基于ψ-(h,r)-凹算子的非线性分数阶(p,q)-差分方程的唯一迭代解  

Unique iterative solution for nonlinear fractional (p,q)-difference equation based on ψ-(h,r)-concave operators

作　　者：王菊芳[1] 王斯 禹长龙[1,2] WANG Jufang;WANG Si;YU Changlong(School of Science,Hebei University of Science and Technology,Shijiazhuang,Hebei 050018,China;Faculty of Science,Beijing University of Technology,Beijing 100124,China)

机构地区：[1]河北科技大学理学院,河北石家庄050018 [2]北京工业大学理学部,北京100124

出　　处：《河北科技大学学报》2022年第5期505-515,共11页Journal of Hebei University of Science and Technology

基　　金：国家自然科学基金(12272011,11772007);河北省自然科学基金(A2015208114)。

摘　　要：为了丰富分数阶(p,q)-差分方程边值问题的基本理论,研究了一类非线性分数阶(p,q)-差分方程非局部问题的可解性。首先,计算线性分数阶(p,q)-差分方程边值问题的Green函数并研究其性质;其次,运用基于定义在有序集上增的ψ-(h,r)-凹算子的不动点定理,证明分数阶(p,q)-差分方程解的存在唯一性定理;再次,通过选取初值,构造单调迭代序列,获得边值问题的唯一迭代解;最后,给出实例,验证本文研究结果的正确性。结果表明,在赋予非线性项f一定的条件下,非线性分数阶(p,q)-差分方程具有唯一非平凡解。研究结果拓展了分数阶量子差分方程的可解性理论,可为分数阶(p,q)-差分方程的进一步应用提供有力的理论基础。In order to enrich the basic theory of boundary value problems of fractional(p,q)-difference equations, the solvability of nonlocal problems for a class of nonlinear fractional(p,q)-difference equations was investigated.Firstly, the Green function of the boundary value problem of linear fractional(p,q)-difference equation was calculated and its properties were studied.Secondly, a fixed point theorem based on the increasing ψ-(h,r)-concave operators defined on ordered sets was used to prove the existence and uniqueness of solutions for fractional(p,q)-difference equations.Thirdly, by selecting the initial value, the monotone iterative sequence was constructed to obtain the unique iterative solution of the boundary value problem.Finally, an example was provided to illustrate the correctness of the research results.The results show that when certain conditions are given to the nonlinear term f,the nonlinear fractional(p,q)-difference equation has a unique non-trivial solution.The research results extend the solvability theory of fractional quantum difference equations and provide a powerful theoretical basis for further application of fractional(p,q)-difference equation.

关 键 词：量子差分方程 分数阶(p q)-差分方程 迭代解 ψ-(h r)-凹算子 不动点定理 

分 类 号：O175.8[理学—数学]