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

Sun, Jian-Ping

Appl. Math. Lett. 19, No. 11, 1244-1248 (2006).

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
\[ f^{0}=\lim_{x\rightarrow 0^{+}}\max_{k\in [ 0,T]}\frac{f(k,x)}{x}, \]
\[ f^{\infty}=\lim_{x\rightarrow +\infty }\max_{k\in [ 0,T]}\frac{f(k,x)}{x}, \]
\[ f_{0}=\lim_{x\rightarrow 0^{+}}\min_{k\in [ 0,T]}\frac{f(k,x)}{x}, \]
\[ f_{\infty}=\lim_{x\rightarrow +\infty }\min_{k\in [ 0,T]}\frac{f(k,x)}{x}. \]

Reviewer: Fozi Dannan (Damascus)

MSC:

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

Keywords:

discrete periodic boundary value problem; positive solution; fixed point; cone

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