Almost periodic solution of an impulsive multispecies logarithmic population model
- Li Wang^{1}Email author and
- Hui Zhang^{1}
https://doi.org/10.1186/s13662-015-0393-y
© Wang and Zhang; licensee Springer. 2015
Received: 9 October 2014
Accepted: 29 January 2015
Published: 25 March 2015
Abstract
In this paper, some easily verifiable conditions are derived for the existence of almost periodic solution of an impulsive multispecies logarithmic population model in terms of the continuous theorem of coincidence degree theory, which is rarely applied to studying the existence of almost periodic solution of an impulsive differential equation. Our results generalize previous results by Alzabut, Stamov and Sermutlu. Besides, our technique used in this paper can be applied to study the existence of almost periodic solution of an impulsive differential equation with linear impulsive perturbations.
Keywords
1 Introduction
By utilizing the continuous theorem of coincidence degree theory, we obtain some sufficient conditions for the existence of almost periodic solution of Eq. (1.1). Our results generalize previous results obtained in [6] and are easier to verify than the conditions obtained in paper [15].
The continuous theorem of coincidence degree theory has been extensively used to study the existence of periodic solution of a differential equation, regardless of the equation being with impulse or without. Using this theorem, papers [6] and [16] investigate the existence of almost periodic solution of a population model without impulse. To our best knowledge, the continuous theorem has not been used to prove the existence of almost periodic solution of an impulsive multispecies logarithmic population model. Besides, our technique used in this paper can be applied to study the existence of almost periodic solution of an impulsive differential equation with linear impulsive perturbations.
The remaining part of this paper is organized as follows. We present some preliminaries in the next section. In Section 3, by employing the continuous theorem of coincidence degree theory, we establish a criterion for the existence of almost periodic solution of system (1.1).
2 Preliminaries
In this section, some lemmas and definitions, which are of importance in proving our main result in Section 3, will be presented.
Definition 2.1
([17])
\(\varphi(\cdot)\in C(R, R^{n})\) is said to be almost periodic in sense of Bohr if \(\forall\varepsilon>0\), there exists a relatively dense set \(T(\varphi,\varepsilon)\) such that if \(\tau\in T(\varphi,\varepsilon)\), then \(|\varphi(t)-\varphi(t+\tau)|<\varepsilon\) for all \(t\in R\). Denote by \(\operatorname{ap}(R, R^{n})\) all such functions.
Definition 2.2
([3])
- (1)
\(\{t_{k}\}\) is equipotentially almost periodic, that is, \(\forall\varepsilon>0\), there exists a relatively dense set of ε-almost periodic common for any sequences \(\{t^{j}_{k}\}\), \(t^{j}_{k}=t_{k+j}-t_{k}\);
- (2)
\(\forall\varepsilon>0\), \(\exists\delta>0\) such that if the points \(t'\), \(t''\) belong to the same interval of continuity and \(|t'-t''|<\delta\), then \(|\varphi(t')-\varphi(t'')|<\varepsilon\);
- (3)
\(\forall\varepsilon>0\), there exists a relatively dense set \(T(\varphi,\varepsilon)\) such that if \(\tau\in T(\varphi,\varepsilon)\), then \(|\varphi(t)-\varphi(t+\tau)|<\varepsilon\) for all \(t\in R\) which satisfy the condition \(|t-t_{i}|>\varepsilon\), \(i=0,\pm1,\pm2,\ldots\) .
- (A1)
\(\prod_{0< t_{k}<t}(1+d_{jk})\), \(\prod_{0< t_{k}<t-\tau_{ij}(t)}(1+d_{jk})\) are positive almost periodic functions, \(\inf_{t\in R}\prod_{0< t_{k}<t}(1+d_{jk})>0\), \(j=1,\ldots, n\);
- (A2)
\(\int_{-\infty}^{t}k_{ij}(t-s)\ln\prod_{0< t_{k}<s}(1+d_{jk})\,ds\) are almost periodic functions, \(i, j=1,\ldots,n\).
Remark
There exist a great deal of functions satisfying assumptions (A1) and (A2). For instance, let \(\{t_{k}\}\) be an arbitrary equipotentially almost periodic sequence, \(\tau_{ij}(t)=\tau>0\), \(d_{j,k}=d_{j,k+T}\), \((1+d_{j,k})>0\), \(k=1,2,\ldots\) and \((1+d_{j1})(1+d_{j2})\ldots(1+d_{jT})=1\), then \(\prod_{0< t_{k}<t}(1+d_{jk})\) is a positive almost periodic function with discontinuous points \(t_{k}\), and \(\inf_{t\in R}\prod_{0< t_{k}<t}(1+d_{jk})>0\). \(\prod_{0< t_{k}<t-\tau_{ij}(t)}(1+d_{jk})\) is also a positive almost periodic function with discontinuous points \(t_{k}+\tau\), \(i,j=1,\ldots,n\).
Lemma 2.3
- (1)
If \(N_{i}(t)\in\mathcal{AP}(R, R)\) is a positive solution of Eq. (1.1), then \(y_{i}(t)=\prod_{0< t_{k}<t}(1+d_{ik})^{-1}N_{i}(t)\) is a positive ap solution of Eq. (2.1), \(i=1,2,\ldots, n\).
- (2)
If \(y_{i}(t)\in\operatorname{ap}(R, R)\) is a positive solution of Eq. (2.1), then \(N_{i}(t)=\prod_{0< t_{k}<t}(1+d_{ik})y_{i}(t)\) is a positive \(\mathcal{AP}\) solution of Eq. (1.1), \(i=1,2,\ldots,n\).
Proof
Since \(\prod_{0< t_{k}<t}(1+d_{ik})\in\mathcal{AP} (R, R)\) and \(\inf_{t\in R}\prod_{0< t_{k}<t}(1+d_{ik})>0\), \(i=1,\ldots, n\), from [3] we know \(\prod_{0< t_{k}<t}(1+d_{ik})^{-1}\in\mathcal{AP}(R, R)\). Therefore, if \(y_{i}(t)\in \operatorname{ap}(R, R) \), then \(N_{i}(t)=\prod_{0< t_{k}<t}(1+d_{ik})y_{i}(t) \in\mathcal{AP}(R, R)\); if \(N_{i}(t)\in \mathcal{AP}(R, R)\), combining \(N_{i}(t_{k}^{+})=(1+d_{ik})N_{i}(t_{k})\), then \(y_{i}(t)=\prod_{0< t_{k}<t}(1+d_{ik})^{-1}N_{i}(t)\in \operatorname{ap}(R, R)\). Similar as [18], the rest of the proof of Lemma 2.3 can be obtained easily, we omit it here. □
Obviously, if Eq. (2.2) has ap solution, then Eq. (2.1) has strictly positive ap solution. It follows from Lemma 2.3 that Eq. (1.1) has strictly positive \(\mathcal{AP}\) solution. Consequently, we mainly study the existence of ap solution of Eq. (2.2). To do so, we firstly summarize a few concepts.
Let X and Z be real Banach spaces, \(L:\operatorname{dom}L\subset X \rightarrow Z\) be a linear mapping, \(N: X \rightarrow Z\) be a continuous mapping. L is called a Fredholm mapping of index zero if \(\operatorname{dim} \operatorname{Ker}L=\operatorname{codim} \operatorname{Im}L<\infty\) and ImL is close in Z. If L is a Fredholm mapping of index zero, there are continuous projects \(P:X\rightarrow X\), \(Q: Z\rightarrow Z\) such that \(\operatorname{Im} P=\operatorname{Ker} L\), \(\operatorname{Im} L=\operatorname{Ker} Q= \operatorname{Im} (I-Q)\). It follows that \(L|\operatorname{dom} L\cap\operatorname{Ker} P: (I-P)X\rightarrow\operatorname{Im} L\) is invertible. We denote the inverse of that map by \(K_{p}\). If Ω is an open subset of X, the mapping N will be called L-compact on Ω if \(QN(\bar{\Omega})\) is bounded and \(K_{p}(I-Q)N: \bar{\Omega}\rightarrow X\) is compact. Since ImQ is isomorphic to KerL, there exists an isomorphism \(J:\operatorname{Im} Q\rightarrow\operatorname{Ker} L\). The following result is proved in [19].
Lemma 2.4
(Continuous theorem)
- (1)
For each \(\lambda\in(0,1)\), every solution x of \(Lx=\lambda Nx\) is such that \(x\notin\partial\Omega\);
- (2)
For each \(x\in\operatorname{Ker}L\cap\partial\Omega\), \(QNx\neq0\);
- (3)
\(\operatorname{deg}(JQN,\operatorname{Ker}L\cap\Omega,0 )\neq0\).
3 Main results
Lemma 3.1
X and Z are Banach spaces equipped with the norm \(\|\cdot\|\).
Proof
Lemma 3.2
Proof
Remark
If \(f\in\operatorname{ap}(R,R^{n})\) and \(\forall\lambda\in \Lambda_{f}\), \(|\lambda|>\alpha>0\), then f has an ap primitive function. It does not hold for an \(\mathcal{AP}\) function. From [20] we know that if \(f\in\mathcal{AP}(R,R^{n})\), \(\forall \lambda\in\Lambda_{f}\), \(\alpha_{1}>|\lambda|>\alpha>0\), \(\sum_{i=1}^{\infty} |a(\lambda_{i},f)|<+\infty\), then f has an \(\mathcal{AP}\) primitive function. That is the reason why in Lemma 3.1 we take \(Z_{1}\) like that.
Lemma 3.3
N is L-compact on \(\bar{\Omega}\) (Ω is an open, bounded subset of X).
Proof
Noticing Lemmas 3.1-3.3, for Eq. (1.1), we have the following result.
Theorem 3.4
If (A1) and (A2) are satisfied, \(m(\sum_{j=1}^{n}(a_{ij}(t)+ b_{ij}(t)+c_{ij}(t)\int_{-\infty}^{t}k_{ij}(t-s)\,ds))\neq0\), \(i=1,2,\ldots, n\), then Eq. (1.1) has at least one strictly positive almost periodic solution.
Proof
From the analysis above, we know that in order to prove the existence of strictly positive \(\mathcal{AP}\) solution of Eq. (1.1), we only need to investigate the existence of ap solution of Eq. (2.2).
Remark
Declarations
Acknowledgements
The authors would like to thank the Editor and the referees for their constructive suggestions on improving the presentation of the paper. This work was supported by the Fundamental Research Funds for the Central Universities (3102014JC-Q01087), the National Natural Science Foundation of China (No. 31300310).
Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.
Authors’ Affiliations
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