Periodicity in general delay nonlinear difference equations using fixed point theory. (English) Zbl 1068.39021

The author considers the system of non-linear functional-difference equations with delay \[ \Delta x(n)=F(n,x_{n}),\quad n\in \mathbb{Z} , \tag{\(*\)} \] where \(F:\mathbb{Z}\times BC\rightarrow \mathbb{R}^{k}\) is continuous in \(x\) and \(T\)-periodic in \(n\), \(BC\) is the space of bounded sequences \(\phi :(-\infty ,0]\rightarrow \mathbb{R}^{k}\) with the maximum norm \(\left\| .\right\|\), and \(x_{n}\) means that \(x_{n}(s)=x(n+s)\) for \(s\leq 0\). Using Schaefer’s fixed point theorem, the author shows that if there is an {a priori} bound on all possible \(T\)-periodic solution of \((\ast)\), then there is a \(T\)-periodic solution. The {a priori} bound is established by different methods including non-negative Lyapunov functionals.
In fact, the considered system \((\ast)\) covers a wide range of difference equations and systems as it is illustrated by several examples that are considered. One of those is the following example: The scalar delay difference equation \(\Delta x(n)=a(n)x(n)+b(n)x(n-h)+p(n), n\in \mathbb{Z}\), where \(a(n) , b(n)\) and \(p(n)\) are \(T\)-periodic sequences , and \(h\in \mathbb{Z}\) with \(h\geq 0\) has a \(T\)-periodic solution if the following conditions hold: (i) \(a(n) >0\) or \(a(n)<0\) for all \(n\in Z.\) (ii) There exists a constant \(N>1\) such that \(\left| a(n)\right| -N\left| b(n+h)\right| \geq 0.\) (iii) \(\rho -\left\| b\right\| -\rho T(\left\| a\right\| +\left\| b\right\| )>0\) where \(\rho =\min_{n\in [ 0,T-1]}\left| a(n)\right|.\)


39A11 Stability of difference equations (MSC2000)
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