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Affineperiodic solutions for nonlinear dynamic equations on time scales
Advances in Difference Equations volumeÂ 2015, ArticleÂ number:Â 286 (2015)
Abstract
The existence of affineperiodic solutions for dynamic equations on time scales is studied. Mainly, via the topological degree theory, a general existence theorem is proved, which provides an effective method in the qualitative theory for nonlinear dynamic equations on time scales.
1 Introduction
The periodicity problem is a very important topic in the study of differential equations, but not all the natural phenomena can be described by periodicity only. We found that some differential equations exhibit a certain symmetry rather than periodicity [1â€“3]. For example, we denote by \(GL_{n}(R)\) the ndimensional general linear group over \(R^{1}\) and consider the system
where \(f: R^{1}\times R^{n} \to R^{n}\) is continuous, and for some \(Q\in GL_{n}(R)\), the following affine symmetry holds:
In the sense of (2), we have the concept of an affineperiodic system (APS for short).
Definition 1.1
The system (1) is said to be a \((Q, T)\)affineperiodic system, if there exists \(Q\in GL_{n}(R)\) and \(T > 0\) such that
holds for all \((t, x)\in R^{1}\times R^{n}\).
For APS (1), we define its affineperiodic solution as follows.
Definition 1.2
If \(x(t)\) is a solution of APS (1) on \(R^{1}\) and
then \(x(t)\) is said to be a \((Q, T)\)affineperiodic solution.
As a structural property of functions, the affineperiodicity is a generalization of pure periodicity. Recently, some existence theorems as regards affineperiodic solutions have been proved for APSs. We refer to [1â€“4]. Particularly, in [1], the existence of affineperiodic solutions for APS (1) was established via topological degree theory. However, the existence of affineperiodic solutions for APSs when â€˜timeâ€™ is not continuous has not been discussed yet. The aim of this paper is to touch on such a topic for APSs on time scales.
A time scale is an arbitrary nonempty closed subset of \(R^{1}\), often denoted by \(\mathbb{T}\). The theory of time scales was first introduced by Hilger [5] in 1990 in order to study differences between discrete and continuous analysis. The time scale calculus offers great help on unifying discrete and continuous dynamic systems and presents a powerful tool for applications to economics and biology models, among others. Hence it has been attracting more and more attention during the past years and the existence of solutions for systems on time scales has become an interesting and popular topic. In this respect, Amster et al. [6] studied boundary value problems for dynamic systems and proved the existence of solutions via topological degree theory; Lizama and Mesquita [7] considered nonautonomous dynamic equations and proved the existence of almost automorphic solutions by assuming that the associated homogeneous equation of the system admits an exponential dichotomy. In the field of second and higherorder periodic problems, the lower and upper solutions technique is a powerful tool and has been used in most of the studies, for example, the work of Akin [8] and Stehlik [9]. In this paper, we will establish a topological degree theory to prove the existence of affineperiodic solutions for APSs on time scales, which is not very common in this area.
To discuss the APSs on time scales, some basic notations and definitions are needed, and most of them can be found in [10]. The first basic and important concept in the theory of time scales is the forward (backward) jump operator.
Definition 1.3
Let \(\mathbb{T}\) be a time scale. For \(t\in\mathbb{T}\) we define the forward jump operator \(\sigma: \mathbb{T} \to\mathbb{T}\) by
while the backward jump operator \(\rho: \mathbb{T}\to\mathbb{T}\) is defined by
In DefinitionÂ 1.3, we put \(\inf\emptyset=\sup\mathbb{T}\) and \(\sup \emptyset=\inf\mathbb{T}\).
Definition 1.4
The graininess function \(\mu:\mathbb{T}\to[0,\infty)\) is defined by
A point \(t\in\mathbb{T}\) is called rightscattered if \(\sigma (t)>t\), while if \(\rho(t)< t\) we say that t is leftscattered. Points that are rightscattered and leftscattered at the same time are called isolated. Also, if \(t<\sup\mathbb{T}\) and \(\sigma(t)=t\), then t is called rightdense, and if \(t>\inf\mathbb{T}\) and \(\rho(t)=t\), then t is called leftdense. Points that are rightdense and leftdense at the same time are called dense. If \(\mathbb{T}\) has a leftscattered maximum m, then \(\mathbb{T}^{\kappa} = \mathbb{T}\setminus \{ m \}\), otherwise \(\mathbb{T}^{\kappa} = \mathbb{T}\). For a time scale \(\mathbb{T}\), we denote by \([a, b]_{\mathbb{T}}\) (\((a, b)_{\mathbb {T}}\)) the set \([a, b]\cap\mathbb{T}\) (\((a, b)\cap\mathbb{T}\)), where \(a, b\in R^{1}\). Then we have the definition of the socalled delta derivative.
Definition 1.5
Assume \(f:\mathbb{T}\to R^{n}\) is a function and let \(t\in\mathbb {T}^{\kappa}\). Then we define \(f^{\Delta}(t)\) to be the vector (provided it exists) with the property that given any \(\epsilon>0\), there is a neighborhood U of t (i.e. \(U = (t\delta, t+\delta)_{\mathbb{T}}\) for some \(\delta>0\)) such that
We call \(f^{\Delta}(t)\) the delta derivative of f at t.
In order to describe classes of functions that are integrable, we need the following definition.
Definition 1.6
A function \(f:\mathbb{T}\to R^{n}\) is called rdcontinuous provided it is continuous at rightdense points in \(\mathbb{T}\) and its leftsided limits exist (finite) at leftdense points in \(\mathbb{T}\). The set of rdcontinuous functions \(f:\mathbb{T}\to R^{n}\) will be denoted by
Let \(f\in C_{\mathrm{rd}}\). If \(F^{\Delta}(t) = f(t)\), we have
Remark 1.1
\(F(t)\) is continuous on \(\mathbb{T}\) when \(f\in C_{\mathrm{rd}}(\mathbb{T})\) (see TheoremÂ 1.16 in [10]).
We also need the notion of periodic time scales which was introduced by Atici et al. [11, 12]. The following definition is borrowed from [11â€“13].
Definition 1.7
We say that a time scale \(\mathbb{T}\) is periodic if there exists \(T>0\) such that if \(t\in\mathbb{T}\) then \(t\pm T\in\mathbb{T}\). For \(\mathbb {T}\neq R^{1}\), the smallest positive T is called the period of the time scale.
Now, we can define the APSs on time scales.
Definition 1.8
Let \(\mathbb{T}\) be a Tperiodic time scale, \(f: \mathbb{T} \times R^{n}\to R^{n}\) a rdcontinuous function. The system
is said to be a \((Q, T)\)affineperiodic system, if there exists \(Q\in GL_{n}(R)\) such that
holds for all \((t, x)\in\mathbb{T}\times R^{n}\).
The solutions of dynamic equations on time scales are defined as follows.
Definition 1.9
Consider the equation
where \(f:\mathbb{T}\times R^{n}\to R^{n}\). A function \(x:\mathbb{T}\to R^{n}\) is called a solution of (5) if
and \(x(t)\) satisfies (5) for all \(t\in\mathbb{T}\). Given \(t_{0}\in\mathbb{T}\) and \(x_{0}\in R^{n}\), the problem
is called an initial value problem, and a solution of (5) with \(x(t_{0}) = x_{0}\) is called a solution to this initial value problem.
Hence we have the definition of affineperiodic solutions.
Definition 1.10
A function \(x:\mathbb{T}\to R^{n}\) is said to be an affineperiodic solution of (4) if \(x(t)\) is a solution of (4) and for any \(t\in\mathbb{T}\),
According to the definitions of APS on time scales and its solutions, we have the following existence theorem. The proof can be found in SectionÂ 3.
Consider the APS
where \(f: \mathbb{T} \times R^{n}\to R^{n}\) is rdcontinuous and ensures the uniqueness of solutions with respect to initial value, \(Q\in O(n)\), \(\mathbb{T}\) is a Tperiodic time scale.
Theorem 1.1
Let \(D\subset{R^{n}}\) be a bounded open set. Assume the following hypotheses hold for the system (6).
 (H_{1}):

For each \(\lambda\in(0, 1]\), every possible affineperiodic solution \(x(t)\) of the auxiliary equation \(x^{\Delta} = \lambda f(t, x)\) satisfies the following: if \(x(t)\in\bar{D}\), then
$$x(t) \notin\partial D\quad \forall t\in[0, T]_{\mathbb{T}}. $$  (H_{2}):

The Brouwer degree
$$\operatorname{deg} \bigl(g, D\cap\operatorname{Ker}(I  Q), 0 \bigr) \neq0, \quad\textit{if } \operatorname{Ker}(I  Q) \neq \{0 \}, $$where
$$g(a) = \frac{1}{T}\int_{0}^{T}Pf(s, a){ \Delta}s, $$\(P: R^{n}\to\operatorname{Ker}(I  Q)\) is an orthogonal projection.
Then (6) has at least one \((Q, T)\)affineperiodic solution \(x_{*}(t) \in D\) for all \(t\in[0, T]_{\mathbb{T}}\).
As an application of TheoremÂ 1.1, we have the following corollary on basis of Lyapunov functions. The proof and two examples can be found in SectionÂ 4.
Corollary 1.1
Consider the system (6). Assume that there exist constants \(M>0\) and \(\delta>0\), such that
If \(\operatorname{Ker}(I  Q) \neq \{0 \}\), for all \(x\in\operatorname{Ker}(I  Q)\) and \( x(t)\geq M\), \(t\in\mathbb{T,}\)
where \(P:R^{n}\to\operatorname{Ker}(I  Q)\) is an orthogonal projection.
Then the system (6) has at least one \((Q, T)\)affineperiodic solution \(x_{\ast}(t)\).
Before starting our proof, we first introduce some basic definitions and theorems in SectionÂ 2.
2 Preliminaries
In this section, we shall recall some useful theorems and prove a lemma which will be used in SectionÂ 3. The following theorem shows some easy and useful relationships concerning the delta derivative.
Theorem 2.1
Assume \(f:\mathbb{T}\to R^{n}\) is a function and let \(t\in\mathbb {T}^{\kappa}\). Then we have the following:

(i)
If f is differentiable at t, then f is continuous at t.

(ii)
If f is continuous at t and t is rightscattered, then f is differentiable at t with
$$f^{\Delta}(t) = \frac{f(\sigma(t))f(t)}{\mu(t)}. $$ 
(iii)
If t is rightdense, then f is differentiable at t iff the limit
$$\lim_{s\to t}\frac{f(t)f(s)}{ts} $$exists as a finite number. In this case
$$f^{\Delta}(t) = \lim_{s\to t}\frac{f(t)f(s)}{ts}. $$ 
(iv)
If f is differentiable at t, then
$$f \bigl(\sigma(t) \bigr) = f(t) + \mu(t)f^{\Delta}(t). $$
To prove CorollaryÂ 1.1, a chain rule is obviously necessary. The following chain rule is due to PÃ¶tzsche [14] and Keller [15].
Theorem 2.2
Let \(f:R^{1}\to R^{1}\) be continuously differentiable and suppose that \(g:\mathbb{T}\to R\) is delta differentiable. Then \(f\circ g:\mathbb {T}\to R^{1}\) is delta differentiable and the formula
holds.
Consider the system (6). Now a basic topic is to investigate the existence of \((Q, T)\)affineperiodic solutions \(x(t)\). The following lemma shows that this problem is equivalent to proving the existence of solutions of a boundary value problem (BVP for short).
Lemma 2.1
Consider the system (6). The existence of \((Q, T)\)affineperiodic solutions of (6) is equivalent to the existence of solutions of the BVP
Proof
For any solution \(x(t)\) of (6), let \(u(t) = Q^{1}x(t+T)\). By TheoremÂ 2.1:

(i)
If t is a rightdense point, we have
$$\begin{aligned} u^{\Delta}(t) &= \lim_{\Delta t\to0}\frac {u(t+\Delta t)u(t)}{\Delta t} \\ &= \lim_{\Delta t\to0}\frac{Q^{1}x(t+\Delta t +T)Q^{1}x(t+T)}{\Delta t} \\ &= Q^{1}x^{\Delta}(t+T) \\ &= Q^{1}f \bigl(t+T, x(t+T) \bigr) \\ &= f \bigl(t, Q^{1}x(t+T) \bigr) \\ &= f \bigl(t, u(t) \bigr). \end{aligned}$$ 
(ii)
If t is a rightscattered point, we have
$$\begin{aligned} u^{\Delta}(t) &= \frac{u(\sigma(t))u(t)}{\mu(t)} \\ &= \frac{Q^{1}x(\sigma(t)+T)Q^{1}x(t+T)}{\mu(t)} \\ &= Q^{1}x^{\Delta}(t+T) \\ &= f \bigl(t, u(t) \bigr). \end{aligned}$$
By (i) and (ii), we see that \(u(t)\) is a solution of (6). Since \(f(t, x)\) ensures the uniqueness of solutions with respect to initial value and \(u(0) = Q^{1}x(T)\), we know that \(u(t)\equiv x(t)\) if and only if \(x(0) = Q^{1}x(T)\).â€ƒâ–¡
Finally, as a useful tool in our proof, we introduce the definition of a retraction map.
Definition 2.1
Let X be a topological space and A a subspace of X. Then a continuous map \(r: X \to A\) is called a retraction if the restriction of r to A is the identity map on A.
3 Proof of Theorem 1.1
By a coordinate transformation, we can always make \(0\in\mathbb{T}\) without loss of generality.
Consider the BVP of the auxiliary equation
where \(\lambda\in[0, 1]\). Let \(x(t)\) be any solution of (10)(11). Then BVP (10)(11) is equivalent to the integral equation
Denote \(x(0)\) by \(x_{0}\). Then
where I is the identity matrix.
Consider (12) in two parts:

(I)
If \(\operatorname{Ker}(I  Q) \neq \{0 \}\).
In this case, \((I  Q)^{1}\) does not exist. By a coordinate transformation, we can just let
without loss of generality, where \((I  Q_{1})^{1}\) exists.
Let \(P: R^{n}\to\operatorname{Ker}(I  Q)\) be the orthogonal projection. Then
where \(x_{\operatorname{Ker}}^{0} \in\operatorname{Ker}(I  Q)\), \(x_{\perp}^{0} \in \operatorname{Im} (I Q)\) and \(x_{0} = x_{\operatorname{Ker}}^{0} + x_{\perp}^{0}\).
Let \(\mathcal{J} = (I  Q)_{\operatorname{Im}(I  Q)}\). It is easy to see that \(\mathcal{J}^{1}\) exists. Thus (13) is equivalent to
Thus we have
Let
and define the norm as \(\ x\= \sup_{t\in[0, T]_{\mathbb {T}}}x(t)\). It is easy to see that X is a Banach space with the norm \(\\cdot\\).
For \(x \in X\) which satisfies \(x(t)\in\bar{D}\) for all \(t\in[0, T]_{\mathbb{T}}\), we define an operator \(\mathcal{T}(x_{\operatorname{Ker}}^{0}, x, \lambda)\) by
where \(\lambda\in[0, 1]\). We claim that each fixed point x of \(\mathcal{T}\) in X is a solution of BVP (10)(11).
In fact, if x is a fixed point of \(\mathcal{T}\), we have
Thus
By (16), we know that
Thus
Since (15) holds, we have
Thus
Then
By (16) and (17), we know that (12) holds. Thus
Then
It means that the fixed point x is a solution of BVP (10)(11).
Now, we need to prove the existence of fixed points of \(\mathcal{T}\).
Take a constant M which satisfies \(M>\sup_{t\in[0, T]_{\mathbb{T}},x\in\bar{D}}f(t, x)\), and let
It is easy to make a retraction \(\alpha_{\lambda}: X \to X_{\lambda}\).
Define an operator \(\widehat{\mathcal{T}} (x_{\operatorname{Ker}}^{0}, x, \lambda)\) by
Obviously,
is a homotopy, where
We claim that
where id is the identity operator.
Suppose, on the contrary, that there exists \((\hat{x}_{\operatorname{Ker}}^{0}, \hat{x}, \hat{\lambda})\in\partial((D\cap\operatorname{Ker}(I  Q))\times\tilde {D}) \times[0, 1]\), such that \((\operatorname{id}  H)(\hat{x}_{\operatorname{Ker}}^{0}, \hat{x}, \hat{\lambda}) = 0\). As \(\hat{x}_{\operatorname{Ker}}^{0} \in\partial(D\cap\operatorname{Ker}(I  Q))\subset\partial{D}\) is contradictory to (H_{1}); we know that \(\hat{x}_{\operatorname{Ker}}^{0}\notin\partial(D\cap\operatorname{Ker}(I  Q))\). In other words, \(\hat{x}\in\partial\tilde{D}\). Then (20) can be proved as follows:
(i) When \(\hat{\lambda} = 0\), by the definition of set \(X_{\lambda}\), we have
Hence \(\alpha_{0}\circ x(t) \equiv\alpha_{0}\circ x(0)\) for all \(t\in [0, T]_{\mathbb{T}}\). Since \((\operatorname{id}  H)(\hat{x}_{\operatorname{Ker}}^{0}, \hat{x}, 0) = 0\), we have
It means that \(\hat{x}(t) \equiv\hat{x}(0)\) for all \(t \in[0, T]_{\mathbb{T}}\). Take \(\hat{x}(t) = p\), we have \(\alpha_{0}\circ\hat {x}_{\operatorname{Ker}}^{0} = \hat{x}(t) = p\). Consequently \(\frac{1}{T}\int_{0}^{T}Pf(\tau, p)\Delta \tau= 0\), and this is equivalent to \(g(p) = 0\) by the definition of \(g(a)\). Notice that \(\hat{x}\in\partial \tilde{D}\) and \(\tilde{D}= \{x\in X: x(t)\in D \ \forall t\in [0, T]_{\mathbb{T}} \}\). Hence there exists \(t_{0}\in[0, T]_{\mathbb{T}}\) such that \(\hat{x}(t_{0})\in\partial{D}\). As \(\hat {x}(t) \equiv p\) for all \(t\in[0, T]_{\mathbb{T}}\), we obtain \(p\in \partial{D}\). Thus we have \(p\in\partial{D}\) and \(g(p) = 0\). It is contradictory to (H_{2}) because the Brouwer degree \(\operatorname{deg}(g, D, 0) \neq0\).
(ii) When \(\hat{\lambda} \in(0, 1]\), as \(0 = (\operatorname{id}  H)(\hat {x}_{\operatorname{Ker}}^{0}, \hat{x}, \hat{\lambda})\), we have
Thus
and
Note that
By the definition of \(X_{\lambda}\), we obtain \(\hat{x}\in X_{\hat {\lambda}}\), which means that \(\alpha_{\hat{\lambda}}\circ\hat{x} = \hat {x}\). Thus (21) is equivalent to
By a similar discussion to (16), we can prove that \(\hat{x}(t)\) is a solution of BVP (10)(11). By hypothesis (H_{1}), we know that \(\hat{x}(t)\notin\partial{D}\) for any \(t\in[0, T]_{\mathbb {T}}\). It is contradictory to \(\hat{x}\in\partial{\tilde{D}}\).
By (i) and (ii), we obtain
Therefore, by the homotopy invariance and the theory of the Brouwer degree, we have
It means that there exists \(\hat{x}_{\ast}\in\tilde{D}\), such that
Similar to the proof in (ii), we get \(\hat{x}_{\ast}\in X_{1}\). Then
By (22) and (23), we see that \(\hat{x}_{\ast}\) is a fixed point of \(\mathcal{T}\) in X. Thus \(\hat{x}_{\ast}(t)\) is a solution of BVP (10)(11).
(II) If \(\operatorname{Ker}(I  Q) = \{0 \}\).
In this case, \((I  Q)^{1}\) exists. Then
Hypothesis (H_{2}) will not be needed anymore. Consider the homotopy
Similar to the proof when \(\operatorname{Ker}(I  Q) \neq \{0 \}\), we have \(0\notin(\operatorname{id}  H)(\partial\tilde{D}\times[0, 1])\). Hence
It means that there exists \(\hat{x}_{\ast}\), which satisfies \(\hat {x}_{\ast}(t)\in D\) for all \(t\in[0, T]_{\mathbb{T}}\), such that
Therefore \(\hat{x}_{\ast}(t)\) is a solution of BVP (10)(11).
By LemmaÂ 2.1 and the proofs in (I) and (II), it is easy to see that APS (6) has a \((Q, T)\)affineperiodic solution \(x_{\ast }(t)\), which is an extension of \(\hat{x}_{\ast}(t)\) on \(\mathbb{T}\). By hypothesis (H_{1}) and \(\hat{x}_{\ast}(t)\in D\) for all \(t\in[0, T]_{\mathbb{T}}\), we know that \(x_{\ast}(t)\in D\) for all \(t\in[0, T]_{\mathbb{T}}\).
4 Proof of Corollary 1.1
Consider the auxiliary equation (10) of the system (6),
Let
Clearly, D is bounded. We claim that for \(\lambda\in(0, 1]\), every possible \((Q, T)\)affineperiodic solution \(x(t)\) of (10) satisfies (H_{1}).
In fact, assume that \(x(t)\) is a \((Q, T)\)affineperiodic solution of (10), \(Q\in O(n)\). Set \(u(t) = V(x(t))\). Then
hence \(u: \mathbb{T}\to R^{1}_{+}\) is Tperiodic. Thereby there exists \(t_{j}\in[0, T)\cap\mathbb{T}\) and \(t_{0}\in[0, T]\cap\mathbb {T}\) with \(t_{j}\nearrow t_{0}\) or \(t_{j}\searrow t_{0}\) such that
By DefinitionÂ 1.9, we know that \(x\in C(\mathbb{T}, R^{n})\). Hence \(u(t_{0}) = \sup_{[0, T]_{\mathbb{T}}} u(t)\). By TheoremÂ 2.2, we have
By (7), this result yields
Consequently, by the definition of D and (24), we have
Thus (H_{1}) holds.
If \(\operatorname{Ker}(I  Q) = \{0 \}\), by the proof of TheoremÂ 1.1, we know that the system (6) admits a \((Q, T)\)affineperiodic solution.
Now we prove that if \(\operatorname{Ker}(I  Q) \neq \{0 \}\),
Indeed, let \(B_{M} = \{p\in R^{n}:  p< M \}\). Consider the homotopy
where \((p, \lambda)\in(B_{M_{0}}\cap\operatorname{Ker}(I  Q))\times[0, 1]\). It follows that
For any \((p, t)\in\partial(B_{M}\cap\operatorname{Ker}(I  Q))\times R^{1}\), by (8), we know that the sign of \(\langle\nabla V(p), Pf(t, p)\rangle\) does not change. By the definition of \(g(a)\), we have
It means that \(\langle\nabla V(p), g(p)\rangle\) always has the same sign with \(\langle\nabla V(p), Pf(t, p)\rangle\). Also, by (8), we know that \(\nabla V(p)\neq0\) when \(p\in \partial(B_{M}\cap\operatorname{Ker}(I  Q))\). Consequently, the right hand side of (25) is nonzero.
Thus
which implies that \(0 \notin H(\partial(B_{M}\cap\operatorname{Ker}(I  Q))\times[0, 1])\).
The homotopy invariance of the Brouwer degree implies
Hence hypothesis (H_{2}) holds. Thus CorollaryÂ 1.1 follows from TheoremÂ 1.1.
5 Examples
As an application of TheoremÂ 1.1, CorollaryÂ 1.1 is very useful and more directly. In this section, we will show two examples and prove the existence of affineperiodic solutions of them by using CorollaryÂ 1.1.
Example 5.1
Let \(\mathbb{T}\) be a 2Ï€periodic time scale. Consider the system
Let M be a constant large enough. Set \(V(x) = \frac{1}{2}x^{2}\), \(f(t, x(t)) = x^{3}(t) + \sin t\). We have
Hence (26) is a 2Ï€affineperiodic system. Then when \( x> M\), it is easy to see that
Also, since \(P = \operatorname{id}\), we have
By CorollaryÂ 1.1, (26) has a 2Ï€periodic solution.
Example 5.2
Let \(\mathbb{T}\) be a 1periodic time scale. Consider the system
Set
Then (27) is a \((Q, 1)\)affineperiodic system. Let
and M a constant large enough. When \( x> M\), similar to ExampleÂ 5.1, it is easy to see that
Also, since \(P: R^{4}\to\operatorname{Ker}(I  Q)\) is an orthogonal projection,
we have
By CorollaryÂ 1.1, (27) has a \((Q, 1)\)affineperiodic solution.
6 Conclusion
In this paper, we considered the affineperiodic systems on time scales and provided TheoremÂ 1.1. This theorem asserts the existence of affineperiodic solutions in a certain topological formalism. As an application of TheoremÂ 1.1, we proved CorollaryÂ 1.1 and gave two examples. The affineperiodicity is a new and attractive topic in this area, and many significant problems remain to be further studied. In our forthcoming papers, we will discuss the affineperiodic solutions for impulsive equations. Besides, affineperiodic solutions of higherorder equations and delay equations etc. are also very interesting problems.
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Acknowledgements
This work was partially supported by NSFC Grant 11171132. We are deeply grateful to the anonymous referee for his/her patience and valuable comments.
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Wang, C., Li, Y. Affineperiodic solutions for nonlinear dynamic equations on time scales. Adv Differ Equ 2015, 286 (2015). https://doi.org/10.1186/s1366201506340
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DOI: https://doi.org/10.1186/s1366201506340