Existence Of Solutions For Several Classes Of Nonlinear Differential Equations

In recent decades, nonlinear analysis has gained much attention by experts and scholars in the mathematics, physics and engineering fields, it is caused by the applications in this fields. But the differential equation theory as the important branch of the nonlinear analysis, has also gained a lot of attention, and there are many papers in this respect, see [1-8]. But many aspects of the theory still need to be explored.Instead of using the fixed point theorems, in this paper we use the first eigenvalue, u0-positive operator and monotone iterative method to investigate the existence of solutions for several classes of nonlinear dif-ferential equations.This paper is divided into four chapters:In Chapter 1, We introduce the latest researches and some definitions and lemmas will be used in the following chapters.In Chapter 2, by using the first eigenvalue, uo-positive operator, we consider the existence and uniqueness of solutions for a class of nonlinear differential equations with integral boundary conditions: where n<?? n+1,n is a real number,D? is the Riemann-Liouville derivative.In Chapter 3, by using the first eigenvalue, u0-positive operator, we consider the existence and uniqueness of solutions for a class of nonlinear differential equations with boundary conditions: where n<??n+1.n?3,n is a real number, D? is the Riemann-Liouville derivative.In Chapter 4, by using the monotone iterative method, we consider the nonlinear differential equations at resonance: where a denotes a linear functional on C(J) given by involving a Stieltjes integral with a suitable function A of bounded vari-ation on J, we discuss such problems at resonance when A(t) is an in-creasing function on J, and also when ?(t) is a decreasing function on [0, h], is a increasing function on [h, T].