Fractional Langevin equations with multi-point and non-local integral boundary conditions

In this paper, we investigate a class of nonlinear Langevin equations involving two fractional orders with nonlocal integral and three-point boundary conditions. Using the Banach contraction principle, Krasnoselskii’s and the nonlinear alternative Leray Schauder theorems, the existence and uniqueness results of solutions are proven. The paper was appended examples which illustrate the applicability of the results.

In this article, we study a neutral fractional integrodifferential equation supplemented with nonlocal flux type integral boundary conditions. The existence and uniqueness results are obtained by using Banach fixed point theorem and Leray-Schauder nonlinear alternative theorem. The obtained results are illustrated by examples at the end.

In this paper, we introduce and thoroughly examine new generalized ψ-conformable fractional integral and derivative operators associated with the auxiliary function ψ(t). We rigorously analyze and confirm the essential properties of these operators, including their semigroup behavior, linearity, boundedness, and specific symmetry characteristics, particularly their invariance under time reversal. These operators not only encompass the well-established Riemann–Liouville and Hadamard operators but also extend their applicability. Our primary focus is on addressing complex fractional boundary value problems, specifically second-order nonlinear implicit ψ-conformable fractional integro-differential equations with nonlocal fractional integral boundary conditions within Banach algebra. We assess the effectiveness of these operators in solving such problems and investigate the existence, uniqueness, and Ulam–Hyers stability of their solutions. A numerical example is presented to demonstrate the theoretical advancements and practical implications of our approach. Through this work, we aim to contribute to the development of fractional calculus methodologies and their applications.

We study the existence and uniqueness of solutions for a fractional boundary value problem involving Hadamard-type fractional differential equations and nonlocal fractional integral boundary conditions. Our results are based on some classical fixed point theorems. Some illustrative examples are also included.

This paper presents some existence and uniqueness results for a boundary value problem of fractional differential equations of order � 2 (1,2] with fourpoint nonlocal fractional integral boundary conditions. Our results are based on some standard tools of fixed point theory and nonlinear alternative of LeraySchauder type. Some illustrative examples are also discussed.

This paper investigates a boundary value problem of Caputo type sequential fractional differential equations supplemented with nonlocal Riemann-Liouville fractional integral boundary conditions. Some existence results for the given problem are obtained via standard tools of fixed point theory and are well illustrated with the aid of examples. Some special cases are also presented.

In this paper, we study a new class of one-dimensional semi-linear problems of fractional differential equations supplemented with nonlocal flux type integral boundary conditions. New existence and uniqueness results are obtained for the given problem by using some standard fixed point theorems. The obtained results are illustrated with the aid of examples.

This paper investigates the existence of solutions for a nonlinear boundary value problem of sequential fractional differential equations with four-point nonlocal Riemann-Liouville type fractional integral boundary conditions. We apply Banach’s contraction principle and Krasnoselskii’s fixed point theorem to establish the existence of results. Some illustrative examples are also presented.

This research paper is about the existence and uniqueness of the coupled system of nonlinear fractional Langevin equations with multipoint and nonlocal integral boundary conditions. The Caputo fractional derivative is used to formulate the fractional differential equations, and the fractional integrals mentioned in the boundary conditions are due to Atangana–Baleanu and Katugampola. The existence of solution has been proven by two main fixed-point theorems: O’Regan’s fixed-point theorem and Krasnoselskii’s fixed-point theorem. By applying Banach’s fixed-point theorem, we proved the uniqueness result for the concerned problem. This research paper highlights the examples related with theorems that have already been proven.

<abstract><p>Fractional Langevin equations play an important role in describing a wide range of physical processes. For instance, they have been used to describe single-file predominance and the behavior of unshackled particles propelled by internal sounds. This article investigates fractional Langevin equations incorporating recent extensive fractional operators of different orders. Nonperiodic and nonlocal integral boundary conditions are assumed for the model. The Hyres-Ulam stability, existence, and uniqueness of the solution are defined and analyzed for the suggested equations. Also, we utilize Banach contraction principle and Krasnoselskii fixed point theorem to accomplish our results. Moreover, it will be apparent that the findings of this study include various previously obtained results as exceptional cases.</p></abstract>

We study a boundary value problem of sequential fractional differential equations equipped with nonlocal integral boundary conditions (strip conditions of finite arbitrary size) involving the first-order derivative of the unknown function. As a variant problem, we discuss a case when nonlocal integral boundary conditions are governed by the unknown function. The existence of solutions for the given problems is investigated by means of some standard tools of fixed point theory. We emphasize that existence results obtained in this paper are new and enrich the existence theory for sequential fractional differential equations. The obtained results are well illustrated with the aid of examples.

In this work, we present the existence, uniqueness, and stability result of solution to the nonlinear fractionaldifferential equations involving Hilfer-Katugampola derivative subject to nonlocal fractional integral bound-ary conditions. The reasoning is mainly based upon properties of Mittag-Leffler functions, and fixed-pointmethods such as Banach contraction principle and Krasnoselskii's fixed point theorem. Moreover, the gener-alized Gornwall inequality lemma is used to analyze different types of stability. Finally, one example is givento illustrate our theoretical results.

In this article, we investigate the existence and uniqueness of solutions for a nonlinear coupled system of Liouville–Caputo type fractional integro-differential equations supplemented with non-local discrete and integral boundary conditions. The nonlinearity relies both on the unknown functions and their fractional derivatives and integrals in the lower order. The consequence of existence is obtained utilizing the alternative of Leray–Schauder, while the result of uniqueness is based on the concept of Banach contraction mapping. We introduced the concept of unification in the present work with varying parameters of the multi-point and classical integral boundary conditions. With the help of examples, the main results are well demonstrated.

PurposeThis paper aims to investigate the existence and uniqueness of solution for generalized Sturm–Liouville and Langevin equations formulated using Caputo–Hadamard fractional derivative operator in accordance with three nonlocal Hadamard fractional integral boundary conditions. With regard to this nonlinear boundary value problem, three popular fixed point theorems, namely, Krasnoselskii’s theorem, Leray–Schauder’s theorem and Banach contraction principle, are employed to theoretically prove and guarantee three novel theorems. The main outcomes of this work are verified and confirmed via several numerical examples.Design/methodology/approachIn order to accomplish our purpose, three fixed point theorems are applied to the problem under consideration according to some conditions that have been established to this end. These theorems are Krasnoselskii's theorem, Leray Schauder's theorem and Banach contraction principle.FindingsIn accordance to the applied fixed point theorems on our main problem, three corresponding theoretical results are stated, proved, and then verified via several numerical examples.Originality/valueThe existence and uniqueness of solution for generalized Sturm–Liouville and Langevin equations formulated using Caputo–Hadamard fractional derivative operator in accordance with three nonlocal Hadamard fractional integral boundary conditions are studied. To the best of the authors’ knowledge, this work is original and has not been published elsewhere.

The primary objective of this manuscript is to investigate the existence and uniqueness of solutions for the Langevin (k,φ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$(\\mathtt{k},\\varphi )$\\end{document}-Hilfer fractional differential equation of different orders with multipoint nonlocal fractional integral boundary conditions. We consider the generalized version of the Hilfer fractional diferential equation called as (k,φ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$(\\mathtt{k},\\varphi )$\\end{document}-Hilfer fractional differential equation. We provide some significant outcomes about (k,φ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$(\\mathtt{k},\\varphi )$\\end{document}-Hilfer fractional Langevin differential equation that requires deriving equivalent fractional integral equation to (k,φ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$(\\mathtt{k},\\varphi )$\\end{document}-Hilfer Langevin fractional differential equation. The existence result is established using the Krasnoselskii’s fixed-point theorem, while the uniqueness is addressed with the help of Banach contraction principle. Additionally, we investigate the different forms of Ulam stability for the solution of the mentioned problem, under specific conditions. To validate our main outcomes, we present a detailed example at the end of the manuscript.