exponential decay for a general class of nonautonomous abstract semilinear evolution equations with time-varying delay feedback

In this paper, we consider a general class of nonautonomous abstract delayed evolution equations with a nonlinear source term. under appropriate assumptions on the time-independent operator and the initial data, we establish global existence using the method of steps and employing classical results from the theory of inhomogeneous evolution problems. then, by assuming that the operator associated with the non-delayed part of the system generates an exponentially stable semigroup, we obtain an exponential decay estimate. this is achieved through a direct proof based on duhamel's formula combined with gronwall's inequality, under lipschitz continuity conditions on the nonlinear source term. finally, we conclude the paper by providing illustrative examples that validate the generalized setting of our system.