几类高阶差分系统周期解的存在性
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摘要
微分方程、差分方程作为现代数学的一个重要分支,广泛应用于计算机科学、经济学、神经网络、生态学及控制论等学科领域中,因此对微分方程、差分方程解的性态的研究不仅有着重要的理论意义,而且具有重要的实用价值.几十年来,许多学者对微分方程周期解的存在性与多重性应用不同的方法进行了深入广泛的研究,这些方法主要有临界点理论(包括极小极大理论、几何指标理论与Morse理论)、不动点理论、重合度理论、Kaplan-Yorke藕合系统法等.在这些方法中,临界点理论已成为处理这类问题的强有力的工具.但是应用临界点理论研究差分方程周期解的存在性的文献很少,其主要原因在于难以找到适当的变分结构.本博士论文应用临界点理论研究了几类高阶差分系统的周期解的存在性和一类椭圆系统的解的存在性,得到了一系列全新的结果,主要内容如下:
首先,简要介绍了变分法的历史,回顾了与所研究问题相关的椭圆方程、哈密尔顿系统的历史背景与发展现状,并对本文的工作进行了简要的陈述.
其次,构建了几类新的高阶差分系统(或方程)模型,并通过构建恰当的变分结构,将两类高阶差分系统(或方程)的周期解和一类椭圆系统的解的存在性问题转化为适当函数空间上对应泛函的临界点的存在性问题,拓展了原有的二阶差分方程(或系统)模型.
在第二章中,我们讨论了一类高阶差分系统.首先,利用Morse理论结合临界群的计算等方法研究了高阶差分系统在非线性项是渐近线性的和超线性的两种情形,得出以下结论:当非线性项在无穷远处是渐近线性时,如果变分泛函在无穷远处的Morse指标和原点处的Morse指标不同,则系统在共振和非共振两种状态下都存在非平凡周期解.当非线性项在无穷远处是超线性时,系统至少存在三个不同的周期解.然后,分别利用环绕定理、对称山路引理得到了该高阶差分系统存在多个和无穷多个非平凡周期解的结论,部分结果推广了已有文献的结论.再利用Morse理论结合Lyapunov-schmidt约化方法、三临界点定理研究该高阶差分系统,将原有的对微分方程的研究方法推广到差分方程,并获得了该高阶差分系统多个和无穷多个非平凡周期解的存在条件.
在第三章中,我们利用环绕定理研究一类高阶泛函差分方程的周期解的存在性,得到了该方程至少存在一个非平凡周期解的若干充分条件.
在第四章中,我们考虑一类高阶差分方程.在非线性项是共振的情形,我们利用临界点理论中的局部环绕及无穷远处的角条件获得了该高阶差分方程多个非平凡周期解的存在条件.
在第五章中,我们结合畴数理论,利用推广的山路引理研究了一种椭圆系统的解的存在性,所得结果推广了某些文献的结论.
As an important banch of modern mathematics, differential equation and difference equation have been widely applied in the area of computer science, economy, neutral net, ecology and control theory, so it is meaningful for the study of the solution of differential equations and difference equations. In the last decades, many researchers have deeply studied the existence and multiplicity of periodic solutions of differential equations with different approaches, such as critical point theory(which includes minimax theory, geometrical index theory and Morse theory), fixed point theory, coincidence degree theory, the Kaplan-Yorke method and so on. Among these approaches, critical point theory is a powerful tool to deal with such problem. However, results on periodic solutions of difference equations by using critical point theory are very scare in the literature because of the difficulty of finding the suitable variation structure. In this dissertation, the existence of periodic solutions for some class of higher order difference equation(system) and of solutions for an elliptic system is studied by using critical point theory, and a series of new results are obtained. The contents of the dissertation are introduced as follows:
Firstly, we sketch the development of methods of variation. The historical background and the recent development of elliptic equations(systems) and Hamil-tonian systems related to our problems are introduced. At the same time, the main contents of the dissertation are outlined.
Secondly, we construct some class of higher order difference system, and chang the existence of periodic solutions of some higher order difference equation (systems) and of solutions for an elliptic system into the existence of critical points of corresponding functional on suitable function space after finding the suitable variation structure. We have developed the second order models.
In chapter 2, we study a higher-order difference system in the case that the nonlinearity is asyptotically linear and superlinear by combining Morse theory with computation of critical point groups at first. We conclude the following results: In the case that the nonlinearity is asymptotically linear, the existence of nontrivial periodic solutions are obtained under the conditions of resonance or nonresonance if the Morse index at infinity different from the one at zero; In the case that the nonlinearity is superlinear, at least three nontrivial periodic solutions are obtained. Then, multiple and infinite many periodic solutions for the higher order difference system are obtained by using linking theory and symmetric mountain pass lemma respectively, and some results improve or extend the related results in the literatures.
At last, by combining Morse theory with Lyapunov-schmidt reduction method and three critical points theory, multiple and infinitely many periodic solutions for the higher order difference system are obtained. The method of studying differential equation has been developed to that of difference equation.
In chapter 3, The existence of nontrivial periodic solutions of a higher order functional difference equation is considered by linking theorem. Nontrivial periodic solutions are obtained.
The existence of nontrivial periodic solutions of a higher order difference equation with resonance is considered in chapter 4. Some sufficient conditions of the existence of at least three or four nontrivial periodic solutions are obtained by using local linking and abstract angle conditions at infinity.
In the last chapter we consider an elliptic system by using the generalized Mountain Pass Lemma, and some results improve or extend the related resultes in the literatures.
首先,简要介绍了变分法的历史,回顾了与所研究问题相关的椭圆方程、哈密尔顿系统的历史背景与发展现状,并对本文的工作进行了简要的陈述.
其次,构建了几类新的高阶差分系统(或方程)模型,并通过构建恰当的变分结构,将两类高阶差分系统(或方程)的周期解和一类椭圆系统的解的存在性问题转化为适当函数空间上对应泛函的临界点的存在性问题,拓展了原有的二阶差分方程(或系统)模型.
在第二章中,我们讨论了一类高阶差分系统.首先,利用Morse理论结合临界群的计算等方法研究了高阶差分系统在非线性项是渐近线性的和超线性的两种情形,得出以下结论:当非线性项在无穷远处是渐近线性时,如果变分泛函在无穷远处的Morse指标和原点处的Morse指标不同,则系统在共振和非共振两种状态下都存在非平凡周期解.当非线性项在无穷远处是超线性时,系统至少存在三个不同的周期解.然后,分别利用环绕定理、对称山路引理得到了该高阶差分系统存在多个和无穷多个非平凡周期解的结论,部分结果推广了已有文献的结论.再利用Morse理论结合Lyapunov-schmidt约化方法、三临界点定理研究该高阶差分系统,将原有的对微分方程的研究方法推广到差分方程,并获得了该高阶差分系统多个和无穷多个非平凡周期解的存在条件.
在第三章中,我们利用环绕定理研究一类高阶泛函差分方程的周期解的存在性,得到了该方程至少存在一个非平凡周期解的若干充分条件.
在第四章中,我们考虑一类高阶差分方程.在非线性项是共振的情形,我们利用临界点理论中的局部环绕及无穷远处的角条件获得了该高阶差分方程多个非平凡周期解的存在条件.
在第五章中,我们结合畴数理论,利用推广的山路引理研究了一种椭圆系统的解的存在性,所得结果推广了某些文献的结论.
As an important banch of modern mathematics, differential equation and difference equation have been widely applied in the area of computer science, economy, neutral net, ecology and control theory, so it is meaningful for the study of the solution of differential equations and difference equations. In the last decades, many researchers have deeply studied the existence and multiplicity of periodic solutions of differential equations with different approaches, such as critical point theory(which includes minimax theory, geometrical index theory and Morse theory), fixed point theory, coincidence degree theory, the Kaplan-Yorke method and so on. Among these approaches, critical point theory is a powerful tool to deal with such problem. However, results on periodic solutions of difference equations by using critical point theory are very scare in the literature because of the difficulty of finding the suitable variation structure. In this dissertation, the existence of periodic solutions for some class of higher order difference equation(system) and of solutions for an elliptic system is studied by using critical point theory, and a series of new results are obtained. The contents of the dissertation are introduced as follows:
Firstly, we sketch the development of methods of variation. The historical background and the recent development of elliptic equations(systems) and Hamil-tonian systems related to our problems are introduced. At the same time, the main contents of the dissertation are outlined.
Secondly, we construct some class of higher order difference system, and chang the existence of periodic solutions of some higher order difference equation (systems) and of solutions for an elliptic system into the existence of critical points of corresponding functional on suitable function space after finding the suitable variation structure. We have developed the second order models.
In chapter 2, we study a higher-order difference system in the case that the nonlinearity is asyptotically linear and superlinear by combining Morse theory with computation of critical point groups at first. We conclude the following results: In the case that the nonlinearity is asymptotically linear, the existence of nontrivial periodic solutions are obtained under the conditions of resonance or nonresonance if the Morse index at infinity different from the one at zero; In the case that the nonlinearity is superlinear, at least three nontrivial periodic solutions are obtained. Then, multiple and infinite many periodic solutions for the higher order difference system are obtained by using linking theory and symmetric mountain pass lemma respectively, and some results improve or extend the related results in the literatures.
At last, by combining Morse theory with Lyapunov-schmidt reduction method and three critical points theory, multiple and infinitely many periodic solutions for the higher order difference system are obtained. The method of studying differential equation has been developed to that of difference equation.
In chapter 3, The existence of nontrivial periodic solutions of a higher order functional difference equation is considered by linking theorem. Nontrivial periodic solutions are obtained.
The existence of nontrivial periodic solutions of a higher order difference equation with resonance is considered in chapter 4. Some sufficient conditions of the existence of at least three or four nontrivial periodic solutions are obtained by using local linking and abstract angle conditions at infinity.
In the last chapter we consider an elliptic system by using the generalized Mountain Pass Lemma, and some results improve or extend the related resultes in the literatures.
引文
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