无穷区间上二阶三点q-差分方程边值问题解的存在性  

Existence of solutions to boundary value problems of second- order three-point q-difference equations on a infinite interval

作　　者：禹长龙[1] 张博雅 韩获德 YU Changlong;ZHANG Boya;HAN Huode(School of Science,Hebei University of Science and Technology,Shijiazhuang,Hebei 050018,China)

机构地区：[1]河北科技大学理学院

出　　处：《河北科技大学学报》2019年第6期469-476,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);河北省高等学校科学技术研究项目(QN2017065)

摘　　要：为了拓展非线性量子差分方程边值问题的基本理论,研究了一类无穷区间上非线性项含有一阶q-微分的二阶三点非线性q-差分方程边值问题解的存在性。首先,给出并证明了含有无穷限广义积分的二重q-积分的交换积分次序公式;其次,计算出了无穷区间上二阶三点线性q-差分方程边值问题的Green函数,并研究了Green函数的性质;再次,在抽象空间上构造积分算子,然后运用Leray-Schauder连续定理,获得了无穷区间上二阶三点非线性q-差分方程边值问题解的存在性结果;最后给出实例。实例验证表明所得结果是正确的。研究结果对量子微积分的发展及其在数学物理等领域的应用都有着重要的意义。In order to extend the basic theory of boundary value problems for nonlinear quantum difference equations,the existence of solutions for a class of second order three-point nonlinear q-differential equations with a first order q-differential on a nonlinear interval is studied.Firstly,changing the order of integration formula of double q-integral with infinite limit generalized integral is given and proved.Secondly,the Green function of the boundary value problem of second-order three-point linear q-difference equation on the infinite interval is calculated and the property of Green function is studied.Next,the integral operator T is constructed on the abstract space,and the Leray-Schauder continuous theorem is used to obtain the existence of the solution of the boundary value problems for the second-order three-point nonlinear q-difference equation on the infinite interval.Finally,an example is given to illustrate the validity of the results.The research results have important significance for the development of quantum calculus and its application in the fields of mathematical physics.

关 键 词：非线性泛函分析 q-差分方程 无穷区间 三点边值问题 Leray-Schauder连续定理 

分 类 号：O175.8[理学—数学]