Nonlinear fractional Caputo-Langevin equation with nonlocal Riemann-Liouville fractional integral conditions

Abstract

In this paper, we study the existence and uniqueness of solution for a problem consisting of a sequential nonlinear fractional Caputo-Langevin equation with nonlocal Riemann-Liouville fractional integral conditions. A variety of fixed point theorems, such as Banach's fixed point theorem, Krasnoselskii's fixed point theorem, Leray-Schauder's nonlinear alternative and Leray-Schauder degree theory, are used. Examples illustrating the obtained results are also presented.

Existence of Solutions for Fractional Differential Equations Involving Two Riemann-Liouville Fractional Orders

“…Now, we prove the existence of solutions of multi-point boundary value problem (1.1) by applying Leray-Schauder nonlinear alternative [25].…”

“…Fractional derivative arises from many physical processes, such as a charge transport in amorphous semiconductors [22], electrochemistry and material science, they are in fact described by differential equations of fractional order [9,10,17,18]. Recently, many studies on fractional differential equations, involving different operators such as Riemann-Liouville operators [19,24], Caputo operators [1,3,13,25], Hadamard operators [23] and q−fractional operators [2], have appeared during the past several years. Moreover, by applying different techniques of nonlinear analysis, many authors have obtained results of the existence and uniqueness of solutions for various classes of fractional differential equations, for example, we refer the reader to [3-8, 11, 12, 14, 15, 19] and the references cited therein.…”

Langevin equation involving one fractional order with three-point boundary conditions

“…Fractional calculus has been extensively studied and developed during the last few decades due to its important application in many areas. It has become a new research field in differential equations [9,16,23,26]. The Langevin equation (first formulated by Langevin in 1908) is found to be an effective tool to describe the evolution of physical phenomena in fluctuating environments [6].…”

Anti-Periodic Boundary Value Problems for Nonlinear Langevin Fractional Differential Equations

“…In recent years, the subject of fractional differential equations is gaining much importance and attention. For details, see [1][2][3][4][6][7][8][9][10][11] and the references therein.…”

“…Recently, there has been a significant development in solving fractional Langevin equation (see [7][8][9][10] and the references therein). To the best of our knowledge, there are few papers dealing with anti-periodic BVP involving fractional Langevin equation with two fractional orders c D α [10].…”