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Advances in Nonlinear Analysis

Editor-in-Chief: Radulescu, Vicentiu / Squassina, Marco


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Asymptotic behavior of evolution systems in arbitrary Banach spaces using general almost periodic splittings

Josef Kreulich
Published Online: 2016-12-02 | DOI: https://doi.org/10.1515/anona-2016-0075

Abstract

We present sufficient conditions on the existence of solutions, with various specific almost periodicity properties, in the context of nonlinear, generally multivalued, non-autonomous initial value differential equations,

dudt(t)A(t)u(t),t0,u(0)=u0,

and their whole line analogues, dudt(t)A(t)u(t), t, with a family {A(t)}t of ω-dissipative operators A(t)X×X in a general Banach space X. According to the classical DeLeeuw–Glicksberg theory, functions of various generalized almost periodic types uniquely decompose in a “dominating” and a “damping” part. The second main object of the study – in the above context – is to determine the corresponding “dominating” part [A()]a(t) of the operators A(t), and the corresponding “dominating” differential equation,

dudt(t)[A()]a(t)u(t),t.

Keywords: Evolution equations; almost periodicity; limiting equation

MSC 2010: 47J35; 37L05; 335B40

1 Introduction and preliminaries

We consider general closed and translation invariant subspaces of BUC(,X), as in [26]. 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,

YBUC(,X),Y=YaY0.

We answer the question of when a solution of the nonlinear evolution equation

dudt(t)A(t)u(t),u(0)=u0,

with A(t) being dissipative, is a member of Y, is asymptotically close to Y, or itself splits in the same manner, u=ua+u0, where uaYa is a generalized solution to

dudt(t)[A()]a(t)u(t),

where [A()]a denotes the almost periodic part of the family of possibly unbounded nonlinear dissipative operators {A(t):t}. We answer questions concerning the perturbations, i.e., of when the following identity holds:

[A()+B()]a(t)=[A()]a(t)+[B()]a(t).

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 f:𝕁×XX to obtain Eberlein weak almost periodicity in Banach spaces. For special cases where the family A(t)A is time-independent, or the family A() itself is periodic, cf. [21, 22]. We also refer to [2], where the result has been proved for the special case of classical almost periodic functions on the real line. In [29] the existence of an EWAP solution on the whole line is proved in finite dimensions for a semilinear system of the form

y=Ay+f(t,y),

where A denotes a special linear operator and f:×nn 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 f:𝕁×XX. 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 fCb(𝕁,X), 𝕁{,+:=[0,),[a,)}, is said to be weakly almost periodic in the sense of Eberlein (EWAP) if the orbit of f with respect to 𝕁, namely,

O𝕁(f):={fr:={tf(t+r)}:r𝕁},

is relatively compact with respect to the weak topology of the sup-normed Banach space (Cb(𝕁,X)) (cf. [9, 22, 21, 24, 27]). The space of all such functions will be denoted by W(𝕁,X). Moreover, W0(𝕁,X) denotes the closed subspace of all fW(𝕁,X) such that some sequence {fsn}n of translates of f is weakly convergent to the zero function (“weakly” referring to the weak topology of (Cb(𝕁,X))).

Results of DeLeeuw and Glicksberg [6, 5] imply the following decomposition:

W(𝕁,X)=AP(,X)|𝕁W0(𝕁,X).

Here, AP(,X) denotes the space of almost periodic functions, i.e.,

AP(,X):={fCb(,X):O(f) is relatively compact in (Cb(,X),)}.

An exposition of this result is given in the book of Krengel [17]. For technical reasons we introduce the following spaces on the real line:

W+(,X):={fBUC(,X):f|+W(+,X)}

and

W0+(,X):={fW+(,X):f|+W0(+,X)}.

The decomposition of W(𝕁,X) and the uniqueness of the almost periodic part gives

W+(,X)=AP(,X)W0+(,X),

and by the uniform continuity of Eberlein weakly almost periodic functions, we obtain that for a given a and fW+(,X), we have f|[a,)W([a,),X). Additionally, some examples are given in order to point to problems in this class of functions.

2 Preliminaries on integral solutions

In [23] the following two types of equations have been discussed. The initial value problem

u(t)A(t)u(t)+ωu(t),t+,u(0)=u0,(2.1)

and the whole line equation on ,

v(t)A(t)v(t)+ωv(t),t,(2.2)

with ω. For these equations we define integral solutions. First, some prerequisites.

Let X be a general Banach space. As the given ω plays a crucial role, let 𝕁{+,}, and throughout the paper assume that 0<λ,μ<1|ω|. To obtain the solutions of equations (2.1) and (2.2), we follow the approach given in [23]. The assumptions for the family {A(t):t𝕁} are as follows.

Assumption 2.1.

The set {A(t):t𝕁} is a family of m-dissipative operators.

In comparison to the assumption given in [15], the uniform continuity on 𝕁 for h is added.

Assumption 2.2.

There exist hBUC(𝕁,X) and L:++, continuous and monotone non-decreasing, such that for λ>0 and t1,t2𝕀, we have

x1-x2x1-x2-λ(y1-y2)+λh(t1)-h(t2)L(x2)

for all [xi,yi]A(ti), i=1,2.

Recalling the study [23], the next assumption is stronger than that in [15] due to the Lipschitz continuity and linearized stability in y2. This becomes important to obtain uniform convergence on the half line depending on the Lipschitz constant on forthcoming g and ω.

Assumption 2.3.

There exist bounded and Lipschitz continuous functions g,h:𝕀X, and a continuous and monotone non-decreasing function L:++ such that for λ>0, and t1,t2𝕀, we have

x1-x2x1-x2-λ(y1-y2)+λh(t1)-h(t2)L(x2)+λg(t1)-g(t2)y2

for all [xi,yi]A(ti), i=1,2.

Remark 2.4.

Due to Assumptions 2.2 and 2.3, from [23, Definition 2.6, Remark 2.8], we read,

D(A(t))D^D(A(t))¯=DA¯.

Due to the fact that we consider (A(t)+ωI) as a perturbation of A(t) by ωI, we have to define the perturbed control functions hω and Lω. We define

hω:𝕀(X×X,1),t(h(t),|ω|g(t)),

and Lω(t)=L(t)+t.

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.

Definition 2.5.

Let 𝕀=+ and assume that either Assumption 2.2 or Assumption 2.3 is satisfied for the family {A(t)+ωI:t𝕀}. Let also 0a<b. A continuous function u:[a,b]X is called an integral solution of (2.1) if u(0)=u0 and

u(t)-x-u(r)-xrt([y,u(ν)-x]++ωu(ν)-x)𝑑ν+Lω(x)rthω(ν)-hω(r)𝑑ν+yrtg(ν)-g(r)𝑑ν

for all artb, and [x,y]A(r)+ωI (with g0 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

(ddt)λuλ(t)A(t)uλ(t)+ωuλ(t),t+,u(0)=u0,(2.3)

with the Yosida approximation

(ddt)λu(s):=1λ(u(s)-u0-1λ0se-τλ(u(s-τ)-u0)𝑑τ).

Thus, we obtain the solution uλ to (2.3) as fixpoint (cf. [23, Lemma 3.1]), i.e.,

uλ(t)=Jλω(t)(e-tλu0+1λ0te-τλuλ(t-τ)𝑑τ),

with (see [23, Remark 2.7])

Jλω(t)x:=Jλ1-λω(t)(11-λωx)

The results for these approximations on + are the following.

Theorem 2.6 ([23]).

Let I=R+, and let A(t) fulfill Assumptions 2.1 and 2.2 with ω<0, or Assumption 2.3 with Lg<-ω, where Lg is the Lipschitz constant of g. Further, let u0D¯A, and let uλ be the corresponding Yosida approximations to equation (2.3). Then uλ converges uniformly on R+ to the integral solution, as λ0+.

Further, we define an integral solution for the whole line problem (2.2).

Definition 2.7.

Let 𝕀=, and assume that either Assumption 2.2 or Assumption 2.3 is satisfied for the family {A(t)+ωI:t𝕀}. A continuous function u:X is called an integral solution on if

u(t)-x-u(r)-xrt([y,u(ν)-x]++ωu(ν)-x)𝑑ν+Lω(x)rthω(ν)-hω(r)𝑑ν+yrtg(ν)-g(r)𝑑ν

for all -<rt<, and [x,y]A(r)+ω (with g0 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:

(ddt)λuλ(t)A(t)uλ(t)+ωuλ(t),t,

with

(ddt)λu(t):=1λ(u(t)-1λ0exp(-sλ)u(t-s)𝑑s).

Again the uλ are derived by a Banach iteration, i.e.,

uλ(t)=Jλ1-λω(t)(11-λω(01λe-sλuλ(t-s)𝑑s)).

For these approximants we have the following result.

Theorem 2.8 ([23]).

Let I=R.

  • (i)

    If Assumption 2.1 , and Assumption 2.2 with ω<0 , are fulfilled, then the Yosida approximants (uλ:λ>0) are Cauchy in BUC(,X) when λ0 . The limit u(t):=limλ0uλ(t) is an integral solution on .

  • (ii)

    Let Assumptions 2.1 and 2.3 be fulfilled. Further, assume that the Lipschitz constant Lg of g in Assumption 2.3 is less than -ω . Then the Yosida approximants (uλ:λ>0) are Cauchy in BUC(,X) when λ0+ . The limit u(t):=limλ0+uλ(t) is an integral solution on .

Remark 2.9 ([22]).

  • (i)

    The construction of the solutions implies u(t)D(A(t))¯=DA¯.

  • (ii)

    In the following, with regard to equations (2.1) and (2.2), we always consider the solutions given by Theorem 2.6 and Theorem 2.8, respectively.

Moreover, we have a comparison result between the solution found on and +.

Corollary 2.10 ([23]).

Let A(t) fulfill Assumption 2.1 and either Assumption 2.2 with ω<0, or Assumption 2.3 with Lg<-ω, where Lg is the Lipschitz constant of g. Then the solution v of (2.2) and the solution u of (2.1) satisfy

u(t)-v(t)exp(ωt)u0-v(0)for all 0t.

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,

{Jλ:D(Jλ)XXλ>0},

are called pseudo-resolvents if

R(μλI+λ-μλJλ)D(Jμ),λ,μ>0,

and

Jλu=Jμ(μλu+λ-μλJλu),λ,μ>0,uD(Jλ).

Lemma 3.2 ([25, Lemma 7.1, pp. 193–194]).

Pseudo-resolvents have the generator

A=1λ(I-Jλ-1)for any λ>0.

Next we motivate a generalization for various types of almost periodicity.

Lemma 3.3.

Let Y be a closed linear subspace of BUC(R,X), and {B(t):tR} a family of m-dissipative and Lipschitz continuous operators defined on X with the common Lipschitz constant K. Further, if for all fY, {tB(t)f(t)}Y, then for all fY, {tJλB(t)f(t)}Y, for small λ>0.

Proof.

For given λ>0 and xY, we define

Tλ:YY,f{tλB(t)(f(t)+x(t))}.

Then, for given f,gY, we have

Tλf-TλgλKf-g.

Consequently, for λK<1, there exists fλY such that

fλ={tλB(t)(fλ(t)+x(t))}

and

(I-λB(t))(x(t)+fλ(t))=x(t)+fλ(t)-λB(t)(x(t)+fλ(t))=x(t).

Thus, we found that for λK<1, (I-λB(t))-1x(t)=x(t)+fλ(t)Y. ∎

Due to the previous results it becomes straightforward to define types of almost periodicity even for multivalued ω-dissipative operators.

Definition 3.4.

Let Y be a closed linear translation-invariant subspace of BUC(,X). An m-dissipative family {A(t):t}X×X is called Y invariant if {tJλ(t)x(t)}Y for all xY and 0<λλ0, and λω<1.

Remark 3.5.

The examples for Y are AP(,X), AAP(+,X), W(,X), WRC(,X) and CAA(,X). In the case of almost automorphy, we have to add uniform continuity (i.e., AA(,X)BUC(,X)). The spaces AAP(+,X), W(+,X) and WRC(+,X) are considered in the following way:

Y={fBUC(,X):f|+AAP(+,X),(WRC(+,X),W(+,X))}.

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 A(t) by

A~:={A(t),t0,A(0),t<0.

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 A(). Therefore, let Y split into the direct sum

Y=YaY0,

with a projection Pa:YYa satisfying Pa1.

Definition 3.6.

A linear, closed and translation invariant subspace YBUC(,X) splits almost periodic if Y=YaY0, where Ya,Y0 are closed translation invariant linear subspaces, the constants are in Ya (XYa), the corresponding projection fulfills

Pa(Y)Ya,with Pa1,

and, finally, Pa commutes with the translation semigroup.

Lemma 3.7.

Let Y split almost periodic, and let {A(t):tR}X×X be m-dissipative and Y-invariant so that

Jλ(t)x=Jλ,a(t)x+ϕ,

with Jλ,a()xYa and ϕY0. If for a given fY,

Pa(Jλ()f())=Jλ,a()Paf(),(3.1)

then the members of the following family of operators are pseudo-resolvents:

Jλ,a(t):XX,xPa(Jλ()x)(t)

If {A(t):tR} is Y-invariant and (3.1) is fulfilled, we say that {A(t):tR} splits almost periodic with respect to Ya, Y0.

Proof.

First we prove that {Jλ,a(t):t} are contractions:

Jλ,a(t)x-Jλ,a(t)y=Pa(Jλ()x)(t)-Pa(Jλ()y)(t)Jλ()x-Jλ()yx-y.(3.2)

Next the range condition, but this is fulfilled, because we assumed A(t) to be m-dissipative, therefore X=D(Jλ(t))=D(Jλ,a(t)).

As Jλ(t) are resolvents, we have the following resolvent equation:

Jλ(t)u=Jμ(t)(μλu+λ-μλJλ(t)u),λ,μ>0,uX.

Using, u(t)uYa, the pseudo-resolvent-equation holds in Y. Hence, we can apply the projection Pa on both sides of the equation. Thus,

PaJλ()u=Pa(Jμ()(μλu()+λ-μλJλ()u)),λ,μ>0,uX.

Applying (3.1), and evaluating at t, serves for the proof. ∎

Definition 3.8.

Let {A(t):t}X×X be m-dissipative and split almost periodic with respect to Y, Ya. The almost periodic part of A() is defined as follows:

[A()]a:=1λ(I-Jλ,a()-1),λ>0.

Remark 3.9.

By the previous definition, for m-dissipative operators {A(t):t}X×X which split almost periodic, we obtain

JλAa(t)x=Jλ,a(t)x:=(Jλ()x)a(t),

where ()a denotes the Ya part of a function given in Y.

Proof.

Using the fact that A=1λ(I-(JλA)-1) for A dissipative, we have

A=BJλA=JλBfor some λ>0,

and

1λ(I-JλAa()-1)=[A()]a=1λ(I-Jλ,aA()-1).

The first equality comes with the dissipativeness of [A()]a due to (3.2), and the second by definition. ∎

Remark 3.10.

Let {A(t):t}X×X split almost periodic with respect to Ya,Y0. Due to (3.2), [A()]a is dissipative. From D(Jλ,a)=X, we derive the m-dissipativeness, and the facts that Pax=x for all xYa, and, together with equation (3.1), that the almost periodic part of A() is Ya-invariant.

Proof.

Given xYa, we apply (3.1):

Jλ,a()x()=Jλ,a()Pax()=PaJλ()x()Ya.

Thus, Jλ,a()x()Ya, and therefore Jλ,a()x()=PaJλ,a()x(). ∎

For an example, we recall the definition of demi-closedness [15, Definition 1.15, p. 18]

Definition 3.11.

An operator AX×X is called demi-closed if A is norm-(weakly-sequentially) closed as a subset of X×X, i.e., xn-x0, w-limnyn=y, and [xn,yn]A for all n implies [x,y]A.

Example 3.12.

Let a: be weakly almost periodic and assume that a(t)q for some q>0. Using the splitting a(t)=aa(t)+ϕ(t), with aaAP(), for an m-dissipative operator AX×X, we define

A(t):=a(t)A.

Let Ya=AP(,X), and either Y=WRC(+,X) or Y=W(+,X). Let also A be demiclosed and Jλ compact. Then A(t) splits almost periodic, and the Ya-part is given by [A()]a(t)=aa(t)A. The same result can be obtained using Y=AAP(+,X), with aAAP(+,X).

Proof.

For a given λ>0, we have to prove that {tJλ(t)x} is weakly almost periodic. Note, that Jλ(t)x=Jλa(t)x. Thus, we obtain the result using the following help function with 0<r<q:

f:×X,(t,p)Jλ(|p-r|+r)x

which is Lipschitz continuous, by the resolvent equation (λ(|p-r|+r)λr>δ>0), f(t,a(t))=Jλa(t)x, and since fW(×,X) (cf. Definition A.4), Corollary A.10 applies. To prove {tJλ(t)x(t)}Y for xY in the case where Y=WRC(+,X), we can use Corollary A.10 with the help function

f:×(×X)X,(t,(p1,p2))Jλ(|p1-r|+r)p2,

and f(t,a(t),x(t))=Jλa(t)x(t)=Jλ(t)x(t). Hence, for xY=W(+,X), by a further use of the previously defined help function and an application of Theorem A.8, it remains to prove that, for K1X weakly compact metrizable, the following function is continuous:

ι:[a,b]×(K1,w)(X,),(s,x)Jλ(|s-r|+r)x.

For a given {xn}nK1, weakly convergent to xK1, and {sn}n[a,b] convergent to s[r,R], we define λn:=λ(|sn-r|+r)μ[λr,R] and claim that

JλnxnJμxwhen n.

To this end, we show that any subsequence of {Jλnxn}n has a subsequence convergent to Jμx. Let such a subsequence be chosen. Then, without loss of generality, we have un:=Jμxnu. This is equivalent to xnun-μAun, and the demiclosedness leads to JμxnJμx. Using the fact that AλxAμx for all 0<μ<λ, by the resolvent equation, we have

Jλnxn-JμxJλnxn-Jμxn+Jμxn-Jμx|λn-μ|Aλrxn+Jμxn-Jμx.

The Lipschitz continuity of Aλr proves the boundedness of Aλrxn, which finishes the proof.

To compute the almost periodic part of {tJλa(t)x}, 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 tB(t)xY=YaY0 (i.e., it splits canonically), and the splitting is obtained from coinciding resolvents.

Proposition 3.13.

Let YBUC(R,X) split almost periodic, and let {B(t):tR} be a family of uniformly Lipschitz operators defined on X, with Lipschitz constant K. Further, let B()xY for all xX. Due to the splitting, we have B(t)x=Pa(B()x)(t)+ϕ(t). We define

Ba(t)x:=Pa(B()x)(t),

and assume that

Pa(B()y())(t)=Ba(t)Pa(y())(t)for all yY.(3.3)

Then, for Ba()x, the almost periodic part in the sense of Definition 3.8, we have

Ba()x=Ba()xfor all xX.

Proof.

To prove the equality we show that the resolvents coincide, i.e.,

(I-λBa(t))-1x=Pa(I-λB(t))-1=Jλ,aB(t)x.(3.4)

The above together with Remark 3.9 will finish the proof. To prove (3.4) we need a representation of the resolvents. In doing this, we consider

Tλa:YaYa,f{tλBa(t)(f(t)+x)}.

For given x,yX, we have

Ba(t)x-Ba(t)y=Pa(B()x)(t)-Pa(B()y)(t)B()x-B()y)Kx-y.

The Lipschitz continuity of Ba leads to

Tλaf-TλagλKf-gfor f,gYa.

Consequently, for λK<1, we find a net of fixpoints fλa satisfying

λBa(t)(fλa(t)+x)=fλa(t),(I-λBa(t))(fλa(t)+x)=fλa(t)+x-fλa(t)=x,fλa(t)+x=(I-λBa(t))-1x.

Next we consider the resolvent of Ba 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:

Tλ:YY,f{tλB(t)(f(t)+x)}.

Using the fact that B is Lipschitz for λK<1, we find a net of fixpoints fλ satisfying

λB(t)(fλ(t)+x)=fλ(t),(I-λB(t))(fλ(t)+x)=fλ(t)+x-fλ(t)=x,fλ(t)+x=(I-λB(t))-1x.

Using the representation, we compute the almost periodic part and apply assumption (3.3), to obtain

Pa(fλ)+x=Pa(fλ+x)=Pa(λB()(fλ+x)+x=λBa()(Pafλ+x)+x.

We found that Pafλ is a fixpoint of Tλa, hence fλa=Pa(fλ). This leads to

(I-λBa(t))-1x=fλa(t)+x=Pa(fλ)(t)+x=Pa(I-λB())-1x=JλB(t)x,

which finishes the proof. ∎

Kenmochi and Otani [16] considered nonlinear evolution equations governed by time dependent subdifferential operators in Hilbert space H:

u(t)-φt(u(t)),u(0)=u0.

They constructed a not necessarily complete metric space of convex functions (Φ,d), whereby, d(φtn,φ)0, implies φtnφ, in the sense of Mosco, cf. [16, Lemma 4.1, p. 75]. For short, we write Φφtnφ. Next we show that the notion of almost periodicity on subdifferentials given in [16] is stronger than the one given via resolvents defined in this study, when viewing -φ(t) as a dissipative operator.

Proposition 3.14.

Let φt:HR be proper, lower semicontinuous and convex for all tR. If

φ():(Φ,d),tφt,

is almost periodic, then

Jλφ()x:H,tJλφ(t)x:=(I+λφt)-1x,

is almost periodic for all xH.

Proof.

We assume that for xH, the mapping {tJλφ(t)x} is not almost periodic. Consequently, we find a sequence {tn}n such that {Jλφ(t+tn)x}n is not uniformly Cauchy. Without loss of generality,

Φφ(t+tn)ψtuniformly for t.

The non-Cauchy assumption on {Jλφ(t+tn)x}n leads to subsequences {tnk}k,{tmk}k{tn}n, and a sequence {sn}n such that

Jλφ(sk+tnk)x-Jλφ(sk+tmk)xεfor all k.

Note that {tφt} is assumed to be almost periodic, O(φ):={{tφt+s}:s}¯ is a compact subset of Cb(,(Φ,d)), and ψ(+s)O(φ) for all s. Consequently, we may assume that

Φψ(t+sk)ϕ(t)uniformly for t.

Clearly,

Φφ(t+sk+tnk)ϕtandΦφ(t+sk+tmk)ϕt

uniformly for t. Applying [16, Lemma 4.1, p. 75], we obtain

φ(0+sk+tnk)ϕ0andφ(0+sk+tmk)ϕ0,

in the sense of Mosco, which implies, by using [1, Theorem 3.26, p. 305],

Jλφ(sk+tnk)xJλϕxandJλφ(sk+tmk)xJλϕx

when k. 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:

dudt(t)A(t)u(t)+ωu(t),t,(4.1)

and the corresponding initial value problem

dudt(t)A(t)u(t)+ωu(t),t+,u(0)=u0,(4.2)

where A(t) is a possibly nonlinear multivalued and dissipative operator satisfying a type of almost periodicity defined above. Throughout this section, we assume Y,Ya, and Y0 to be closed and translation invariant subspaces of BUC(,X), with XYa. In the case where A(t) is classically almost periodic and Y=AP(,X), the result in case of Assumption 2.2 is due to [2]. 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 (ω<0) is needed to obtain the uniform convergence of the approximants. Moreover, when ω=0 and A(t) only m-dissipative, there exists a counterexample for classical almost periodicity in the case of dimension two, see [13, Remark 1.3 (2)].

Definition 4.1.

A solution u of (4.2) is called asymptotically Y if there exists vY such that

limtu(t)-v(t)=0.

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.

Theorem 4.2.

Let, for a given YBUC(R,X), A() be Y-invariant, and {A(t):tR} fulfill Assumption 2.1 and either Assumption 2.2 with ω<0, or Assumption 2.3 with Lg<-ω, where Lg denotes the Lipschitz constant for the control function g. Then there exists a solution uY to (4.1), and all integral solutions of (4.2) are asymptotically Y.

Let Y and A() split almost periodic, with Y=YaY0. Then the almost periodic part ua of the solution uY fulfills uaYa, and ua is a generalized solution to the evolution equation

duadt(t)[A()]a(t)ua(t)+ωua(t),t.(4.3)

Proof.

The first part is a direct consequence of [22, Theorem 4.10]. For the second part, note that

uλ(t)=Jλ1-λω(t)(11-λω(01λe-sλu(t-s)𝑑s)),

and due to the assumption of an almost periodic splitting, we apply Pa and obtain

uλa=Jλ1-λωa(t)(11-λω(01λe-sλuλa(t-s)𝑑s)).(4.4)

Thus,

(ddt)λuλa(t)[A()]a(t)uλa(t)+ωuλa(t),t.(4.5)

Applying Theorem 2.8, uλu uniformly on , and by the continuity of the projection Pa, we have uλaua. We call ua the generalized solution to (4.3). ∎

Remark 4.3.

  • (i)

    The notion of an integral solution in the case of a general splitting is not possible due to the missing control functions for {[A()]a(t):t}. Thus, in general we cannot construct a solution applying existence results of [23] to the equation

    dudt(t)[A()]a(t)u(t)+ωu(t),

    due to lack of Assumptions 2.2 and 2.3.

  • (ii)

    If {[A()]a(t):t} fulfills Assumption 2.2 with a function haBUC(,X), La non-decreasing and ωa<0, then ua is an integral solution, in the sense of Definition 2.7, with respect to haω,La.

  • (iii)

    If {[A()]a(t):t} fulfills Assumption 2.3 with ha,ga:X bounded and Lipschitz, La non-decreasing and Lga<-ωa, then the generalized solution is an integral solution, in the sense of Definition 2.7, with respect to the control functions haω,ga,La.

Proof.

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. ∎

Proposition 4.4.

In the context of Theorem 4.2, let Y=WRC(R,X), Ya=AP(R,X) and Y0=WRC0(R,X). If the control functions h,g belong to WRC(R,X), then the generalized-(almost periodic)-solution is an integral solution with respect to Lω, and the almost periodic parts of hω,g.

Proof.

Clearly, hωWRC(,X×X). From Corollary A.14, we find a sequence {sn}n, with sn, such that, due to the compact ranges,

hω(+sn)haω,g(+sn)gaandu(+sn)ua,

pointwise in the norm of X. As the solution on is an integral solution (see Definition 2.7), we have

u(t)-x-u(r)-xrt([y,u(ν)-x]++ωu(ν)-x)𝑑ν+Lω(x)rthω(ν)-hω(r)𝑑ν+yrtg(ν)-g(r)𝑑ν.

Therefore, for t:=t+sn, r:=r+sn, we have

u(t+sn)-x-u(r+sn)-xrt([y,u(ν+sn)-x]++ωu(ν+sn)-x)𝑑ν   +Lω(x)rthω(ν+sn)-hω(r+sn)𝑑ν+yrtg(ν+sn)-g(r+sn)𝑑ν.(4.6)

Using the fact that [y,]+ is upper semicontinuous, we have

lim supn[y,u(ν+sn)-x]+[y,ua(ν)-x]+.

Consequently, we may pass to the limit superior on both sides of (4.6), and by Fatou’s lemma we obtain the inequality. ∎

Corollary 4.5.

In the context of Theorem 4.2, let

Y=AAP+(,X):={fBUC(,X):f|+AAP(+,X)},Ya=AP(,X)𝑎𝑛𝑑Y0=C0+(,X):={fBUC(,X):f|+C0(+,X)}.

If the control functions h, g belong to AAP+(R,X), then the generalized-(almost periodic)-solution is an integral solution with respect to Lω, and the almost periodic parts of hω, g.

Proof.

Apply inequality (4.6), the existence of a sequence {sn}n with sn, and the pointwise-norm convergence. ∎

Theorem 4.6.

Let {A(t):tR},{B(t):tR}X×X split almost periodic to Y=YaY0. Consider the equations

dudt(t)A(t)u(t)+ωu(t),t+,u(0)=u0DA¯,

and

dvdt(t)B(t)v(t)+ωv(t),t+,v(0)=v0DB¯.

If the families {A(t):tR}, {B(t):tR} satisfy Assumption 2.1 and either Assumption 2.2 with ω<0, or Assumption 2.3 with Lg<-ω, where Lg denotes the Lipschitz constant for the control function g, then

{tJλA(t)x-JλB(t)x}Y0for all xX,(4.7)

implies that

{tUA+ωI(t,s)x-UB+ωI(t,s)y}

is asymptotically Y0 for all xDA¯, yDB¯.

Proof.

It is sufficient to prove that the almost periodic parts of UA+ωI(,s)x and UB+ωI(,s)y 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 uλ,vλ be the approximants to the whole line solution given by [23, Theorem 2.18]. Due to condition (4.7),

(JλA())ax=(JλB())ax.

Thus, using the fixpoint equations for the approximants uλ and vλ, we have

(uλ(t))a=(Jλ1-λωA)a(t)(11-λω(01λe-sλuλa(t-s)𝑑s))=(Jλ1-λωB)a(t)(11-λω(01λe-sλuλa(t-s)𝑑s)),

Thus, (uλ(t))a is the fixpoint of the strict contraction defining the solution (vλ(t))a. Consequently, by the uniqueness of the fixpoint, (uλ(t))a=(vλ(t))a. 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 ([22]).

Let

Y=AAP+(,X):={fBUC(,X):f|+AAP(+,X)},Ya=AP(,X),

and let {A(t):tR},{B(t):tR}X×X split almost periodic with Y=YaY0. Consider the equations

dudt(t)A(t)u(t)+ωu(t),t+,u(0)=u0DA¯,

and

dvdt(t)B(t)v(t)+ωv(t),t+,u(0)=v0DB¯.

Let also the families {A(t):tR}, {B(t):tR} satisfy Assumption 2.1 and either Assumption 2.2 with ω<0, or Assumption 2.3 with Lg<-ω, where Lg denotes the Lipschitz constant for the control function g. If

limtJλA(t)x-JλB(t)x=0for all xX,λ small,

then, for the corresponding evolution systems UA+ωI(t,s) and UB+ωI(t,s), we have

limtUA+ωI(t,0)x-UB+ωI(t,0)y=0for all xDA¯,yDB¯,

i.e., they are asymptotically equivalent.

In the case of Eberlein weak almost periodicity, we obtain the following corollary.

Corollary 4.8.

Let Y=W(R,X), Ya=AP(R,X), Y0=W0(R,X), and let {A(t):tR},{B(t):tR}X×X split almost periodic with Y=YaY0. Consider the equations

dudt(t)A(t)u(t)+ωu(t),t+,u(0)=u0DA¯,

and

dvdt(t)B(t)v(t)+ωv(t),t+,u(0)=v0DB¯.

Let also the families {A(t):tR}, {B(t):tR} satisfy Assumption 2.1 and either Assumption 2.2 with ω<0, or Assumption 2.3 with Lg<-ω, where Lg denotes the Lipschitz constant for the control function g. If for all αR,

limT1T0Texp(-iατ)(JλA(t+τ)x-JλB(t+τ)x)𝑑τ=0

for all λ small, xX, uniformly in tR+, then

limT1T0Texp(-iατ)(UA+ωI(t+τ,0)x-UB+ωI(t+τ,0)y)𝑑τ=0

for all xDA¯,yDB¯, and αR, uniformly in tR+.

Proof.

From [6, 5] we learn that Y=YaY0, and that Y0 is characterized by a zero mean. ∎

In the case of ordinary differential equations the result of Theorem 4.6 extends as follows.

Proposition 4.9.

Let YBUC(R,X), let f,g:R×XX be Lipschitz, and let

{tf(t,x(t)},{tg(t,x(t)}Yfor all xY.

Further, let

fa(,z)=Paf(,z),ga(,z)=Pag(,z)for all zX,

and

Paf(,x())=fa(,Pax()),Pag(,x())=ga(,Pax())for all xY.

Then

{tf(t,x)-g(t,x)}Y0for all xX,(4.8)

if and only if

{tJλf(t)x-Jλg(t)x}Y0for all xX.(4.9)

Proof.

Due (4.8), the almost periodic parts of f,g coincide. Therefore, with respect to the proof of Proposition 3.13, the resolvents come with a common fixpoint of Tλ, which proves (4.9). If (4.9) holds we have

Jλ,af=Jλ,ag,

which gives fa=ga. An application of Proposition 3.13 concludes the proof. ∎

5 Perturbations

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 A(t)+f(t,), as they are considered in [3], and in [31] for the case of almost automorphy. We start with a Lipschitz perturbation.

Theorem 5.1.

Let {A(t):tR} split almost periodic with respect to Y,Ya. Further, let {B(t):tR} be uniformly Lipschitz with constant K, satisfying Pa(B()f=Ba()Paf for all fY, where Ba()x:=PaB()x for all xX. Then A(t)+B(t) is ω-dissipative with at least ω=K, and splits almost periodic with respect to Ya,Y0. Moreover,

{A()+B()}a=Aa()+Ba().(5.1)

Proof.

We start with proving the ω-dissipativeness. Let t and yiA(t)xi for i=1,2. Then

x1-x2xy-x2-λ(y1-y2)x1-x2-λ(y1+B(t)x1-y2-B(t)x2)+λKx1-x2.

Thus, A(t)+B(t) is dissipative with ω=K. From the uniform Lipschitz condition of {B(t):t}, we obtain the uniform Lipschitz condition of {Ba(t):t}. For given x,yX, we have

Ba(t)x-Ba(t)yBa()x-Ba()yB()x-B()yKx-y.

Consequently, we obtain that Aa(t)+Ba(t) is ω-dissipative with ω=K.

To prove the m-dissipativeness and to obtain a representation of the resolvent of A()+B(), for given ϕY, we define

Tλ:YY,f{tJλA(t){ϕ(t)+λB(t)(f(t)+ϕ(t))}-ϕ(t)},

with JλA(t)=(I-λA(t))-1.

An estimation gives that Tλ is a strict contraction for λK<1. Thus, the Banach fixpoint principle leads to a fλY such that

fλ(t)=JλA(t){ϕ(t)+λB(t)(fλ(t)+ϕ(t))}-ϕ(t).

We have

ϕ(t)+fλ(t)=JλA(t){ϕ(t)+λB(t)(fλ(t)+ϕ(t))},(I-λA(t))(ϕ(t)+fλ(t))ϕ(t)+λB(t)(ϕ(t)+fλ(t)),(I-λA(t)-λB(t))(ϕ(t)+fλ(t))ϕ(t),(ϕ(t)+fλ(t))=(I-λA(t)-λB(t))-1ϕ(t).

As the constants are contained in YaY, the m-dissipativeness of A(t)+B(t) is proved. We also proved (I-λ(A(t)+B(t)))-1ϕ(t)=ϕ(t)+fλ(t)Y. Therefore, it remains to prove the perturbation result (5.1), which is equivalent to

Jλ,aA+B(t)x=JλAa+Ba(t)x.

From the previous step, we have (I-λ(A(t)+B(t))-1x=x+fλ(t) for ϕx. This leads to

Pa(JλA+B()x)(t)=Pa(x+fλ)(t)=x+Pa(fλ)(t)=x+Pa(JλA(t)(x+λB(t)(fλ(t)+x))-x)=x+Jλ,aA(t)(x+λBa(t)(fλa(t)+x))-x.

Noting that JλAa(t)x=Jλ,aA(t)x, we define

Tλa:YaYa,f{tJλ,aA(t)(x+λBa(t)(f(t)+x))-x}.

Let fλa be the net of fixpoints for λK<1. Then we have

(I-λ(Aa(t)+Ba(t)))-1x=x+fλa=Jλ,aA(t)(x+λBa(t)(fλa(t)+x))=Pa(JλA+B()x)(t).

As a consequence, we obtain the same contraction mapping, and the uniqueness of the fixpoint concludes the proof. ∎

Remark 5.2.

In many studies the existence of solutions to equations like

y(t)A(t)y(t)+B(t)y(t),t,(5.2)

need the precondition L+ωA<0, 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., R(I+λ(A(t)+B(t))=X) and the dissipativity constant ωA+B of {A(t)+B(t)+ωI:t}X×X. 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 ωA+B<0 in case of Assumption 2.2, or Lg<ωA+B in case of Assumption 2.3, is sufficient. Consequently, the direct connection between the Lipschitz constant on xB(t)x and ωA to obtain a bounded solution on is cut.

If X* is strictly convex, then the duality mapping

F:X{0}X,x{y : y(x)=x2=y2},

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 X* is uniformly convex.

Lemma 5.3.

Let X* be uniformly convex, let IR, and let the families {A(t):tI},{B(t):tI}X×X be m-dissipative. If there exist R>0 and x1(t):IX such that for all tI,

x1(t)D(A(t))D(B(t))K(0,R)𝑎𝑛𝑑|A(t)x1(t)|,|B(t)x1(t)|R,(5.3)

then, for {y(t)}tIX bounded, there exist C>0 and a unique xλ(t)D(A(t)), xλ(t)C, such that

xλ(t)-A(t)xλ(t)-Bλ(t)xλ(t)y(t).(5.4)

Further, if for C10, Bλ(t)xλ(t)C1 uniformly in λ>0, tI, then, there exists a function x:IX such that x(t)D(A(t))D(B(t))K(0,C) and

x(t)-A(t)x(t)-B(t)x(t)y(t)for all t𝕀.(5.5)

Additionally,

limλ0xλ(t)=x(t)uniformly in 𝕀.

Proof.

As A(t)+Bλ(t) is m-dissipative, the solution xλ(t) of (5.4) is unique for every t. As we need a representation for xλ(t), from the proof of [15, Lemma 1.23, Theorem 1.24], we read

xλ(t)=(I-λλ+1A(t))-1(λλ+1y(t)+1λ+1(I-λB(t))-1xλ(t)).

Using Banach’s fixpoint theorem, we will find xλ(t)D(A(t)). From the precondition (5.3), we find uniformly bounded pairs (x1(t),y1,λ(t)) such that

y1,λ(t)x1(t)-A(t)x1(t)-Bλ(t)x1(t).

Using Bλ(t)x1(t)|B(t)x1(t)|, we can choose y1,λ so that

xλ(t)-x1(t)2y(t)-y1,λ(t),F(xλ(t)-x1(t))y(t)-y1,λ(t)xλ(t)-x1(t).(5.6)

Thus, xλ(t) is uniformly bounded. Using (5.4) for λ,μ>0, and the fact that A(t) is dissipative, we have

xλ(t)-xμ(t)2Bλ(t)xλ(t)-Bμ(t)xμ(t),F(xλ(t)-xμ(t)).

Using Bλ(t)xλ(t)B(t)JλB(t)xλ(t) and the dissipativeness of B, we have

Bλ(t)xλ(t)-Bμ(t)xμ(t),F(JλB(t)xλ(t)-JμB(t)xμ(t))0.

In sum this leads with the boundedness of {Bλ(t)xλ(t)}t𝕀,λ>0 to some C1>0, so that

xλ(t)-xμ(t)2Bλ(t)xλ(t)-Bμ(t)xμ(t),F(xλ(t)-xμ(t))-F(JλB(t)xλ(t)-JμB(t)xμ(t))2C1F(xλ(t)-xμ(t))-F(JλB(t)xλ(t)-JμB(t)xμ(t)).(5.7)

Recalling that λBλ(t)xλ(t)=JλB(t)xλ(t)-xλ(t), the uniform continuity on bounded sets of the duality map yields that {xλ}λ>0 is uniformly Cauchy when λ0. The completeness of X gives uniform convergence in B(𝕀,X), cf. [8, p. 258]. We define

x(t)=limλ0xλ(t).

For a given t𝕀, 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 Bλ(s)xλ(s)C1 for all s𝕀, λ>0, we have x(t)D(B(t)). The reflexivity gives Bλn(t)xλn(t)w(t) weakly, with w(t)B(t)x(t). Applying (5.4), we find a bounded selection yλ(t)A(t)xλ(t) satisfying

xλ(t)-yλ(t)-Bλ(t)xλ(t)=y(t).

Thus, by the demiclosedness of A(t) and convergence of {yλn(t):n} we have

yλn(t)z(t)=y(t)-x(t)+w(t)

weakly, with z(t)A(t)x(t). Summarizing the previous, we have

y(t)=xλn(t)-yλn(t)-Bλn(t)xλn(t)x(t)-z(t)-w(t)

weakly in X, which concludes the proof. ∎

Theorem 5.4.

Let X* be uniformly convex, let IR, and let the families {A(t):tI},{B(t):tI}X×X be m-dissipative. Assume that there exist R>0 and x1(t):IX such that for all tI,

x1(t)D(A(t))D(B(t))K(0,R)𝑎𝑛𝑑|A(t)x1(t)|,|B(t)x1(t)|R.

Further, let y,F(Bλ(t)x)0 for all [x,y]A(t), tI, λ>0. Then A(t)+B(t) is m-dissipative and the solution of (5.4) converges uniformly on I to the solution of (5.5).

Proof.

The proof is analogous to that of [15, Corollary 1.25], proving the boundedness of Bλ(t)xλ(t). ∎

Remark 5.5.

To study the resolvents JμA+B(t)=(I+μ(A(t)+B(t)))-1, the same methods, as in Lemma 5.3, apply. Therefore, assume the condition (5.3). This selection gives a bounded pair (x1(t),y1,λ,μ(t)) satisfying

y1,λ,μ(t):=(1-μ)x1(t)+μy1,λ(t)x1(t)-μA(t)x1(t)-μBλ(t)x1(t).

Considering, yB(𝕀,X) and

y(t)xλμ(t)-μA(t)xλμ(t)-μBλ(t)xλμ(t),

we find that the solutions are fixpoints of the contractions

xλμ(t)=(I-λμλ+μA(t))-1(λλ+μy(t)+μλ+μJλB(t)xλμ(t)).

Moreover, applying the bounded selection (x1(t),y1,λ,μ(t)), we have, similar to (5.6), some C2>0 such that xλμ(t)C2 uniformly in λ>0, t𝕀. Thus, if for a given μ>0,

Bλ(t)xλμ(t)C1for all t𝕀,λ>0,(5.8)

then, by an observation similar to (5.7), we have

xμ(t):=limλ0xλμ(t)uniformly for t𝕀.

The uniform convexity of X* and the demiclosedness of A(t) and B(t) imply that xμ(t) is a bounded solution of

xμ(t)-μA(t)xμ(t)-μB(t)xμ(t)y(t),

which gives

JμA+B(t)y(t)=limλ0xλμ(t)uniformly for t𝕀.

Theorem 5.6.

Let X be uniformly convex, and let YBUC(R,X), Y=YaY0. Further, let {A(t):tR} and {B(t):tR} be families of dissipative operators, which split almost periodic with respect to Ya, Y0. Under the assumptions (5.3) and (5.8) of Remark 5.5, we have

(A()+B())a=Aa()+Ba().

Proof.

We have to show that the almost periodic part of JλA+B(t)x equals JλAa+Ba(t)x. To prove that the fixpoints are in Y, note that XYaY, and that JλA and JλB leave Y invariant. Thus, for yX, we have the strict contraction

T:YY,x{tJλμλ+μA(t)(λλ+μy+μλ+μJλB(t)x(t))}.

From the Remark 5.5, for a given μ>0, we obtain the convergence when λ0 uniformly for t. In doing so, we use the representations of the fixpoints given in Remark 5.5,

Paxλμ()=Pa(Jλμλ+μA()(λλ+μy+μλ+μJλB()xλμ()))=(Jλμλ+μA())a(λλ+μy+μλ+μ(JλB())aPaxλμ()).

As Pa:YY is bounded, we obtain the uniform convergence of Paxλμ as well. For Aa+Ba, we have the strict contraction

Ta:YaYa,x{t(Jλμλ+μA())a(t)(λλ+μy+μλ+μ(JλB())a(t)x(t))}.

Thus, the projection gives the fixpoint xλ,aμ:=Paxλμ and uniform convergence when λ0. From (5.8), we conclude

(Bλ())a(t)xλ,aμ(t)PaBλ()xλμ()Bλ()xλμ()C1uniformly in λ>0.

Hence, we are in the situation of Remark 5.5, and can redo the final steps, similar to the proof of Lemma 5.3. Consequently, for a given t, there exists a sequence {λn:n} such that

y=xλn,aμ(t)-μyλn,aμ(t)-μBλn,a(t)xλn,aμ(t)xaμ(t)-μzaμ(t)-μwaμ(t)

weakly when n, with zaμ(t)Aa(t)xμ(t) and waμ(t)Ba(t)xμ(t). This concludes the proof. ∎

In the next example we show how the previous results apply to perturbations to the linear Dirichlet problem (Δ0) on bounded domains Ω.

Example 5.7.

Let Ωn be open, bounded and with smooth boundary, let H=L2(Ω),

Δ:H01(Ω)H2(Ω)L2(Ω)L2(Ω),

with ω its dissipativity constant, and let φ:𝕁× be a proper, lower semicontinuous and convex function, satisfying φ(t)00 for all t𝕁. Defining, for all t𝕁,

ϕ(t):={[u,v]L2(Ω)×L2(Ω):u(x)D(φ(t)) a.e., φ(t)(u())L1(Ω), and v(x)φ(t)(u(x)) a.e.},

we have that Δ-ϕ(t) is m-dissipative and ϕ(t)00. If additionally, -φ and -ϕ split almost periodic, with respect to Ya,,Y0,BUC() and Ya,L2,Y0,L2BUC(,L2(Ω)), respectively, with

Pa,YL2(I+λϕ(t))-1u={xPa,Y(I+λφ(t))-1u(x)}a.e. for all uL2(Ω),(5.9)

then we have

(Δ-ϕ(t))a=Δ-ϕa(t)=Δ-(φa(t)),(5.10)

where -(φ())a and -ϕ()a are the generalized almost periodic parts. Moreover, if the Dirichlet problem with respect to Δ-φ(t)() fulfills either Assumption 2.2 (see [4, Example, pp. 88–90]), or Assumption 2.3, with Lg<-ω, then the solution of equation

u(t)=Δu(t)-ϕ(t)u(t)+f(t),t,

with fYL2, has an almost periodic splitting as well.

Proof.

From [15, Example 1.60 and Lemma 1.61, pp. 53–54], we obtain that ϕ is a subdifferential, and therefore -ϕ is m-dissipative. Further, [15, Proposition 1.63, p. 55] proves the m-dissipativeness of Δ-ϕ on L2(Ω). As f(t)Δ0+ϕ(t)(0)+f(t), 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 fY, we use the fact that the mollified fεY is Lipschitz and approximates f uniformly due to the uniform continuity of f. Applying [23, Proposition 6.1], we obtain, with fε, 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.

Example 5.8.

Let X* be uniformly convex and AX×X an m-dissipative operator with a compact resolvent. Further, let {B(t)}t𝕁X×X be m-dissipative, and let JλB()zW(𝕁,X) for all zX. For a given vX, we define

Fλ,μv:𝕁×XX,(t,x)Jλμλ+μA(λλ+μv+μλ+μJλB(t)x).

Then Fλ,μv fulfills Definition A.4 with D=(X,). If A,B(t) fulfill the assumption of Theorem 5.4, then

JλA+B()w()WRC(𝕁,X)for all wW(𝕁,X).

Proof.

Following the Definition A.4 and Theorem A.10, for given gW(𝕁,X), we have to verify that

{tJλμλ+μAg(t)}WRC(𝕁,X).

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 JλA. For given wW(𝕁,X), choosing

g(t)=λλ+μw(t)+μλ+μJλB(t)x,

it remains to prove that

ι:KWRC(𝕁,X),x(tJλμλ+μA(λλ+μw(t)+μλ+μJλB(t)x)),

is continuous, which comes with the inequality

ι(x)-ι(y)μλ+μx-y.

From Lemma 5.3 and Remark 5.5, we obtain the resolvent JμA+B()w(t) as uniform limits of fixpoints defined by WRC(𝕁,X)xλ,μ(t)=Fλ,μv(t,xλ,μ(t)), i.e.,

Jλμλ+μA(λλ+μw(t)+μλ+μJλB(t)xλ,μ(t))JμA+Bw(t)when λ0,

uniformly for t𝕁, and the proof is finished. ∎

Remark 5.9.

From Appendices B and C, we can obtain similar results for asymptotically almost periodic or continuous almost automorphic functions.

From Remark 3.9 and the previous perturbation results, we obtain the following corollary.

Corollary 5.10.

Let X be uniformly convex, and let {A(t):tR} and {C(t):tR} split almost periodic with respect to Y=YaY0, and fulfill the assumptions of Theorem 5.6. Consider the equations

dudt(t)A(t)u(t)+ωu(t),t+,u(0)=u0DA¯,

and

dvdt(t)A(t)v(t)+C(t)v(t)+ωv(t),t+,u(0)=v0DA+C¯.

If the families {A(t):tR},{A(t)+C(t):tR} satisfy either Assumption 2.2 with ω<0, or Assumption 2.3 with Lg<-ω, where Lg denotes the Lipschitz constant for the control function g, then {tCa(t)}=0 implies that {tUA+C+ω(t,s)y-UA+ω(t,s)x} is asymptotically Y0 for all xDA¯ and yDA+C¯.

Proof.

From the assumptions of Theorem 5.6 we obtain the m-dissipativeness for every operator {A(t)+C(t)}, i.e., Assumption 2.1. Next we apply Remark 3.9 and the perturbation result for A+C, which give

Pa(JλA+C()x-JλA()x)(t)=JλAa+Ca(t)x-JλAa(t)x=0.

Finally, an application of Theorem 4.6 concludes the proof. ∎

A Eberlein weak almost periodicity

This and the remaining appendices discuss a number of closed and translation invariant subspaces of BUC(,X) and (BUC(+,X)), which under certain conditions satisfy the invariance conditions for the resolvent Jλ(t). 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.

Proposition A.1.

  • (i)

    If fCb(𝕁,X) , then fW(𝕁,X) if and only if for {(tm,xm*)}m𝕁×extBX* and for all {ωn}n𝕁 , the following double limits condition holds:

    limnlimmf(ωn+tm),xm*=limmlimnf(ωn+tm),xm*,

    whenever the iterated limits exist, see [ 27 ].

  • (ii)

    If fW(𝕁,X) , then

    -limR1R0Rf(t+r)𝑑t=zX

    exists uniformly over r𝕁 . Moreover, if fW0(+,X) , then the ergodic limit z is equal to 0X (cf. [ 28 ] ).

Proposition A.2.

Let {an,m}n,mNK with K compact metrizable. Then there exist subsequences {nk}kN and {ml}lN such that the following limits exist:

limklimlank,ml,limllimkank,ml.

Proof.

Let m=1. Then we find a subsequence {n(1,k)}k such that limkan(1,k),1 exists. Thus, we may assume that for a given m, we can find {n(m,k)}k such that limkan(m,k),l exist for 1lm. For the step “mm+1”, choose an appropriate subsequence {n(m+1,k)}k{n(m,k)}k such that limkan(m+1,k),m+1 exists. After this recursive construction, we define nk:=n(k,k), and since n(k,k)<n(k,k+1)n(k+1,k+1), we have {n(k,k)}k,km{n(m,k)}k,km for all m. Computing the limits, we obtain bm:=limkank,m. Passing to an appropriate {ml}l, we are done for the first double limit. Redoing the previous steps on the found subsequence {ank,ml}k,l, for the interchanged limits we will find {mlj}j and {nki}i, 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 ([29]).

If X has finite dimension, gW(R,X) and fC(R×Rn,Rn), with the property that

ι:KW(,n),pf(,p),

is continuous for every compact KRn, then {tf(t,g(t))}W(R,Rn).

Traple’s proof does not extend to the infinite dimensional case, for he used both the algebraic structure of W() and the fact that the coordinate-wise polynomials are dense in C(K,n) for Kn 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 (D,τ) is assumed to be a Hausdorff topological space. Moreover, in this study, D is X with the weak topology or X with norm topology.

Definition A.4.

A function f:𝕁×DX is called DEWAP if

  • (i)

    f(,p)W(𝕁,X) for all pD,

  • (ii)

    for every compact and metric subset KD, the following map is continuous:

    ι:KW(𝕁,X),pf(,p).

We denote these functions by

W(𝕁×D,X):={fC(𝕁×D,X):f is DEWAP}.

With the aim to find a representation for the almost periodic part, we recall some technical results.

Proposition A.5 ([28]).

Eberlein weakly almost periodic functions are uniformly continuous.

Thus, W(,X) fulfills the assumptions made on Y in the previous sections.

Remark A.6.

Let f:𝕁×DX fulfill Definition A.4, and let KD be compact metric. Then fBUC(𝕁×K,X).

Proof.

As K is compact metric, for given ε>0, we find δ>0 such that for all x,yK with dK(x,y)<δ, we have ι(x)-ι(y)<ε. From the compactness of K we find {xi}i=1n with the following properties:

  • (i)

    supzKinf1kndK(z,xk)<δ.

  • (ii)

    There exists δ1>0 such that for |t-s|<δ1, we have sup1knf(t,xk)-f(s,xk)<ε.

  • (iii)

    From Definition A.4 (i), we obtain some K>0 with max1knf(,xk)<K.

First we prove the boundedness. For zK,

f(t,z)inf1knf(t,z)-f(t,xk)+max1knf(t,xk)inf1knι(z)-ι(xk)+max1knf(,xk)ε+K.

To prove the uniform continuity, let t,s𝕁, |t-s|<δ1 and x,yK, with d(x,y)<δ. Then, we obtain

f(t,x)-f(s,y)f(t,x)-f(s,x)+f(s,x)-f(s,y)f(t,x)-f(t,xk)+f(t,xk)-f(s,xk)+f(s,xk)-f(s,x)+ι(x)-ι(y)2supzKinf1knι(z)-ι(xk)+ι(x)-ι(y)+sup1knf(t,xk)-f(s,xk)2ε+ε+ε,

which finishes the proof. ∎

Remark A.7.

From the previous remark we conclude that if f:𝕁×DX fulfills Definition A.4, and KD is compact metric, then the following map is continuous:

S:𝕁×KW(𝕁,X),(t,x){sf(t+s,x)}.

Note that the continuity of ι does not imply a norm compact range for every f(,z) if z is given. For example, one may take f(,z)g for some g, which is Eberlein weakly almost periodic. Then ι is clearly continuous, but the range of g is not necessarily compact. If D=(X,weak), then for every gC(𝕁×D,X), lipschitzian in the second variable, with g(,x) Eberlein weakly almost periodic for all xX, and C:XX compact linear,

f:𝕁×DX,(t,x)g(t,Cx),

fulfills the previous definition. Consequently, for each AW(,L(X)) and C:XX compact linear,

f:𝕁×DX,(t,x)A(t)Cx,

will satisfy the assumptions of Definition A.4. An example for AW(+,L(X)) is given by A(t)=S(t)CT(t), provided the underlying space is reflexive, {S(t)}t+,{T(t)}t+ are bounded C0-semigroups, and C is a compact linear operator. A proof of this fact can be found in [19].

The restriction on K to be compact metric comes with the range of a function fW(𝕁,X), since f is continuous, span¯{f(𝕁)} is separable and f(𝕁)¯ 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.

Theorem A.8.

If D=(X,𝑤𝑒𝑎𝑘), gW(J,X) and fW(J×D,X), then {tf(t,g(t))}W(J,X).

For the proof, the following technical lemma is needed.

Lemma A.9.

Let (D,τ) be a topological Hausdorff space, let f:J×DX with f(,z)W(J,X) for all zD, and let

ι:DW(𝕁,X),pf(,p),

be continuous. Further, let a given sequence {xn,m}n,mND satisfy the following double limits condition:

limnlimmxn,m=limmlimnxn,m=z.

Then the interchanged limits are equal for f(tm+ωn,xn,m),xm*, whenever

{(tm,xm*)}m𝕁×BX*,{ωn}n𝕁,

and the iterated limits exist.

Proof.

Applying Proposition A.2 to bounded set on we may assume that the following iterated limits in the situation of the above lemma exist:

bn,m:=f(tm+ωn,xn,m),xm*,b:=limnlimmbn,m,b~:=limmlimnbn,m,an,m:=f(tm+ωn,z),xm*.

By our hypothesis, we have {tf(t,z)} is EWAP, thus {an,m}n,m satisfies the double limits condition, i.e.,

a=limnlimman,m=limmlimnan,m.

Now,

|b-a||b-bn,m|+|bn,m-an,m|+|an,m-a||b-bn,m|+ι(xn,m)-ι(z)+|an,m-a|.

As the double limits on the right-hand side exist and are equal to 0, we proved b=a. As the same routine works for b~, we finished the proof. ∎

Proof of Theorem A.8.

To verify the double limits condition, we apply the previous lemma for given

{(tm,xm*)}m𝕁×BX*and{ωn}n𝕁,

to

D:=(X,w)andK:={g(tm+ωn)}¯n,mw.

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 Y:=span¯{g(𝕁)}, for gW(𝕁,X) is separable, hence the weak topology on K:=g(𝕁)¯w is metric, where w denotes the weak topology. By an application of Proposition A.2, we may assume that for xn,m=g(tkm+ωln), the double limits exist, and gW(𝕁,X) 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

WRC(𝕁,X):={fW(𝕁,X):f(𝕁) is relative compact in X},

which was introduced by Goldberg and Irwin [12].

Corollary A.10.

Let, for a Banach space Y, D=(Y,), and fW(J×D,X). Then, for any given g:JY, Eberlein weakly almost periodic with a relatively compact range, we have {tf(t,g(t))}W(J,X). Moreover, if {tf(t,x)}WRC(J,X) for all xY, then {tf(t,g(t))}WRC(J,X).

Proof.

The reader will have no difficulty to apply the previous theorem to K:=g(𝕁)¯, since (K,w)=(K,), hence obtain the first part.

For the second part, it remains to prove the compactness of {f(t,g(t)):t𝕁}. Thus, for a given sequence {tn}n, we have to find a subsequence {tnk}nk such that {f(tnk,g(tnk))}nk is convergent in X. Since g has compact range, without loss of generality, g(tn)x. For this xY, we may choose a subsequence such that f(tnk,x)y for some yX. From the continuity of ι, we obtain f(,g(tnk))f(,x) uniformly on 𝕁. Thus,

f(tnk,g(tnk))-yf(tnk,g(tnk))-f(tnk,x)+f(tnk,x)-yf(,g(tnk))-f(,x)+f(tnk,x)-y,

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.

Corollary A.11.

Let AWRC(J,L(X)). Then {tA(t)g(t)} is Eberlein weakly almost periodic for all gW(J,X).

Proof.

Letting D:=(L(X),), gW(𝕁,X), and

f:𝕁×DX,(t,B)f(t,B):=Bg(t),

the previous corollary serves for the proof. ∎

From Corollary A.10, we also obtain the next result of Goldberg and Irwin [12].

Corollary A.12.

If fWRC(J,X), then f()W(J).

Theorem A.8 gives a condition on f such that {tf(t,x(t))} is Eberlein weakly almost periodic. Noting that every fW0(𝕁,X) satisfies

limT1T0Tfτdτ=0,

it is also of interest when {tf(t,x(t))}W0(𝕁,X) for a given xW0(𝕁,X). More generally, we have,

limT1T0T(fτ-fτa)dτ=0,

where fa denotes the almost periodic part of f. Thus, the question arises how the almost periodic part of the map {tf(t,x(t)} looks like. In order to discuss these problems, we introduce the projection on the almost periodic part:

Pa:W(𝕁,X)W(𝕁,X).

Proposition A.13.

For the decomposition

W(+,X)=AP(,X)|+W0(+,X),

we have that the projection Pa onto AP(R,X)|R+ has norm less than or equal to one.

Proof.

For fW(+,X), we find {sn}n+ such that fsnfa. Consequently, for given x*BX* and t+, we have

|x*(fa(t))|=limsn|x*(f(t+sn)|f,

which leads to the claim. ∎

Corollary A.14.

Any two functions f,gW(J,X) have a common sequence {tn}nN, such that the translates {ftn}nN and {gtn}nN are weakly convergent to the almost periodic part of f and g, respectively.

Proof.

First we consider the case where the almost periodic parts of f and g are equal to zero, and let {un}n and {sn}n, be chosen such that

fum0andgsn0.

By the proposition above, we have that every Eberlein weakly almost periodic function is uniformly continuous, hence the semigroup of translations {T(t)}t𝕁 is strongly continuous. Since O(f)=T(𝕁)f, O(f)¯ is a weakly compact closure of translates of a uniformly continuous function, hence O(f)¯ is compact metrizable in the weak topology of BUC(𝕁,X). As a consequence of Proposition A.2, we may pass to subsequences of {um}m and {sn}n, such that the iterated limits of {fsn+um}n,m exist in the weak topology of BUC(𝕁,X), and, without loss of generality, the sequences are chosen in this way. From the interchangeable double limits condition, we obtain

w-limmw-limnfsn+um=w-limnw-limmfsn+um.

Thus, if H:=T(𝕁)fT(𝕁)g, and d:H¯×H¯[0,) denotes the metric which induces the weak topology, then we can repeat the arguments on g, and, without loss of generality, we have

limnlimm(d(fsn+um,0)+d(gsn+um,0))=0.

Thus, the desired result is a consequence of the classical diagonal process for the double sequence

kn,m=d(fsn+um,0)+d(gsn+um,0).

If f,gW(𝕁,X), then for the function h, given by

h:𝕁X×X,t(f(t),g(t)),

we find, by the double limits criterion and the representation for the dual of X×X, that it is Eberlein weakly almost periodic if for given sequences {tm}m,{ωn}n𝕁 and {xm*}m,{ym*}mBX*,

limnlimm{f(tm+ωn),xm*+g(tm+ωn),ym*}=limmlimn{f(tm+ωn),xm*+g(tm+ωn),ym*},

whenever the iterated limits exist. However, by a successive diagonalisation of the double sequences

{f(tm+ωn),xm*}n,mand{g(tm+ωn),ym*}n,m,

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 W0 part:

hap=(fap,gap),h0=(f0,g0).

Clearly, from the decomposition of f and g, we obtain

h=(fap,gap)+(f0,g0).

Further, by the observation in the first part of this proof, we have that (f0,g0) is W0, and choosing subsequences two times will prove that (fap,gap) is almost periodic, hence the claim follows from the uniqueness of the decomposition. Hence, the sequence {tn}n, for which

htnhapweakly as n,

is the desired one. ∎

Lemma A.15.

For every KD compact metric, and fW(J×D,X), there exists a sequence {sn}nN such that

fsn(,x)=f(+sn,x)fa(,x)

for every xK, where fa(,x) denotes the almost periodic part of f(,x).

Proof.

Given any ϵ>0, we find an n(ϵ) and {xi}i=1n(ϵ), such that

Ki=1n(ϵ){x:f(,x)-f(,xi)<ϵ}.

Since ι(K) is compact metric (therefore separable),

S:𝕁×KW(𝕁,X),(t,x)T(t)f(,x),

is continuous, and S(𝕁,K) separable, by Remark A.7, where {T(t)}t𝕁 denotes the semigroup of translations. Therefore,

L:={f(t+,x):t𝕁,xK}S(𝕁,K).

Consequently, L is a subset of a closed and separable subspace Y of Cb(𝕁,X).

By the fact that

Li=1n(ϵ)H(f(,xi))+ϵBCb(𝕁,X),

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

d(f,g)=i=02-i|f-g,xi*|1+|f-g,xi*|,with {xi*}iBCb(𝕁,X)*.

Choosing ϵ=1k, we obtain, by the way of the first observation, elements {x1k,,xn(1k)k}, and by setting

{y1,y2,}:={(x11,,xn(1)1,x12,,xn(12)2,},

we construct a dense sequence {yi}i. As a consequence of Corollary A.14 and by a simple induction, we find, for all n, a sequence {skn}k such that

f(skn+,yi)fa(,yi)

converges weakly as k for all 1in in BUC(𝕁,X). This, together with the existence of a metric, implies that for all n, there exists ln such that

d(f(sln+,yi),fa(,yi))<1n

for all lln and 1in.

Now, for a given xK and yk, we have

d(f(+slnn,x),fa(,x))d(f(+slnn,x),f(+slnn,yk))+d(f(+slnn,yk),fa(,yk))+d(fa(,yk),fa(,x))2f(,x)-f(,yk)+d(f(+slnn,yk),fa(,yk)),

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. ∎

Theorem A.16.

Let D:=(X,w), yW(J,X) and fW(J×D,X). Then the following identity holds:

Paf(,y())(t)={tfa(t,ya(t))},

where fa(,x) and ya denote the almost periodic parts of f(,x) and y, respectively.

Proof.

From the notation of the theorem, we find that K:={ya(t):t𝕁}¯ is a norm compact set. Now let K1:=K{v,w}, vw, v,wK. Since v,w are two discrete points attached to K1 with a positive distance to K we have that K1 is still norm compact. Let

F(t,z):={y(t),z=v,f(t,y(t)),z=w,f(t,z),zK.

By Theorem A.8, F fulfills the hypothesis of Lemma A.15. Note that F is a function on discrete points {x,y} plus f(t,z) on K, hence ι is continuous on K1. Consequently, we find a sequence {sn}n such that

fsn(,x)fa(,x)for every xK,ysnya,f(+sn,y(+sn))Paf(,y()).

Using that all the sequences are convergent, it remains to compute the limit, which can be done in the pointwise weak topology as follows:

x*f(t+sn,y(t+sn))=x*{f(t+sn,y(t+sn))-f(t+sn,ya(t))}+x*f(t+sn,ya(t)).

Since fW(𝕁×D,X), the first term on the right-hand side tends to zero as n tends to infinity, and the theorem is proved. ∎

Corollary A.17.

If for a Banach space Y, D=(Y,) and fW(J×D,X), then, for every given gWRC(J,Y),

Pa(f(,g()))=fa(,ga()).

Proof.

Using that on K:=g(𝕁)¯ the norm and the weak topology coincide leads to the given result. ∎

Remark A.18.

We consider the context of Example 5.7 with X=L2(Ω),

YL2=W(,L2(Ω)),Ya,L2=AP(,L2(Ω)),Y0,L2=W0(,L2(Ω))

and

Y=W(),Ya,=AP(),Y0,=W0().

Then, for uL2(Ω), we have

Pa,YL2(I+λϕ(t))-1u={xPa,Y((I+λφ(t))-1u(x))}a.e.

Proof.

We give a proof for uC0(Ω), and note that C0(Ω)¯=L2(Ω). If uC0(Ω), we have that u(Ω)¯ is compact. As ((I+λφ(t))-1 is a contraction, we can apply Lemma A.15 to

f:×u(Ω)¯×L2(Ω),(ty)((I+λφ(t))-1y(I+λϕ(t))-1u),

which leads to a single sequence {sn}s with

(I+λφ(+sn))-1yJλ,a-φ()yfor all yu(Ω)¯,(I+λϕ(+sn))-1uJλ,a-ϕ()u

weakly in BUC() and BUC(,L2(Ω)), respectively. Applying the weaker pointwise and pointwise weak topology, we have, for t and vL2(Ω),

limn(I+λφ(t+sn))-1u(x)=Jλ,a-φ(t)u(x)a.e.,|(I+λφ(t+sn))-1u(x)||u(x)|a.e.,

and

Jλ,a-ϕ(t)u,v=limnJλ-ϕ(t+sn)u,v=Ω(I+λφ(t+sn))-1u(x)v(x)𝑑x=ΩJλ,a-φ(t)u(x)v(x)𝑑x,

which concludes the proof. ∎

Remark A.19.

Note that the methods apply in a similar way, when W(𝕁,X) is substituted by W+(,X) and W0(𝕁,X) by W0+(,X), since the weak relative compactness of orbit on a positive half line serves for W(𝕁,X),W0(𝕁,X)BUC(𝕁,X). Thus, W+(,X)|𝕁=W(𝕁,X) and W0+(,X)|𝕁=W0(𝕁,X), hence the proofs are similar.

B Asymptotically almost periodic functions

The classical concepts are due to Frechet [11, 10]. By [11, 10, 6, 5], we have

AAP(+,X):={fBUC(+,X):O+(f) is relative compact in BUC(+,X)}={fBUC(+,X):f=g|++ϕ,gAP(,X) and ϕC0(+,X)}.

In consequence, we have the following proposition.

Proposition B.1.

For the decomposition

AAP(+,X)=AP(,X)|+C0(+,X)

we have that the projection Pa onto AP(R,X)R+ has norm less than or equal to one.

Using compactness methods we have the following theorem.

Theorem B.2.

Let f:R×XX be such that f(,x)AAP(R+,X), with f(t,) being uniformly Lipschitz with a constant L, and fa(,x) its almost periodic part. Further, let gAAP(R+,X), with ga its the almost periodic part. Then

{tf(t,g(t)}AAP(+,X),{tf(t,g(t))}a={tfa(t,ga(t))}.

Proof.

As g(+) is relative compact, Lemma A.15 serves for the needed norm convergent subsequence, so that for all xg(+)¯, f(+sn,x)fa(,x) weakly. The relative compactness O(f(,x)) serves for the norm convergence. The rest of the proof is straightforward. ∎

C Almost automorphic functions

Bochner introduced the notion of almost automorphy.

Definition C.1.

A function fC(,X) is said to be almost automorphic if for any real sequence {sn}n, there exists a subsequence {snk}k such that

limkf(t+snk)=g(t)for all t,

and

limkg(t-snk)=f(t)for all t.

We define

AA(,X)={fC(,X):f almost automorphic}.

If the limit g is continuous, then f is called continuous (Bochner)-almost automorphic. We define

CAA(,X)={fC(,X):f continuous almost automorphic}.

Noting, that for fAA(,X), f() is relatively compact, clearly, we have AA(,X)Cb(,X), and that AA(,X) is translation invariant.

The following Theorem is due to [30, Lemma 4.1.1, p. 742].

Theorem C.2.

Continuous almost automorphic functions are uniformly continuous, i.e.,

CAA(,X)BUC(,X).

Remark C.3.

In [7], the asymptotically almost automorphic functions AAA(+,X) are discussed. By definition, we have

AAA(+,X)=AA(,X)|+C0(+,X).

For suitable f:×XX, the almost automorphic part

Pa{tf(t,x(t))}={tfa(t,xa(t))}

was computed. Thus, the underlying study becomes applicable, when switching from almost automorphy to continuous almost automorphy, and adding for f(t,x) the uniform continuity and the Lipschitz continuity, in the first and second variable, respectively. As for u=ua+u0,

{ua(t):t}¯{u(t):t+}¯,

and the projection has a norm less than one.

References

  • [1]

    H. Attouch, Variational Convergence for Functions and Operators, Pitman, Boston, 1984.  Google Scholar

  • [2]

    B. Aulbach and N. Van Minh, A sufficient condition for almost periodicity of nonautonomous nonlinear evolution equations, Nonlinear Anal. 51 (2002), 145–153.  CrossrefGoogle Scholar

  • [3]

    P. Cieuta, S. Fatajou and G. N’Guérékata, Bounded and almost automorphic solutions of some semilinear differential equations in Banach spaces, Nonlinear Anal. 71 (2009), 74–684.  Google Scholar

  • [4]

    M. G. Crandall and A. Pazy, Nonlinear evolution equations in Banach spaces, Israel J. Math. 11 (1972), 57–94.  CrossrefGoogle Scholar

  • [5]

    K. de Leeuw and I. Glicksberg, Almost periodic functions on semigroups, Acta Math. 105 (1961), 99–140.  CrossrefGoogle Scholar

  • [6]

    K. de Leeuw and I. Glicksberg, Applications of almost periodic compactifications, Acta Math. 105 (1961), 63–97.  CrossrefGoogle Scholar

  • [7]

    H. S. Ding, J. Liang and T. J. Xiao, Asymptotically almost automorphic solutions for some integrodifferential equations with nonlocal initial conditions, J. Math Anal. Appl. 338 (2008), 1188–1199.  Web of ScienceGoogle Scholar

  • [8]

    N. Dunford and J. T. Schwartz, Linear Operators. Part I, John Wiley & Sons, New York, 1957.  Google Scholar

  • [9]

    W. F. Eberlein, Abstract ergodic theorems and weak almost periodic functions, Trans. Amer. Math. Soc. 67 (1949), 217–240.  CrossrefGoogle Scholar

  • [10]

    M. Frechet, Les fonctions asymptotiquement presque-périodiques, Rev. Sci 79 (1941), 341–354.  Google Scholar

  • [11]

    M. Frechet, Les fonctions asymptotiquement presque-périodiques continues, C. R. Acad. Sci. Paris 213 (1941), 520–522.  Google Scholar

  • [12]

    S. Goldberg and P. Irwin, Weakly almost periodic vector valued functions, Diss. Math. 157 (1979), 1–42.  Google Scholar

  • [13]

    A. Haraux, Asymptotic behavior of trajectories for some nonautonomous, almost periodic processes, J. Differential Equations 49 (1983), 473–483.  CrossrefGoogle Scholar

  • [14]

    Y. Hino, T. Naito, N. V. Minh and J. S. Shin, Almost Periodic Solutions of Differential Equations in Banach Spaces, Taylor & Francis, London, 2002.  Google Scholar

  • [15]

    K. Ito and F. Kappel, Evolution Equations and Approximations, Ser. Adv. Math. Appl. Sci. 61, World Scientific, Singapore, 2002.  Google Scholar

  • [16]

    N. Kenmochi and M. Otani, Nonlinear evolution equations governed by subdifferential operators with almost periodic time dependence, Rend. Accad. Naz. Sci XL V. Ser. Mem. Mat. 10 (1986), 65–91.  Google Scholar

  • [17]

    U. Krengel, Ergodic Theorems, De Gruyter Stud. Math. 6, De Gruyter, Berlin, 1985.  Google Scholar

  • [18]

    J. Kreulich, Eberlein weak almost periodicity and differential equations in Banach spaces, Ph.D. thesis, Universität Essen, 1992.  Google Scholar

  • [19]

    J. Kreulich, The RAGE theorem in Banach spaces, Semigroup Forum 49 (1994), 151–163.  CrossrefGoogle Scholar

  • [20]

    J. Kreulich, Eberlein weakly almost periodic solutions of evolution equations in Banach spaces, Differential Integral Equations 9 (1996), 1005–1027. Google Scholar

  • [21]

    J. Kreulich, Bounded solutions of nonlinear Cauchy problems, Abstr. Appl. Anal. 7 (2002), no. 12, 637–661.  CrossrefGoogle Scholar

  • [22]

    J. Kreulich, Asymptotic equivalence of nonlinear evolution equations in Banach spaces, J. Evol. Equ. 14 (2014), 969–1000.  CrossrefWeb of ScienceGoogle Scholar

  • [23]

    J. Kreulich, Asymptotic behaviour of nonlinear evolution equations in Banach spaces, J. Math. Anal. Appl. 424 (2015), no. 2, 1054–1102.  CrossrefGoogle Scholar

  • [24]

    P. Milnes, On vector-valued weakly almost periodic functions, J. Lond. Math. Soc. (2) 22 (1980), 467–472.  Google Scholar

  • [25]

    I. Miyadera, Nonlinear Semigroups, Transl. Math. Monogr. 109, American Mathematical Society, Providence, 1992.  Google Scholar

  • [26]

    J. Prüss, Bounded solutions of Volterra equations, SIAM J. Math. Anal. 19 (1988), 133–149.  CrossrefGoogle Scholar

  • [27]

    W. M. Ruess and W. H. Summers, Integration of asymptotically almost periodic functions and weak asymptotic almost periodicity, Diss. Math. 279 (1989), 1–35.  Google Scholar

  • [28]

    W. M. Ruess and W. H. Summers, Ergodic theorems for semigroups of operators, Proc. Amer. Math. Soc. 114 (1992), 423–432.  CrossrefGoogle Scholar

  • [29]

    J. Traple, Weakly almost periodic solutions of differential equations, J. Differential Equations 45 (1982), 199–206.  CrossrefGoogle Scholar

  • [30]

    W. A. Veech, Automorphic functions on groups, Amer. J. Math. 87 (1965), no. 3, 719–751.  CrossrefGoogle Scholar

  • [31]

    T-J. Xiao, J. Liang and J. Zhang, Pseudo almost automorphic solutions to semilinear differental equations in Banach spaces, Semigroup Forum 76 (2008), 518–524.  CrossrefGoogle Scholar

About the article

Received: 2016-03-31

Revised: 2016-08-24

Accepted: 2016-09-11

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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© 2019 Walter de Gruyter GmbH, Berlin/Boston. This work is licensed under the Creative Commons Attribution 4.0 Public License. BY 4.0

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