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On multipoint nonlocal boundary value problems for hyperbolic differential and difference equations. (English) Zbl 1201.65128

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.

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
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