记忆依赖微分方程组解的存在唯一性

The Existence and Uniqueness of Solution for Memory-Dependent Differential Equations

摘要: 记忆依赖性导数与分数阶导数相比，其核函数可以根据实际情况进行选择，具有更强的表现力。本文主要讨论一阶记忆依赖微分方程组解的存在性与唯一性。在特殊情况下找到一阶线性记忆依赖型微分方程组的准确解。首先利用分部积分对等式进行一系列的变换，然后通过构造皮卡迭代序列，进一步论证向量级数一致收敛，从而证明了当其时滞足够小，核函数一阶可微时，该方程组的解存在；再通过Grownwall不等式论证其解唯一。找到了当核函数取特定形式时，一阶线性记忆赖型微分方程组的准确解。并将其与常微分方程组的解进行比较，发现当时滞越大时两者的解越接近。

Abstract: The weighted function of the memory dependent derivative can be selected according to the actual situation compared with the fractional derivative, which has stronger expressiveness. This paper mainly discusses the existence and uniqueness of the solution of the first-order memory dependent differential equations. Find the exact solution of the first-order linear memory dependent dif-ferential equations in special cases. Firstly, a series of transformations are performed by using the partial integral-pair equation, and then the vector series convergence is further demonstrated by constructing the Picard iteration sequence, which proves that the solution of the equation system exists when its delay is small enough and the weighted function is first-order differentiable; and then the solution is demonstrated uniquely by the Grownwall inequality. Find the exact solution of the first-order linear memory dependent differential equations when the weighted function takes a specific form. By comparing it with the solution of the ordinary differential equations, it is found that the solution is more and more close when the delay is larger.