## 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 $$\rho =\min_{n\in [ 0,T-1]}\left| a(n)\right|.$$

### MSC:

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

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