Weakly nonlinear hyperbolic differential equation in Hilbert space

We consider nonlinear perturbations of the hyperbolic equation in the Hilbert space. Necessary and sufficient conditions for the existence of solutions of boundary-value problem for the corresponding equation and iterative procedures for their finding are obtained in the case when the operator in linear part of the problem hasn’t inverse and can have nonclosed set of values. As an application we consider boundary-value problems for countable systems of differential equations and van der Pol equation in a separable Hilbert space.