无穷区间上二阶三点q-差分方程边值问题解的存在性

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　　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。
　　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
　　中图分类号：O175.8   文献标志码：A   doi：10.7535/hbkd.2019yx06003
　　Abstract：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.
　　Keywords：nonlinear functional analysis; q-difference equation; infinite interval; three-point boundary value problem; Leray-Schauder continuation theorem
　　 最早起源于20世纪初，由JACKSON提出的量子微积分，又名q-微积分，是一类无极限的微积分，参见文献＼[1—2＼]。由量子力学的知识可知，时间和空间是不连续的，不能任意分割，也不存在小于普朗克尺度的量，這足以说明用经典微积分描述的物理现象与真实世界必然会存在偏差。此时，量子微积分应运而生。q-微积分被广泛地应用于数学、物理等科学领域，如宇宙弦与黑洞、适形量子力学、核和高能物理、数值理论、组合、正交多项式、基本超几何函数和其他科学的量子理论、力学和相对论等领域[3-9]。
　　
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