非线性差分方程的同宿轨、周期解与边值问题
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摘要
本篇博士论文主要应用临界点理论研究非线性差分方程的同宿轨、周期解及边值问题的存在性。本文对差分方程定性理论的发展有重要的促进作用。全文共分五章,主要内容如下。
第一章简述了问题产生的历史背景、问题的研究状态、最新进展、预备知识以及本文的主要工作。
第二章利用山路引理结合周期逼近的方法讨论了周期离散非线性薛定谔方程同宿轨的存在性,并在一定条件下部分地解决了Alexander Pankov提出的两个公开问题。
第三章利用山路引理研究高阶非线性差分方程同宿轨的存在性。我们分别在有周期假设条件和无周期假设条件下建立了同宿轨存在的若干判别准则,而且得到了一些在无穷远处指数退化的同宿轨。所得结果推广了某些文献的结论。
第四章考虑二阶非线性差分方程周期解的存在性。应用鞍点定理对次线性情形及环绕定理对非超线性和非次线性情形进行了系统的分析,得到了全新的研究结果。
第五章讨论了二阶非线性差分方程在不同条件下的边值问题,通过建立变分框架,运用临界点理论,获得了几类在有限区间上的边值问题解的存在性的若干充分条件。
The existence of homoclinic orbits, periodic solutions and boundary value problems for nonlinear difference equations is studied by using critical point theory in this dissertation. It will motivate the development of qualitative theory of difference equations. This dissertation is composed of five chapters. The content of the dissertation is as follows.
Chapter 1 gives a brief introduction to the historical background, status and the up-to-date progress for all the investigated problems together with preliminary tools and main results in this dissertation.
The existence of homoclinic orbits for periodic discrete nonlinear Schrodinger equations is obtained in Chapter 2 by using Mountain Pass Lemma in combination with periodic approximations. Two open problems proposed by Alexander Pankov are partially solved under certain hypotheses.
In Chaper 3, the existence of homoclinic orbits for higher order nonlinear difference equations is studied by using Mountain Pass Lemma. Some criteria for the existence of homoclinic orbits of these equations with periodic assumptions and without periodic assumptions are worked out, respectively. Moreover, some homoclinic orbits decaying exponentially at infinity are obtained. Our results extend some known results in the literature.
The existence of periodic solutions to second order nonlinear difference equations is investigated in Chapter 4. The solutions to second order sublinear difference equations by using Saddle Point Theorem and to second order neither superlinear nor sublinear difference equations by using Linking Theorem are discussed. Some new results are obtained.
In Chapter 5, boundary value problems to a class of second order nonlinear difference equations are studied. By establishing variational structure and applying critical point method, the existence of solutions of boundary value problems for second order nonlinear difference equations on a finite discrete segment with various boundary value conditions is considered. Some new sufficient conditions are obtained.
第一章简述了问题产生的历史背景、问题的研究状态、最新进展、预备知识以及本文的主要工作。
第二章利用山路引理结合周期逼近的方法讨论了周期离散非线性薛定谔方程同宿轨的存在性,并在一定条件下部分地解决了Alexander Pankov提出的两个公开问题。
第三章利用山路引理研究高阶非线性差分方程同宿轨的存在性。我们分别在有周期假设条件和无周期假设条件下建立了同宿轨存在的若干判别准则,而且得到了一些在无穷远处指数退化的同宿轨。所得结果推广了某些文献的结论。
第四章考虑二阶非线性差分方程周期解的存在性。应用鞍点定理对次线性情形及环绕定理对非超线性和非次线性情形进行了系统的分析,得到了全新的研究结果。
第五章讨论了二阶非线性差分方程在不同条件下的边值问题,通过建立变分框架,运用临界点理论,获得了几类在有限区间上的边值问题解的存在性的若干充分条件。
The existence of homoclinic orbits, periodic solutions and boundary value problems for nonlinear difference equations is studied by using critical point theory in this dissertation. It will motivate the development of qualitative theory of difference equations. This dissertation is composed of five chapters. The content of the dissertation is as follows.
Chapter 1 gives a brief introduction to the historical background, status and the up-to-date progress for all the investigated problems together with preliminary tools and main results in this dissertation.
The existence of homoclinic orbits for periodic discrete nonlinear Schrodinger equations is obtained in Chapter 2 by using Mountain Pass Lemma in combination with periodic approximations. Two open problems proposed by Alexander Pankov are partially solved under certain hypotheses.
In Chaper 3, the existence of homoclinic orbits for higher order nonlinear difference equations is studied by using Mountain Pass Lemma. Some criteria for the existence of homoclinic orbits of these equations with periodic assumptions and without periodic assumptions are worked out, respectively. Moreover, some homoclinic orbits decaying exponentially at infinity are obtained. Our results extend some known results in the literature.
The existence of periodic solutions to second order nonlinear difference equations is investigated in Chapter 4. The solutions to second order sublinear difference equations by using Saddle Point Theorem and to second order neither superlinear nor sublinear difference equations by using Linking Theorem are discussed. Some new results are obtained.
In Chapter 5, boundary value problems to a class of second order nonlinear difference equations are studied. By establishing variational structure and applying critical point method, the existence of solutions of boundary value problems for second order nonlinear difference equations on a finite discrete segment with various boundary value conditions is considered. Some new sufficient conditions are obtained.
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