Multiple Solutions for a Class of Fractional Boundary Value Problems

Abstract

We study the multiplicity of solutions for the following fractional boundary value problem: $\mathrm{(d}/d\mathrm{t)}((1/\mathrm{2)}{\hspace{0.17em}}_0{D}_t^{-\beta }(u\text{'}(t))+\mathrm{(1}/\mathrm{2)}{\hspace{0.17em}}_0{D}_T^{-\beta }(u\text{'}(t)))+\lambda \nabla F(t,u(t))=0,\text{a.e.}t\in [0,T],u(0)=u(T)=0,$ where ${\hspace{0.17em}}_0{D}_t^{-\beta }$ and ${\hspace{0.17em}}_0{D}_T^{-\beta }$ are the left and right Riemann-Liouville fractional integrals of order , respectively, $\lambda >0$ is a real number, $F:[0,T]\times {ℝ}^N\to ℝ$ is a given function, and $\nabla F(t,x)$ is the gradient of $F$ at $x$. The approach used in this paper is the variational method. More precisely, the Weierstrass theorem and mountain pass theorem are used to prove the existence of at least two nontrivial solutions.