Existence and Uniqueness of Solution of Fractional FitzHugh-Nagumo System

Abstract

The research of infinite dimensional dynamic system arose in the 1980s. With Mandelbrot's fractal theory, the theory of fractional calculus, as the basic tool of fractal theory, has been widely concerned and applied, which makes the theory of fractional calculus develop rapidly. In past decades, the fractional calculus theory has been widely used in many fields. The fractional differential equation model can more accurately simulate practical problems than the integer order model, which makes the fractional differential equation become the current research hotspot. However, it is difficult for us to obtain the explicit solution of most Nonlinear Fractional Ordinary Differential Equations. Therefore, the focus of the research on Fractional Ordinary differential equations has shifted to the geometric and topological properties of solutions. As an important part of studying lattice systems, attractors are used to describe the geometric and topological properties of solutions of lattice systems. At present, the research on the solutions of most fractional order lattice systems is only limited to discussing the existence of solutions in finite intervals. However, there have been few relevant results on the existence of solutions in the whole space of fractional order lattice systems. Therefore, it is meaningful to study the existence of solutions in the whole space of fractional order Fitzhugh Nagumo lattice systems.