Some Existence Results on Implicit Fractional Differential Equations

Abstract

In this paper, we study the existence and uniqueness of a solution for the nonlinear implicit fractional differential equation boundary value problem
$$D^{\alpha} u(t) = f\left(t,u(t),D^{\alpha}u(t)\right),$$
with Riemann-Liouville fractional derivative via the different boundary conditions \(u(0)=u(T)\), and the three point boundary conditions \(u(0)=\beta_1u(\eta)\) and \(u(T)=\beta_2u(\eta)\), where \(T > 0, t \in I=[0, T], 0< \alpha< 1, 0 < \eta < T,\) \(0 < \beta_1 < \beta_2 < 1.\)