Existence and uniqueness results for nonlinear fractional Langevin integro-differential equations with boundary conditions

Abstract

This paper is devoted to the study of nonlinear fractional Langevin integro differential equations with boundary conditions. Some effective results concerning the existence and uniqueness are obtained by applying the Banach contraction mapping principle and the Schauder fixed point theorem. An example is presented illustrating the effectiveness of the theoretical results.

A Study of an IBVP of Fractional Differential Equations in Banach Space via the Measure of Noncompactness

“…Then, our existence results extend the main results in [19]. In addition, if τ = ς − 1 our problem and study reduces to the problem studied in [10], which is a generalization of [20].…”

“…In this part, based on the examples given into the papers [10,19], we give an example to make Theorem 3 more clear to the reader. Therefore, we propose the following problem:…”

“…Darbo [4] was the first researcher that employed the measure of noncompactness to study the relationship between that of compact and contraction mappings. On the other hand, Darbo's, Sadovski's [5], and Mönch's [6,7] fixed-point theorems were considered effective tools for studying the existence of solutions of several classes and types of differential equations, especially for fractional differential equations (see [8][9][10][11][12][13][14][15][16][17][18] and references therein).…”

“…In [20], Xu and Sun studied problems (1) and (2) when Ψ(1, r) ≡ 0, which means τ = ς − 1, and they gave the existence and uniqueness of a positive solution using the same method in [19]. On the other hand, Lachouri, Ardjouni, and Djoudi [10] generalized problems (1) and (2) in the case of τ = ς − 1 (see [20]) and gave interesting results for the existence of a solution in Banach space by using the Mönch fixed-point theorem and the measure of noncompactness. In view of the above works, we take ς ∈ (1, 2], τ ∈ (0, ς] and study problem roblems (1) and ( 2) in Banach space.…”

A generalized neutral-type inclusion problem in the frame of the generalized Caputo fractional derivatives

“…where with these data, we find = -0.58012 = 0. Define H : [0, 1] × R → P(R) by ϕ → H(t, ϕ) = 0, 2 sin(ϕ) (t 2 + 30) + 1 20 , (18) and the function K similar to above in (14). From (16)…”