無窮區(qū)間上二階三點q-差分方程邊值問題解的存在性

禹長龍 張博雅 韓獲德

摘 要：為了拓展非線性量子差分方程邊值問題的基本理論，研究了一類無窮區(qū)間上非線性項含有一階q-微分的二階三點非線性q-差分方程邊值問題解的存在性。首先，給出并證明了含有無窮限廣義積分的二重q-積分的交換積分次序公式;其次，計算出了無窮區(qū)間上二階三點線性q-差分方程邊值問題的Green函數(shù)，并研究了Green函數(shù)的性質(zhì);再次，在抽象空間上構(gòu)造積分算子，然后運用Leray-Schauder連續(xù)定理，獲得了無窮區(qū)間上二階三點非線性q-差分方程邊值問題解的存在性結(jié)果;最后給出實例。實例驗證表明所得結(jié)果是正確的。研究結(jié)果對量子微積分的發(fā)展及其在數(shù)學(xué)物理等領(lǐng)域的應(yīng)用都有著重要的意義。

關(guān)鍵詞：非線性泛函分析;q-差分方程;無窮區(qū)間;三點邊值問題;Leray-Schauder連續(xù)定理

中圖分類號：O175.8 ? 文獻標志碼：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世紀初，由JACKSON提出的量子微積分，又名q-微積分，是一類無極限的微積分，參見文獻＼[1—2＼]。由量子力學(xué)的知識可知，時間和空間是不連續(xù)的，不能任意分割，也不存在小于普朗克尺度的量，這足以說明用經(jīng)典微積分描述的物理現(xiàn)象與真實世界必然會存在偏差。此時，量子微積分應(yīng)運而生。q-微積分被廣泛地應(yīng)用于數(shù)學(xué)、物理等科學(xué)領(lǐng)域，如宇宙弦與黑洞、適形量子力學(xué)、核和高能物理、數(shù)值理論、組合、正交多項式、基本超幾何函數(shù)和其他科學(xué)的量子理論、力學(xué)和相對論等領(lǐng)域[3-9]。

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