Existence Of Periodic Solutions Of Dynamic Equations On Time Scales

As we know,many phenomena in the real world show periodic behaviors.The problem of periodic solution of continuous dynamical system has been one of the central topics since Poincare and Lyapunov.However,not all the natural phenomena can be described by continuous systems.Recently,some theories and methods are developed to dynamic equations on time scales.A time scale is an arbitrary non-empty closed subset of R,generally denoted by T.The theory of time scales was in order to study continuous-discrete hybrid processes.For instance,if T = Z,dynamic equations are just usual difference equations,while,taking T = R,they are usual differential equations.Thus,the theories of time scales unify and generalize the existing theories of both d-ifference equations and differential equations.This theory becomes a powerful tool to economics,populations models,biology models and so on.It has been attracting more and more attentions during the past years and the existence of solutions for systems on time scales has been extensively investigated,spe-cially concerning periodicity.The existence of periodic solutions of dynamic equation with a small parameter ? when "time" is continuous-discrete hybrid has gradually gained attention.The aim of this thesis is to study the existence of periodic solutions for the perturbed dynamic equations on time scales of the type where fi:T × U ? Rn for i = 0,1,…,k and r:T x U x(-?0,?0)? Rn are rd-continuous,and T-periodic in the first argument,U is an open subset of Rn,and ? is a small parameter.The main approaches of this thesis are coincidence degree theory and the averaging method.Further,we extend some averaging theorem to k-th order in ?.And using the averaging theorem to prove the existence of periodic solutions of dynamic equations on time scales.More precisely,results of finding periodic solutions are given via the topological degree theory.The proof is inspired by the classical one,but certain technical details on time scales are more complicated.This paper is organized as follows.In the first chapter,we talk about the origin of periodic solutions of d-ifferential equations,the method of averaging and the theory of time scales.Moreover,we introduce the backgrounds of our problems.At last,we present some basic definitions,concepts and results concerning time scales which are essential to prove our main results.A summary of this thesis is also made in this chapter.In the second chapter,we present the first order existence theorem of periodic solutions for dynamic equations on time scales in Section 2.1.Then we extend the averaging theory for dynamic equations on time scales to arbitrary order in ?.Recently,Llibre,Novaes,and Teixeira extended the averaging theory up to any order in ? by using the Brouwer degree.Comparing with their results,we extend continuous system to the system on time scale as well as we give a new condition.Moreover,our averaging functions F(·,?)are much easier to calculate,which do not need smoothness.This main result offers a topological method to study the existence of periodic solutions for dynaamic equation on time scales in theory.Next we give the result about n-order existence theorem of periodic solu-tions as follows.Theorem 0.0.1 Assume that T is a T-periodic time scale,U(?)Rn is an open bounded set.We consider the following dynamic equation where fi:T×U ? Rn fori = 1,…,k,r:TxUx(-?0,?0)are rd-continuous functions,T-periodic in the first argument and locally Lipschitz with respect'to x.Moreover,we assume that the follouwing conditions hold:(i)For each t ?T,p ?(?)U,there exists a neighborhood Np of p,a constant?>0 independent of ? and integers 1 ? j ?n such that for any q ?Np,t ?[0,T]]T,and ? ?[-?0,?0]\{0}.(ii)Suppose that for each ? ?[-?0,?0]\{0},where the averaged function F(·,?)is defined on time scales by Then,there exists a T-periodic solution x(t)of equation(0.0.2)such that x(t)? U,for|?|>0 suf|?|>0 sufficiently small.We proveTheorem 0.0.1 via topological degree theory.Finally,to illustrate our main result,the last section is devoted to some examples.In the third chapter,we extend the averaging theory of Llibre to dynamic equation on time scales.In previous studies,Llibre has proved the averaging theory of low order(up to three)for a differential system which depends on a small parameter ?.They prove this problem by using the averaging method.Our main results extend the differential equation to the dynamic equation on time scales.First of all,we state the first order averaging theorem and the second order averaging theorem for dynamic equations on time scales in Section 3.1 and Section 3.2.Then,we give the averaging theory for finding periodic solutions to an arbitrary order in ? for dynamic equations on time scales in Section 3.3.We define the i-th order averaged function Fi on time scale as Fi(z)=yi(T,z)/i!,(0.0.3)where yi:T × U ? Rn are defined by the following integral equation,for i = 1,2,…,k-1,where M = b1? + b2 + … + bl and the sum is over all non-negative integer solutions of Diophantine equation b1 + 2b2 + … lbl = l and Sl is the set of all 1-triples of non-negative integers(b1,b2,…,bl).In the following,we state our main results:Theorem 0.0.2 when f0 = 0 and Theorem 0.0.3 when f0 ? 0.Theorem 0.0.2 Suppose that f0 ? 0.We consider the following dynamic equationwhere fi:T×U ?Rn for i = 1,…,k,r:T×U×(-?0,?0)are rd-continuous functions,T-periodic in the first argument and locally Lipschitz with respect to x.In addition,we assume the following conditions.(?)fi(t,·)? Ck for all t ?T,(?)k fi and r are locally Lipschitz in the second argument for i = 1,2,…,k.(?)Assume that Fi = 0 for i = 0,1,2,…,r-1 and Fr ? 0 with r ?{1,2,…,k}(here we are taking F0 = 0).(?)Moreover,we assume that for an open and bounded set V(?)U,deg(F,(z),V,0)?0.Then,for |?|>0 sufficiently small,there exists a T-periodic solution x(·,?)such that x(·,?)? a when ? ? 0.Theorem 0.0.3 Suppose that f0 ? 0.We consider the following dynamic equationwhere fi:T×U ? Rn for i = 0,1,…,k,r:T×U ×(-?0,?0)are continuous functions,T-periodic in the first argument and locally Lipschitz with respect to x.In addition,we assume the following conditions.(?')There exists an open subset W of U such that ?(t,z)is T-periodic in the first argument t.(?')fi(t ·)? Ck for all t ? T,(?)kfi and r are locally Lipschitz in second argument for i = 1,2,…,k.(?')Assume that Fi = 0 for i = 1,2,…,r-1 and Fr ? 0 with r ?{1,2,…,k}.Moreover,we assume that for an open and bounded set V(?)W,deg(Fr(z),V,0)?0.Then,for |?|>0 sufficiently small,there exists a T-periodic solution x(·,?)such that x(·,?)? a when ??0.