Asymptotic Constancy for the Solutions of Caputo Fractional Differential Equations with Delay

In this paper, we aim to study the neutral-type delayed Caputo fractional differential equations of the form CDαxt−gt,xt=ft,xt,t∈t0,∞,t0≥0 with order 0<α<1, which can be used to describe the growth processes in real-life sciences at which the present growth depends on not only the past state but also the past growth rate. Our ultimate goal in this study is to concentrate on the convergence of the solutions to a predetermined constant by establishing a linkage between the delayed fractional differential equation and an integral equation. In our analysis, the sufficient conditions for the asymptotic results are obtained due to fixed point theory. The utilization of the contraction mapping principle is a convenient approach in obtaining technical conditions that guarantee the asymptotic constancy of the solutions.