一类非线性四元数分数阶差分方程解的存在唯一性

Existence and Uniqueness of Solutions for a Class of Nonlinear Quaternion Fractional Di?erence Equations

摘要: 由于四元数代数的非交换性, 所以四元数分数阶差分方程与经典的实、复分数阶差分方程存在着本 质的不同. 本文首先引入了关联辅助函数和部分有界序列空间的定义. 然后, 基于完备空间的定义, 得出了部分有界序列空间的完备性结论. 在此基础上, 基于压缩映射原理给出了一类非线性四元数 分数阶差分方程解的存在唯一性定理, 并给出了解的具体形式. 此外, 给出示例说明了所得结果的 可行性.

Abstract: Due to the non-commutativity of quaternion algebra, quaternion fractional difference equations exhibit fundamental differences compared to classical real and complex frac- tional difference equations. In this thesis, we first introduce the definitions of partial- ly bounded sequence spaces and associated auxiliary functions. Subsequently, based on the definition of complete metric spaces, the completeness of partially bounded sequence spaces is established. Building upon this foundation, the existence and u- niqueness theorem for solutions to a class of nonlinear quaternion fractional difference equations is derived using the contraction mapping principle, along with the explic- it formulation of the solution. Furthermore, an illustrative example is provided to demonstrate the feasibility of the obtained results.