Second-order nonlinear impulsive integro-differential equations of mixed type with time-varying generating operators and optimal controls on Banach spaces

In this paper, a class of second-order nonlinear impulsive integro-differential equations of mixed type whose principle is time-varying generating operators with unbounded perturbation on Banach spaces is considered. Discussing the perturbation of time-varying operator matrix and constructing corresponding the evolution system generated by operator matrix, we introduce the reasonable mild solution of second-order nonlinear impulsive integro-differential equations of mixed type and prove the existence of mild solutions. The existence of optimal controls for a Lagrange problem of systems governed by the second-order nonlinear impulsive integro-equations of mixed type is also presented. An example is given for demonstration.