一类非线性分数阶微分方程边值问题正解的存在性
|
摘要
本文运用Schauder不动点定理和Krasnoselskii’s不动点定理获得了非线性分数阶微分方程边值问题~CD■u(t)=f(t,u(t),u′(t),u″(t)),t∈(0,1),u′(0)+u″(0)=0,u′(1)+u″(1)=0,u(0)=0正解的存在性,其中2<α≤3,~CD■是Caputo分数阶导数.
In this paper, by using the Schauder fixed point theorem and Krasnoselskii's fixed point theorem, the existence of positive solutions for the boundary value problem of the nonlinear fractional differential equation ~CD■u(t)=f(t,u(t),u′(t),u″(t)),t∈(0,1),u′(0)+u″(0)=0,u′(1)+u″(1)=0,u(0)=0 is obtained, where 2<α≤3,~CD■is the Caputo fractional derivative.
In this paper, by using the Schauder fixed point theorem and Krasnoselskii's fixed point theorem, the existence of positive solutions for the boundary value problem of the nonlinear fractional differential equation ~CD■u(t)=f(t,u(t),u′(t),u″(t)),t∈(0,1),u′(0)+u″(0)=0,u′(1)+u″(1)=0,u(0)=0 is obtained, where 2<α≤3,~CD■is the Caputo fractional derivative.
引文
[1]Yang X,Wei Z L.Existene of positive solutions for the boundary value problem of nonlinear fractional differential equations[J].Commun Nonlinear Sci,2012,17:85.
[2]岳俊瑞.非线性分数阶微分方程边值问题的正解[D].兰州:兰州理工大学,2016.
[3]Ahmad B,Nieto J J.Sequential fractional differential equations with three-point boundary value problem[J].Comp Math Appl,2012,64:3046.
[4]Ahmad B.Sharp estimates for the unique solution of two-point fractional-order boundary value problem[J].Appl Math Lett,2017,65:77.
[5]Ai-Refai M,Ali-Hajji M.Monotone iterative sequences for nonlinear boundary value problem of fractional order[J].Nonlinear Appl,2011,74:3531.
[6]Zhao X K,Ge W G.Positive solutions for fractional four-point boundary value problem[J].Commun Nonlinear Sci,2011,16:3665.
[7]Abdulla A B,Al-Refai M.On the existece and uniqueness of solutions for a class of nonlinear fractional boundary value problem[J].King Saud University-Sci,2016,28:103.
[8]Rehman M,Ali-Khan R,Asif N A.Three-point boundary value problem for nonlinear fractional differential equations[J].Acta Math Sci,2011,31:1337.
[9]Bai Z B.Eigenvalue intervals for a class of fractional boundary value problem[J].Comp Math Appl,2012,64:3253.
[10]Bai Z B,Sun W C.Existence and multiplicity of positive solutions for singular fractional boundary value problem[J].Comp Appl Math,2012,63:1369.
[11]Zhao Y G,Sun S R,Han Z L.The existence of multiple positive soiutions for boundary value problem of nonlinear fractional differential equations[J].Commun Nonlinear Sci,2011,16:2086.
[12]Liang S H,Zhang J H.Positive solutions for boundary value problem of nonlinear fractional differential equations[J].Nonl-Theor Analinear,2009,71:5545.
[13]Zhang S Q.Positive solutions for boundary value problems of nonlinear fractional differential equatins[J].Electron J Differ Eq,2006,36:1.
[14]Liang S H,Zhang J H.Existence and uniqueness of strictly nondecreasing and positive solutions for a fractional three-point boundary value problem[J].Elec J Diff Eqns,2011,62:1333.
[15]Kilbas A A,Sriuastave H M,Trujillo J J.Theory and applications of fractional differentrial equations[M].Amsterdarn:North-Holland,2006.
[2]岳俊瑞.非线性分数阶微分方程边值问题的正解[D].兰州:兰州理工大学,2016.
[3]Ahmad B,Nieto J J.Sequential fractional differential equations with three-point boundary value problem[J].Comp Math Appl,2012,64:3046.
[4]Ahmad B.Sharp estimates for the unique solution of two-point fractional-order boundary value problem[J].Appl Math Lett,2017,65:77.
[5]Ai-Refai M,Ali-Hajji M.Monotone iterative sequences for nonlinear boundary value problem of fractional order[J].Nonlinear Appl,2011,74:3531.
[6]Zhao X K,Ge W G.Positive solutions for fractional four-point boundary value problem[J].Commun Nonlinear Sci,2011,16:3665.
[7]Abdulla A B,Al-Refai M.On the existece and uniqueness of solutions for a class of nonlinear fractional boundary value problem[J].King Saud University-Sci,2016,28:103.
[8]Rehman M,Ali-Khan R,Asif N A.Three-point boundary value problem for nonlinear fractional differential equations[J].Acta Math Sci,2011,31:1337.
[9]Bai Z B.Eigenvalue intervals for a class of fractional boundary value problem[J].Comp Math Appl,2012,64:3253.
[10]Bai Z B,Sun W C.Existence and multiplicity of positive solutions for singular fractional boundary value problem[J].Comp Appl Math,2012,63:1369.
[11]Zhao Y G,Sun S R,Han Z L.The existence of multiple positive soiutions for boundary value problem of nonlinear fractional differential equations[J].Commun Nonlinear Sci,2011,16:2086.
[12]Liang S H,Zhang J H.Positive solutions for boundary value problem of nonlinear fractional differential equations[J].Nonl-Theor Analinear,2009,71:5545.
[13]Zhang S Q.Positive solutions for boundary value problems of nonlinear fractional differential equatins[J].Electron J Differ Eq,2006,36:1.
[14]Liang S H,Zhang J H.Existence and uniqueness of strictly nondecreasing and positive solutions for a fractional three-point boundary value problem[J].Elec J Diff Eqns,2011,62:1333.
[15]Kilbas A A,Sriuastave H M,Trujillo J J.Theory and applications of fractional differentrial equations[M].Amsterdarn:North-Holland,2006.