Solvability of Boundary Value Problems for the Inclusion $u_{tt}-u_{zz}\in g(t,x,u)$ via the Theory of Multi-Valued A-Proper Maps

Abstract

The existence of a coincidence point $x$ for the inclusion

is studied where $L:D(L)\subset E\to F$ is a linear hyperbolic operator and $\Gamma :E\to 2^F$ is a convex-valued map. It is shown that any such monotone demi-continuous map $\Gamma$ is weakly $A$-proper. Some existence theorems for ($\ast$) are established and the reults are applicated to a boundary value problem for the inclusion $u_{tt}-u_{zz}\in g(t,x,u)$.