# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Admissibility for discrete Volterra equations. (English) Zbl 1107.39012

The existence and uniqueness of solutions of the so called discrete Volterra summation equation

$x\left(n\right)=h\left(n\right)+\sum _{j=0}^{n}B\left(n,j\right)f\left(j,x\left(j\right)\right),\phantom{\rule{1.em}{0ex}}n\in {ℤ}^{+}=\left\{0,1,2,\cdots \right\},\phantom{\rule{2.em}{0ex}}\left(1\right)$

is investigated, where ${\left\{h\left(n\right)\right\}}_{n=0}^{\infty }$ are real $d$-vector sequences, $B\left(n,j\right)$ is a real $d$ by $d$ matrix for each pair $\left(j,n\right)\in {ℤ}^{+}×{ℤ}^{+}$ such that $j\le n,$ and $f:{ℤ}^{+}×{ℝ}^{d}\to {ℝ}^{d}·$ ‘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

$\left(Bx\right)\left(n\right)=\sum _{j=0}^{n}B\left(n,j\right)x\left(j\right)$

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. 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.

##### MSC:
 39A11 Stability of difference equations (MSC2000) 39A12 Discrete version of topics in analysis 46B45 Banach sequence spaces