Discontinuous wave equations and a topological degree for some classes of multi-valued mappings. (Q1775155)

The authors extend the Leray-Schauder degree to certain classes of multi-valued admissible mappings on separable Hilbert spaces. This extension cover quasi- and pseudomonotone multi-valued mappings. They apply the results to the semilinear wave equation \(u_{tt}-u_{xx}+g(u)+f(x,t,u)=h(x,t)\) where \(g\) is bounded nondecreasing, \(h\in L^2((0,\pi )\times (0,2\pi ))\) and \(f\) is sublinear in \(u\). Under some additional conditions, an existence of weak \(2\pi \)-periodic solutions is proved and the same result is obtained for the nonlinear discontinuous vibrating string equation.