Almost Automorphic Functions of Order n and Applications to Dynamic Equations on Time Scales

Definition 1. A time scale is an arbitrary nonempty closed subset of real numbers.

Definition 2. Let $$ be a time scale. The forward and backward jump operators $$ and the graininess $$ are defined, respectively, by

Definition 3. Let $$ be time scales.

• (i)

  A point $$ is called right-dense if $$ and σ(t) = t.

• (ii)

  A point $$ is called left-dense if $$ and ρ(t) = t.

Definition 4 (see [9].)A function $$ is called regulated provided its right-sided limits exist (finite) at all right-dense points in $$ and its left-sided limits exist (finite) at all left-dense points in $$.

Definition 5 (see [9].)A function $$ is called rd⁡-continuous provided it is continuous at right-dense points in $$ and its left-sided limits exist (finite) at left-dense points in $$.

Definition 6 (see [9].)Let $$ be a function and $$. We define f^Δ(t) to be the number (provided it exists) with the property that, given any ϵ > 0, there exists a neighborhood U of t such that

We call f^Δ(t) the delta (or Hilger) derivative of f at t.

Moreover, we say that f is delta (or Hilger) differentiable (or in short differentiable) on $$ provided f^Δ(t) exists for all $$. The function $$ is called the (delta) derivative of f on $$.

Theorem 7 (see [9].)Assume $$ is a function and let $$. Then one has the following.

• (i)

  If f is differentiable at t, then f is continuous at t.

• (ii)

  If f is continuous at t and t is right-scattered, then f is differentiable at t with

  $$ (6)

• (iii)

  If  t is right-dense, then f is differentiable at t if and only if the limit

  $$ (7)
  exists as a finite number. In this case
  $$ (8)

• (iv)

  If f is differentiable at t, then

  $$ (9)

Theorem 8 (see [9].)Assume $$ are differentiable at $$. Then

• (i)

  the sum $$ is differentiable at t with

  $$ (10)

• (ii)

  for any constant α, $$ is differentiable at t with

  $$ (11)

• (iii)

  the product $$ is differentiable at t with

  $$ (12)

Definition 9 (see [9].)For a function $$ one will talk about the second derivative f^ΔΔ provided f^Δ is differentiable on $$ with derivative $$. Similarly one defines higher order derivatives $$. Finally, for $$, one denotes σ²(t): = σ(σ(t)) and ρ²(t): = ρ(ρ(t)), and σⁿ(t) and ρⁿ(t) are defined accordingly. For convenience one also puts ρ⁰(t) = σ⁰(t) = t and $$.

Theorem 10 (see [9] Leibniz formula.)Let $$ be the set consisting of all possible strings of length n, containing exactly k times σ and n − k times Δ. If f^Λ exists for all $$, then

holds for all $$.

Theorem 11 (chain rule). Let $$ be continuously differentiable and suppose $$ is delta differentiable on $$. Then $$ is delta differentiable and the formula

holds for all $$.

Theorem 12 (chain rule). Assume that $$ is strictly increasing and $$ is a time scale. Let $$, where $$ is a Banach space. If ν^Δ(t) and $$ exist for $$, then

where $$ denote the  ^Δ in $$.

Definition 13. If $$, $$, and f is a C_rd⁡ function on [a, +∞), then the improper integral of f is defined by

provided this limit exists. In this case, the improper integral is said to converge.

Lemma 14 (see [4].)Let $$, $$, and assume that $$ is continuous at (t, t), where $$ with t > a. Assume also that f^Δ(t, ·) is rd⁡-continuous on [a, σ(t)]. Suppose that, for each ϵ > 0, there exists a neighborhood U of τ ∈ [a, σ(t)] such that

where f^Δ denotes the derivative of f with respect to the first variable. Then

Definition 15 (see [9].)One says that a function $$ is regressive provided

The set of all regressive functions will be denoted by $$.

Definition 16. One defines the set $$ of all positively regressive elements of $$ by

Definition 17 (see [9].)If $$, then one defines the generalized exponential function by

where the cylinder transformation $$ is given by where log⁡ is the principal logarithm function. For h = 0, we define ξ₀(z) = z for all $$.

Lemma 18 (see [9].)Assume that $$ are two regressive functions. Then

• (i)

  e₀(t, s) ≡ 1 and e_p(t, t) ≡ 1;

• (ii)

  e_p(t, s) = 1/e_p(s, t) = e_⊖p(s, t);

• (iii)

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

• (iv)

  [e_p(t, s)] ^Δ = p(t)e_p(t, s).

Lemma 19 (see [9].)Assume that $$. Then

• (i)

  e_p(t, s) > 0, for all $$;

• (ii)

  if p(t) ≤ q(t) for all t ≥ s, $$, then e_p(t, s) ≤ e_q(t, s) for all t ≥ s.

Lemma 20 (see [9].)If $$ and $$, then

• (i)

  [e_p(c, ·)] ^Δ = −p[e_p(c, ·)] ^σ,

• (ii)

  $$ for all t ≥ s.

Proposition 21 (see [9].)Let $$ be rd⁡-continuous and regressive, $$, and $$. Then the unique solution of the initial value problem

is given by

Definition 22. Let A be an m × m matrix-valued function on $$. One says that A is rd⁡-continuous on $$ if each entry of A is rd⁡-continuous on $$. One denotes by $$ the class of all rd⁡-continuous m × m matrix-valued functions on $$.

We say that A is delta differentiable on $$ if each entry of A is delta differentiable on $$. And in this case, we have

Definition 23. An m × m matrix-valued function A is called regressive if

The class of all such regressive rd⁡-continuous functions is denoted by

Definition 24 (see [11].)A time scale $$ is called invariant under translations if

Lemma 25. Let $$ be an invariant under translations time scale. Then one has

• (i)

  $$;

• (ii)

  $$.

Proof.

• (i)

  In view of the definition of Π, if $$, then for all τ ∈ Π we have $$. Thus, $$. Conversely, assume that $$. Then, for any τ ∈ Π, we have $$. Thus $$.

• (ii)

  It is clear that if $$ then $$. Now assume that $$. If $$, then we have $$ for any $$; particularly, for t = τ, we have $$. This contradicts the fact that $$. Thus $$.

Lemma 26 (see [2].)Let $$ be an invariant under translation time scale. If t is right-dense (resp., right-scattered), then for every h ∈ Π, t + h is right-dense (resp., right-scattered).

Lemma 27 (see [2].)Let $$ be an invariant under translations time scale and h ∈ Π. Then

• (i)

  σ(t + h) = σ(t) + h and σ(t − h) = σ(t) − h, for every $$;

• (ii)

  μ(t + h) = μ(t) = μ(t − h), for every $$.

Remark 28. As we pointed out in Section 1 time scales invariant under translations are not automatically symmetric. Since almost automorphic functions are defined on symmetric domains, some definitions and results in [11] on these functions will be given with additional assumption on the time scale. More precisely we will assume that the time scale is symmetric and invariant under translations.

Definition 29. Let $$ be Banach space and let $$ be a symmetric time scale which is invariant under translations. Then the rd⁡-continuous function $$ is called almost automorphic on $$ if for every sequence (s_n) on Π, there exists a subsequence (τ_n)⊂(s_n) such that

is well defined for each $$ and for each $$.

Remark 30. In view of Lemma 27, if $$ is a symmetric time scale which is invariant under translations then the graininess function $$ is an almost automorphic function.

Theorem 31. Let $$ be a symmetric time scale which is invariant under translation. Assume that the rd⁡-continuous functions $$ are almost automorphic on $$. Then the following assertions hold:

• (i)

  f + g is almost automorphic on time scales;

• (ii)

  λf is almost automorphic on $$ for every scalar λ;

• (iii)

  for each l ∈ Π, the function $$ defined by f_l(t) = f(t + l) is almost automorphic on time scales;

• (iv)

  $$ defined by $$ is almost automorphic on time scales;

• (v)

  $$; that is, f is a bounded function;

• (vi)

  $$, where

  $$ (31)

Proof. See [11].

Remark 32. Notice that

• (i)

  in order to give a sense to (iii), we consider l as an element of Π instead of $$ as in [11];

• (ii)

  we need the symmetry of the time scale $$ to obtain that $$ if $$, that is, to give a sense to the definition of $$ in (iv).

Remark 33. The space $$ equipped with the norm $$ is a Banach space (see [11] pp. 2280).

Lemma 34. If $$, the range $$ is relatively compact in $$.

Proof. Let $$ be fixed and let $$ be a sequence in R_f. Then, for any $$, there is $$ such that $$. By invariance under translations of $$, for each $$, we can find $$ such that $$. Hence, for all $$, we have $$. Since f is almost automorphic on time scale, there exists a subsequence (α_n) _n of $$ such that

Thus, the subsequence (x_n = f(a + α_n)) converges to $$. Therefore, R_f is relatively compact in $$.

Theorem 35 (see [11].)Let $$ be a symmetric time scale which is invariant under translations. Let also $$ and $$ be two almost automorphic functions on time scales. Then the function $$ defined by (uf)(t) = u(t)f(t) is almost automorphic on time scales.

Theorem 36 (see [11].)Let $$ be a symmetric time scale which is invariant under translations and let (f_n) be a sequence of almost automorphic functions such that lim⁡_n→+∞f_n(t) = f(t) converges uniformly for each $$. Then, f is an almost automorphic function.

Theorem 37 (see [11].)Let $$ be a symmetric time scale which is invariant under translations and let $$ and $$ be Banach spaces. Suppose $$ is an almost automorphic function on time scales and $$ is a continuous function; then the composite function $$ is almost automorphic on time scales.

Definition 38 (see [11].)Let $$ be a (real or complex) Banach space and let $$ be a symmetric time scale which is invariant under translations. Then a rd⁡-continuous function $$ is called almost automorphic on $$ for each $$ if for every sequence (s_n) on Π, there exists a subsequence (τ_n)⊂(s_n) such that

is well defined for each $$, $$ and for each $$ and $$.

Theorem 39 (see [11].)Let $$ be a symmetric time scale which is invariant under translations and let $$ be almost automorphic functions on time scale in t for each x in $$. Then the following assertions hold:

• (i)

  f + g is almost automorphic on time scale in t for each x in $$;

• (ii)

  λf is almost automorphic function on time scale in t for each x in $$, where λ is an arbitrary scalar;

• (iii)

  $$, for each $$;

• (iv)

  $$, for each $$, where $$ is the function in Definition 38.

Theorem 40. Let $$ be a symmetric time scale which is invariant under translations and let $$ be almost automorphic functions on time scale in t for each x in $$ and satisfy Lipschitz condition in x uniformly in t; that is,

for all $$. Assume that $$ almost automorphic on time scale. Then the function $$ defined by U(t) = f(t, ϕ(t)) is almost automorphic on time scale.

Proposition 41. $$ equipped with the norm defined above is a Banach space.

Proof. It’s clear that $$ is a linear space and that (37) is a norm on $$. Now, let (f_j) _j be a Cauchy sequence in $$. Then for any ϵ > 0, there exists $$ such that, for all $$, j > N, we have

In other words, given ϵ > 0, there is $$ such that, for all $$, j > N, In particular, for all $$, $$, i = 0,1, …, n, are Cauchy sequences in $$ which is a Banach space. Thus if we denote by f the limit of $$, we have that

Since for i = 0,1, …, n,

passing to the limit in these above n + 1 relations as p → +∞, we obtain for i = 0,1, …, n, This means that, for all $$, which on the one hand proves that $$ since for any $$, and on the other hand shows that

Definition 42. Let $$ be a (real or complex) Banach space and let $$ be a symmetric time scale which is invariant under translations. Then a rd⁡-continuous function $$ is called almost automorphic if f(t, x) is almost automorphic in $$ uniformly for each x ∈ B, where B is any bounded subset of $$.

Definition 43. A function $$ is said to be $$-almost automorphic (briefly $$.), if $$ belong to $$ for all i = 1, …, n.

Lemma 44. We have $$.

Proof. It is straightforward from the definition of an almost automorphic function on time scales (see Theorem 31).

Proposition 45. A linear combination of $$-a.a. functions is a $$-a.a. function. Moreover, let $$ be a Banach space over the field $$. Let $$, $$, and $$. Assume that ν^Λ exists for all $$ and is almost automorphic on time scale. Then the following functions are also in $$:

• (i)

  f + g,

• (ii)

  λf,

• (iii)

  νf,

• (iv)

  f_a(t): = f(t + a), where a ∈ Π is fixed.

Proof. For the proof of (i) and (ii), one proceeds as in [11]. To prove (iii), we use the Leibnitz formula on time scales, the definition, and the properties of an almost automorphic function; we get the result easily.

Now, let us prove (iv). For any a ∈ Π, if we consider the function $$ defined by v(t) = a + t, then we have f_a(t) = (f∘v)(t), for all t in $$. It is clear that v is strictly increasing, $$, and v^Δ(t) = 1, for all $$. Using Theorem 12, we obtain $$ for each $$. Hence, for f^Δ being an almost automorphic function on time scale, we deduce that $$ is almost also automorphic on time scale. Similarly, we prove that $$ is almost automorphic for i = 1,2, …, n. Thus, $$.

Theorem 46. If a sequence $$ of $$-a.a. functions is such that $$ as k → +∞, then f is $$-a.a. function.

Proof. From the assumption, it is clear that $$. Moreover, $$ uniformly on $$ for each i = 0, …, n. Thus Theorem 36 allows us to say that f is a $$. function.

Corollary 47. $$ considered with norm (37) turns out to be a Banach space.

Proof. In view of Proposition 45 and Lemma 44, $$ is a linear subspace of $$. Let (f_p), $$, $$, be a Cauchy sequence. Then there exists $$ such that $$. By Theorem 46, $$, so it is a Banach space.

Theorem 48. Let $$ be a symmetric time scale which is invariant under translations. Let also $$ be a Banach space and $$ an almost automorphic function on $$. Assume that f is Δ-differentiable on $$ and f^Δ is uniformly continuous. Then f^Δ is also almost automorphic on time scales.

Proof. Assume that the points of $$ are right-dense. Then for $$ being invariant under translations, we obtain $$. Hence f^Δ = f^′ and since f^Δ is uniformly continuous, it follows from Theorem  2.4.1 in [12] that f^Δ is almost automorphic on time scale.

Now, let us suppose that $$ has at least a right-scattered point t₀; then we have

Given a sequence $$, since f is almost automorphic, there is a subsequence (α_n) _n such that since Lemma 27 holds. On the other hand, we have The proof is completed.

Theorem 49. If $$ and $$ is uniformly continuous, then $$.

Proof. In view of Theorem 48, we have $$. This means that f is in $$. Then it follows that $$.

Notation 1. Let $$ be a symmetric time scale which is invariant under translations. If $$ is a function and a sequence α = (α_n) ⊂ Π is such that

we will write T_sf = g.

Remark 50 (see [12].)Consider the following.

• (i)

  T_s is a linear operator.

•  

  Given a fixed sequence α = (α_n) ⊂ Π, the domain of T_s is $$. D(T_s) is a linear set.

• (ii)

  Let us write −s = (−α_n) and suppose that f ∈ D(T_s) and T_sf ∈ D(T_−s). The product operator A_s = T_−sT_sf is well defined. It is also a linear operator.

• (iii)

  A_s maps bounded functions into bounded functions, and for an almost authorphic function on time scale f, we get A_sf = f.

Theorem 51. Let $$ be a symmetric time scale containing zero and invariant under translations. Let also $$. One considers the function $$ defined by $$. Then $$ if and only if R_F is relatively compact in $$.

Proof. In view of Lemma 34, it suffices to prove that $$ if R_F is relatively compact in $$.

Assume that R_F is relatively compact in $$ and let $$. Then there exists a subsequence $$ of $$ such that

pointwise on $$, and for some $$.

We get, for every $$,

Making the change of variable $$, we obtain If we apply the Lebesgue dominated theorem to this latter identity, we get for each $$. Let us observe that the range of the function $$ is also relatively compact and so that we can extract a subsequence (s_n) of $$ such that for some $$. Now we can write so that Let us prove now that α₂ = 0. Using Notation 1 we get where s = (s_n). Now it is easy to observe that F and α₂ belong to the domain of A_s; therefore A_sF is also in the domain of A_s and we deduce the equation We can continue indefinitely the process to get But we have and F(t) is a bounded function.

This leads to contradiction if α₂ ≠ 0. Hence, α₂ = 0, and using Remark 50, we have A_sF = F, so F is almost automorphic. The proof is complete.

Theorem 52. If $$ and the range R_F is relatively compact, then $$.

Proof. If $$ and the range R_F is relatively compact, then in view of Theorem 51, $$. Since $$, F^Δ(t) = f(t), for $$. Thus $$ and, consequently, $$.

Theorem 53. If $$ and the range R_F is relatively compact then $$.

Proof. If $$ and the range R_F is relatively compact then in view of Theorem 51, $$. Therefore, $$. This means that $$.

Corollary 54. If $$ and the range R_f is relatively compact, then $$.

Proof. We know that $$, for each $$. In view of Theorem 53, we have $$.

Proposition 55. Let $$ be a bounded linear operator and $$. Then we have $$.

Proof. Since A is a bounded linear operator, we have $$. Therefore, observing the fact that $$ for each i = 0,1, …, n because $$, it follows from Theorem 37 that $$, for all i = 0,1, …, n. Hence $$.

Corollary 56. Let $$ be a bounded linear operator with a relatively compact range. Then $$-valued function G defined above is in $$.

Proof. According to Proposition 55, $$ for every $$. Since A is a bounded linear operator, the range R_A of the operator A contains the range R_G of G. Hence R_G is relatively compact and it follows from Theorem 53 that $$.

Remark 57. In Corollary 56, if operator A is compact (or of the finite rank), the stated result holds.

Theorem 58. If $$ and $$, then $$.

Proof. First, we observe that the result holds if for n = 0, in view of Theorem 37, we have that $$ if $$ and $$. So, it suffices to show that $$ to complete the proof of the theorem.

By Theorem 11, for each $$, we have

Since $$ and $$ we have that, for any h ∈ [0,1], the function t ↦ f(t) + hμ(t)f^Δ(t) belongs to $$. Therefore for ϕ^′ being continuous, Theorem 37 allows us to say that the function t ↦ ϕ^′(f(t) + hμ(t)f^Δ(t)) is also in $$ for any h ∈ [0,1] and consequently the function $$ is $$. It then follows from Theorem 35 that $$ since $$.

Definition 59 (see [5].)Let A(t) be m × mrd⁡-continuous matrix-valued function on $$. The linear system

is said to admit an exponential dichotomy on $$ if there exist positive constants K, α, projection P, and the fundamental solution matrix X(t) of (65) satisfying where $$ is the matrix norm on $$. This means that if A = (a_ij) _m×m then we can take $$.

Theorem 60 (see [11].)Let $$ be a symmetric time scale which is invariant under translation and let $$ be almost automorphic and nonsingular on $$ and $$ and $$ are bounded. Assume that linear system (65) admits an exponential dichotomy and $$ is an almost automorphic function on time scales. Then system (67) has an almost automorphic solution as follows:

where X(t) is the fundamental solution matrix of  (65).

Lemma 61 (see [3].)Let $$ be an almost automorphic function, $$, and $$. Then the linear system

admits an exponential dichotomy on $$.

Lemma 62. Let $$ be a symmetric time scale which is invariant under translation. Let A(t) = diag⁡(−c₁(t), −c₂(t), …, −c_m(t)) be such that the functions $$ are almost automorphic, $$, and $$, i = 1,2, …, m. Assume also that $$ is an almost automorphic function on time scales. Then system (67) has an almost automorphic solution as follows:

Lemma 63. Let $$ be a symmetric time scale which is invariant under translation. Let A(t) = diag⁡(−c₁(t), −c₂(t), …, −c_m(t)) be such that the functions $$ are $$-almost automorphic, $$, and $$, i = 1,2, …, m. Assume also that f is a $$-almost automorphic function on time scales. Then system (67) has a $$-almost automorphic solution as follows:

Proof. By Lemma 62,

is a C_rd⁡-almost automorphic solution to system (67). Now, since using on the one hand the fact that functions $$, t ↦ c_i(t), and t ↦ f_i are C_rd⁡-almost automorphic and on the other hand the fact that Theorem 35 holds, we deduce that $$, i = 1,2, …, m, are also C_rd⁡-almost automorphic. The proof of Lemma 63 is then complete.

Definition 64. Let $$ be a $$-almost automorphic solution of (2) with initial value $$. Assume that there exists a positive constant λ with $$ such that for $$, there exists M > 1 such that for an arbitrary solution z(t) = (x₁(t), x₂(t), …, x_m(t)) ^T of (2) with initial value φ(t) = (φ₁(t), φ₂(t), …, φ_m(t)) ^T, z satisfies

where

Then, the solution z^* is said to be exponentially stable.

Theorem 65. Assume that (H₁) and (H₃) hold. Assume also that

Then, (2) has a unique $$-almost automorphic solution in E₀ = {ϕ(t) ∈ E∣‖ϕ‖_E ≤ L}, where L is a positive constant satisfying

Proof. For any φ ∈ E, we consider the following $$-almost automorphic system:

Since (H₁) holds, it follows from Lemma 61 that the system admits an exponential dichotomy on $$. Thus by Lemma 63, system (2) has a $$-almost automorphic solution:

Now, we prove that the following mapping is a contraction on E₀:

where To this end we proceed in two steps.

Step 1. We prove that if φ ∈ E₀ then (Γφ) ∈ E₀.

Let φ ∈ E₀; then using (H₂) and the fact that $$, we have

Consequently,

Step 2. We prove that Γ is a contraction on E₀.

Let φ and ϕ be in E₀. Then using (H₃) and the fact that $$, we obtain

Hence, Because $$, $$, Γ is a contraction on E₀. We then deduce by the fixed point theorem of Banach that Γ has a unique solution in E₀. Consequently system (2) has a unique $$-almost automorphic solution in E₀:

Theorem 66. Assume that (H₁) and (H₃) hold. Assume also that

Then, the $$-almost automorphic solution of (2) is globally exponentially stable.

Proof. According to Theorem 65, system (2) has a $$-almost automorphic solution $$ with initial value

Assume that z(t) = (x₁(t), x₂(t), …, x_m(t)) ^T is an arbitrary solution of (2) with initial value φ(t) = (φ₁(t), φ₂(t), …, φ_m(t)) ^T. Let $$, i = 1,2, …, m. Then it follows from (2) that Multiplying both sides of the first equation in (93) by $$ and integrating on $$, where $$, we obtain, for i = 1,2, …, m, This means that, for i = 1,2, …, m, Then choose $$ and One proves by contradiction as in [8] that This means that the $$-almost automorphic solution x^* of (2) is globally exponentially stable.