## Positive solution for first-order discrete periodic boundary value problem.(English)Zbl 1180.39023

Consider the following first order discrete periodic boundary value problem (PBVP)
$\Delta x(k)+f(k,x(k+1))=0, \quad x(0)=x(T+1), \;k\in [ 0,T],\tag{$$*$$}$
where $$T$$ is a fixed positive integer and $$f:[0,T]\times [ 0,\infty )\rightarrow \mathbb{R}$$ is continuous and there exists a number $$M>1$$ such that
$(M-1)x-f(k,x)\geq 0 \quad\text{for }x\in [ 0,\infty ),\;k\in [ 0,T].$
The main result of the article is the following: Theorem. PBVP ($$*$$) has at least one solution if one of the following conditions is satisfied: (i) $$f_{0}>0$$, and $$f^{\infty }<0$$; or (ii) $$f_{\infty }>0$$, and $$f^{0}<0$$, where
$f^{0}=\lim_{x\rightarrow 0^{+}}\max_{k\in [ 0,T]}\frac{f(k,x)}{x},$
$f^{\infty}=\lim_{x\rightarrow +\infty }\max_{k\in [ 0,T]}\frac{f(k,x)}{x},$
$f_{0}=\lim_{x\rightarrow 0^{+}}\min_{k\in [ 0,T]}\frac{f(k,x)}{x},$
$f_{\infty}=\lim_{x\rightarrow +\infty }\min_{k\in [ 0,T]}\frac{f(k,x)}{x}.$

### MSC:

 39A23 Periodic solutions of difference equations 39A12 Discrete version of topics in analysis 34B15 Nonlinear boundary value problems for ordinary differential equations
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### References:

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