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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: $$\align & d^2 u(t)/dt^2+Au(t)=f(t),\quad 0 \le t \le 1, \\ & u(0)=\sum_{r=1}^n \alpha_r u(\lambda_r)+\varphi, \,\, u_t(0)=\sum_{r=1}^n \beta_r u(\lambda_r)+\psi, \endalign$$ where $$0 < \lambda_1 \le \lambda_2 \le \dots \lambda_n \le 1,$$ 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.

65L10Boundary value problems for ODE (numerical methods)
65M12Stability and convergence of numerical methods (IVP of PDE)
35L10Second order hyperbolic equations, general
47D09Operator sine and cosine functions and higher-order Cauchy problems
34B10Nonlocal and multipoint boundary value problems for ODE
34G10Linear ODE in abstract spaces
39A12Discrete version of topics in analysis
35L20Second order hyperbolic equations, boundary value problems
35L90Abstract hyperbolic equations