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Pseudo almost periodic functions and pseudo almost periodic solutions to dynamic equations on time scales

Advances in Difference Equations20122012:77

https://doi.org/10.1186/1687-1847-2012-77

  • Received: 27 October 2011
  • Accepted: 7 June 2012
  • Published:

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.

Keywords

  • dynamic equations on time scales
  • pseudo almost periodic functions
  • exponential dichotomy
  • pseudo almost periodic solutions

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 [114]. 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 [1534]. 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 . The forward and backward jump operators σ , ρ : T T and the graininess μ : T + are defined, respectively, by
σ ( t ) = inf { s T : s > t } , ρ ( t ) = sup { s T : s < t } , μ ( t ) = σ ( t ) - t .

A point t T is called left-dense if t > inf T and ρ(t) = t, left-scattered if ρ(t) < t, right-dense if t < sup T and σ(t) = t, and right-scattered if σ(t) > t. If has a left-scattered maximum m, then T k = T \ { m } ; otherwise T k = T . If has a right-scattered minimum m, then T k = T \ { m } ; otherwise T k = T .

A function f : T 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 y : T and t T k , 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
| [ y ( σ ( t ) ) - y ( s ) ] - y Δ ( t ) [ σ ( t ) - s ] | < ε | σ ( t ) - s |

for all s U.

Let y be right-dense continuous, if Y Δ(t) = y(t), then we define the delta integral by
a t y ( s ) Δ s = Y ( t ) - Y ( a ) .

A function p : T is called regressive provided 1+µ(t)p(t) ≠ 0 for all t T k . The set of all regressive and rd-continuous functions p : T will be denoted by R = R ( T ) = R ( T , ) . We define the set R + = R + ( T , ) = { p R : 1 + μ ( t ) p ( t ) > 0 , t T } .

A n × n-matrix-valued function A on a time scale is called regressive provided I + µ(t)A(t) is invertible for all t T , and the class of all such regressive and rd-continuous functions is denoted, similar to the above scalar case, by R = ( T ) = R ( T , n × n ) .

If r is a regressive function, then the generalized exponential function e r is defined by
e r ( t , s ) = exp s t ξ μ ( τ ) ( r ( τ ) ) Δ τ
for all s, t T , with the cylinder transformation
ξ h ( z ) = Log ( 1 + hz ) h , if h 0 , z , if h = 0 .
Definition 2.1 ([1, 3]). Let p, q : T be two regressive functions, define
p q = p + q + μ p q , p = - p 1 + μ p , p q = p ( q ) .
Lemma 2.1 ([1, 3]). Assume that p, q : T are two regressive functions, then
  1. (i)

    e0(t, s) 1 and e p (t, t) 1;

     
  2. (ii)

    e p (σ(t), s) = (1 + µ(t)p(t))e p (t, s);

     
  3. (iii)

    e p ( t , s ) = 1 e p ( s , t ) = e p ( s , t ) ; e p ( t , s ) e p ( s , r ) = e p ( t , r ) ;

     
  4. (iv)

    (ep(t, s))Δ = (p)(t)ep(t, s);

     
  5. (v)

    If a, b, c T , then a b p ( t ) e p ( c , σ ( t ) ) Δ t = e p ( c , a ) - e p ( c , b ) .

     
Definition 2.2 ([36]). For every x, y , [ x , y ) = { t : x t < y } , define a countably additive measure m1 on the set
F 1 = { [ ã , b ̃ ) T : ã , b ̃ T , ã b ̃ } ,
that assigns to each interval [ ã , b ̃ ) T its length, that is,
m 1 ([ ã , b ̃ )) =  b ̃ - ã .
The interval [ ã , a ) is understood as the empty set. Using m1, they generate the outer measure m 1 * on P ( T ) , defined for each E P ( T ) as
m 1 * ( E ) = inf R ̃ i I R ̃ ( b ̃ i - ã i ) + , b E , + , b E ,
with
R ̃ = { [ ã i , b ̃ i ) T F 1 } i I R ̃ : I R ̃ N , E i I R ̃ ( [ a i , b i ) T ) .
A set A T is said to be Δ-measurable if the following equality:
m 1 * ( E ) = m 1 * ( E A ) + m 1 * ( E ( T \ A ) )
holds true for all subset E of . Define the family
M ( m 1 * ) = { A T : A i s Δ - m e a s u r a b l e } ,

the Lebesgue Δ-measure, denoted by µΔ, is the restriction of m 1 * to M ( m 1 * ) .

Definition 2.3 ([35]). A time scale is called an almost periodic time scale if
Π : = { τ : t ± τ T , t T } { 0 } .

Remark 2.1. In the following, we always use to denote an almost periodic time scale.

Throughout this paper, E n denotes n or n , D denotes an open set in E n or D = E n , S denotes an arbitrary compact subset of D.

Definition 2.4 ([35]). Let be an almost periodic time scale. A function f C ( T × D , E n ) is called an almost periodic function in t T uniformly for x D if the ε-translation set of f
E { ε , f , S } = { τ Π : | f ( t + τ , x ) - f ( t , x ) | < ε , f o r a l l ( t , x ) T × S }
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
| f ( t + τ , x ) - f ( t , x ) | < ε , f o r a l l t T × S .

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 f C ( T × D , E n ) and f(t, x) be almost periodic in t uniformly for x D, we denote
a ( f , λ , x ) : = lim T + 1 T t 0 t 0 + T f ( t , x ) e - i λ t Δ t , where t 0 T , T Π ,
(3.1)

where λ , i = - 1 . Obviously, for a fixed (f, λ, x), a ( f , λ , x ) E n .

Definition 3.1. a(f(t, 0, x)) is called mean-value of f(t, x) if
0 < a ( f , 0 , x ) = lim T 1 T t 0 t 0 + T f ( t , x ) Δ t < + .

Theorem 3.1. For any λ , 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 { t i } i + , where t i = it1. We will prove that the sequence { 1 t i t 0 t 0 + t i f ( t , x ) Δ t } i + converges uniformly with respect to x S.

For any integers m, n and x S, taking t m , t n , we have
1 t n t 0 t 0 + t n f ( t , x ) Δ t - 1 t m t 0 t 0 + t m f ( t , x ) Δ t 1 t m t n t m t 0 t 0 + t n f ( t , x ) Δ t - t 0 t 0 + t m n f ( t , x ) Δ t + 1 t m t n t 0 t 0 + t m n f ( t , x ) Δ t - t n t 0 t 0 + t m f ( t , x ) Δ t t 1 t m t n k = 1 m t 0 t 0 + t n f ( t , x ) Δ t - t ( k - 1 ) n t 0 + t k n f ( t , x ) Δ t + k = 1 n t ( k - 1 ) m t 0 + t k m f ( t , x ) Δ t - t 0 t 0 + t m f ( t , x ) Δ t .
(3.2)
Consider the following integral form:
t a t 0 + t a + s f ( t , x ) Δ t - t 0 t 0 + t s f ( t , x ) Δ t ,
(3.3)

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 l = l ( ε 4 , S ) be an inclusion length of E ( f , ε 4 , S ) and τ E ( f , ε 4 , S ) [ t a - t 0 , t a - t 0 + l ] , then, for all x S, we get***
t a t 0 + t a + s f ( t , x ) Δ t - t 0 t 0 + t s f ( t , x ) Δ t = t 0 + τ t 0 + τ + t s - t 0 t 0 + t s + t 0 + τ + t s t 0 + t a + s + t a t 0 + τ f ( t , x ) Δ t t 0 t 0 + t s | f ( t + τ , x ) - f ( t , x ) | Δ t + t 0 + τ + t s t 0 + t a + s | f ( t , x ) | Δ t + t a t 0 + τ | f ( t , x ) | Δ t ε t s 4 + 2 l G ,
(3.4)
where G = sup ( t , x ) T × S | f ( t , x ) | . According to (3.4), we can reduce (3.2) to the following:
1 t n t 0 t 0 + t n f ( t ) Δ t - 1 t m t 0 t 0 + t m f ( t ) Δ t < t 1 t m t n m ε t n 4 + 2 l G + n ε t m 4 + 2 l G = ε 2 + 2 l G t 1 1 m + 1 n 0 , t m , t n + .

By the Cauchy convergence criterion, the sequence 1 t i t 0 t 0 + t i f ( t , x ) Δ t i 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 < Ttn+1, so for all x S, we have
t 0 t 0 + T f ( t , x ) Δ t - t 0 t 0 + t n f ( t , x ) Δ t G ( T - t n ) G t 1 .
Therefore,
1 T t 0 t 0 + T f ( t , x ) Δ t - 1 t n t 0 t 0 + t n f ( t , x ) Δ t < 1 T t 0 t 0 + T f ( t , x ) Δ t - t 0 t 0 + t n f ( t , x ) Δ t + 1 t n - 1 T t 0 t 0 + t n | f ( t , x ) | Δ t G t 1 T + 1 t n - 1 T t n G < 2 G n 0 , t n + .
Hence,
a ( f , 0 , x ) = lim n + 1 t n t 0 t 0 + t n f ( t , x ) Δ t  uniformly for x S .

Besides, for 1 T t 0 t 0 + T f ( t , x ) Δ t is continuous with respect to x S, where S is an arbitrary compact set in E n , 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 f ( t , x ) C ( T × D , E n ) is almost periodic in t uniformly for x D, then
lim T + 1 T α α + T f ( t , x ) e - i λ t Δ t : = m ( f ( t , x ) , λ , x )
uniformly exists for α T and
m ( f ( t , x ) , λ , x ) = a ( f ( t + α , x ) e - i λ α , λ , x ) .
Proof. For m(f, λ, x) = m(f(t, x)e −iλt , 0, x), it suffices to show that, for x S, α T , the following uniformly exists:
m ( f , 0 , x ) = lim T + 1 T α α + T f ( t , x ) Δ t .
(3.5)
Take l = l ( ε 4 , S ) and τ E { ε 4 , f , S } [ α - t 0 , α - t 0 + l ] , G = sup ( t , x ) T × S | f ( t , x ) | , for x S, we obtain
1 T α α + T f ( t , x ) Δ t - 1 T t 0 t 0 + T f ( t , x ) Δ t = 1 T t 0 + τ t 0 + τ + T - t 0 t 0 + T + t 0 + τ + T α + T + α t 0 + τ f ( t , x ) Δ t 1 T t 0 t 0 + T | f ( t + τ , x ) - f ( t , x ) | Δ t + t 0 + τ + T α + T | f ( t , x ) | Δ t + α t 0 + τ | f ( t , x ) | Δ t 1 T ε T 4 + 2 l G T = ε 4 + 2 l G T
(3.6)
and
1 n T t 0 t 0 + n T f ( t , x ) Δ t - 1 T t 0 t 0 + T f ( t , x ) Δ t = 1 n k = 1 n 1 T t 0 + ( k - 1 ) T t 0 + k T f ( t , x ) Δ t - t 0 t 0 + T f ( t , x ) Δ t ε 4 + 2 l G T .
(3.7)
From (3.7), let n, for x S, we have
a ( f , 0 , x ) - 1 T t 0 t 0 + T f ( t , x ) Δ t ε 4 + 2 l G T .
(3.8)
Using trigonometric inequality, according (3.6) and (3.8), we can take T > 8 l G ε such that
1 T α α + T f ( t , x ) Δ t - a ( f , 0 , x ) ε 2 + 4 l G T < ε .
Hence, we can easily obtain that (3.5) uniformly exists for α T and m(f, 0, x) = a(f, 0, x) = a(f(t, x), 0, x). Furthermore,
1 T α α + T f ( t , x ) Δ t = 1 T t 0 t 0 + T f ( t + α , x ) Δ t .
Therefore, a(f(t + α, x), 0, x) uniformly exists for α T 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
m ( f ( t , x ) , λ , x ) = m ( f ( t , x ) e - i λ t , 0 , x ) = a ( f ( t + α , x ) e - i λ ( t + α ) , 0 , x ) = a ( f ( t + α , x ) e - i λ α , λ , x ) .

Hence, m(f(t, x), λ, x) uniformly exists for α T . This completes the proof. □

In Theorem 3.1 and Theorem 3.2, if we take λ = 0, then we have
a ( f ( t , x ) , 0 , x ) = lim T + 1 T t 0 t 0 + T f ( t , x ) Δ t : = m t ( f ( t , x ) )
(3.9)
and
a ( f ( t + α ) , 0 , x ) = lim T + 1 T t 0 t 0 + T f ( t + α , x ) Δ t : = m t ( f ( t + α , x ) )
(3.10)

uniformly converge for x S and for x S, α T , 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,
m t ( e i λ t ) = lim T + 1 T t 0 t 0 + T e i λ t Δ t = 0 , w h e r e t 0 T .
(3.11)
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 [ t 0 , t 0 + T ] = i = 0 n [ σ ( t i ) , t i + 1 ] , where
t 0 < t 1 < t 2 < < t n = t 0 + T
are right scattered points. Then
m t ( e i λ t ) = lim T + 1 T t 0 t 0 + T e i λ t Δ t = lim T + 1 T i = 0 n - 1 t i σ ( t i ) e i λ t Δ t + σ ( t i ) t i + 1 e i λ t d t = lim T + 1 T i = 0 n - 1 μ ( t i ) ( cos λ t i + i sin λ t i ) + 1 λ [ ( sin λ t i + 1 - sin λ σ ( t i ) ) + i ( cos λ σ ( t i ) - cos λ t i + 1 ) ] ,

since sin t and cos t are bounded for t , one can easily see that (3.11) holds. The proof is complete. □

Theorem 3.3. Let f ( t , x ) C ( T × D , E n ) 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 ,
m t f ( t , x ) - k = 1 N b k e i λ k t 2 = m t ( | f ( t , x ) | 2 ) - k = 1 N | a ( f , λ k , x ) | 2 + k = 1 N | b k - a ( f , λ k , x ) | 2 .
(3.12)
Proof. Note that | f ( t , x ) | 2 = f ( t , x ) , f ( t , x ) ¯ is almost periodic in t for x D where 〈,〉 denotes the usual inner product in E n and f t , x ¯ denotes the conjugate of f(t, x), so it has a mean-value, thus
m t f ( t , x ) - k = 1 N b k e i λ k t 2 = m t f ( t , x ) - k = 1 N b k e i λ k t , f ( t , x ) ¯ - k = 1 N b k e - i λ k t = m t ( | f ( t , x ) | 2 ) - k = 1 N b k ¯ , a ( f , λ k , x ) - k = 1 N b k , a ( f , λ k , x ) ¯ + l = 1 N j = 1 N b l , b j ¯ m t ( e i ( λ l - λ j ) t ) ,
by Lemma 3.1, it is easy to obtain that
m t f ( t , x ) - k = 1 N b k e i λ k t 2 = m t ( | f ( t , x ) | 2 ) - k = 1 N b k ¯ , a ( f , λ k , x ) - k = 1 N b k , a ( f , λ k , x ) ¯ + j = 1 N b j , b j = m t ( | f ( t , x ) | 2 ) - k = 1 N | a ( f , λ k , x ) | 2 + k = 1 N | b k - a ( f , λ k , x ) | 2 .

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:
m t f ( t , x ) - k = 1 N b k e i λ k t 2 = m t ( | f ( t , x ) | 2 ) - k = 1 N | a ( f , λ k , x ) | 2 .

By Corollary 3.1, one can easily get the following corollary:

Corollary 3.2. Let f ( t , x ) C ( T × D , E n ) be almost periodic in t uniformly for x D, then
k = 1 N | a ( f , λ k , x ) | 2 m t ( | f ( t , x ) | 2 ) .

Theorem 3.4. Let f ( t , x ) C ( T × D , E n ) 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 t , x T × S , there exists M > 0 such that |f(t, x)|M. Therefore, for any n , the real number set { λ : | a f , λ , x | > 1 n } is finite (If it is infinite, then k = 1 | a ( f , λ k , x ) | 2 > k = 1 1 n + , this contradicts Corollary 3.2). Hence, for any fixed x S, one can obtain the real number set {λ : a(f, λ, x) ≠ 0} is countable. Furthermore, by Corollary 3.2, one can see that
k = 1 N sup x S | a ( f , λ k , x ) | 2 M 2 .

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 f : T × D n is a non-negative almost periodic function in t uniformly for x D and f 0, then a(f, 0, x) > 0.

Proof. Let f ( t 0 , x ) = M > 0 and pick δ > 0 so that f ( t , x ) 2 M 3 on ( t 0 - δ , t 0 + δ ) × S . Let l Π be an inclusion length of E { M 3 , f , S } and take l > 2δ (In fact, one can choose 0 < τ0 Π such that 0 = l Π, n is some positive integer). If h Π, t 0 T , find τ E { M 3 , f , S } [ h + δ - t 0 , h + δ - t 0 + l ] . Then t 0 - δ + τ [ h , h + l ] . Either t 0 + τ or t 0 - 2 δ + τ [ h , h + l ] since l > 2δ. In the first case if t ( t 0 - δ + τ , t 0 + τ ) then
| f ( t , x ) | | f ( t + τ , x ) | - | f ( t + τ , x ) - f ( t , x ) | 2 M 3 - M 3 = M 3 .
The second case can be handled similarly. In either case t 0 + h t 0 + h + l f ( t , x ) Δ t > M 3 δ since on a subinterval of length δ , f ( t , x ) M 3 . Now write h = (n -1) l to get t 0 + ( n - 1 ) l t 0 + n l f ( t , x ) Δ t > M 3 δ . Hence
1 N l t 0 t 0 + N l f ( t , x ) Δ t = 1 N l n = 1 N t 0 + ( n - 1 ) l t 0 + n l f ( t , x ) Δ t > M δ 3 l .

Letting N one can get a ( f , 0 , x ) M δ 3 l > 0 . The proof is complete. □

4. Pseudo almost periodic functions on time scales

Let B C ( T × D , E n ) denote the space of all bounded continuous functions from T × D to E n . Set
A P ( T × D ) n = { g C ( T × D , E n ) : g  is almost periodic in  t  uniformly for  x D } , A P ( T ) n = { g C ( T , E n ) : g  is almost periodic} ,
P ̃ A P 0 ( T ) n = φ B C ( T , E n ) : φ  is  Δ  - measurable such that  lim r + 1 2 r t 0 - r t 0 + r | φ ( s ) | Δ s = 0 , where  t 0 T , r Π
and
P ̃ A P 0 ( T × D ) n = φ B C ( T × D , E n ) : φ ( , x ) P ̃ A P 0 ( T )  for each  x D  and lim r + 1 2 r t 0 - r t 0 + r | | φ ( s , x ) | | Δ s = 0  uniformly for  x D ,  where  t 0 T , r Π .
Remark 4.1. φ P ̃ A P 0 ( T ) does not require lim | t | φ ( t ) exists. Consider, for example, let T = n = 1 [ n , n + 1 n ] and
φ ( t ) = 1 n , n t n + 1 n , 0 , t e l s e w h e r e .
Obviously, for any fixed n 0 and t T , one can easily see that t ± n 0 T , thus n0 ∏, that is, is an almost periodic time scale. It is clear that lim|t|→∞φ(t) does not exist, noting that { n + 1 n } n are right scattered points, so
lim r 1 2 r t 0 - r t 0 + r | φ ( s ) | Δ s = lim n 1 n k = 1 n k k + 1 k 1 k d s + k = 1 n μ ( k + 1 k ) 1 k = lim n 1 n k = 1 n 1 k 1 k + 1 - 1 k 1 k = lim n 1 n k = 1 n 1 k = 0 .

Hence φ P ̃ A P 0 ( T ) .

Definition 4.1. A function f C ( T × D , E n ) is called pseudo almost periodic in t uniformly for x D if f = g + φ, where g A P ( T × D ) n and φ P ̃ A P 0 ( T × D ) n .

Remark 4.2. Note that g and φ are uniquely determined. Indeed, since
N ( φ ) = lim r + 1 2 r t 0 - r t 0 + r | | φ ( s , x ) | | Δ s = 0 ,

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 P ̃ A P ( T × D ) n the set of pseudo almost periodic functions uniformly for x D.

Example 4.1. Let T = k = 1 [ 2 k , 2 k + 1 ] ,

f(t) = g(t) + φ(t), where g(t) = sin t + sin πt, φ ( t ) = - 1 t σ ( t ) , t T

and

F (t, x) = f(t) cos x, t T .

Since
lim r + 1 2 r t 0 - r t 0 + r | φ ( s ) | Δ s = lim r + 1 2 r t 0 - r t 0 + r 1 s σ ( s ) Δ s = lim r + 1 2 r 1 s t 0 - r t 0 + r = 0 ,

so, φ P ̃ A P 0 ( T ) . Therefore, f P ̃ A P ( T ) , F P ̃ A P ( T × D ) .

Theorem 4.1. If f P ̃ A P ( T × D ) n , then
lim r + 1 2 r t 0 - r t 0 + r f ( s , x ) Δ s : = M ( f )

exists and is finite. It is the mean value of f. Moreover M(f) = M(g).

Proof. Indeed
lim r + 1 2 r t 0 - r t 0 + r f ( s , x ) Δ s = lim r + 1 2 r t 0 - r t 0 + r g ( s , x ) Δ s + lim r + 1 2 r t 0 - r t 0 + r φ ( s , x ) Δ s .
Since g A P ( T × D ) n then
lim r + t 0 - r t 0 + r g ( s , x ) Δ s
exists and is finite by Theorem 3.1. Furthermore, one has
- | φ ( s , x ) | φ ( s , x ) | φ ( s , x ) | .
Then
- lim r + 1 2 r t 0 - r t 0 + r | φ ( s , x ) | Δ s lim r + 1 2 r t 0 - r t 0 + r φ ( s , x ) Δ s lim r + 1 2 r t 0 - r t 0 + r | φ ( s , x ) Δ s .
Since φ P ̃ A P 0 ( T × D ) n ,
lim r + 1 2 r t 0 - r t 0 + r | φ ( s , x ) | Δ s = 0 = M ( φ ) .
Hence
lim r + 1 2 r t 0 - r t 0 + r φ ( s , x ) Δ s = 0 .

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 f P ̃ A P ( T × D ) n then a(f, λ, x) = a(g, λ, x).

Furthermore, from Definition 4.1, one can easily show that

Theorem 4.2. If f P ̃ A P ( T × D ) n and g is the almost periodic component of f, then we have
g ( T × S ) f ( T × S ) ¯
and
f g inf ( t , x ) T × S g ( t , x ) inf ( t , x ) T × S f ( t , x ) ,

where f T × S and g T × S denote the value field of f and g on T × S , respectively, f T × S ¯ denotes the closure of f T × S , 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
μ Δ ( C ( [ t 0 - r , t 0 + r ] T ) ) 2 r 0 a s r , w h e r e t 0 T .

By the definition of P ̃ A P 0 ( T × D ) n , the proof of the following theorem is straightforward.

Theorem 4.3. A function φ P ̃ A P 0 ( T × D ) n if and only if for ε > 0, the set C ε = { t T : φ ( t , x ) ε } is an ergodic zero subset in .

Theorem 4.4. (i) A function φ P ̃ A P 0 ( T × D ) if and only if φ 2 P ̃ A P 0 ( T × D ) .

(ii) Φ P ̃ A P 0 ( T × D ) n if and only if the norm function Φ ( , x ) P ̃ A P 0 ( T × D ) .

Proof. (i) The sufficiency follows since
1 2 r t 0 - r t 0 + r φ ( s , x ) Δ s 1 2 r t 0 - r t 0 + r φ ( t , x ) 2 Δ s 1 / 2 t 0 - r t 0 + r 1 Δ s 1 / 2 = 1 2 r t 0 - r t 0 + r φ ( s , x ) 2 Δ s 1 / 2 .
The necessity follows from the fact that
1 2 r t 0 - r t 0 + r φ ( s , x ) 2 Δ s φ 1 2 r t 0 - r t 0 + r φ ( t , x ) Δ s ,

since φ is bounded on . Therefore, one can easily see that (i) is satisfied.

(ii) By (i), Φ = ( φ 1 , φ 2 , , φ n ) P A P 0 ( T × D ) n if and only if φ i φ ̄ i P ̃ A P 0 ( T × D ) , i = 1 , 2 , , n . The latter is equivalent to Φ ( , x ) 2 = i = 1 n φ ( , x ) 2 P ̃ A P 0 ( T × D ) , which again by (i), is equivalent to Φ ( , x ) P ̃ A P 0 ( T × D ) . □

The proof is complete.

For H = ( h 1 , h 2 , , h n ) E n , suppose that H(t) D for all t T . Define H × ι : T T × D by
H × ι ( t ) = ( t , h 1 ( t ) , h 2 ( t ) , , h n ( t ) ) .

For F = ( f 1 , f 2 , , f n ) P ̃ A P ( T × D ) n , 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 f C ( T × D , E n ) is said to be continuous in x S uniformly for t T 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 t T .

Theorem 4.5. Suppose that the function f P ̃ A P ( T × D ) n is continuous in x S uniformly for t T and F P ̃ A P ( T ) n such that F ( T ) D , then f ( F × ι ) P ̃ A P ( T ) n , where F ( T ) 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 G = ( g 1 , g 2 , , g n ) A P ( T ) n as above. Note that
f ( F × ι ) = g ( F × ι ) + φ ( F × ι ) = g ( G × ι ) + [ g ( F × ι ) - g ( G × ι ) + φ ( F × ι ) ] .

It follows from Theorem 4.2 that G T F T ¯ D . By Theorem 3.15 in [35], we have g ( G × ι ) A P ( T ) n . 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 P ̃ A P 0 ( T ) n .

First we show that g ( F × ι ) - g ( G × ι ) P ̃ A P 0 ( T ) n .

It is trivial in the case that g = 0. So we assume that g ≠ 0. Set D 1 = F ( T ) ¯ . By Theorem 3.1 in [35], the function g is uniformly continuous on T × D 1 . For ε > 0, there exists a δ > 0 such that
g ( t , x 1 ) - g ( t , x 2 ) < ε 2 , x 1 , x 2 D 1 , x 1 - x 2 < δ , t T .
(4.1)
Set
C δ = { t T : F ( t ) - G ( t ) = Φ ( t ) δ } .
(4.2)
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 rT,
μ Δ ( ( [ t 0 - r , t 0 + r ] T ) C δ ) 2 r < ε 4 g .
(4.3)
By (4.1), (4.2) and (4.3), we have
1 2 r t 0 - r t 0 + r g ( s , F ( s ) ) - g ( s , G ( s ) ) Δ s = 1 2 r ( [ t 0 - r , t 0 + r ] T ) \ C δ + ( [ t 0 - r , t 0 + r ] T ) C δ g ( s , F ( s ) ) - g ( s , G ( s ) ) Δ s ε 2 + 2 g μ Δ ( ( [ t 0 - r , t 0 + r ] T ) C δ ) 2 r < ε .

Therefore, g ( F × ι ) - g ( G × ι ) P ̃ A P 0 ( T ) n .

Finally, we show that φ ( F × ι ) P ̃ A P 0 ( T ) n . Note that f = g + φ and g is uniformly continuous on T × D 1 . By the hypothesis, f is continuous in S D1 uniformly for t T ; so is φ. Since D1 is compact in E n , one can find, say m, open balls O k with center x k D1, k = 1, 2, ..., m, and radius δ(x k , ε/2) such that D 1 k = 1 m O k and
φ ( t , x ) - φ ( t , x k ) < ε 2 , x O k , t T .
(4.4)
The set
B k = { t T : F ( t ) O k }
(4.5)

is open and T = k = 1 m B k . Let E k = B k \ j = 1 k - 1 B j , then E k E j = when kj, 1 ≤ k, jm.

Since for each φ ( , x ( k ) ) P ̃ A P 0 ( T ) n , there is a number T0 > 0 such that
k = 1 m 1 2 r t 0 - r t 0 + r φ ( s , x ( k ) ) Δ s < ε 2 , r T 0 .
(4.6)
It follows from (4.4), (4.5) and (4.6) that
1 2 r t 0 - r t 0 + r φ ( s , F ( s ) ) Δ s 1 2 r k = 1 m E k ( [ t 0 - r , t 0 + r ] T ) φ ( s , F ( s ) ) - φ ( s , x ( k ) ) + φ ( s , x ( k ) + Δ s ε 2 + k = 1 m 1 2 r t 0 - r t 0 + r φ ( s , x ( k ) ) Δ s < ε .

This shows that φ ( F × ι ) P ̃ A P 0 ( T ) n . The proof is complete. □

Define
E 0 ( T × D ) n = { f C ( T × D , E n ) : f ( t , x ) 0 , uniformly in x D , as | t | } .
E 0 ( T ) n = { f C ( T , E n ) : f ( t ) 0 , as | t | } .

Definition 4.4. Let A A P ( T × D ) n denote all the functions f of the form f = g + φ, where g A P ( T × D ) n and φ E 0 ( T × D ) n . The members of A A P ( T × D ) n are called asymptotically almost periodic functions in t uniformly for x D.

It is obvious that E 0 ( T × D ) n P ̃ A P 0 ( T × D ) n and A A P ( T × D ) n P ̃ A P ( T × D ) n .

Corollary 4.2. If f A A P ( T × D ) n and F A A P ( T ) n such that F ( T ) D , then f ( F × ι ) A A P ( T ) n .

Proof. Obliviously,
f ( F × ι ) = g ( F × ι ) + φ ( F × ι ) = g ( G × ι ) + [ g ( F × ι ) - g ( G × ι ) + φ ( F × ι ) ] ,

where g ( G × ι ) A P ( T ) n . By the hypothesis that Φ = F - G E 0 ( T ) n and φ E 0 ( T × D ) n , it follows that g ( F × ι ) - g ( G × ι ) E 0 ( T ) n since the uniform continuity of g and φ ( F × ι ) E 0 ( T ) n since φ ( t , F ( t ) ) sup x D φ ( t , x ) . The proof is complete. □

Theorem 4.6. Suppose that g A P ( T × D ) n satisfies that for every ε > 0,
μ Δ { t : g ( t , x ) > - ε , t [ t 0 - r , t 0 + r ] T } 2 r 1 , a s r + , w h e r e t 0 T , r Π .

Then g ≥ 0 for all T × S , where S is an arbitrary compact subset of D.

Proof. Suppose that the conclusion does not hold. This implies that g ( t 0 , x ) < 0 for some t 0 . Choose ε > 0, ε < - g ( t 0 , x ) .

By continuity, there exists δ > 0 so that t - t 0 δ 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 ε 2 -almost period τ with the property that
g ( t + τ , x ) - g ( t , x ) < ε 2 .
Choose a sequence τ k of almost periods, τ k [ t 0 + k l , t 0 + ( k + 1 ) l ] , we have
g ( t + τ k , x ) < - ε 2 , and t [ t 0 - δ , t 0 + δ ] T and every k .
Denote M = | t 0 | + δ , we have
μ Δ { t [ t 0 - k l - M , t 0 + k l + M ] T : g ( t , x ) < - ε 2 } 2 k δ .
Therefore,
μ Δ { t [ t 0 - k l - M , t 0 + k l + M ] T : g ( t , x ) < - ε 2 } 2 k l + 2 M 2 k δ 2 k l + 2 M .

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 f C ( T × D , E n ) , f = g + φ , where g A P ( T × D ) n and φ P ̃ A P 0 ( T × D ) n , then
  1. (i)

    If lim | t | φ ( t , x ) exists, then lim | t | φ ( t , x ) = 0 .

     
  2. (ii)

    For all ( t , x ) T × S , 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 α ̃ > 0 and c Π such that φ ( t , x ) > α ̃ for tc, which yields
1 r t 0 t 0 + r φ ( s , x ) Δ s = 1 r t 0 t 0 + c φ ( s , x ) Δ s + t 0 + c t 0 + r φ ( s , x ) Δ s 1 r α ̃ ( r - c ) .
Passing to the limit as r, we obtain
lim r 1 2 r t 0 - r t 0 + r φ ( s , x ) Δ s α ̃ ,

which contradicts the fact that φ P ̃ A P 0 ( T × D ) n .

(ii) Assuming f ≥ 0, we want to show that g ≥ 0. We have f = g + φ with
lim r 1 2 r t 0 - r t 0 + r φ ( s , x ) Δ s = 0 .
Thus, there exists {c n }n Π, c n → + as n such that g(t + c n , x) → g(t, x) for all ( t , x ) T × S . Furthermore, for any ε > 0 and r > 0, one can easily get
μ Δ { t [ t 0 - r , t 0 + r ] T : φ ( t , x ) > ε } 0 , as r ,
which implies that
μ Δ { t : g ( t , x ) > - ε , t [ t 0 - r , t 0 + r ] T } 2 r 1 , as r + , where t 0 T , r Π .

By Theorem 4.6, one can have g(t, x) ≥ 0 for all ( t , x ) T × S .

The proof is complete. □

5 Pseudo almost periodic solutions of dynamic equations on time scales

Consider the non-autonomous equation
x Δ = A ( t ) x + F ( t )
(5.1)
and its associated homogeneous equation
x Δ = A ( t ) x ,
(5.2)

where the n × n coefficient matrix A(t) is continuous on and column vector F = (f1, f2, ..., f n ) T is in E n . Define F = sup t T F ( t ) . 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
X ( t ) P X - 1 ( s ) K e α ( t , s ) , s , t T , t s , X ( t ) ( I - P ) X - 1 ( s ) K e α ( s , t ) , s , t T , t s .
(5.3)
Lemma 5.1. Let α > 0, then for any fixed s T and s = −∞, one has the following:
e α ( t , s ) 0 , t + .
Proof. If µ(t) > 0, since α R + , we have
1 + μ ( t ) α = 1 + μ ( t ) - α 1 + μ ( t ) α = 1 1 + μ ( t ) α < 1 .
Thus, α R + and it is easy to have
Log ( 1 + μ ( t ) α ) for all t T .
So
ξ μ ( t ) ( α ) = Log ( 1 + μ ( t ) α ) μ ( t ) < 0 .
Hence
e α ( t , s ) = exp s t ξ μ ( t ) ( α ) Δ t 0 as t + .

If μ(t) = 0, one can easily get the conclusion. If s = -∞, it is easy to see that s t ξ μ ( t ) ( α ) Δ t - as t → +∞, thus, eα(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 F P ̃ A P 0 ( T ) n . Then (5.1) has a unique bounded solution x P ̃ A P 0 ( T ) n .

Proof. Similar to the proof of Theorem 4.1 in [35], by checking directly, one can see that the function:
x ( t ) = - t X ( t ) P X - 1 ( σ ( s ) ) F ( s ) Δ s - t + X ( t ) ( I - P ) X - 1 ( σ ( s ) ) F ( s ) Δ s
(5.4)
is a solution of (5.1). Now, we show that the solution is bounded. It follows from (5.4) that
x ( t ) = sup t T - t X ( t ) P X - 1 ( σ ( s ) ) F ( s ) Δ s - t + X ( t ) ( I - P ) X - 1 ( σ ( s ) ) F ( s ) Δ s sup t T - t e α ( t , σ ( s ) ) s + t + e α ( σ ( s ) , t ) Δ s K F 1 α - 1 α K F = 2 + μ α α K F ,

where = sup t T . 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 x P ̃ A P 0 ( T ) n . Let I ( t ) = - t X ( t ) P X - 1 ( σ ( s ) ) F ( s ) Δ s and H ( t ) = t + X ( t ) ( I - P ) X - 1 ( σ ( s ) ) F ( s ) Δ s . Then x = I + H. It follows from (5.3) and Theorem 2.15 in [38] that
1 2 r t 0 - r t 0