Research On The Existence Of Solutions Of Initial Value Problems For Langevin Equations Involving Two Fractional Orders

Fractional calculus is a theory which has been widely used in modern mathematics,the development history has more than 300 years,it can describe arbitrary differential and integral operator characteristics,and overcome the locality of integer order calculus,thus,the historical dependence process of system function development is well demonstrated.Recently,fractional Langevin equations have wide applications such as fractional Langevin equations for modeling of single-file diffusion and for a free particle driven by power law type of noise.So fractional Langevin equations have been paid more and more attention and the existence results of solutions have been widely studied by a great number of scholars.This thesis is mainly studies the existence and uniqueness of solutions and nonnegative solutions of initial value problems for Langevin equations involving two fractional orders.The thesis is divided into four chapters.The main contents are as follows:Chapter 1,introduction.We summarize the studying background of the subject.In chapter 2,we consider the following initial value problem for Langevin equations involving two fractional orders:where(?)and(?)are the Caputo fractional derivatives,f:[0,1]× R?R is a continuous function and 0<?<?(?+1),m-1<??m,n-1<??n,l=max{m,n},n,m ? N+,k?N,?j?0,j?[0,m-1],vi-??i?0,i?[0,n-1].We get the existence and uniqueness of nonnegative solutions by fixed point theorems in partially ordered metric spaces.In chapter 3,we investigate the following initial value problem:where(?)and(?)denote the Caputo fractional derivatives,f:[0,1]×R?R is continuous function and m-1<??m,n-1<??n,l=max{m,n},k?N,n,m? N+,?>0,?j?0,j?{0,...,m-1},vi-??i?0,i ? {0,...,n-1}.By using single upper-solution or lower-solution method and monotone iterative approach,several existence and uniqueness results of nonnegative solutions are obtained.In chapter 4,we investigate the existence and uniqueness of solutions for Langevin equations with two fractional orders:where(?)and(?)denote the Caputo fractional derivatives,f:[0,1]×R?R is a continuous function and m-1<??m,n-1<??n,?>0,l=max{m,n},n,m?N+,k?N,?k,vk?R.By using e-positive operators and Altman fixed point theory,several existence and uniqueness results of solutions are obtained.