MULTIPLE SOLUTIONS FOR FRACTIONAL THREE-POINT BOUNDARY VALUE PROBLEMS: A VARIATIONAL APPROACH

We are concerned with the following fractional boundary value problem: $\begin{cases} {}_tD_T^\alpha ({}_0^CD_t^\alpha u(t))=\lambda f(u(t)), \quad t \in (0, T), \hfill \\ u(0) = 0, \quad u(T) = \beta u(\eta), \end{cases}$ where $ \lambda{>}0$ is a parameter, $ \beta\in \mathbb{R}$, $ \eta \in (0, T)$. By proving the appropriate fractional derivative space, we establish the variational structure of the proposed problem and derive the existence result for multiple weak solutions using a variant of the mountain pass theorem. To demonstrate the applicability of the main results, we present an example. This paper explores the application of the variational approach in solving fractional three-point BVPs. Our findings are novel and distinct from the conclusions of existing literature, holding significant theoretical value.