) − 𝛥2𝑚 (0, 𝑥02 , 𝑦02 ) ∥ ≤ 𝑁𝛾 [𝐺3 ‖𝑥01 − 𝑥02 ‖ + 𝐺4 ‖𝑦01 − 𝑦02 ‖ … (4.14) rewrite (4.13) and (4.14) by the vector form, we get (4.2). Remark 4.1. Theorem (4.1), confirms the stability of the solution of (1.1), that is when a slight change happens in the points 𝑥0 , 𝑦0, then a slight change will happen in the function ∆(0, 𝑥0 , 𝑦0 ). For this remark see [10]. Remark 4.2. In the general case the initial values of the periodic solutions of (1.1) should be sought by a numerical method. It is possible to use the properties of the 𝛥 −constant expressed by the following theorem. http://www.iaeme.com/IJMET/index.asp 1693 editor@iaeme.com
Periodic Solution of Integro-Differential Equations Depended on Special Function with Singular Kernels Theorem 4.2. Suppose that the system (1.1) be defined on the domain (1.2). Then there exists a sequence of functions Δ1𝑚 (0, 𝑥0 , 𝑦0 ) and Δ2𝑚 (0, 𝑥0 , 𝑦0 ) which are defined by (3.1) and (3.2) the following inequality holds :- ∥ 𝛥1𝑚 (0, 𝑥0 , 𝑦0 ) ∥ 𝑏1𝑚 𝑀𝛾 𝑀 ( )≤( )+( ) … (4.15) (0, ∥ 𝛥2𝑚 𝑥0 , 𝑦0 ∥ ) 𝑏2𝑚 𝑁𝛾 𝑁 for all 𝑚 ≥ 0. Proof. From (3.10), we get ∥ 𝛥1𝑚 (0, 𝑥0 , 𝑦0 ) ∥ ≤ ∥ 𝛥1𝑚 (0, 𝑥0 , 𝑦0 )−𝛥1 (0, 𝑥0 , 𝑦0 ) + 𝛥1 (0, 𝑥0 , 𝑦0 ) ∥ ≤ ∥ 𝛥1𝑚 (0, 𝑥0 , 𝑦0 )−𝛥1 (0, 𝑥0 , 𝑦0 ) ∥ +∥ 𝛥1 (0, 𝑥0 , 𝑦0 ) ∥ By using the inequalities (3.3) and (4.1), we get 𝑀𝛾 𝐶1 ∥ 𝛥1𝑚 (0, 𝑥0 , 𝑦0 ) ∥ ≤ 〈( ) , Λ𝑚 (𝐸 − Λ)−1 Ω1 〉 + 𝑀𝛾 𝑀 … (4.16) 𝑀𝛾 𝐶2 And from (3.11), we get ∥ 𝛥2𝑚 (0, 𝑥0 , 𝑦0 ) ∥ ≤ ∥ 𝛥2𝑚 (0, 𝑥0 , 𝑦0 )−𝛥2 (0, 𝑥0 , 𝑦0 ) + 𝛥2 (0, 𝑥0 , 𝑦0 ) ∥ ≤ ∥ 𝛥2𝑚 (0, 𝑥0 , 𝑦0 )−𝛥2 (0, 𝑥0 , 𝑦0 ) ∥ +∥ 𝛥2 (0, 𝑥0 , 𝑦0 ) ∥ By using the inequalities (3.3) and (4.1), we get 𝑁𝛾 𝐶3 ∥ 𝛥2𝑚 (0, 𝑥0 , 𝑦0 ) ∥ ≤ 〈( ) , Λ𝑚 (𝐸 − Λ)−1 Ω1 〉 + 𝑁𝛾 𝑁 … (4.17) 𝑁𝛾 𝐶4 Rewrite the inequalities (4.16) and (4.17) in a vector form as :- ∥ 𝛥1𝑚 (0, 𝑥0 , 𝑦0 ) ∥ 𝑀𝛾 𝐶1 〈( ) , Λ𝑚 (𝐸 − Λ)−1 Ω1 〉 𝑀𝛾 𝐶2 𝑀𝛾 𝑀 ≤ +( ) … (4.18) 𝑁𝛾 𝐶3 𝑚 (𝐸 𝑁 𝑁 〈( (0, 𝑥0 , 𝑦0 ) ∥) ( 𝑁𝛾 𝐶4 ) , Λ − Λ)−1 Ω1 〉 𝛾 ( ∥ 𝛥2𝑚 ) from the inequality (4.18), we get (4.15). REFERENCES [1] Aziz, M. A., Periodic solutions for some systems of non-linear ordinary differential equations, M. Sc. Thesis, College of Education, University of Mosul, (2006). [2] Butris, R. N., Ava, SH. R. and Hewa, S. F., Existence, uniqueness and stability of periodic solution for nonlinear system of integro-differential equations science Journal of University of Zakho Vol. 5.No. 1 pp. 120-127, March- (2017).. [3] Butris, R. N., Periodic solution of non-linear system of integro-differential equations depending the Gamma distribution, India, Gen. Math. Notes V0l 1.15, No. 1, (2013). [4] Kigurodze and Puza, M., On periodic solutions of non-linear functional differential equations, Georgian Math. J. Vol. 6, No. 1.(1999). [5] Narjanov, O. D., On a Periodic solution for integro-differential equation, Math. J. , Kiev, Ukraine, Tom.2. (1977). http://www.iaeme.com/IJMET/index.asp 1694 editor@iaeme.com
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