Existence And Uniqueness Of Periodic Solutions For Second Order Ordinary Differential Equations

The study on periodic solutions for second order ordinary differential equationsis a very important branch in the modern mathematics. It can be applied to many aspects in scientific research as well. People on periodic solutions for differential equations, has been a strong interest, not only as periodic solutions characterized some of the periodic movements, but also because it is universal, can be similar to portray some non-periodic movements. Due to its physical backgroundand its extensive application in the real situation, periodic solutions for second order ordinary differential equations drive many attentions from scholars in mathematics field. We can discuss the solutions of the periodic equationx″+ f(t,x,x′) = 0in a given interval [0, T], so that periodic problem converts to two points boundary value problem.Two points boundary value problem for second order differential equations can be traced back to I. Newton and G. W. Leibniz Calculus established at the initial stage. In the beginning of 19th century, French mathematician J. Fourier using the method of separation of variables with heat conduction problems, concludedtwo points boundary value problem for second order differential equationsx″(t) +λk~2x(t) = 0, x(0) = x(l) = 0,of whichλis the parameter. Because whether the solutions of the problem are existing depends on the value of A, it concludes the concept of eigenvalue. From 1930s, France Paris university professor Charles Sturm and French professorJoseph Liouville did research about two points boundary value problem for second order differential equations together. They convert the second order linear differential equation into(p(t)x′)′+λq(t)x = 0, p(t), q(t) > 0.The term of the transformed equation need to meet can write as a regular form x′(a) -αx(a) = x′(b) +βx(b) = 0,α,β≥0,now known as Sturm - Liouville boundary conditions. Their research turned out to a series results of eigenvalue, then formed Sturm - Liouville theory.In the 20th century, functional analysis become a key fundamental theory in researching boundary value problem for second order differential equations gradually.In fact, the common of differential operation and integral operation is that they can be abstracted to functional operator, based on which functional analysisis established and developed. In the mid-30s, French mathematician J. Leray and J. Schauder established Leray-Schauder Degree theory. Their method has achieved tremendous success when they operate on researching linear differential, integral and functional equations; Especially for application of boundary value problem for ordinary differential equations. The core is all fixed point theorem for the establishment and application. In the last century, periodic problem gets a lot of important results as well. B. Mehri and G. G. Hamedani did the research of the existence of the periodic solution for the equationwith certain conditions [6]; S. Sedziwy get the result of the existence of the periodic solutions for the equationx″= f(t,x,x′),where he assumes that the upper and lower solution exists and f is caratheodary function[9]. K.Schmitt did the study onx″+ f(t,x,x′)=0 with f satisfying local Lipschitz and the existence of a constantα,βsatisfyingf(t,α,0)≤0≤f(t,β,0),then he get the conclusion on the existence of periodic solutions[8]. In addition,through Leray-Schauder Degree theory, J. R. Ward gave out the sufficient condition[11] of the existence of periodic solutions for the equationx″+ cx′+ g(x) = f(t).These conclusions are significant; meanwhile the methods of research are worth to use as references. In [4], the author gives a simple method to discuss the existence and uniqueness of solution for two-point boundary value problem. In this paper, we will extend this method to periodic problem. We consider the periodic problemx′+f(t,x,x′) = 0,f(t,x,x′) = f(t + 2π,x,x′),(1) Throughout this pater, we shall study this problem under the following assumptions: (H) f, f_x, f_x′are continuous in R×R×R, andwhere N is some positive integer,Firstly, we consider the homogeneous equationWe getlemma 3.1 : Suppose that p, q are L~2-integrable 2π-periodic function,α,β,γsatisfy the condition (H), withThen (2) has only the trivial 2π-periodic solution x(t)≡0.Secondly, considering (1) we get the main resulttheorem 3.1 Assume that (H) holds. Then the problem (1) has a unique 2π-periodic solution.