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中国工业与应用数学学会会刊
主管：中华人民共和国教育部
主办：西安交通大学
ISSN 1005-3085  CN 61-1269/O1
工程数学学报

工程数学学报 ›› 2024, Vol. 41 ›› Issue (5): 973-979.doi: 10.3969/j.issn.1005-3085.2024.05.013

• • 上一篇    下一篇

具有周期系数的Kirchhoff-型差分方程同宿解的存在性

王振国1,   丁廉业2   

  1. 1. 太原学院数学系，太原  030032
    2. 黄淮学院数学与统计学院，驻马店  463000
  • 收稿日期:2022-12-02 接受日期:2023-06-07 出版日期:2024-10-15
  • 基金资助:
    河南省自然科学基金 (232300420127)；太原学院一般科研项目 (23TYYB04).

Homoclinic Solutions for the Kirchhoff-type Difference Equations with Periodic Coefficients

WANG Zhenguo1,   DING Lianye2   

  1. 1. Department of Mathematics, Taiyuan University, Taiyuan 030032
    2. School of Mathematics and Statistics, Huanghuai University, Zhumadian 463000
  • Received:2022-12-02 Accepted:2023-06-07 Online:2024-10-15
  • Supported by:
    The Natural Science Foundation of Henan Province (232300420127); the Scientific Research Project of Taiyuan University (23TYYB04).

摘要：

运用临界点理论研究一类具有周期系数的Kirchhoff-型差分方程同宿解问题。首先，构造了差分方程所对应的能量泛函。在文中的假设条件下，保证了能量泛函具有山路几何结构，从而得到了一个Palais-Smale序列。然后，利用一个可变号的全局性条件证明了该Palais-Smale序列的有界性。进一步，借助于$l^{2}$空间的紧支撑子集的性质和系数的周期性得到了该差分方程的一个非平凡同宿解。最后，给出两个例子说明了主要结论的正确性。

关键词: Kirchhoff-型差分方程, 同宿解, Palais-Smale序列, 山路引理

Abstract:

By means of critical point theory, we investigate homoclinic solution problems for the Kirchhoff-type difference equations with periodic coefficients. First, we verify that the graph of the energy functional satisfies the mountain pass geometrical properties. Such mountain pass geometry produces a Palais-Smale sequence. Second, we exploit one global property condition to guarantee that this Palais-Smale sequence is bounded. Further, by using the subset of $l^{2}$ consisting of functions with compact support and periodicity of coefficients, we obtain the existence of one nontrivial homoclinic solution for the Kirchhoff-type difference equations with periodic coefficients. Finally, two examples are given to illustrate our main results.

Key words: Kirchhoff-type difference equations, homoclinic solutions, Palais-Smale sequence, mountain pass lemma

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