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Admissibility for discrete Volterra equations. (English) Zbl 1107.39012
The existence and uniqueness of solutions of the so called discrete Volterra summation equation $$ x(n)=h(n)+\sum_{j=0}^{n}B(n,j)f(j,x(j)),\quad n\in \Bbb Z^{+}=\{0,1,2,\dots\}, \tag1$$ is investigated, where $\left\{ h(n)\right\} _{n=0}^{\infty }$ are real $d$-vector sequences, $B(n,j)$ is a real $d$ by $d$ matrix for each pair $ (j,n)\in \Bbb Z^{+}\times \Bbb Z^{+}$ such that $j\leq n,$ and $f:\Bbb Z^{+}\times \Bbb R^{d}\rightarrow \Bbb R^{d}.$ `Stable’ solutions such as convergent, zero convergent, bounded, subexponential, dominated solutions are sought. Instead of case by case search, the author considers sequence spaces which are Banach spaces containing all respective stable sequences. Then conditions are found such that an operator $B$ maps all elements in one sequence space $ X$ into another sequence space $Y$ (such a concept is called admissibility). In particular, the operator $B$ defined by $$ (Bx)(n)=\sum_{j=0}^{n}B(n,j)x(j) $$ is considered and explicit (admissibility) conditions found so that it maps all convergent sequences to convergent sequences, etc. Then under additional admissibility conditions on $f$ and $h,$ stable solutions of (1) can be found. We remark that the idea of the abstract setting is not new [see e.g. {\it M. Kwapisz}, Aequationes Math. 43, No. 2/3, 191--197 (1992; Zbl 0758.39001)]. But the abstract approach clears up what more need to be done in each specific case and what not.

39A11Stability of difference equations (MSC2000)
39A12Discrete version of topics in analysis
46B45Banach sequence spaces
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