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

Kiyamehr Zohre, Baghani Hamid

Journal of Applied Analysis

2021

Vol. 27, nr 1

47--55

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.