Existence Of Solutions For Anti-periodic Boundary Value Problems Of Fractional Langevin Equations With Quasilinear Operators

Fractional differential equations have a wide range of applications in many fields,such as physics,polymer rheology,blood flow phenomena,viscoelasticity,control fields,and so on,which have attracted scholars’ attention.Langevin equation describes the dynamic behavior of Brownian motion as a result of the collision of fluid molecules,particles moving irregularly in fluids,considered powerful work on the evolution of physical phenomena in fluctuating environments.Based on the characteristics of fractional order differential operators,that is,fractional order differential operators have long-range correlation and memory,some scholars introduce fractional order differential operators to describe the memory resistance in the process of Brownian particle motion,thus obtaining fractional order Langevin equation.Therefore,it is of great theoretical significance to study the boundary value problem of fractional Langevin equation.In this paper,we mainly discuss three kinds of anti-periodic boundary value problems of fractional Langevin equation with quasilinear operators based on the inspiration of existing literatures.By using Schaefer fixed point theorem,Leray-Schauder nonlinear selection theorem,Krasnoselskii fixed point theorem and Banach contraction mapping principle,we obtain some results on the existence and uniqueness of solutions.The obtained results generalize and enrich the existing literature on the research of boundary value problems for fractional Langevin equations.The full text is divided into five chapters.This first chapter introduces the background and application of fractional calculus,Langevin equation and quasilinear operator,as well as the current research status of boundary value problems of fractional Langevin equation at home and abroad and the main research content of this paper.The second chapter studies the problem of antiperiodic boundary values prbblem of the coupled Sturm-Liouville-Langevin equation with p-Laplacian operator in the framework of Caputo-Hadamard fractional calculus.The Sturm-Liouville-Langevin equation is a generalized form of the Sturm-Liouville and Langevin equations.This chapter uses Leray-Schauder nonlinear alternative and Krasnoselskii’s fixed-point theorem to obtain existence results for the solution,respectively.The third chapter discusses the problem of the dual antiperiodic boundary values problem of the fractional order Langevin equations with p-Laplacian operator.The dual antiperiodic boundary value condition is a nonlocal boundary value condition and is a generalization of the classical antiperiodic boundary value condition.Based on the Krasnoselskii’s fixed point theorem and the Banach contractive mapping principle,the existence and uniqueness results of the solutions are obtained,respectively.The forth chapter studies the antiperiodic boundary value problem of fractional La-ngevin equation with p(t)-Laplacian operator.The p(t)-Laplacian operator is a generali-zed form of the p-Laplacian operator,and when p(t)=p>1(p is constant),the p(t)-Laplacian operator is degenerate to the p-Laplacian operator.Using Schaefer’s fixed point theorem to obtain the solution existence results.The fifth chapter summarizes the work of this paper and prospects for future research work.