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Open Mathematics

formerly Central European Journal of Mathematics

Editor-in-Chief: Gianazza, Ugo / Vespri, Vincenzo


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Volume 16, Issue 1

Issues

Volume 13 (2015)

Stepanov-like pseudo almost periodic functions on time scales and applications to dynamic equations with delay

Chao-Hong Tang
  • Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China
  • School of Mathematics and Physics, Mianyang Teachers’ College, Mianyang, Sichuan 621000, China
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  • De Gruyter OnlineGoogle Scholar
/ Hong-Xu Li
Published Online: 2018-07-25 | DOI: https://doi.org/10.1515/math-2018-0073

Abstract

In this paper, we introduce the concept of Sp-pseudo almost periodicity on time scales and present some basic properties of it, including the translation invariance, uniqueness of decomposition, completeness and composition theorem. Moreover, we prove the seemingly simple but nontrivial result that pseudo almost periodicity implies Stepanov-like pseudo almost periodicity. As an application of the abstract results, we present some existence and uniqueness results on the pseudo almost periodic solutions of dynamic equations with delay.

Keywords: Sp-pseudo almost periodicity; Time scales; Dynamic equations

MSC 2010: 42A75; 34N05

1 Introduction

The theory of time scales was established by S. Hilger in 1988 (see [1]). This theory unifies continuous and discrete problems and provides a powerful tool for applications to economics, populations models, quantum physics among others, and hence has been attracting the attention of many mathematicians (see [2, 3] and the references therein). In 2011, Li and Wang [4, 5] introduced the concept of almost periodic functions on time scales. Since then, many generalized forms of almost periodicity have been introduced on time scales, such as pseudo almost periodicity [6], almost automorphy[7], weighted pseudo almost periodicity [8] etc.

To consider the almost periodicity of integrable functions on the real line, Stepanov [9] and Wiener [10] introduced Stepanov almost periodicity in 1926. Then this concept was extended to Stepanov-like pseudo almost periodicity by Diagana [11] in 2007. On the other hand, Li and Wang [12] extended Stepanov almost periodicity on time scales in 2017. Motivated by the above works, the main purpose of this paper is to consider Stepanov-like pseudo almost periodicity on time scales.

The definition and some basic properties of Stepanov-like pseudo almost periodicity are given in Section 3, including the translation invariance, uniqueness of decomposition, completeness and composition theorem. Moreover, we prove the seemingly simple but nontrivial result that pseudo almost periodicity implies Stepanov-like pseudo almost periodicity. As an application of the abstract results, we present some results on the existence and uniqueness of pseudo almost periodic solutions of dynamic equations with delay in Section 4.

2 Preliminaries

The concepts and results in this section can be found in [2, 3, 5, 6, 7, 12, 13, 14, 15] or deduced simply from the results given there. Throughout this paper, we denote by ℕ, ℤ, ℝ and ℝ+ the sets of positive integers, integers, real numbers and nonnegative real numbers, respectively. 𝔼n denotes the Euclidian space n or ℂn with Euclidian norm ‖⋅‖, and n×n the space of all n × n real-valued matrices with matrix norm ‖⋅‖.

Let 𝕋 be a time scale, that is, a closed and nonempty subset of ℝ. The forward and backward jump operators σ, ρ:𝕋 → 𝕋 and the graininess μ:𝕋 → ℝ+ are defined, respectively, by

σ(t)=inf{sT:s>t},ρ(t)=sup{sT:s<t},μ(t)=σ(t)t.

If σ(t) > t, we say that t is right-scattered. Otherwise, t is called right-dense. Analogously, if ρ(t) > t, then t is called left-scattered. Otherwise, t is left-dense. We always denote 𝕋 a time scale from now on.

Let a, b ∈ 𝕋 with ab, [a,b], [a,b), (a,b], (a,b) being the usual intervals on the real line. The intervals [a,a),(a,a], (a,a) are understood as the empty set, and we use the following symbols:

[a,b]T=[a,b]T,[a,b)T=[a,b)T,(a,b]T=(a,b]T,(a,b)T=(a,b)T.

Note that in this paper we use the above symbols only if a, b ∈ 𝕋.

Denote

C(T;En)={f:TEn:f is continuous},C(T×D;En)={f:T×DEn:f is continuous},BC(T;En)={f:TEn:f is bounded and continuous},Llocp(T;En)={f:TEn:f is locallyLpΔintegrable}.

It is easy to see that BC(𝕋;𝔼n) is a Banach space with supremum norm.

If 𝕋 has a left-scattered maximum m, then 𝕋κ = 𝕋 {m}; otherwise 𝕋κ = 𝕋.

Definition 2.1

For f : 𝕋 → 𝔼n and t ∈ 𝕋κ, fΔ(t) ∈ 𝔼n is called the delta derivative of f(t) if for a given ε > 0, there exists a neighborhood U of t such that

|f(σ(t))f(s)fΔ(t)(σ(t)s)|<ε|σ(t)s|

for all sU. Moreover, f is said to be delta differentiable on 𝕋 if fΔ(t) exists for all t ∈ 𝕋.

We note that the integral ab f(t)Δ t always means ∫[a,b)𝕋 f(t)Δ t in this paper, and all the theorems of the general Lebesgue integration theory also hold for the Δ-integrals on 𝕋. For more details of continuity, differentiable, Δ-measure and Δ-integral on 𝕋, Jordan Δ-measure and multi-Riemann Δ-integrable on 𝕋2, we refer the readers to [2, 3, 13, 15, 16].

2.1 Almost periodicity and pseudo almost periodicity on 𝕋

Definition 2.2

([5, 7]). A time scale 𝕋 is called invariant under translations if

Π:={τR:t±τT,tT}{0}.

From now on, we always assume that 𝕋 is invariant under translations.

Definition 2.3

([5, 14])

  1. A function fC(𝕋;𝔼n) is called almost periodic on 𝕋 if for every ε > 0, the set

    T(f,ε)={τΠ:|f(t+τ)f(t)|<ε,tT}

    is relatively dense in Π. T(f, ε) is called the ε-translation set of f, τ is called the ε-translation number of f. Denote by AP(𝕋;𝔼n) the set of all almost periodic functions.

  2. Let Ω ⊂ 𝔼n be open. The set AP(𝕋 × Ω; 𝔼n) consists of all continuous functions f : 𝕋 ×Ω → 𝔼n such that f(⋅, x) ∈ AP(𝕋;𝔼n) uniformly for each xS, where S is any compact subset of Ω. That is, for ε > 0, ⋂xS T(f(⋅,x),ε) is relatively dense in Π.

Let fBC(𝕋;𝔼n). Then set

PAP0(T;En)=fBC(T;En):limr+12rt0rt0+r|f(s)|Δs=0, where t0T,rΠ.

Definition 2.4

([6]). A function fBC(𝕋;𝔼n) is called pseudo almost periodic if f = g + ϕ, where gAP(𝕋;𝔼n) and ϕPAP0(𝕋;𝔼n). We denote by PAP(𝕋;𝔼n) the set of all pseudo almost periodic functions.

Proposition 2.5

f(𝕋) is relatively compact if fAP(𝕋;𝔼n).

Proof

It follows from [17, Theorem 3.4] that for fC(𝕋;𝔼n), fAP(𝕋;𝔼n) if and only if there exists gAP(ℝ;𝔼n) such that f(t) = g(t) for t ∈ 𝕋. Meanwhile, it is well known that g(ℝ) is relatively compact if g ∈ AP(ℝ;𝔼n). Therefore f(𝕋) is relatively compact if fAP(𝕋;𝔼n). □

Proposition 2.6

([6]).

  1. If fPAP(𝕋;𝔼n) and ϕPAP0(𝕋;𝔼n), then for any τΠ, f(⋅ + τ) ∈ PAP(𝕋;𝔼n) and ϕ(⋅ + τ) ∈ PAP0(𝕋;𝔼n).

  2. PAP(𝕋,𝔼n) and PAP0(𝕋,𝔼n) are Banach spaces under the sup norm.

2.2 Sp-almost periodic functions on 𝕋

We always assume that p ≥ 1 afterwards without any further comments. Let

K:=inf{|τ|:τΠ,τ0}, if TR,1, if T=R.

Define ‖⋅‖Sp: Llocp(𝕋;𝔼n) → ℝ+ as

fSp:=suptT1Ktt+K|f(s)|pΔs1pfor fLlocp(T;En).

fLlocp(𝕋; 𝔼n) is called Sp-bounded if ∥fSp < ∞. Denote by BSp(𝕋;𝔼n) the space of all these functions.

Definition 2.7

  1. ([12]). A function fBSp(𝕋;𝔼n) is called Sp-almost periodic on 𝕋 if for every ε > 0, the ε-translation set of f

    T(f,ε)={τΠ:f(+τ)fSp<ε}

    is relatively dense in Π. Denote the set of all these functions by SpAP(𝕋,𝔼n).

  2. A function f: 𝕋 × Ω → 𝔼n with Ω ⊂ 𝔼n is called Sp-almost periodic in t ∈ 𝕋 if f(⋅, x) ∈ SpAP(𝕋;𝔼n) uniformly for each xS, where S is an arbitrary compact subset of Ω. That is, for ε > 0, ⋂xS T(f(⋅,x),ε) is relatively dense in Π. Denote the set of all such functions by SpAP(𝕋 ×Ω;𝔼n).

Proposition 2.8

  1. ([12, 18]). If fSpAP(𝕋;𝔼n), then for any τΠ, f(⋅ + τ) ∈ SpAP(𝕋;𝔼n).

  2. BSp(𝕋;𝔼n), SpAP(𝕋,𝔼n) are Banach spaces under the Sp-norm ∥⋅∥Sp.

  3. Let 1 ≤ qp < ∞. Then BSp(𝕋;𝔼n)⊂ BSq(𝕋;𝔼n), SpAP(𝕋;𝔼n)⊂ SqAP(𝕋;𝔼n) andfSq ≤ ∥fSp for fBSp(𝕋;𝔼n).

  4. AP(𝕋;𝔼n)⊂ SpAP(𝕋;𝔼n).

3 Sp-pseudo almost periodic functions

Now we introduce the concept of Sp-pseudo almost periodicity on time scales and present the main properties of it.

3.1 Definitions

We define the norm operator 𝓝 on BSp(𝕋;𝔼n) as follows:

N(f)(t):=1Ktt+K|f(s)|pΔs1pfor fBSp(T;En),tT.

Lemma 3.1

The norm operator 𝓝 maps BSp(𝕋;𝔼n) into BC(𝕋;ℝ+) and maps SpAP(𝕋;𝔼n) into AP(𝕋;ℝ+). Moreover, for f,gBSp(𝕋;𝔼n), t ∈ 𝕋,

N(f)||=||f||Sp,|N(f)(t)N(g)(t)|N(f±g)(t)N(f)(t)+N(g)(t).(1)

Proof

It is obvious that ∥𝓝(f)∥ = ∥fSp. Then 𝓝(f) is bounded when fBSp(𝕋;𝔼n). The second part of (1) can be got from Minkowski inequality immediately.

Let fBSp(𝕋;𝔼n). Then fLlocp(𝕋;𝔼n), and the absolute continuity of integral follows that for ε > 0, there exists δ = δ(ε) > 0 such that for any Δ-measurable set e with μΔ(e) < δ,

e|f(s)|pΔs<Kε2.

Thus, for t1, t2 ∈ 𝕋, t1 < t2, |t1t2| < δ,

|(N(f))p(t1)(N(f))p(t2)|1Kt1t2|f(s)|pΔs1Kt1+Kt2+K|f(s)|pΔs1Kt1t2|f(s)|pΔs+1Kt1+Kt2+K|f(s)|pΔs<ε.

This implies that (𝓝(f))p is continuous, and 𝓝(f) is continuous. So 𝓝:BSp(𝕋;𝔼n) → BC(𝕋;ℝ+).

Let fSpAP(𝕋;𝔼n). Then 𝓝(f) ∈ BC(𝕋;ℝ+) by the proof above. For τT(f,ε), by (1),

N(f)(+τ)N(f)||=suptT|N(f)(t+τ)N(f)(t)|=suptT|N(f(+τ))(t)N(f)(t)|suptTN(f(+τ)f)(t)=||N(f(+τ)f)||=||f(+τ)f||Sp<ε,

which implies that T(f,ε)⊂ T(𝓝(f),ε). Hence 𝓝(f) ∈ AP(𝕋;ℝ+). □

Definition 3.2

A function fBSp(𝕋;𝔼n) is said to be ergodic if 𝓝(f) ∈ PAP0(𝕋;ℝ+), i.e.

limr+12rt0rt0+r1Ktt+K|f(s)|pΔs1pΔt=0,wheret0T,rΠ.

We denote by SpPAP0(𝕋;𝔼n) the set of all ergodic functions from 𝕋 to 𝔼n.

Definition 3.3

  1. A function fBSp(𝕋;𝔼n) is called Stepanov-like pseudo almost periodic (Sp-pseudo almost periodic) if f = g + ϕ, where gSpAP(𝕋;𝔼n) and ϕSpPAP0(𝕋;𝔼n). g and ϕ are called the almost periodic component and the ergodic perturbation of f, respectively. We denote by SpPAP(𝕋;𝔼n) the set of all such functions f.

  2. A function f: 𝕋 × Ω → 𝔼n with Ω ⊂ 𝔼n is called Stepanov-like pseudo almost periodic (Sp-pseudo almost periodic) in t ∈ 𝕋 if f(⋅,x) = g(⋅,x) + ϕ(⋅,x) ∈ SpPAP(𝕋;𝔼n) for each xΩ and g(⋅,x) ∈ SpAP(𝕋;𝔼n) uniformly for each xS, where S is an arbitrary compact subset of Ω. Denote the set of all these functions by SpPAP(𝕋 ×Ω;𝔼n).

Remark 3.4

  1. We note that Definition 3.3 is a generalization of Sp-pseudo almost periodicity on ℝ introduced by Diagana [19].

  2. It is easy to check that SqPAP(𝕋;𝔼n)⊂ SpPAP(𝕋;𝔼n) for 1 ≤ pq.

3.2 Some basic properties

From now on, we write f = g + ϕSpPAP(𝕋;𝔼n) implies gSpAP(𝕋;𝔼n) and ϕSpPAP0(𝕋;𝔼n). In this subsection, we give some basic properties of Sp-pseudo almost periodicity, including the uniqueness of decomposition, the translation invariance and the completeness.

Proposition 3.5

The decomposition of Sp-pseudo almost periodic functions is unique.

Proof

If f = g1 + ϕ1 = g2 + ϕ2SpPAP(𝕋;𝔼n), then g1g2 = ϕ2ϕ1. This implies 𝓝(g1g2) = 𝓝(ϕ1ϕ2). Note that 𝓝(ϕ1ϕ2) ≤ 𝓝(ϕ1) + 𝓝(ϕ2) by (1), it follows that 𝓝(ϕ1ϕ2) ∈ PAP0(𝕋;ℝ+), and then 𝓝(g1g2) ∈ PAP0(𝕋;ℝ+). Meanwhile, g1g2SpAP(𝕋;𝔼n) since g1, g2SpAP(𝕋;𝔼n). Thus 𝓝(g1g2) ∈ AP(𝕋;ℝ+) by Lemma 3.1. Now it follows from [6, Theorem 3.5] that 𝓝(g1g2) = 0. This yields that g1 = g2 in SpAP(𝕋;𝔼n), and consequently, ϕ1 = ϕ2 in SpPAP0(𝕋;𝔼n). □

By Proposition 2.6 (i), Proposition 2.8 (i) and Lemma 3.1, we can easily obtain the following translation invariance of Sp-pseudo almost periodic functions. Here we omit the details.

Proposition 3.6

Let fSpPAP(𝕋;𝔼n). Then f(⋅+τ) ∈ SpPAP(𝕋;𝔼n) for τΠ.

Proposition 3.7

(SpPAP0(𝕋;𝔼n), ∥⋅∥Sp) is a Banach space.

Proof

By Proposition 2.8 (ii), we only need to prove the closedness of SpPAP0(𝕋;𝔼n) in BSp(𝕋;𝔼n). In fact, let {ϕk} ⊂ SpPAP0(𝕋;𝔼n) and ϕBSp(𝕋;𝔼n) with ∥ϕkϕSp → 0 as k → ∞. By Lemma 3.1, {𝓝(ϕk)} ⊂ PAP0(𝕋;ℝ+) and

||N(ϕk)N(ϕ)||||N(ϕkϕ)||=||ϕkϕ||Sp0 as k.

This implies 𝓝(ϕ) ∈ PAP0(𝕋;ℝ+) since PAP0(𝕋;ℝ+) is a Banach space by Proposition 2.6 (ii). Hence ϕSpPAP0(𝕋;𝔼n) and SpPAP0(𝕋;𝔼n) is closed. □

To prove the completeness of the space SpPAP(𝕋;𝔼n) we need the following lemma.

Lemma 3.8

If f = g + ϕSpPAP(𝕋;𝔼n), thengSp ≤ ∥fSp.

Proof

Let Q(t) := 𝓝(g)(t) – 𝓝(ϕ)(t), t ∈ 𝕋. Then by (1),

|Q(t)|N(g+ϕ)(t)=N(f)(t),tT.

This implies that ∥Q ≤ ∥𝓝(f)∥. On the other hand, 𝓝(g) ∈ AP(𝕋;ℝ+) by Lemma 3.1, and clearly, – 𝓝(ϕ) ∈ PAP0(𝕋;ℝ). Then Q = 𝓝(g) – 𝓝(ϕ) ∈ PAP(𝕋;ℝ). Thus, by [6, Theorem 4.2] and (1),

||g||Sp=||N(g)||||Q||||N(f)||=||f||Sp.

Proposition 3.9

(SpPAP(𝕋;𝔼n), ∥⋅∥Sp) is a Banach space.

Proof

It suffices to prove that SpPAP(𝕋;𝔼n) is closed in BSp(𝕋;𝔼n). Let {fk} = {gk + ϕk} ⊂ SpPAP(𝕋;𝔼n) and fBSp(𝕋;𝔼n) with ∥fkfSp → 0 as k → ∞. Then ∥fkfjSp → 0 as k, j → ∞. It follows from Lemma 3.8 that ∥gkgjSp ≤ ∥fkfjSp → 0 as k, j → ∞. This together with Proposition 2.8 (ii) implies that there exists gSpAP(𝕋;𝔼n) such that ∥gkgSp → 0 as k → ∞.

Meanwhile, by Lemma 3.8,

||ϕkϕj||Sp=||fkfj+gjgk||Sp||fkfj||Sp+||gkgj||Sp0 as k,j,

which implies that ϕkϕ as k → ∞ for some ϕSpPAP0(𝕋;𝔼n) by Proposition 3.7. Let f = g + ϕ. Then fSpPAP(𝕋;𝔼n) and fkf as k → ∞. That is SpPAP(𝕋;𝔼n) is closed in BSp(𝕋;𝔼n). □

3.3 PAP(𝕋;𝔼n)⊂ Sp PAP(𝕋;𝔼n)

We prove the seemingly simple but nontrivial result that PAP(𝕋;𝔼n)⊂ Sp PAP(𝕋;𝔼n) in this subsection.

For t0 ∈ 𝕋, let

EP={(t,s)T×T:t0t<t0+K,ts<t+K},

Lemma 3.10

  1. EP is Jordan Δ-measurable.

  2. If f : EP → ℝ is bounded continuous, then f is Riemann Δ-integrable over EP, and

    t0t0+Ktt+Kf(t,s)ΔsΔt=t0t0+Kt0sf(t,s)ΔtΔs+t0+Kt0+2KsKt0+Kf(t,s)ΔtΔs.(2)

Proof

By a fundamental calculation, we can prove that EP is Jordan Δ-measurable and f is Riemann Δ-integrable over EP. Here we omit the details, and we only prove that (2) holds. Let R = [t0,t0 + 𝓚)𝕋 ×[t0,t0 + 2𝓚)𝕋 and F : R → ℝ be defined as

F(t,s)=f(t,s),if (t,s)EP,0,if (t,s)REP.

Then for t ∈ [t0,t0+𝓚)𝕋,

F(t,s)=f(t,s),if s[t,t+K)T,0,if s[t0,t)T[t+K,t0+2K)T,

and F(t,⋅) can only be discontinuous at t and t + 𝓚 since f : EP → ℝ is bounded continuous. It follows from [20, Theorem 5.8] that F(t,⋅) is Δ-integrable on [t0,t0 + 2𝓚)𝕋. Thus, by [21, Theorem 2.15],

RF(t,s)ΔtΔs=t0t0+Kt0t0+2KF(t,s)ΔsΔt=t0t0+Ktt+Kf(t,s)ΔsΔt.(3)

On the other hand, for s ∈ [t0,t0 + 𝓚)𝕋,

F(t,s)=f(t,s),if t[t0,s)T,0,if t[s,t0+K)T,

and for s ∈ [t0+𝓚,t0 + 2𝓚)𝕋,

F(t,s)=0,if t[t0,sK)T,f(t,s),if t[sK,t0+K)T.

Similarly, we can get that for every s ∈ [t0,t0 + 2𝓚)𝕋, F(⋅,s) is Δ-integrable on [t0,t0 + 𝓚)𝕋. Thus, by [21, Remark 2.16],

RF(t,s)ΔtΔs=t0t0+2Kt0t0+KF(t,s)ΔtΔs

=t0t0+Kt0sf(t,s)ΔtΔs+t0+Kt0+2KsKt0+Kf(t,s)ΔtΔs.

This together with (3) leads to the conclusion. □

Proposition 3.11

PAP(𝕋;𝔼n)⊂ Sp PAP(𝕋;𝔼n).

Proof

Let f = g + ϕPAP(𝕋;𝔼n) with gAP(𝕋;𝔼n) and ϕPAP0(𝕋;𝔼n). Then fBSp(𝕋;𝔼n), AP(𝕋;𝔼n)⊂ SpAP(𝕋;𝔼n) by Proposition 2.8 (iii), and we need only to prove that ϕPAP0(𝕋;𝔼n), i.e. 𝓝(ϕ) ∈ PAP0(𝕋;ℝ). In fact, let q > 0 with 1/p + 1/q = 1, for fixed t0 ∈ 𝕋 and m ∈ ℕ,

1mKt0t0+mKN(ϕ)(t)Δt(mK)1q1t0t0+mK(N(ϕ)(t))pΔt1p=1mK2t0t0+mKtt+K|ϕ(s)|pΔsΔt1p||ϕ||p1p1mK2t0t0+mKtt+K|ϕ(s)|ΔsΔt1p=||ϕ||1q1mK2i=0m1t0t0+Ktt+K|ϕ(s+iK)|ΔsΔt1p.

Meanwhile, by Lemma 3.10 and the fact that |ϕ| + |ϕ(⋅ + 𝓚)| ∈ PAP0(𝕋;n),

1mK2i=0m1t0t0+Ktt+K|ϕ(s+iK)|ΔsΔt=1mK2i=0m1t0t0+Kt0s|ϕ(s+iK)|ΔtΔs+t0+Kt0+2KsKt0+K|ϕ(s+iK)|ΔtΔs1mKi=0m1t0t0+K|ϕ(s+iK)|Δs+t0+Kt0+2K|ϕ(s+iK)|Δs=1mKt0t0+mK(|ϕ(s)|+|ϕ(s+K)|)Δs0as m+.

Then

1mKt0t0+mKN(ϕ)(t)Δt0as m+.

This implies that 𝓝(ϕ) ∈ PAP0(𝕋;ℝ). □

3.4 Composition theorems

We will use the following Sp-Lipschitz condition for fSpAP(𝕋 ×𝔼n;𝔼n):

(H) There exists a constant Lf > 0 such that for any x,yBSp(𝕋;𝔼n) and t ∈ 𝕋,

N(f(,x())f(,y()))(t)LfN(xy)(t).

Remark 3.12

Obviously, (H) implies thatf(⋅, x(⋅))– f(⋅, y(⋅))∥SpLfxySp. Moreover, f satisfies (H) if f(t,x) is Lipschitz continuous in x ∈ 𝔼n uniformly in t ∈ 𝕋, i.e.|f(t,x) – f(t,y)| ≤ L|xy| for every x,y ∈ 𝔼n, t ∈ 𝕋 and some constant L.

Theorem 3.13

Assume that fSpAP(𝕋 ×𝔼n;𝔼n) satisfies (H), and uSpAP(𝕋;𝔼n) with u(𝕋) compact. Then f(⋅,u(⋅)) ∈ SpAP(𝕋;𝔼n).

Proof

By (H),

||f(,u())f(,0)||SpLf||u||Sp.

Then

||f(,u())||Sp||f(,0)||Sp+Lf||u||Sp<.

That is

f(,u())BSp(T;En).(4)

Since u(𝕋) is compact, for ε> 0, there exist finite open balls Ok, k = 1, 2, …, m, with center uku(𝕋) and radius ε8Lf such that u(𝕋) ⊂ k=1mOk. Set Bk:= {s ∈ 𝕋: u(s) ∈ Ok}, k = 1,2,…, m. Then 𝕋 = k=1m Bk. Moreover, let E1 := B1, Ek := Bk(i=1k1Bi), k = 2, …, m. Then EiEj = ∅ for ij and 𝕋 = k=1m Ek. Define a step function û:𝕋 → 𝔼n by û(s) := uk, sEk, k = 1,2,…, m. It is clear that |u(s) – û(s)| < ε8Lf for all s ∈ 𝕋. Then by (H), for τk=1m T(f(⋅,uk),ε4m),

||f(+τ,u())f(,u())||Sp||f(+τ,u())f(+τ,u^())||Sp+||f(+τ,u^())f(,u^())||Sp+||f(,u^())f(,u())||Sp2Lf||uu^||Sp+suptT1Ktt+K|f(s+τ,u^(s))f(s,u^(s))|pΔs1p<ε4+suptT1Kk=1m[t,t+K)TEk|f(s+τ,uk)f(s,uk)|pΔs1pε4+k=1m||f(+τ,uk)f(,uk)||Spε4+mε4m=ε2.

Notice that 𝒢 := k=1mT(f(,uk),ε4m)T(u,ε2Lf) is relatively dense by [22, Lemma 4.9]. Thus, for τ𝒢, by (H),

||f(+τ,u(+τ))f(,u())||Sp||f(+τ,u(+τ))f(+τ,u())||Sp+||f(+τ,u())f(,u())||Sp<Lf||u(+τ)u||Sp+ε2<ε.

This implies that 𝒢T(f(⋅,u(⋅)), ε), and T(f(⋅,u(⋅)), ε) is relatively dense. Therefore, f(⋅,u(⋅)) ∈ SpAP(𝕋;𝔼n). □

Theorem 3.14

Let f = g + ϕSpPAP(𝕋 ×𝔼n;𝔼n) and u = x + ySpPAP(𝕋;𝔼n) with x(𝕋) compact. Assume that f and g satisfy (H) with Lipschitz constants Lf and Lg, respectively. Then f(⋅,u(⋅)) ∈ SpPAP(𝕋;𝔼n).

Proof

Let I1(t) = g(t,x(t)), I2(t) = f(t,u(t))–f(t,x(t)) and I3(t) = ϕ(t,x(t)), t ∈ 𝕋. Then

f(t,u(t))=I1(t)+I2(t)+I3(t),tT.

By Theorem 3.13, we have I1SpAP(𝕋;𝔼n). So it suffices to prove that I2,I3SpPAP0(𝕋;𝔼n).

Since f satisfies (H) with Lf,

||I2||Sp=||f(,u())f(,x())||SpLf||ux||Sp=Lf||y||Sp<.

which implies that BSp(𝕋; 𝔼n). Moreover, since ySpPAP0(𝕋;𝔼n), for given t0∈𝕋,

12rt0rt0+rN(I2)(s)Δs=12rt0rt0+rN(f(,u())f(,x()))(s)ΔsLf2rt0rt0+rN(y)(s)Δs0, as r+.

This shows that I2SpPAP0(𝕋;𝔼n).

By the same arguments as to get (4), we can get I3BSp(𝕋;𝔼n). Since x(𝕋) is compact, for any ε>0, as the proof of Theorem 3.13, we can find xix(𝕋), Ei⊂𝕋, i = 1,2,…,l and :𝕋→𝔼n such that (s): = xi, sEi, i = 1,2,…, l, EiEj = ∅ for ij, 𝕋 = i=1lEi, and |x(s)–(s)|< ε2(Lf+Lg) for all s ∈𝕋. Since ϕ(⋅,x)∈ SpPAP0(𝕋;𝔼n) for every x∈𝔼n, there exists r0 >0 such that for r>r0, 1≤ il,

12rt0rt0+rN(ϕ(,xi))(s)Δs<ε2l.(5)

Note that ϕ satisfies (H) with Lf+Lg since f and g satisfy (H) with Lf and Lg, respectively, then by (H) and (5), for r>r0,

12rt0rt0+rN(ϕ(,x()))(t)Δt12rt0rt0+rN(ϕ(,x())ϕ(,x^()))(t)Δt+12rt0rt0+rN(ϕ(,x^()))(t)Δt12rt0rt0+r(Lf+Lg)N(xx^)(t)Δt+12rt0rt0+r1Ki=1l[t,t+K)TEi|ϕ(s,xi)|pΔs1pΔtε2+i=1l12rt0rt0+rN(ϕ(,xi))(t)Δt<ε.

This yields that I3SpPAP0(𝕋;𝔼n).□

By Proposition 2.5, x(𝕋) is compact for xAP(𝕋;𝔼n). Then by Theorem 3.11 and 3.14, we have the following corollary.

Corollary 3.15

Let f = g + ϕSpPAP(𝕋×𝔼n;𝔼n) and uPAP(𝕋;𝔼n). Assume that f and g satisfy (H). Then f(⋅,u(⋅))∈ SpPAP(𝕋;𝔼n).

4 Dynamic equations

4.1 Exponential functions

A function p:𝕋→ℝ is called regressive provided 1+μ(t)p(t)≠0 for all t ∈𝕋κ. The set of all regressive and rd-continuous functions p :𝕋 → ℝ will be denoted by 𝓡 = 𝓡(𝕋) = 𝓡(𝕋;ℝ). We define the set 𝓡+ = 𝓡+(𝕋;ℝ) = {p ∈𝓡: 1+ μ(t)p(t)>0 for t ∈𝕋}. The set of all regressive functions on time scales forms an Abelian group under the addition ⊕ defined by pqp +q + μ(t)pq. Meanwhile, the additive inverse in this group is denoted by ⊖pp1+μ(t)p.

Definition 4.1

([2]).If p ∈ 𝓡 then the exponential function is defined by

ep(t,s)=expstξμ(τ)(p(τ))Δτ,

for s,t ∈ 𝕋, with the cylinder transformation

ξh(z)=1hLog(1+hz),ifh0,z,ifh=0,

where Log is the principal logarithm.

Definition 4.2

[2] A matrix-valued function A: 𝕋 → ℝ n×n is called regressive if I + μ(t)A(t) is invertible for all t ∈𝕋 κ, and the class of all such regressive and rd-continuous functions is denoted, similarly to the scalar case, by 𝓡 =𝓡(𝕋) = 𝓡(𝕋;ℝn×n).

Definition 4.3

([2]). Let t0 ∈𝕋 and A ∈𝓡(𝕋;ℝn×n). The unique matrix-valued solution of the initial value problem (IVP)

XΔ(t)=A(t)X(t),X(t0)=I,(6)

where I denotes the n ×n identity matrix, is called the matrix exponential function (at t0), which is denoted by eA(⋅,t0).

We note that the existence and uniqueness of IVP (6) can be obtained by [2, Theorem 5.8].

Lemma 4.4

([2]) Let t,s ∈ 𝕋.

  1. ep(t,t) = 1, eA(t,t) = I.

  2. ep(σ(t),s) = (1+μ(t)p(t))ep(t,s).

  3. ep(t,s)ep(s,r) = ep(t,r), eA (t,s)eA(s,r) = eA(t,r).

Lemma 4.5

Let a >0 be a constant and t,s ∈ 𝕋.

  1. ea(t,s)≤ 1 if ts.

  2. ea(t +τ,s + τ) = ea(t,s) for τΠ.

  3. There exists N >0 such that(ts)ea(t,s)≤ N for ts.

  4. For t0 ∈ 𝕋, ea(t0,⋅) is increasing on (–∞,t0]𝕋.

  5. The series j=1 ea(t,σ(t)–(j1)𝓚) converges uniformly for t ∈ 𝕋. Moreover, for all t∈ 𝕋,

    j=1ea(t,σ(t)(j1)K)λa:=11eaK,T=R,2+aμ¯+1aμ¯,TR,

    where μ¯:=suptTμ(t).

Proof

(i) is obvious. (ii) can be readily obtained by the fact that μ(t + τ) = μ(t) for all t ∈ 𝕋 and τΠ. If 𝕋 = ℝ, (ts)ea(t,s) = (ts)ea(ts)≤ 1ae. That is (iii) holds with N = 1ae. If 𝕋≠ ℝ, let {ti}i∈I, I ⊆ℝ, be all right-scattered points in 𝕋. By [13, Theorem 5.2],

ea(t,s)=exp[s,t)T(a)dτti[s,t)TLog(1+aμ(ti))=eaμL([s,t)T)ti[s,t)T11+aμ(ti)ti[s,t)T11+aμ(ti),(7)

where μL denotes the Lebesgue measure. For t> s, there exists a unique nts ∈ ℝ such that t ∈[s + (nts-1)𝓚,s + nts 𝓚)𝕋. Let t0∈ 𝕋 be right-scattered, then for every n ∈ℤ, t0 + n 𝓚 is right-scattered and μ(t0 + n𝓚) = μ(t0). Moreover, for s ∈ 𝕋, [s,s + (nts–1)𝓚)𝕋 contains nts –1 right-scattered points with form t0 + n 𝓚. Denote Γ = 1 + (t0) for the convenient of writing. Then by (7),

(ts)ea(t,s)ntsKti[s,s+(nts1)K)T11+aμ(ti)ntsK11+aμ(t0)nts1=ntsKΓ1ntsKΓ11/lnΓlnΓ.

So (iii) holds with N = 𝓚 Γ1–1 /lnΓ/lnΓ.

(iv) can be verified easily by the definition.

If 𝕋 = ℝ, for t ∈ 𝕋,

j=1ea(t,σ(t)(j1)K)=j=1ea(j1)=11ea.

That is (v) holds for 𝕋 = ℝ. If 𝕋 ≠ ℝ, then 𝓚 ≥ μ = suptTμ(t) >0, and it is easy to see that there exists a right-scattered point t0 such that μ(t0) = μ. In addition, for t ∈ 𝕋 and j ≥3, [t, σ(t)-(j–1)𝓚)𝕋 contains at least j–2 right-scattered points with forms t0 + nt 𝓚, nt ∈ℤ, μ(t0 + nt 𝓚) = μ(t0) = μ, and

ea(t,σ(t)(j1)K)(ea(σ(t0),t0))j2=(1+aμ¯)2j.

Then for any t ∈ 𝕋,

j=1ea(t,σ(t)(j1)K)ea(t,σ(t))+ea(t,σ(t)K)+j=3(1+aμ¯)2j(1+aμ¯)+1+1aμ¯=2+aμ¯+1aμ¯.

That is (v) holds for 𝕋 ≠ ℝ.

Lemma 4.6

Assume that A ∈𝓡(𝕋;ℝn×n) is almost periodic and

eA(t,s)Ceα(t,s),ts,(8)

where C and α are positive real numbers. Let M = (1 + α 𝓚)C2N with N the constant in Lemma 4.5 (iii), and for ε >0,

Υ(ε)={rΠ:eA(t+r,σ(s)+r)eA(t,σ(s))<ε,t,sT,tσ(s)}.

Then T(A, ε/M) ⊂ Υ(ε), which implies that Υ(ε) is relatively dense in Π.

Proof

For ε>0, let rT(A, ε/M) and U(t,σ(s)): = eA(t + r,σ(s) + r)–eA(t,σ(s)). Differentiate U with respect to t and denote by ΔUΔt the partial derivative, then

ΔUΔt=A(t+r)eA(t+r,s+r)A(t)eA(t,σ(s))=A(t)U(t,σ(s))+(A(t+r)A(t))eA(t+r,σ(s)+r).

Note that U(σ(s),σ(s)) = 0, then by the variation of constants formula ([2, Theorem 5.24]),

U(t,σ(s))=σ(s)teA(t,σ(τ))(A(τ+r)A(τ))eA(τ+r,σ(s)+r)Δτ.

Therefore, by (8), Lemma 4.4, 4.5 and the fact that μ(τ)≤ 𝓚, τ∈ 𝕋, for t,s ∈ 𝕋 with tσ(s),

U(t,σ(s))σ(s)teA(t,σ(τ))(A(τ+r)A(τ))eA(τ+r,σ(s)+r)ΔτεMC2σ(s)teα(t,σ(τ))eα(τ+r,σ(s)+r)Δτ=εMC2eα(t,σ(s))σ(s)teα(τ,σ(τ))Δτ=εMC2eα(t,σ(s))σ(s)t(1+αμ(τ))ΔτεMC2(1+αK)(tσ(s))eα(t,σ(s))εMC2(1+αK)N=ε.

This implies that T(A, ε/M) ⊂ Υ(ε), and Υ(ε) is relatively dense in Π. □

4.2 Dynamic equations with delay

As an application of the results obtained in the above sections, we consider the following nonlinear dynamic equation with delay:

xΔ(t)=A(t)x(t)+f(t,x(tω)),tT,(9)

where A(t) is an n×n almost periodic matrix function, ωΠ, ω>0 and fSpPAP(𝕋×𝔼n;𝔼n)∩ C(𝕋×𝔼n;𝔼n).

To consider (9), we first consider its corresponding linear equation:

xΔ(t)=A(t)x(t)+f(t),tT,(10)

where f = g + ϕSp PAP(𝕋;𝔼n)∩ C(𝕋;𝔼n).

Lemma 4.7

Assume that A∈ 𝓡(𝕋;ℝn×n) with (8) satisfied. Then (10) admits a unique bounded continuous solution u(t) given by

u(t)=teA(t,σ(s))f(s)Δs,tT.(11)

Proof

For t ∈ 𝕋, j≥1, by Hölder inequality,

tjKt(j1)K|f(s)|ΔsK1/qtjKt(j1)K|f(s)|pΔs1/p=KN(f)(tjK)KfSp.(12)

Then by (8), (11) and Lemma 4.5 (iv), (v), for t ∈ 𝕋,

|u(t)|j=1tjKt(j1)KeA(t,σ(s))|f(s)|ΔsCj=1tjKt(j1)Keα(t,σ(s))|f(s)|ΔsCj=1eα(t,σ(t)(j1)K)tjKt(j1)K|f(s)|ΔsCλαKfSp.

Thus u is well defined and bounded continuous. Moreover, by Lemma 4.4, fix t0 ∈𝕋,

u(t)=teA(t,σ(s))f(s)Δs=eA(t,t0)teA(t0,σ(s))f(s)Δs.

Then by Lemma 4.4 and [2, Theorem 5.3 (iii)],

uΔ(t)=A(t)eA(t,t0)teA(t0,σ(s))f(s)Δs+eA(σ(t),t0)eA(t0,σ(t))f(t)=A(t)u(t)+f(t),

which implies that u is a solution of (10). Assume that v:𝕋→𝔼n is another bounded solution of (10). For r∈𝕋, by the variation of constants formula ([2, Theorem 5.24]),

v(t)=eA(t,r)v(r)+rteA(t,σ(s))f(s)Δs,tT.

Since v is bounded, (8) implies that eA(t,r)v(r)→0 as r→–∞. Letting r →–∞,

v(t)=teA(t,σ(s))f(s)Δs=u(t).

That is the bounded solution of (10) is unique.□

Theorem 4.8

Assume that A∈ 𝓡(𝕋;ℝn×n) is almost periodic and (8) holds. Then (10) admits a unique pseudo almost periodic solution u(t) given by (11).

Proof

By Lemma 4.7, it suffices to prove that uPAP(𝕋;𝔼n). In fact, for t ∈ 𝕋, let

u(t)=teA(t,σ(s))f(s)Δs=j=1uj(t),

where

uj(t)=tjKt(j1)KeA(t,σ(s))f(s)Δs=tjKt(j1)KeA(t,σ(s))g(s)Δs+tjKt(j1)KeA(t,σ(s))ϕ(s)Δs=φj(t)+ψj(t),jN.

For ε> 0, it follows from [22, Lemma 4.9] that TA,ε2MK(1+gSp)Tg,ε2CK is relatively dense in Π. Denote

G1=Υε2K(1+gSp)Tg,ε2CK,

where Υ the one given in Lemma 4.6. Then 𝒢1 is relatively dense in Π by Lemma 4.6. Let τ𝒢1, t,s ∈ 𝕋 with tσ(s),

|eA(t+τ,σ(s+τ))g(s+τ)eA(t,σ(s))g(s)|eA(t+τ,σ(s)+τ)eA(t,σ(s))|g(s+τ)|+eA(t,σ(s))|g(s+τ)g(s)|ε|g(s+τ)|2K(1+gSp)+Ceα(t,σ(s))|g(s+τ)g(s)|ε|g(s+τ)|2K(1+gSp)+C|g(s+τ)g(s)|.

Now by the same calculation of (12), we can get for j∈ℕ,

|φj(t+τ)φj(t)|=tjKt(j1)KeA(t+τ,σ(s+τ))g(s+τ)eA(t,σ(s))g(s)Δsε2K(1+gSp)tjKt(j1)K|g(s+τ)|Δs+CtjKt(j1)K|g(s+τ)g(s)|ΔsεKgSp2K(1+gSp)+CKg(+τ)gSp<ε2+CKε2CK=ε.

This implies that 𝒢1T(φj,ε). Then T(φj,ε) is relatively dense in Π and φj is almost periodic for j ∈ ℕ. Meanwhile, by Lemma 4.5 (iv), for j∈ℕ and t ∈ 𝕋,

eα(t,σ(t)(j1)K)eα(t,σ(t))=1+αμ(t)1+αμ¯.

Then by (8) and the same calculation of (12),

|ψj(t)|tjKt(j1)KeA(t,σ(s))|ϕ(s)|ΔsCtjKt(j1)Keα(t,σ(s))|ϕ(s)|ΔsCeα(t,σ(t)(j1)K)tjKt(j1)K|ϕ(s)|ΔsC(1+αμ¯)KN(ϕ)(tjK),

which implies that ψjBC(𝕋;𝔼n). Notice that ϕSpPAP0(𝕋;𝔼n). Thus for a fixed t0∈𝕋,

limr+12rt0rt0+r|ψj(t)|ΔtC(1+αμ¯)Klimr+12rt0rt0+rN(ϕ)(tjK)Δt=0.

Hence ψjPAP0(𝕋;𝔼n) and uj = φj + ψj∈ PAP(𝕋; 𝔼n). Consequently, uPAP(𝕋;𝔼n).□

Remark 4.9

When 𝕋 = ℕ, μ(t) = 0 for all t ∈ 𝕋 and hence A𝓡 automatically. Then Theorem 4.8 is an extension of [19, Theorem 3.2] to time scales.

For nonlinear dynamics equation (9), we have the following result.

Theorem 4.10

Assume that A ∈ 𝓡(𝕋;ℕn×n) with (8) satisfied, and f = g + ϕSpPAP(𝕋×𝔼n;𝔼n)∩ C(𝕋×𝔼n;𝔼n) with f and g satisfying (H) with Lipschitz constants Lf and Lg, respectively. Then (9) has a unique pseudo almost periodic solution u satisfying

u(t)=teA(t,σ(s))f(s,u(sω))Δs,tT,(13)

provided that C 𝓚 Lfλα<1, where λ α is as in Lemma 4.5.

Proof

Let φPAP(𝕋;𝔼n). It follows from Proposition 2.6 (i) and Corollary 3.15 that f(⋅, φ(⋅–ω))∈ SpPAP(𝕋;𝔼n). Let

T(φ)(t):=teA(t,σ(s))f(s,φ(sω))Δs,tT.

Then T(φ)∈ PAP(𝕋;𝔼n) by Theorem 4.8. That is T: PAP(𝕋;𝔼n)→ PAP(𝕋;𝔼n). By (H), (8), Lemma 4.5 and the same calculation of (12), for φ, θPAP(𝕋;𝔼n), t ∈ 𝕋,

|T(φ)(t)T(θ)(t)|teA(t,σ(s))|f(s,φ(sω))f(s,θ(sw))|ΔsCj=1tjKt(j1)Keα(t,σ(s))|f(s,φ(sω))f(s,θ(sω))|ΔsCj=1eα(t,σ(t)(j1)K)tjKt(j1)K|f(s,φ(sω))f(s,θ(sω))|ΔsCKj=1eα(t,σ(t)(j1)K)N(f(,φ(ω))f(,θ(ω)))(tjK)CKLfj=1eα(t,σ(t)(j1)K)N(φ(ω)θ(ω))(tjK)CKLfλαφθ.

This implies that

T(φ)T(θ)CKLfλαφθ.

Thus T is a contraction operator since C 𝓚 Lfλα <1, and then T has a unique fixed point uPAP(𝕋;𝔼n). This means that (9) has a unique pseudo almost periodic solution u satisfying (13).□

Acknowledgement

This work is supported by National Natural Science Foundation of China ( Grant No. 11471227, 11561077).

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About the article

Received: 2018-02-13

Accepted: 2018-05-22

Published Online: 2018-07-25


Citation Information: Open Mathematics, Volume 16, Issue 1, Pages 826–841, ISSN (Online) 2391-5455, DOI: https://doi.org/10.1515/math-2018-0073.

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© 2018 Tang and Li, published by De Gruyter. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. BY-NC-ND 4.0

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