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.


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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