Introduction and Preliminaries

Abstract

This monograph presents several recent developments on the theory of almost automorphic and almost periodic functions (in the sense of Bohr) with values in an abstract space and its application to abstract differential equations. We suppose that the reader is familiar with the fundamentals of Functional Analysis. However, to facilitate the understanding of the exposition, we give in the beginning, without proofs, some facts of the theory of topological vector spaces and operators which will be used later in the text.

This monograph presents several recent developments on the theory of almost automorphic and almost periodic functions (in the sense of Bohr) with values in an abstract space and its application to abstract differential equations. We suppose that the reader is familiar with the fundamentals of Functional Analysis. However, to facilitate the understanding of the exposition, we give in the beginning, without proofs, some facts of the theory of topological vector spaces and operators which will be used later in the text.

1 Banach Spaces

We denote by $\mathbb R$ and $\mathbb C$ the fields of real and complex numbers, respectively. We will consider a (real or complex) normed space $\mathbb X$, that is a vector space over the field $\Phi =\mathbb R$ or $\mathbb C$ (respectively) with norm ∥⋅∥.

Definition 1.1

A sequence of vectors (x _n) in $\mathbb X$ is said to be a Cauchy sequence if for every 𝜖 > 0, there exists a natural number N such that ∥x _n − x _m∥ < 𝜖 for all n, m > N.

Proposition 1.2

The following are equivalent:

• (i) (x _n) is a Cauchy sequence.

• (ii) $\|x_{n_{k+1}}-x_{n_{k}}\|\to 0$ as k →∞, for every increasing subsequence of positive integers (n _k).

Proposition 1.3

If (x _n) is a Cauchy sequence in a normed space $\mathbb X$ , the sequence of reals (∥x _n∥) is convergent.

Definition 1.4

A Banach space $\mathbb X$ is a complete normed space, that is, a normed space $\mathbb X$ in which every Cauchy sequence is convergent to an element of $\mathbb X$.

Definition 1.5

A Banach space $\mathbb X$ is said to be uniformly convex if for every α, 0 < α < 2, there exists a number δ = δ(α) > 0 such that for every $x,y\in \mathbb X$ with ∥x∥ < 1, ∥y∥ < 1, ∥x − y∥ > α, we have ∥x + y∥≤ 2(1 − δ).

Now if $x,y\in \mathbb X$ (not necessarily in the open unit ball), the conditions become

$$\displaystyle \begin{aligned}\left\|\frac{x+y}{2}\right\|\leq (1-\delta)\cdot \mbox{max}\{\|x\|,\|y\|\}\end{aligned}$$

if

$$\displaystyle \begin{aligned}\|x-y\|\geq \alpha\cdot \mbox{max} \{\|x\|,\|y\|\}.\end{aligned}$$

We observe that Hilbert spaces are examples of uniformly convex Banach spaces.

Definition 1.6

A subset S of a normed space $\mathbb X$ is said to be open if for every x ∈ S, there exists 𝜖 > 0 such that the open ball

$$\displaystyle \begin{aligned}B(x,\epsilon):=\{y\in \mathbb X \;:\;\|x-y\|<\epsilon\}\end{aligned}$$

is included in S. S is said to be closed if its complement in $\mathbb X$ is open.

Proposition 1.7

A subset S of a normed space $\mathbb X$ is closed if and only if every sequence of elements of S which converges in $\mathbb X$ , has its limit in S.

Definition 1.8

The closure of a subset S in a normed space $\mathbb X$, denoted $\overline {S}$, is the intersection of all closed sets containing S.

It is easy to verify the following:

Proposition 1.9

Let S be a subset of a normed space $\mathbb X$ ; then

$$\displaystyle \begin{aligned}\overline{S}=\{x\in\mathbb X\;:\;\exists (x_n)\subset S,\;\displaystyle\lim_{n\to\infty}x_n=x\}.\end{aligned}$$

Definition 1.10

A subset S of a normed space $\mathbb X$ is said to be

• (i) Dense in $\mathbb X$ if $\overline {S}=\mathbb X$;

• (ii) Bounded in $\mathbb X$ if it is either empty or included in a closed ball;

• (iii) Relatively compact in $\mathbb X$ if $\overline {S}$ is compact. Equivalently S is relatively compact if and only if every sequence in S contains a convergent sequence. It is observed that every relatively compact set is bounded.

Definition 1.11

Let $\mathbb X$ be a Banach space over the field $\Phi =\mathbb R$ or $\mathbb C$. The (continuous) dual space of $\mathbb X$ is the normed space of all bounded linear functionals $\varphi :\mathbb X\to \Phi$ which we denote $\mathbb X^*$.

We can rewrite Definition 1.10 (ii) as follows:

Definition 1.12

A subset S of a Banach space $\mathbb X$ is said to be bounded if φ(S) is bounded in Φ for every $\varphi \in \mathbb X^*$.

Proposition 1.13 ([54])

Weakly bounded sets are bounded in any Banach space $\mathbb X$ . In particular every weakly convergent sequence is bounded in $\mathbb X$.

We refer to $(\mathbb X^*)^*=\mathbb X^{**}$ , the bidual of $\mathbb X$. $\mathbb X$ can be considered as embedded in $\mathbb X^{**}$ as follows:

For $x\in \mathbb X$ , let

$$\displaystyle \begin{aligned}J(x):\mathbb X^*\to\Phi(=\mathbb R \;or\; \mathbb C)\end{aligned}$$

be defined by

$$\displaystyle \begin{aligned}J(x)[\varphi]=\varphi (x),\;\varphi \in \mathbb X^*.\end{aligned}$$

Then J(x) is a linear form. It is continuous since

$$\displaystyle \begin{aligned}|J(x)[\varphi]|=|\varphi(x)|\leq \|\varphi\|\|x\||,\;\forall \varphi\in\mathbb X^*.\end{aligned}$$

Hence $J(x)\in \mathbb X^{**}$ for all $x\in \mathbb X$ . The map $J:\mathbb X\to \mathbb X^{**}$ defined this way is also linear and isometric. It is called the canonical embedding of $\mathbb X$ into its bidual $\mathbb X^{**}$.

Definition 1.14

If the canonical embedding $J:\mathbb X\to \mathbb X^{**}$ is surjective, i.e. $\mathbb X=\mathbb X^{**}$, we say that $\mathbb X$ is reflexive.

Proposition 1.15

If $\mathbb X$ is a reflexive Banach space and (x _n) is a bounded sequence, then we can extract a subsequence $(x^{\prime }_n)$ which will converge weakly to an element of $\mathbb X$.

2 L ^p Spaces

Let I be an open interval of $\mathbb R$ and denote by $C_c(I,\mathbb X)$ the Banach space of all continuous functions $I\to \mathbb X$ with compact support.

Definition 1.16

A function $f: I\to \mathbb X$ is said to be measurable if there exists a set S ⊂ I of measure 0 and a sequence $(f_n)\subset C_c(I,\mathbb X)$ such that f _n(t) → f(t) for all t ∈ I ∖ S.

It is clear that if $f: I\to \mathbb X$ is measurable, then $\|f\|: I\to \mathbb R$ is measurable too.

Theorem 1.17

Let $f_n: I\to \mathbb X,\;\;n=1,2,\ldots$ be a sequence of measurable functions and suppose that $f: I\to \mathbb X$ and f _n(t) → f(t), as n →∞, for almost all t ∈ I. Then f is measurable.

Proof

We have f _n → f on I ∖ S, where S is a set of measure 0. Let (f _n.k) be a sequence of functions in $C_c(I,\mathbb X)$ such that f _n.k → f almost everywhere on I as k →∞. By Egorov’s Theorem (cf. [69, p. 16]) applied to the sequence of functions ∥f _n,k − f _n∥, there exists a set S _n ⊂ I of measure less that $\frac {1}{2^n}$ such that f _n,k − f → f _n uniformly on I ∖ S _n, as k →∞.

Now let k(n) be such that $\|f_{n,k(n)}\|<\frac {1}{n}$ on I ∖ S _n and F _n := f _n,k(n). Also let B := S ∪ (∩_m≥1⋃_n>m S _n). Then it is clear that B is a subset of I of measure 0. Take t ∈ I ∖ B. So we get f _n(t) → f(t), as n →∞. On the other hand if n is large enough, t ∈ I ∖ S _n. It follows that $\|F_n-f\|<\frac {1}{n}$, which means that F _n(t) → f(t), as n →∞, and consequently, f is measurable. □

Remark 1.18

It is easy to observe that if $\phi :I\to \mathbb R$ and $f:I\to \mathbb X$ are measurable, then the product $\phi f:I\to \mathbb X$ is measurable too.

Theorem 1.19 (Pettis Theorem)

A function $f:I\to \mathbb X$ is measurable if and only if the following conditions hold:

• (a) f is weakly measurable (i.e. for every x ^∗∈ X ^∗ , the dual space of X, the function $x^*f:I\to \mathbb X$ is measurable).

• (b) There exists a set S ⊂ I of measure 0 such that f(I ∖ S) is separable.

Proof

See [69, p. 131]. □

We also have the following:

Theorem 1.20

If $f:I\to \mathbb X$ is weakly continuous, then it is measurable.

Theorem 1.21 (Bochner’s Theorem)

Assume that $f:I\to \mathbb X$ is measurable. Then f is integrable if and only if ∥f∥ is integrable. Moreover, we have

$$\displaystyle \begin{aligned}\left\|\int_I f\right\|\leq \int_I\|f\|.\end{aligned}$$

Proof

Let $f: I\to \mathbb X$ be integrable. Then there exists a sequence of functions $f_n\in C_c(I,\mathbb X)$, n = 1, 2, … such that ∫_I∥f _n(t) − f(t)∥dt → 0, as n →∞. Using the inequality ∥f∥≤∥f − f _n∥ + ∥f _n∥, for all n, we see that ∥f∥ is integrable.

Conversely assume that ∥f∥ is integrable. Let $F_n \in C_c(I,\mathbb R),\;\;n=1,2,\ldots$ be a sequence of continuous functions such that ∫_I|F _n −∥f∥|→ 0 as n →∞ and |F _n|≤ F almost everywhere for some $F:I\to \mathbb R$ with ∫_I|F| < ∞.

Since f is measurable, there exists $f_n\in C_c(I,\mathbb X),\;\;n=1,2,\ldots$ such that f _n → f almost everywhere.

We now let

$$\displaystyle \begin{aligned}u_n:=\frac{|F_n|}{\|f_n\|+\frac{1}{n}},\;\;n=1,2,\ldots\end{aligned}$$

Then it is obvious that u _n ≤ F, n = 1, 2, … and u _n → f almost everywhere on I. Therefore ∫_I∥u _n − f∥→ 0 as n →∞ and consequently f is integrable.

Using the Lebesgue–Fatou Lemma (cf. [69]), we get

$$\displaystyle \begin{aligned}\left\|\int_If\right\|\leq \displaystyle\lim_{n\to\infty}\left\|\int_Iu_n\right\|\leq \int_I\|f\|.\end{aligned}$$

This completes the proof. □

Theorem 1.22 (Lebesgue’s Dominated Convergence Theorem)

Let $f_n:I\to \mathbb X,\;\;n=1,2,\ldots$ be a sequence of integrable functions and $g:I\to \mathbb R^+$ be an integrable function. Let also $f:I\to \mathbb X$ and assume that:

• (i) for all n = 1, 2, …, ∥f _n∥≤ g, almost everywhere on I.

• (ii) f _n(t) → f(t), as n →∞ for all t ∈ I.

Then f is integrable on I and

$$\displaystyle \begin{aligned}\int_I f=\displaystyle\lim_{n\to\infty}\int_I f_n.\end{aligned}$$

Definition 1.23

Let 1 ≤ p ≤∞. We will denote by $L^p(I,\mathbb X)$ the space of all classes of equivalence (with respect to the equality on I) of measurable functions $f:I\to \mathbb X$ such that ∥f∥^p is integrable. If we equip $L^p(I,\mathbb X)$ with the norm

$$\displaystyle \begin{aligned}\|f\|{}_p:=\left(\int_I\|f(t)\|{}^pdt\right)^{\frac{1}{p}},\;\;1\leq p<\infty\end{aligned}$$

and

$$\displaystyle \begin{aligned}\|f\|{}_{\infty}:=ess \sup_{I}\|f(t)\|,\;\;p=\infty,\end{aligned}$$

then $L^p(I,\mathbb X)$ turns out to be a Banach space.

We shall denote by $L^p_{loc}(I,\mathbb X)$ the space of all (equivalence classes of) measurable functions $f:I\to \mathbb X$ such that the restriction of f to every bounded subinterval of I is in $L^p(I,\mathbb X)$.

3 Linear Operators

Let us consider a normed space $\mathbb X$ and a linear operator $A:\mathbb X\to \mathbb X$. We define the norm of A by

$$\displaystyle \begin{aligned}|||A|||:=\sup_{\|x\|=1}\|Ax\|.\end{aligned}$$

Definition 1.24

A linear operator $A:\mathbb X\to \mathbb X$ is said to be continuous at $x\in \mathbb X$ if for any sequence $(x_n)\subset \mathbb X$ such that x _n → x, we have Ax _n → Ax, that is, ∥Ax _n − Ax∥→ 0 as ∥x _n − x∥→ 0.

If A is continuous at each $x\in Y\subset \mathbb X$, we say that A is continuous on Y .

Proposition 1.25

A linear operator $A:\mathbb X\to \mathbb X$ is continuous (on $\mathbb X$ ) if and only if it is continuous at a point of $\mathbb X$.

Based on the above Proposition, we generally prove continuity of a linear operator by checking its continuity at the zero vector.

Definition 1.26

A linear operator $A:\mathbb X\to \mathbb X$ is said to be bounded if there exists M > 0 such that ∥Ax∥≤ M∥x∥ for all $x\in \mathbb X$.

We observe that a linear operator $A:\mathbb X\to \mathbb X$ is continuous if and only if it is bounded.

Proposition 1.27 (The Uniform Boundedness Principle)

Let $\mathcal {F}$ be a nonempty family of bounded linear operators over a Banach space $\mathbb X$ . If $\sup \limits _{A\in \mathcal {F}}\|Ax\|<\infty$ for each $x\in \mathbb X$ , then $\sup \limits _{A\in \mathcal {F}}|||A|||<\infty$.

Definition 1.28

A linear operator A in a normed space $\mathbb X$ is said to be compact if AU is relatively compact, where U is the closed unit ball

$$\displaystyle \begin{aligned}U:=\{x\in \mathbb X \;:\;\|x\|\leq 1\}.\end{aligned}$$

Proposition 1.29

If $\mathbb X$ is a Banach space, the linear operator $A:\mathbb X\to \mathbb X$ is compact if and only if for every bounded sequence $(x_n)\subset \mathbb X$ , the sequence $(Ax_n)\subset \mathbb X$ has a convergent subsequence; in other words, AS is relatively compact for every bounded subset S of $\mathbb X$.

4 Functions with Values in a Banach Space

We shall consider functions $x:I\to \mathbb X$ where I is an interval of the real number set $\mathbb R$ and $\mathbb X$ a Banach space.

Definition 1.30

A function x(t) is said to be (strongly) continuous at a point t ₀ ∈ I if ∥x(t) − x(t ₀)∥→ 0 as t → t ₀ and strongly continuous on I if it is (strongly) continuous at each point of I. If t ₀ is an end point of I, t → t ₀ (from the right or from the left), accordingly.

x(t) is said to be weakly continuous on I if for any $\varphi \in \mathbb X^*$, the dual space of $\mathbb X$, the numerical function $(\varphi x)(t):I\to \mathbb R$ is continuous. It is obvious that the strong continuity of x implies its weak continuity. The converse is not true in general.

In fact we have

Proposition 1.31

If $x(t):I\to \mathbb X$ is weakly continuous and has a range with a compact closure in $\mathbb X$ , then x(t) is strongly continuous on I.

In this monograph, continuity will always denote strong continuity, unless otherwise explicitly specified.

Proposition 1.32

Let I = [a, b]. Then the set $C(I,\mathbb X)$ of all continuous functions $x(t):I\to \mathbb X$ is a Banach space when equipped with the norm

$$\displaystyle \begin{aligned}\|x\|{}_{C(I,\mathbb X)}:=\sup_{t\in I}\|x(t)\|.\end{aligned}$$

Definition 1.33

A function $x(t):I\to \mathbb X$ is said to be differentiable at an interior point t ₀ of I if there exists some $y\in \mathbb X$ such that $\|\frac {x(t_0+\Delta t)-x(t_0)}{\Delta t}-y\|\to 0$ as Δt → 0 and differentiable on an open subinterval of I if it is differentiable at each point of I. Such $y\in \mathbb X$, when it exists at t ₀ is denoted x′(t ₀) and called the derivative of x(t) at t ₀.

Definition 1.34

If the function $x(t):I\to \mathbb X$ is continuous on I = [a, b], we define its integral on I (in the sense of Riemann) as the following limit:

$$\displaystyle \begin{aligned}\displaystyle\lim_{n\to\infty}\displaystyle\sum_{k=1}^{n}x(t_k)\Delta t_k,\end{aligned}$$

where the diameter of the partition a = t ₀ < t ₁ < … < t _n = b of I tends to zero. When the limit exists we denote it by $\int _{a}^{b}x(t)dt$.

One can easily establish the estimate

$$\displaystyle \begin{aligned}\left\|\int_{a}^{b}x(t)dt\right\|\leq \int_{a}^{b}\|x(t)\|dt.\end{aligned}$$

Improper integrals are defined as in the case of classical calculus. For instance, if the function is continuous on the interval [a, ∞), then we define its integral on [a, ∞) as follows:

$$\displaystyle \begin{aligned}\int_{a}^{\infty}x(t)dt=\displaystyle\lim_{b\to\infty}\int_{a}^{b}x(t)dt\end{aligned}$$

if the limit exists in $\mathbb X$. This integral is said to be absolutely convergent if $\displaystyle \int _{a}^{\infty }\|x(t)\|dt<\infty$.

5 Semigroups of Linear Operators

Definition 1.35

Let $A:\mathbb X\to \mathbb X$ be a linear operator with domain $D(A)\subset \mathbb X$, a Banach space. The family T = (T(t))_t≥0 of bounded linear operators on $\mathbb X$ is said to be a C ₀-semigroup if

• (i) For all $x\in \mathbb X$, the mapping $T(t)x:\mathbb R^+\to \mathbb X$ is continuous.

• (ii) T(t + s) = T(t)T(s) for all $t,s\in \mathbb R^+$ (semigroup property).

• (iii) T(0) = I, the identity operator.

The operator A is called the infinitesimal generator (or generator in short) of the C ₀-semigroup T if

$$\displaystyle \begin{aligned}Ax=\displaystyle\lim_{t\to 0^+}\frac{T(t)x-x}{t}\end{aligned}$$

and

$$\displaystyle \begin{aligned}D(A):=\left\{x\in\mathbb X\;/\; \displaystyle\lim_{t\to 0^+}\frac{T(t)x-x}{t} \;\;exists\right\}.\end{aligned}$$

It is observed that S commutes with T(t) on D(A). We define a C ₀-group in a similar way, by replacing $\mathbb R^+$ by $\mathbb R$.

For a bounded operator A, we have

$$\displaystyle \begin{aligned}T(t):=e^{tA}=\displaystyle\sum_{n=0}^{\infty}\frac{t^nA^n}{n!}.\end{aligned}$$

Theorem 1.36

Let T = (T(t))_t≥0 be a C ₀ -semigroup. Then there exists K ≥ 1 and $\alpha \in \mathbb R$ such that

$$\displaystyle \begin{aligned}\|T(t)\|\leq Ke^{\alpha t},\;\;\forall t\geq 0.\end{aligned}$$

If α < 0, we say that T is exponentially stable.

Proposition 1.37

• (a) The function t →∥T(t)∥ from $\mathbb R^+\to \mathbb R^+$ is measurable and bounded on any compact interval of $\mathbb R^+$.

• (b) The domain D(A) of its generator A is dense in $\mathbb X.$

• (c) The generator A is a closed operator.

For more details, cf. [35] and [69].

6 Topological Vector Spaces

Let E be a vector space over the field Φ ($\Phi =\mathbb R$ or $\mathbb C$). We say that E is a topological vector space, which we denote E = E(τ), if E is equipped with a topology τ which is compatible to the algebraic structure of E.

It is easy to check that for all a ∈ E, the translation f : E → E defined by f(x) = x + a is a homeomorphism. Thus if $\mho$ is a base of neighborhoods of the origin, $\mho +a$ is a base of neighborhoods of a. Consequently the whole topological structure of E will be determined by a base of neighborhoods of the origin.

In this book, we will mainly use neighborhoods of the origin, which we sometimes call neighborhoods in short.

Another interesting fact is that for every λ ∈ Φ, λ ≠ 0, the mapping f : E → E defined by f(x) = λx is a homeomorphism, so that λU will be a neighborhood (of the origin) if U is a neighborhood (of the origin), λ ≠ 0.

Let us also recall the following:

Proposition 1.38

If $\mho$ is a base of neighborhoods, then for each $U\in \mho$ , we have:

• (i) U is absorbing, that is for each x ∈ U, there exists λ > 0 such that x ∈ αU for all α with |α|≥ λ;

• (ii) There exists $W\in \mho$ such that W + W ⊂ U;

• (iii) There exists a balanced neighborhood V  such that V ⊂ U (A balanced or symmetric set is a set V  such that αV = V  if |α| = 1).

A consequence of the above proposition is that every topological space E possesses a base of balanced neighborhood.

We will call a locally convex topological vector space (or shortly a locally convex space), every topological vector space which has a base of convex neighborhoods. It follows that in a locally convex space, any open set contains a convex, balanced, and absorbing open set.

A locally convex space whose topology is induced by an invariant complete metric is called a Fréchet space.

Proposition 1.39

Let E be a vector space over the field Φ ( $\Phi =\mathbb R$ or $\mathbb C$ ). A function $p:E\to \mathbb R^+$ is called a seminorm if

• (i) p(x) ≥ 0 for every x ∈ E;

• (ii) p(λx) = |λ|p(x), for every x ∈ E and λ ∈ Φ;

• (iii) p(x + y) ≤ p(x) + p(y), for every x, y ∈ E.

It is noted that if p is a seminorm on E, then the sets {x  :  p(x) < λ} and {x ∕ p(x) ≤ λ}, where λ > 0, are absorbing. They are also absolutely convex. We recall that a set B ⊂ E is said to be absolutely convex if for every x, y ∈ E and λ, μ ∈ Φ, with |λ| + |μ|≤ 1, we have λx + μy ∈ B.

Theorem 1.40

For every set Q of seminorms on a vector space E, there exists a coarsest topology on E compatible with its algebraic structure and in which each seminorm in Q is continuous. Under this topology, E is a locally convex space and a base of neighborhoods is formed by the closed sets

$$\displaystyle \begin{aligned}\{x\in E\;:\;\sup_{1\leq i\leq n}p_i(x)\leq \epsilon\},\end{aligned}$$

where 𝜖 > 0 and p _i ∈ Q, i = 1, 2, …n.

Also E will be separated if and only if for each x ∈ E, x ≠ 0, there exists a seminorm p ∈ Q such that p(x) > 0.

An important fact that will be used is the following consequence of the Hahn–Banach Extension Theorem:

Proposition 1.41 ([69, page 107])

For each non-zero a in a locally convex space E, there exists a linear functional φ ∈ E ^∗ , the dual space of E, such that φ(a) ≠ 0.

A subset S of a locally convex space is called totally bounded if, for every neighborhood U, there are a _i ∈ S, i = 1, 2, …n, such that

$$\displaystyle \begin{aligned}S\subset \cup_{i=1}^{n}(a_i+U).\end{aligned}$$

It is clear that every totally bounded set is bounded. Also, the closure of a totally bounded set is totally bounded.

We observe [69, page 13] that in a complete metric space, total boundedness and relatively compactness are equivalent notions.

Now for functions of the real variable with values in a locally convex space E, we define continuity, differentiability, and integration as in [54, 56, 69].

We finally revisit Proposition 1.27 in the context of locally convex spaces as follows (cf. [45, page 199]):

Proposition 1.42 (Uniform Boundedness Principle)

Let φ = {A _α  :  α ∈ Γ} where each A _α : E → F is a bounded linear operator and E, F are Fréchet spaces. Suppose that {A _α x  :  α ∈ Γ} is bounded for each x ∈ E. Then φ is uniformly bounded.

Notes

Details on this topic can be found in [66].

7 The Exponential of a Bounded Linear Operator

Let E be a complete, Hausdorff locally convex space.

Definition 1.43

A family of continuous linear operators B _α : E → E, α ∈ Γ is said to be equicontinuous if for any seminorm p, there exists a seminorm q such that

$$\displaystyle \begin{aligned}p(B_{\alpha}x)\leq q(x),\;\;\mbox{for any}\; x\in E, \;\; \mbox{any}\;\alpha\in \Gamma.\end{aligned}$$

Theorem 1.44

Let A : E → E be a continuous linear operator such that the family {A ^k : k = 1, 2, …} is equicontinuous. Then for each x ∈ E, t ≥ 0, the series

$$\displaystyle \begin{aligned} \displaystyle\sum_{k=0}^{\infty}\frac{t^k}{k!}A^kx\end{aligned} $$

(where A ⁰ = I, the identity operator on E) is convergent.

Proof

Let p be a seminorm on E. By equicontinuity of {A ^k : k = 1, 2, …}, there exists a seminorm q on E such that

$$\displaystyle \begin{aligned} p(A^kx)\leq q(x),\;\;\mbox{for all}~~ k,\;\mbox{and}\;x\in E.\end{aligned} $$

Therefore we have

$$\displaystyle \begin{aligned}p\left(\displaystyle\sum_{k=n}^{m}\frac{t^k}{k!}A^kx\right)\leq \displaystyle\sum_{k=n}^{n}\frac{t^k}{k!}p(A^kx)\leq q(x)\displaystyle\sum_{k=n}^m\frac{t^k}{k!},\end{aligned}$$

which proves that the sequence $\displaystyle \sum \limits _{k=0}^{n}\frac {t^k}{k!}A^kx$ is a Cauchy sequence in E. It is then convergent and we denote the limit by

$$\displaystyle \begin{aligned}e^{tA}x:=\displaystyle\sum_{k=0}^{\infty}\frac{t^k}{k!}A^kx.\end{aligned}$$

□

Theorem 1.45

The mapping x → e ^tA x, t ≥ 0, defines a continuous linear operator E → E.

Proof

Consider the linear operators $A_n:=\displaystyle \sum \limits _{k=0}^{n}\frac {t^k}{k!}A^k,\;n=0,1,2,\ldots$ The family {A _n :  n = 0, 1, 2, …} is equicontinuous on any compact interval of $\mathbb R^+$.

Indeed, by equicontinuity of {A ^k :  k = 1, 2, …}, if p is a given seminorm, then there exists a seminorm q such that

$$\displaystyle \begin{aligned}p(A_nx)\leq \displaystyle\sum_{k=0}^{n}\frac{t^k}{k!}p(A^kx)\leq q(x)\displaystyle\sum_{k=0}^{n}\frac{t^k}{k!}\leq q(x)e^t\end{aligned}$$

for every n = 0, 1, 2, …. It follows that

$$\displaystyle \begin{aligned}p(e^{tA}x)\leq q(x)e^t,\end{aligned}$$

for every t ≥ 0 and x ∈ E. This completes the proof. □

Theorem 1.46

Let A and B be two continuous linear operators E → E such that {A ⁿ;n = 1, 2, …} and {B ⁿ;n = 1, 2, …} are equicontinuous. Assume that A and B commute, that is AB = BA; then

$$\displaystyle \begin{aligned}e^{tA}\cdot e^{tB}=e^{t(A+B)},\;\;t\geq 0.\end{aligned}$$

Proof

The proof is similar to the numerical case, that is for any real numbers a and b, we have

$$\displaystyle \begin{aligned}\displaystyle\sum_{n=0}^{\infty}\frac{(ta)^n}{n!}.\displaystyle\sum_{n=0}^{\infty}\frac{(tb)^n}{n!}=\displaystyle\sum_{n=0}^{\infty}\frac{(t(a+b))^n}{n!}.\end{aligned}$$

Indeed for any integer k and x ∈ E, we have

$$\displaystyle \begin{aligned}(A+B)^k x=\displaystyle\sum_{j=0}^{k}\binom{k}{j}A^j B^{k-j}x=\displaystyle\sum_{j=0}^{k}\binom{k}{j}B^{k-j}A^jx,\end{aligned}$$

where $\binom {k}{j}=\frac {k!}{j!(k-j)!}$.

In the last equality, we used the fact that AB = BA. Let p be a given seminorm on E. Then there exists a seminorm q such that

$$\displaystyle \begin{aligned}\begin{array}{rlll} \displaystyle p((A+B)^k x)&\leq& \displaystyle\sum_{j=0}^{k}\binom{k}{j}p(B^{k-j}A^jx)\\ &\leq&\displaystyle\displaystyle\sum_{j=0}^{k}\binom{k}{j}q(A^jx)\\ &\leq&\displaystyle 2^k \sup_{j\geq 0}q(A^jx) \end{array} \end{aligned}$$

since $\displaystyle \sum \limits _{j=0}^{k}\binom {k}{j}=2^k$.

This last inequality shows that the family $\left \{\frac {(A+B)^k}{2^k}:\;k=1,2,\ldots \right \}$ is equicontinuous, so by Theorem (1.44), we can define e ^t(A+B) by

$$\displaystyle \begin{aligned}e^{t(A+B)}x:=\displaystyle\sum_{n=0}^{\infty}\frac{(t(A+B))^n x}{n!}.\end{aligned}$$

Now using the Cauchy product formula, we obtain

$$\displaystyle \begin{aligned}e^{tA}\cdot e^{tB}=\displaystyle\sum_{n=0}^{\infty}\frac{(tA)^n}{n!}\cdot \displaystyle\sum_{n=0}^{\infty}\frac{(tB)^n}{n!}=\displaystyle\sum_{n=0}^{\infty}C_n,\end{aligned}$$

where

$$\displaystyle \begin{aligned} \begin{array}{rlll} C_n&=&\displaystyle\sum_{k=0}^{n}\frac{(tA)^k}{k!}\cdot \frac{(tB)^{n-k}}{(n-k)!}\\ &=&\displaystyle\sum_{k=0}^{n}\frac{t^n}{k!(n-k)!}A^kB^{n-k}\\ &=&\displaystyle\sum_{k=0}^{n}\binom{n}{k}\frac{t^n}{n!}A^kB^{n-k}\\ &=&\displaystyle\sum_{k=0}^{n}\frac{(t(A+B))^n}{n!}. \end{array} \end{aligned}$$

That means $\displaystyle \sum \limits _{n=0}^{\infty }C_n=e^{t(A+B)}.$ The proof is complete. □

Theorem 1.47

Suppose that A is a continuous linear operator E → E such that {A ⁿ;n = 1, 2, …} is equicontinuous. Then for every x ∈ E, we have

$$\displaystyle \begin{aligned}\displaystyle\lim_{h\to 0^+}\left(\frac{e^{hA}-I}{h}\right)x=Ax.\end{aligned}$$

Proof

Let p be a seminorm. Then there exists a seminorm q such that

$$\displaystyle \begin{aligned} \begin{array}{rllll} p((\frac{e^{hA}-I}{h})x-Ax)&=&p(\frac{1}{h}(\displaystyle\sum_{n=0}^{\infty}\frac{h^n}{n!}A^n-I)x-Ax)\\ &\leq& p(\frac{1}{h}(\displaystyle\sum_{n=2}^{\infty}\frac{h^n}{n!}A^n x)\\ &\leq& \displaystyle\sum_{n=2}^{\infty}\frac{h^{n-1}}{n!}p(A^n x)\\ &\leq& q(x)\displaystyle\sum_{n=2}^{\infty}\frac{h^{n-1}}{n!}\\ &=&q(x)\left(\frac{e^h-1}{h}-1\right). \end{array} \end{aligned}$$

And since $\displaystyle \lim _{h\to ^+}\frac {e^h-1}{h}=1$, we get the result. □

From the above, we can deduce that

$$\displaystyle \begin{aligned}\frac{d}{dt}e^{tA}x=e^{tA}\cdot Ax=Ae^{tA}x,\end{aligned}$$

Using the semigroup property above, we get also

$$\displaystyle \begin{aligned}e^{(t+s)A}=e^{tA}\cdot e^{sA}.\end{aligned}$$

We can use the same technique to prove similar results if t ≤ 0 and establish e ^tA for $t\in \mathbb R$.

We are now ready to prove the following:

Theorem 1.48

The function $e^{tA}x_0:\mathbb R\to E$ is the unique solution of the differential equation

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

satisfying x(0) = x ₀.

Proof

Suppose there were another solution y(t) with y(0) = x ₀. Consider the function v(s) = e ^(t−s)A y(s), with t fixed in $\mathbb R$; then we have

$$\displaystyle \begin{aligned} \begin{array}{rlll} v'(s)&=&-Ae^{(t-s)A}y(s)+e^{(t-s)A}y'(s)\\ &=&-Ae^{(t-s)A}y(s)+e^{(t-s)A}Ay(s)\\ &=&0, \end{array} \end{aligned}$$

for every $s\in \mathbb R$. Therefore, v′(s) = 0 on $\mathbb R$, so that

$$\displaystyle \begin{aligned}v(t)=v(0),\;\;t\in\mathbb R\end{aligned}$$

or

$$\displaystyle \begin{aligned}y(t)=e^{tA}y(0)=e^{tA}x_0,\;\;t\in \mathbb R.\end{aligned}$$

Since t is arbitrary, this completes the proof. □

Let us recall the following fixed point theorem from [15]:

Theorem 1.49

Let D be a closed and convex subset of a Hausdorff locally convex space such that 0 ∈ D, and let G be a continuous mapping of D into itself. If the implication

$$\displaystyle \begin{aligned}(V=conv G(V),\mathit{\mbox{or}}\; V=G(V)\cup \{0\})\Rightarrow V \mathit{\mbox{is relatively compact}}\end{aligned}$$

holds for every subset V  of D, then G has a fixed point.

8 Non-locally Convex Spaces

It is well known that an F-space (X, +, ⋅, ||⋅||) is a linear space (over the field $\Phi =\mathbb {R}$ or $K=\mathbb {C}$) such that ||x + y||≤||x|| + ||y|| for all x, y ∈ X, ||x|| = 0 if and only if x = 0, ||λx||≤||x||, for all scalars λ with |λ|≤ 1, x ∈ X, and with respect to the metric D(x, y) = ||x − y||, X is a complete metric space (see e.g. [25, p. 52], or [37]). Obviously D is invariant to translations.

In addition, if there exists 0 < p < 1 with ||λx|| = |λ|^p||x||, for all λ ∈ K, x ∈ X, then ||⋅|| will be called a p-norm and X will be called p-Fréchet space. (This is only a slight abuse of terminology. Note that in e.g. [10] these spaces are called p-Banach spaces). In this case, it is immediate that D(λx, λy) = |λ|^p D(x, y), for all x, y ∈ X and λ ∈ Φ.

It is known that the F-spaces are not necessarily locally convex spaces. Three classical examples of p-Fréchet spaces, non-locally convex, are the Hardy space H ^p with 0 < p < 1 that consists in the class of all analytic functions $f:\mathbb {D}\to \mathbb {C}$, $\mathbb {D}=\{z\in \mathbb {C} ;|z|<1\}$ with the property

$$\displaystyle \begin{aligned}||f||=\frac{1}{2\pi}\sup\left\{\int_{0}^{2\pi}|f(re^{it})|{}^{p}dt ;r\in [0,1)\right\}<+\infty,\end{aligned}$$

the sequences space

$$\displaystyle \begin{aligned}l^{p}=\left\{x=(x_n)_{n} ; ||x||=\displaystyle\sum_{n=1}^{\infty}|x_{n}|{}^{p}<\infty\right\}\end{aligned}$$

for 0 < p < 1, and the L ^p[0, 1] space, 0 < p < 1, given by

$$\displaystyle \begin{aligned}L^{p}=L^{p}[0,1]=\left\{f:[0,1]\to \mathbb{R} ;||f||=\int_{0}^{1}|f(t)|{}^{p}dt<\infty\right\}.\end{aligned}$$

More generally, we may consider L ^p( Ω,  Σ, μ), 0 < p < 1, based on a general measure space ( Ω,  Σ, μ), with the p-norm given by $||f||=\int _{\Omega }^{ }|f|{ }^{p}d\mu$.

Some important characteristics of the F-spaces are given by the following remarks:

Remark 1.50

• (1) Three of the basic results in Functional Analysis hold in F-spaces too : the Principle of Uniform Boundedness (see e.g. [25, p. 52]), the Open Mapping Theorem, and the Closed Graph Theorem (see e.g. [37, p. 9–10]).

  But on the other hand, the Hahn–Banach Theorem fails in non-locally convex F-spaces. More exactly, if in an F-space the Hahn–Banach theorem holds, then that space is necessarily locally convex space (see e.g. [37, Chapter 4]).

• (2) If (X, +, ⋅, ||⋅||) is a p-Fréchet space over the field Φ, 0 < p < 1, then its dual X ^∗ is defined as the class of all linear functionals h : X → Φ which satisfy |h(x)|≤|||h|||⋅||x||^1∕p, for all x ∈ X, where $|||h|||=\sup \{|h(x)| ; ||x||\le 1\}$ (see e.g. [10, pp. 4–5]). Note that |||⋅||| in fact is a norm on X ^∗.

For 0 < p < 1, while (L ^p)^∗ = 0, we have that (l ^p)^∗ is isometric to l ^∞—the Banach space of all bounded sequences (see e.g. [37, p. 20–21]), therefore (l ^p)^∗ becomes a Banach space. Also, if ϕ ∈ (H ^p)^∗ then there exists a unique g, analytic on $\mathbb {D}$ and continuous on the closure of $\mathbb {D}$, such that

$$\displaystyle \begin{aligned}\phi(f)=\frac{1}{2\pi}\displaystyle\lim_{r\to 1}\int_{0}^{2\pi}f(re^{it})g(e^{-it})dt,\end{aligned}$$

for all f ∈ H ^p (see e.g. [26, p. 115, Theorem 7.5]). Moreover, (H ^p)^∗ becomes a Banach space with respect to the usual norm $|||\phi |||=\sup \left \{|\phi (f)| ; ||f||\le 1\right \}$ (see the same paper [4]).

In both cases of l ^p and H ^p, 0 < p < 1, their dual spaces separate the points of corresponding spaces.

• (3) The spaces l ^p and H ^p, 0 < p < 1, have Schauder bases (see e.g. [37, p. 20], for l ^p and [37, 64] for H ^p). It is also worth to note that according to e.g. [28], every linear isometry T of H ^p onto itself has the form

  $$\displaystyle \begin{aligned} T(f)(z)=\alpha[\phi^{\prime}(z)]^{1/p}f(\phi(z)), \end{aligned} $$

  (8.1)

  where α is some complex number of modulus one and ϕ is some conformal mapping of the unit disc onto itself.