Order Superlinear Multiple Positive Periodic Solutions To Exclude The Singular Equation

Second-order singular nonlinear ordinary differential equations with superlin-ear repulsive forces arise from celestial mechanics. Such type of equations has high values not only in the research fields but also in practice, and therefore have become a very important part in the theory of ordinary differential equations. Many mathematicians have paid much attention to them. In this paper, we are devoted to study the existence of single and multiple positive periodic solutions of the pertubation of the Hill equationx" + a(t)x = f(t,x).The type of perturbations f(t,x) we are mainly interested in is that f(t,x) has a repulsive singularity near x = 0 and f(t,x) is superlinear near x = +00.This paper is composed of two parts. In the first chapter, we introduce the historical backgroud of the problems which will be investigated and the main results of this paper. Also, we will state some preliminary results which will be used in our paper.In the second chapter, first we are devoted to study the existence of twin positive periodic solutions to the positone case, i. e., f(t,x) > 0, V (t, x) e [0,1] x (0, +00). In this case, we prove that the weak singularity of f(t,x) at x = 0 is allowed, as revealed in [18] and [20].In chapter 2, the semipositone case, i. e., f(t,x) + M > 0 for some M > 0, is also studied. In this case, some strong force conditions are needed to obtain the existence and multiplicity of positive periodic solutions.In the second chapter, we will use Leray-Schaulder alternative theorem and Krasnoselskii fixed point theorem to obtain our existence results.Some illustrating examples will be given in chapter 2.