The existence of two positive periodic solutions for the delay differential neoclassical growth model
- Zijun Ning^{1} and
- Wentao Wang^{1}Email author
https://doi.org/10.1186/s13662-016-0995-z
© Ning and Wang 2016
Received: 15 August 2016
Accepted: 11 October 2016
Published: 21 October 2016
Abstract
By using the Krasnoselskii fixed point theorem in a cone, we investigate the existence of two positive periodic solutions of the generalized delay differential neoclassical growth model with periodic coefficients and delays. Moreover, we give an example to demonstrate the theoretical result.
Keywords
neoclassical growth model positive periodic solution delay Krasnoselskii’s fixed point theorem1 Introduction
Let \(g(x)=x^{\gamma}e^{-\delta_{M} x}\). Then it is obvious that \(g(x)\) increases strictly on \([0, \frac{\gamma}{\delta_{M}}]\) and decreases strictly on \([\frac{\gamma}{\delta_{M}},+\infty]\). Thus, there exists unique \(r_{0}\in(\frac{\gamma}{\delta_{M}},+\infty )\) such that \(g(r_{0})=g(\sigma\frac{\gamma}{\delta_{M}})\).
- (S)
\(A\omega\beta_{m} g(r_{0})>r_{0}\).
Definition 1.1
- (i)
\(ax+by\in K\) for all \(x,y\in K\) and \(a,b>0\);
- (ii)
\(x, -x\in K\) implies \(x=0\).
The Krasnoselskii fixed point theorem in a cone (see [26]) is as follows.
Lemma 1.1
- (i)
\(\Vert \Psi x\Vert \geq \Vert x\Vert , \forall x\in K\cap\partial\Omega_{1}\) and \(\Vert \Psi x\Vert \leq \Vert x\Vert , \forall x\in K\cap\partial \Omega_{2}\);
- (ii)
\(\Vert \Psi x\Vert \geq \Vert x\Vert , \forall x\in K\cap\partial\Omega_{2}\) and \(\Vert \Psi x\Vert \leq \Vert x\Vert , \forall x\in K\cap\partial \Omega_{1}\).
Then Ψ has a fixed point in \(K\cap(\overline{\Omega }_{2}\setminus\Omega_{1})\).
2 Existence of two positive periodic solutions
In this section, we establish some sufficient conditions on the existence of two positive periodic solutions of model (1.3).
Lemma 2.1
\(\Psi(K)\subset K\).
Proof
Theorem 2.1
Suppose that \((S)\) holds. Then (1.3) has at least two ω-positive periodic solutions.
Proof
Clearly, \(\overline{\Omega}_{1}\subset\Omega_{2}\), \(\overline{\Omega }_{2}\subset\Omega_{3}\), and \(\overline{\Omega}_{3}\subset\Omega_{4}\). Since \(\Psi(K)\subset K\) and Ψ is a completely continuous operator on X, it follows from Lemma 1.1 that Ψ has one fixed point \(x_{1}\in K\cap(\overline{\Omega }_{2}\setminus\Omega_{1})\) and another fixed point \(x_{2}\in K\cap(\overline{\Omega}_{4}\setminus \Omega_{3})\), which are obviously different. Moreover, \(x_{1}(t)\geq \sigma r_{1}>0\) and \(x_{2}(t)\geq\sigma r_{0}>0\). Therefore, \(x_{1}\) and \(x_{2}\) are two positive periodic solutions of (1.3). The proof of Theorem 2.1 is completed. □
Remark 2.1
It is worth mentioning that although the authors of [27] considered the existence of two positive periodic solutions for equation (1.3) with \(\gamma=2\), it is neglected that the two positive periodic solutions might merge into one when they are on the boundary of \(\Omega_{2}\). Here, we have proved that there exist two distinct positive periodic solutions for equation (1.3), which generalizes and improves the result of [27].
3 An example
In this section, we give an example to support the results obtained in the previous section.
Example 3.1
Remark 3.1
To the best of our knowledge, few authors have considered the existence of two positive periodic solutions for the generalized delay differential neoclassical growth model (1.3). It is clear that all the results in [1, 2, 10, 11, 24] and the references therein cannot be applicable to prove that there exist two 2π-positive periodic solutions for model (3.1). So the results of this paper are essentially new and complement some existing ones in [1, 2, 10, 11, 24].
4 Conclusions
In this paper, we study the generalized delay differential neoclassical growth model with periodic coefficients and delays. By using the Krasnoselskii fixed point theorem, we have derived conditions on the existence of two positive periodic solutions, which gives a satisfying answer to the open problem mentioned in the Introduction. In the future, we will consider the stability of the two positive periodic solutions for model (1.3), which is an interesting and challenging work.
Declarations
Acknowledgements
This work was supported by the Natural Scientific Research Fund of Zhejiang Province of China (Grant No. LY16A010018) and the National Natural Science Foundation of P.R. China (Grant No. 11301341).
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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