# 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)
Existence and comparison results for first order periodic implicit difference equations with maxima. (English) Zbl 1005.39012

Let $\varphi :{ℝ}^{T+1}\to {ℝ}^{T+1}$ denote the operator defined by

${\left(\varphi u\right)}_{k}=\left\{\begin{array}{cc}\underset{i\in \left\{0,\cdots ,k\right\}}{max}{u}_{i},\hfill & 0\le k\le h,\hfill \\ \underset{i\in \left\{k-h,\cdots ,k\right\}}{max}{u}_{i},\hfill & h

Consider the so-called quasi-linear difference equation with maxima

${\Delta }{u}_{k}+M{u}_{k+1}+N{\left(\varphi u\right)}_{k+1}={\sigma }_{k},\phantom{\rule{4pt}{0ex}}k\in \left\{0,1,\cdots ,T-1\right\}·\phantom{\rule{2.em}{0ex}}\left(*\right)$

The authors obtain uniqueness and comparison results for solution of (*) satisfying the boundary condition ${u}_{\delta }={u}_{T}$. Next the similar problem is studied for nonlinear version of (*) including the proof of existence of solution on the basis of concept of lower and upper solutions.

A lot of examples illustrates the obtained results.

##### MSC:
 39A11 Stability of difference equations (MSC2000)