Application of Topological Degree Method for Solutions of Coupled Systems of Multipoints Boundary Value Problems of Fractional Order Hybrid Differential Equations
We established the theory to coupled systems of multipoints boundary value problems of fractional order hybrid differential equations with nonlinear perturbations of second type involving Caputo fractional derivative. The proposed problem is as follows: D cαxt-ft,xt=gt,yt,Iαyt, t∈J=[0,1],D cαyt-ft,y...
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| Format: | Article |
| Language: | English |
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Wiley
2017-01-01
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| Series: | Complexity |
| Online Access: | http://dx.doi.org/10.1155/2017/7676814 |
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| author | Muhammad Iqbal Yongjin Li Kamal Shah Rahmat Ali Khan |
| author_facet | Muhammad Iqbal Yongjin Li Kamal Shah Rahmat Ali Khan |
| author_sort | Muhammad Iqbal |
| collection | DOAJ |
| description | We established the theory to coupled systems of multipoints boundary value problems of fractional order hybrid differential equations with nonlinear perturbations of second type involving Caputo fractional derivative. The proposed problem is as follows: D cαxt-ft,xt=gt,yt,Iαyt, t∈J=[0,1],D cαyt-ft,yt=gt,xt,Iαxt, t∈J=0,1, D cpx0=ψxη1, x′0=0,…,xn-20=0, D cpx1=ψxη2, D cpy0=ψyη1, y′0=0,…,yn-20=0, D cpy1=ψyη2, where p,η1,η2∈0,1, ψ is linear, D cα is Caputo fractional derivative of order α, with n-1<α≤n, n∈N, and Iα is fractional integral of order α. The nonlinear functions f, g are continuous. For obtaining sufficient conditions on existence and uniqueness of positive solutions to the above system, we used the technique of topological degree theory. Finally, we illustrated the main results by a concrete example. |
| format | Article |
| id | doaj-art-31725949c4e54334a5c5adb1ad653121 |
| institution | Kabale University |
| issn | 1076-2787 1099-0526 |
| language | English |
| publishDate | 2017-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Complexity |
| spelling | doaj-art-31725949c4e54334a5c5adb1ad6531212025-02-03T01:12:08ZengWileyComplexity1076-27871099-05262017-01-01201710.1155/2017/76768147676814Application of Topological Degree Method for Solutions of Coupled Systems of Multipoints Boundary Value Problems of Fractional Order Hybrid Differential EquationsMuhammad Iqbal0Yongjin Li1Kamal Shah2Rahmat Ali Khan3Department of Mathematics, University of Malakand, Dir (L), Khyber Pakhtunkhwa, PakistanDepartment of Mathematics, Sun Yat-Sen University, Guangzhou, ChinaDepartment of Mathematics, University of Malakand, Dir (L), Khyber Pakhtunkhwa, PakistanDepartment of Mathematics, University of Malakand, Dir (L), Khyber Pakhtunkhwa, PakistanWe established the theory to coupled systems of multipoints boundary value problems of fractional order hybrid differential equations with nonlinear perturbations of second type involving Caputo fractional derivative. The proposed problem is as follows: D cαxt-ft,xt=gt,yt,Iαyt, t∈J=[0,1],D cαyt-ft,yt=gt,xt,Iαxt, t∈J=0,1, D cpx0=ψxη1, x′0=0,…,xn-20=0, D cpx1=ψxη2, D cpy0=ψyη1, y′0=0,…,yn-20=0, D cpy1=ψyη2, where p,η1,η2∈0,1, ψ is linear, D cα is Caputo fractional derivative of order α, with n-1<α≤n, n∈N, and Iα is fractional integral of order α. The nonlinear functions f, g are continuous. For obtaining sufficient conditions on existence and uniqueness of positive solutions to the above system, we used the technique of topological degree theory. Finally, we illustrated the main results by a concrete example.http://dx.doi.org/10.1155/2017/7676814 |
| spellingShingle | Muhammad Iqbal Yongjin Li Kamal Shah Rahmat Ali Khan Application of Topological Degree Method for Solutions of Coupled Systems of Multipoints Boundary Value Problems of Fractional Order Hybrid Differential Equations Complexity |
| title | Application of Topological Degree Method for Solutions of Coupled Systems of Multipoints Boundary Value Problems of Fractional Order Hybrid Differential Equations |
| title_full | Application of Topological Degree Method for Solutions of Coupled Systems of Multipoints Boundary Value Problems of Fractional Order Hybrid Differential Equations |
| title_fullStr | Application of Topological Degree Method for Solutions of Coupled Systems of Multipoints Boundary Value Problems of Fractional Order Hybrid Differential Equations |
| title_full_unstemmed | Application of Topological Degree Method for Solutions of Coupled Systems of Multipoints Boundary Value Problems of Fractional Order Hybrid Differential Equations |
| title_short | Application of Topological Degree Method for Solutions of Coupled Systems of Multipoints Boundary Value Problems of Fractional Order Hybrid Differential Equations |
| title_sort | application of topological degree method for solutions of coupled systems of multipoints boundary value problems of fractional order hybrid differential equations |
| url | http://dx.doi.org/10.1155/2017/7676814 |
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