A fixed point result with applications in the study of viscoplastic frictionless contact problems

Let C(R+; X) denote the Frechet space of continuous functions defined on R+ = [0, infinity) with values on a real Banach space (X, || . || x). We prove a fixed point theorem for operators Lambda : C(R+; X) -> C(R+; X) which satisfy a sequence of inequalities involving an integral term. Then we consider a mathematical model which describes the frictionless contact between a viscoplastic body and a deformable foundation. The process is quasistatic and is studied on the unbounded interval of time [0, infinity). We provide the variational formulation of the problem, then we use the abstract fixed point theorem to prove the existence of a unique weak solution to the model. We complete our study with a regularity result.