Multiple positive solutions for nonlocal boundary value problems of singular fractional differential equations

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• 作者：Xingqiu Zhang ; Qiuyan Zhong
• 关键词：fractional differential equations ; nonlocal boundary value problem ; singularity ; multiple positive solutions
• 刊名：Boundary Value Problems
• 出版年：2016
• 出版时间：December 2016
• 年：2016
• 卷：2016
• 期：1
• 全文大小：1,417 KB
• 刊物主题：Difference and Functional Equations; Ordinary Differential Equations; Partial Differential Equations; Analysis; Approximations and Expansions; Mathematics, general;
• 出版者：Springer International Publishing
• ISSN：1687-2770

文摘

We consider the existence of multiple positive solutions for the following nonlinear fractional differential equations of nonlocal boundary value problems: $$ \left \{ \textstyle\begin{array}{l} D_{0{+}}^{\alpha}u(t)+f(t,u(t))=0, \quad 0< t< 1, \\ u(0)=0,\qquad D_{0{+}}^{\beta}u(0)=0,\qquad D_{0{+}}^{\beta}u(1)=\sum_{i=1}^{\infty} \xi_{i} D_{0{+}}^{\beta}u(\eta_{i}), \end{array}\displaystyle \right . $$ where \(2<\alpha\leq3\), \(1\leq\beta\leq2\), \(\alpha-\beta\geq1\), \(0<\xi_{i}, \eta_{i}<1\) with \(\sum_{i=1}^{\infty} \xi_{i}\eta_{i}^{\alpha -\beta-1}<1\). Existence result of at least two positive solutions is given via fixed point theorem on cones. The nonlinearity f may be singular both on the time and the space variables. Keywords fractional differential equations nonlocal boundary value problem singularity multiple positive solutions MSC 26A33 34B15 34B18 1 IntroductionThe purpose of this paper is to investigate the multiplicity of positive solutions for the following nonlocal boundary value problems of singular fractional differential equations: $$ \left \{ \textstyle\begin{array}{l} D_{0{+}}^{\alpha}u(t)+f(t,u(t))=0, \quad 0< t< 1, \\ u(0)=0, \qquad D_{0{+}}^{\beta}u(0)=0,\qquad D_{0{+}}^{\beta}u(1)=\sum_{i=1}^{\infty} \xi_{i} D_{0{+}}^{\beta}u(\eta_{i}), \end{array}\displaystyle \right . $$ (1) where \(2<\alpha\leq3\), \(1\leq\beta\leq2\), \(\alpha-\beta\geq1\), \(0<\xi_{i}, \eta_{i}<1\) with \(\sum_{i=1}^{\infty} \xi_{i}\eta_{i}^{\alpha -\beta-1}<1\), \(f\in C(J\times\mathbb{R}_{++}, \mathbb{R}_{+})\), \(J=(0,1)\), \(\mathbb{R}_{+}=[0,+\infty)\), \(\mathbb{R}_{++}=(0,+\infty)\), \(D_{0{+}}^{\alpha}\) is the standard Riemann-Liouville’s fractional derivative of order α. The nonlinearity f permits singularities at \(t=0,1\) and \(u=0\). A function \(u\in C[0,1]\) is said to be a positive solution of BVP (1) if \(u(t)>0\) on \((0,1)\) and u satisfies (1) on \([0,1]\).Recently, much attention has been paid on the study of nonlocal boundary value problems of fractional differential equations; see [1–16] and [17–25]. By virtue of the contraction map principle and the fixed point index theory, Bai [1] investigated the existence and uniqueness of positive solutions for the following fractional differential equation: $$ D_{0{+}}^{\alpha}u(t)+f\bigl(t,u(t)\bigr)=0, \quad 0< t< 1, 1< \alpha \leq2, $$ (A) subject to three point boundary value conditions $$ u(0)=0, u(1)=\mu u(\xi), $$ where \(0<\mu\xi^{\alpha-1}<1\), \(0\leq\mu\leq1\), \(0<\xi<1\), f is continuous on \([0, 1]\times[0,+\infty)\). When \(f : [0, 1] \times[0,+\infty)\to[0,+\infty)\) satisfies Carathéodory type conditions, by using some fixed point theorems, like Leray-Schauder nonlinear alternative and a mixed monotone method, Li et al. [2], Xu et al. [3, 4] obtained the existence and multiplicity results of positive solutions for the fractional differential equation (A) with fractional derivative in boundary conditions $$ u(0)=0,\qquad D_{0{+}}^{\beta}u(1)=\mu D_{0{+}}^{\beta}u( \xi),\quad 0\leq\beta\leq 1. $$ In 2011, Lv [5] and Yang et al. [6] discussed the existence of minimal and maximal and the uniqueness of positive solutions for fractional differential equation (A) under multi-point boundary value conditions, $$ u(0)=0, \qquad D_{0{+}}^{\beta}u(1)=\sum _{i=1}^{m-2} \xi_{i} D_{0{+}}^{\beta}u( \eta_{i}),\quad 0\leq\beta\leq1. $$ Motivated by the above papers, when \(2<\alpha\leq3\) and f is continuous, Li et al. [7] obtained the existence results of at least one and unique solutions for fractional differential equation (A) subject to more general multi-point boundary value conditions $$ u(0)=0,\qquad D_{0{+}}^{\beta}u(0)=0, \qquad D_{0{+}}^{\beta }u(1)= \sum_{i=1}^{m-2} \xi_{i} D_{0{+}}^{\beta}u(\eta_{i}). $$ The tools to obtain the main results are the nonlinear alternative of the Leray-Schauder and the Banach contraction mapping principle.Compared with the existing literature, this paper has the following two new features. First, different from [7], infinite-point boundary value conditions are considered in this paper. At the same time, the nonlinearity f in this paper permits singularities with respect to both the time and the space variables which is seldom considered at present. Second, the purpose of this paper is to investigate the existence of multiple positive solutions for BVP (1). As to multiple positive solutions, it is worth pointing out that conditions imposed on f are different from that in [4]. To achieve this goal, first we convert the expression of the unique solution into an integral form and then get the Green function BVP (1). After further discussion of the properties of the Green function, a suitable cone is constructed to obtain the main result in this paper by means of the Guo-Krasnoselskii fixed point theorem.