关于分数阶q-差分方程正解的存在性  

THE EXISTENCE OF POSITIVE SOLUTIONS TO THE FRACTIONAL q-DIFFERENCE EQUATION

作　　者：田春平 顾海波[1] 孙会贤 Tian Chunping;Gu Haibo;Sun Huixian(School of Mathematics Science, Xinjiang Normal University, 830017, Urumqi, China)

机构地区：[1]新疆师范大学数学科学学院,乌鲁木齐830017

出　　处：《山东师范大学学报（自然科学版）》2022年第1期32-43,共12页Journal of Shandong Normal University(Natural Science)

基　　金：国家自然科学基金资助项目(11961069,11861068);新疆优秀青年科技人才培训计划资助项目(2019Q022);新疆维吾尔自治区自然科学基金资助项目(2019D01A71,2018D01A27);新疆维吾尔自治区高校科研计划资助项目(XJEDU2018Y033).

摘　　要：本文研究了一类分数阶q-差分方程边值问题正解的存在性,在前人研究成果的基础上,基于薛定谔方程探究了一类高阶分数阶带有扰动项的问题.首先,运用迭代方法研究了λ=0时特殊解的存在性.然后,利用格林函数的性质,以及锥拉伸和锥压缩不动点定理,研究了当λ>0时参数的变化对边值问题解的影响,并讨论了正解的存在性以及正解的存在区间,给出了至少存在两个正解的充分条件,得到了两个推论性结果.最后通过两个算例说明了所得结果的适用性.In this paper,we study the existence of positive solutions to a class of fractional q-difference equation the boundary value problems,and explore a class of higher order fractional with perturbation terms based on the Schrodinger equation.We first study the existence of solutions in special cases through an iterative approach.Secondly,using the properties of the Green′s function and the cone stretching and cone compression fixed point theorem,we discussed the influence on the solution of the boundary value problem when the parameters changed,demonstrated the existence and the existence interval of positive solutions,and also give sufficient conditions for the existence of at least two positive solutions and obtain two corollary results.Finally,two examples are presented to illustrate the applicability of the results.

关 键 词：分数阶q-差分方程 迭代方法 锥拉伸 锥压缩 正解的存在性 

分 类 号：O175.14[理学—数学]