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Bounds for solutions of a three-point partial difference equation. (English) Zbl 0909.39002

The authors obtain exponential type bounds for the solutions of the first order partial difference boundary value problem: \[ u_{m+1,n} +a_{m,n} u_{m,n+1} +b_{m,n}u_{m,n} =c_{m,n},\quad u_{0,n}= \psi_n,\quad m,n=0,1,2,\dots. \tag{*} \] Supposing that the sequences \(\{a_{m,n}\}\), \(\{b_{m,n}\}\) are uniformly bounded: \(| a_{m,n} |\leq \alpha\), \(| b_{m,n} |\leq \beta\), the forcing term \(\{c_{m,n}\}\) is exponentially bounded: \(| c_{m,n} |\leq M\gamma^m \delta^n\), and the boundary sequence \(\{\psi_n\}\) is exponentially bounded: (**) \(|\psi_n | \leq P\xi^n\), then the solution of (*) is exponentially bounded: \(| u_{m,n} |\leq Q\lambda^m \omega^n\), \(m,n=0,1,2, \dots\), (for some positive constants \(Q,\lambda, \omega)\). Omitting the assumption (**) is supposing \(\alpha+ \beta\leq 1\), \(\gamma,\delta \in(0,1)\) the authors give another bound for the solution of (*), namely \(| u_{m,n} |\leq\| \psi\|_{n,n+m} +Q \lambda^m \omega^n\), where \(\|\psi \|_{s,t} =\max_{s\leq k\leq t}| \psi_k|\) and the constant \(Q\) is given in the terms of \(M,\alpha,\beta, \lambda,\omega\). In the case: \(| c_{m,n} |\leq M\) (the forcing term is only bounded), \(0<\alpha +\beta<1\), the bound for the solution of (*) has the form \(| u_{m,n} |\leq \eta^m\| \psi\|_{n,n+m} +M/(1-\eta)\), where \(\eta\in (0,1)\) is some constant.

MSC:

39A10 Additive difference equations
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