A coupled system describing the interaction of a differential subsystem with nonlinearities of a sector type and a linear difference subsystem is considered. It is assumed that the system is positive. A diagonal Lyapunov–Krasovskii functional is constructed and conditions are determined under which the absolute stability of the investigated system can be proved with the aid of such a functional. In the case of nolinearities of the power form, estimates for the convergence rate of solution to the origin are derived. The stability analysis of the corresponding system with parameter switching is fulfiled. Sufficient conditions guaranteeing the asymptotic stability of the zero solution for any admissible switching law are obtained.