A double step Rothe scheme for hyperbolic Clarke subdifferential inclusions controlled by evolution equations

Abstract

In this paper we deal with a coupled system which consists of a hyperbolic Clarke subdifferential inclusion and an evolution equation in Banach spaces. Using temporally semidiscrete method based on the double step scheme, we construct a discrete approximate system. The existence of solutions and its a-priori estimates for the discrete approximate system are provided by the surjectivity of multivalued pesudomonotone operators and discrete Gronwall’s inequality. Finally, we show that the solution sequence of the discrete approximate system converges weakly to a limit element, which is a solution of the coupled original system.