1 Introduction and preliminaries
We consider general closed and translation invariant subspaces of , as in . This will give a framework on how to work with various forms of almost periodicity. Thus, we consider a general splitting of a closed translation invariant subspace,
We answer the question of when a solution of the nonlinear evolution equation
with being dissipative, is a member of Y, is asymptotically close to Y, or itself splits in the same manner, , where is a generalized solution to
where denotes the almost periodic part of the family of possibly unbounded nonlinear dissipative operators . We answer questions concerning the perturbations, i.e., of when the following identity holds:
The underlying study is divided into sections where the nonlinearity is studied, which is even new in finite dimensions, and the appendices which show how Eberlein weak almost periodicity (Appendix A) and asymptotic almost periodicity (Appendix B) come into play, or give a hint on how continuous almost automorphy (Appendix C) becomes applicable. In Appendix A we provide the analysis around Eberlein weakly almost periodic functions (EWAP). We show, how Appendices A and B apply to obtain the splittings in Section 4. We give certain conditions on to obtain Eberlein weak almost periodicity in Banach spaces. For special cases where the family is time-independent, or the family itself is periodic, cf. [21, 22]. We also refer to , where the result has been proved for the special case of classical almost periodic functions on the real line. In  the existence of an EWAP solution on the whole line is proved in finite dimensions for a semilinear system of the form
where A denotes a special linear operator and is Lipschitz.
Applying general existence results, even in the finite dimensional case, more general results on sums of dissipative operators are obtained while dispensing with the relationship between the ω-dissipativeness and the Lipschitz continuity of the nonlinear perturbation . In this study general perturbation results for dissipative operators are used (Section 5). Moreover, for these general perturbation results, we also show the connections between the differential equation and the almost periodic part of the weakly almost periodic solution, and, even more abstractly, how the generally defined splitting carries over from the solution to the differential equation.
Given a Banach space X, a function , , is said to be weakly almost periodic in the sense of Eberlein (EWAP) if the orbit of f with respect to , namely,
is relatively compact with respect to the weak topology of the sup-normed Banach space (cf. [9, 22, 21, 24, 27]). The space of all such functions will be denoted by . Moreover, denotes the closed subspace of all such that some sequence of translates of f is weakly convergent to the zero function (“weakly” referring to the weak topology of ).
Here, denotes the space of almost periodic functions, i.e.,
An exposition of this result is given in the book of Krengel . For technical reasons we introduce the following spaces on the real line:
The decomposition of and the uniqueness of the almost periodic part gives
and by the uniform continuity of Eberlein weakly almost periodic functions, we obtain that for a given and , we have . Additionally, some examples are given in order to point to problems in this class of functions.
2 Preliminaries on integral solutions
In  the following two types of equations have been discussed. The initial value problem
and the whole line equation on ,
with . For these equations we define integral solutions. First, some prerequisites.
Let X be a general Banach space. As the given plays a crucial role, let , and throughout the paper assume that . To obtain the solutions of equations (2.1) and (2.2), we follow the approach given in . The assumptions for the family are as follows.
The set is a family of m-dissipative operators.
In comparison to the assumption given in , the uniform continuity on for h is added.
There exist and , continuous and monotone non-decreasing, such that for and , we have
for all , .
Recalling the study , the next assumption is stronger than that in  due to the Lipschitz continuity and linearized stability in . This becomes important to obtain uniform convergence on the half line depending on the Lipschitz constant on forthcoming g and .
There exist bounded and Lipschitz continuous functions , and a continuous and monotone non-decreasing function such that for , and , we have
for all , .
Due to the fact that we consider as a perturbation of by , we have to define the perturbed control functions and . We define
Now we are in the situation to define the integral solution to the initial value problem. As the integral solution is defined with the help of Assumptions 2.2 and 2.3, it looks slightly different from the one in [15, Definition 6.18, pp. 217–218], which is due to Benilan.
for all , and (with in case of Assumption 2.2).
In order to solve the initial value problem (2.1), we use the Yosida approximation of the derivative. This leads to the equation
with the Yosida approximation
with (see [23, Remark 2.7])
The results for these approximations on are the following.
Theorem 2.6 ().
Let , and let fulfill Assumptions 2.1 and 2.2 with , or Assumption 2.3 with , where is the Lipschitz constant of g. Further, let , and let be the corresponding Yosida approximations to equation (2.3). Then converges uniformly on to the integral solution, as .
Further, we define an integral solution for the whole line problem (2.2).
for all , and (with in case of Assumption 2.2).
Similar to the initial value case, the Yosida approximation of the derivative was considered and led to the following result:
Again the are derived by a Banach iteration, i.e.,
For these approximants we have the following result.
Theorem 2.8 ().
Let Assumptions 2.1 and 2.3 be fulfilled. Further, assume that the Lipschitz constant of g in Assumption 2.3 is less than . Then the Yosida approximants are Cauchy in when . The limit is an integral solution on .
Remark 2.9 ().
Moreover, we have a comparison result between the solution found on and .
3 Main definitions and context
We start with the notion of pseudo-resolvents:
Definition 3.1 ([25, Definition 7.1, p. 193]).
The members of a family of contractions,
are called pseudo-resolvents if
Lemma 3.2 ([25, Lemma 7.1, pp. 193–194]).
Pseudo-resolvents have the generator
Next we motivate a generalization for various types of almost periodicity.
Let Y be a closed linear subspace of , and a family of m-dissipative and Lipschitz continuous operators defined on X with the common Lipschitz constant K. Further, if for all , , then for all , , for small .
For given and , we define
Then, for given , we have
Consequently, for , there exists such that
Thus, we found that for , . ∎
Due to the previous results it becomes straightforward to define types of almost periodicity even for multivalued ω-dissipative operators.
Let Y be a closed linear translation-invariant subspace of . An m-dissipative family is called Y invariant if for all and , and .
The examples for Y are , , , and . In the case of almost automorphy, we have to add uniform continuity (i.e., ). The spaces , and are considered in the following way:
This makes sense even for the splitting, as the almost periodic part is uniquely defined on while the weak almost periodic function is only given on . This approach allows to consider equations on the real line, which are given only on , while extending by
The control functions are extended in the same manner.
Due to Lemma 3.3, we can define the almost periodic part of a generally multivalued operator . Therefore, let Y split into the direct sum
with a projection satisfying .
A linear, closed and translation invariant subspace splits almost periodic if , where are closed translation invariant linear subspaces, the constants are in (), the corresponding projection fulfills
and, finally, commutes with the translation semigroup.
Let Y split almost periodic, and let be m-dissipative and Y-invariant so that
with and . If for a given ,
then the members of the following family of operators are pseudo-resolvents:
If is Y-invariant and (3.1) is fulfilled, we say that splits almost periodic with respect to , .
First we prove that are contractions:
Next the range condition, but this is fulfilled, because we assumed to be m-dissipative, therefore .
As are resolvents, we have the following resolvent equation:
Using, , the pseudo-resolvent-equation holds in Y. Hence, we can apply the projection on both sides of the equation. Thus,
Applying (3.1), and evaluating at , serves for the proof. ∎
Let be m-dissipative and split almost periodic with respect to Y, . The almost periodic part of is defined as follows:
By the previous definition, for m-dissipative operators which split almost periodic, we obtain
where denotes the part of a function given in Y.
Using the fact that for A dissipative, we have
The first equality comes with the dissipativeness of due to (3.2), and the second by definition. ∎
Let split almost periodic with respect to . Due to (3.2), is dissipative. From , we derive the m-dissipativeness, and the facts that for all , and, together with equation (3.1), that the almost periodic part of is -invariant.
Given , we apply (3.1):
Thus, , and therefore . ∎
For an example, we recall the definition of demi-closedness [15, Definition 1.15, p. 18]
An operator is called demi-closed if A is norm-(weakly-sequentially) closed as a subset of , i.e., , , and for all implies .
Let be weakly almost periodic and assume that for some . Using the splitting , with , for an m-dissipative operator , we define
Let , and either or . Let also A be demiclosed and compact. Then splits almost periodic, and the -part is given by . The same result can be obtained using , with .
For a given , we have to prove that is weakly almost periodic. Note, that . Thus, we obtain the result using the following help function with :
and . Hence, for , by a further use of the previously defined help function and an application of Theorem A.8, it remains to prove that, for weakly compact metrizable, the following function is continuous:
For a given , weakly convergent to , and convergent to , we define and claim that
To this end, we show that any subsequence of has a subsequence convergent to . Let such a subsequence be chosen. Then, without loss of generality, we have . This is equivalent to , and the demiclosedness leads to . Using the fact that for all , by the resolvent equation, we have
The Lipschitz continuity of proves the boundedness of , which finishes the proof.
To compute the almost periodic part of , apply Corollary A.17. ∎
To have a comparison between the classical almost periodicity and the generalized one, we provide the following proposition. It shows that under certain conditions (i.e., it splits canonically), and the splitting is obtained from coinciding resolvents.
Let split almost periodic, and let be a family of uniformly Lipschitz operators defined on X, with Lipschitz constant K. Further, let for all . Due to the splitting, we have We define
and assume that
Then, for , the almost periodic part in the sense of Definition 3.8, we have
To prove the equality we show that the resolvents coincide, i.e.,
For given , we have
The Lipschitz continuity of leads to
Consequently, for , we find a net of fixpoints satisfying
Next we consider the resolvent of which is defined by the almost periodic part of the resolvent of B. Hence, we need a representation for the resolvent of B and we have to compute the almost periodic part. We define the fixpoint mapping with respect to B:
Using the fact that B is Lipschitz for , we find a net of fixpoints satisfying
Using the representation, we compute the almost periodic part and apply assumption (3.3), to obtain
We found that is a fixpoint of , hence . This leads to
which finishes the proof. ∎
Kenmochi and Otani  considered nonlinear evolution equations governed by time dependent subdifferential operators in Hilbert space H:
They constructed a not necessarily complete metric space of convex functions , whereby, , implies , in the sense of Mosco, cf. [16, Lemma 4.1, p. 75]. For short, we write . Next we show that the notion of almost periodicity on subdifferentials given in  is stronger than the one given via resolvents defined in this study, when viewing as a dissipative operator.
Let be proper, lower semicontinuous and convex for all . If
is almost periodic, then
is almost periodic for all .
We assume that for , the mapping is not almost periodic. Consequently, we find a sequence such that is not uniformly Cauchy. Without loss of generality,
The non-Cauchy assumption on leads to subsequences , and a sequence such that
Note that is assumed to be almost periodic, is a compact subset of , and for all . Consequently, we may assume that
uniformly for . Applying [16, Lemma 4.1, p. 75], we obtain
in the sense of Mosco, which implies, by using [1, Theorem 3.26, p. 305],
when . A contradiction to the non-Cauchy assumption. ∎
4 Main results
In this section we show how the previous results apply to evolution equations of the following type:
and the corresponding initial value problem
where is a possibly nonlinear multivalued and dissipative operator satisfying a type of almost periodicity defined above. Throughout this section, we assume , and to be closed and translation invariant subspaces of , with . In the case where is classically almost periodic and , the result in case of Assumption 2.2 is due to . Even in the book [14, equation (C1), p. 153], Hino et al. considered Assumption 2.2. Thus, they were not able to consider operators coming from Example 3.12. The problem is considered in the infinite dimensional case, and is even new for finite dimensions. The ω-dissipativeness is needed to obtain the uniform convergence of the approximants. Moreover, when and only m-dissipative, there exists a counterexample for classical almost periodicity in the case of dimension two, see [13, Remark 1.3 (2)].
A solution u of (4.2) is called asymptotically Y if there exists such that
The difference to earlier splitting results is that a splitting of the solution is found, but in the case of general dissipative and time dependent operators, the equation fulfilled by the almost periodic part was unknown.
Let, for a given , be Y-invariant, and fulfill Assumption 2.1 and either Assumption 2.2 with , or Assumption 2.3 with , where denotes the Lipschitz constant for the control function g. Then there exists a solution to (4.1), and all integral solutions of (4.2) are asymptotically Y.
Let Y and split almost periodic, with . Then the almost periodic part of the solution fulfills , and is a generalized solution to the evolution equation
The first part is a direct consequence of [22, Theorem 4.10]. For the second part, note that
and due to the assumption of an almost periodic splitting, we apply and obtain
The notion of an integral solution in the case of a general splitting is not possible due to the missing control functions for . Thus, in general we cannot construct a solution applying existence results of  to the equation
If fulfills Assumption 2.3 with bounded and Lipschitz, non-decreasing and , then the generalized solution is an integral solution, in the sense of Definition 2.7, with respect to the control functions .
Equation (4.5) and the fixpoint equation (4.4) show that we are in the situation of approximations used in the study [23, equation (31), p. 1076]. The uniqueness of fixpoints and the corresponding uniform limits (cf. Theorem 2.8) conclude the proof. ∎
In the context of Theorem 4.2, let , and . If the control functions belong to , then the generalized-(almost periodic)-solution is an integral solution with respect to , and the almost periodic parts of .
Clearly, . From Corollary A.14, we find a sequence , with , such that, due to the compact ranges,
pointwise in the norm of X. As the solution on is an integral solution (see Definition 2.7), we have
Therefore, for , , we have
Using the fact that is upper semicontinuous, we have
Consequently, we may pass to the limit superior on both sides of (4.6), and by Fatou’s lemma we obtain the inequality. ∎
In the context of Theorem 4.2, let
If the control functions h, g belong to , then the generalized-(almost periodic)-solution is an integral solution with respect to , and the almost periodic parts of , g.
Apply inequality (4.6), the existence of a sequence with , and the pointwise-norm convergence. ∎
Let split almost periodic to . Consider the equations
is asymptotically for all , .
It is sufficient to prove that the almost periodic parts of and coincide. Due to the uniform convergence of the approximants, it suffices to prove it for the approximants. Moreover, asymptotically they are close to their corresponding bounded solution on the whole line [23, Corollary 2.16]. Therefore, let be the approximants to the whole line solution given by [23, Theorem 2.18]. Due to condition (4.7),
Thus, using the fixpoint equations for the approximants and , we have
Thus, is the fixpoint of the strict contraction defining the solution . Consequently, by the uniqueness of the fixpoint, Thus, the almost periodic parts of the solutions of (2.2) coincide. Thanks to Corollary 2.10, the solutions of the initial value problem (2.1) are asymptotically close. ∎
Corollary 4.7 ().
and let split almost periodic with . Consider the equations
then, for the corresponding evolution systems and , we have
i.e., they are asymptotically equivalent.
In the case of Eberlein weak almost periodicity, we obtain the following corollary.
Let , , , and let split almost periodic with . Consider the equations
for all λ small, , uniformly in , then
for all , and , uniformly in .
In the case of ordinary differential equations the result of Theorem 4.6 extends as follows.
Let , let be Lipschitz, and let
if and only if
In this section we put some general perturbation theorems for dissipative operators into the context of almost periodic splittings. This extends the theory of semilinear operators like , as they are considered in , and in  for the case of almost automorphy. We start with a Lipschitz perturbation.
Let split almost periodic with respect to . Further, let be uniformly Lipschitz with constant K, satisfying for all , where for all . Then is ω-dissipative with at least , and splits almost periodic with respect to . Moreover,
We start with proving the ω-dissipativeness. Let and for . Then
Thus, is dissipative with . From the uniform Lipschitz condition of , we obtain the uniform Lipschitz condition of . For given , we have
Consequently, we obtain that is ω-dissipative with .
To prove the m-dissipativeness and to obtain a representation of the resolvent of , for given , we define
An estimation gives that is a strict contraction for . Thus, the Banach fixpoint principle leads to a such that
As the constants are contained in , the m-dissipativeness of is proved. We also proved . Therefore, it remains to prove the perturbation result (5.1), which is equivalent to
From the previous step, we have for . This leads to
Noting that , we define
Let be the net of fixpoints for . Then we have
As a consequence, we obtain the same contraction mapping, and the uniqueness of the fixpoint concludes the proof. ∎
In many studies the existence of solutions to equations like
need the precondition , where L denotes the Lipschitz constant of B. Theorem 5.1 can be viewed as a split of the assumptions on m-dissipativity (i.e., ) and the dissipativity constant of . For the m-dissipativity the Lipschitz constant on B is needed. To obtain a bounded solution to (5.2) by the methods of Section 2, we obtain that in case of Assumption 2.2, or in case of Assumption 2.3, is sufficient. Consequently, the direct connection between the Lipschitz constant on and to obtain a bounded solution on is cut.
If is strictly convex, then the duality mapping
is single-valued and, in case of a uniform convex dual, uniformly continuous on bounded subsets of X, cf. [15, Proposition 1.1, p. 2]. Note that a dissipative operator, in this case, is always strictly dissipative (i.e., ), due to the single valuedness of F. Consequently, a sum of dissipative operators is dissipative. Next we consider the case where is uniformly convex.
Let be uniformly convex, let , and let the families be m-dissipative. If there exist and such that for all ,
then, for bounded, there exist and a unique , , such that
Further, if for , uniformly in , , then, there exists a function such that and
Using Banach’s fixpoint theorem, we will find . From the precondition (5.3), we find uniformly bounded pairs such that
Using , we can choose so that
Thus, is uniformly bounded. Using (5.4) for , and the fact that is dissipative, we have
Using and the dissipativeness of B, we have
In sum this leads with the boundedness of to some , so that
Recalling that , the uniform continuity on bounded sets of the duality map yields that is uniformly Cauchy when . The completeness of X gives uniform convergence in , cf. [8, p. 258]. We define
For a given , we can conclude the proof with the arguments used in the proof of [15, Theorem 1.24, pp. 25–26, and Theorem 1.17 (ii), p. 18]. Using for all , , we have . The reflexivity gives weakly, with . Applying (5.4), we find a bounded selection satisfying
Thus, by the demiclosedness of and convergence of we have
weakly, with . Summarizing the previous, we have
weakly in X, which concludes the proof. ∎
Let be uniformly convex, let , and let the families be m-dissipative. Assume that there exist and such that for all ,
The proof is analogous to that of [15, Corollary 1.25], proving the boundedness of . ∎
we find that the solutions are fixpoints of the contractions
Moreover, applying the bounded selection , we have, similar to (5.6), some such that uniformly in , . Thus, if for a given ,
then, by an observation similar to (5.7), we have
The uniform convexity of and the demiclosedness of and imply that is a bounded solution of
Let be uniformly convex, and let , . Further, let and be families of dissipative operators, which split almost periodic with respect to , . Under the assumptions (5.3) and (5.8) of Remark 5.5, we have
We have to show that the almost periodic part of equals . To prove that the fixpoints are in Y, note that , and that and leave Y invariant. Thus, for , we have the strict contraction
As is bounded, we obtain the uniform convergence of as well. For , we have the strict contraction
Thus, the projection gives the fixpoint and uniform convergence when . From (5.8), we conclude
weakly when , with and . This concludes the proof. ∎
In the next example we show how the previous results apply to perturbations to the linear Dirichlet problem on bounded domains Ω.
Let be open, bounded and with smooth boundary, let ,
with ω its dissipativity constant, and let be a proper, lower semicontinuous and convex function, satisfying for all . Defining, for all ,
we have that is m-dissipative and . If additionally, and split almost periodic, with respect to and , respectively, with
then we have
where and are the generalized almost periodic parts. Moreover, if the Dirichlet problem with respect to fulfills either Assumption 2.2 (see [4, Example, pp. 88–90]), or Assumption 2.3, with , then the solution of equation
with , has an almost periodic splitting as well.
From [15, Example 1.60 and Lemma 1.61, pp. 53–54], we obtain that is a subdifferential, and therefore is m-dissipative. Further, [15, Proposition 1.63, p. 55] proves the m-dissipativeness of on . As , we verified condition (5.3). The claim (5.10) is an application of Theorem 5.6, Theorem 4.2 and assumption (5.9). To consider right-hand sides , we use the fact that the mollified is Lipschitz and approximates f uniformly due to the uniform continuity of f. Applying [23, Proposition 6.1], we obtain, with , an additional Lipschitz and bounded control function. Finally, [23, inequality (43), p. 1084] finishes the proof. ∎
To view the case of Eberlein weak almost periodicity we have the following.
Let be uniformly convex and an m-dissipative operator with a compact resolvent. Further, let be m-dissipative, and let for all . For a given , we define
The Eberlein weak almost periodicity is a consequence of the proof of Example 3.12, and the relative compact range comes with the compactness of . For given , choosing
it remains to prove that
is continuous, which comes with the inequality
uniformly for , and the proof is finished. ∎
From Remark 3.9 and the previous perturbation results, we obtain the following corollary.
Let be uniformly convex, and let and split almost periodic with respect to , and fulfill the assumptions of Theorem 5.6. Consider the equations
A Eberlein weak almost periodicity
This and the remaining appendices discuss a number of closed and translation invariant subspaces of and , which under certain conditions satisfy the invariance conditions for the resolvent . Moreover, we give some examples for almost periodic splittings. We mainly restrict our attention to weak almost periodicity, in the sense of Eberlein, and to asymptotically almost periodic functions. These results demonstrate the use of the above general results in a variety of special cases.
Let with K compact metrizable. Then there exist subsequences and such that the following limits exist:
Let . Then we find a subsequence such that exists. Thus, we may assume that for a given , we can find such that exist for . For the step “”, choose an appropriate subsequence such that exists. After this recursive construction, we define , and since , we have for all . Computing the limits, we obtain . Passing to an appropriate , we are done for the first double limit. Redoing the previous steps on the found subsequence , for the interchanged limits we will find and , the desired subsequences. ∎
The proposition above allows to assume, without loss of generality, that the limits in Proposition A.1 exist. Recall that only the equality of the limits has to be proven.
In this section we provide an extension of Traple’s result to infinite dimensions.
Theorem A.3 ().
If X has finite dimension, and , with the property that
is continuous for every compact , then .
Traple’s proof does not extend to the infinite dimensional case, for he used both the algebraic structure of and the fact that the coordinate-wise polynomials are dense in for closed and bounded. Clearly, for general Banach spaces such structure is missing.
In order to give sufficient conditions for the extension of Traple’s result, hence existence of Eberlein weakly almost periodic solutions to nonlinear differential equations in infinite dimensions, we need the following definition, where is assumed to be a Hausdorff topological space. Moreover, in this study, D is X with the weak topology or X with norm topology.
A function is called DEWAP if
for all ,
for every compact and metric subset , the following map is continuous:
We denote these functions by
With the aim to find a representation for the almost periodic part, we recall some technical results.
Proposition A.5 ().
Eberlein weakly almost periodic functions are uniformly continuous.
Thus, fulfills the assumptions made on Y in the previous sections.
Let fulfill Definition A.4, and let be compact metric. Then .
As K is compact metric, for given , we find such that for all with , we have From the compactness of K we find with the following properties:
There exists such that for , we have .
From Definition A.4 (i), we obtain some with .
First we prove the boundedness. For ,
To prove the uniform continuity, let , and , with . Then, we obtain
which finishes the proof. ∎
From the previous remark we conclude that if fulfills Definition A.4, and is compact metric, then the following map is continuous:
Note that the continuity of ι does not imply a norm compact range for every if z is given. For example, one may take for some g, which is Eberlein weakly almost periodic. Then ι is clearly continuous, but the range of g is not necessarily compact. If , then for every , lipschitzian in the second variable, with Eberlein weakly almost periodic for all , and compact linear,
fulfills the previous definition. Consequently, for each and compact linear,
will satisfy the assumptions of Definition A.4. An example for is given by , provided the underlying space is reflexive, are bounded -semigroups, and C is a compact linear operator. A proof of this fact can be found in .
The restriction on K to be compact metric comes with the range of a function , since f is continuous, is separable and is weakly compact. The fact that the weak topology on weakly compact sets in separable Banach spaces is metrizable, see [8, Section V.6.3., pp. 434], motivates the restriction.
If , and , then .
For the proof, the following technical lemma is needed.
Let be a topological Hausdorff space, let with for all , and let
be continuous. Further, let a given sequence satisfy the following double limits condition:
Then the interchanged limits are equal for , whenever
and the iterated limits exist.
Applying Proposition A.2 to bounded set on we may assume that the following iterated limits in the situation of the above lemma exist:
By our hypothesis, we have is EWAP, thus satisfies the double limits condition, i.e.,
As the double limits on the right-hand side exist and are equal to 0, we proved . As the same routine works for , we finished the proof. ∎
Proof of Theorem A.8.
To verify the double limits condition, we apply the previous lemma for given
From [8, Section V.6.3, p. 434], we recall that the weak topology on weakly compact subsets in separable B-spaces is a metric topology. Noting that continuous images of separable spaces are separable, we obtain that , for is separable, hence the weak topology on is metric, where w denotes the weak topology. By an application of Proposition A.2, we may assume that for , the double limits exist, and implies that they have to coincide.
Thus, we are in the situation of the previous lemma and our claim is proved. ∎
From the proof of Lemma A.9, and using that only local continuity is needed, we give the corollary for
which was introduced by Goldberg and Irwin .
Let, for a Banach space Y, , and . Then, for any given , Eberlein weakly almost periodic with a relatively compact range, we have . Moreover, if for all , then .
The reader will have no difficulty to apply the previous theorem to , since , hence obtain the first part.
For the second part, it remains to prove the compactness of . Thus, for a given sequence , we have to find a subsequence such that is convergent in X. Since g has compact range, without loss of generality, . For this , we may choose a subsequence such that for some . From the continuity of ι, we obtain uniformly on . Thus,
and the proof is complete. ∎
In [18, Example 2.17, p. 17], it is shown that the compactness assumption on the range of g is essential.
Let . Then is Eberlein weakly almost periodic for all .
Letting , , and
the previous corollary serves for the proof. ∎
If , then .
Theorem A.8 gives a condition on f such that is Eberlein weakly almost periodic. Noting that every satisfies
it is also of interest when for a given . More generally, we have,
where denotes the almost periodic part of f. Thus, the question arises how the almost periodic part of the map looks like. In order to discuss these problems, we introduce the projection on the almost periodic part:
For the decomposition
we have that the projection onto has norm less than or equal to one.
For , we find such that . Consequently, for given and , we have
which leads to the claim. ∎
Any two functions have a common sequence , such that the translates and are weakly convergent to the almost periodic part of f and g, respectively.
First we consider the case where the almost periodic parts of f and g are equal to zero, and let and , be chosen such that
By the proposition above, we have that every Eberlein weakly almost periodic function is uniformly continuous, hence the semigroup of translations is strongly continuous. Since , is a weakly compact closure of translates of a uniformly continuous function, hence is compact metrizable in the weak topology of . As a consequence of Proposition A.2, we may pass to subsequences of and , such that the iterated limits of exist in the weak topology of , and, without loss of generality, the sequences are chosen in this way. From the interchangeable double limits condition, we obtain
Thus, if , and denotes the metric which induces the weak topology, then we can repeat the arguments on g, and, without loss of generality, we have
Thus, the desired result is a consequence of the classical diagonal process for the double sequence
If , then for the function h, given by
we find, by the double limits criterion and the representation for the dual of , that it is Eberlein weakly almost periodic if for given sequences and ,
whenever the iterated limits exist. However, by a successive diagonalisation of the double sequences
we may assume that their individual iterated limits exist and are equal, hence h is Eberlein weakly almost periodic. Thus, h has a unique decomposition into an almost periodic and a part:
Clearly, from the decomposition of f and g, we obtain
Further, by the observation in the first part of this proof, we have that is , and choosing subsequences two times will prove that is almost periodic, hence the claim follows from the uniqueness of the decomposition. Hence, the sequence , for which
is the desired one. ∎
For every compact metric, and , there exists a sequence such that
for every , where denotes the almost periodic part of .
Given any , we find an and , such that
Since is compact metric (therefore separable),
is continuous, and separable, by Remark A.7, where denotes the semigroup of translations. Therefore,
Consequently, L is a subset of a closed and separable subspace Y of .
By the fact that
we obtain the relative weak compactness for L. Hence, the weak topology on L is metrizable, and we may choose a metric of the form
Choosing , we obtain, by the way of the first observation, elements , and by setting
we construct a dense sequence . As a consequence of Corollary A.14 and by a simple induction, we find, for all , a sequence such that
converges weakly as for all in . This, together with the existence of a metric, implies that for all , there exists such that
for all and .
Now, for a given and , we have
where the last inequality follows from the definition of the metric and the fact that the norm of the projection on the almost periodic part is less or equal to one. This completes the proof. ∎
Let , and . Then the following identity holds:
where and denote the almost periodic parts of and y, respectively.
From the notation of the theorem, we find that is a norm compact set. Now let , , . Since are two discrete points attached to with a positive distance to K we have that is still norm compact. Let
Using that all the sequences are convergent, it remains to compute the limit, which can be done in the pointwise weak topology as follows:
Since , the first term on the right-hand side tends to zero as n tends to infinity, and the theorem is proved. ∎
If for a Banach space Y, and , then, for every given ,
Using that on the norm and the weak topology coincide leads to the given result. ∎
We consider the context of Example 5.7 with ,
Then, for , we have
We give a proof for , and note that . If , we have that is compact. As is a contraction, we can apply Lemma A.15 to
which leads to a single sequence with
weakly in and , respectively. Applying the weaker pointwise and pointwise weak topology, we have, for and ,
which concludes the proof. ∎
Note that the methods apply in a similar way, when is substituted by and by , since the weak relative compactness of orbit on a positive half line serves for . Thus, and , hence the proofs are similar.
B Asymptotically almost periodic functions
In consequence, we have the following proposition.
For the decomposition
we have that the projection onto has norm less than or equal to one.
Using compactness methods we have the following theorem.
Let be such that , with being uniformly Lipschitz with a constant L, and its almost periodic part. Further, let , with its the almost periodic part. Then
As is relative compact, Lemma A.15 serves for the needed norm convergent subsequence, so that for all , weakly. The relative compactness serves for the norm convergence. The rest of the proof is straightforward. ∎
C Almost automorphic functions
Bochner introduced the notion of almost automorphy.
A function is said to be almost automorphic if for any real sequence , there exists a subsequence such that
If the limit g is continuous, then f is called continuous (Bochner)-almost automorphic. We define
Noting, that for , is relatively compact, clearly, we have , and that is translation invariant.
The following Theorem is due to [30, Lemma 4.1.1, p. 742].
Continuous almost automorphic functions are uniformly continuous, i.e.,
In , the asymptotically almost automorphic functions are discussed. By definition, we have
For suitable , the almost automorphic part
was computed. Thus, the underlying study becomes applicable, when switching from almost automorphy to continuous almost automorphy, and adding for the uniform continuity and the Lipschitz continuity, in the first and second variable, respectively. As for ,
and the projection has a norm less than one.
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About the article
Published Online: 2016-12-02
Citation Information: Advances in Nonlinear Analysis, Volume 8, Issue 1, Pages 1–28, ISSN (Online) 2191-950X, ISSN (Print) 2191-9496, DOI: https://doi.org/10.1515/anona-2016-0075.
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