Pseudo-Almost Periodic and Pseudo-Almost Automorphic Solutions to Some Evolution Equations ...

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CUBO A Mathematical Journal
o
Vol.16, N¯02, (01–31). June 2014
Pseudo-Almost Periodic and Pseudo-Almost Automorphic
Solutions to Some Evolution Equations Involving
Theoretical Measure Theory
Toka Diagana Khalil Ezzinbi Mohsen Miraoui
Howard University, Universit´e Cadi Ayyad, Institut sup´erieur des Etudes
2441 6th Street N.W., Facult´e des Sciences Semlalia, technologiques de Kairouan,
Washington, D.C. 20059, D´epartement de Rakkada-3191 Kairouan,
USA. Math´ematiques, Tunisie.
[email protected] BP 2390, Marrakesh, Maroc
ABSTRACT
Motivatedbytherecentworksbythefirstandthesecondnamedauthors,inthispaper
we introduce the notion of doubly-weighted pseudo-almost periodicity (respectively,
doubly-weighted pseudo-almost automorphy) using theoretical measure theory. Basic
properties of these new spaces are studied. To illustrate our work, we study, under
Acquistapace–Terreni conditions and exponential dichotomy, the existence of (µ,ν)-
pseudo almost periodic (respectively, (µ,ν)-pseudo almost automorphic) solutions to
some nonautonomous partial evolution equations in Banach spaces. A few illustrative
examples will be discussed at the end of the paper.
RESUMEN
Motivado por los trabajos recientes del primer y segundo autor, en este art´ıculo intro-
ducimoslanoci´ondeseudo-casiperiodicidadcondoblepeso(seudo-casiautomorf´ıacon
doble peso respectivamente) usando Teor´ıa de la Medida. Se estudian las propiedades
b´asicasde estos espaciosnuevos. Parailustrar nuestrotrabajo,bajo las condicionesde
Acquistapace-Terreni y dicotom´ıa exponencial estudiamos la existencia de soluciones
(respectivamente, (µ,ν) seudo-casi peri´odicas (µ,ν) seudo-casi autom´orficas) para al-
gunas ecuaciones parciales de evoluci´on aut´onomas en espacios de Banach. Algunos
ejemplos ilustrativos se discutir´an al final del art´ıculo.
Keywords and Phrases: Evolution family; exponential dichotomy; Acquistapace–Terreni con-
ditions; pseudo-almost periodic; pseudo-almost automorphic; evolution equation; nonautonomous
equation; doubly-weighted pseudo-almost periodic; doubly-weighted pseudo-almost automorphy;
(µ,ν)-pseudo-almostperiodicity;(µ,ν)-pseudo-almostautomorphy;neutralsystems;positivemea-
sure.
2010 AMS Mathematics Subject Classification: 34C27; 34K14; 34K30; 35B15; 43A60;
47D06; 28Axx; 58D25; 65J08.
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1 Introduction
Motivated by the recent works by Ezzinbi et al. [12, 13] and Diagana [30], in this paper we make
extensiveuseoftheoreticalmeasuretheoryto introduceandstudythe conceptofdoubly-weighted
pseudo almost periodicity (respectively, doubly-weighted pseudo almost automorphy). Obviously,
thesenewnotionsgeneralizeallthedifferentnotionsofweightedpseudo-almostperiodicity(respec-
tively,weightedpseudo-almostautomorphy)recentlyintroducedintheliterature. Incontrastwith
[12, 13],here the idea consistsofusing twopositivemeasuresinsteadofone. Doingso willprovide
us alargerandricherclassofweightedergodicspaces. Basicpropertiesofthesenew functionswill
be studied including their translation invariance and compositions etc.
To illustrate our study, we study the existence of (µ,ν)-pseudo-almost periodic (respectively,
(µ,ν)-pseudo-almostautomorphic)solutionstothefollowingnonautonomousdifferentialequations,
d
u(t)=A(t)u(t)+F(t,u(t)), t R, (1.1)
dt ∈
and
d
u(t)−G(t,u(t)) =A(t) u(t)−G(t,u(t)) +F(t,u(t)), t R, (1.2)
dt! " ! " ∈
where A(t) : D(A(t)) X X for t R is a family of closed linear operators on a Banach
⊂ # ∈
space X, satisfying the well-known Acquistapace–Terreni conditions, and F,G : R X X are
× #
jointly continuous functions →satisfying some additional conditions. One should indicate that the
autonomous case, i.e., A(t) = A for all t R, and the periodic case, that is, A(t+θ) =→A(t) for
∈
some θ > 0, have been extensively studied, see [8, 10, 40, 41, 53, 56] for the almost periodic case
and [18, 22, 39, 42, 50, 51] for the almost automorphic case. Recently, Diagana [24, 25, 26, 32]
studied the existence and uniqueness of weighted pseudo-almost periodic and weighted pseudo-
almost automorphic solutions to some classes of nonautonomous partial evolution equations of
type Eq. (1.1). Similarly,inDiagana[33],the existence ofpseudo-almostperiodic solutionsto Eq.
(1.2)hasbeenstudiedintheparticularcasewhenG=0. Inthispaperitgoesbacktostudyingthe
existenceofdoubly-weightedpseudo-almostperiodic(respectively,doubly-weightedpseudo-almost
automorphic) solutions in the general case as outlined above using theoretical measure theory.
The existence and uniqueness of almost periodic, almost automorphic, pseudo-almost peri-
odic and pseudo-almost automorphic solutions is one of the most attractive topics in the quali-
tative theory of ordinary or functional differential equations due to applications in the physical
sciences, mathematical biology, and control theory. The concept of almost automorphy, which
was introduced by Bochner [15], is an important generalization of the classical almost period-
icity in the sense of Bohr. For basic results on almost periodic and almost automorphic func-
tions we refer the reader to [7, 59, 61], where the authors give an important overview about
their applications to differential equations. In recent years, the existence of almost periodic,
pseudo-almost periodic, almost automorphic, and pseudo-almost automorphic solutions to dif-
ferent kinds of differential equations have been extensively investigated by many people, see, e.g.,
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[3, 4, 5, 16, 17, 19, 20, 30, 23, 24, 32, 33, 34, 35, 36, 37, 39, 43, 44, 45, 46, 48, 57, 58, 60] and the
references therein.
The concept of weighted pseudo-almost periodicity, which was introduced by Diagana [25,
26, 27, 29] is a natural generalization of the classical pseudo-almost periodicity due to Zhang
[59, 60, 61]. A few years later, Blot et al. [11], introduced the concept of weighted pseudo-almost
automorphy as a generalization of weighted pseudo-almost periodicity. More recently, Ezzinbi
et al. [12, 13] presented a new approach to study weighted pseudo-almost periodic and weighted
pseudo-almostautomorphicfunctionsusingtheoreticalmeasuretheory,whichturnsouttobemore
general than Diagana’s approach.
Let us explain the meaning of this notion as introduced by Ezzinbi et al.et al. [12, 13]. Let
µ be a positive measure on R. We say that a continuous function f : R X is µ-pseudo-almost
#
periodic (respectively, µ-pseudo almost automorphic) if f = g+ϕ, where g is almost periodic
(respectively, almost automorphic) and ϕ is ergodic with respect to the→measure µ in the sense
that
1
lim ϕ(s) dµ(s)=0,
r→∞µ(Qr)"Qr∥ ∥
where Q :=[−r,r] and µ(Q ):= dµ(t).
r r
"Qr
One can observe that a ρ-weighted pseudo almost automorphic function is µ-pseudo almost
automorphic, where the measure µ is absolutely continuous with respect to the Lebesgue measure
and its RadonNikodym derivative is ρ,
dµ(t)
=ρ(t).
dt
Here we generalize the above-mentioned notion of µ-pseudo-almost periodicity. Fix two pos-
itive measures µ,ν in R. We say that a function f : R X is (µ,ν)-pseudo-almost periodic
#
(respectively, (µ,ν)-pseudo-almost automorphic) if
→
f=g+ϕ,
where g is almost periodic (respectively, almost automorphic) and ϕ is (µ,ν)-ergodic in the sense
that
1
lim ϕ(s) dµ(s)=0,
r→∞ν(Qr)"Qr∥ ∥
Clearly,the(µ,µ)-pseudo-almostperiodicitycoincideswiththeµ-pseudo-almostperiodicity. More
generally,the(µ,ν)-pseudo-almostperiodicitycoincideswiththeµ-pseudo-almostperiodicitywhen
the measures µ and ν are equivalent.
In this paper, we introduce and study properties of (µ,ν)-pseudo-almost periodic functions
andmakeuseofthesenewfunctionstostudytheexistenceanduniquenessof(µ,ν)-pseudo-almost
periodic (respectively,(µ,ν)- pseudo-almostautomorphic)solutionsofthe nonautonomouspartial
evolution equations Eq. (1.1) and Eq. (1.2) in a Banach space.
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4 Toka Diagana, Khalil Ezzinbi & Mohsen Miraoui
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The organization of this paper is as follows. In Section 2, we recall some definitions and
lemmas of (µ,ν)-pseudo almost periodic functions, (µ,ν)-pseudo-almost automorphic functions,
and the basic notations of evolution family and exponential dichotomy. In Section 3, we study
the existence and uniqueness of (µ,ν)-pseudo almost periodic (respectively, (µ,ν)-pseudo almost
automorphic) solutions to both Eq. (1.1) and Eq. (1.2). In Section 4, we give some examples to
illustrate our abstract results.
2 Preliminaries
2.1 (µ,ν)-Pseudo-Almost Periodic and (µ,ν)-Pseudo-Almost Automor-
phic Functions
Let (X, ), (Y, ) be two Banach spaces and let BC(R,X) (respectively, BC(R Y,X)) be
∥·∥ ∥·∥ ×
the space of bounded continuous functions f : R X (respectively, jointly bounded continuous
−
functions f:R Y X). Obviously, the space BC(R,X) equipped with the super norm
× −
→
→ f ∞ :=sup f(t)
∥ ∥ t∈R∥ ∥
is a Banach space. Let B(X,Y) denote the Banach spaces of all bounded linear operator from X
into Y equipped with natural topology with B(X,X)=B(X).
Definition 2.1. [21] A continuous function f : R X is said to be almost periodic if for every
#
ε>0 there exists a positive number l(ε) such that every interval of length l(ε) contains a number
τ such that →
f(t+τ)−f(t) <ε for t R.
∥ ∥ ∈
Let AP(R,X) denote the collection of almost periodic functions from R to X. It can be easily
shown that (AP(R,X), ∞) is a Banach space.
∥·∥
Definition 2.2. [38] A jointly continuous function f:R Y X is said to be almost periodic in
× #
t uniformly for y Y, if for every ε > 0, and any compact subset K of Y, there exists a positive
∈
number l(ε) such that every interval of length l(ε) contains a→number τ such that
f(t+τ,y)−f(t,y) <ε for (t,y) R K.
∥ ∥ ∈ ×
We denote the set of such functions as APU(R Y,X).
×
Let µ,ν . If f : R X is a bounded continuous function, we define its doubly-weighted
∈ M #
mean, if the limit exists, by
→
1
(f,µ,ν):= lim f(t)dµ(t).
M r→∞ν(Qr)"Qr
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It is well-known that if f AP(R,X), then its mean defined by
∈
1
(f):= lim f(t)dt
M r→∞2r
"Qr
exists [15]. Consequently, for every λ R, the following limit
∈
1
a(f,λ):= lim f(t)e−iλtdt
r→∞2r
"Qr
exists and is called the Bohr transform of f.
Itiswell-knownthata(f,λ)isnonzeroatmostatcountablymanypoints[15]. Thesetdefined
by
σ (f):= λ R:a(f,λ)=0
b
∈ ̸
# $
is called the Bohr spectrum of f [47].
Theorem2.3. [47]Letf AP(R,X). Thenforeveryε>0thereexistsatrigonometricpolynomial
∈
n
Pε(t)= akeiλkt
k=1
%
where a X and λ σ (f) such that f(t)−P (t) <ε for all t R.
k k b ε
∈ ∈ ∥ ∥ ∈
µ(Q )
Theorem 2.4. Let µ,ν and suppose that lim r = θ . If f : R X is an almost
∈ M r→∞ν(Qr) µν #
periodic function such that
→
1
lim eiλtdµ(t) =0 (2.1)
r→∞###ν(Qr)"Qr ###
# #
for all 0=λ σ (f), then the doubl#y-weighted mean of f,#
b
̸ ∈
1
(f,µ,ν)= lim f(t)dµ(t)
M T→∞ν(QT)"QT
exists. Furthermore, (f,µ,ν)=θ (f).
µν
M M
Proof. The proof of this theorem was given in [30] in the case of measures of the form ρ(t)dt.
For the sake of completeness we reproduce it here for positive measures. If f is a trigonometric
polynomial, say, f(t) = nk=0akeiλkt where ak ∈ X−{0} and λk ∈ R for k = 1,2,...,n, then
σ (f)={λ : k=1,2,...,n}. Moreover,
b k &
1 µ(Q ) 1 n
f(t)dµ(t) = a r + a eiλkt dµ(t)
ν(Q ) 0ν(Q ) ν(Q ) k
r "Qr r r "Qr$k=1 %
%
µ(Q ) n 1
= a r + a eiλktdµ(t)
0ν(Q ) k ν(Q )
r k=1 $ r "Qr %
%
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6 Toka Diagana, Khalil Ezzinbi & Mohsen Miraoui
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and hence
n
1 µ(Q ) 1
&&&ν(Qr)"Qrf(t)dµ(t)−a0ν(Qrr)&&& ≤ k%=1∥ak∥###ν(Qr)"Qreiλktdµ(t)###
& &
which by Eq. (2.1) yields
1
f(t)dµ(t)−a θ 0 as r
&ν(Q ) 0 µν&
& r "Qr &
& &
and therefore (f,µ,ν)&=a0θµν =θµνM(f). &→ →∞
M
If in the finite sequence of λ there exist λ = 0 for k = 1,2,...l with a X−{0} for all
k nk m ∈
m=n (k=1,2,...,l), it can be easily shown that
k
̸
l
(f,µ,ν)=θ a =θ (f).
M µν nk µνM
k=1
%
Now if f:R X is an arbitrary almost periodic function, then for every ε>0 there exists a
#
trigonometric polynomial (Theorem 2.3) P defined by
ε
→
n
Pε(t)= akeiλkt
k=1
%
where a X and λ σ (f) such that
k k b
∈ ∈
f(t)−P (t) <ε (2.2)
ε
∥ ∥
for all t R.
∈
Proceeding as in Bohr [15] it follows that there exists r such that for all r ,r >r ,
0 1 2 0
1 1
P (t)dµ(t)− P (t)dµ(t) =θ (P )− (P ) =0<ε.
&&ν(Qr1)"Qr1 ε ν(Qr2)"Qr2 ε && µν&&M ε M ε &&
& & & &
In view of the above it follows that for all r ,r >r ,
1 2 0
1 1 1
f(t)dµ(t) − P (t)dµ(t) f(t)−P (t) dµ(t)
&&ν(Qr1)"Qr1 ν(Qr2)"Qr2 ε &&≤ ν(Qr1)"Qr1 ∥ ε ∥
& &
1 1
+ P (t)dµ(t)− P (t)dµ(t) <ε.
&&ν(Qr1)"Qr1 ε ν(Qr2)"Qr2 ε &&
& &
Now for all r>r ,
0
1 1
f(t)dµ(t)− P (t)dµ(t) <ε
&&ν(Qr)"Qr ν(Qr)"Qr ε &&
& &
and hence (f,µ,ν)= (P ,µ,ν)=θ (P )=θ (f). The proof is complete.
ε µν ε µν
M M M M
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Definition 2.5. [51] A continuous function f : R X is called almost automorphic if for every
sequence (σn)n∈N there exists a subsequence (sn)n∈N (σn)n∈N such that
⊂
→
lim f(t+s −s )=f(t) for each t R.
n,m→∞ n m ∈
Equivalently,
g(t):= lim f(t+s ) and f(t)= lim g(t−s )
n→∞ n n→∞ n
are well defined for each t R.
∈
Let AA(R,X) denote the collection of all almost automorphic functions from R to X. It can
be easily shown that (AA(R,X), . ∞) is a Banach space.
∥∥
Definition 2.6. [13] A function f : R X Y is said to be almost automorphic in t uniformly
×
with respect to x in X if the following two conditions hold:
(i) for all x X, f(.,x) AA(R,Y), →
∈ ∈
(ii) f is uniformly continuous on each compact set K in X with respect to the second variable x,
namely, for each compact set K in X, for all ε>0, there exists δ>0 such that for all x , x K,
1 2
∈
one has
x −x δ sup f(t,x )−f(t,x ) ε.
1 2 1 2
∥ ∥≤ t∈R∥ ∥≤
Denote by AAU(R X,Y) the set of all such functions.
⇒
×
Remark2.7. [13]Notethatintheabovelimitthefunctiongisjustmeasurable. Iftheconvergence
in both limits is uniform in t R, then f is almost periodic. The concept of almost automorphy is
∈
then larger than almost periodicity. If fis almost automorphic, thenits range is relatively compact,
thus bounded in norm.
Example 2.8. [49] Let k: R R be such that
1
k(t)→=sin , t R.
!2+cos(t)+cos(√2t)" ∈
Then k is almost automorphic, but it is not uniformly continuous on R. Then, it is not almost
periodic.
In what follows, we introduce a new concept of ergodicity, which will generalize those given
in [12] and [29, 31].
Let denote the Lebesque σ-field of R and let be the set of all positive measures µ on
B M B
satisfying µ(R)=+ and µ([a,b])< , for all a,b R (a b).
∈ ≤
Definition 2.9. [12] Let µ,ν . The measures µ and ν are said to be equivalent there exist
∞ ∈ M ∞
constants c ,c >0 and a bounded interval Ω R (eventually ) such that
0 1
⊂ ∅
c ν(A) µ(A) c ν(A)
0 1
≤ ≤
for all A satisfying A Ω= .
∈B ∩ ∅
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8 Toka Diagana, Khalil Ezzinbi & Mohsen Miraoui
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We introduce the following new space.
Definition 2.10. Let µ,ν . A bounded continuous function f : R X is said to be (µ,ν)-
∈ M
ergodic if
1
lim f(s) dµ(s)=0. →
r→∞ν(Qr)"Qr∥ ∥
We then denote the collection of all such functions by (R,X,µ,ν).
E
We are now ready to introduce the notion of (µ,ν)-pseudo-almost periodicity (respectively,
(µ,ν)-pseudo-almost automorphy) for two positive measures µ,ν .
∈M
Definition 2.11. Let µ,ν . A continuous function f : R X is said to be (µ,ν)-pseudo
∈ M
almost periodic if it can be written in the form
→
f=g+h,
where g AP(R,X) and h (R,X,µ,ν). The collection of such functions is denoted by
∈ ∈ E
PAP(R,X,µ,ν).
Definition 2.12. Let µ,ν . A continuous function f : R X is said to be (µ,ν)-pseudo
∈ M
almost automorphic if it can be written in the form
→
f=g+h,
where g AA(R,X) and h (R,X,µ,ν). The collection of such functions will be denoted by
∈ ∈ E
PAA(R,X,µ,ν).
We formulate the following hypotheses.
(M.1) Let µ,ν such that
∈M
µ(Q )
limsup r < . (2.3)
r→∞ ν(Qr)
∞
(M.2) For all τ R, there exist β>0 and a bounded interval I such that
∈
µ({a+τ: a A}) βµ(A) when A satisfies A I= .
∈ ≤ ∈B ∩ ∅
Theorem2.13. Letµ,ν satisfy(M.2). ThenthespacesPAP(R,X,µ,ν)andPAA(R,X,µ,ν)
∈M
are translation invariants.
Proof. We show that (R,X,µ,ν) is translationinvariant. Let f (R,X,µ,ν), we will show that
E ∈E
t f(t+s) belongs to (R,X,µ,ν) for each s R.
# E ∈
→
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Indeed, letting µ = µ({t+s : t A}) for A it follows from (M.2) that µ and µ are
s s
∈ ∈ B
equivalent (see [12]).
Now
1 ν(Q ) 1
f(t+s) dµ(t) = r+|s| . f(t+s) dµ(t)
ν(Q ) ∥ ∥ ν(Q ) ν(Q ) ∥ ∥
r "Qr r r+|s| "Qr
ν(Q ) 1
= r+|s| . f(t) dµ (t)
ν(Q ) ν(Q ) ∥ ∥ −s
r r+|s| "Qr+|s|
ν(Q ) cst.
r+|s| . f(t) dµ(t).
≤ ν(Q ) ν(Q ) ∥ ∥
r r+|s| "Qr+|s|
Since ν satisfies (M.2) and f (R,X,µ,ν), we have
∈E
1
lim f(t+s) dµ(t)=0.
r→∞ν(Qr)"Qr∥ ∥
Therefore, (R,X,µ,ν) is translation invariant. Since AP(R,X) and AA(R,X) are translation
E
invariants, then PAP(R,X,µ,ν) and PAA(R,X,µ,ν) are translation invariants.
Theorem 2.14. Let µ,ν satisfy (M.1), then ( (R,X,µ,ν), . ∞) is a Banach space.
∈M E ∥∥
Proof. It is clear that ( (R,X,µ,ν) is a vector subspace of BC(R,X). To complete the proof, it is
E
enough to prove that ( (R,X,µ,ν) is closed in BC(R,X). If (f ) be a sequence in ( (R,X,µ,ν)
n n
E E
such that
lim f =f
n→∞ n
uniformly in R.
From ν(R)= , it follows ν(Q )>0 for r sufficiently large. Using the inequality
r
f(t) dµ(t) f(t)−f (t) dµ(t)+ f (t) dµ(t)
∞ ∥ ∥ ≤ ∥ n ∥ ∥ n ∥
"Qr "Qr "Qr
we deduce that
1 µ(Q ) 1
ν(Q ) ∥f(t)∥dµ(t)≤ ν(Qr)∥f−fn∥∞+ ν(Q ) ∥fn(t)∥dµ(t),
r "Qr r r "Qr
then from (M.1) we have
1
lirm→s∞upν(Qr)"Qr∥f(t)∥dµ(t)≤cst.∥f−fn∥∞
for all n∈N. Since nl→im∞∥f−fn∥∞ =0, we deduce that
1
lim f(t) dµ(t)=0.
r→∞ν(Qr)"Qr∥ ∥
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Lemma 2.15. [13] Let g AA(R,X) and ε > 0 be given. Then there exist s ,...,s R such
1 m
∈ ∈
that
i=1
R= (s +C ), whereC :={t R: g(t)−g(0) <ε}.
i ε ε
∈ ∥ ∥
’m
Theorem 2.16. Let µ,ν and f PAA(R,X,µ,ν) be such that
∈M ∈
f=g+φ,
where g AA(R,X) and φ (R,X,µ,ν). If PAA(R,X,µ,ν) is translation invariant, then
∈ ∈E
{g(t); t R} {f(t); t R},(theclosureoftherangeoff). (2.4)
∈ ⊂ ∈
Proof. The proofis similar to the one givenin [13]. Indeed, if we assume that (2.4) does not hold,
then there exists t R such that g(t ) is not in {f(t); t R}. Since the spaces AA(R,X) and
0 0
∈ ∈
(R,X,µ,ν) are translation invariants, we can assume that t = 0, then there exists ε > 0 such
0
E
that f(t)−g(0) >2ε for all t R. Then we have
∥ ∥ ∈
φ(t) = f(t)−g(t) f(t)−g(0) − g(t)−g(0) ε
∥ ∥ ∥ ∥≥∥ ∥ ∥ ∥≥
for all t C . Therefore,
ε
∈
φ(t−s ) ε, foralli {1,...,m}, and t s +C .
i i ε
∥ ∥≥ ∈ ∈
Let φ be the function defined by
i=m
φ(t):= φ(t−s ) .
i
∥ ∥
i=1
%
From Lemma 2.15, we deduce that
φ(t) ε forallt R. (2.5)
∥ ∥≥ ∈
Since (R,X,µ,ν) is translationinvariant,then [t φ(t−s )] (R,X,µ,ν) for all i {1,...,m},
i
E ∈E ∈
then φ (R,X,µ,ν) which is a contradiction. Consequently (2.4) holds.
∈E
→
Theorem 2.17. Let µ,ν satisfy (M.2), then the decomposition of a (µ,ν)-pseudo almost
∈ M
automorphic function in the form f=g+h, where g AA(R,X) and h (R,X,µ,ν), is unique.
∈ ∈E
Proof. Suppose that f=g +φ =g +φ , where g ,g AA(R,X) and φ ,φ (R,X,µ,ν).
1 1 2 2 1 2 1 2
∈ ∈E
Then 0 = (g −g )+(φ −φ ) PAA(R,X,µ,ν) where g −g AA(R,X) and φ −φ
1 2 1 2 1 2 1 2
∈ ∈ ∈
(R,X,µ,ν). From Theorem 2.16, we obtain (g −g )(R) {0}, therefore we have g = g and
1 2 1 2
E ⊂
φ =φ .
1 2
From Theorem 2.17, we deduce