Existence and Uniqueness of Positive Solutions of Boundary-Value Problems for Fractional Differential Equations with p-Laplacian Operator and Identities on the Some Special Polynomials
We consider the following boundary-value problem of nonlinear fractional differential equation with p-Laplacian operator D0+β(ϕp(D0+αu(t)))+a(t)f(u)=0, 0<t<1, u(0)=γu(h)+λ, u′(0)=μ, ϕp(D0+αu(0))=(ϕp(D0+αu(1)))′=(ϕp(D0+αu(0)))′′=(ϕp(D0+αu(0)))′′′=0, where 1<α⩽2, 3<β⩽4 are real numbers, D0...
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| Format: | Article |
| Language: | English |
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Wiley
2013-01-01
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| Series: | Journal of Function Spaces and Applications |
| Online Access: | http://dx.doi.org/10.1155/2013/753171 |
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| author | Erdoğan Şen Mehmet Acikgoz Jong Jin Seo Serkan Araci Kamil Oruçoğlu |
| author_facet | Erdoğan Şen Mehmet Acikgoz Jong Jin Seo Serkan Araci Kamil Oruçoğlu |
| author_sort | Erdoğan Şen |
| collection | DOAJ |
| description | We consider the following boundary-value problem of nonlinear fractional differential equation with p-Laplacian operator D0+β(ϕp(D0+αu(t)))+a(t)f(u)=0, 0<t<1, u(0)=γu(h)+λ, u′(0)=μ, ϕp(D0+αu(0))=(ϕp(D0+αu(1)))′=(ϕp(D0+αu(0)))′′=(ϕp(D0+αu(0)))′′′=0, where 1<α⩽2, 3<β⩽4 are real numbers, D0+α,D0+β are the standard Caputo fractional derivatives, ϕp(s)=|s|p-2s, p>1, ϕp-1=ϕq, 1/p+1/q=1,0⩽γ<1, 0⩽h⩽1, λ,μ>0 are parameters, a:(0,1)→[0,+∞), and f:[0,+∞)→[0,+∞) are continuous. By the properties of Green function and Schauder fixed point theorem, several existence and nonexistence results for positive solutions, in terms of the parameters λ and μ are obtained. The uniqueness of positive solution on the parameters λ and μ is also studied. In the final section of this paper, we derive not only new but also interesting identities related special polynomials by which Caputo fractional derivative. |
| format | Article |
| id | doaj-art-bfe5272374b3499fafa02506afde6837 |
| institution | OA Journals |
| issn | 0972-6802 1758-4965 |
| language | English |
| publishDate | 2013-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Journal of Function Spaces and Applications |
| spelling | doaj-art-bfe5272374b3499fafa02506afde68372025-08-20T02:19:27ZengWileyJournal of Function Spaces and Applications0972-68021758-49652013-01-01201310.1155/2013/753171753171Existence and Uniqueness of Positive Solutions of Boundary-Value Problems for Fractional Differential Equations with p-Laplacian Operator and Identities on the Some Special PolynomialsErdoğan Şen0Mehmet Acikgoz1Jong Jin Seo2Serkan Araci3Kamil Oruçoğlu4Department of Mathematics, Faculty of Science and Letters, Namik Kemal University, 59030 Tekirdağ, TurkeyDepartment of Mathematics, Faculty of Science and Arts, University of Gaziantep, 27310 Gaziantep, TurkeyDepartment of Applied Mathematics, Pukyong National University, Busan 608-737, Republic of KoreaAtatürk Street, 31290 Hatay, TurkeyDepartment of Mathematics Engineering, Istanbul Technical University, Maslak, 34469 Istanbul, TurkeyWe consider the following boundary-value problem of nonlinear fractional differential equation with p-Laplacian operator D0+β(ϕp(D0+αu(t)))+a(t)f(u)=0, 0<t<1, u(0)=γu(h)+λ, u′(0)=μ, ϕp(D0+αu(0))=(ϕp(D0+αu(1)))′=(ϕp(D0+αu(0)))′′=(ϕp(D0+αu(0)))′′′=0, where 1<α⩽2, 3<β⩽4 are real numbers, D0+α,D0+β are the standard Caputo fractional derivatives, ϕp(s)=|s|p-2s, p>1, ϕp-1=ϕq, 1/p+1/q=1,0⩽γ<1, 0⩽h⩽1, λ,μ>0 are parameters, a:(0,1)→[0,+∞), and f:[0,+∞)→[0,+∞) are continuous. By the properties of Green function and Schauder fixed point theorem, several existence and nonexistence results for positive solutions, in terms of the parameters λ and μ are obtained. The uniqueness of positive solution on the parameters λ and μ is also studied. In the final section of this paper, we derive not only new but also interesting identities related special polynomials by which Caputo fractional derivative.http://dx.doi.org/10.1155/2013/753171 |
| spellingShingle | Erdoğan Şen Mehmet Acikgoz Jong Jin Seo Serkan Araci Kamil Oruçoğlu Existence and Uniqueness of Positive Solutions of Boundary-Value Problems for Fractional Differential Equations with p-Laplacian Operator and Identities on the Some Special Polynomials Journal of Function Spaces and Applications |
| title | Existence and Uniqueness of Positive Solutions of Boundary-Value Problems for Fractional Differential Equations with p-Laplacian Operator and Identities on the Some Special Polynomials |
| title_full | Existence and Uniqueness of Positive Solutions of Boundary-Value Problems for Fractional Differential Equations with p-Laplacian Operator and Identities on the Some Special Polynomials |
| title_fullStr | Existence and Uniqueness of Positive Solutions of Boundary-Value Problems for Fractional Differential Equations with p-Laplacian Operator and Identities on the Some Special Polynomials |
| title_full_unstemmed | Existence and Uniqueness of Positive Solutions of Boundary-Value Problems for Fractional Differential Equations with p-Laplacian Operator and Identities on the Some Special Polynomials |
| title_short | Existence and Uniqueness of Positive Solutions of Boundary-Value Problems for Fractional Differential Equations with p-Laplacian Operator and Identities on the Some Special Polynomials |
| title_sort | existence and uniqueness of positive solutions of boundary value problems for fractional differential equations with p laplacian operator and identities on the some special polynomials |
| url | http://dx.doi.org/10.1155/2013/753171 |
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