The Langevin Equation in Terms of Generalized Liouville–Caputo Derivatives with Nonlocal Boundary Conditions Involving a Generalized Fractional Integral

Abstract

In this paper, we establish sufficient conditions for the existence of solutions for a nonlinear Langevin equation based on Liouville-Caputo-type generalized fractional differential operators of different orders, supplemented with nonlocal boundary conditions involving a generalized integral operator. The modern techniques of functional analysis are employed to obtain the desired results. The paper concludes with illustrative examples.

On a Generalized Langevin Type Nonlocal Fractional Integral Multivalued Problem

“…Here ρ = 1/3, α = 5/4, β = 1/4, λ = 1/5, a = 1, T = 2 η = 3/2, µ = 2/7, γ = 3/4, ξ = 7/4. Using the given data, we find that ζ 1 ≈ 0.082260, ζ 2 ≈ 0.232036, |Ω| ≈ 0.293634, Λ 1 ≈ 1.336009 and Λ 2 ≈ 0.673563, where ζ 1 , ζ 2 , Λ 1 and Λ 2 are given by (16), (17), (14) and (15) respectively.…”

“…x(a) = 0, x(η) = 0, x(T) = µ ρ I γ a+ x(ξ), a < η < ξ < T, µ ∈ R, (1) where ρ c D α a+ , ρ c D β a+ denote the Caputo-type generalized fractional differential operators of order 1 < α ≤ 2, 0 < β < 1, ρ > 0, respectively, F : [a, T] × R → P (R) is a multi-valued map (P (R) is the family of all nonempty subsets of R), ρ I γ a+ is the generalized fractional integral operator of order γ > 0 and ρ > 0. Here we emphasize that the single-valued analogue of the problem (1) was discussed in [14].…”

Measure of noncompactness for an infinite system of fractional Langevin equation in a sequence space

“…In our paper, we deal with the generalized Liouville-Caputo derivative which is considered a generalization for many known fractional derivatives [15]. Historically, in 2011, Katugampola [30] introduced a new version of fractional integral given by…”

“…Lemma 2.1 ( [15,32]) Suppose that n ∈ N, n -1 < ν ≤ n, 0 < ρ ≤ 1, and f ∈ X p c (a, b). Then we have…”

“…It worth pointing out that the Langevin equations (formulated by Langevin in 1908) present an accurate way to describe the evolution of physical phenomena in a fluctuating environment [14]. There is a clear progress on fractional Langevin equations in physics [15][16][17][18][19]. New results on the existence of solutions for fractional Langevin equations under variety of boundary value conditions have been published; see [20][21][22][23][24][25][26][27][28][29] and the references mentioned therein.…”