Anti-periodic Solutions To Higher Order Ordinary Differential Equations

There are uncountably many examples of problems taken from the natural sciences (in particular physics and biology) for which a mathematical model may be constructed in terms of a differential equationIn general, if x represents the size of some quantity (such as the size of a biological population, or the amount of current generated by an electrical system), the rate at which x changes in time will be denoted by (?), called the derivative of x. The functional notation x(t) denotes the size of x at a given time t, and the differential equation x = f(t,x) states that the rate at x is changing depends, not only upon the time at which this rate of change is measured, but alsoupon the size of x at that time.In trying to solve such a mathematical model (that is, find a function x(t) whose derivative is actually given by the date function f(t,x)),there are basically two avenues of approach, referred to as quantitative analysis and qualitative analysis. In quantitative analysis, the goal is to find explicit numerical values of the solution by some method determined by the form of the differential equation. Such an analysis will rely heavily upon numerical approximation techniques, for which powerful computer programs will need to be used; this method of investigating solutions for differential equations will not be discussed in this thesis.Just as for quantitative analysis, the role of qualitative analysis has been tremendously developed during the twentieth century. In this branch of the theory of differential equations, the question is not "What is the solution?" but rather "What is the solution doing?", which involes questions of how possible solutions will behave as time goes on. Important properties of the quantative behavior of solutions involve boundedness (does the size of the solutions have to be constrained by some given maxmal value, or are solutions allowed to increase without bound?), stability (if two different solutions start out as being close to each other, will they remain close to each other, or will they get farther and farther apart? ), and periodicity.A periodic solution is one which, while it may not constantly have the same value x₀ as it started out with, will always keep returning to that value after it has moved away from it. Moer precisely, a solution x(t) is periodic with T > 0 if its behavior on the time interval 0≤t≤T is exactlythe same as its behavioe on the intervals T≤t≤2T, 2T≤t≤3T, et cetera; this is denoted by x(t + T) = x(t) for all t≥0.There are many examples of physical phenmena, modeled by differential equations, for which the periodicity of solutions is either guaranteed or desired. An excellent discussion of such types of problems is given in T. Burton's textbook on periodic solutions for differential equations. As one example, consider the orbit of the earth about the sun, and let x denote the distance between the earth and the sun. This distance is not constant(the earth is about three million miles farther from the sun in June than it is in January), and changes according to some differential equation x = f(t,x) (the explicit from of which is derived using Newton's law of gravitation). To preserve life on the earth, the variation of x must not be too extreme (being too close to or too far away from from the sun would have devastating effects), and it is to be hoped that x will continue to periodically oscillate about some average distance from the sun. Fortunately, it is indedd true that this the case.A number of authors (H. Amann, E. Zehnder [7, 8], G. Birkhoff [11, 12], M. Bottkol, S. Chow, F. Clarke [24, 25], W. Gordon [35], A. Lyapunov [42], J. Moser [44], P.Rabinowitz [49, 50], A. Weinstein [56], and others) have given many refined results on the periodic solutions of differential equations. Recently, anti-periodic solutions have been paid more attention by mathematicians. Anti-periodic problems have been studied extensively in the last ten years. For example, for first-order ordinary differential equations, a Massera's type criterion is presented in [17]; LaSalle oscillation theorem for anti-periodic solutions is presented in [57]; the validity of the monotone iterative technique is shown in [32, 61, 63]. Also for higher-orderordinary differential equations existence and uniqueness results based on a Leray-Schauder type argument are presented in [1, 2]; anti-periodic boundary conditions for evolution equations are considered in [4, 5, 20, 37]; anti-periodic boundary conditions for wave equations are considered in [26, 45]; anti-periodic boundary conditions for parabolic equations are considered in [47, 51, 62]; anti-periodic boundary conditions have been considered for the Schr(o|¨)dinger and Hill differential operator[29, 30]; also anti-periodic boundary conditions appear in the study of difference equations [15, 55]; existence results were extended to anti-periodic boundary value problems for impulsive differential equations [32, 40]. For recent developments involving the existence of anti-periodic solutions of differential equations, inequalities, and other interesting results on anti-periodic boundary value problems, the reader is referred to [3, 21, 18, 19, 27, 28, 34, 33, 38, 40, 48, 58, 59].In this thesis, we consider the existence and uniqueness of anti-periodic solutions to higher order ordinary differential equations. In Chapter 2, we transform the problem into the fixed points prblem of a operator first, then prove the existence of the fixed points. Thus, the existence and uniqueness of anti-periodic solutions to odd order ordinary differential equations are obtained. In Chapter 3, we prove the the existence and uniqueness of anti-periodic solutions to even order ordinary differential equations by using Schauder's fixed point theorem. At last, we generalize the results to the equations case. Now let us introduce our main results.In Chapter 2, we consider the existence and uniqueness of the anti-periodic solution to the following 2n+l-order nonlinear ordinary differential equationwhere f:R×R is continuous, and satisfies f(t + T,x)= -f(t,-x).we have:Theorem 1 Assume that there areα> 0,β>0,such either of the following conditions holds:Then (1) admits a unique T-anti-periodic solution.In Chapter 3, we prove the existence and uniqueness of the anti-periodic solution to the following 2n-order nonlinear ordinary differential equationwhere f:R×R→R is continuous, and satisfies f(t + T,x) = -f(t,-x). We obtain the following theorem:Theorem 2 Assume that there are nonnegtive integer N andδ>0,such thatThen (2) admits a unique T- anti-periodic solution.Our main results are obtained by using fixed point theory. Though there already exist some results [1, 2] on certain special class of higher order ordinary differential equations, ours are more general.