Ashyralyev, Allaberen; Yildirim, Ozgur On multipoint nonlocal boundary value problems for hyperbolic differential and difference equations. (English) Zbl 1201.65128 Taiwanese J. Math. 14, No. 1, 165-194 (2010). 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. Reviewer: Rolf Dieter Grigorieff (Berlin) Cited in 24 Documents MSC: 65L10 Numerical solution of boundary value problems involving ordinary differential equations 65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs 35L10 Second-order hyperbolic equations 47D09 Operator sine and cosine functions and higher-order Cauchy problems 34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations 34G10 Linear differential equations in abstract spaces 39A12 Discrete version of topics in analysis 35L20 Initial-boundary value problems for second-order hyperbolic equations 35L90 Abstract hyperbolic equations Keywords:hyperbolic equation in a Hilbert space; boundary value problem; nonlocal conditions in time; self-adjoint operator; semigroup framework; fixed point theorem; existence of solutions; time discretization; stability; divided difference scheme PDFBibTeX XMLCite \textit{A. Ashyralyev} and \textit{O. Yildirim}, Taiwanese J. Math. 14, No. 1, 165--194 (2010; Zbl 1201.65128) Full Text: DOI