Existence Results for a Fractional Boundary Value Problem via Critical Point Theory

Boucenna, A. ; Moussaoui, T.

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica, Tome 54 (2015) no. 1, pp. 47-64

Résumé

In this paper, we consider the following boundary value problem \[ \left\lbrace \begin{array}{lll} D_{T^{-}}^{\alpha } (D_{0^{+}}^{\alpha } (D_{T^{-}}^{\alpha }(D_{0^{+}}^{\alpha } u(t))) ) = f(t, u(t)), \quad t \in [0, T], \\ u(0)= u(T)= 0\\ D_{T^{-}}^{\alpha }(D_{0^{+}}^{\alpha }u(0))= D_{T^{-}}^{\alpha }(D_{0^{+}}^{\alpha }u(T))= 0, \end{array} \right. \] where $0 \alpha \le 1$ and $f\colon [0, T]\times \mathbb {R} \rightarrow \mathbb {R} $ is a continuous function, $D_{0^{+}}^{\alpha }$, $D_{T^{-}}^{\alpha }$ are respectively the left and right fractional Riemann–Liouville derivatives and we prove the existence of at least one solution for this problem.

Détail

Classification : 26A33, 34B15, 58E05
Keywords: Existence results; fractional differential equation; boundary value problem; critical point theory; minimization principle; Mountain pass theorem; Third order; nonlinear differential equation; uniform stability; uniform ultimate boundedness; periodic solutions