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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: $$\cases \frac{d}{dt}\left(\frac 12{_0D_t^{-\beta}}(u'(t))+\frac 12{_tD_T^{-\beta}}(u'(t))\right)+\nabla F(t,u(t))=0\quad\text{a.e. }t\in[0,T],\\ u(0)=u(T)=0,\endcases$$ where $_0D_t^{-\beta}$ and $_tD^{-\beta}_T$ are the left and right Riemann-Liouville fractional integrals of order $0\le\beta<1$ respectively, $F:[0,T]\times\bbfR^N\to\bbfR$ 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.

34A08Fractional differential equations
34B15Nonlinear boundary value problems for ODE
58E05Abstract critical point theory
Full Text: DOI
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