Existence and Multiplicity of Solutions for Some Fractional Boundary Value Problem via Critical Point Theory

Abstract

We study the existence and multiplicity of solutions for the following fractional boundary value problem: $\mathrm{(d}/d\mathrm{t)}((1/\mathrm{2)}{}_0{D}_t^{-\beta }\left(u\text{'}\right(t\left)\right)+\mathrm{(1}/\mathrm{2)}{}_t{D}_T^{-\beta }\left(u\text{'}\right(t\left)\right))+\nabla F(t,u\left(t\right))=0,\text{a}.\text{e}.t\in [0,T],u(0)=u(T)=0$, where $F(t,\cdot )$ are superquadratic, asymptotically quadratic, and subquadratic, respectively. Several examples are presented to illustrate our results.