MMN-5218
Existence of solutions for fractional boundary value problem via global minimization theorem
Souad Ayadi; Amina Boucenna; Meltem Erden Ege; Ozgur Ege;Abstract
The aim of this work is to use a global minimization theorem to prove the existence of at least a nontrivial solution for the problem:
\begin{equation*}\label{pb1}
\begin{cases}
D^{\alpha}_{T^{-}}\left( D^{\alpha}_{0^{+}} \left( D^{\alpha}_{T^{-}}\left( D^{\alpha}_{0^{+}}u(x)\right)\right)\right)= f\left( x,u(x)\right), \quad x \in \left[0, T \right],\\
u(0)= u(T) = 0,\\
D^{\alpha}_{T^{-}}\left( D^{\alpha}_{0^{+}}u(0)\right)= D^{\alpha}_{T^{-}}\left( D^{\alpha}_{0^{+}}u(T)\right) = 0,
\end{cases}
\end{equation*}
where $0 <\alpha\leq 1$ and $ f:\; \left[0, T \right]\times \mathbb R \rightarrow \mathbb R $ a Carath\'{e}odory function, $D^{\alpha}_{0^{+}},D^{\alpha}_{T^{-}}$ are the left and right fractional Riemann-Liouville derivatives of order $ \alpha$, respectively.
Vol. 26 (2025), No. 2, pp. 579-590