Existence Of Solutions Of Boundary Value Problems For Nonlinear Differential Equations

In recent years,it has obtained new breakthrough of fixed point the-ories and applications for nonlinear operators.In this paper,using the cone theory,lattice theory and the fixed point theory,we discuss the integer and the fractional differential equation boundary value problem and some examples are given.This paper is divided into the following four chapters:Chapter 1 Preference,we introduce the main contents of this paper.Chapter 2 In this chapter,we use the lattice structure and its related fixed point theorems to investigate the following forms of nonlinear third-order two-point boundary value problem where f(f,x(t))∈C([0,1]× R,R),and we obtain at least three non-trivial solutions which is a positive solution of a sign-changing solution of a negative solution.We generalize th results in[8].Chapter 3 In this chapter,we investigate the following form of frac-tional boundary value problem where t∈[0,1],n>3,α∈(n-1,n],β∈[1,n-2]and D0+α is the Riemann-Liouville’s fractional derivative. By using upper and lower solutions combining the Schauder fixed point theorem,we solve that f(t,u(t))may have a singularity at t=1.Chapter 4 This chapter gives the Green’s function,by employing the fixed point theory,we study the existence of solutions for the integral boundary value problems where n-1<α≤n,n≥2,α-i>1,0≤i≤n-1.cD0+α is Caputo fractional derivative.