Existence Of Solutions For Initial Value And Boundary Value Problem Of Fractional Langevin Equations

In recent years,the initial value and boundary value problem of nonlinear fractional Langevin differential equation has aroused extensive research worldwide.Compared with ordinary integral differential equation,Fractional Langevin differential equation can better explain some complex phenomena and has important applications in solving practical problems.In this paper,we mainly discuss the existence and uniqueness of solutions to initial and boundary value problems for two types of fractional order nonlinear Langevin differential equations,and then obtain some meaningful new results.This paper consists of the following two chapters:In the first chapter,we mainly study the initial value problem of a class of fractional integro-differential Langevin equations（?）where ^cD^α,^cD^β,^cD^θ are Caputo fractional derivatives,α∈(m-1,m],β∈（n-1,n）,n,m∈N+,m≥2,l=max{m,n},0<θ<1,γ,μκ,υκ∈R,f∈C（[0,1]× R × R × R ×R,R）,x（α+κ）（0）=DκcDαx（0）.T and S are linear operators defined as follows:（?）where k₁∈C（D,R）,k₂∈C（[0,1]×[0,1],R）,D={（t,s）∈[0,1]×[0,1]:t≥s}.By using Schauder fixed point theorem and Banach contraction mapping theory,we obtain the existence and uniqueness of the solution of the initial value problem.The second chapter,we mainly study a class of nonlocal boundary value problems for fractional Langevin equation（?） where λ∈R,0<θ≤1,1<α,β≤2,A,B are increase function,^cD^β,^cD^α,^cD^θ are Caputo fractional derivatives,∫₀¹u（s）dA（s）,∫₀¹cDαu（s）dB（s）are Riemann-Stieltjes integrals.g:C（[0,1],R）→R is a given continuous functional,f:[0,1]× R×R→R is a given continuous function.The uniqueness and existence of the solution are obtained by applying Banach contraction mapping principle、Krasnoselskii fixed point theorem、Leray-Schauder alternative theorem and Schauder fixed point theorem.