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A Variational approach for a problem involving a $\psi$-Hilfer fractional operator
José Vanterler da C. Sousa,Leandro S Tavares,Cesar Torres Ledesma
Keywords:$\psi$-fractional derivative space; variational structure; fractional differential equations; boundary value problem; mountain pass theorem.
      Boundary value problems driven by fractional operators has drawn the attention of several researchers in the last decades due to its applicability in several areas of Science and Technology. The suitable definition of the fractional derivative and its associated spaces is a natural problem that arise on the study of this kind of problem. A manner to avoid of such problem is to consider a general definition of fractional derivative. The purpose of this manuscript is to contribute, in the mentioned sense, by presenting the $\psi-$fractional spaces $\mathbb{H}_{p}^{\alpha,\beta;\psi}([0,T],\mathbb{R})$. As an application we study a problem, by using the Mountain Pass Theorem, which includes an wide class of equations.