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Weighted piecewise pseudo almost automorphic functions with applications to abstract impulsive -dynamic equations on time scales

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Abstract

In the present paper, by introducing the concept of equipotentially almost automorphic sequence, the concept of weighted piecewise pseudo almost automorphic functions on time scales is proposed. Some first results about their basic properties are obtained and some composition theorems are established. Then we apply these to investigate the existence of weighted piecewise pseudo almost automorphic mild solutions to abstract impulsive -dynamic equations on time scales. In addition, the exponential stability of weighted piecewise pseudo almost automorphic mild solutions is also considered. Finally, the obtained results are applied to the study of a class of -partial differential equations on time scales.

MSC:34N05, 35B15, 43A60, 12H20, 35R12.

1 Introduction

Almost automorphic functions, which are more general than almost periodic functions, were introduced by Bochner in relation to some aspects of differential geometry (see [13]). For more details as regards this topic we refer to the recent books [46], where the authors gave important overviews about the theory of almost automorphic functions and their applications to differential equations. Almost automorphic and pseudo almost automorphic solutions in the context of differential equations had been studied by several authors [721]. N’Guérékata [13] and Xiao [15, 21] with their collaborators established the existence and uniqueness theorems of pseudo almost automorphic solutions to some semilinear abstract differential equations. Recently, Blot et al. [22] introduced the concept of weighted pseudo almost automorphic functions, which generalizes the concept of weighted pseudo almost periodicity [2326], and the author proved some interesting properties of the space of weighted pseudo almost automorphic functions like the completeness and the composition theorem, which have many applications in the context of differential equations. For other contributions to the study of weighted pseudo almost automorphy, we refer the reader to [2730] and references therein.

On the other hand, the theory of time scales, which has recently received a lot of attention, was introduced by Hilger in his PhD thesis in 1988 [31] in order to unify continuous and discrete analysis. This theory represents a powerful tool for applications to economics, population models, and quantum physics among others. In fact, the progressive field of dynamic equations on time scales contains links to and extends the classical theory of differential and difference equations. For instance, by choosing the time scale to be the set of real numbers, the general result yields a result for differential equations. In a similar way, by choosing the time scale to be the set of integers, the same general result yields a result for difference equations. However, since there are many other time scales than just the set of real numbers or the set of integers, one has a much more general result. For these reasons, based on the concept of almost periodic time scales proposed in [32, 33], the concept of weighted pseudo almost automorphic functions on almost periodic time scales was formally introduced by Wang and Li (2013) in [34]. Moreover, some first results were proven which concern the weighted pseudo almost automorphic mild solution to abstract Δ-dynamic equations on time scales. In addition, by using the results obtained in [32, 33], Lizama and Mesquita [35] presented some new results about basic properties of almost automorphic functions on time scales and proved the existence and uniqueness of an almost automorphic solution to a class of Δ-dynamic equations.

For another thing, many phenomena in nature are characterized by the fact that their states are subject to sudden changes at certain moments and therefore can be described by impulsive system (see [36, 37]). Many evolution processes, optimal control models in economics, stimulated neural networks, population models, artificial intelligence, and robotics are characterized by the fact that at certain moments of time they undergo abrupt changes of state. The existence of almost periodic solutions of abstract impulsive differential equations has been considered by many authors; see [3841].

However, to the best of our knowledge, the concept of weighted piecewise pseudo almost automorphic functions on time scales has not been introduced in any literature until now, so there was no work on discussing weighted piecewise pseudo almost automorphic problems of impulsive dynamic equations on time scales before. Therefore, in this paper, by introducing the concept of equipotentially almost automorphic sequence, the concept of weighted piecewise pseudo almost automorphic functions on time scales is proposed. The first results about their basic properties are obtained and some composition theorems are established. Then we apply these composition theorems to investigate the existence of weighted piecewise pseudo almost automorphic mild solutions to abstract impulsive -dynamic equations as follows:

{ x ( t ) = A ( t ) x ϱ + f ( t , x ( t ) ) , t T , t t i , i Z , Δ x ( t i ) = x ( t i + ) x ( t i ) = I i ( x ( t i ) ) , t = t i ,
(1)

where A PC ld (T,X) is a linear operator in the Banach space X and f PC ld (T×X,X), x ϱ =x(ϱ(t)). f, I i , t i satisfy suitable conditions that will be established later and T is an almost periodic time scale. In addition, the notations x( t i + ) and x( t i ) represent the right-hand and the left-hand side limits of x() at t i , respectively. In addition, some useful lemmas are obtained and the exponential stability of weighted piecewise pseudo almost automorphic mild solutions is also considered. Finally, we apply these obtained results to study a class of -partial differential equations on time scales.

2 Preliminaries

In the following, we will introduce some basic knowledge of time scales which is very useful to the proof of our relative results.

A time scale T is a closed subset of . It follows that the jump operators σ,ϱ:TT defined by σ(t)=inf{sT:s>t} and ϱ(t)=sup{sT:s<t} (supplemented by infϕ:=supT and supϕ:=infT) are well defined. The point tT is left-dense, left-scattered, right-dense, right-scattered if ϱ(t)=t, ϱ(t)<t, σ(t)=t, σ(t)>t, respectively. If T has a right-scattered minimum m, define T k :=Tm; otherwise, set T k =T. By the notations [ a , b ] T , [ a , b ) T and so on, we will denote time scale intervals

[ a , b ] T ={tT:atb},

where a,bT with a<ϱ(b).

The graininess function is defined by ν:T[0,): ν(t):=tϱ(t), for all tT.

Definition 2.1 ([42])

The function f:TR is called ld-continuous provided that it is continuous at each left-dense point and has a right-sided limit at each point, write f C ld (T)= C ld (T,R). Let t T k , the Delta derivative of f at t such that

|f ( ϱ ( t ) ) f(s) f (t) [ ϱ ( t ) s ] |ε|ϱ(t)s|

for all sU, at fixed t. Let F be a function, it is called the antiderivative of f:TR provided F (t)=f(t) for each t T k . If F (t)=f(t), then we define the delta integral by

a t f(s)s=F(t)F(a).

Definition 2.2 ([42])

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

Definition 2.3 ([42])

If r is a regressive function, then the generalized exponential function e ˆ r is defined by

e ˆ r (t,s)=exp { s t ξ ˆ ν ( τ ) ( r ( τ ) ) τ }

for all s,tT, where the ν-cylinder transformation is as in

ξ ˆ h (z):= 1 h Log(1zh).

Lemma 2.1 ([42])

Assume that p,q:TR are two ν-regressive functions, then

  1. (i)

    e ˆ 0 (t,s)1 and e ˆ p (t,t)1;

  2. (ii)

    e ˆ p (ϱ(t),s)=(1ν(t)p(t)) e ˆ p (t,s);

  3. (iii)

    e ˆ p (t,s)= 1 e ˆ p ( s , t ) = e ν p (s,t);

  4. (iv)

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

  5. (v)

    ( e ˆ ν p ( t , s ) ) =( ν p)(t) e ˆ ν p (t,s).

Lemma 2.2 ([43])

For each t 0 T in T T k the single-point set { t 0 } is -measurable and its -measure is given by μ ({ t 0 })= t 0 ϱ( t 0 ).

Lemma 2.3 ([43])

If a,bT and ab, then

μ ( ( a , b ] T ) =ba, μ ( ( a , b ) T ) =ϱ(b)a.

If a,bT T k and ab, then

μ ( [ a , b ) T ) =ϱ(b)ϱ(a), μ ( [ a , b ] T ) =bϱ(a).

For more details of time scales and -measurability, one is referred to [42, 43]. For more on time scales, see [4449].

Definition 2.4 ([3234])

A time scale T is called an almost periodic time scale if

Π:={τR:t±τT,tT}{0}.

Remark 2.4 Definition 3.1 introduced in [35] is the same as the concept of almost periodic time scales proposed in [32, 34], and T is also called an invariant time scale under translations in [35].

After these preparations, in the next section, we will introduce the concept of weighted piecewise pseudo almost automorphic functions on time scales in a Banach space and some of their basic properties are investigated.

3 Weighted piecewise pseudo almost automorphic functions on time scales

In the following, we will give the definition of ld-piecewise continuous functions on time scales.

Definition 3.1 We say φ:TX is ld-piecewise continuous with respect to a sequence { τ i }T which satisfy τ i < τ i + 1 , iZ, if φ(t) is continuous on ( τ i , τ i + 1 ] T and ld-continuous on T{ τ i }. Furthermore, ( τ i , τ i + 1 ] T are called intervals of continuity of the function φ(t).

For convenience, PC ld (T,X) denotes the set of all ld-piecewise continuous functions with respect to a sequence { τ i }, iZ. Similar to Definition 3.1, we can also introduce the concept of functions which belong to PC rd (T,X).

Throughout the paper, we denote by X a Banach space; let B be the set consisting of all sequences { t i } i Z such that θ= inf i Z ( t i + 1 t i )>0. For { t i } i Z B, let BPC ld (T,X) be the space formed by all bounded ld-piecewise continuous functions ϕ:TX such that ϕ() is continuous at t for any t { t i } i Z and ϕ( t i )=ϕ( t i ) for all iZ; let Ω be a subset of X and let BPC ld (T×Ω,X) be the space formed by all bounded piecewise continuous functions ϕ:T×ΩX such that, for any xΩ, ϕ(,x) BPC ld (T×X,X). For any tT, ϕ(t,) is continuous at xΩ.

Let UPC(T,X) be the space of all functions φ PC ld (T,X) such that ϕ satisfies the condition: for any ε>0, there exists a positive number δ=δ(ε) such that if the points t , t belong to the same interval of continuity of φ and | t t |<δ implies φ( t )φ( t )<ε.

Now, we introduce the set

B= { { t k } : t k T , t k < t k + 1 , k Z , lim t ± = ± } ,

which denotes all unbounded increasing sequences of real numbers. Let T,PB and let s(TP):BB be a map such that the set s(TP) forms a strictly increasing sequence. For DR and ε>0, we introduce the notations θ ε (D)={t+ε:tD}, F ε (D)= ε { θ ε (D)}. Denote by ϕ ˜ =(φ(t),T) the element from the space PC ld (T,X)×B. For every sequence of real numbers { s n }, n=1,2, with θ s n ϕ ˜ :=(φ(t+ s n ),T s n ), we shall consider the sets {φ(t+ s n ),T s n } PC ld ×B, where

T s n ={ t k s n :kZ,n=1,2,}.

Definition 3.2 Let { t i }B, iZ. We say { t i j } is a derivative sequence of { t i } and

t i j = t i + j t i ,i,jZ.

Definition 3.3 Let t i j = t i + j t i , i,jZ. We say { t i j }, i,jZ, is equipotentially almost automorphic on an almost periodic time scale T if, for any sequence { s n }Z, there exists a subsequence { s n } such that

lim n t k s n = γ k

is well defined for each kZ and

lim n γ k s n = t k

for each kZ.

Definition 3.4 A function ϕ PC ld (T,X) is said to be ld-piecewise almost automorphic if the following conditions are fulfilled:

  1. (i)

    T={ t k } is an equipotentially almost automorphic sequence.

  2. (ii)

    Let φ PC ld (T,X) be a bounded function with respect to a sequence T={ t k }. We say that φ is piecewise almost automorphic if from every sequence { s n } n = 1 Π, we can extract a subsequence { τ n } n = 1 such that

    ϕ ˜ = ( φ ( t ) , T ) = lim n ( φ ( t + τ n ) , T τ n ) = lim n θ τ n ϕ ˜

is well defined for each tT and

ϕ ˜ = ( φ ( t ) , T ) = lim n ( φ ( t τ n ) , T + τ n ) = lim n θ τ n ϕ ˜

for each tT. Denote by AA(T,X) the set of all such functions.

  1. (iii)

    A bounded function f PC ld (T×X,X) with respect to a sequence T={ t k } is said to be piecewise almost automorphic if f(t,x) is piecewise automorphic in tT uniformly in xB, where B is any bounded subset of X. Denote by AA(T×X,X) the set of all such functions.

Similarly, we can also introduce the concept of piecewise almost automorphic functions which belong to PC rd (T,X).

Let U be the set of all functions ρ:T(0,) which are positive and locally -integrable over T. For a given r[0,)Π and t 0 T, set

m(r,ρ, t 0 ):= t 0 r t 0 + r ρ(s)s
(2)

for each ρU.

Remark 3.1 In (2), if T=R, t 0 =0, one can easily get

m(r,ρ, t 0 ):= r r ρ(s)ds

if T=Z, t 0 =0, one has the following:

m(r,ρ)= k = r + 1 r ρ(k).

Define

U : = { ρ U : lim r m ( r , ρ , t 0 ) = } , U B : = { ρ U : ρ  is bounded and  inf s T ρ ( s ) > 0 } .

It is clear that U B U U. Now for ρ U define

PAA 0 ( T , ρ ) : = { ϕ BPC ld ( T , X ) : lim r 1 m ( r , ρ , t 0 ) t 0 r t 0 + r ϕ ( s ) ρ ( s ) s = 0 , t 0 T , r Π } .

Similarly, we define

PAA 0 ( T × X , ρ ) : = { Φ BPC ld ( T × Ω , X ) : lim r 1 m ( r , ρ , t 0 ) t 0 r t 0 + r Φ ( s , x ) ρ ( s ) s = 0 uniformly with respect to  x K , t 0 T , r Π } .

We are now ready to introduce the sets WPAA(T,ρ) and WPAA(T×X,ρ) of weighted pseudo almost periodic functions:

WPAA ( T , ρ ) = { f = g + ϕ PC ld ( T , X ) : g AA ( T , X )  and  ϕ PAA 0 ( T , ρ ) } , WPAA ( T × X , ρ ) = { f = g + ϕ PC ld ( T × X , X ) : g AA ( T × X , X ) WPAA ( T × X , ρ ) =  and  ϕ PAA 0 ( T × X , ρ ) } .

Lemma 3.2 Let ϕ BPC ld (T,X). Then ϕ PAA 0 (T,ρ) where ρ U B if and only if, for every ε>0,

lim r 1 m ( r , ρ , t 0 ) μ ( M r , ε , t 0 ( ϕ ) ) =0,

where rΠ and M r , ε , t 0 (ϕ):={t [ t 0 r , t 0 + r ] T :ϕ(t)ε}.

Proof (a) Necessity. For contradiction, suppose that there exists ε 0 >0 such that

lim r 1 m ( r , ρ , t 0 ) μ ( M r , ε 0 , t 0 ( ϕ ) ) 0.

Then there exists δ>0 such that, for every nN,

1 m ( r n , ρ , t 0 ) μ ( M r n , ε 0 , t 0 ( ϕ ) ) δfor some  r n >n,where  r n Π.

So we get

1 m ( r n , ρ , t 0 ) t 0 r t 0 + r ϕ ( s ) ρ ( s ) s = 1 m ( r n , ρ , t 0 ) M r n , ε 0 , t 0 ( ϕ ) ϕ ( s ) ρ ( s ) s + 1 m ( r n , ρ , t 0 ) × [ t 0 r , t 0 + r ] T M r n , ε 0 , t 0 ( ϕ ) ϕ ( s ) ρ ( s ) s 1 m ( r n , ρ , t 0 ) M r n , ε 0 , t 0 ( ϕ ) ϕ ( s ) ρ ( s ) s ε 0 m ( r n , ρ , t 0 ) M r n , ε 0 , t 0 ( ϕ ) ϕ ( s ) ρ ( s ) s ε 0 δ γ ,

where γ= inf s T ρ(s). This contradicts the assumption.

  1. (b)

    Sufficiency. Assume that lim r 1 m ( r , ρ , t 0 ) μ ( M r , ε , t 0 (ϕ))=0. Then for every ε>0, there exists r 0 >0 such that, for every r> r 0 ,

    1 m ( r , ρ , t 0 ) μ ( M r , ε , t 0 ( ϕ ) ) < ε K M ,

where M:= sup t T ϕ(t)< and K:= sup t T ρ(t)<.

Now, we have

1 m ( r , ρ , t 0 ) t 0 r t 0 + r ϕ ( s ) ρ ( s ) s = 1 m ( r , ρ , t 0 ) ( M r , ε , t 0 ( ϕ ) ϕ ( s ) ρ ( s ) s + [ t 0 r , t 0 + r ] T M r , ε , t 0 ( ϕ ) ϕ ( s ) ρ ( s ) s ) M K m ( r , ρ , t 0 ) μ ( M r , ε , t 0 ( ϕ ) ) + ε m ( r , ρ , t 0 ) [ t 0 r , t 0 + r ] T M r , ε , t 0 ( ϕ ) ρ ( s ) s 2 ε .

Therefore, lim r 1 m ( r , ρ , t 0 ) t 0 r t 0 + r ϕ(s)ρ(s)s=0, that is, ϕ PAA 0 (T,ρ). This completes the proof. □

Lemma 3.3 PAA 0 (T,ρ) is a translation invariant set of BPC ld (T,X) with respect to Π if ρ U B , i.e., for any sΠ, one has ϕ(t+s):= θ s ϕ PAA 0 (T,ρ) if ρ U B .

Proof For any sΠ, ϕ PAA 0 (T,ρ), ε>0, r>0, we have

M r , ε , t 0 ( T s ϕ ) = { t [ t 0 r , t 0 + r ] T : T s ( t ) ε } = { t [ t 0 r , t 0 + r ] T : ϕ ( t + s ) ε } = { t [ t 0 r + s , t 0 + r + s ] T : ϕ ( t ) ε } { t [ t 0 r | s | , t 0 + r + | s | ] T : ϕ ( t ) ε } .

Hence

1 m ( r , ρ , t 0 ) μ ( M r , ε , t 0 ( T s ϕ ) ) 1 m ( r , ρ , t 0 ) μ ( M r + | s | , ε , t 0 ( T s ϕ ) ) = m ( r + | s | , ρ , t 0 ) m ( r , ρ , t 0 ) 1 m ( r + | s | , ρ , t 0 ) μ ( M r + | s | , ε , t 0 ( ϕ ) ) .

Since ϕ PAA 0 (T,ρ), by Lemma 3.2, we have

1 m ( r + | s | , ρ , t 0 ) ( M r + | s | , ε , t 0 ( ϕ ) ) 0,r.

Furthermore, lim r m ( r + | s | , ρ , t 0 ) m ( r , ρ , t 0 ) =1, thus

1 m ( r , ρ , t 0 ) μ ( M r , ε , t 0 ( T s ( ϕ ) ) ) 0,r.

Again, using Lemma 3.2, one can get θ s ϕ PAA 0 (T,ρ) for any sΠ. This completes the proof. □

By Definition 3.4, one can easily get the following lemma.

Lemma 3.4 Let ϕAA(T,X), then the range of ϕ, ϕ(T), is a relatively compact subset of X.

Lemma 3.5 If f=g+ϕ with gAA(T,X), and ϕ PAA 0 (T,ρ), where ρ U B , then g(T) f ( T ) ¯ .

Proof (1) For any tT{ t i }, g(t)g(T), one has g(t)=f(t)ϕ(t). Since gAA(T,X), there exists a sequence { α n }Π such that g(t+ α n )g(t), n.

Furthermore, by Lemma 3.3, ϕ(t+ α n ) PAA 0 (T,X), so there exists βΠ such that ϕ(t+ α n +β)0, n. Hence, let s=t+β, and one has

f(s+ α n β)ϕ(t+ α n +β)g(t)for each tT as n,

i.e. f(s+ α n β)g(t) for each tT as n.

  1. (2)

    If { t i }B, noting that Definition 3.4, the above sequence { α n }Π and the number βΠ is suitable for the increasing sequence { t i }, so the proof process is the same as (1). This completes the proof. □

Lemma 3.6 The decomposition of a weighted piecewise pseudo almost automorphic function according to AA PAA 0 is unique for any ρ U B .

Proof Assume that f= g 1 + ϕ 1 and f= g 2 + ϕ 2 . Then ( g 1 g 2 )+( ϕ 1 ϕ 2 )=0. Since g 1 g 2 AA(T,X), and ϕ 1 ϕ 2 PAA 0 (T,ρ), in view of Lemma 3.5, we deduce that g 1 g 2 =0. Consequently, ϕ 1 ϕ 2 =0, i.e. ϕ 1 = ϕ 2 . This completes the proof. □

Theorem 3.7 For ρ U B , (WPAA(T,ρ), ) is a Banach space.

Proof Assume that { f n } n N is a Cauchy sequence in WPAA(T,ρ). We can write uniquely f n = g n + ϕ n . Using Lemma 3.5, we see that g p g q f p f q , from which we deduce that { g n } n N is a Cauchy sequence in AA(T,X). Hence, ϕ n = f n g n is a Cauchy sequence in PAA 0 (T,ρ). We deduce that g n gAA(T,X), ϕ n ϕ PAA 0 (T,ρ), and finally f n g+ϕWPAA(T,ρ). This completes the proof. □

Definition 3.5 Let ρ 1 , ρ 2 U . One says that ρ 1 is equivalent to ρ 2 , written ρ 1 ρ 2 if ρ 1 / ρ 2 U B .

Theorem 3.8 Let ρ 1 , ρ 2 U . If ρ 1 ρ 2 , then WPAA(T, ρ 1 )=WPAA(T, ρ 2 ).

Proof Assume that ρ 1 ρ 2 . There exist a,b>0 such that a ρ 1 ρ 2 b ρ 1 . So

am(r, ρ 1 , t 0 )m(r, ρ 2 , t 0 )bm(r, ρ 1 , t 0 ),

where rΠ and

a b 1 m ( r , ρ 1 , t 0 ) t 0 r t 0 + r ϕ ( s ) ρ 1 ( s ) s 1 m ( r , ρ 2 , t 0 ) t 0 r t 0 + r ϕ ( s ) ρ 2 ( s ) s b a 1 m ( r , ρ 1 , t 0 ) t 0 r t 0 + r ϕ ( s ) ρ 1 ( s ) s .

This completes the proof. □

Lemma 3.9 If gAA(T×X,X) and αAA(T,X), then G(t):=g(,α())AA(T,X).

Proof Let T={ t i }, ϕ ˜ =(g(t,x),T)AA(T×X,X)×B, from every sequence { s n } n = 1 Π, we can extract a subsequence { τ n } n = 1 such that

ϕ ˜ := ( g ( t , x ) , T ) = lim n θ τ n ϕ ˜ = lim n ( g ( t + τ n , x ) , T τ n )

uniformly exists on PC ld (T×X,X)×B. Since αAA(T,X), one can extract { τ n }{ τ n } such that

lim n θ τ n ϕ ˜ = lim n ( g ( t + τ n , α ( t + τ n ) ) , T τ n ) = lim n ( g ( t + τ n , α ( t ) ) , T τ n ) = ( g ( t , α ( t ) ) , T ) .

Hence, GAA(T,X). This completes the proof. □

Theorem 3.10 Let f=g+ϕWPAA(T×X,ρ), where gAA(T×X,X), ϕ PAA 0 (T×X,ρ), ρ U B , and the following conditions hold:

  1. (i)

    {f(t,x):tT,xK} is bounded for every bounded subset KΩ.

  2. (ii)

    f(t,), g(t,) are uniformly continuous in each bounded subset of Ω uniformly in tT.

Then f(,h())WPAA(T,ρ) if hWPAA(T,ρ) and h(T)Ω.

Proof We have f=g+ϕ, where gAA(T×X,X) and ϕ PAA 0 (T×X,ρ) and h= ϕ 1 + ϕ 2 , where ϕ 1 AA(T,X) and ϕ 2 PAA 0 (T,ρ). Hence, the function f(,h()) can be decomposed as

f ( , h ( ) ) = g ( , ϕ 1 ( ) ) + f ( , h ( ) ) g ( , ϕ 1 ( ) ) = g ( , ϕ 1 ( ) ) + f ( , h ( ) ) f ( , ϕ 1 ( ) ) + ϕ ( , ϕ 1 ( ) ) .

By Lemma 3.9, g(, ϕ 1 ())AA(T,X). Now, consider the function

Ψ():=f ( , h ( ) ) f ( , ϕ 1 ( ) ) .

Clearly, Ψ BPC ld (T,X). For Ψ to be in PAA 0 (T,ρ), it is sufficient to show that

lim r 1 m ( r , ρ , t 0 ) μ ( M r , ε , t 0 ( Ψ ) ) =0.

Let K be a bounded subset of Ω such that ϕ(T)K, ϕ 1 (T)K. By (ii), f(t,) is uniformly continuous in ϕ(T) uniformly in tT, and we see that, for given ε>0, there exists δ>0 such that y 1 , y 2 K and y 1 y 2 <δ implies that

f ( t , y 1 ) f ( t , y 2 ) <ε,tT.

Thus, for each tT, ϕ 2 (t)<δ implies for all tT,

f ( t , h ( t ) ) f ( t , ϕ 1 ( t ) ) <ε,

where ϕ 2 (t)=h(t) ϕ 1 (t). For r>0 and any fixed t 0 T, let M r , δ , t 0 ( ϕ 2 )={t [ t 0 r , t 0 + r ] T : ϕ 2 δ}, we can obtain

1 m ( r , ρ , t 0 ) μ ( M r , ε , t 0 ( Ψ ( t ) ) ) = 1 m ( r , ρ , t 0 ) μ ( M r , ε , t 0 ( f ( t , h ( t ) ) f ( t , ϕ 1 ( t ) ) ) ) 1 m ( r , ρ , t 0 ) μ ( M r , δ , t 0 ( h ( t ) ϕ 1 ( t ) ) ) = 1 m ( r , ρ , t 0 ) μ ( M r , δ , t 0 ( ϕ 2 ( t ) ) ) .

Now since ϕ 2 PAA 0 (T,ρ), Lemma 3.2 yields

lim r 1 m ( r , ρ , t 0 ) μ ( M r , ε , t 0 ( ϕ 2 ( t ) ) ) =0,

and this implies that Ψ PAA 0 (T,ρ).

Finally, we need to show ϕ(, ϕ 1 ()) PAA 0 (T,ρ). Note that f=g+ϕ and g(t,) is uniformly continuous in ϕ 1 (T) uniformly in tT. By the assumption (ii), f(t,) is uniformly continuous in ϕ 1 (T) uniformly in tT, so is ϕ. Since ϕ 1 (T) is relatively compact in X, for ε>0, there exists δ>0 such that ϕ 1 (T) k = 1 m B k , where B k ={xX:x x k <δ} for some x k ϕ 1 (T) and

ϕ ( t , ϕ 1 ( t ) ) ϕ ( t , x k ) < ε 2 , ϕ 1 (t) B k ,tT.
(3)

It is easy to see that the set U k :={tT: ϕ 1 (t) B k } is open and ϕ 1 (T)= k = 1 m U k . Define

V 1 = U 1 , V k = U k i = 1 k 1 U i ,2km.

Then it is clear that V i V j if ij, 1i,jm. So we get

{ t [ t 0 r , t 0 + r ] T : ϕ ( t , ϕ 1 ( t ) ) ε 2 } k = 1 m { t V k : ϕ ( t , ϕ 1 ( t ) ) ϕ ( t , x k ) + ϕ ( t , x k ) ε } k = 1 m ( { t V k : ϕ ( t , ϕ 1 ( t ) ) ϕ ( t , x k ) ε 2 } { t V k : ϕ ( t , x k ) ε 2 } ) .

In view of (3), it follows that

{ t V k : ϕ ( t , ϕ 1 ( t ) ) ϕ ( t , x k ) ε 2 } =,k=1,2,,m.

Thus we get

1 m ( r , ρ , t 0 ) μ ( M r , ε , t 0 ( ϕ ( t , ϕ 1 ( t ) ) ) ) k = 1 m 1 m ( r , ρ , t 0 ) μ ( M r , ε , t 0 ( ϕ ( t , x k ) ) ) .

Since ϕ PAA 0 (T×X,ρ) and lim r 1 m ( r , ρ , t 0 ) μ ( M r , ε , t 0 (ϕ(t, x k )))=0, it follows that

lim r 1 m ( r , ρ , t 0 ) μ ( M r , ε , t 0 ( ϕ ( t , ϕ 1 ( t ) ) ) ) =0,

by Lemma 3.2, ϕ(, ϕ 1 ()) PAA 0 (T,ρ). This completes the proof. □

Theorem 3.10 has the following consequence.

Corollary 3.11 Let f=g+ϕWPAP(T,ρ), where ρ U B . Assume that f and g are Lipschitzian in xX uniformly in tT. Then f(,h())WPAA(T,ρ) if hWPAA(T,ρ).

Next, we will show the following two lemmas, which are useful in the proof of our results.

Lemma 3.12 If φ PC ld (T,X) is an almost automorphic function with respect to the sequence T and { t k }T is equipotentially almost automorphic satisfying inf i Z t i q =θ>0, qZ, then {φ( t k )} is an almost automorphic sequence in X.

Proof Let t i j = t i + j t i , i,jZ. Obviously, from the definition of Π, it is easy to know that t i j Π. Since φ PC ld (T,X) is an almost automorphic function and { t k }T is equipotentially almost automorphic, from Definition 3.3 and Definition 3.4, for any sequence { s n }Z, we find that there exists a subsequence { s n } such that

lim n ( φ ( t k + s n ) , T t k s n ) = lim n ( φ ( t k + t k s n ) , T t k s n ) = ( φ ( t k ) , T ) = ( φ ( t k + γ k ) , T γ k )

and

lim n ( φ ( t k s n ) , T + t k s n ) = lim n ( φ ( t k s n + γ k s n ) , T γ k s n + t k s n ) = ( φ ( t k ) , T ) .

Hence, {φ( t k )} is an almost automorphic sequence in X. This completes the proof. □

Lemma 3.13 A necessary and sufficient condition for a bounded sequence { a n } to be in PAA 0 (Z,ρ) is that there exists a uniformly continuous function f PAA 0 (T,ρ) such that f( t n )= a n , t n T, nZ, ρ U B .

Proof Necessity. We define a function

f(t)= a n +(t t 0 nr)( a n + 1 a n ), t 0 +nrt< t 0 +(n+1)r,tT,nZ, t 0 T,

where rΠ. It is obviously uniformly continuous on T. f PAA 0 (T,ρ) since

1 m ( k r , ρ , t 0 ) t 0 k r t 0 + k r f ( s ) ρ ( s ) s = 1 m ( k r , ρ , t 0 ) j = k k 1 t 0 + j r t 0 + ( j + 1 ) r a j + ( s t 0 j r ) ( a j + 1 a j ) ρ ( s ) s 1 m ( k r , ρ , t 0 ) ρ ̲ j = k k 1 ( a j ρ ( t j ) r + a j + 1 a j t 0 + j r t 0 + ( j + 1 ) r ( s t 0 j r ) ρ ( s ) s ) 1 ρ ̲ m ( k r , ρ , t 0 ) j = k k 1 r a j ρ ( t j ) + ( a k + a k ) r 2 m ( k r , ρ , t 0 ) ρ ¯ 1 ρ ̲ t j [ t 0 k r , t 0 + k r ] T ν ( t j ) ρ ( t j ) j = k k 1 r f ( t j ) ρ ( t j ) + a k + a k m ( k r , ρ , t 0 ) r 2 ρ ¯ = 1 ρ ̲ j = k k 1 ν ( t j ) ρ ( t j ) j = k k 1 r f ( t j ) ρ ( t j ) + a k + a k m ( k r , ρ , t 0 ) r 2 ρ ¯ 0 as  k ,

where ρ ̲ = inf t T ρ(t), ρ ¯ = sup t T ρ(t).

Sufficiency. Let 0<ε<1, there exists δ>0 such that, for t ( t n δ , t n ) T , nZ, we have

f ( t ) ρ(t)(1ε) f ( t n ) ρ( t n ),nZ.

Without loss of generality, let t n 0, t n <0, nZ, there exist r n , r n Π R + such that t 0 + r n = t n , t 0 r n = t n . Let r n =max{ r n , r n }Π. Therefore,

t 0 r n t 0 + r n f ( t ) ρ ( t ) t t 0 r n t 0 + r n f ( t ) ρ ( t ) t = t n t n f ( t ) ρ ( t ) t j = n + 1 n t 0 + t j 1 t 0 + t j f ( t ) ρ ( t ) t j = n + 1 n t 0 + t j δ t 0 + t j f ( t ) ρ ( t ) t j = n + 1 n δ ( 1 ε ) f ( t j ) ρ ( t j ) δ ( 1 ε ) j = n + 1 n f ( t j ) ρ ( t j ) ,

so one can obtain

1 m ( r n , ρ , t 0 ) t 0 r n t 0 + r n f ( t ) ρ(t)tδ(1ε) 1 m ( r n , ρ , t 0 ) j = n + 1 n f ( t j ) ρ( t j ),
(4)

it is easy to see that r n is increasing with respect to n Z + , one can find some n 0 >n such that

m ( r n , ρ , t 0 ) = t 0 r n t 0 + r n ρ(s)s t j [ t 0 r n 0 , t 0 + r n 0 ] T ν( t j )ρ( t j )= j = n 0 + 1 n 0 ν( t j )ρ( t j ),
(5)

from (4) and (5), we have

1 m ( r n , ρ , t 0 ) t 0 r n t 0 + r n f ( t ) ρ(t)tδ(1ε) 1 j = n 0 + 1 n 0 ν ( t j ) ρ ( t j ) j = n 0 + 1 n 0 ϕ 2 ( t j ) ρ( t j ),
(6)

noting that n implies n 0 , since f PAA 0 (T,ρ), it follows from the inequality (6) that f( t n )= a n PAA 0 (Z,ρ). This completes the proof. □

By Lemma 3.13, we can straightforwardly get the following theorem.

Theorem 3.14 A necessary and sufficient condition for a bounded sequence { a n } to be in WPAA(Z,ρ) is that there exists a uniformly continuous function fWPAA(T,ρ) such that f( t n )= a n , t n T, nZ, ρ U B .

Theorem 3.15 Assume that ρ U B and the sequence of vector-valued functions { I i } i Z is weighted pseudo almost automorphic, i.e., for any xΩ, { I i (x),iZ} is weighted pseudo almost automorphic sequence. Suppose { I i (x):iZ,xK} is bounded for every bounded subset KΩ, I i (x) is uniformly continuous in xΩ uniformly in iZ. If hWPAA(T,ρ)UPC(T,X) such that h(T)Ω, then I i (h( t i )) is a weighted pseudo almost automorphic sequence.

Proof Fix hWPAA(T,ρ)UPC(T,X), first we show h( t i ) is weighted pseudo almost automorphic. Since h= ϕ 1 + ϕ 2 , where ϕ 1 AA(T,X), ϕ 2 PAA 0 (T,ρ). It follows from Lemma 3.12 that the sequence ϕ 1 ( t i ) is almost automorphic. To show that h( t i ) is weighted pseudo almost automorphic, we need to show that ϕ 2 ( t i ) PAA 0 (Z,ρ). By the assumption, h, ϕ 1 UPC(T,X), so is ϕ 2 . Let 0<ε<1, there exists δ>0 such that, for t ( t i δ , t i ) T , iZ, we have

ϕ 2 ( t ) ρ(t)(1ε) ϕ 2 ( t i ) ρ( t i ),iZ.

Without loss of generality, let t i 0, t i <0, iZ; there exists r i , r i Π R + such that t 0 + r i = t i , t 0 r i = t i . Let r i =max{ r i , r i }Π. Therefore,

t 0 r i t 0 + r i ϕ 2 ( t ) ρ ( t ) t t 0 r i t 0 + r i ϕ 2 ( t ) ρ ( t ) t = t i t i ϕ 2 ( t ) ρ ( t ) t j = i + 1 i t 0 + t j 1 t 0 + t j ϕ 2 ( t ) ρ ( t ) t j = i + 1 i t 0 + t j δ t 0 + t j ϕ 2 ( t ) ρ ( t ) t j = i + 1 i δ ( 1 ε ) ϕ 2 ( t j ) ρ ( t j ) δ ( 1 ε ) j = i + 1 i ϕ 2 ( t j ) ρ ( t j ) ,

so one can obtain

1 m ( r i , ρ , t 0 ) t 0 r i t 0 + r i ϕ 2 ( t ) ρ(t)tδ(1ε) 1 m ( r i , ρ , t 0 ) j = i + 1 i ϕ 2 ( t j ) ρ( t j ),
(7)

it is easy to see that r i is increasing with respect to i Z + , and one can find some i 0 >i such that

m ( r i , ρ , t 0 ) = t 0 r i t 0 + r i ρ(s)s t j [ t 0 r i 0 , t 0 + r i 0 ] T ν( t j )ρ( t j )= j = i 0 + 1 i 0 ν( t j )ρ( t j ),
(8)

from (7) and (8), we have

1 m ( r i , ρ , t 0 ) t 0 r i t 0 + r i ϕ 2 ( t ) ρ(t)tδ(1ε) 1 j = i 0 + 1 i 0 ν ( t j ) ρ ( t j ) j = i 0 + 1 i 0 ϕ 2 ( t j ) ρ( t j ),
(9)

noting that i implies i 0 , since ϕ 2 PAA 0 (T,ρ), it follows from the inequality (9) that ϕ 2 ( t i ) PAA 0 (Z,ρ). Hence, h( t i ) is weighted pseudo almost automorphic.

Now, we show that I i (ϕ( t i )) is weighted pseudo almost automorphic. Let

I ( t , x ) = I n ( x ) + ( t t 0 n r ) [ I n + 1 ( x ) I n ( x ) ] , t 0 + n r t < t 0 + ( n + 1 ) r , n Z , r Π , Φ 0 ( t ) = h ( t n ) + ( t t 0 n r ) [ h ( t n + 1 ) h ( t n ) ] , t 0 + n r t < t 0 + ( n + 1 ) r , n Z , r Π .

Since I n , h( t n ) both are pseudo almost automorphic, by Lemma 3.13 and Theorem 3.14, we know that IWPAA(T×Ω,ρ), Φ 0 WPAA(T,ρ). For every tT, there exists a number nZ such that |t t 0 nr|r,

I ( t , x ) I n ( x ) + | t t 0 n r | [ I n + 1 ( x ) + I n ( x ) ] ( 1 + r ) I n ( x ) + r I n + 1 ( x ) .

Since { I n (x):nZ,xK} is bounded for every bounded set KΩ, {I(t,x):tT,xK} is bounded for every bounded set KΩ. For every x 1 , x 2 Ω, we have

I ( t , x 1 ) I ( t , x 2 ) I n ( x 1 ) I n ( x 2 ) + | t t 0 n r | [ I n + 1 ( x 1 ) I n + 1 ( x 2 ) + I n ( x 1 ) I n ( x 2 ) ] ( 1 + r ) I n ( x 1 ) I n ( x 2 ) + r I n + 1 ( x 1 ) I n + 1 ( x 2 ) .

Noting that I i (x) is uniformly continuous in xΩ uniformly in iZ, we then find that I(t,x) is uniformly in xΩ uniformly in tT. Then by Theorem 3.10, I(, Φ 0 ())WPAA(T,X). Again, using Lemma 3.13 and Theorem 3.14, we find that I(i, Φ 0 (i)) is a weighted pseudo almost automorphic sequence, that is, I i (h( t i )) is weighted pseudo almost automorphic. This completes the proof. □

From Theorem 3.15, one can easily get the following corollary.

Corollary 3.16 Assume the sequence of vector-valued functions { I i } i Z is weighted pseudo almost automorphic, ρ U B , if there is a number L>0 such that

I i ( x ) I i ( y ) Lxy

for all x,yΩ, iZ, if hWPAA(T,ρ)UPC(T,ρ) such that h(T)Ω, then I i (h( t i )) is a weighted pseudo almost automorphic sequence.

4 Weighted piecewise pseudo almost automorphic mild solutions to abstract impulsive -dynamic equations

In this section, we investigate the existence and exponential stability of a weighted piecewise pseudo almost automorphic mild solution to Eq. (1). Before starting our investigation, we will show a lemma which will be used in our proofs.

Lemma 4.1 Let ν ω R ν + , for all tT, αΠ, there exist constants β 1 , β 2 >0 such that

β 1 ν(t)ν(t+α) β 2 ν(t).
(10)

Then there exist positive constants K and ω such that

e ˆ ν ω (t+α,s+α) K e ˆ ν ω (t,s),ts.

Proof Obviously, if ν=0, T=R, the result holds. Assume that ν0. Since ν ω R ν + , one has

e ν ω ( t + α , s + α ) = exp { s + α t + α 1 ν ( τ ) ln 1 1 ν ( τ ) ω τ } = exp { s t 1 ν ( τ + α ) ln 1 1 ν ( τ + α ) ω τ } .

Since T is an almost periodic time scale, μ is bounded. Hence, by the inequality (10), we can obtain

e ˆ ν ω ( t + α , s + α ) exp { s t 1 β 2 ν ( τ ) ln 1 1 β 1 ν ( τ ) ω τ } = { exp { s t ln ( 1 ν ( τ ) ( ν β 1 ω ) ) ν ( τ ) } } 1 β 2 .

Therefore, there exists a positive constant K >0 such that

e ˆ ν ω (t+α,s+α)= [ e ˆ ν β 1 ω ( t , s ) ] 1 β 2 K e ˆ ν ω (t,s),

where ω = β 1 ω. This completes the proof. □

Remark 4.2 It is easy to see that if T is almost periodic, then μ(t) is bounded, so there exist a sufficiently small constant β 1 >0 and a sufficiently large constant β 2 >0 such that (10) is valid. Therefore, Lemma 4.1 holds when T is an almost periodic time scale.

Let T be an almost periodic time scale, and consider the impulsive -dynamic equation

x =A(t) x ϱ ,tT,
(11)

where A:TB(X) is a linear operator in the Banach space T. We denote by B(X,Y) the Banach space of all bounded linear operators from T to Y. This is simply denoted as B(X) when X=Y.

Definition 4.1 T(t,s):T×TB(X) is called the linear evolution operator associated to (11) if T(t,s) satisfies the following conditions:

  1. (1)

    T(s,s)=Id, where Id denotes the identity operator in X;

  2. (2)

    T(t,s)T(s,r)=T(t,r);

  3. (3)

    the mapping (t,s)T(t,s)x is continuous for any fixed xX.

Definition 4.2 An evolution system T(t,s) is called exponentially stable if there exist K 0 1 and ω>0 such that

T ( t , s ) B ( X ) K 0 e ˆ ν ω (t,s),ts.

Definition 4.3 A function x:TX is called a mild solution of Eq. (1) if, for any tT, t>c, c t i , iZ, one has

x(t)=T(t,c)x(c)+ c t T(t,s)f ( s , x ( s ) ) s+ c < t i < t T(t, t i ) I i ( x ( t i ) ) .

In fact, using the semigroup theory, we can easily see that

x(t)=T(t,c)x(c)+ c t T(t,s)f ( s , x ( s ) ) s,t>c,

is a mild solution to

x =A(t) x ϱ +f ( t , x ( t ) ) .

For any cT, we can find iZ, t i 1 <c t i , for t ( c , t i ] T ,

x(t)=T(t,c)x(c)+ c t T(t,s)f ( s , x ( s ) ) s,

by using x( t i + )x( t i )= I i (x( t i )), we have

x ( t i + ) =T( t i ,c)x(c)+ c t i T( t i ,s)f ( s , x ( s ) ) s+ I i ( x ( t i ) ) ,

then we have

x ( t ) = T ( t , t i ) x ( t i + ) + t i t T ( t , s ) f ( s , u ( s ) ) s = T ( t , t i ) [ T ( t i , c ) x ( c ) + c t i T ( t i , s ) f ( s , x ( s ) ) s + I i ( x ( t i ) ) ] + t i t T ( t , s ) f ( s , x ( s ) ) s = T ( t , c ) x ( c ) + c t i T ( t , s ) f ( s , x ( s ) ) s + T ( t , t i ) I i ( x ( t i ) ) + t i t T ( t , s ) f ( s , x ( s ) ) s = T ( t , c ) x ( c ) + c t T ( t , s ) f ( s , x ( s ) ) s + T ( t , t i ) I i ( x ( t i ) ) .

Repeating this procedure, we get

x(t)=T(t,c)x(c)+ c t T(t,s)f ( s , x ( s ) ) s+ c < t i < t T(t, t i ) I i ( x ( t i ) ) .

In the following, consider the abstract differential system (1) with the following assumptions:

(H1) The family {A(t):tT} of operators in X generates an exponentially stable evolution system {T(t,s):ts}, i.e., there exist K 0 >1 and ω>0 such that

T ( t , s ) B ( X ) K 0 e ω (t,s),ts,

and for any sequence { s n }Π, there exists a subsequence { s n }{ s n } such that

lim n T ( t + s n , s + s n ) = T (t,s) is well defined for each t,sT,ts.

(H2) f=g+ϕWPAP(T,ρ), where ρ U and f(t,) is uniformly continuous in each bounded subset of Ω uniformly in tT; I i is a weighted pseudo almost periodic sequence, I i (x) is uniformly continuous in xΩ uniformly in iZ, inf i Z t i 1 =θ>0.

To investigate the existence and uniqueness of a weighted piecewise pseudo almost automorphic mild solution to Eq. (1), we need the following lemma.

Lemma 4.3 Let vAA(T,X), νAA(T, R + ), ω R ν + and (H1)-(H2) are satisfied. If u:TX is defined by

u 0 (t)= t T(t,s)v(s)s+ t i < t T(t, t i ) I i ( v ( t i ) ) ,ts,

then u 0 ()AA(T,X).

Proof Let { s n } n = 1 Π. Since v is almost automorphic, there exists a subsequence { τ n } n = 1 { s n } n = 1 such that h(t):= lim n v(t+ τ n ) is well defined for each tT.

Now, we consider

u ( t + τ n ) = t + τ n T ( t + τ n , s ) v ( s ) s = t T ( t + τ n , s + τ n ) v ( s + τ n ) s = t T ( t + τ n , s + τ n ) v n ( s ) s ,

where v n (s)=v(s+ τ n ), n=1,2, .

Since ω R ν + , one can choose sufficiently small constant β 1 >0 such that ω = β 1 ω is ν-positive regressive. Further, noting that e ˆ ν ω (t,s)(1ν(s) ω )= e ˆ ν ω (t,ϱ(s)), by (H1) and Lemma 4.1, we have

u ( t + τ n ) t T ( t + τ n , s + τ n ) v n ( s ) s t K 0 e ˆ ν ω ( t + τ n , s + τ n ) v n ( s ) s K 0 K t e ˆ ν ω ( t , s ) v n ( s ) s 1 1 ν ¯ ω K 0 K v t e ˆ ν ω ( t , ϱ ( s ) ) s = K 0 K v ( 1 ν ¯ ω ) ν ω [ e ˆ ν ω ( t , ) e ˆ ν ω ( t , t ) ] K 0 K ( 1 ν ̲ ω ) v ( 1 ν ¯ ω ) ω ,

where ν ¯ = sup t T ν(t), ν ̲ = inf t T ν.

Therefore, by the condition (H1), we have

T(t+ τ n ,s+ τ n ) T (t,s),n.

Furthermore, it is easy to see that v n (s)h(s) as n, sT and for any ts, by Lebesgue’s dominated convergence theorem, we get

lim n u(t+ τ n )= t T (t,s)h(s)s.

Moreover, we consider

u ( t + τ n ) = t i < t + τ n T ( t + τ n , t i ) I i ( v i ( t i ) ) = t i < t T ( t + τ n , t i + τ n ) I i ( v ( t i + τ n ) ) = t i < t T ( t + τ n , t i + τ n ) I i ( v i n ) ,

where v( t i + τ n ):= v i n . By Lemma 4.1, we can get

u ( t + τ n ) = t i < t + τ n T ( t + τ n , t i ) I i ( v i ( t i ) ) = t i < t T ( t + τ n , t i + τ n ) I i ( v i n ) I K 0 t i < t e ˆ ν ω ( t + τ n , t i + τ n ) I K 0 K t i < t e ˆ ν ω ( t , t i ) I K 0 K 1 e ˆ ν ω ( θ , 0 ) .

Since vAA(T,X), v i n h( t i ), n, iZ. Hence, for any t> t i , iZ, by Lebesgue’s dominated convergence theorem, we get

lim n u (t+ τ n )= t i < t T (t, t i ) I i ( h ( t i ) ) .

So we have

lim n u 0 (t+ τ n )= lim n u(t+ τ n )+ lim n u (t+ τ n )

is well defined for each tT. Therefore, u 0 ()AA(T,X). This completes the proof. □

Theorem 4.4 Let f(,ϑ())WPAA(T,ρ), where ϑWPAA(T,ρ) and {T(t,s),ts} is exponentially stable, ρ U B . Then

F():= ( ) T(,s)f ( s , ϑ ( s ) ) s+ t i < T(, t i ) I i ( ϑ ( t i ) ) WPAA(T,ρ).

Proof Fix ϑWPAA(T,X), then we have f(,ϑ())= ϕ 1 ()+ ϕ 2 (), where ϕ 1 AA(T,X), ϕ 2 PAA 0 (T,X), so

t T(t,s)f ( s , ϑ ( s ) ) s= t T(t,s) ϕ 1 (s)s+ t T(t,s) ϕ 2 (s)s:= I 1 (t)+ I 2 (t)

and

t i < t T(t, t i ) I i ( ϑ ( t i ) ) = t i < t T(t, t i ) β i + t i < t T(t, t i ) γ i := ϒ 1 (t)+ ϒ 2 (t).

By Lemma 4.3, we can easily see that I 1 , ϒ 1 AA(T,X).

Moreover, we have

1 m ( r , ρ , t 0 ) t 0 r t 0 + r I 2 ( t ) t = 1 m ( r , ρ , t 0 ) t 0 r t 0 + r t T ( t , s ) ϕ 2 ( s ) s t 1 m ( r , ρ , t 0 ) t 0 r t 0 + r t t K 0 e ˆ ν ω ( t , s ) ϕ 2 ( s ) s = 1 m ( r , ρ , t 0 ) t 0 r t 0 + r t ( t 0 r K 0 e ˆ ν ω ( t , s ) ϕ 2 ( s ) s + t 0 r t K 0 e ˆ ν ω ( t , s ) ϕ 2 ( s ) s ) = 1 m ( r , ρ , t 0 ) t 0 r ϕ 2 ( s ) s t 0 r t 0 + r K 0 e ˆ ν ω ( t , s ) s + 1 m ( r , ρ , t 0 ) t 0 r t 0 + r ϕ 2 ( s ) s s t 0 + r K 0 e ˆ ν ω ( t , s ) t : = I 1 0 + I 2 0 .

Then

I 1 0 = 1 m ( r , ρ , t 0 ) t 0 r ϕ 2 ( s ) s t 0 r t 0 + r K 0 e ˆ ν ω ( t , s ) t = 1 m ( r , ρ , t 0 ) t 0 r ϕ 2 ( s ) s t 0 r t 0 + r K 0 1 ν ( ν ω ) e ˆ ν ω ( ϱ ( t ) , s ) t 1 m ( r , ρ , t 0 ) K 0 ( 1 ν ̲ ω ) t 0 r ϕ 2 ( s ) s t 0 r t 0 + r e ˆ ω ( s , ϱ ( t ) ) t = 1 m ( r , ρ , t 0 ) K 0 ( 1 ν ̲ ω ) ω t 0 r ϕ 2 ( s ) [ e ˆ ω ( s , t 0 r ) e ˆ ω ( s , t 0 + r ) ] s 1 m ( r , ρ , t 0 ) K 0 ( 1 ν ̲ ω ) ω ϕ 2 ( t 0 r e ˆ ν ω ( t 0 r , s ) s t 0 r e ˆ ν ω ( t 0 + r , s ) s ) = 1 m ( r , ρ , t 0 ) K 0 ( 1 ν ̲ ω ) 2 ω 1 ν ω ( e ˆ ν ω ( t 0 r , ) e ˆ ν ω ( t 0 r , t 0 r ) e ˆ ν ω ( t 0 + r , ) + e ˆ ν ω ( t 0 + r , t 0 r ) ) 0 as  r

and

I 2 0 = 1 m ( r , ρ , t 0 ) t 0 r t 0 + r ϕ 2 ( s ) s s t 0 + r K 0 e ˆ ν ω ( t , s ) t = 1 m ( r , ρ , t 0 ) t 0 r t 0 + r ϕ 2 ( s ) s s t 0 + r 1 1 ν ( ν ω ) e ˆ ν ω ( ϱ ( t ) , s ) t 1 m ( r , ρ , t 0 ) K 0 ( 1 ν ̲ ω ) t 0 r t 0 + r ϕ 2 ( s ) s s t 0 + r e ˆ ω ( s , ϱ ( t ) ) t = 1 m ( r , ρ , t 0 ) K 0 ( 1 ν ̲ ω ) ω t 0 r t 0 + r ϕ 2 ( s ) [ e ˆ ω ( s , s ) e ˆ ω ( s , t 0 + r ) ] s 1 m ( r , ρ , t 0 ) K 0 ( 1 ν ̲ ω ) ω t 0 r t 0 + r ϕ 2 ( s ) s .

Since ϕ 2 PAA 0 (T,ρ), we have lim r 1 m ( r , ρ , t 0 ) t 0 r t 0 + r ϕ 2 (s)s=0. Hence, lim r I 2 0 =0.

It remains to show ϒ 2 PAA 0 (T,ρ). For any r>0, there exist i(r), j(r) such that

t i ( r ) 1 < t 0 r t i ( r ) << t j ( r ) t 0 +r< t j ( r ) + 1 .

Since γ i PAA 0 (Z,ρ), M γ i = sup i Z γ i <, and noting that, for aT, e ˆ ν ω (t,a)=(1ν(t)ω) e ˆ ω (a,ϱ(t)), we have

1 m ( r , ρ , t 0 ) t 0 r t 0 + r ϒ 2 ( t ) t = 1 m ( r , ρ , t 0 ) t 0 r t 0 + r t i < t T ( t , t i ) γ i t 1 m ( r , ρ , t 0 ) t 0 r t 0 + r t i < t K 0 e ˆ ν ω ( t , t i ) γ i t 1 m ( r , ρ , t 0 ) t i < t 0 r K 0 e ˆ ν ω ( t 0 r , t i ) γ i × t 0 r t 0 + r e ˆ ν ω ( t , t 0 r ) t + 1 m ( r , ρ , t 0 ) t 0 r < t i < t 0 + r γ i t 0 r t 0 + r K 0 e ˆ ν ω ( t , t i ) t 1 m ( r , ρ , t 0 ) t i < t 0 r K 0 ( 1 ν ̲ ω ) 2 ω M γ i e ˆ ν ω ( t 0 r , t i ) + 1 m ( r , ρ , t 0 ) t 0 r < t i < t 0 + r K 0 ( 1 ν ̲ ω ) 2 ω γ i 1 m ( r , ρ , t 0 ) K 0 M γ i ( 1 ν ̲ ω ) 2 ω 1 1 e ˆ ν ω ( θ , 0 ) + K 0 ( 1 ν ̲ ω ) 2 ω 1 m ( r , ρ , t 0 ) k = i ( r ) j ( r ) γ k .

Since γ i PAA 0 (Z,ρ), for r, m(r,ρ), we have

lim r 1 m ( r , ρ , t 0 ) k = i ( r ) j ( r ) γ k = lim r 1 k = i ( r ) j ( r ) ρ ( t k ) ν ( t k ) k = i ( r ) j ( r ) γ k =0.

Clearly, as r, one has

1 m ( r , ρ , t 0 ) K 0 M γ i ( 1 ν ̲ ω ) ω 1 1 e ˆ ν ω ( θ , 0 ) 0.

Hence

lim r 1 m ( r , ρ , t 0 ) t 0 r t 0 + r ϒ 2 ( t ) t=0.

Thus, t i < T(, t i ) I i (ϑ( t i )) PAA 0 (T,ρ), then F()WPAA(T,ρ). This completes the proof. □

Lemma 4.5 If x PC ld (T, R + ) satisfies the following inequality:

x(t)α+ ( a , t ] T p 1 (τ)x(τ)τ+ t k < t β k x( t k ),tT,

then

x(t)α t k < t (1+ β k ) e ˆ p 1 (t,a),tT.

Proof Define

r(t)=α+ ( a , t ] T p 1 (τ)x(τ)τ+ t k < t β k x( t k ),tT.

Consider

{ r ( t ) = p 1 ( t ) x ( t ) p 1 ( t ) r ( t ) , t t k , r ( a ) = α , r ( t k + ) = ( 1 + β k ) r ( t k ) .

For t ( a , t 1 ] T , we can calculate that

[ r e ˆ ν p 1 ( , t 0 ) ] = r ( t ) e ν p 1 ( ϱ ( t ) , t 0 ) + r ( t ) ( ν p 1 ) ( t ) e ˆ ν p 1 ( t , t 0 ) = r ( t ) e ν p 1 ( ϱ ( t ) , t 0 ) + r ( t ) ( p 1 ) ( t ) 1 ν ( t ) ( p 1 ) ( t ) e ˆ ν p 1 ( ϱ ( t ) , t 0 ) = [ r ( t ) ( ν ( ν p 1 ) ) ( t ) r ( t ) ] e ˆ ν p 1 ( ϱ ( t ) , t 0 ) = [ r ( t ) p 1 ( t ) r ( t ) ] e ˆ ν p 1 ( ϱ ( t ) , t 0 ) 0 .

This implies that x(t)α e ˆ p 1 (t,a). Further we have

r ( t ) r ( t i + ) e ˆ p 1 ( t , t i ) α ( 1 + β i ) t k < t i ( 1 + β k ) e ˆ p 1 ( t i , a ) e ˆ p 1 ( t , t i ) = α t k < t ( 1 + β k ) e ˆ p 1 ( t , a ) , t ( t i , t i + 1 ] T .

Thus

x(t)α t k < t (1+ β k ) e ˆ p 1 (t,a),tT.

This completes the proof. □

The following existence result is based on the contraction principle.

Theorem 4.6 Assume the following conditions hold:

(A1) The family {A(t):tT} of operators in X generates an exponentially stable evolution system {T(t,s):ts}, i.e., there exist K 0 >1 and ω>0 such that

T ( t , s ) B ( X ) K 0 e ˆ ν ω (t,s),ts,

and for any sequence { s n }Π, there exists a subsequence { s n }{ s n } such that

lim n T ( t + s n , s + s n ) = T (t,s) is well defined for each t,sT,ts.

(A2) fWPAA(T×Ω,ρ) and f satisfies the Lipschitz condition with respect to the second argument, i.e.,

f ( t , x ) f ( t , y ) L 1 xy,tT,x,yΩ.

(A3) I i is a weighted pseudo almost periodic sequence, and there exists a number L 2 >0 such that

I i ( x ) I i ( y ) L 2 xy

for all x,yΩ, iZ.

Assume that

K 0 L 1 ( 1 ν ̲ ω ) ω + K 0 L 2 1 e ˆ ν ω ( θ , 0 ) <1,

then Eq. (1) has a unique weighted piecewise pseudo almost automorphic mild solution.

Proof Consider the nonlinear operator Γ given by

Γφ= t T(t,s)f ( s , φ ( s ) ) s+ t i < t T(t, t i ) I i ( φ ( t i ) ) .

By Theorem 4.4, we see that Γ maps WPAA(T,ρ) into WPAA(T,ρ).

It suffices now to show that the operator Γ has a fixed point in WPAA(T,ρ). For φ 1 , φ 2 WPAA(T,ρ), one has the following:

Γ φ 1 ( t ) Γ φ 2 ( t ) = t