Existence and multiplicity of solutions for fractional differential equations with parameters

Abstract

In this paper, we study boundary value problems for the following nonlinear fractional differential equations involving the Caputo fractional derivative

$$\begin{aligned}&D^{\alpha }_{0+}u(t)-Mu(t)-\lambda g(t)f(u)=0,\quad t\in [0,1],\nonumber \\&u(0)=\eta u(1),\;\;u'(0)=\gamma u'(1), \end{aligned}$$

where $1<\alpha <2$ is a real number, $D^{\alpha }_{0+}$ is the Caputo fractional derivative, $M,\eta$ and $\gamma$ are all positive constants, $\lambda$ is a positive parameter, $f$: $[0,\infty )\rightarrow [0,\infty )$ and $g$: $[0,1]\rightarrow [0,\infty )$ are continuous. By means of Krasnosel’skii’s and Schaefer’s fixed point theorems, the existence of solutions is obtained, respectively. Finally, we present some examples to illustrate our main results.