Existence and asymptotic behavior results of positive periodic solutions for discretetime logistic model
 Bo Du^{1, 2} and
 Shouli Zhu^{1}Email author
https://doi.org/10.1186/s1366201505272
© Du and Zhu 2015
Received: 2 February 2015
Accepted: 3 June 2015
Published: 17 June 2015
Abstract
A discretetime logistic model with delay is studied. The existence of a positive periodic solution for a discretetime logistic model is obtained by a continuation theorem of coincidence degree theory, and a sufficient condition is given to guarantee the global exponential stability of a periodic solution. Finally, an example is given to show the effectiveness of the results in this paper.
Keywords
1 Introduction
Examples of the discrete phenomena in nature abound and somehow the continuous version have commandeered all our attention  perhaps owing to that special mechanism in human nature that permits us to notice only what we have been conditioned to. The theory of difference equations has grown at an accelerated rate in the past decades; see [1–3].
For a complicated dynamical system, we note that discretetime neural networks have been studied by many authors; see e.g. Hu and Wang [9], Wang and Xu [10], Xiong and Cao [11], Yuan et al. [12], Zhao and Wang [13] and Zou and Zhou [14] for DNNs without time delays and Chen et al. [15], Liang et al. [16], Liang et al. [17] and Xiang et al. [18] for DNNs with discrete time delays. For more related results, see [19–28].
So far, to the best of the authors knowledge, there are few results for the existence and stability of positive periodic solutions to (1.1). The major challenges are as follows: (1) In order to obtain existence of positive periodic solutions, we must change (1.1) to the proper form by a variable transformation. How can we choose the above variable transformation, which is the key to the study of (1.1)? (2) Since it is very difficult to construct a Lypunov functional to (1.1), how can we choose a proper special function for obtaining the stability results, which is significant to our proof? (3) It is nontrivial to establish a unified framework.
It is, therefore, the main purpose of this paper to make the first attempt to handle the listed challenges.
Remark 1.1
The following sections are organized as follows: In Section 2, the existence of positive periodic solution to (1.1) is obtained. In Section 3, sufficient conditions are established for the global exponential stability of (1.1). In Section 4, an example is given to show the feasibility of our results.
2 Existence of positive periodic solution
Let X and Y be real Banach spaces and let \(L:D(L)\subset X\rightarrow Y\) be a Fredholm operator with index zero, here \(D(L)\) denotes the domain of L. This means that ImL is closed in Y and \(\operatorname{dim}\operatorname{Ker}L=\operatorname{codim}\operatorname{Im}L<+\infty\). If L is a Fredholm operator with index zero, then there exist continuous projectors \(P:X\rightarrow X\), \(Q:Y\rightarrow Y\) such that \(\operatorname{Im}P=\operatorname{Ker}L\), \(\operatorname{Im}L=\operatorname{Ker}Q=\operatorname{Im}(IQ)\). It follows that \(L_{D(L)\cap \operatorname{Ker}P}:(IP)X\rightarrow \operatorname{Im}L\) is invertible. Denote by \(K_{p}\) the inverse of \(L_{P}\).
Let Ω be an open bounded subset of X, a map \(N :\bar{\Omega}\rightarrow Y\) is said to be Lcompact in \(\bar{\Omega}\) if \(QN(\bar{\Omega})\) is bounded and the operator \(K_{p}(IQ)N(\bar{\Omega})\) is relatively compact. Because ImQ is isomorphic to KerL, there exists an isomorphism \(J:\operatorname{Im}Q\rightarrow \operatorname{Ker}L\). We first recall the famous Mawhin continuation theorem.
Lemma 2.1
[33]
 (1)
\(Lx\neq\lambda Nx\), \(\forall x\in\partial\Omega\cap D(L)\), \(\forall\lambda\in(0,1)\),
 (2)
\(Nx\notin \operatorname{Im}L\), \(\forall x\in\partial\Omega\cap \operatorname{Ker}L\),
 (3)
\(\operatorname{deg}\{JQN,\Omega\cap \operatorname{Ker}L,0\}\neq0\),
Lemma 2.2
[34]
Now, we state the main results and give its proof.
Theorem 2.1
 (H_{1}):

there exists a constant \(C>0\) such that if \(x(n)\) is a Nperiodic sequences and satisfiesthen we have$$\sum_{n=0}^{N1} \bigl[\ln\bigl(1+ \beta(n)e^{y(n\tau)}\bigr)+\ln\alpha(n) \bigr]=0, $$$$\sum_{n=0}^{N1}\bigl\vert \ln\bigl(1+ \beta(n)e^{y(n\tau)}\bigr)+\ln\alpha(n)\bigr\vert \leq C; $$
 (H_{2}):

there exists a constant \(D>0\) such that when \(y>D\),and$$\ln\bigl(1+\beta(n)e^{y(n)}\bigr)>\ln\alpha(n) $$uniformly hold for \(n\in Z\).$$\ln\bigl(1+\beta(n)e^{y(n)}\bigr)< \ln\alpha(n) $$
Proof
From Lemma 2.1, we know that \(Lx(n)=Nx(n)\) has at least one periodic solution in \(\bar{\Omega}\). That is, (1.1) has at least one positive Nperiodic solution. The proof is completed. □
Corollary 2.1
 (i)
\(y\mathcal{F}(n,y)>0\) for \(y>D_{1}\), \(n\in I_{N}\),
 (ii)one of the following two conditions holds:
 (a)
\(y\mathcal{F}(n,y)\leq D_{2}\) for \(y\geq D_{1}\), \(n\in I_{N}\),
 (b)
\(y\mathcal{F}(n,y)\geq D_{2}\) for \(y\leqD_{1}\), \(n\in I_{N}\).
 (a)
3 Global exponential stability of periodic solution
In this section, we establish some results for exponential stability of the Nperiodic solution of (1.1).
Definition 3.1
Theorem 3.1
 (i)where \(\alpha^{+}=\max\{\alpha(n),n\in Z\}\), \(\beta^{+}=\max\{\beta(n),n\in Z\}\), \(\lambda_{0}>1\) with$$\alpha^{+}+\lambda_{0}^{\tau+1}L\beta^{+}< 1, $$$$\lambda_{0}\bigl(\alpha^{+}+\lambda_{0}^{\tau+1}L\beta^{+} \bigr)< 1. $$
 (ii)
If \(f(x,y)=xy\), \(x,y\in R\), then \(f(x,y)f(x^{*},y*)\leq Lyy^{*}\), where L is a positive constant.
Proof
Remark 3.1
Because (3.2) contains the nonlinear term \(x^{*}(n+1)x^{*}(n\tau)x(n+1)x(n\tau)\), which results in great difficulty in obtaining exponential stability, we add condition (ii).
Remark 3.2
In general, the Lyapunov functional method is crucial for studying stability problems. In the present paper, due to the stronger nonlinearity of (1.1), the Lyapunov functional method is not valid. We overcome these difficulties by constructing a novel functional, which is different for the corresponding ones of past work.
4 Numerical simulations
In this paper, we discussed the existence and stability of positive periodic solutions for (1.1). First, the sufficient conditions that ensure the existence of a positive periodic solution were obtained by using the continuation theorem and some inequality techniques. Then a nonLyapunov method was used to establish the criteria for the global exponential stability of the periodic solution. Finally, a numerical example was presented to demonstrate the effectiveness of our theoretical results. The proposed criteria in this paper are easy to verify. The proposed analysis method is also easy to extend to the case of other differential equations.
Declarations
Acknowledgements
This paper is supported by the Postdoctoral Foundation of Jiangsu (1402113C) and the Postdoctoral Foundation of China (2014M561716).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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