# zbMATH — the first resource for mathematics

On the Dirichlet problem for the nonlinear wave equation in bounded domains with corner points. (English) Zbl 0863.35062
Summary: Using Mawhin’s coincidence topological degree arguments and fixed point theory for non-expansive mappings results, we discuss the solvability of the Dirichlet problem for the semilinear equation of the vibrating string $$u_{xx}-u_{yy}+f(x,y,u)=0$$ in a bounded domain with corner points. When the winding number associated to the domain is rational, we improve and extend some results of A. A. Lyashenko [J. Math. Kyoto Univ. 33, No. 2, 543-570 (1993; Zbl 0797.35124)] and A. A. Lyashenko and M. W. Smiley [J. Math. Anal. Appl. 189, No. 3, 872-896 (1995; Zbl 0821.35099)]. The case where the winding number is irrational is also examined.

##### MSC:
 35L70 Second-order nonlinear hyperbolic equations 35B10 Periodic solutions to PDEs 35L05 Wave equation
##### Keywords:
topological degree; non-expansive mappings; winding number
Full Text: