On multipoint nonlocal boundary value problems for hyperbolic differential and difference equations.

*(English)*Zbl 1201.65128The paper is concerned with the existence of solutions of the following boundary value problem with nonlocal conditions with respect to time:

where

$$0<{\lambda}_{1}\le {\lambda}_{2}\le \cdots {\lambda}_{n}\le 1,$$

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.

Reviewer: Rolf Dieter Grigorieff (Berlin)

##### MSC:

65L10 | Boundary value problems for ODE (numerical methods) |

65M12 | Stability and convergence of numerical methods (IVP of PDE) |

35L10 | Second order hyperbolic equations, general |

47D09 | Operator sine and cosine functions and higher-order Cauchy problems |

34B10 | Nonlocal and multipoint boundary value problems for ODE |

34G10 | Linear ODE in abstract spaces |

39A12 | Discrete version of topics in analysis |

35L20 | Second order hyperbolic equations, boundary value problems |

35L90 | Abstract hyperbolic equations |