Positive solutions to boundary-value problems of p-Laplacian fractional differential equations with a parameter in the boundary
In this article, we consider the following boundary-value problem of nonlinear fractional differential equation with $p$-Laplacian operator $$displaylines{ D_{0+}^eta(phi_p(D_{0+}^alpha u(t)))+a(t)f(u)=0, quad 0<t<1, cr u(0)=gamma u(xi)+lambda, quad phi_p(D_{0+}^alpha u(0))=(phi_...
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Texas State University
2012-11-01
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Online Access: | http://ejde.math.txstate.edu/Volumes/2012/213/abstr.html |
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doaj-538114d03eb84818a74def4a6ccf237d2020-11-24T22:47:51ZengTexas State UniversityElectronic Journal of Differential Equations1072-66912012-11-012012213,114Positive solutions to boundary-value problems of p-Laplacian fractional differential equations with a parameter in the boundaryZhenlai HanHongling LuShurong SunDianwu YangIn this article, we consider the following boundary-value problem of nonlinear fractional differential equation with $p$-Laplacian operator $$displaylines{ D_{0+}^eta(phi_p(D_{0+}^alpha u(t)))+a(t)f(u)=0, quad 0<t<1, cr u(0)=gamma u(xi)+lambda, quad phi_p(D_{0+}^alpha u(0))=(phi_p(D_{0+}^alpha u(1)))' =(phi_p(D_{0+}^alpha u(0)))''=0, }$$ where $0<alphaleqslant1$, $2<etaleqslant 3$ are real numbers, $D_{0+}^alpha, D_{0+}^eta$ are the standard Caputo fractional derivatives, $phi_p(s)=|s|^{p-2}s$, $p>1$, $phi_p^{-1}=phi_q$, $1/p+1/q=1$, $0leqslantgamma<1$, $0leqslantxileqslant1$, $lambda>0$ is a parameter, $a:(0,1)o [0,+infty)$ and $f:[0,+infty)o[0,+infty)$ 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 parameter $lambda$ are obtained. The uniqueness of positive solution on the parameter $lambda$ is also studied. Some examples are presented to illustrate the main results. http://ejde.math.txstate.edu/Volumes/2012/213/abstr.htmlFractional boundary-value problempositive solutionconeSchauder fixed point theoremuniquenessp-Laplacian operator |
collection |
DOAJ |
language |
English |
format |
Article |
sources |
DOAJ |
author |
Zhenlai Han Hongling Lu Shurong Sun Dianwu Yang |
spellingShingle |
Zhenlai Han Hongling Lu Shurong Sun Dianwu Yang Positive solutions to boundary-value problems of p-Laplacian fractional differential equations with a parameter in the boundary Electronic Journal of Differential Equations Fractional boundary-value problem positive solution cone Schauder fixed point theorem uniqueness p-Laplacian operator |
author_facet |
Zhenlai Han Hongling Lu Shurong Sun Dianwu Yang |
author_sort |
Zhenlai Han |
title |
Positive solutions to boundary-value problems of p-Laplacian fractional differential equations with a parameter in the boundary |
title_short |
Positive solutions to boundary-value problems of p-Laplacian fractional differential equations with a parameter in the boundary |
title_full |
Positive solutions to boundary-value problems of p-Laplacian fractional differential equations with a parameter in the boundary |
title_fullStr |
Positive solutions to boundary-value problems of p-Laplacian fractional differential equations with a parameter in the boundary |
title_full_unstemmed |
Positive solutions to boundary-value problems of p-Laplacian fractional differential equations with a parameter in the boundary |
title_sort |
positive solutions to boundary-value problems of p-laplacian fractional differential equations with a parameter in the boundary |
publisher |
Texas State University |
series |
Electronic Journal of Differential Equations |
issn |
1072-6691 |
publishDate |
2012-11-01 |
description |
In this article, we consider the following boundary-value problem of nonlinear fractional differential equation with $p$-Laplacian operator $$displaylines{ D_{0+}^eta(phi_p(D_{0+}^alpha u(t)))+a(t)f(u)=0, quad 0<t<1, cr u(0)=gamma u(xi)+lambda, quad phi_p(D_{0+}^alpha u(0))=(phi_p(D_{0+}^alpha u(1)))' =(phi_p(D_{0+}^alpha u(0)))''=0, }$$ where $0<alphaleqslant1$, $2<etaleqslant 3$ are real numbers, $D_{0+}^alpha, D_{0+}^eta$ are the standard Caputo fractional derivatives, $phi_p(s)=|s|^{p-2}s$, $p>1$, $phi_p^{-1}=phi_q$, $1/p+1/q=1$, $0leqslantgamma<1$, $0leqslantxileqslant1$, $lambda>0$ is a parameter, $a:(0,1)o [0,+infty)$ and $f:[0,+infty)o[0,+infty)$ 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 parameter $lambda$ are obtained. The uniqueness of positive solution on the parameter $lambda$ is also studied. Some examples are presented to illustrate the main results. |
topic |
Fractional boundary-value problem positive solution cone Schauder fixed point theorem uniqueness p-Laplacian operator |
url |
http://ejde.math.txstate.edu/Volumes/2012/213/abstr.html |
work_keys_str_mv |
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