非线性(p,q)-差分方程非局部问题的正解  

Positive solutions for nonlocal problems of nonlinear(p,q)-difference equations

作　　者：禹长龙[1] 韩获德 王菊芳[1] 邢厚民 YU Changlong;HAN Huode;WANG Jufang;XING Houmin(School of Science,Hebei University of Science and Technology,Shijiazhuang,Hebei 050018,China;College of Letter and Science,University of California,Berkeley,California 94720,USA)

机构地区：[1]河北科技大学理学院,河北石家庄050018 [2]美国加州大学伯克利分校文理学院,加州伯克利94720

出　　处：《河北科技大学学报》2021年第4期352-359,共8页Journal of Hebei University of Science and Technology

基　　金：国家自然科学基金(11201112);河北省自然科学基金(A201520811);河北省教育厅基金(ON2017065)。

摘　　要：为了完善非线性量子差分方程边值问题的基本理论,研究了二阶非线性(p,q)-差分方程非局部问题的可解性。首先,计算线性(p,q)-差分方程边值问题的Green函数,研究Green函数的性质;其次,运用Banach压缩映像原理和Guo-Krasnoselskii不动点定理,获得二阶三点非线性(p,q)-边值问题正解的存在性和唯一性定理;再次,给出线性(p,q)-差分方程非局部问题的Lyapunov不等式;最后,给出2个实例,证明所得结果是正确的。结果表明,在赋予非线性项f一定的增长条件下,非线性(p,q)-差分方程非局部问题正解具有存在性和唯一性。研究结果丰富了量子差分方程可解性的理论,对(p,q)-差分方程在数学、物理等领域的应用提供了重要的理论依据。In order to improve the basic theory of boundary value problems for nonlinear quantum difference equations,in this paper,we study the solvability of nonlocal problems for second order three-point nonlinear(p,q)-difference equations.Firstly,the Green function of the boundary value problem of linear(p,q)-difference equation is calculated and the property of Green function is studied.Secondly,we obtain the existence and uniqueness of the positive solution for the problem by the Banach contraction mapping principle and the Guo-Krasnoselskii fixed point theorem in a cone.Next,we get the Lyapunov inequality for nonlocal problems of linear(p,q)-difference equations.Finally,two examples are given to illustrate the validity of the results.The results show that the existence and uniqueness of positive solutions for nonlocal problems of nonlinear(p,q)-difference equations are obtained,under the condition of nonlinear term f certain growth.The research results enrich the theory of solvability of quantum difference equations and provide important theoretical basis for the application of(p,q)-difference equation in mathematics,physics and other fields.

关 键 词：非线性泛函分析 非线性(p q)-差分方程 非局部问题 Banach压缩映像原理 Guo-Krasnoselskii不动点定理 正解 

分 类 号：O175.8[理学—数学]