# 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)
The existence of periodic and subharmonic solutions of subquadratic second order difference equations. (English) Zbl 1046.39005

Of concern is the nonlinear second order difference equation

${x}_{n+1}-2{x}_{n}+{x}_{n-1}+f\left(n,{x}_{n}\right)=0,\phantom{\rule{4pt}{0ex}}n\in ℤ,$

where $f={\left({f}_{1},\cdots ,{f}_{m}\right)}^{T}\in C\left(ℝ×{ℝ}^{m},{ℝ}^{m}\right)$ and $f\left(t+M,z\right)=f\left(t,z\right)$ for some positive integer $M$ and for all $\left(t,z\right)\in ℝ×{ℝ}^{m}$. One supposes there exists a function $F\left(t,z\right)\in {C}^{1}\left(ℝ×{ℝ}^{m},{ℝ}^{m}\right)$ such that the gradient of $F\left(t,z\right)$ in $z$ coincides with $f\left(t,z\right)$. Let $p$ be a given positive integer. In this paper, the existence of $pM$-periodic solutions of the above difference equation is studied, under different hypotheses on $f$ and $F$. The method used here is from the critical point theory. These results are the discrete analogues of some theorems obtained in the continuous case for the second order differential equation ${x}^{\text{'}\text{'}}+f\left(t,x\right)=0$, $t\in ℝ$.

##### MSC:
 39A11 Stability of difference equations (MSC2000) 39A12 Discrete version of topics in analysis