Abstract
We establish several criteria for the existence of positive periodic solutions of the multi-parameter differential systems
where the functions \(g_{1}, g_{2}:[0,\infty)\to[0,\infty)\) are assumed to be unbounded. The analysis in the paper relies on the classical fixed point index theory. Our main findings improve and complement some existing results in the literature.
Similar content being viewed by others
1 Introduction
Let \(\omega>0\) be a constant. In this article we shall seek some criterion to guarantee that the multiparameter system
admits a positive ω-periodic solution, where the functions \(a_{i}, b_{i}, \tau_{i}, \zeta_{i}\in C(\mathbb{R},\mathbb{R})\) are ω-periodic, and \(g_{i}\in C([0,\infty),[0,\infty))\) are unbounded, \(i=1,2\). In addition, we assume that the nonlinear terms \(f, g\in C([0,\infty)\times[0,\infty),[0,\infty))\) and λ, μ are positive parameters.
Here a positive periodic solution of (1.1) means a solution \((u,v)\in E:=X^{2}\) of (1.1) satisfying \(u>0\), \(v>0\) on \([0,\omega]\), where
is a Banach space, and the norm of \(x\in X\) is
Moreover, for \((x,y)\in E\), we denote \(\|(x,y)\|=\|x\|+\|y\|\), and write \((x,y)\geq(0,0)\) if \((x,y)\in E\) fulfills \(x(t)\geq0\), \(y(t)\geq0\), \(t\in[0,\omega]\).
Obviously, the first equation of (1.1) reduces in some special circumstances to
and when \(\lambda=0\), \(g(u)\equiv1\), Eq. (1.2) becomes \(u'(t)=a(t)u(t)\), which is famous in Malthusian population dynamics. In recent decades, (1.2) has also been extensively applied to describe various physiological processes emerging in practical applications, for instance, the production of blood cells, respiration, cardiac arrhythmias, etc. One may refer to [1–6] and references therein. Nevertheless, the research work in the above mentioned papers is mainly dependent on the condition that \(g(u)\) is positive and bounded, that is, there are constants \(L>l>0\) such that \(0< l\leq g(u)\leq L\), \(u\in[0,\infty)\). Jin and Wang [7] have recently studied the spectral problem
and they obtained some existence results on positive periodic solutions by means of the fixed point theory. It is worth noting the function \(e^{u}\) is unbounded on \([0,\infty)\). Since then, Eq. (1.2) has been extensively investigated under the more general case that \(g(u)\) is unbounded on \([0,\infty)\), by applying the lower and upper solutions method, fixed point theory, and so on. See, for example, [7–10].
Besides, researchers have focused on the differential systems associated to (1.2), namely,
One can see [11–14] for some related results. However, in [11–13], the authors have only dealt with the special case \(g_{i}(u_{i})\equiv1\), \(i=1,2,\dots,n\). Indeed in that case, the Green’s function corresponding to \(u'_{i}(t)=a_{i}(t)u_{i}(t)\) is simple, and some suitable cones could be easily constructed. Furthermore, system (1.3) investigated in above papers includes only one positive parameter λ. Hence, it will be interesting to study the multiparameter systems (1.1) with \(g_{i}\) (\(i=1,2\)) being unbounded. On the other hand, what is worth mentioning is that Zhang et al. [14] considered system (1.1) for the special case \(g_{i}\equiv1\), \(i=1,2\), where nonlinearities \(f(u,v)\) and \(g(u,v)\) were assumed to be nondecreasing, and only the case \(f(0,0)>0\), \(g(0,0)>0\) was treated. Therefore, we want to know whether or not (1.1) has a positive periodic solution under more relaxed assumption \(f(0,0)=0\), \(g(0,0)=0\). In view of above reasons, we shall concentrate on the existence of positive periodic solutions for system (1.1) in the current paper, to further improve and generalize tho results in the literature. For this purpose, we assume
- (C1)
\(a_{i}, b_{i}, \tau_{i}, \zeta_{i}\in C(\mathbb{R},[0,\infty))\) are ω-periodic with \(\int_{0}^{\omega}a_{i}(t)\,dt>0\), \(\int_{0}^{\omega}b_{i}(t)\,dt>0\), \(i=1,2\).
- (C2)
There is \(l_{i}>0\) such that \(0< l_{i}\leq g_{i}(s)<\infty\), \(s\in [0,\infty)\).
- (C3)
\(f, g\in C([0,\infty)\times[0,\infty),[0,\infty))\) with \(f(u,v)>0, g(u,v)>0\) for \((u,v)\neq(0,0)\).
Remark 1.1
For other research work on periodic solutions of functional differential equations and systems, we refer the readers to [15–17] and references therein.
The remainder of the paper is arranged as follows. In Sect. 2, we introduce some preliminaries needed in our proof. Section 3 is devoted to stating and proving our main findings. Meanwhile, some related results and remarks will be given.
2 Preliminaries
Recall that \(E=X^{2}\) is the Banach space defined as in Sect. 1. We first give the following lemma.
Lemma 2.1
Assume (C1)–(C3). If \((u,v)\in E\)is a solution of (1.1), then
where
Proof
Multiplying the both sides of the first equation of (1.1) with \(e^{-\int_{0}^{t} a_{1}(s)g_{1}(u(s))\,ds}\), we can obtain
Integrating above equation from t to \(t+\omega\) and by elementary calculation, we can easily get
Similar evaluation shows
□
Let \(q>0\) be a fixed constant. Then we can establish a series of lemmas required in the subsequent discussion.
Lemma 2.2
Assume (C1)–(C3). Let \(\sigma_{i}=e^{-\int_{0}^{\omega}a_{i}(\theta)\,d\theta}\), \(i=1,2\). Then for any \((u,v)\in E\)satisfying \((u,v)\geq(0,0)\)and \(\|(u,v)\|\leq q\),
where
Proof
Clearly, for \((u,v)\in E\) with \((u,v)\geq(0,0)\) and \(\| (u,v)\|\leq q\), we have \(0\leq u\leq\|u\|\leq q\). Thus,
and then simple estimation shows (2.1) holds for \(i=1\). The case \(i=2\) is similar. □
Defining for \(i=1,2\),
Then it is not hard to verify \(\eta_{i}(q)\in(0,1)\), and accordingly,
Set
and for \(r>0\),
Then P and \(K_{q}\) are cones in E.
Lemma 2.3
Assume (C1)–(C3). Let \(0< r\leq q\). Then for any \((u,v)\in\bar{\varOmega}_{r}\),
Proof
Similar to the proof of Lemma 2.2, we obtain for \(t\leq s\leq t+\omega\),
Moreover, since \(\varphi(t):=\frac{\sigma_{i}^{t}}{1-\sigma_{i}^{t}}\) and \(\psi (t):=\frac{1}{1-\sigma_{i}^{t}}\) are strictly decreasing on \([0,\infty)\), one can easily see that (2.2) holds true. □
Define, for given \((u,v)\in E\),
where
and
Then we have
Lemma 2.4
Assume (C1)–(C3) and \(0< r\leq q\). Then \(T_{\lambda ,\mu}(\bar{\varOmega}_{r})\subseteq K_{q}\)and \(T_{\lambda,\mu}:\bar{\varOmega }_{r}\to K_{q}\)is completely continuous.
Proof
For \((u,v)\in\bar{\varOmega}_{r}\), we can deduce from Lemma 2.3 that
which yields
Meanwhile, (2.2) implies
In an analogous manner, we get
Hence \(T_{\lambda,\mu}(\bar{\varOmega}_{r})\subseteq K_{q}\). The completely continuity of \(T_{\lambda,\mu}\) is obvious. □
It is obvious that if \((u,v)\) is a fixed point of the completely continuous operator \(T_{\lambda,\mu}\) in \(K_{q}\), then \((u,v)\) is a positive periodic solution of (1.1). We conclude this section by giving the main tool employed in proving our main results.
Lemma 2.5
AssumeEis a Banach space and \(K\subseteq E\)is a cone. For \(r>0\), let \(K_{r}=\{u\in K: \|u\|< r\}\)and \(\partial K_{r}=\{u\in K: \|u\|=r\}\). Suppose \(T: \bar{K}_{r}\to K\)is a completely continuous operator satisfying \(Tu\neq u\), \(u\in\partial K_{r}\). Then
- (i)
If \(\|Tu\|<\|u\|\), \(u\in\partial K_{r}\), then \(i(T, \bar{K}_{r}, K)=1\);
- (ii)
If \(\|Tu\|>\|u\|\), \(u\in\partial K_{r}\), then \(i(T, \bar{K}_{r}, K)=0\).
3 Main results
Let
Theorem 3.1
Assume (C1)–(C3) hold and \(f_{0}=0=g_{0}\). Then for every \(q>0\), there is a constant \(\gamma_{q}>0\)such that for all \(\lambda , \mu>\gamma_{q}\), system (1.1) admits a positive periodic solution \((u,v)\)satisfying \(\|(u,v)\|\leq q\).
Proof
Choose \(r_{1}=q\) and define
Take
By Lemma 2.4, we know \(T_{\lambda,\mu}(\bar{\varOmega}_{q})\subseteq K_{q}\) and \(T_{\lambda,\mu}:\bar{\varOmega}_{q}\to K_{q}\) is completely continuous. Fix \(\lambda, \mu>\gamma_{q}\). Then for \((u,v)\in\partial\varOmega_{q}\), we have \(\eta(q)q\leq u+v\leq q\), and so
which implies
Similarly,
Hence \(\|T_{\lambda,\mu}(u,v)\|>\|(u,v)\|\) on \(\partial\varOmega_{q}\), and then Lemma 2.5 gives \(i(T_{\lambda,\mu}, \bar{\varOmega}_{q}, K_{q})=0\).
On the other hand, since \(f_{0}=g_{0}=0\), there exists a constant \(r_{2}\) with \(0< r_{2}< q\), such that for \((u,v)\) satisfying \(0< u+v\leq r_{2}\),
where \(\varepsilon>0\) is a constant satisfying
For \((u,v)\in\partial\varOmega_{r_{2}}\), we can deduce by (2.2) and (3.1) that
and hence
In an analogous way, we get
Thus \(\|T_{\lambda,\mu}(u,v)\|<\|(u,v)\|\) on \(\partial\varOmega_{r_{2}}\). Lemma 2.5 ensures \(i(T_{\lambda,\mu}, \bar{\varOmega}_{r_{2}}, K_{q})=1\).
Consequently, \(i(T_{\lambda,\mu}, \bar{\varOmega}_{q}\setminus\varOmega _{r_{2}}, K_{q})=-1\). Therefore, \(T_{\lambda,\mu}\) possesses a fixed point \((u,v)\) in \(\bar{\varOmega}_{q}\setminus\varOmega_{r_{2}}\), and system (1.1) has a positive periodic solution \((u,v)\) with \(\| (u,v)\|\leq q\). □
Theorem 3.2
Assume (C1)–(C3) hold and \(f_{0}=\infty\). Then for every \(q>0\), there is a constant \(\gamma_{q}>0\)such that for all \(\lambda , \mu<\gamma_{q}\), system (1.1) admits a positive periodic solution \((u,v)\)satisfying \(\|(u,v)\|\leq q\).
Proof
Fix \(r_{1}=q\) and set
Define
By Lemma 2.4, \(T_{\lambda,\mu}(\bar{\varOmega}_{q})\subseteq K_{q}\) and \(T_{\lambda,\mu}:\bar{\varOmega}_{q}\to K_{q}\) is completely continuous. Thus, for fixed \(\lambda, \mu<\gamma_{q}\) and \((u,v)\in\partial\varOmega_{q}\),
and then
By a similar argument, we can also obtain
Therefore, \(\|T_{\lambda,\mu}(u,v)\|<\|(u,v)\|\) for \((u,v)\in\partial \varOmega_{q}\). Using Lemma 2.5 again, we can easily get \(i(T_{\lambda,\mu}, \bar{\varOmega}_{q}, K_{q})=1\).
By the assumption \(f_{0}=\infty\), there exists a constant \(r_{2}\in(0,q)\), such that for \((u,v)\) satisfying \(0< u+v\leq r_{2}\),
where \(\varUpsilon>0\) satisfies
Thus for \((u,v)\in\partial\varOmega_{r_{2}}\), we get by (2.2) and (3.2) that
which means \(\|A_{\lambda}(u,v)\|>\|(u,v)\|\) on \(\partial\varOmega_{r_{2}}\). Hence
and Lemma 2.5 again implies \(i(T_{\lambda,\mu}, \bar{\varOmega}_{r_{2}}, K_{q})=0\).
Consequently, \(i(T_{\lambda,\mu}, \bar{\varOmega}_{q}\setminus\varOmega _{r_{2}}, K_{q})=1\). Thus, \(T_{\lambda,\mu}\) has a fixed point \((u,v)\) in \(\bar{\varOmega}_{q}\setminus\varOmega_{r_{2}}\), and (1.1) has a positive periodic solution \((u,v)\) with \(\|(u,v)\|\leq q\). □
Similarly to Theorems 3.1 and 3.2, we can prove the following
Theorem 3.3
Assume (C1)–(C3) and \(g_{0}=\infty\). Then for every \(q>0\), there is a constant \(\gamma_{q}>0\)such that for all \(\lambda , \mu<\gamma_{q}\), system (1.1) admits a positive periodic solution \((u,v)\)satisfying \(\|(u,v)\|\leq q\).
Remark 3.1
Clearly, the results of Theorems 3.1–3.3 generalize and complement the corresponding ones in [7, 9, 12–14].
To illustrate our main findings, we may choose \(\omega=2\pi\) and \(\tau _{i}\equiv0\), \(\zeta_{i}\equiv0\) (\(i=1,2\)) in the subsequent discussion. Let
Then it is not hard to check that (C1) is satisfied. Moreover, define
then there are constants \(l_{1}=1\) and \(l_{2}=2\) such that
Hence (C2) is also satisfied.
Example 3.1
For \((u,v)\in[0,\infty)\times[0,\infty)\), let
Then \(f, g\in C([0,\infty)\times[0,\infty),[0,\infty))\) with \(f(u,v)>0\), \(g(u,v)>0\) for \((u,v)\neq(0,0)\). Thus (C3) holds true. Furthermore, simple calculation gives \(f_{0}=0=g_{0}\). Consequently, the results of Theorem 3.1 are valid.
Example 3.2
We shall follow the same notations and definitions as before. Let us redefine
Clearly, f verifies (C3). Moreover, it is not difficult to see \(f_{0}=\infty\), and accordingly the results of Theorem 3.2 are also valid.
At the end of the section, we list some related results and remarks.
Let us consider the multiparameter differential systems
where \(\lambda, \mu>0\) are parameters. Under the same assumptions as before, one can check that system (3.3) is equivalent to
where
Furthermore, by a similar argument as above, it is not difficult to see that the results of Theorems 3.1–3.3 remain true for system (3.3).
Remark 3.2
It is worth remarking that, under some reasonable assumptions, the results of the paper are still valid for the more general coupled systems
and
References
Chow, S.N.: Existence of periodic solutions of autonomous functional differential equations. J. Differ. Equ. 15, 350–378 (1974)
Wazewska-Czyzewska, M., Lasota, A.: Mathematical problems of the dynamics of a system of red blood cells. Mat. Stosow. 6, 23–40 (1976)
Gurney, W.S., Blythe, S.P., Nisbet, R.N.: Nicholson’s blowflies revisited. Nature 287, 17–21 (1980)
Freedman, H.I., Wu, J.: Periodic solutions of single-species models with periodic delay. SIAM J. Math. Anal. 23, 689–701 (1992)
Kuang, Y.: Delay Differential Equations with Applications in Population Dynamics. Academic Press, New York (1993)
Mackey, M.C., Glass, L.: Oscillations and chaos in physiological control systems. Science 197, 287–289 (1997)
Jin, Z.L., Wang, H.Y.: A note on positive periodic solutions of delayed differential equations. Appl. Math. Lett. 23, 581–584 (2010)
Graef, J., Kong, L.J.: Existence of multiple periodic solutions for first order functional differential equations. Math. Comput. Model. 54, 2962–2968 (2011)
Ma, R.Y., Chen, R.P., Chen, T.L.: Existence of positive periodic solutions of nonlinear first-order delayed differential equations. J. Math. Anal. Appl. 384, 527–535 (2011)
Ma, R.Y., Lu, Y.Q.: One-signed periodic solutions of first-order functional differential equations with a parameter. Abstr. Appl. Anal. 2011, Article ID 843292 (2011)
Wang, H.Y.: Positive periodic solutions of functional differential systems. J. Differ. Equ. 202, 354–366 (2004)
Wang, H.Y.: Positive periodic solutions of singular systems of first order ordinary differential equations. Appl. Math. Comput. 218, 1605–1610 (2011)
Chen, R.P., Ma, R.Y., He, Z.Q.: Positive periodic solutions of first-order singular systems. Appl. Math. Comput. 218, 11421–11428 (2012)
Zhang, G., Cheng, S.S.: Positive periodic solution of coupled delay differential systems depending on two parameters. Taiwan. J. Math. 4, 639–652 (2004)
Kiguradze, I., Puza, B.: On boundary value problems for systems of linear functional differential equations. Czechoslov. Math. J. 47, 341–373 (1997)
Kiguradze, I., Puza, B.: Boundary Value Problems for Systems of Linear Functional Differential Equations. Folia, Masaryk University, Brno (2003)
Domoshnitsky, A., Hakl, R., Sremr, J.: Component-wise positivity of solutions to periodic boundary value problem for linear functional differential systems. J. Inequal. Appl. 2012, Article ID 112 (2012). https://doi.org/10.1186/1029-242X-2012-112
Deimling, K.: Nonlinear Functional Analysis. Springer, Berlin (1985)
Guo, D.J., Lakshmikantham, V.: Nonlinear Problems in Abstract Cones. Academic Press, New York (1988)
Acknowledgements
Not applicable.
Availability of data and materials
Not applicable.
Funding
The first author is supported by National Natural Science Foundation of China (Grant No. 61761002), First-Class Disciplines Foundation of Ningxia (No. NXYLXK2017B09), and the Key Project of North Minzu University (No. ZDZX201804).
Author information
Authors and Affiliations
Contributions
RC analyzed and proved the main results, and was a major contributor in writing the manuscript. XL checked the English grammar and typing errors in the full text. All authors read and approved the final manuscript.
Corresponding author
Ethics declarations
Competing interests
The authors declare that they have no competing interests.
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Chen, R., Li, X. Positive periodic solutions for multiparameter nonlinear differential systems with delays. J Inequal Appl 2020, 24 (2020). https://doi.org/10.1186/s13660-020-2294-1
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13660-020-2294-1