A method of establishing sufficient conditions for the stability of a system consisting of a time-varying linear process and a first-order nonlinear time-varying feedback is presented, Rate of decay of unforced response and BIBO stability can also be determined, The governing equation of the system is represented in the form of an integro-differential equation, so that a modified form of the Lyapunov-Razumikhin-type condition can be applied. The first part of the note deals with stability of vector functional equations, The results obtained are applied to feedback systems, An example is worked out to show the applicability of the results obtained.