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).
