Existence of solutions of BVPs for fractional Langevin equations involving Caputo fractional derivatives (Q2043452)

This article investigates the existence and uniqueness of solutions for the following boundary value problem of the Langevin equation with two different fractional orders: \[ D^\beta(D^\alpha + \lambda)x(t) = f(t, x(t)),\,\, 0 < t < 1,\, 0 < a \leq 1, \,1 < \beta \leq 2, \] subject to the multi-point boundary conditions \[ x(0) = 0, \;D^{2\alpha}x(1) + \lambda D^\alpha x(1) = 0,\; x(1) =\int_0^\eta x(\tau)d\tau,\quad \text{for some } 0<\eta<1, \] where \(D^a\) is the Caputo fractional derivative of order \(a\), \(f : [0, 1] \times \mathbb{R} \rightarrow \mathbb{R}\) is a given continuous function, and \(\lambda\) is a real number. The corresponding nonlinear integral equation is considered and a fixed point theorem isn applied to show the existence and uniqueness results. At last two examples are considered to prove the applicability of the results.