## 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
Full Text:

### References:

 [1] Lakshmikantham, V.; Leela, S., Existence and monotone method for periodic solutions of first-order differential equations, J. Math. Anal. Appl., 91, 237-243 (1983) · Zbl 0525.34031 [2] Lakshmikantham, V.; Leela, S., Remarks on first and second order periodic boundary value problems, Nonlinear Anal., 8, 281-287 (1984) · Zbl 0532.34029 [3] Lakshmikantham, V., Periodic boundary value problems of first and second order differential equations, J. Appl. Math. Simulat., 2, 131-138 (1989) · Zbl 0712.34058 [4] Leela, S.; Oguztoreli, M. N., Periodic boundary value problem for differential equations with delay and monotone iterative method, J. Math. Anal. Appl., 122, 301-307 (1987) · Zbl 0616.34062 [5] Haddock, J. R.; Nkashama, M. N., Periodic boundary value problems and monotone iterative methods for functional differential equations, Nonlinear Anal., 22, 267-276 (1994) · Zbl 0804.34062 [6] Liz, E.; Nieto, J. J., Periodic boundary value problems for a class of functional differential equations, J. Math. Anal. Appl., 200, 680-686 (1996) · Zbl 0855.34080 [7] Peng, S., Positive solutions for first order periodic boundary value problem, Appl. Math. Comput., 158, 345-351 (2004) · Zbl 1082.34510 [8] Guo, D.; Lakshmikantham, V., Nonlinear Problems in Abstract Cones (1988), Academic Press: Academic Press New York · Zbl 0661.47045
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.