1 The Nemytskii’s Operator

Definition 3.1

A continuous function \(f:\mathbb R\times \mathbb X\to \mathbb X\) is said to be almost automorphic if f(t, x) is almost automorphic in \(t\in \mathbb R\) uniformly for all x ∈ K, where K is any bounded subset of \(\mathbb X\). In other words for every sequence of real numbers \((s^{\prime }_n)\) there exists a subsequence (s n) such that

$$\displaystyle \begin{aligned}g(t,x)=\displaystyle\lim_{n\to\infty}f(t+s_n,x)\end{aligned}$$

is well-defined in \(t\in \mathbb R\) for all K and

$$\displaystyle \begin{aligned}\displaystyle\lim_{n\to\infty}g(t-s_n,x)=f(t,x)\end{aligned}$$

for all \(t\in \mathbb R\) and x ∈ K.

We denote by \(AA(\mathbb R\times \mathbb X,\mathbb X)\) the set of all such functions.

Theorem 3.2

If \(f,f_1,f_2\in AA(\mathbb R\times \mathbb X,\mathbb X)\) , then we have

  1. (i)

    \(f_1+f_2\in AA(\mathbb R\times \mathbb X,\mathbb X)\).

  2. (ii)

    \(\lambda f\in AA(\mathbb R\times \mathbb X,\mathbb X)\) , for any scalar λ.

Proof

Obvious. □

Theorem 3.3

If \(f\in AA(\mathbb R\times \mathbb X,\mathbb X)\) ,then

$$\displaystyle \begin{aligned}\sup_{t\in\mathbb R}\|f(t,x)\|=\sup_{t\in\mathbb R}\|g(t,x)\|=C_x<\infty\end{aligned}$$

for x in any bounded set \(K\subset \mathbb X\) where g is the function in Definition 3.1.

Proof

It is analogous to the proof of Remark 2.6. □

Theorem 3.4

If \(f\in AA(\mathbb R\times \mathbb X,\mathbb X)\) is lipschitzian in x uniformly in \(t\in \mathbb R\) , then the function g as in Definition 3.1 is also lipschitzian with the same Lipschitz constant.

Proof

Let L be a Lipschitz constant for the function f, i.e.

$$\displaystyle \begin{aligned}\|f(t,x)-f(t,y)\|<L\|x-y\|\end{aligned}$$

for x, y in any bounded subset K of \(\mathbb X\) uniformly in \(t\in \mathbb R\).

Let \(t\in \mathbb R\) be arbitrary and ε > 0 and K a bounded set in \(\mathbb X\) be given. Then for any sequence of real numbers \((s^{\prime }_n)\), there exists a subsequence (s n) such that

$$\displaystyle \begin{aligned}\|f(t+s_n,x)-g(t,x)\|<\frac{\varepsilon}{2}\end{aligned}$$

and

$$\displaystyle \begin{aligned}\|g(t-s_n,x)-f(t,x)\|<\frac{\varepsilon}{2}\end{aligned}$$

for n sufficiently large and uniformly in x ∈ K.

Let us write for x, y ∈ K

$$\displaystyle \begin{aligned}g(t,x)-g(t,y) & =g(t,x)-f(t+s_n,x)+f(t+s_n,x)-f(t+s_n,y)\\ & +f(t+s_n,y)-g(t,y).\end{aligned} $$

For n sufficiently large we get

$$\displaystyle \begin{aligned}\|g(t,x)-g(t,y)\|<\varepsilon+L\|x-u\|.\end{aligned}$$

And since ε is arbitrary we obtain

$$\displaystyle \begin{aligned}\|g(t,x)-g(t,y)\|\leq \varepsilon\end{aligned}$$

uniformly for x, y ∈ K, which completes the proof. □

Theorem 3.5 ([39])

Let \(f\in AA(\mathbb R\times \mathbb X,\mathbb X)\) and assume that f(t, ⋅) is uniformly continuous on each bounded set \(K\subset \mathbb X\) uniformly for \(t\in \mathbb R\) ; in other words, for any ε > 0 there exists δ > 0 such that if x, y ∈ K with ∥x − y∥ < δ, then ∥f(t, x) − f(t, y)∥ < ε for all \(t\in \mathbb R\) . Let \(\varphi \in AA(\mathbb X)\).

Then the Nemytskii operator \(\mathcal {N}:\mathbb R\to \mathbb X\) defined by \(\mathcal {N}(\cdot ):=f(\cdot ,\varphi (\cdot ))\) is in \(AA(\mathbb X)\).

Proof

Let \((s^{\prime }_n)\) be a sequence of real numbers. Then there exists a subsequence \((s_n)\subset (s^{\prime }_n)\) such that

  1. (i)

    limn→∞ f(t + s n, x) = g(t, x), for each \(t\in \mathbb R\) and \(x\in \mathbb X\),

  2. (ii)

    limn→∞ g(t − s n, x) = f(t, x), for each \(t\in \mathbb R\) and \(x\in \mathbb X\),

  3. (iii)

    limn→∞ φ(t + s n) = γ(t) for each \(t\in \mathbb R\),

  4. (iv)

    limn→∞ γ(t + s n) = φ(t) for each \(t\in \mathbb R\).

Let us define \(G:\mathbb R\to \mathbb X\) by G(t) = g(t, γ(t)). Then we obtain

$$\displaystyle \begin{aligned}\displaystyle\lim_{n\to\infty}\mathcal{N}(t+s_n)=G(t)\end{aligned}$$

and

$$\displaystyle \begin{aligned}\displaystyle\lim_{n\to\infty}G(t-s_n)=\mathcal{N}(t)\end{aligned}$$

for each \(t\in \mathbb R\).

Consider the inequality

$$\displaystyle \begin{aligned} \|\mathcal{N}(t+s_n)-G(t)\|&\leq \|f(t+s_n,\varphi(t+s_n))-f(t+s_n,\gamma(t))\|\\ &\quad +\|f(t+s_n,\gamma(t))-g(t,\gamma(t))\|. \end{aligned} $$

Since \(\varphi \in AA(\mathbb X)\), then φ and γ are bounded. Let us choose \(K\in \mathbb X\) such that φ(t), γ(t) ∈ K for all \(t\in \mathbb R\). In view of (iii) and the uniform continuity of f(t, x) in x ∈ K, we will have

$$\displaystyle \begin{aligned}\displaystyle\lim_{n\to\infty} \|f(t+s_n,\varphi(t+s_n))-f(t+s_n,\gamma(t))\|=0.\end{aligned}$$

Now by (i), we get

$$\displaystyle \begin{aligned}\displaystyle\lim_{n\to\infty} \|f(t+s_n,\gamma(t))-g(t,\gamma(t))\|=0,\end{aligned}$$

which proves that for each \(t\in \mathbb R\)

$$\displaystyle \begin{aligned}\displaystyle\lim_{n\to\infty}\mathcal{N}(t+s_n)=G(t).\end{aligned}$$

Similarly, we can prove that

$$\displaystyle \begin{aligned}\displaystyle\lim_{n\to\infty}G(t-s_n)=\mathcal{N}(t)\end{aligned}$$

for each \(t\in \mathbb R\). The proof is now complete. □

Theorem 3.6 ([39])

Let \(f\in AAA(\mathbb R^+\times \mathbb X,\mathbb X)\) with principal term g(t, x) and corrective term h(t, x). Assume that g(t, x) is uniformly continuous on any bounded set \(K\subset \mathbb X\) uniformly for \(t\in \mathbb R\) . Assume also that \(\varphi \in AAA(\mathbb X)\) . Then the Nemytskii operator \(\mathcal {N}:\mathbb R\to \mathbb X\) defined by \(\mathcal {N}(\cdot ):=f(\cdot ,\varphi (\cdot ))\) is in \(AAA(\mathbb X)\).

Proof

Let α(t) and β(t) be the principal and corrective terms of φ(t), respectively. Let us write

$$\displaystyle \begin{aligned} f(t,\varphi(t))&=g(t,\alpha(t))+f(t,\varphi(t))-g(t,\alpha(t))=g(t,\alpha(t))+g(t,\varphi(t))\\ &\quad -g(t,\alpha(t))+h(t,\varphi(t)). \end{aligned} $$

In view of Theorem 3.5, \(g(t,\alpha (t))\in AA(\mathbb R\times \mathbb X,\mathbb X)\).

On the other hand, the uniform continuity of g(t, φ(t)) implies that for any ε > 0, there exists δ > 0 such that

$$\displaystyle \begin{aligned}\|g(t,\varphi(t))-g(t,\alpha(t))\|<\varepsilon\end{aligned}$$

if φ(t), α(t) ∈ K for any \(t\in \mathbb R^+\) and a given bounded set \(K\subset \mathbb X\) and ∥φ(t) − α(t)∥ < δ. Moreover since \(\beta (t)\in C_0(\mathbb R,\mathbb X)\), there exists T > 0 such that

$$\displaystyle \begin{aligned}\|\varphi(t)-\alpha(t)\|=\|\beta(t)\|<\delta,\end{aligned}$$

for t > T. Consequently, we get

$$\displaystyle \begin{aligned}\displaystyle\lim_{t\to\infty}\|g(t,\varphi(t))-g(t,\alpha(t))\|=0.\end{aligned}$$

We know also that

$$\displaystyle \begin{aligned}\displaystyle\lim_{t\to\infty}\|h(t,\varphi(t))\|=0.\end{aligned}$$

This proves that

$$\displaystyle \begin{aligned}g(t,\varphi(t))-g(t,\alpha(t))+h(t,\varphi(t))\in C_0(\mathbb R^+,\mathbb X),\end{aligned}$$

and consequently

$$\displaystyle \begin{aligned}\mathcal{N}(\cdot):=f(\cdot,\varphi(\cdot))\in AAA(\mathbb X)\end{aligned}$$

□

Bibliographical Notes

Most of this chapter are contained in the first edition of this book. It is noted that C. Lizama and J.G Mesquita [41,42,43] and Milcé et al. [46, 47, 50, 62, 63] studied almost automorphy on time scales and its application to dynamic equations on time scales. This is another growing field which needs further investigation.