Abstract
In this paper, we first introduce a concept of the mean-value of uniformly almost periodic functions on time scales and give some of its basic properties. Then, we propose a concept of pseudo almost periodic functions on time scales and study some basic properties of pseudo almost periodic functions on time scales. Finally, we establish some results about the existence of pseudo almost periodic solutions to dynamic equations on time scales.
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1 Introduction
The theory of dynamic equations on time scales has been developed over the last several decades, it has been created in order to unify the study of differential and difference equations. Many papers have been published on the theory of dynamic equations on time scales [1–14]. In addition, the existence of almost periodic, asymptotically almost periodic, pseudo-almost periodic solutions is among the most attractive topics in the qualitative theory of differential equations and difference equations due to their applications, especially in biology, economics and physics [15–34]. Recently, in [14, 35], the almost periodic functions and the uniformly almost periodic functions on time scales were presented and investigated, as applications, the existence of almost periodic solutions to a class of functional differential equations and neural networks were studied effectively (see [13, 14, 35]). However, there is no concept of pseudo-almost periodic functions on time scales so that it is impossible for us to study pseudo almost periodic solutions for dynamic equations on time scales.
Motivated by the above, our main purpose of this paper is firstly to introduce a concept of mean-value of uniformly almost periodic functions and give some useful and important properties of it. Then we propose a concept of pseudo almost periodic functions which is a new generalization of uniformly almost periodic functions on time scales and present some relative results. Finally, we establish some results about the existence and uniqueness of pseudo almost periodic solutions to dynamic equations on time scales.
The organization of this paper is as follows: In Section 2, we introduce some notations, definitions and state some preliminary results needed in the later sections. In Section 3, we introduce a concept of mean-value of uniformly almost periodic functions and establish some useful and important results. In Section 4, we propose a concept of pseudo almost periodic functions on time scales and present some relative results. In Section 5, we establish some results about the existence and uniqueness of pseudo almost periodic solutions to dynamic equations on time scales. As applications of our results, in Section 6, we study the existence of pseudo almost periodic solutions to quasi-linear dynamic equations on time scales.
2 Preliminaries
Let be a nonempty closed subset (time scale) of ℝ. The forward and backward jump operators and the graininess are defined, respectively, by
A point is called left-dense if and ρ(t) = t, left-scattered if ρ(t) < t, right-dense if and σ(t) = t, and right-scattered if σ(t) > t. If has a left-scattered maximum m, then ; otherwise . If has a right-scattered minimum m, then ; otherwise .
A function is right-dense continuous provided that it is continuous at right-dense point in and its left-side limits exist at left-dense points in . If f is continuous at each right-dense point and each left-dense point, then f is said to be a continuous function on .
For and , we define the delta derivative of y(t), yΔ(t), to be the number (if it exists) with the property that for a given ε > 0, there exists a neighborhood U of t such that
for all s ∈ U.
Let y be right-dense continuous, if Y Δ(t) = y(t), then we define the delta integral by
A function is called regressive provided 1+µ(t)p(t) ≠ 0 for all . The set of all regressive and rd-continuous functions will be denoted by . We define the set .
A n × n-matrix-valued function A on a time scale is called regressive provided I + µ(t)A(t) is invertible for all , and the class of all such regressive and rd-continuous functions is denoted, similar to the above scalar case, by .
If r is a regressive function, then the generalized exponential function e r is defined by
for all s, , with the cylinder transformation
Definition 2.1 ([1, 3]). Let p, be two regressive functions, define
Lemma 2.1 ([1, 3]). Assume that p, are two regressive functions, then
-
(i)
e0(t, s) ≡ 1 and e p (t, t) ≡ 1;
-
(ii)
e p (σ(t), s) = (1 + µ(t)p(t))e p (t, s);
-
(iii)
;
-
(iv)
(e⊖p(t, s))Δ = (⊖p)(t)e⊖p(t, s);
-
(v)
If a, b, , then .
Definition 2.2 ([36]). For every x, , , define a countably additive measure m1 on the set
that assigns to each interval its length, that is,
The interval is understood as the empty set. Using m1, they generate the outer measure on , defined for each as
with
A set is said to be Δ-measurable if the following equality:
holds true for all subset E of . Define the family
the Lebesgue Δ-measure, denoted by µΔ, is the restriction of to .
Definition 2.3 ([35]). A time scale is called an almost periodic time scale if
Remark 2.1. In the following, we always use to denote an almost periodic time scale.
Throughout this paper, denotes ℝn or , D denotes an open set in or , S denotes an arbitrary compact subset of D.
Definition 2.4 ([35]). Let be an almost periodic time scale. A function is called an almost periodic function in uniformly for x ∈ D if the ε-translation set of f
is a relatively dense set in for all ε > 0 and for each compact subset S of D; that is, for any given ε > 0 and each compact subset S of D, there exists a constant l(ε, S) > 0 such that each interval of length l(ε, S) contains a τ(ε, S) ∈ E{ε, f, S} such that
τ is called the ε-translation number of f and and l(ε, S) is called the inclusion length of E{ε, f, S}.
3 The mean-value of uniformly almost periodic functions on time scales
Let and f(t, x) be almost periodic in t uniformly for x ∈ D, we denote
where , Obviously, for a fixed (f, λ, x), .
Definition 3.1. a(f(t, 0, x)) is called mean-value of f(t, x) if
Theorem 3.1. For any , a(f, λ, x) defined by (3.1) exists uniformly for x ∈ S and is uniformly continuous on S with respect to x, where S is an arbitrary compact subset of D.
Proof. For any t1 ∈ Π, t1 > 0, we can make a sequence , where t i = it1. We will prove that the sequence converges uniformly with respect to x ∈ S.
For any integers m, n and x ∈ S, taking t m , t n , we have
Consider the following integral form:
where s = n, a = (k − 1)n, k = 1, 2, ..., m or s = m, a = (k − 1)m, k = 1, 2, ..., n. For arbitrary a, s, we can evaluate (3.3):
For any ε > 0, let be an inclusion length of and , then, for all x ∈ S, we get***
where . According to (3.4), we can reduce (3.2) to the following:
By the Cauchy convergence criterion, the sequence converges uniformly with respect to x ∈ S.
For any sufficiently large 0 < T ∈ Π, there exist 0 < t n ∈ Π such that 0 < t n < T ≤ tn+1, so for all x ∈ S, we have
Therefore,
Hence,
Besides, for is continuous with respect to x ∈ S, where S is an arbitrary compact set in , a(f, 0, x) is uniformly continuous on S.
It is oblivious that f(t, x)e−iλt is almost periodic in t uniformly for x ∈ D and a(f, λ, x) = a(f(t, x)e−iλt, 0, x), so it is easy to see that a(f, λ, x) exists uniformly for x ∈ S and is uniformly continuous on S with respect to x. This completes the proof. □
Theorem 3.2. Assume that T ∈ Π and is almost periodic in t uniformly for x ∈ D, then
uniformly exists for and
Proof. For m(f, λ, x) = m(f(t, x)e−iλt, 0, x), it suffices to show that, for x ∈ S, , the following uniformly exists:
Take and , , for x ∈ S, we obtain
and
From (3.7), let n → ∞, for x ∈ S, we have
Using trigonometric inequality, according (3.6) and (3.8), we can take such that
Hence, we can easily obtain that (3.5) uniformly exists for and m(f, 0, x) = a(f, 0, x) = a(f(t, x), 0, x). Furthermore,
Therefore, a(f(t + α, x), 0, x) uniformly exists for and m(f(t, x), 0, x) = a(f(t + α, x), 0, x). It is easy to see that f(t, x)e−iλt is almost periodic in t uniformly for x ∈ D, thus, we have
Hence, m(f(t, x), λ, x) uniformly exists for . This completes the proof. □
In Theorem 3.1 and Theorem 3.2, if we take λ = 0, then we have
and
uniformly converge for x ∈ S and for x ∈ S, , respectively.
Definition 3.2. (3.9) and (3.10) are called the mean value and the strong mean-value of f(t, x), respectively.
Lemma 3.1. Let T ∈ Π, then for any real number λ ≠ 0,
Proof. First note that for any fixed T > 0, by Lemma 3.1 and Theorem 5.2 in [36], [t0, t0 + T ] contains only finitely many right scattered points. Assume that , where
are right scattered points. Then
since sin t and cos t are bounded for t ∈ ℝ, one can easily see that (3.11) holds. The proof is complete. □
Theorem 3.3. Let be almost periodic in t uniformly for x ∈ D, then for any finite set of distinct real numbers λ1, λ2, ..., λ N and any finite set of real or complex n-dimensional vectors b1, b2, ..., b N ,
Proof. Note that is almost periodic in t for x ∈ D where 〈,〉 denotes the usual inner product in and denotes the conjugate of f(t, x), so it has a mean-value, thus
by Lemma 3.1, it is easy to obtain that
The proof is complete. □
In Theorem 3.3, if we take b k = a(f, λ k , x)(k = 1, 2, ..., N), then we have the following corollary:
Corollary 3.1. The best approximation of uniformly almost periodic function f(t, x) on time scales satisfies the following:
By Corollary 3.1, one can easily get the following corollary:
Corollary 3.2. Let be almost periodic in t uniformly for x ∈ D, then
Theorem 3.4. Let be almost periodic in t uniformly for x ∈ D, then there is a countable set of real numbers Λ such that a(f, λ, x) ≡ 0 on S if λ ∉ Λ.
Proof. Since f(t, x) is uniformly almost periodic, then for all , there exists M > 0 such that |f(t, x)| ≤ M. Therefore, for any n ∈ ℕ, the real number set is finite (If it is infinite, then , this contradicts Corollary 3.2). Hence, for any fixed x ∈ S, one can obtain the real number set {λ ∈ ℝ: a(f, λ, x) ≠ 0} is countable. Furthermore, by Corollary 3.2, one can see that
Thus, there is a countable set of real numbers Λ such that a(f, λ, x) ≡ 0 on S if λ ∉ Λ. The proof is complete. □
Theorem 3.5. If is a non-negative almost periodic function in t uniformly for x ∈ D and f ≢ 0, then a(f, 0, x) > 0.
Proof. Let and pick δ > 0 so that on . Let l ∈ Π be an inclusion length of and take l > 2δ (In fact, one can choose 0 < τ0 ∈ Π such that nτ0 = l ∈ Π, n is some positive integer). If h ∈ Π, , find . Then . Either or since l > 2δ. In the first case if then
The second case can be handled similarly. In either case since on a subinterval of length . Now write h = (n -1) l to get . Hence
Letting N → ∞ one can get . The proof is complete. □
4. Pseudo almost periodic functions on time scales
Let denote the space of all bounded continuous functions from to . Set
and
Remark 4.1. does not require exists. Consider, for example, let and
Obviously, for any fixed and , one can easily see that , thus n0 ∈ ∏, that is, is an almost periodic time scale. It is clear that lim|t|→∞φ(t) does not exist, noting that are right scattered points, so
Hence .
Definition 4.1. A function is called pseudo almost periodic in t uniformly for x ∈ D if f = g + φ, where and .
Remark 4.2. Note that g and φ are uniquely determined. Indeed, since
if f = g1 + φ1 = g2 + φ2, then one has N(g1 − g2) = 0, which implies that g1 = g2, thus, φ1 = φ2. g and φ are called the almost periodic component and the ergodic perturbation of the function f, respectively. Denote by the set of pseudo almost periodic functions uniformly for x ∈ D.
Example 4.1. Let
f(t) = g(t) + φ(t), where g(t) = sin t + sin πt,
and
F (t, x) = f(t) cos x, .
Since
so, . Therefore, , .
Theorem 4.1. If , then
exists and is finite. It is the mean value of f. Moreover M(f) = M(g).
Proof. Indeed
Since then
exists and is finite by Theorem 3.1. Furthermore, one has
Then
Since ,
Hence
Therefore M(f) = M(g). The proof is complete. □
By Definition 4.1 and the definition of a(·, λ, x), one can easily have
Corollary 4.1. If then a(f, λ, x) = a(g, λ, x).
Furthermore, from Definition 4.1, one can easily show that
Theorem 4.2. If and g is the almost periodic component of f, then we have
and
where and denote the value field of f and g on , respectively, denotes the closure of , where S is an arbitrary compact subset of D.
Definition 4.2. A closed subset C of is said to be an ergodic zero set in if
By the definition of , the proof of the following theorem is straightforward.
Theorem 4.3. A function if and only if for ε > 0, the set is an ergodic zero subset in .
Theorem 4.4. (i) A function if and only if .
(ii) if and only if the norm function .
Proof. (i) The sufficiency follows since
The necessity follows from the fact that
since φ is bounded on . Therefore, one can easily see that (i) is satisfied.
(ii) By (i), if and only if . The latter is equivalent to , which again by (i), is equivalent to . □
The proof is complete.
For , suppose that H(t) ∈ D for all . Define by
For , let G = (g1, g2, ..., g n ) and Φ = (φ1, φ2, ..., φ n ), where g i and φ i are the almost periodic component and the ergodic perturbation of f i (i = 1, 2, ..., n), respectively.
Definition 4.3. Let S be a compact subset of D. A function is said to be continuous in x ∈ S uniformly for if for given x ∈ S and ε > 0, there exists a δ(x, ε) > 0 such that x' ∈ S and |x − x' | < δ(x, ε) imply that |f(t, x') − f(t, x)| < ε for all .
Theorem 4.5. Suppose that the function is continuous in x ∈ S uniformly for and such that , then , where denotes the value field of F and S is an arbitrary compact subset of D.
Proof. Let f = g + φ and F = G + Φ with as above. Note that
It follows from Theorem 4.2 that . By Theorem 3.15 in [35], we have . To finish the proof, we need to show that the function h = g ο (F × ι) - g ο (G × ι) + φ ο (F × ι) is in .
First we show that .
It is trivial in the case that g = 0. So we assume that g ≠ 0. Set By Theorem 3.1 in [35], the function g is uniformly continuous on . For ε > 0, there exists a δ > 0 such that
Set
It follows from Theorem 4.3 and (ii) of Theorem 4.4 that C δ is an ergodic zero subset of . We can find T > 0 such that when r ≥ T,
By (4.1), (4.2) and (4.3), we have
Therefore, .
Finally, we show that . Note that f = g + φ and g is uniformly continuous on . By the hypothesis, f is continuous in S ⊂ D1 uniformly for ; so is φ. Since D1 is compact in , one can find, say m, open balls O k with center xk ∈ D1, k = 1, 2, ..., m, and radius δ(xk, ε/2) such that and
The set
is open and . Let , then E k ∩ E j = ∅ when k ≠ j, 1 ≤ k, j ≤ m.
Since for each , there is a number T0 > 0 such that
It follows from (4.4), (4.5) and (4.6) that
This shows that . The proof is complete. □
Define
Definition 4.4. Let denote all the functions f of the form f = g + φ, where and . The members of are called asymptotically almost periodic functions in t uniformly for x ∈ D.
It is obvious that and .
Corollary 4.2. If and such that , then .
Proof. Obliviously,
where . By the hypothesis that and , it follows that since the uniform continuity of g and since . The proof is complete. □
Theorem 4.6. Suppose that satisfies that for every ε > 0,
Then g ≥ 0 for all , where S is an arbitrary compact subset of D.
Proof. Suppose that the conclusion does not hold. This implies that for some . Choose ε > 0, .
By continuity, there exists δ > 0 so that implies g(t, x) < −ε. In view of Definition 2.4, there exists l(ε, S) > 0 so that in each interval I of length l, one can find -almost period τ with the property that
Choose a sequence τ k of almost periods, , we have
Denote , we have
Therefore,
The right hand side does not tend to zero as k → +∞. This contradicts the assumption made in the lemma. Therefore, g ≥ 0. The proof is complete. □
Theorem 4.7. If , where and , then
-
(i)
If exists, then .
-
(ii)
For all , if f ≥ 0 then g ≥ 0, where S is an arbitrary compact subset of D.
Proof. (i) Suppose that the property does not hold, then there exist a constant and c ∈ Π such that for t ≥ c, which yields
Passing to the limit as r → ∞, we obtain
which contradicts the fact that .
(ii) Assuming f ≥ 0, we want to show that g ≥ 0. We have f = g + φ with
Thus, there exists {c n }n∈ℕ⊂ Π, c n → +∞ as n → ∞ such that g(t + c n , x) → g(t, x) for all . Furthermore, for any ε > 0 and r > 0, one can easily get
which implies that
By Theorem 4.6, one can have g(t, x) ≥ 0 for all .
The proof is complete. □
5 Pseudo almost periodic solutions of dynamic equations on time scales
Consider the non-autonomous equation
and its associated homogeneous equation
where the n × n coefficient matrix A(t) is continuous on and column vector F = (f1, f2, ..., f n )T is in . Define . We will call A(t) almost periodic if all the entries are almost periodic.
Definition 5.1 ([37]). Equation (5.2) is said to admit an exponential dichotomy on if there exist positive constants K, α, projection P and the fundamental solution matrix X(t) of (5.2), satisfying
Lemma 5.1. Let α > 0, then for any fixed and s = −∞, one has the following:
Proof. If µ(t) > 0, since , we have
Thus, and it is easy to have
So
Hence
If μ(t) = 0, one can easily get the conclusion. If s = -∞, it is easy to see that as t → +∞, thus, e⊖α(t, s) → 0. The proof is complete. □
Theorem 5.1. Suppose that A(t) is almost periodic, (5.2) admits an exponential dichotomy and the function . Then (5.1) has a unique bounded solution .
Proof. Similar to the proof of Theorem 4.1 in [35], by checking directly, one can see that the function:
is a solution of (5.1). Now, we show that the solution is bounded. It follows from (5.4) that
where . The solution x is bounded since F is bounded. By Lemma 4.13 in [35], the bounded solution is unique since the homogeneous equation (5.2) has no nontrivial bounded solution.
In the following, we show that . Let and . Then x = I + H. It follows from (5.3) and Theorem 2.15 in [38] that
To show that , we only need to show that both I1 → 0 and I2 → 0 when r → ∞. By Lemma 5.1, one can obtain
Therefore, by (ii) of Theorem 4.4, , so I2 → 0 as r → +∞.
Similarly, one can show that . The proof is complete. □
Theorem 5.2. Suppose that A(t) is almost periodic and (5.2) admits an exponential dichotomy. Then, for every . (5.1) has a unique bounded solution . The mapping F → x F is bounded and linear with
Proof. Since , where and .
According to the proof of Theorem 5.1, the function
is the unique solution of (5.1), where
By Theorem 4.1 in [35], . By Theorem 5.1, . Therefore, . Obviously, the mapping F → x F is linear. The proof is complete. □
Lemma 5.2. Let be an almost periodic function, and
Then the following linear system
admits an exponential dichotomy on , where m(c i ) denote the mean-value of c i , i = 1, 2, ..., n.
Proof. According to Theorem 2.77 in [3], the linear system (5.7) has a unique solution
where x(t0) = x0, −c = diag(−c1(t), −c2(t), ..., −c n (t)).
Now, we prove that x(t) admits an exponential dichotomy on .
According to Theorem 3.2 and Theorem 3.5, one has
So there exists T0 > 0, when T > T0, one has
that is
thus, for T > T0, we have .
Case 1. If , we have
then
thus
that is,
Case 2. If , one cane easily obtain
Set P = I, we have
where K ≥ 1, . Therefore, x(t) admits an exponential dichotomy with P = I on . This completes the proof. □
Example 5.1. Consider the following dynamic equation on an almost periodic time scale :
where
Obviously, . By Lemma 5.2, it is easy to know that the homogeneous equation of (5.8) admits an exponential dichotomy with P = I on . Similar to Example 4.1, one easily to see that . By Theorem 5.2 and Theorem 2.77 in [3], one can obtain that (5.8) has a unique pseudo almost periodic solution:
6 Applications
Application 1. Consider the following quasi-linear system
where is a n × n almost periodic matrix, and . We call the system
the generating system of (6.1).
By Theorem 5.2, system (6.2) has a unique solution if (5.2) admits an, exponential dichotomy. Now we have the following theorem about (6.1).
Theorem 6.1. If , A(t) be almost periodic and (5.2) admits an exponential dichotomy. Let be the unique solution of system (6.2) and denote , where a > 0. Assume that
(i) and L > 0 such that
(ii) , where K and α are the same as those in Theorem 5.2, .
Then system (6.1) has a unique solution such that x ∈ D for all . Furthermore, ||x - x0|| → 0 as μ0 → 0.
Proof. We construct a sequence of approximations by induction, starting with x0 and taking xk to be the bounded solution of the system
First, we show that xk exists, and where denotes the value field of xk. Obliviously, the conclusion holds for k = 0 by the hypothesis. Assume that the conclusion holds for k − 1. Then we shall show that the conclusion also holds for k. By Theorem 4.5, one can see that , so from Theorem 5.2, (6.4) has a unique solution . It follows from (6.2) and (6.4) that,
By (5.5), we have
Therefore, , since
Next, we show that {xk} is Cauchy sequence in . Since
it follows from (5.5) and (6.3) that
where . This shows that {xk} is a Cauchy sequence in . Since is a Banach space, there is an such that ||xk − x|| → 0, when k → ∞. It follows from (6.4) that x is a solution of (6.1). It is clear that ||x − x0|| → 0, as µ0 → 0.
To show the uniqueness, let x* be another solution of (6.1). Similar to the discussion above, we have
this is a contradiction. The proof is complete. □
Application 2. Let D be a ball in with center at origin and radius r0. Consider the following system
where A(t) is a n × n almost periodic matrix and . Set , where denotes the value field of F.
is a closed subset of . Therefore, is a complete metric space.
Theorem 6.2. Let D, G, A(t), and be as those in the previous paragraph. Assume that, (5.2) admits an exponential dichotomy and the function G satisfies,
and
with . Then (6.5) has a unique solution in .
Proof. By Theorem 5.2, one can define the mapping by the fact that, for , is the unique pseudo almost periodic solution of the system
We claim that since by (5.5),
The mapping is a contraction on . In fact, for F1, , it follows from (5.5) and (6.6) that
Therefore, has a unique fixed point in , which is the unique pseudo almost periodic solution of (6.5). The proof is complete. □
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This work is supported by the National Natural Sciences Foundation of People's Republic of China under Grant 10971183.
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Li, Y., Wang, C. Pseudo almost periodic functions and pseudo almost periodic solutions to dynamic equations on time scales. Adv Differ Equ 2012, 77 (2012). https://doi.org/10.1186/1687-1847-2012-77
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DOI: https://doi.org/10.1186/1687-1847-2012-77