Recently, multi-time stepping methods have become very popular among scientistdue to their high stability in problems with critical conditions. One importantshortcoming of these methods backs to their high amount of uncontrolled amplitudedecay. This study proposes a new multi-time stepping method in which the time stepis split into two sub-steps. The first sub-step is solved using the well-knownNewmark method and for the second sub-step an extended version of Newmarkmethod is applied. In fact, similarity in basic formulas of the mentioned methodsmakes it available to control the amount of amplitude decay in responses obtainedby the proposed method; in other words, the amplitude decay in the proposed methodis controlled through constant parameters of the two methods applied on each substep. The precision assessment of the proposed method is performed using numericalapproaches and revealed the minor period elongation error of the proposed methodin comparison with other existing methods. In addition to this, the unconditionalstability region of constant parameters is also determined through computation ofspectral radius of the proposed method. Finally, practical assessment of the proposedmethod is performed through several numerical examples.

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.