We consider the initial value problem
\displaylines{
\frac{d^{2}u}{dt^{2}}+Au(t)=f(u(t),u(t-w)), \quad t>0, \cr
u(t)=\varphi (t),\quad -w\leq t\leq 0
}
for a nonlinear hyperbolic equation with time delay in a Hilbert space
with the self adjoint positive definite operator A.
We establish the existence and uniqueness of a bounded solution, and
show application of the main theorem for four nonlinear partial
differential equations with time delay. We present first and second order
accuracy difference schemes for the solution of one dimensional nonlinear
hyperbolic equation with time delay. Numerical results are also given