On positive periodic solutions of second-order difference equations with attractive-repulsive singularities
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RESEARCH
Open Access
On positive periodic solutions of second-order difference equations with attractive-repulsive singularities Yanqiong Lu* and Ruyun Ma *
Correspondence: [email protected] Department of Mathematics, Northwest Normal University, Lanzhou, 730070, P.R. China
Abstract In this paper, sufficient conditions for the existence of positive periodic solutions of the second-order difference equation
2 u(t – 1) =
h(t) g(t) – λ + f (t), μ u (t) u (t)
t ∈ Z,
are established, where g, h : Z → [0, ∞) and f : Z → R are T-periodic functions,
λ, μ > 0.
MSC: 34B15 Keywords: positive periodic solutions; difference equations; singular equation
1 Introduction The theory of nonlinear difference equations has been widely used to study discrete models appearing in many fields such as computer science, economics, neural network, ecology, and cybernetics; see, for example, []. In recent years, there have been many papers to study the existence of positive periodic solutions for second-order difference equations. By using various methods and techniques, for example, fixed point theorems, the method of upper and lower solutions, coincidence degree theory, and critical point theory, a series of existence results of periodic solutions have been obtained; we refer the reader to [–] and references therein. However, there are few techniques for studying the existence of positive solutions of difference equations with singularity, and thus, the results in the field are very rare; see [–]. At the same time, we also find that difference equations are closely related to differential equations in the sense that (i) a differential equation model is usually derived from a difference equation, and (ii) numerical solutions of a differential equation have to be obtained by discretizing the differential equation (thus resulting in difference equations). Therefore, it is worthwhile to explore this topic. Let Z denote the integer set for a, b ∈ Z with a < b, [a, b]Z := {a, a + , . . . , b}. In this paper, we are concerned with the existence of positive periodic solutions of the second-order difference equation u(t – ) =
h(t) g(t) – + f (t), uμ (t) uλ (t)
t ∈ Z,
(.)
© 2012 Lu and Ma; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Lu and Ma Advances in Difference Equations 2012, 2012:186 http://www.advancesindifferenceequations.com/content/2012/1/186
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where g, h : Z → [, ∞) and f : Z → R are T-periodic functions, λ, μ > . By a solution to (.) we understand a function u ∈ E := {u : Z → R|u(t) = u(t + T)} satisfying (.). Special cases of Eq. (.) are u(t – ) =
g(t) h(t) – , uμ (t) uλ (t)
t ∈ Z,
(.)
h(t) + f (t), uλ (t)
t ∈ Z,
(.)
u(t – ) = – u(t – ) =
g(t) + f (t), uμ (t)
t ∈ Z.
(.)
In the related literature, it is said that (.) has an attractive si
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