## Existence of solutions for a class of fractional boundary value problems via critical point theory.(English)Zbl 1235.34017

Summary: By the critical point theory, a new approach is provided to study the existence of solutions to the following fractional boundary value problem: $\begin{cases} \frac{d}{dt}\left(\frac 12{_0D_t^{-\beta}}(u'(t))+\frac 12{_tD_T^{-\beta}}(u'(t))\right)+\nabla F(t,u(t))=0\quad\text{a.e. }t\in[0,T],\\ u(0)=u(T)=0,\end{cases}$ where $$_0D_t^{-\beta}$$ and $$_tD^{-\beta}_T$$ are the left and right Riemann-Liouville fractional integrals of order $$0\leq\beta<1$$ respectively, $$F:[0,T]\times\mathbb{R}^N\to\mathbb{R}$$ is a given function and $$\nabla F(t,x)$$ is the gradient of $$F$$ at $$x$$. The variational structure is established and various criteria on the existence of solutions are obtained.

### MSC:

 34A08 Fractional ordinary differential equations 34B15 Nonlinear boundary value problems for ordinary differential equations 58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
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