On multipoint nonlocal boundary value problems for hyperbolic differential and difference equations (Q971516)

The paper is concerned with the existence of solutions of the following boundary value problem with nonlocal conditions with respect to time: \[ \begin{aligned} & d^2 u(t)/dt^2+Au(t)=f(t),\quad 0 \leq t \leq 1, \\ & u(0)=\sum_{r=1}^n \alpha_r u(\lambda_r)+\varphi, \,\, u_t(0)=\sum_{r=1}^n \beta_r u(\lambda_r)+\psi, \end{aligned} \] where \[ 0 < \lambda_1 \leq \lambda_2 \leq \dots \lambda_n \leq 1, \] in a Hilbert space with a self-adjoint positive definite operator \(A\). The existence of a unique solution is proved by rewriting the given equation with the aid of the cosine and the sine operator-function of \(A\) and applying Banach's fixed point theorem under apropriate smallness conditions for the coefficients \(\alpha_r\) and \(\beta_r\). Also corresponding stability estimates are given. In a second part the time discretization of the problem by the standard implicit second order divided difference scheme is considered and corresponding results as in the frist part are derived.