Controllability of Nonlinear Neutral Implicit $\mathbb{ABC}$ Non-integer Integro-differential Equations with State and Control Delay

Abstract

Fractional order modeling is required to analyze the parameters of the dynamical systems involved in real-world phenomena. Controllability of the problem is essential for the practical application. The controllability of ($\mathbb{ABC}$) nonlinear neutral implicit integro-differential equation of fractional order $0<\alpha<1$ with delay in state and control has been examined in this work. To obtain the controllability results of the defined problem, we specifically apply the k-set contraction mapping and Darbo fixed point theorem. The provided example is used to verify the findings.