Existence of solutions of BVPs for fractional Langevin equations involving Caputo fractional derivatives

This article investigates a nonlinear fractional Caputo-Langevin equation Dβ(Dα + λ)x(t) = f(t, x(t)), 0 < t < 1, 0 < α ≤ 1, 1 < β ≤ 2, subject to the multi-point boundary conditions x(0) = 0, D2αx(1) + λDαx(1) = 0, x(1) =η∫0 x(τ) dτ for some 0 < η < 1, where Dα is the Caputo fractional derivative of order α, f : [0, 1] × ℝ → ℝ is a given continuous function, and λ is a real number. Some new existence and uniqueness results are obtained by applying an interesting fixed point theorem.

---

Opracowanie rekordu ze środków MNiSW, umowa Nr 461252 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2021)