无穷区间上非线性q-差分方程共振问题的可解性  

Solvability of resonance problems for nonlinear q-difference equations on infinite intervals

作　　者：禹长龙[1,2] 李双星 李静 王菊芳[1] YU Changlong;LI Shuangxing;LI Jing;WANG Jufang(School of Sciences,Hebei University of Science and Technology,Shijiazhuang,Hebei 050018,China;Interdisciplinary Research Institute,School of Mathematics,Statistics and Mechanics,Beijing University of Technology,Beijing 100124,China)

机构地区：[1]河北科技大学理学院,河北石家庄050018 [2]北京工业大学数学统计学与力学学院交叉科学研究院,北京100124

出　　处：《河北科技大学学报》2024年第2期168-175,共8页Journal of Hebei University of Science and Technology

基　　金：国家重点研发计划项目(2022YFB3806000);国家自然科学基金(12272011);河北省自然科学基金(A2015208114);河北省教育厅基金(QN2017063)。

摘　　要：为了拓展非线性量子差分方程共振边值问题的基本理论,研究了一类无穷区间上非线性量子差分方程共振边值问题。首先,通过构造合适的Banach空间,定义Fredholm算子,计算其核域和值域;其次,定义其他恰当的算子,并运用Mawhin重合度理论,建立该问题解的存在性定理;再次,运用反证法获得该问题解的唯一性结果;最后,给出一个例子说明主要结果的有效性。结果表明,在非线性项满足一定增长的条件下,非线性量子差分方程共振边值问题至少存在一个解。研究结果丰富了量子差分方程的可解性理论,为量子差分方程在数学、物理等领域的应用提供了理论参考。In order to develop the basic theory of resonance boundary value problems for nonlinear quantum difference equations,a class of nonlinear quantum difference equation boundary value problems at resonance on infinite intervals was studied.Firstly,by constructing a suitable Banach space,the Fredholm operator was defined,and its kernel and value domains were obtained.Secondly,by defining other appropriate operators,and using Mawhin's coincidence degree theory,the existence theorem of solutions to this problem was established.Thirdly,the uniqueness of the solution was obtained by means of proof by contradiction.Finally,an example was given to illustrate the validity of the main results.The results show that under certain growth conditions of nonlinear terms,the boundary value problems for nonlinear quantum difference equation at resonance has at least one solution.The research results enrich the solvability theory of quantum difference equations,and provide important theoretical basis for the application of quantum difference equations in mathematics,physics and other fields.

关 键 词：非线性泛函分析 量子差分方程 Mawhin重合度理论 无穷区间 共振 

分 类 号：O175.7[理学—数学]