两类基于向前差分的分数阶差分方程的解的存在唯一性

Existence and Uniqueness of Solutions for two Classes of Fractional Difference Equations Based on Forward Difference

主要研究了Riemann-Liouvile型和Caputo型分数阶差分方程解的存在唯一性.结合分数阶差分方程已有的研究理论,以向前差分为出发点,参考已有的向后差分的研究方法及相关的结论,推导出适用于向前差分方程的结论,运用相关结论及两个特殊函数的收敛性,证明了两类类分数阶差分方程解的存在唯一性.向后差分是就想要得到的目标状态推算出当前状态,而向前差分可以由目前状态推算出未来的目标状态.然而,迄今已有一系列以向后差分为出发点分数阶差分方程理论的专著问世,而鲜见以向前差分为出发点的分数阶差分方程理论.经过推导总结,得出了以向前差分为出发点的两类分数阶差分方程解的存在唯一性.

In this paper,we study the existence and uniqueness of solutions for Riemannliouvile and Caputo fractional difference equations.Combining the theory of fractional order differential equation of the existing research,in order to forward difference as the starting point,the reference of existing backward difference methods and relevant conclusions,applicable to the forward difference equation is deduced,using the relevant conclusions and convergence of the two special function,proved the existence of two kinds of fractional differential equations only.Backward difference calculates the current state based on the desired target state,while forward difference calculates the future target state based on the current state.However,so far,there have been a series of treatises on fractional difference equation theory with backward difference as the starting point,but few fractional difference equation theory with forward difference as the starting point.In this paper,the existence and uniqueness of solutions of two kinds of fractional difference equations based on forward difference are obtained.