Almost periodic functions with values in p-frechet spaces, 0 < p < 1.

Abstract

In this paper we develop a theory of almost periodic functions with values in the non-locally convex spaces called p-Frechet spaces, 0 < p < 1, including the [l.sup.p], [L.sup.p] spaces and the Hardy space [H.sup.p]. Although the p-norm, 0 < p < 1, has not all the properties of an usual norm, the majority of main properties of almost periodic functions with values in Banach spaces are extended to this case. Some applications to differential equations and to dynamical systems in p-Frechet spaces, 0 < p < 1, are given.

1991 Mathematics Subject Classification: 43A60, 34C35, 34G10.

Key words and phrases: p-Frechet space, almost periodic functions, functions with values in p-Frechet spaces, 0 < p < 1.

1. Introduction

Harald Bohr's interest in which functions could be represented by a Dirichlet series, i.e. of the form [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] where [a.sub.n], z [member of] C and [([[lambda].sub.n]).sub.n[member of]N] is a monotone increasing sequence of real numbers (series which play an important role in complex analysis and analytic number theory), led him to develop a theory of almost periodic real (and complex) functions, between the years 1923 and 1926. One of the main applications is to the theory of ordinary and partial differential equations. For example, in oscillation theory, frequently appear differential equations of the form y"(t) + Ay'(t) + By(t) = f(t), where A,B [member of] R and f(t) depends on sine and cosine functions whose periods are not commensurable (e.g. f(t) = cos t/2 + cos [square root 2t], i.e. f is almost periodic), fact which implies the almost periodicity of y(t). The idea of propagation of almost periodicity from the input data of a (differential) equation to its solutions, which represents the main area of applications for the concept of almost periodic functions, generated and continues to generate a huge specialized literature, see e.g. the books [1], [11], [15], [16], [21], to mention only a few. For example, the partial differential equation [u.sub.t](x, t) = D[u.sub.xx](x, t) + f(u(x, t), t); where D > 0, x [member of] R, t [member of] R and f(u, t) is almost periodic in t, uniformly with respect to u, may model a variety of physical and biological phenomena, such as the signal propagation along bistable transmission lines, the propagation of nerve pulse, or the pattern formation in reacting-diffusing media (equations of Fisher or of Kolmogorov type). The time almost periodic dependence reflects the effects of certain "seasonal" variations which are roughly but not exactly periodic. In the proof of existence, uniqueness and stability for almost periodic solutions of differential (partial differential) equations (or systems) basic properties of almost periodic functions are used, for instance the boundedness, the relatively compactness of the range, the property of normality and the property of closure with respect to uniform convergence. (see e.g. [6], [20]).

Other applications of almost periodic functions are, for example, in Statistics, Number Theory, the Theory of Spectrum of Bounded Functions, the Theory of Semigroups of Bounded Linear Operators, Dynamical Systems and Ergodic Theory.

The theory of almost periodic functions was strongly extended to abstract spaces, see for example the monographs [6], [16], [17] for Banach space valued functions and [5], [16], [18] for complete locally convex (Frechet) space valued functions. Also, in the paper [4] (see also Chapter 3 in the book [17]), the theory has been extended to the case of fuzzy-number-valued functions.

The purpose of this paper is to extend in Section 3 the main properties of almost periodic functions with values in Banach spaces, to the class of almost periodic functions with values in other important abstract spaces in Functional Analysis, namely the p-Frechet spaces, 0 < p < 1, which are non-locally convex spaces.

In Section 4 we present some applications to differential equations and dynamical systems in p-Frechet spaces, with 0 < p < 1.

2. Preliminaries

It is well known that an F-space (X, +, x, [parallel] x [parallel]) is a linear space (over the field K = R or K = C) such that [parallel] x + y [parallel] [less than or equal to] [parallel] x [parallel] + [parallel]y[parallel] for all x, y [member of] X, [parallel] x [parallel] = 0 if and only if x = 0, [parallel][lambda] x [parallel] [less than or equal to] [parallel] x [parallel], for all scalars [lambda] with |[lambda]| [less than or equal to] 1, x [member of] X, and with respect to the metric D(x, y) = [parallel]x - y[parallel], X is a complete metric space (see e.g. [9, p. 52] or [14]). Obviously D is invariant under translations.

In addition, if there exists 0 < p < 1 with [parallel][lambda] x [parallel] = [parallel][lambda][|.sup.p][parallel]x[parallel], for all [lambda] [member of] K, x 2 X, then [parallel] x [parallel] will be called a p-norm and X will be called p-Frechet space. (This is only a slight abuse of terminology. Note that in e.g. [2] these spaces are called p-Banach spaces). In this case, it is immediate that D([lambda]x, [lambda]y) = |[lambda][|.sup.p]D(x, y), for all x, y [member of] X and [lambda] [member of] K.

It is known that F-spaces are not necessarily locally convex spaces. Three classical examples of p-Frechet spaces, non-locally convex, are the Hardy space [H.sup.p] with 0 < p < 1 that consists in the class of all analytic functions f : D [right arrow] C, D = {z [member of] C; |z| < 1} with the property

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.],

the sequences space

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

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

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

More generally, we may consider [L.sup.p]([ohm], [summation] [micro]), 0 < p < 1, based on a general measure space ([ohm], [summation], [micro]), with the p-norm given by [parallel]f[parallel] = [[integral].sub.[ohm]] |f[|.sup.p]d[micro].

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

Remarks. 1) Three fundamental results in Functional Analysis hold for F-spaces too : the Principle of Uniform Boundedness (see e.g. [9, p. 52]), the Open Mapping Theorem and the Closed Graph Theorem (see e.g. [14, 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. [14, Chapter 4]).

2) If (X, +, x, [parallel] x [parallel]) is a p-Frechet space over the field K, 0 < p < 1, then its dual [X.sup.*] is defined as the class of all linear functionals h : X [right arrow] K which satisfy |h(x)| [less than or equal to] [??]h[??] x [parallel] x [[parallel].sup.1/p], for all x [member of] X, where [??]h[??] = sup{|h(x)|; [parallel]x[parallel] [less than or equal to] 1} (see e.g. [2], pp. 4-5). Note that [??] x [??] in fact is a norm on [X.sup.*].

For 0 < p < 1, while ([[L.sup.p]).sup.*] = 0, we have that [([l.sup.p]).sup.*] is isometric to [l.sup.[infinity]]--the Banach space of all bounded sequences (see e.g. [14, p. 20-21]); therefore [([l.sup.p]).sup.*] becomes a Banach space. Also, if [phi] [member of] [([H.sup.p]).sup.*] then there exists a unique g, analytic on D and continuous on the closure of D, such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

for all f [member of] [H.sup.p] (see e.g. [10, p. 115, Theorem 7.5]). Moreover, [([H.sup.p]).sup.*] becomes a Banach space with respect to the usual norm [??][pi][??] = sup{|[phi](f)|; [parallel]f[parallel] [less than or equal to] 1} (see the same paper [10]).

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

3) The spaces [l.sup.p] and [H.sup.p], 0 < p < 1, have Schauder bases (see e.g. [14, p. 20] for [l.sup.p] and [14], [19] for [H.sup.p]). It is also worth to note that according to e.g. [12], every linear isometry T of [H.sup.p] onto itself has the form

T(f)(z) = [alpha][[phi]'[(z)].sup.1/p]f([phi](z)), (2.1)

where [alpha] is some complex number of modulus one and [phi] is some conformal mapping of the unit disc onto itself.

3. Basic Definitions and Properties

Everywhere in this section, (X, +, x, [parallel] x [parallel]) will be a p-Frechet space with 0 < p < 1 (over the field K = R or K = C). Also, denote D(x, y) = [parallel] x - y[parallel].

Similarly to [6], p. 137, a trigonometric polynomial of degree [less than or equal to] n with coefficients (and values) in the p-Frechet space X, is defined as a finite sum of the form [T.sub.n](t) = [n.[summation]k=1] [c.sub.k.sup.ei[lambda]nt], where [c.sub.k] [member of] X, k = 1, ... n.

Also, recall that f : R [right arrow] X is said to be continuous at [x.sub.0] [member of] R if: [for all][epsilon] > 0, [there exists][theta] > 0 such that [parallel]f (x) - f ([x.sub.0]) [parallel] < [epsilon], whenever x [member of] R, |x - [x.sub.0]| < [delta]. From the triangle inequality satisfied by the p-norm [parallel] x [parallel], it easily follows the inequality [??]x[parallel] - [parallel]y[??] [less than or equal to] [parallel]x - y[parallel], which immediately implies that if f is continuous at [x.sub.0] as above, then the real valued function [parallel]f(t)[parallel] also is continuous at [x.sub.0].

In this section, starting from a Bohr-kind definition for the almost periodicity, we develop a theory of almost periodic functions with values in a p-Frechet space, 0 < p < 1, similar to that for functions with values in Banach space.

The following three points in Definition 3.1 represent the basic concepts in the theory of almost periodic functions with values in the p-Frechet space X.

Definition 3.1. Let f : R [right arrow] X be continuous on R.

(i) We say that f is B-almost periodic if : [for all][epsilon] > 0, [there exists]l ([epsilon]) > 0 such that any interval of length l ([epsilon]) of the real line contains at least one point [xi] with

[parallel]f (t + [xi]) - f (t) [parallel] < [epsilon], [for all]t [member of] R.

(ii) We say that f is normal if for any sequence [F.sub.n] : R [right arrow] X of the form [F.sub.n] (x) = f (x + [h.sub.n]), n [member of] N, where [([h.sub.n]).sub.n] is a sequence of real numbers, one can extract a subsequence of [([F.sub.n]).sub.n], converging uniformly on R (i.e. [for all] [([h.sub.n]).sub.n], [there exists] ([h.sub.n.sub.k]), [there exists]F : R [right arrow] X (which may depend on [([h.sub.n]).sub.n]), such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] [parallel][F.sub.n.sub.k] (x) - F (x) [parallel] = 0, uniformly with respect to x [member of] R)

(iii) We say that f has the approximation property, if [for all][epsilon] > 0, there exists some trigonometric polynomial T with coefficients in X, such that [parallel]f (x) - T (x) [parallel] < [epsilon], [for all]x [member of] R.

Let us denote AP (X) = {f : R [right arrow] X; f is B-almost periodic}. The next two theorems show that AP (X) is a subclass of uniformly continuous bounded functions.

Remark. 1) A set E [subset] R is called relatively dense (in R), if there exists a number l > 0 such that every interval (a, a + l) contains at least one point of E. By using this concept, we can reformulate Definition 3.1, (i), as follows : f : R [right arrow] X is called B-almost periodic if for every [espilon] > 0, there exists a relatively dense set {[tau]}[epsilon], such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Also, each [tau] [member of] {[tau]}[epsilon] is called [epsilon]-almost period of f.

Theorem 3.2. If f : R [right arrow] X is (continuous) B-almost periodic then f is bounded, that is there exists M > 0 such that [parallel]f (t) [parallel] [less than or equal to] M, [for all]t [member of] R.

Proof. Since in the proof for functions with values in Banach spaces in [6], Theorem 6.1, p. 138, is used only the triangular inequality of the norm and the fact that the real valued function [parallel] f(t) [parallel] is continuous on R, the present proof is then identical.

Theorem 3.3. If f : R [right arrow] X is B-almost periodic then f is uniformly continuous on R.

Proof. Firstly, as in the case of real valued functions, it easily can be proved that a continuous function defined on a compact interval with values in a p-Frechet space is uniformly continuous on that interval. Then, since in the proof for Banach space valued functions, only the triangular inequality of the norm is used, it follows that the present proof is similar to the proof of Theorem 6.2, pp. 138-139 in [6].

Theorem 3.4 If f : R [right arrow] X is B-almost periodic then [lambda] x f, [lambda] [member of] K, [F.sub.h] (x) = f (x + h), x [member of] R and G(x) = [parallel]f (x) [parallel], x [member of] R are B-almost periodic.

Proof. Similar to that for Banach space valued functions in [6], proof of Theorem 6.3, p. 139.

The next result shows that AP (X) is closed with respect to uniform convergence.

Theorem 3.5. If [f.sub.n] : R [right arrow] X, n [member of] N are B-almost periodic and [f.sub.n] [right arrow] f for n [right arrow] 1, uniformly on R (i.e. [for all][epsilon] > 0, there exists [n.sub.0] [member of] N, such that [parallel][f.sub.n] (x) - f (x) [parallel] < [epsilon], [for all]n [greater than or equal to] [n.sub.0], [for all]x [member of] R), then f is B-almost periodic.

Proof. Since in the case of Banach space valued functions, the proof use only the triangular inequality of the norm, we get that the proof of our Theorem 3.5 is similar to the proof of Theorem 6.4 in [6].

Theorem 3.6. The set of values of f : R [right arrow] X supposed to be B-almost periodic, is relatively compact in the complete metric space (X,D), with D(x, y) = [parallel]x - y[parallel].

Proof. Because a well-known result states that in complete metric spaces, relatively compact sets coincide with pre-compact sets, it is sufficient to show that for any [epsilon] > 0, the set of values of the function can be embedded in a finite number of spheres of radius [epsilon]. From here the proof is similar to the proof of Theorem 6.5, p. 140 in [6], since only the triangular inequality is used.

Remark. Let f : R [right arrow] X be B-almost periodic and let us consider the sequence of values (f [([t.sub.n])).sub.n[member of]N]. Denote A = {f ([t.sub.n]); n [member of] N} and take the closure [bar.A] [subset] [bar.f (R)] [subset] X, it follows that [bar.A] is compact, so [bar.A] is sequentially compact too (since (X, D) is a metric space), which by A [subset] [bar.A] implies that the sequence (f[([t.sub.n])).sub.n] has convergent subsequence in X.

The following result shows that the concepts in Definition 3.1, (i) and (ii), in fact are equivalent.

Theorem 3.7. A function f : R [right arrow] X is B-almost periodic if and only if it is normal.

Proof. It is similar to the proof in the case of Banach space valued functions, of Theorem 6.6, pp. 140-141 in [6], since only the triangular properties of the norm is used in that proof.

We also have the following.

Theorem 3.8. (i) The sum of two B-almost periodic functions with values in the p-Frechet space X, with 0 < p < 1, is B-almost periodic;

(ii) If f : R [right arrow] X is B-almost periodic and g : X [right arrow] Y is continuous on [bar.f(R)], where Y is a q-Frechet space, 0 < q < 1, (with q not necessarily equal to p), then h : R [right arrow] Y , defined by h(t) = g[f(t)] is B-almost periodic.

(iii) If [f.sub.1], [f.sub.2] : R [right arrow] X are B-almost periodic, then [for all][epsilon] > 0, there exist common [espilon]-translation numbers for [f.sub.1] and [f.sub.2].

Proof. (i) Similar to the proof of Theorem 6.7, p. 142 in [6] (it is in fact based on Theorem 3.7).

(ii) It is similar to the proof in the case of Banach space valued functions, see e.g. [1], Theorem VII, pp. 6-7.

(iii) It is similar to the proof of Corollary 3.1.10 [16].

Theorem 3.9. If f : R [right arrow] X has the approximation property in Definition 3.1, (iii), then f is B-almost periodic.

Proof. A function f : R [right arrow] X is called [s.sub.0]-periodic if f (t + [s.sub.0]) = f (t), [for all]t [member of] R. Obviously a [s.sub.0]-periodic function is B-almost periodic.

It follows by Theorems 3.4 and 3.8 that any trigonometric polynomial with values in X, is B-almost periodic, which combined with Theorem 3.5 completes the proof.

Remark. Let us denote AP (X) = {f : R [right arrow] X; f is B-almost periodic} and for f [member of] AP (X), let us define [parallel]k[[parallel].sub.b] = sup {[parallel]f (t) [parallel]; t [member of] R}. By Theorem 3.2 we get [parallel]f[[parallel].sub.b] < + [infinity]. It easily follows that [parallel] x [[parallel].sub.b] also is a p-norm on the space

[C.sub.b] (R, X) = {f : R [right arrow] X; is continuous and bounded on R}.

In addition, since (X,D) is a complete metric space, by standard reasonings it follows that [C.sub.b](R,X) turns out to be a complete metric space with respect to the metric [D.sub.b](f; g) = [parallel]f - g[[parallel].sub.b], that is ([C.sub.b](R,X), [parallel] x [[parallel].sub.b]) is a p-Frechet space.

So Theorems 3.2 and 3.5 show that AP (X) is a closed subset of [C.sub.b] (R,X), i.e. (AP (X), [D.sub.b]) is a complete metric space and therefore (AP(X), [parallel]x [[parallel].sub.b]) turns out to be a p-Frechet space. By similar reasonings with those in the proofs of Theorems 6.9 and 6.10 in [6], pp. 142-143], (where we define on [X.sup.m] the p-norm [parallel]x[[parallel].sub.m] = [m.summation over (k=1)] [parallel][x.sub.k][parallel] and the metric [D.sub.m] (x, y) = [m.summation over (i=1) D([x.sub.i], [y.sub.i]), [for all]x = ([x.sub.1]; ..., [x.sub.m]), y = ([y.sub.1]; ..., [y.sub.m]) [member of] [X.sup.m]), we can state the following compactness criterion.

Theorem 3.10. The necessary and sufficient condition that a family A [subset] AP(X) be relatively compact is that the following properties hold true:

(i) A is equi-continuous ;

(ii) A is equi-almost periodic ;

(iii) for any t [member of] R, the set of values of functions from A be relatively compact in X.

In what follows we consider the concept of Bochner's transform. Thus, the Bochner's transform of f in [C.sub.b] (R, X) is denoted by B(f) := [??] and is defined by [??] : R [right arrow] [C.sub.b](R,X), [??](s) [member of] [C.sub.b](R,X), [??](s)(t) = f(t + s), for all t [member of] R.

The properties of Bochner's transform can be summarized as follows.

Theorem 3.11. (i) [parallel][??](s) [[parallel].sub.b] = [parallel]f(x + s) [[parallel].sub.b] = [parallel][??](0) [[parallel].sub.b], for all s [member of] R; (ii) [parallel] [??](s + [tau]) - [??](s)[[parallel].sub.b] = sup{[parallel]f(t + [tau]) - f(t) [parallel]; t [member of] R} = [parallel][??]([tau]) - [??](0) [[parallel].sub.b] for all s, [tau] [member of] R;

(iii) f is B-almost periodic if and only if, [??] is B-almost periodic, with the same set of [epsilon]-almost periods {[tau]}[epsilon];

(iv) [??] is B-almost periodic, if and only if there exists a relatively dense sequence in R, denoted by {[s.sub.n]; n [member] N}, such that the set of functions {[??]([s.sub.n]); n [member of] N}, is relatively compact in ([C.sub.b](R X),[D.sub.b]);

(v) [??] is B-almost periodic, if and only [??](R) is relatively compact in ([C.sub.b](R,X),[D.sub.b]);

(vi) (Bochner's criterion) f is B-almost periodic if and only if [??] (R) is relatively compact in ([C.sub.b](R,X),[D.sub.b]).

Proof. It is similar to the proof for Banach space valued functions, see e.g. [1], pp. 7-9.

Now, we are ready to prove the following sufficient condition for B-almost periodicity in p-Frechet spaces, 0 < p < 1.

Theorem 3.12. Let f [member of] [C.sub.b](R,X). Let us suppose that there exists a relatively dense set of real numbers ([s.sub.n]), such that

(i) The set {f([s.sub.n]); n [member of] N} is relatively compact in the metric space (X,D) and

(ii) for any n, m [member of] N, the relation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

holds with c > 0 independent of n,m.

Then, f is B-almost periodic.

Proof. The inequality in statement together with Theorem 3.11, (ii), obviously implies

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

Since by hypothesis, the set {f([s.sub.n]); n [member of] N} is relatively compact in the metric space (X,D), it has a convergent subsequence (f[([s'.sub.n])).sub.n]. The above inequality implies that also ([??][f([s'.sub.n])).sub.n], is a Cauchy sequence in the complete metric space (C(R,X), [D.sub.b]), so it is convergent. Combined with Theorem 3.11, (iv), it follows that [??] f is B-almost periodic, which combined with Theorem 3.11, (iii), implies that f is B-almost periodic. The theorem is proved.

Weakly Almost Periodic Functions

In what follows we will consider the concept of weakly almost periodicity, at least in the cases of [l.sup.p] and [H.sup.p] spaces, 0 < p < 1. Indeed, according to Remark 2 in Section 2, the dual spaces ([l.sup.p])* and ([H.sup.p])* are non-null. In addition, since {[e.sub.i]; i [member of] N}, with [e.sub.i] = ([[delta].sub.i,n])n [member of] [l.sup.p], [[delta].sub.i,n]-the Kronecker's symbol, is a basis in [l.sup.p] (see e.g. [14], p. 20) and since any [e.sup.*.sub.i] : [l.sup.p] [right arrow] R is linear and continuous (see e.g. [14], p. 12, Theorem 1.8), it easily follows that {p[psi]; [psi][member of] [([l.sup.p]).sup.*]}, with p[psi](x) = |[psi](x)|, for all x [member of] [l.sup.p], defines a sufficient family of semi-norms on [l.sup.p], which evidently induces a weak topology on [l.sup.p], namely a locally convex Hausdorff topology on [l.sup.p].

Also, since according to e.g. [14], p. 35, the point evaluations [[psi].sub.z](f) = f(z); z [member of] D, satisfy [[psi].sub.z] [member of] [([H.sup.p]).sup.*], for all z [member of] D, again it easily follows that [{p[psi](x); [psi] [member of] ([H.sup.p]).sup.*]} with ]p.sub.[psi]](x) = |[psi](x)|, for all x [member of] [H.sup.p], defines a sufficient family of semi-norms on [H.sup.p], which evidently induces a locally convex Hausdorff (weak) topology on [H.sup.p].

Definition 3.13. Let X = [l.sup.p] or X = [H.sup.p] with 0 < p < 1. A function f : R [right arrow] X is said to be weakly almost periodic, if f : R [right arrow] X is continuous and almost periodic, considering X endowed with the (weak) locally convex topology as above (see e.g. [6], pp. 159-160, or [16], [18]).

Remark. Obviously Definition 3.13 has no sense for the p-Frechet space [L.sup.p][0; 1], 0 < p < 1, whose dual is {0}.

Theorem 3.14. Let X = [l.sup.p] or X = [H.sup.p], 0 < p < 1. The necessary and sufficient condition that the function f : R [right arrow] X be weakly almost periodic is that for any [psi] [member of] [X.sup.*], the numerical function h : R [right arrow] R, defined by h(t) = [psi][f(t)] be almost periodic.

Proof. It is similar to the proof for Banach space valued functions (see Theorem 6.1.7, p. 160 in [6]).

Theorem 3.15. Let X = [l.sup.p] or X = [H.sup.p], 0 < p < 1. If a function f : R [right arrow] X is almost periodic, then it is weakly almost periodic and f(R) is relatively compact.

Proof. Since for any [psi] [member of] [X.sup.*] and all t, [tau] [member of] R, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

the usual (strong) almost periodicity and continuity of f immediately implies that f is weakly almost periodic and weakly continuous. This along with Theorem 3.6, proves the necessity of theorem.

Remark. It is known in the case of locally convex spaces that the condition in Theorem 3.15 is sufficient (see for instance [16] p. 58). The proof uses the Hahn-Banach theorem which does not necessarily hold in p-Frechet spaces as mentioned earlier in this paper.

Remarks. 1) The concept of integral was introduced in a p-Frechet space X endowed with the p-norm [parallel] x [parallel], 0 < p < 1, in [3, p. 102] (see also [1, p. 158]), as follows. First, for a = [a.sub.0] < [a.sub.1] < ... < [a.sub.n] = b a partition of [a, b], a step function on [a, b] is of the form s(x) =[n-1.summation over (k=0)][y.sub.k] x [[chi].sub.[[a.sub.k,a.sub.k+1])](x), (where [chi][[a.sub.k,a.sub.k+1]) is the characteristic function of [[a.sub.k, a.sub.k+1]) and [y.sub.k] [member of] X, k = 0, 1, ..., n - 1) and its integral on [a, b] is defined by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] Then, since any continuous function f : [a, b] [right arrow] X is uniformly continuous on [a, b], it is easy to show that is the uniform limit on [a, b] of the sequences of step functions [s.sub.n](x), n [member of] N defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.],

and the integral of f will be defined by [[integral].sup.b.sub.a] f(x)dx [member] X, where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

(It is easy to see that the above [[integral].sup.b.sub.a]f(x)dx does not depend on the sequence of step functions uniformly convergent to f.) Unfortunately, the fundamental theorem of calculus stated in [3, Theorem 2, pp. 104-105] (see also [2, pp. 161-162]), seems to be not valid in general, since for a continuous function f : [a, b] [right arrow] X, with integral

F(t) = [[integral].sup.t.sub.a] f(x)dx we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

but we do not get, in general, the estimate

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.],

with u between t and t + h, as claimed in [2, p. 162] (which would imply that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] As a first consequence, it follows that the the implication [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] uniformly continuous and f' B-almost periodic imply f0 B-almost periodic", does not hold, although in the case of locally convex space valued functions it is valid (see e.g. [16], Theorem 3.2.7, p. 63).

2) Of course that also we could adopt the more particular definition (of Riemann-type) for the integral on [a, b] of a function f : [a, b] [right arrow] X, as unique limit of all the Riemann sums [n-1.summation over (k=0)]f([[xi].sub.k])([a.sub.k+1] - [a.sub.k]), with [[xi].sub.k] [member of] [[a.sub.k], [a.sub.k+1]]. Unfortunately, for this kind of integral too, the property [parallel][lambda]x[parallel] = |[lambda]|,|[[lambda].sup.p][parallel]x[parallel], where 0 < p < 1, produces a bad estimate for the difference between the Riemann sums attached to two functions f, g : [a, b] [right arrow] X, namely

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.],

(in fact, since 0 < p < 1, for [a.sub.k+1-ak] [less than or equal to] 1, we have [([a.sub.k+1-ak]).sup.p], ([a.sub.k+1-ak]), which is the case for n sufficiently large). This fact which has the similar effect as for the first integral, namely the fundamental theorem of calculus for this second integral also does not hold.

3) From the above Remarks 1) and 2), it is evident that for a continuous f : [a, b] [right arrow] X, the inequality

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

does not hold.

Now, if we introduce (as in the case of Banach space valued functions) the mean value

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.],

where the limit is considered in the metric space (X,D), (i.e. there exists M (f) [member of] X with [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] then because of the above Remark 3, it seems that M(f) does not exist for any f [member of] AP(X), since in the proof for the case of Banach space valued functions, the inequality mentioned in Remark 3 is essential. This has as an effect the fact that, in general, one cannot attach Fourier series to a function f [member of] AP(X) and the fact that the almost periodicity of f does not imply the approximation property mentioned in Definition 3.1, (iii).

4) In [13], a theory of semigroups of linear and continuous operators is developed. As one of the applications, it was obtained that the initial value problem in the p-Frechet space X, 0 < p < 1,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.],

(where A : X [right arrow] X is linear and continuous) has as unique solution u(t) = T(t)(x), with T(t)(x) = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (x), the limit being in the the p-norm in X. On the other hand, taking into account the above Remarks 1 and 2, it follows that the inhomogeneous Cauchy problem

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.],

in general seems to not have mild solution, in the sense that, even if we can define it as usual, it does not satisfy the differential equation.

5) It is easy to construct some almost periodic functions f : R [right arrow] X for which there exists M(f) and the fundamental theorem of calculus holds. Indeed, any f of the form c x g, where c [member of] X and g : R [right arrow] R is almost periodic, satisfies these two requirements.

4. Applications

Firstly, we illustrate the idea of propagation of almost periodicity from the input data to the solutions of a simple differential equation in a p-Frechet space (X, [parallel] x [parallel]). In this sense, we present

Theorem 4.1. Let f : R [right arrow] R be an usual almost periodic function and c [member of] X. Then the function y : R [right arrow] X given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

is B-almost periodic and satisfies the differential equation

y'(t) + y(t) = c x f(t),

for all t [member of] R.

Here y'(t) is defined as usual, that is the limit in the metric D(x; y) = [parallel] x - y[parallel], given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Proof. Let us denote F(t) = [[integral].sup.t.sub.-[infinity]] [e.sup.u-t]f(u)du, t [member of] R. By the classical theory, F is an usual almost periodic function. Then by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

it is immediate that y(t) = c x F(t) is B-almost periodic in the sense of Definition 3.1, (i).

Since

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

it easily follows y'(t) = c x [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] and that y(t) satisfies the differential equation, which proves the theorem.

In what follows we apply some classical results in dynamical systems to study the stability of solutions of differential equations with almost periodic input data. Firstly, let us recall some definitions and a theorem from the theory of dynamical systems.

Definition 4.2. ([8])Let (X, d) be a metric space. A dynamical system is a mapping [pi] : X x R [right arrow] X. ([pi](p, >) is called motion).

Let [gamma](p) [equivalent to] ([pi](p, t) R} t [[gamma].sup.+] (p) [equivalent to] {[pi] (p,t):t [greater than or equal to] denote the trajectory and positive semitrajectory of the motion through p; respectively. Let [[ohm].sub.p] = {y [member of] X; [there exists][t.sub.n] [right arrow] [infinity], lim d([pi](p; [t.sub.n]); y) [equivalent to] 0}, be the [omega]-limit set of the point p [member of] X. Some known concepts of stability are given below.

Definition 4.3. ([8])(i) The motion [pi](p, t) is Lagrange-stable if [bar.[gamma](p)] is compact, and it is positively Lagrange-stable if [bar.[gamma]+(p)] is compact.

(ii) The motion [pi](p, t) is uniformly positively Lyapunov-stable with respect to a set D [[subset].bar] X if for every [epsilon] > 0 there exists [delta] > 0 such that d[pi](q, t), [pi]([??], t)) < [epsilon], [for all] t [greater than or equal to] 0, whenever q [member of] [[gamma].sup.+](p); bq 2 D and d(q, [??]) < [delta].

(iii) The motion [pi](p, t) is positively Poisson-stable [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

The following important result holds.

Theorem 4.4. ([8]) If X is a complete metric space and [pi](p, t) is positively Lagranges-table then the following statements are equivalent:

(A) [pi](p, t) is an almost periodic motion(with respect to t).

(B) [pi](p, t) is positively Poisson-stable and uniformly positively Lyapunov-stable with respect to [bar.[[gamma].sup.+]](p):

Let us apply these results to differential equations in p-Frechet spaces. Let

x' [equivalent to] f(x, t)

be a differential equation in the p-Frechet space X, where f : X x R [right arrow] X is continuous, almost periodic in t.

Let [pi] : X x R [right arrow] X, where p [member of] X is fixed and [pi](p, t) [member of] AP(X) is the unique almost periodic solution (supposed to exist) of equation x'(t) [equivalent to] f[x(t),t] which passes through p, i.e. [pi] (p, [t.sub.p]) [equivalent to] [x.sub.p], for certain [t.sub.p] [member of] R, [x.sub.p] [member of] X. If the motion [pi](p, x) is (positively) Lagrange-stable then we say that the equation has a Lagrange-stable solution. If [pi] (p, x) is uniformly (positively) Lyapunov-stable then we say that the equation has an uniformly (positively) Lyapunov-stable solution. Finally, if [pi](p, x) is positively Poisson-stable then we say that the equation has an uniformly (positively) Poisson-stable solution.

The following theorem concerning the existence of stable almost periodic solutions of differential equations in a p-Frechet space holds true.

Theorem 4.5. If the above differential equation has a unique almost periodic solution, then this solution is positively Poisson-stable and uniformly positively Lyapunov-stable.

Proof. Firstly, it is known that (X,D) is a complete metric space. We observe that an almost periodic solution is Lagrange stable. Indeed [bar.[gamma]+(p)] = bar.[pi](p, t) : t [greater than or equal to] 0}. Then by Theorem 3.6, [bar.[gamma](p)] is compact, and so is also [bar.[gamma]+(p)] as a closed subset of a compact space. Now, we apply Theorem 4.4, and the proof is complete.

As a conclusion, if one finds some almost periodic solution of a differential equation in a p-Frechet space, it is also a stable solution of that equation.

References

[1] Amerio L. and G. Prouse, Almost Periodic Functions and Functional Equations, Van Nostrand Reinhold, New York, 1971.

[2] Bayoumi A., Foundations of Complex Analysis in Non-Locally Convex Spaces, North-Holland Mathematics Studies, vol. 193, Elsevier (Amsterdam, 2003).

[3] Bayoumi A., 1999, Fundamental theorem of calculus for locally bounded spaces, J. Nat. Geom., 15, pp. 101-108.

[4] Bede B. and Gal S.G., 2004, Almost periodic fuzzy-number-valued functions, Fuzzy Sets and Systems, 147(3), pp. 385-403.

[5] Bugajewski D. and N'Guerekata G.M., 2004, Almost periodicity in Frechet spaces, J. Math. Anal. Appl., 299, pp. 534-549.

[6] Corduneanu C., Almost Periodic Functions, Intersciences Publishers, John Wiley & Sons, New York-London-Sydney-Toronto, 1968.

[7] Corduneanu C. and Golsdtein J.A., 1984, Almost periodicity of bounded solutions to nonlinear abstract equations, Diff. Equ., North-Holland Math. Studies, 92, pp. 115-121.

[8] Deysach L.G. and Sell G.R., 1965, On the existence of almost periodic motions, Michigan Math. J., 12, pp. 87-95.

[9] Dunford N. and Schwartz J.T., Linear Operators, Part I, Interscience, New York, 1964.

[10] Duren P.L., Romberg B.W. and Shields A.L., 1969, Linear functionals on [H.sup.p] spaces with 0 < p < 1, J. Reine Angew. Math., 238, pp. 32-60.

[11] Fink A.M., Almost Periodic Differential Equations, Springer-Verlag, Berlin, Heidelberg, New York, 1974.

[12] Forelli F., 1964, The isometries of [H.sup.p], Canad. J. Math., 16, pp. 721-728.

[13] Gal S.G. and Goldstein J.A., Semigroups of linear operators on p-Frechet spaces, 0 < p < 1, Acta Math. Hungarica, 114(1-2), pp. 13-36.

[14] Kalton N.J., Peck N.T. and Roberts J.W., An F-Space Sampler, London Mathematical Society Lecture Notes Series, vol. 89, Cambridge University Press, Cambridge, 1984.

[15] Levitan B.M. and Zhikov V.V., Almost Periodic Functions and Differential Equations, Cambridge University Press, Cambridge, London, New York, 1982.

[16] N'Guerekata G.M., Almost Automorphic and Almost Periodic Functions in Abstract Spaces, Kluwer Academic/Plenum Publishers, New York, 2001.

[17] N'Guerekata G.M., Topics in Almost Automorphy, Springer-Verlag, New York, 2005.

[18] N'Guerekata G.M., 1984, Almost periodicity in linear topological spaces and applications to abstract differential equations, Int'l. J. Math. and Math. Sci., 7, pp. 529-541.

[19] Oswald P., (1982/83), On Schauder bases in Hardy spaces, Proc. Roy. Soc. Edinburg, Sect. A, 93(3-4), pp. 259-263.

[20] Shen Wenxian, 1999, Travelling waves in time almost periodic structures governed by bistable nonliniarities, J. Diff. Eq., 159, pp. 1-54.

[21] Yoshizawa T., Stability Theory and the Existence of Periodic and Almost Periodic Solutions, Springer-Verlag, New York, Heidelberg, Berlin, 1975.

Sorin G. Gal

Department of Mathematics

University of Oradea, Romania

Str. Universitatii No. 1

410087 Oradea, Romania

E-mail: [e-mail]

Gaston M. N'Guerekata

Department of Mathematics

Morgan State University

Baltimore, MD 21251, USA

E-mail: [e-mail]