Monotone iterative technique for the elastic systems with structural damping in Banach spaces

In this paper, we use a monotone iterative technique in the presence of lower and upper solutions to discuss the existence of mild solutions to a second order evolution equation initial value problem in an ordered Banach space, which can model an elastic system with structural damping. Under monotone condition and the noncompactness measure condition on nonlinearity, we obtain the existence of extremal mild solutions and positive mild solutions. Moreover, some applications are given to illustrate our theoretical results.