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Zbl 1201.65128
Ashyralyev, Allaberen; Yildirim, Ozgur
On multipoint nonlocal boundary value problems for hyperbolic differential and difference equations. (English)
[J] Taiwanese J. Math. 14, No. 1, 165-194 (2010). ISSN 1027-5487

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.
[Rolf Dieter Grigorieff (Berlin)]

MSC 2000:: *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 cosine functions and higher-order Cauchy problems
34B10 Multipoint boundary value problems
34G10 Linear ODE in abstract spaces
39A12 Discrete version of topics in analysis
35L20 Second order hyperbolic equations, boundary value problems
35L90 Abstract hyperbolic evolution 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

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