The existence of almost periodic solution: via coincidence degree theory
- Sanfu Wang^{1, 2}Email author
Received: 13 December 2015
Accepted: 13 March 2016
Published: 29 March 2016
Abstract
In the present paper, a new method is developed to study the existence of an almost periodic solution for the ordinary or functional differential equations. The approaches are based on topological degree and novel estimation techniques for the a priori bounds of unknown solutions for \(Lx=\lambda Nx\). To investigate the existence of an almost periodic solution, a few good methods have been presented in the previous literature (such as using the Lyapunov function, averaging, exponential dichotomy, stability, separate conditions, and so on). But topological degree theory was never employed to study the almost periodic differential equations. Though Mawhin’s coincidence degree is employed to study the existence of periodic differential equations extensively, it cannot be applied to study the almost periodic systems immediately. Some essentially new and interesting lemmas should be proved before applying topological degree theory to almost periodic systems. To the best knowledge of the authors’, it is the first time that topological degree theory is employed to study the existence of almost periodic solution and this method can be seen as a good supplement to the known methods. Therefore, it will be of great significance to study the almost periodic systems by using this method. The approach followed in the paper could be further generalized to investigate the existence of almost periodic oscillatory in some other nonlinear dynamical systems. It is believed that it can be applied to image patterns, digital image processing, data processing, signal sparse decomposition and information technology, etc.
Keywords
MSC
1 Introduction
The existence of almost periodic solutions of ordinary differential equations has been discussed extensively in theory and in practice (for example, see [1–48] and the references cited therein). In particularly, many useful methods are developed to study the almost periodic differential equations in the classical references such as Hale [1–4], Fink [5–7], Yoshizawa [8, 9], Hino et al. [10], Seifert [11–14], Copple [15], Kato [16], Sell [17, 18], He [19], Favard [20], Bohr and Neugebauer [21], and Lakshmikantham and Leela [22]. We summarize their methods to study the existence of an almost periodic solution for the differential equations as six big categories:
(I) By using the semi-separated condition or the separated condition (considering the hull system), including the famous theorem (each hull system has a unique solution in S, then these solutions are all almost periodic, see He [19], Theorem 32, and Fink [5], Theorem 10.1).
(II) By using the Lyapunov function.
(III) By using the stability, including (weakly) quasi-uniformly asymptotic stability, total stability, stability under the perturbation of the hull (see Fink [5], Yoshizawa [8], and Kato [16]), relatively weakly uniformly asymptotic stability, and relatively total stability (see Hino et al. [10]).
(IV) By using the relations between asymptotically almost periodic function and the almost periodic functions.
(V) By combining the exponential dichotomy and fixed point theory including the Banach fixed point, Schauder fixed point, Lerry-Schauder fixed point approaches, and so on.
(VI) By the average method.
(VII) By using the smallest solution with respect to the norm (see Favard [20]).
(VIII) By the comparison method (see Fink [5] and Lakshmikantham and Leela [22]).
These methods have great significance in the study of the almost periodic differential equations and thus have many application in the specific systems arising from biology, neural networks, physics, chemistry, engineering, and so on (one can refer to [23–48]).
Though so many good methods were developed and applied to study the almost periodic equations, there is no paper studying the existence of almost periodic solutions by using topological degree theory. It is well known that topological degree theory is a powerful tool to study the periodic differential equations. It is usually used to study the existence of periodic solutions for the boundary value problem, the initial value problem, the two-point boundary value problem and so on (for examples, one can refer to [41, 47, 48] and references cited therein). Since topological degree theory plays such a great role in the periodic differential equations, it is natural to ask the question:
Can topological degree theory be employed to study the almost periodic differential equations? If so, how should one use topological degree theory to study the almost periodic systems?
Therefore, the present paper is devoted to giving an affirmative reply to the question. Topological degree theory can be applied to study the almost periodic nonlinear systems including the ordinary differential equations and functional differential equations. However, the method used to study the periodic systems cannot be applied to an almost periodic system directly. The method used in previous work (e.g. [41]) cannot be applied to the almost periodic system directly. Therefore, many definitions should be modified and many indispensable and essentially new lemmas should be proved to suit for the almost periodic systems.
2 Preliminary
In this section, in order to obtain the existence of almost periodic solutions of the differential equations, we shall make some preparation. For convenience, we first summarize in the following a few concepts and prove some preliminary results on almost periodic functions and topological degree theory that will be basic for the next section.
Definition 2.1
Definition 2.2
Definition 2.3
Let \(f\in C(R,R^{n})\), if the limit \(\lim_{T\rightarrow \infty}T^{-1}\int_{0}^{T}f(t)\,dt\) exists, then the limit is called the mean value of f. Denote \(m(f(t))=\lim_{T\rightarrow \infty}T^{-1}\int_{0}^{T}f(t)\,dt\).
Obviously, \(m(a)=a\) and \(m (m(f(t)) )=m(f(t))\), where \(a\in R^{n}\) is a constant vector.
Lemma 2.1
- (a)
Let \(f(t)\in C(R,R^{n})\), \(f(t)\) is almost periodic if only if \(x(t)\) is bounded and uniformly continuous on R.
- (b)
Let \(f:R\times S\rightarrow R^{n}\) be almost periodic in t uniformly with respect to \(x\in S\subset R^{n}\), where S is any compact set of \(R^{n}\). Then \(f(t,x)\) is bounded on \(R\times S\) and uniformly continuous.
- (c)
Let \(f(t)\) and \(g(t)\) be all almost periodic, then \(f(t)+g(t)\) is almost periodic. Moreover, if \(\inf_{t\in R}|g(t)|>0\), then \(f(t)/g(t)\) is also almost periodic.
Lemma 2.2
Lemma 2.3
[5, 8, 19] (mean value theorem)
Suppose that \(f\in C(R,R^{n})\) is an almost periodic function, then \(m(f(t))\) exists, i.e. \(\|m(f(t))\|<+\infty\). Moreover, if \(f(t)\geq0\) and \(f(t)\not\equiv0\), then \(m(f(t))>0\).
Lemma 2.4
[47] (continuation theorem)
- (a)
for each \(\lambda\in(0,1)\), \(x\in\partial\Omega\cap\operatorname{Dom}L\), \(Lx\neq\lambda Nx\);
- (b)
for each \(x\in\partial\Omega\cap\operatorname{Ker}L\), \(QNx\neq0\);
- (c)
\(\operatorname{deg}\{JQN, \Omega\cap\operatorname{Ker}L,0\}\neq0\).
Then \(Lx=Nx\) has at least one solution in \(\overline{\Omega}\cap\operatorname{Dom}L\).
To proceed our study of the existence of almost periodic solutions, we need to prove the following important lemmas.
Lemma 2.5
If \(x(t)\in AP(R,R^{n})\), then there exists \(t_{0}\in R\) such that \(x(t_{0})=\sup_{t\in R}x(t)\).
Proof
Lemma 2.6
Supposing that \(x(t)\in AP(R,R^{n})\), \(m(x(t))=0\), then there exists \(\eta_{0}>0\) such that, for every fixed \(\eta_{0}\leq\eta<+\infty\), \(\Phi(t)=\int_{0}^{t}{\mathrm{e}}^{-\eta(t-s)}x(s)\,d s\) is an almost periodic function and \(\dot{\Phi}(t)\) is also an almost periodic function satisfying \(\dot{\Phi}(t)=x(t)\).
Proof
To \(\Phi(t)\) is an almost periodic function, it suffices to show that \(\Phi(t)\) is bounded. To this end, we just need show that \(\int_{-\infty}^{t}{\mathrm{e}}^{-\eta(t-s)}x(s)\,d s\) is bounded.
3 Existence of an almost periodic solution in general case
Then we have the following results.
Lemma 3.1
- (i)
\(\operatorname{Im}P=R^{n}=\operatorname{Ker}L\);
- (ii)
\(\operatorname{Im}L=\operatorname{Ker}Q=\operatorname{Im}(I-Q)\);
- (iii)
\(\operatorname{dimKer}L=\operatorname{codimIm}L=n<+\infty\).
Proof
By the definition of P and KerL, (i) follows immediately. Noting that \(\operatorname{dimKer}L+\operatorname{dimIm}L=n\), it is easy to check that (iii) holds. Now we devote ourselves to proving (ii).
First we show \(\operatorname{Im}L=\operatorname{Ker}Q\). To this end, we proceed in two steps.
Step 2. \(\forall z(t)\in\operatorname{Ker}Q\), that is, \(z(t)\in AP(R,R^{n})\) and \(m(z(t))=0\). \(z(t)\in AP(R,R^{n})\) implies \(z(t)\in C(R,R^{n})\). Thus, there exists an original function of \(z(t)\), denoted by \(\Phi(t)\) such that \(\dot{\Phi}(t)=z(t)\). On the other hand, it is easy to check that, for any fixed \(0<\eta_{0}\leq\eta<\infty\), \(\Phi_{0}(t)=\int_{0}^{t}{\mathrm{e}}^{-\eta(t-s)}z(s)\,d s\) is also an original function of \(z(t)\) such that \(\dot{\Phi}_{0}(t)=z(t)\). In fact, \(\Phi(t)=\Phi_{0}(t)+C\), where C is an arbitrary constant.
Since \(m(z(t))=0\), by Lemma 2.6, \(\Phi_{0}(t)=\int_{0}^{t}{\mathrm{e}}^{-\eta(t-s)}z(s)\,d s\) is an almost periodic function and \(\dot{\Phi}_{0}(t)\) is also an almost periodic function satisfying \(\dot{\Phi}_{0}(t)=z(t)\). Therefore, \(\Phi(t)=\Phi_{0}(t)+C\) is almost periodic and \(\dot{\Phi}(t)=z(t)=L\Phi(t)\) is also almost periodic. This implies that \(z(t)=L\Phi(t)\in\operatorname{Im}L\). Hence, \(\operatorname{Ker}Q\subset \operatorname{Im}L\).
It follows from the above steps that \(\operatorname{Im}L= \operatorname{Ker}Q\).
Now we show that \(\operatorname{Ker}Q=\operatorname{Im}(I-Q)=\{z(t)-m(z(t))| z(t)\in Z\}\). In fact,
In a word, \(\operatorname{Im}L=\operatorname{Ker}Q=\operatorname{Im}(I-Q)\). That is, (ii) holds. Thus, the proof of Lemma 3.1 is complete. □
Lemma 3.2
Proof
Step 2. By similar arguments to [41], we can prove that \((K_{P}(I-Q)Nx )(\overline{\Omega})\) is equicontinuous.
Therefore, by generalizing the famous Arzela-Ascoli theorem, \((K_{P}(I-Q)Nx )(\overline{\Omega})\) is relatively compact in the space \((X,\|\cdot\|_{1})\). The proof of this lemma is complete. □
Then by Lemma 3.1, Lemma 3.2, and Lemma 2.4, we have the following results on the existence of almost periodic solutions.
Theorem 3.1
- (i)
for each \(\lambda\in(0,1)\), \(x\in\partial\Omega\cap\operatorname{Dom}L\), \(Lx\neq\lambda Nx\);
- (ii)
for each \(x\in\partial\Omega\cap\operatorname{Ker}L\), \(QNx\neq0\);
- (iii)
\(\operatorname{deg}\{JQN, \Omega\cap\operatorname{Ker}L,0\}\neq0\).
Then system (2) has at least one almost periodic solution in \(\overline {\Omega}\cap\operatorname{Dom}L\).
4 Conclusion
In the present paper, a new method is developed to study the existence of an almost periodic solution for the ordinary or functional differential equations. The approach is based on topological degree and novel estimation techniques for the a priori bounds of unknown solutions for \(Lx=\lambda Nx\). This new method can be seen as a supplement of the other classical methods. The approach performed in the paper could be further generalized to investigate the existence of almost periodic oscillatory cases in some other nonlinear dynamical systems. It is believed that it can be applied to the image pattern, digital image processing, data processing, signal sparse decomposition and information technology, etc.
It should be noted that there are particular differences between this paper and previous work [41].
1. This paper considered the almost periodic solution. However, the authors in [41] studied the periodic solutions. In fact, periodic solutions are a special case of almost periodic solutions. For a periodic function, it is defined by \(f(t+T)=f(t)\) for some T. However, for the almost periodic function, it is defined in a more complicated way. For the detailed definition of an almost periodic function, one can refer to Definition 2.1.
2. Due to the big difference between the concepts of almost periodic and periodic, the method used in [41] cannot be applied to the almost periodic case directly. So we need to prove a lot of preliminary results in Section 2. From Lemma 2.1 to Lemma 2.6, all these results are very original, they have never been studied. This is the big contribution of this paper. In fact, these results are very interesting. For periodic solutions, it is not necessary to prove these results. It is obvious for the periodic case.
3. In Section 3, it seems that the method is similar to [41]. But please note that my aim of this paper is to generalize the periodic results in [41] to the almost periodic case. In fact, the detailed proof is different. Thus, it seems similar, but it is a very different problem. For example, for the periodic case, the operators P and Q are defined by \(Px(t)=\frac{1}{T}\int_{0}^{T} x(t)\,dt\), however, for the almost periodic case, we have \(Px(t)=m(x(t))\).
Declarations
Acknowledgements
The author would like to acknowledge the financial support from the National Natural Science Foundation of China under Grant (No. 61572395 and No. 11561060), Tianshui Normal University Key Construction Subject Project (Big data processing in dynamic image).
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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