Existence, uniqueness and synchronization of a fractional tumor growth model in discrete time with numerical results

A mathematical model of discrete fractional equations with initial condition is constructed to study the tumor-immune interactions in this research. The model is a system of two nonlinear difference equations in the sense of Caputo fractional operator. The applications of Banach’s and Leray–Schauder’s fixed point theorems are used to analyze the existence results for the proposed model. Additionally, we developed several kinds of Ulam’s stability results for the suggested model. The tumor-immune fractional map’s dynamic behavior is numerical analyzed for some special cases. Further, adaptive control law is proposed to stabilize the fractional map and a control scheme is introduced to enhance the synchronization of the fractional model.