Positive solutions for a class of fractional 3-point boundary value problems at resonance
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- 作者：Yongqing Wang ; Lishan Liu
- 关键词：fractional differential equation ; positive solution ; resonance ; fixed point index
- 刊名：Advances in Difference Equations
- 出版年：2017
- 出版时间：December 2017
- 年：2017
- 卷：2017
- 期：1
- 全文大小：1534KB
- 刊物主题：Difference and Functional Equations; Mathematics, general; Analysis; Functional Analysis; Ordinary Differential Equations; Partial Differential Equations;
- 出版者：Springer International Publishing
- ISSN：1687-1847
- 卷排序：2017
文摘
In this paper, we study the nonlocal fractional differential equation: $$\left \{ \textstyle\begin{array}{@{}l} D^{\alpha}_{0+}u(t)+f(t,u(t))=0 ,\quad 0< t< 1,\\ u(0)=0,\qquad u(1)=\eta u(\xi), \end{array}\displaystyle \right . $$ where \(1 < \alpha< 2\), \(0 < \xi< 1\), \(\eta\xi^{\alpha-1}= 1\), \(D^{\alpha}_{0+}\) is the standard Riemann-Liouville derivative, \(f:[0,1]\times[0,+\infty)\rightarrow\mathbb{R}\) is continuous. The existence and uniqueness of positive solutions are obtained by means of the fixed point index theory and iterative technique.Keywordsfractional differential equationpositive solutionresonancefixed point index1 IntroductionIn this paper, we consider the following fractional differential equation: $$ \left \{ \textstyle\begin{array}{@{}l} D^{\alpha}_{0+}u(t)+f(t,u(t))=0 ,\quad 0< t< 1,\\ u(0)=0,\qquad u(1)=\eta u(\xi), \end{array}\displaystyle \right . $$ (1.1) where \(1 < \alpha< 2\), \(0 < \xi< 1\), \(\eta\xi^{\alpha-1}= 1\), \(D^{\alpha}_{0+}\) is the standard Riemann-Liouville derivative, \(f:[0,1]\times[0,+\infty)\rightarrow\mathbb{R}\) is continuous. Problem (1.1) happens to be at resonance, since \(\lambda=0\) is an eigenvalue of the linear problem $$ \left \{ \textstyle\begin{array}{@{}l} -D^{\alpha}_{0+}u=\lambda u , \quad 0< t< 1,\\ u(0)=0,\qquad u(1)=\eta u(\xi), \end{array}\displaystyle \right . $$ (1.2) and \(ct^{\alpha-1},c\in\mathbb{R,}\) is the corresponding eigenfunction.Fractional differential equations occur frequently in various fields such as physics, chemistry, engineering and control of dynamical systems, etc. During the last few decades, many papers and books on fractional calculus and fractional differential equations have appeared (see [1–22] and the references therein).