Constructing a solution in time semidiscretization method to an equation of vibrating string with an obstacle

Abstract

This article treats a problem which describes the movement of a string that hits an obstacle. In [M. Schatzman, A hyperbolic problem of second order with unilateral constraints: The vibrating string with a concave obstacle, J. Math. Anal. Appl. 73 (1980) 138–191] M. Schatzman solves this problem in a slightly classical way. In [K. Maruo, Existence of solutions of some nonlinear wave equation, Osaka J. Math. 22 (1985) 21–30; On certain nonlinear differential equations of second order in time, Osaka J. Math. 23 (1986) 1–53] K. Maruo constructs a solution to this problem by the use of the Yosida approximation. The purpose of this article is to construct a solution to this problem in a time semidiscretization method. In general approximation by the time semidiscretization method is different from the Yosida approximation. A simple example is presented in Appendix.