The existence of a positive solution for nonlinear fractional differential equations with integral boundary value conditions

In this paper, first we consider the existence of a positive solution for the nonlinear fractional differential equation boundary--value problem \begin{eqnarray*} &&^C\D^{\alpha}u(t)+f(t,u(t))=0, \quad 0< t<1,\quad 2<\alpha\leq 3,\\ &&u'(0)=u''(0)=0, \quad u(1...

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Detalles Bibliográficos
Autores: Cabada Fernández, Alberto, Dimitrijevic, Sladjana, Tomovic, Tatjana, Aleksic, Suzana
Tipo de recurso: artículo
Fecha de publicación:2017
País:España
Institución:Universidad de Santiago de Compostela (USC)
Repositorio:Minerva. Repositorio Institucional de la Universidad de Santiago de Compostela
Idioma:inglés
OAI Identifier:oai:minerva.usc.gal:10347/45460
Acceso en línea:https://hdl.handle.net/10347/45460
Access Level:acceso abierto
Palabra clave:Fractional differential equation
Integral boundary value condition
Positive solution
Green’s function
Fixed-point theorem
1202 Análisis y análisis funcional
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spelling The existence of a positive solution for nonlinear fractional differential equations with integral boundary value conditionsCabada Fernández, AlbertoDimitrijevic, SladjanaTomovic, TatjanaAleksic, SuzanaFractional differential equationIntegral boundary value conditionPositive solutionGreen’s functionFixed-point theorem1202 Análisis y análisis funcionalIn this paper, first we consider the existence of a positive solution for the nonlinear fractional differential equation boundary--value problem \begin{eqnarray*} &&^C\D^{\alpha}u(t)+f(t,u(t))=0, \quad 0< t<1,\quad 2<\alpha\leq 3,\\ &&u'(0)=u''(0)=0, \quad u(1)=\lambda\int_0^1u(s)\id s, \end{eqnarray*} where $0\leq\lambda<1$, and $^C\D^\alpha$ is the Caputo's differential operator of order $\alpha$, and $f:[0,1]\times[0,\infty)\rightarrow[0,\infty)$ is a continuous function. Using some cone theoretic techniques, we deduce a general existence theorem for this problem. Then, we consider two following more general problems for arbitrary $\alpha$, $1\leq n<\alpha\leq n+1$: Problem 1: \begin{eqnarray*} &&^C\D^{\alpha}u(t)+f(t,u(t))=0, \quad 0< t<1,\\ &&u^{(i)}(0)=0,\; 0\leq i\leq n,\; i\neq k, \quad u(1)=\lambda\int_0^1u(s)\id s, \end{eqnarray*} where $k\in\{0, 1, \dots, n-1\}$, $0\leq\lambda<k+1$; Problem 2: \begin{eqnarray*} &&D^{\alpha}u(t)+f(t,u(t))=0, \quad 0< t<1,\label{problemn2}\\ &&u^{(i)}(0)=0,\; 0\leq i\leq n-1,\; \quad u(1)=\lambda\int_0^1u(s)\id s,\label{uslovin2} \end{eqnarray*} where $0\leq\lambda \leq \alpha$ and $D^{\alpha}$ is the Riemann--Liouville fractional derivative of order $\alpha$. For these problems we give existence results, which improve recent results in the literature.WileyUniversidade de Santiago de Compostela. Departamento de Análise Matemática20172017-01-0120172017-01-01journal articlehttp://purl.org/coar/resource_type/c_6501AMhttp://purl.org/coar/version/c_ab4af688f83e57aainfo:eu-repo/semantics/articleapplication/pdfhttps://hdl.handle.net/10347/45460reponame:Minerva. Repositorio Institucional de la Universidad de Santiago de Compostelainstname:Universidad de Santiago de Compostela (USC)Inglésengopen accesshttp://purl.org/coar/access_right/c_abf2Attribution-NonCommercial-NoDerivatives 4.0 Internationalhttp://creativecommons.org/licenses/by-nc-nd/4.0/info:eu-repo/semantics/openAccessoai:minerva.usc.gal:10347/454602026-06-15T12:47:27Z
dc.title.none.fl_str_mv The existence of a positive solution for nonlinear fractional differential equations with integral boundary value conditions
title The existence of a positive solution for nonlinear fractional differential equations with integral boundary value conditions
spellingShingle The existence of a positive solution for nonlinear fractional differential equations with integral boundary value conditions
Cabada Fernández, Alberto
Fractional differential equation
Integral boundary value condition
Positive solution
Green’s function
Fixed-point theorem
1202 Análisis y análisis funcional
title_short The existence of a positive solution for nonlinear fractional differential equations with integral boundary value conditions
title_full The existence of a positive solution for nonlinear fractional differential equations with integral boundary value conditions
title_fullStr The existence of a positive solution for nonlinear fractional differential equations with integral boundary value conditions
title_full_unstemmed The existence of a positive solution for nonlinear fractional differential equations with integral boundary value conditions
title_sort The existence of a positive solution for nonlinear fractional differential equations with integral boundary value conditions
dc.creator.none.fl_str_mv Cabada Fernández, Alberto
Dimitrijevic, Sladjana
Tomovic, Tatjana
Aleksic, Suzana
author Cabada Fernández, Alberto
author_facet Cabada Fernández, Alberto
Dimitrijevic, Sladjana
Tomovic, Tatjana
Aleksic, Suzana
author_role author
author2 Dimitrijevic, Sladjana
Tomovic, Tatjana
Aleksic, Suzana
author2_role author
author
author
dc.contributor.none.fl_str_mv Universidade de Santiago de Compostela. Departamento de Análise Matemática
dc.subject.none.fl_str_mv Fractional differential equation
Integral boundary value condition
Positive solution
Green’s function
Fixed-point theorem
1202 Análisis y análisis funcional
topic Fractional differential equation
Integral boundary value condition
Positive solution
Green’s function
Fixed-point theorem
1202 Análisis y análisis funcional
description In this paper, first we consider the existence of a positive solution for the nonlinear fractional differential equation boundary--value problem \begin{eqnarray*} &&^C\D^{\alpha}u(t)+f(t,u(t))=0, \quad 0< t<1,\quad 2<\alpha\leq 3,\\ &&u'(0)=u''(0)=0, \quad u(1)=\lambda\int_0^1u(s)\id s, \end{eqnarray*} where $0\leq\lambda<1$, and $^C\D^\alpha$ is the Caputo's differential operator of order $\alpha$, and $f:[0,1]\times[0,\infty)\rightarrow[0,\infty)$ is a continuous function. Using some cone theoretic techniques, we deduce a general existence theorem for this problem. Then, we consider two following more general problems for arbitrary $\alpha$, $1\leq n<\alpha\leq n+1$: Problem 1: \begin{eqnarray*} &&^C\D^{\alpha}u(t)+f(t,u(t))=0, \quad 0< t<1,\\ &&u^{(i)}(0)=0,\; 0\leq i\leq n,\; i\neq k, \quad u(1)=\lambda\int_0^1u(s)\id s, \end{eqnarray*} where $k\in\{0, 1, \dots, n-1\}$, $0\leq\lambda<k+1$; Problem 2: \begin{eqnarray*} &&D^{\alpha}u(t)+f(t,u(t))=0, \quad 0< t<1,\label{problemn2}\\ &&u^{(i)}(0)=0,\; 0\leq i\leq n-1,\; \quad u(1)=\lambda\int_0^1u(s)\id s,\label{uslovin2} \end{eqnarray*} where $0\leq\lambda \leq \alpha$ and $D^{\alpha}$ is the Riemann--Liouville fractional derivative of order $\alpha$. For these problems we give existence results, which improve recent results in the literature.
publishDate 2017
dc.date.none.fl_str_mv 2017
2017-01-01
2017
2017-01-01
dc.type.none.fl_str_mv journal article
http://purl.org/coar/resource_type/c_6501
AM
http://purl.org/coar/version/c_ab4af688f83e57aa
dc.type.openaire.fl_str_mv info:eu-repo/semantics/article
format article
dc.identifier.none.fl_str_mv https://hdl.handle.net/10347/45460
url https://hdl.handle.net/10347/45460
dc.language.none.fl_str_mv Inglés
eng
language_invalid_str_mv Inglés
language eng
dc.rights.none.fl_str_mv open access
http://purl.org/coar/access_right/c_abf2
Attribution-NonCommercial-NoDerivatives 4.0 International
http://creativecommons.org/licenses/by-nc-nd/4.0/
dc.rights.openaire.fl_str_mv info:eu-repo/semantics/openAccess
rights_invalid_str_mv open access
http://purl.org/coar/access_right/c_abf2
Attribution-NonCommercial-NoDerivatives 4.0 International
http://creativecommons.org/licenses/by-nc-nd/4.0/
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
dc.publisher.none.fl_str_mv Wiley
publisher.none.fl_str_mv Wiley
dc.source.none.fl_str_mv reponame:Minerva. Repositorio Institucional de la Universidad de Santiago de Compostela
instname:Universidad de Santiago de Compostela (USC)
instname_str Universidad de Santiago de Compostela (USC)
reponame_str Minerva. Repositorio Institucional de la Universidad de Santiago de Compostela
collection Minerva. Repositorio Institucional de la Universidad de Santiago de Compostela
repository.name.fl_str_mv
repository.mail.fl_str_mv
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