The Existence of Positive Solutions for Boundary Value Problem of the Fractional Sturm-Liouville Functional Differential Equation
We study boundary value problems for the following nonlinear fractional Sturm-Liouville functional differential equations involving the Caputo fractional derivative: CDβ(p(t)CDαu(t)) + f(t,u(t-τ),u(t+θ))=0, t∈(0,1), CDαu(0)= CDαu(1)=( CDαu(0))=0, au(t)-bu′(t)=η(t), t∈[-τ,0], cu(t)+du′(t)=ξ(t), t∈...
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| Format: | Article |
| Language: | English |
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Wiley
2013-01-01
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| Series: | Abstract and Applied Analysis |
| Online Access: | http://dx.doi.org/10.1155/2013/301560 |
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| author | Yanan Li Shurong Sun Zhenlai Han Hongling Lu |
| author_facet | Yanan Li Shurong Sun Zhenlai Han Hongling Lu |
| author_sort | Yanan Li |
| collection | DOAJ |
| description | We study boundary value problems for the following nonlinear fractional Sturm-Liouville functional differential equations involving the Caputo fractional derivative: CDβ(p(t)CDαu(t)) + f(t,u(t-τ),u(t+θ))=0, t∈(0,1), CDαu(0)= CDαu(1)=( CDαu(0))=0, au(t)-bu′(t)=η(t), t∈[-τ,0], cu(t)+du′(t)=ξ(t), t∈[1,1+θ], where CDα, CDβ denote the Caputo fractional derivatives, f is a nonnegative continuous functional defined on C([-τ,1+θ],ℝ), 1<α≤2, 2<β≤3, 0<τ, θ<1/4 are suitably small, a,b,c,d>0, and η∈C([-τ,0],[0,∞)), ξ∈C([1,1+θ],[0,∞)). By means of the Guo-Krasnoselskii fixed point theorem and the fixed point index theorem, some positive solutions are obtained, respectively. As an application, an example is presented to illustrate our main results. |
| format | Article |
| id | doaj-art-9d2bb59eb1484811ab047fbb45a2479a |
| institution | Kabale University |
| issn | 1085-3375 1687-0409 |
| language | English |
| publishDate | 2013-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Abstract and Applied Analysis |
| spelling | doaj-art-9d2bb59eb1484811ab047fbb45a2479a2025-02-03T01:22:11ZengWileyAbstract and Applied Analysis1085-33751687-04092013-01-01201310.1155/2013/301560301560The Existence of Positive Solutions for Boundary Value Problem of the Fractional Sturm-Liouville Functional Differential EquationYanan Li0Shurong Sun1Zhenlai Han2Hongling Lu3School of Mathematical Sciences, University of Jinan, Jinan, Shandong 250022, ChinaSchool of Mathematical Sciences, University of Jinan, Jinan, Shandong 250022, ChinaSchool of Mathematical Sciences, University of Jinan, Jinan, Shandong 250022, ChinaSchool of Mathematical Sciences, University of Jinan, Jinan, Shandong 250022, ChinaWe study boundary value problems for the following nonlinear fractional Sturm-Liouville functional differential equations involving the Caputo fractional derivative: CDβ(p(t)CDαu(t)) + f(t,u(t-τ),u(t+θ))=0, t∈(0,1), CDαu(0)= CDαu(1)=( CDαu(0))=0, au(t)-bu′(t)=η(t), t∈[-τ,0], cu(t)+du′(t)=ξ(t), t∈[1,1+θ], where CDα, CDβ denote the Caputo fractional derivatives, f is a nonnegative continuous functional defined on C([-τ,1+θ],ℝ), 1<α≤2, 2<β≤3, 0<τ, θ<1/4 are suitably small, a,b,c,d>0, and η∈C([-τ,0],[0,∞)), ξ∈C([1,1+θ],[0,∞)). By means of the Guo-Krasnoselskii fixed point theorem and the fixed point index theorem, some positive solutions are obtained, respectively. As an application, an example is presented to illustrate our main results.http://dx.doi.org/10.1155/2013/301560 |
| spellingShingle | Yanan Li Shurong Sun Zhenlai Han Hongling Lu The Existence of Positive Solutions for Boundary Value Problem of the Fractional Sturm-Liouville Functional Differential Equation Abstract and Applied Analysis |
| title | The Existence of Positive Solutions for Boundary Value Problem of the Fractional Sturm-Liouville Functional Differential Equation |
| title_full | The Existence of Positive Solutions for Boundary Value Problem of the Fractional Sturm-Liouville Functional Differential Equation |
| title_fullStr | The Existence of Positive Solutions for Boundary Value Problem of the Fractional Sturm-Liouville Functional Differential Equation |
| title_full_unstemmed | The Existence of Positive Solutions for Boundary Value Problem of the Fractional Sturm-Liouville Functional Differential Equation |
| title_short | The Existence of Positive Solutions for Boundary Value Problem of the Fractional Sturm-Liouville Functional Differential Equation |
| title_sort | existence of positive solutions for boundary value problem of the fractional sturm liouville functional differential equation |
| url | http://dx.doi.org/10.1155/2013/301560 |
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