無窮區間上二階三點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，計算出了無窮區間上二階三點線性q-差分方程邊值問題的Green函數，并研究了Green函數的性質;再次，在抽象空間上構造積分算子，然后運用Leray-Schauder連續定理，獲得了無窮區間上二階三點非線性q-差分方程邊值問題解的存在性結果;最后給出實例。實例驗證表明所得結果是正確的。研究結果對量子微積分的發展及其在數學物理等領域的應用都有著重要的意義。

關鍵詞：非線性泛函分析;q-差分方程;無窮區間;三點邊值問題;Leray-Schauder連續定理

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＼]。由量子力學的知識可知，時間和空間是不連續的，不能任意分割，也不存在小于普朗克尺度的量，這足以說明用經典微積分描述的物理現象與真實世界必然會存在偏差。此時，量子微積分應運而生。q-微積分被廣泛地應用于數學、物理等科學領域，如宇宙弦與黑洞、適形量子力學、核和高能物理、數值理論、組合、正交多項式、基本超幾何函數和其他科學的量子理論、力學和相對論等領域[3-9]。

參考文獻/References：

[1] JACKSON F H. On q-functions and a certain difference operator[J]. Transactions of the Royal Society of Edinburgh， 1908， 46： 253-281.

[2] JACKSON F H. On q-definite integrals[J]. The Quarterly Journal of Pure and Applied Mathematics， 1910， 41： 193-203.

[3] JACKSON F H. On q-difference equations[J]. American Journal of Mathematics，1910， 32： 305-314.

[4] CARMICHAEL R D. The general theory of linear q-difference equations[J]. American Journal of Mathematics， 1912， 34： 147-168.

[5] MASON T E. On properties of the solutions of linear q-difference equations with entire function coefficients[J]. American Journal of Mathematics，1915， 37： 439-444.

[6] ADAMS C R. On the linear ordinary q-difference equation[J]. Annals of Mathematics，1928， 30：195-205.

[7] PAGE D N. Information in black hole radiation[J].Physical Review Letters，1993， 71（23）： 3743-3746.

[8] YOUM D. q-deformed conformal quantum mechanics[J]. Physical Review D，2000， 62（9）： 276-284.

[9] ANNABY M H， MANSOUR Z S. q-Fractional Calculus and Equations[M]. Berlin： Springer， 2012.

[10] AHMAD B. Boundary value problems for nonlinear third-order q-difference equations[J]. Electronic Journal of Differential Equations，2011， 94：1-7.

[11] AHMAD B， NIETO J J. On nonlocal boundary value problem of nonlinear q-difference equations[J]. Advances in Difference Equation，2012：2012-81.

[12] YU Changlong， WANG Jufang. Existence of solutions for nonlinear second-order q-difference equations with first-order q-derivatives[J]. Advances in Difference Equation，2013：2013-124.

[13] EL-SHAHED M， HASSAN H A. Positive solutions of q-difference equation[J]. Proceedings of the American Mathematical Society，2010， 138： 1733-1738.

[14] AHMAD B， NTOUYAS S K. Boundary value problems for q-difference inclusions[J]. Abstract and Applied Analysis，2011（3/4）：292860.

[15] AHMAD B， NIETO J J. Basic theory of nonlinear third-order q-difference equations and inclusions[J]. Mathematical Modelling and Analysis，2013， 18（1）： 122-135.

[16] O'REGAN D. Theory of Singular Boundary Value Problems[M]. Singapore：World Scientific，1994.

[17] BAXLEY J V. Existence and uniqueness for nonlinear boundary value problems on infinite interval[J]. Journal of Mathematical Analysis and Applications，1990， 147： 127-133.

[18] GUO D. Second order impulsive integro-differential equations on unbounded domains in Banach spaces[J]. Nonlinear Analysis，1999， 35： 413-423.

[19] AGARWAL R P， O′REGAN D. Fixed point theory for self maps between Fréchet spaces[J]. Journal of Mathematical Analysis and Applications，2001，256（2）：498-512.

[20] FRIGON M， O'REGAN D. Fixed point of cone-compressing and cone-extending operators in Fréchet spaces[J]. Bulletin of the London Mathematical Society，2003， 35（5）：672-680.

[21] LIAN Hairong， GE Weigao. Solvability for second-order three-point boundary value problems on a half-line[J].Applied Mathematics Letters， 2006， 19（10）：1000-1006.

[22] KAC V， CHEUNG P. Quantum Calculus[M]. New York：Springer，2002.

[23] AGARWAL R P， O'REGAN D. Infinite Interval Problems for Differential， Difference and Integral Equations[M]. Netherlands：Kluwer Academic Publisher，2001.