Almost Periodic Solutions of the Differential Equation in Locally Convex Spaces

Abstract

As an application of results obtained in Chap. 3, we will study conditions for almost periodicity of solutions of the linear differential equation $x'(t)=Ax(t)+f(t),\;t\in \mathbb R$ and the associated homogeneous equation in locally convex spaces. We will start with the case of a bounded linear operator A and then study the general case of a (eventually unbounded) linear operator A, which generates an equicontinuous C ₀-semigroup of linear operators.

As an application of results obtained in Chap. 3, we will study conditions for almost periodicity of solutions of the linear differential equation $x'(t)=Ax(t)+f(t),\;t\in \mathbb R$ and the associated homogeneous equation in locally convex spaces. We will start with the case of a bounded linear operator A and then study the general case of a (eventually unbounded) linear operator A which generates an equicontinuous C ₀-semigroup of linear operators.

1 Linear Equations

Definition 11.1

A Fréchet space E is said to be perfect if it satisfies the following property:

(F) Every function $f:\mathbb R\to E$ with a bounded range $\{f(t):\;\;t\in \mathbb R\}$ and f′(t) almost periodic is necessarily almost periodic.

Example 11.2

Denote by s the linear space of all real numbers $s=\{x=(x_n):\;\;x_n\in \mathbb R,\;\;n=1,2,\ldots \}$. For each n, define

$$\displaystyle \begin{aligned} p_n(x):=|x_n|,\;\;x\in s. \end{aligned}$$

Obviously p _n is a seminorm defined on s. Define

$$\displaystyle \begin{aligned}q_n:=p_1\vee p_2\vee\ldots\vee p_n,\;\;n=1,2,\ldots\end{aligned}$$

We have q _n ≤ q _n+1, n = 1, 2, … The space s considered with the family of seminorms (q _n) is a Fréchet space. Moreover, it can be proved (see [1], 17.7, p. 210) that each closed and bounded subset of s is compact. Thus, in particular, s is a Banach space. Moreover, in view of Theorem 8.12, s is perfect.

1.1 The Homogeneous Equation x′ = Ax

Consider in a complete Hausdorff locally convex space E the equation

$$\displaystyle \begin{aligned} x'(t)=Ax(t),\;t\in\mathbb R. \end{aligned} $$

(1.1)

Theorem 11.3 ([51])

Let E be a perfect Fréchet space. Assume that

• (i) A is a compact linear operator on E;

• (ii) {A ^k : k = 1, 2, …} is equicontinuous;

• (iii) for every seminorm p, there exists a seminorm q such that

  $$\displaystyle \begin{aligned}p(e^{tA}x)\leq q(x),\;\;\mathit{\mbox{for every}}\;t\in \mathbb R,\;x\in E.\end{aligned}$$

  Then, the unique solution of Eq.( 1.48 ) is almost periodic in E.

Proof

Let x(t) = e ^tA x ₀ be the unique solution of Eq. (1.1). Then, by (iii), it is bounded. Since E is a perfect Fréchet space, it suffices to prove that x′(t) is almost periodic (cf. Property (F) above).

Now, assumption (i) implies that the set $\{x'(t):\;\in \mathbb R\}$ is also relatively compact in E.

Let $(s^{\prime }_n)$ be an arbitrary sequence of real numbers; then, we can extract a subsequence (s _n) such that (x′(s _n)) is convergent and thus is a Cauchy sequence in E.

But we have

$$\displaystyle \begin{aligned}\begin{array}{rlll} x'(t+s_n)&=&Ax(t+s_n)\\ &=&A e^{(t+s_n)A}x_0\\ &=&Ae^{tA}e^{s_nA}x_0\\ &=&Ae^{tA}x(s_n)\\ &=&e^{tA}Ax(s_n)\\ &=&e^{tA}x'(s_n) \end{array} \end{aligned}$$

for every n = 1, 2, … and every $t\in \mathbb R$.

If p is a seminorm on E, then there exists a seminorm q on E such that

$$\displaystyle \begin{aligned}\begin{array}{rllll} p(x'(t+s_n)-x'(t+s_m))&=&p(e^{tA}(x'(s_n)-x'(s_m)))\\ &\leq& q(x'(s_n)-x'(s_m)) \end{array} \end{aligned}$$

for every n, m and every $t\in \mathbb R$.

It follows that (x′(t + s _n)) is uniformly Cauchy in t; therefore, it is uniformly convergent in t.

We conclude by Bochner’s criterion that x′(t) is almost periodic. This completes the proof. □

1.2 The Inhomogeneous Case

Now, let us consider in a perfect Fréchet space E the inhomogeneous equation:

$$\displaystyle \begin{aligned} x'(t)=Ax(t)+f(t),\;\;t\in\mathbb R, \end{aligned} $$

(1.2)

with the following assumptions:

• (H1) A is a compact linear operator on E;

• (H2) {A ^k; k = 1, 2, …} is equicontinuous;

• (H3) for every seminorm p _n, there exists a seminorm q _n such that

  $$\displaystyle \begin{aligned}p_n(e^{tA}x)\leq q_n(x),\;\;\mbox{for every}\;t\in\mathbb R,\;x\in E;\end{aligned}$$

• (H4) f ∈ AP(E) and for each $p_n\in \mathcal {P}$, there exists a function $\psi _{p_{n}}:\mathbb R\to \mathbb R^+$ with

  $$\displaystyle \begin{aligned}p_n(f(s))\leq \psi_{p_{n}}(s)\;\;\mbox{and}\;\; \int_{-\infty}^{\infty}\psi_{p_{n}}(s)ds<\infty.\end{aligned}$$

  Now, let us state and prove the following.

Theorem 11.4 ([16])

Under assumptions (H1)–(H4), every solution of Eq.(1.2) is almost periodic in E.

Proof

By Theorem 11.3, the function e ^tA x(0) ∈ AP(E).

Now, let $F(t):=\int _{0}^{t}e^{(t-s)A}f(s)ds$. It is also immediate that s → e ^−sA f(s) is in AP(E) based on Theorem 8.26. In view of assumptions (H5) and (H8), F(t) is bounded over $\mathbb R$. We then deduce that $\int _{0}^{t}e^{-sA}f(s)ds\in AP(E)$ since E is a perfect Fréchet space. But this last integral is equal to e ^−tA F(t). Applying again Theorem 8.26, we obtain e ^tA(e ^−tA F(t)) = F(t) is almost periodic. The proof is now complete. □

Let us now consider a Fréchet space in which property (F) may not hold.

Theorem 11.5 ([51, 55])

Let E be a Fréchet space (not necessarily perfect), and assume that assumptions (i)–(iii) of Theorem 11.3 are satisfied. Assume also that the range $\mathcal {R}(A)$ of the operator A is dense in E.

Then, every solution of Eq.(1.1) is almost periodic.

Proof

Let us observe that the first part of the proof of Theorem 11.3 tells us that if x(t) = e ^tA x ₀ is a solution of Eq. (1.48) with x ₀ ∈ D(A) the domain of A, then x′(t) will be almost periodic. □

Lemma 11.6

Every solution of Eq.( 1.48 ) with initial data in $\mathcal {R}(A)$ , the range of A, is almost periodic.

Proof

Let $a\in \mathcal {R}(A)$ and consider the unique solution y(t) with y(0) = a. There exists x ₀ ∈ D(A) such that Ax ₀ = a. We have

$$\displaystyle \begin{aligned}y(t)=e^{tA}a=e^{tA}Ax_0=Ae^{tA}x_0=Ax(t)=x'(t),\end{aligned}$$

where x(t) = e ^tA x ₀. Therefore, x′(t), and consequently y(t), is almost periodic. □

Proof (of Theorem 11.5 (continued))

Consider the solution x(t) of Eq. (1.48) with x(0) ∈ E. Since $\mathcal {R}(A)$ is dense in E, there exists a sequence (a _n) in $\mathcal {R}(A)$ such that

$$\displaystyle \begin{aligned}\displaystyle\lim_{n\to\infty}a_n=x(0).\end{aligned}$$

Consider a sequence of solutions (y _n(t)) with y _n(0) = a _n, n = 1, 2, … By the above, each y _n(t) is almost periodic. □

Now, to prove almost periodicity of x(t), it suffices to prove that (y _n(t)) converges to x(t) uniformly in $t\in \mathbb R$. We have

$$\displaystyle \begin{aligned}x(t)=e^{tA}x(0),\;\;y_n(t)=e^{tA}a_n.\end{aligned}$$

So, given a seminorm p, there exists a seminorm q by (iii) such that

$$\displaystyle \begin{aligned}p(y_n(t)-x(t))=p(e^{tA}(a_n-x(0)))\leq q(a_n-x(0))\end{aligned}$$

for every n = 1, 2, … and every $t\in \mathbb R$. The conclusion follows immediately.

Bibliographical Notes

The contributions in this chapter are due to D. Bugajewski and G.M. N’Guérékata [16, 55].