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

Definition 2.2

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

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

Definition 2.3

([5, 14])

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

   T(f,ε)={τ∈Π:|f(t+τ)−f(t)|<ε,∀t∈T}

   is relatively dense in Π. T(f, ε) is called theε-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 → 𝔼ⁿsuch that f(⋅, x) ∈ AP(𝕋;𝔼ⁿ) uniformly for each x ∈ S, where S is any compact subset of Ω. That is, for ε > 0, ⋂_{x ∈ S}T(f(⋅,x),ε) is relatively dense in Π.

Definition 2.4

([6]). A function f ∈ BC(𝕋;𝔼ⁿ) is called pseudo almost periodic if f = g + ϕ, where g ∈ AP(𝕋;𝔼ⁿ) and ϕ ∈ PAP₀(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 f ∈ AP(𝕋;𝔼ⁿ).

Proof

It follows from [17, Theorem 3.4] that for f ∈ C(𝕋;𝔼ⁿ), f ∈ AP(𝕋;𝔼ⁿ) if and only if there exists g ∈ AP(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 f ∈ AP(𝕋;𝔼ⁿ). □

Proposition 2.6

([6]).

1. If f ∈ PAP(𝕋;𝔼ⁿ) and ϕ ∈ PAP₀(𝕋;𝔼ⁿ), then for any τ ∈ Π, 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.

Definition 2.7

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

   T(f,ε)={τ∈Π:∥f(⋅+τ)−f∥Sp<ε}

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

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

Proposition 2.8

1. ([12, 18]). If f ∈ S^pAP(𝕋;𝔼ⁿ), then for any τ ∈ Π, f(⋅ + τ) ∈ S^pAP(𝕋;𝔼ⁿ).

2. BS^p(𝕋;𝔼ⁿ), S^pAP(𝕋,𝔼ⁿ) are Banach spaces under the S^p-norm ∥⋅∥_S^p.

3. Let 1 ≤ q ≤ p < ∞. Then BS^p(𝕋;𝔼ⁿ)⊂ BS^q(𝕋;𝔼ⁿ), S^pAP(𝕋;𝔼ⁿ)⊂ S^qAP(𝕋;𝔼ⁿ) and ∥f∥_S^q ≤ ∥f∥_S^pfor f ∈ BS^p(𝕋;𝔼ⁿ).

4. AP(𝕋;𝔼ⁿ)⊂ S^pAP(𝕋;𝔼ⁿ).

Lemma 3.1

The norm operator 𝓝 maps BS^p(𝕋;𝔼ⁿ) into BC(𝕋;ℝ⁺) and maps S^pAP(𝕋;𝔼ⁿ) into AP(𝕋;ℝ⁺). Moreover, for f,g ∈ BS^p(𝕋;𝔼ⁿ), t ∈ 𝕋,

Proof

It is obvious that ∥𝓝(f)∥_∞ = ∥f∥_S^p. Then 𝓝(f) is bounded when f ∈ BS^p(𝕋;𝔼ⁿ). The second part of (1) can be got from Minkowski inequality immediately.

Let f ∈ BS^p(𝕋;𝔼ⁿ). Then f ∈ Llocp(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 μ_Δ(e) < δ,

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

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

Let f ∈ S^pAP(𝕋;𝔼ⁿ). Then 𝓝(f) ∈ BC(𝕋;ℝ⁺) by the proof above. For τ ∈ T(f,ε), by (1),

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

Definition 3.2

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

We denote by S^pPAP₀(𝕋;𝔼ⁿ) the set of all ergodic functions from 𝕋 to 𝔼ⁿ.

Definition 3.3

1. A function f ∈ BS^p(𝕋;𝔼ⁿ) is called Stepanov-like pseudo almost periodic (S^p-pseudo almost periodic) iff = g + ϕ, where g ∈ S^pAP(𝕋;𝔼ⁿ) and ϕ ∈ S^pPAP₀(𝕋;𝔼ⁿ). g and ϕ are called the almost periodic component and the ergodic perturbation of f, respectively. We denote by S^pPAP(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 (S^p-pseudo almost periodic) in t ∈ 𝕋 if f(⋅,x) = g(⋅,x) + ϕ(⋅,x) ∈ S^pPAP(𝕋;𝔼ⁿ) for each x ∈ Ω and g(⋅,x) ∈ S^pAP(𝕋;𝔼ⁿ) uniformly for each x ∈ S, where S is an arbitrary compact subset of Ω. Denote the set of all these functions by S^pPAP(𝕋 ×Ω;𝔼ⁿ).

Remark 3.4

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

2. It is easy to check that S^qPAP(𝕋;𝔼ⁿ)⊂ S^pPAP(𝕋;𝔼ⁿ) for 1 ≤ p ≤ q.

Proposition 3.5

The decomposition of S^p-pseudo almost periodic functions is unique.

Proof

If f = g₁ + ϕ₁ = g₂ + ϕ₂ ∈ S^pPAP(𝕋;𝔼ⁿ), then g₁ – g₂ = ϕ₂ – ϕ₁. This implies 𝓝(g₁ – g₂) = 𝓝(ϕ₁ – ϕ₂). Note that 𝓝(ϕ₁ – ϕ₂) ≤ 𝓝(ϕ₁) + 𝓝(ϕ₂) by (1), it follows that 𝓝(ϕ₁ – ϕ₂) ∈ PAP₀(𝕋;ℝ⁺), and then 𝓝(g₁ – g₂) ∈ PAP₀(𝕋;ℝ⁺). Meanwhile, g₁ – g₂ ∈ S^pAP(𝕋;𝔼ⁿ) since g₁, g₂ ∈ S^pAP(𝕋;𝔼ⁿ). Thus 𝓝(g₁ – g₂) ∈ AP(𝕋;ℝ⁺) by Lemma 3.1. Now it follows from [6, Theorem 3.5] that 𝓝(g₁ – g₂) = 0. This yields that g₁ = g₂ in S^pAP(𝕋;𝔼ⁿ), and consequently, ϕ₁ = ϕ₂ in S^pPAP₀(𝕋;𝔼ⁿ). □

Proposition 3.6

Let f ∈ S^pPAP(𝕋;𝔼ⁿ). Then f(⋅+τ) ∈ S^pPAP(𝕋;𝔼ⁿ) for τ ∈ Π.

Proposition 3.7

(S^pPAP₀(𝕋;𝔼ⁿ), ∥⋅∥_S^p) is a Banach space.

Proof

By Proposition 2.8 (ii), we only need to prove the closedness of S^pPAP₀(𝕋;𝔼ⁿ) in BS^p(𝕋;𝔼ⁿ). In fact, let {ϕ_k} ⊂ S^pPAP₀(𝕋;𝔼ⁿ) and ϕ ∈ BS^p(𝕋;𝔼ⁿ) with ∥ϕ_k – ϕ∥_S^p → 0 as k → ∞. By Lemma 3.1, {𝓝(ϕ_k)} ⊂ PAP₀(𝕋;ℝ⁺) and

This implies 𝓝(ϕ) ∈ PAP₀(𝕋;ℝ⁺) since PAP₀(𝕋;ℝ⁺) is a Banach space by Proposition 2.6 (ii). Hence ϕ ∈ S^pPAP₀(𝕋;𝔼ⁿ) and S^pPAP₀(𝕋;𝔼ⁿ) is closed. □

Lemma 3.8

If f = g + ϕ ∈ S^pPAP(𝕋;𝔼ⁿ), then ∥g∥_S^p ≤ ∥f∥_S^p.

Proof

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

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),

□

Proposition 3.9

(S^pPAP(𝕋;𝔼ⁿ), ∥⋅∥_S^p) is a Banach space.

Proof

It suffices to prove that S^pPAP(𝕋;𝔼ⁿ) is closed in BS^p(𝕋;𝔼ⁿ). Let {f_k} = {g_k + ϕ_k} ⊂ S^pPAP(𝕋;𝔼ⁿ) and f ∈ BS^p(𝕋;𝔼ⁿ) with ∥f_k – f∥_S^p → 0 as k → ∞. Then ∥f_k – f_j∥_S^p → 0 as k, j → ∞. It follows from Lemma 3.8 that ∥g_k – g_j∥_S^p ≤ ∥f_k – f_j∥_S^p → 0 as k, j → ∞. This together with Proposition 2.8 (ii) implies that there exists g ∈ S^pAP(𝕋;𝔼ⁿ) such that ∥g_k – g∥_S^p → 0 as k → ∞.

Meanwhile, by Lemma 3.8,

which implies that ϕ_k → ϕ as k → ∞ for some ϕ ∈ S^pPAP₀(𝕋;𝔼ⁿ) by Proposition 3.7. Let f = g + ϕ. Then f ∈ S^pPAP(𝕋;𝔼ⁿ) and f_k → f as k → ∞. That is S^pPAP(𝕋;𝔼ⁿ) is closed in BS^p(𝕋;𝔼ⁿ). □

Lemma 3.10

1. E_P is Jordan Δ-measurable.

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

   ∫t0t0+K∫tt+Kf(t,s)ΔsΔt=∫t0t0+K∫t0sf(t,s)ΔtΔs+∫t0+Kt0+2K∫s−Kt0+Kf(t,s)ΔtΔs.(2)

Proof

By a fundamental calculation, we can prove that E_P is Jordan Δ-measurable and f is Riemann Δ-integrable over E_P. 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

Then for t ∈ [t₀,t₀+𝓚)_𝕋,

and F(t,⋅) can only be discontinuous at t and t + 𝓚 since f : E_P → 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],

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

and for s ∈ [t₀+𝓚,t₀ + 2𝓚)_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],

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

Proposition 3.11

PAP(𝕋;𝔼ⁿ)⊂ S^p PAP(𝕋;𝔼ⁿ).

Proof

Let f = g + ϕ ∈ PAP(𝕋;𝔼ⁿ) with g ∈ AP(𝕋;𝔼ⁿ) and ϕ ∈ PAP₀(𝕋;𝔼ⁿ). Then f ∈ BS^p(𝕋;𝔼ⁿ), AP(𝕋;𝔼ⁿ)⊂ S^pAP(𝕋;𝔼ⁿ) by Proposition 2.8 (iii), and we need only to prove that ϕ ∈ PAP₀(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,

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

Then

This implies that 𝓝(ϕ) ∈ PAP₀(𝕋;ℝ). □

Remark 3.12

Obviously, (H) implies that ∥f(⋅, x(⋅))– f(⋅, y(⋅))∥_S^p ≤ L_f ∥x – y∥_S^p. 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|x – y| for every x,y ∈ 𝔼ⁿ, t ∈ 𝕋 and some constant L.

Theorem 3.13

Assume that f ∈ S^pAP(𝕋 ×𝔼ⁿ;𝔼ⁿ) satisfies (H), and u ∈ S^pAP(𝕋;𝔼ⁿ) withu(𝕋)compact. Then f(⋅,u(⋅)) ∈ S^pAP(𝕋;𝔼ⁿ).

Proof

By (H),

Then

That is

Since u(𝕋) is compact, for ε> 0, there exist finite open balls O_k, k = 1, 2, …, m, with center u_k ∈ u(𝕋) and radius ε8Lf such that u(𝕋) ⊂ ⋃k=1mO_k. Set B_k:= {s ∈ 𝕋: u(s) ∈ O_k}, k = 1,2,…, m. Then 𝕋 = ⋃k=1mB_k. Moreover, let E₁ := B₁, E_k := B_k ∖(⋃i=1k−1Bi),k = 2, …, m. Then E_i ∩ E_j = ∅ for i ≠ j and 𝕋 = ⋃k=1mE_k. Define a step function û:𝕋 → 𝔼ⁿ by û(s) := u_k, s ∈ E_k, k = 1,2,…, m. It is clear that |u(s) – û(s)| < ε8Lf for all s ∈ 𝕋. Then by (H), for τ ∈ ⋂k=1mT(f(⋅,u_k),ε4m),

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

This implies that 𝒢⊂ T(f(⋅,u(⋅)), ε), and T(f(⋅,u(⋅)), ε) is relatively dense. Therefore, f(⋅,u(⋅)) ∈ S^pAP(𝕋;𝔼ⁿ). □

Theorem 3.14

Let f = g + ϕ ∈ S^pPAP(𝕋 ×𝔼ⁿ;𝔼ⁿ) and u = x + y ∈ S^pPAP(𝕋;𝔼ⁿ) withx(𝕋)compact. Assume that f and g satisfy (H) with Lipschitz constants L_f and L_g, respectively. Then f(⋅,u(⋅)) ∈ S^pPAP(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

By Theorem 3.13, we have I₁ ∈ S^pAP(𝕋;𝔼ⁿ). So it suffices to prove that I₂,I₃ ∈ S^pPAP₀(𝕋;𝔼ⁿ).

Since f satisfies (H) with L_f,

which implies that BS^p(𝕋; 𝔼ⁿ). Moreover, since y∈ S^pPAP₀(𝕋;𝔼ⁿ), for given t₀∈𝕋,

This shows that I₂ ∈ S^pPAP₀(𝕋;𝔼ⁿ).

By the same arguments as to get (4), we can get I₃ ∈ BS^p(𝕋;𝔼ⁿ). Since x(𝕋) is compact, for any ε>0, as the proof of Theorem 3.13, we can find x_i ∈ x(𝕋), E_i⊂𝕋, i = 1,2,…,l and x̂:𝕋→𝔼ⁿ such that x̂(s): = x_i, s ∈ E_i, i = 1,2,…, l, E_i∩ E_j = ∅ for i≠ j, 𝕋 = ⋃i=1lEi, and |x(s)–x̂(s)|< ε2(Lf+Lg) for all s ∈𝕋. Since ϕ(⋅,x)∈ S^pPAP₀(𝕋;𝔼ⁿ) for every x∈𝔼ⁿ, there exists r₀ >0 such that for r>r₀, 1≤ i≤ l,

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

This yields that I₃ ∈ S^pPAP₀(𝕋;𝔼ⁿ).□

Corollary 3.15

Let f = g + ϕ∈ S^pPAP(𝕋×𝔼ⁿ;𝔼ⁿ) and u∈ PAP(𝕋;𝔼ⁿ). Assume that f and g satisfy (H). Then f(⋅,u(⋅))∈ S^pPAP(𝕋;𝔼ⁿ).

Definition 4.1

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

for s,t ∈ 𝕋, with the cylinder transformation

where Log is the principal logarithm.

Definition 4.2

[2] A matrix-valued function A: 𝕋 → ℝ ^n×nis 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;ℝ^n×n).

Definition 4.3

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

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

Lemma 4.4

([2]) Let t,s ∈ 𝕋.

1. e_p(t,t) = 1, e_A(t,t) = I.

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

3. e_p(t,s)e_p(s,r) = e_p(t,r), e_A (t,s)e_A(s,r) = e_A(t,r).

Lemma 4.5

Let a >0 be a constant and t,s ∈ 𝕋.

1. e_{⊖ a}(t,s)≤ 1 if t≥ s.

2. e_⊖a(t +τ,s + τ) = e_⊖a(t,s) for τ ∈Π.

3. There exists N >0 such that(t–s)e_⊖a(t,s)≤ N for t ≥ s.

4. For t₀ ∈ 𝕋, e_⊖a(t₀,⋅) is increasing on (–∞,t₀]_𝕋.

5. The series∑j=1∞e_{⊖ a}(t,σ(t)–(j–1)𝓚) converges uniformly for t ∈ 𝕋. Moreover, for all t∈ 𝕋,

   ∑j=1∞e⊖a(t,σ(t)−(j−1)K)≤λa:=11−e−aK,T=R,2+aμ¯+1aμ¯,T≠R,

   where μ¯:=supt∈Tμ(t).

Proof

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

where μ_L denotes the Lebesgue measure. For t> s, there exists a unique n_ts ∈ ℝ such that t ∈[s + (n_ts-1)𝓚,s + n_ts 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 + (n_ts–1)𝓚)_𝕋 contains n_ts –1 right-scattered points with form t₀ + n 𝓚. Denote Γ = 1 + aμ(t₀) for the convenient of writing. Then by (7),

So (iii) holds with N = 𝓚 Γ^{1–1 /lnΓ}/lnΓ.

(iv) can be verified easily by the definition.

If 𝕋 = ℝ, for t ∈ 𝕋,

That is (v) holds for 𝕋 = ℝ. If 𝕋 ≠ ℝ, then 𝓚 ≥ μ = supt∈Tμ(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 + n_t 𝓚, n_t ∈ℤ, μ(t₀ + n_t 𝓚) = μ(t₀) = μ, and

Then for any t ∈ 𝕋,

That is (v) holds for 𝕋 ≠ ℝ.

Lemma 4.6

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