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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: \[ \begin{cases} \frac{d}{dt}\left(\frac 12{_0D_t^{-\beta}}(u'(t))+\frac 12{_tD_T^{-\beta}}(u'(t))\right)+\nabla F(t,u(t))=0\quad\text{a.e. }t\in[0,T],\\ u(0)=u(T)=0,\end{cases} \] 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 and fractional differential inclusions
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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