Existence of Positive Solutions for a Class of Higher-Order Caputo Fractional Differential Equation
Existence of Positive Solutions for a Class of Higher-Order Caputo Fractional Differential Equation
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DOI：
10.1007/s12346-014-0121-0
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发表时间：
2014-09
影响因子：
1.4
通讯作者：
Kaihong Zhao;Ping Gong
中科院分区：
文献类型：
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作者：
Kaihong Zhao;Ping Gong
本文研究了非线性分数阶常微分边值问题{D_0+^αu(T)+λf(t，u(T))+q(T)=0，&0<；t<；1，u(0)=u‘(1)=u’‘(0)=\dots=u^(n-1)(0)=0，&.D0+αu(T)+λf(t，u(T))+q(T)=0，0<；t<；1，u(0)=u‘(1)=u’‘(0)=⋯=u(n-1)(0)=0，其中α(n-1<；α≤n)α(n-1<；α≤n)为实数。λ>；0λ>；0是一个参数。D_0+^αD_0+α是标准的卡普托导数。利用锥上的Leray-Schauder型非线性选择和Guo-Krasnoselskii不动点定理，建立了这类非线性分数阶常微分方程边值问题正解存在的充分条件。作为应用，给出了一些例子来说明我们的主要结果。
In this paper, the following nonlinear fractional ordinary differential boundary value problem {D_ 0+^ α u (t)+ λ f (t, u (t))+ q (t)= 0, & 0< t< 1,\u (0)= u'(1)= u''(0)=\dots= u^(n-1)(0)= 0, &\. D 0+ α u (t)+ λ f (t, u (t))+ q (t)= 0, 0< t< 1, u (0)= u′(1)= u′′(0)=⋯= u (n-1)(0)= 0, is considered, where α (n-1< α ≤ n) α (n-1< α≤ n) is a real number. λ> 0 λ> 0 is a parameter. D_ 0+^ α D 0+ α is the standard Caputo differentiation. Some sufficient conditions for the existence of positive solutions to this boundary value problem of nonlinear fractional differential equation are established by nonlinear alternative of Leray–Schauder type and Guo–Krasnoselskii fixed point theorem on cones. As applications, some examples are provided to illustrate our main results.