A generalization for a finite family of functions of the converse of Browder's fixed point theorem

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by  Radu Miculescu

A generalization for a finite family of functions of the converse of
Browder’s fixed point theorem
Radu MICULESCU and Alexandru MIHAIL
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Bucharest University, Faculty of Mathematics and Computer Science
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2 Str. Academiei 14, 010014 Bucharest, Romania; [email protected]

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Bucharest University, Faculty of Mathematics and Computer Science
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J Str. Academiei 14, 010014 Bucharest, Romania; mihail [email protected]

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Abstract. Taking as model the attractor of an iterated function system con-
]
A
sisting of ϕ-contractions on a complete and bounded metric space, we introduce
C
the set-theoretic concept of family of functions having attractor. We prove that,
.
h given such a family, there exist a metric on the set on which the functions are
t
a defined and take values and a comparison function ϕ such that all the family’s
m
functions are ϕ-contractions. In this way we obtain a generalization for a finite
[ family of functions of the converse of Browder’s fixed point theorem. As byprod-

1 ucts we get a particular case of Bessaga’s theorem concerning the converse of the
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contraction principle and a companion of Wong’s result which extends the above
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6 mentionedBessaga’sresultforafinitefamilyofcommutingfunctionswithcommon
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fixed point.
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0
. 2010 Mathematics Subject Classification: 28A80, 37B10, 37C25,
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0 54H20
7 Key words and phrases: family of functions having attractor, comparison
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: function, ϕ-contractions, iterated function system, topological self-similar system
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X 1. INTRODUCTION
r
a
The problem of the converse of Banach-Picard-Caccioppoli principle was
treated by several mathematicians each of them concentrating on different
assumptions. C. Bessaga (see [3], [10] and [13]) was the first one to treat the
problem by using only set-theoretic assumptions. J. S. W. Wong (see [24])
extended Bessaga’s result for a finite family of commuting functions with
common unique fixed point. Other results on this direction are due to L.
Janoˇs (see [12]), P.R. Meyers (see [18]) and S. Leader (see [15]).
The idea of replacing the contractivity condition imposed on the function
f : X → X considered in the Banach-Picard-Caccioppoli principle by a
1
weaker one described by the inequality d(f(x),f(y)) ≤ ϕ(d(x,y)) for all
x,y ∈ X, where ϕ has certain properties defining the so called comparison
function, was treated, among others, by D.W. Boyd and J.S. Wong (see [4]),
F. Browder (see [5]), J. Matkowski (see [17]) and I. A. Rus (see [21]). A
function f satisfying the previous inequality is called ϕ-contractions. From
thepoint of view of theproblem treated inthis paper a special placeis played
by Browder’s result concerning ϕ-contractions (see Theorem 2.5). For more
details about this result one can consult [11].
Iterated function systems, introduced by J. Hutchinson (see [9]) and pop-
ularized by M. Barnsley (see [1]), represent one of the most general way to
generate fractals. The large variety of their applications is the background
of the current effort to extend the classical Hutchinson’s theory. One line of
research in this direction is to weaken the usual contraction condition by con-
sidering iterated function systems consisting of ϕ-contractions. For results
in this direction one can consult [7], [8], [22] and [23].
By selecting some properties of the attractor of an iterated function sys-
temconsisting ofϕ-contractionsonacomplete andbounded metricspace, we
introduced the set-theoretic concept of family of functions having attractor
(Definition 3.3). We prove that, given such a family, there exist a complete
and bounded metric on the set on which the functions are defined and take
values and a comparison function ϕ such that all the family’s functions are
ϕ-contractions(see Theorem 3.21). In this way we obtaina generalization for
a finite family of functions of the converse of Browder’s fixed point theorem.
If F = (f ) is a family of functions having attractor A, where f : X →
i i∈I i
X and I is finite, we obtain the result tracking the following steps:
- the construction (based on the main result from [19]) of a metric d on A
and a comparison function ϕ such that d(f (x),f (y)) ≤ ϕ(d(x,y)) for every
i i
i ∈ I and every x,y ∈ A, i.e. f ’s are ϕ-contractions on the attractor with
i
respect to d (Theorem 3.4)
- the construction of a semi-metric dµ on X, associated to F and to a
sequence µ, such that dµ(f (x),f (y)) ≤ dµ(x,y) for every x,y ∈ X, i.e. f ’s
i i i
are nonexpansive on X with respect to dµ (Proposition 3.8)
- the construction of a complete and bounded metric d on X (Proposition
3.16)
- the construction of a comparison function ϕ such that d(f (x),f (y)) ≤
i i
ϕ(d(x,y)) for every i ∈ I and every x,y ∈ X, i.e. f ’s are ϕ-contractions
i
with respect to d (Lemma 3.20).
Finally we present a result which removes the boundedness condition on
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the metric d, we point out that one can obtain from our result a particular
caseofBessaga’stheoremconcerningtheconverse ofthecontractionprinciple
(seeTheorem5from[10])andwepresent acompanionofWong’sresultwhich
extendstheabovementionedBessaga’sresultforafinitefamilyofcommuting
functions with common fixed point (see [24]).
2. PRELIMINARIES
For a function f : X → X and n ∈ N, by f[n] we mean the composition
of f by itself n times.
Definition 2.1 (comparison function). A function ϕ : [0,∞) → [0,∞)
is called a comparison function if it has the following three properties:
i) ϕ is increasing;
ii) ϕ(t) < t for every t > 0;
iii) ϕ is right-continuous.
Remark 2.2.
i) Any function ϕ : [0,∞) → [0,∞) satisfying ii) and iii) from the above
definition has the following property: limϕ[n](t) = 0 for every t > 0 (see
n→∞
Remark 1 from [16]).
ii) ϕ(0) = 0 for every comparison function.
Definition 2.3 (ϕ-contraction). Let (X,d) be a metric space and a func-
tion ϕ : [0,∞) → [0,∞). A function f : X → X is called a ϕ-contraction if
d(f(x),f(y)) ≤ ϕ(d(x,y)) for all x,y ∈ X.
Remark 2.4. Every ϕ-contraction is Lipschitz, so it is continuous.
The next result is known as Browder’s Theorem.
Theorem 2.5 (see Theorem 1 from [5], Theorem 1 from [11] or Example
2.9., 1) from [2]). Let (X,d) be a complete and bounded metric space and
ϕ : [0,∞) → [0,∞) a comparison function. Then every ϕ-contraction f :
X → X has a unique fixed point x and limf[n](x) = x for every x ∈ X.
0 0
n→∞
Given a metric space (X,d) and a subset Y of X, by d(Y) we denote
the diameter of Y and by K(X) we denote the family of non-empty compact
subsets of X.
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For a nonempty set I, by Λ(I) we mean the set IN∗ and by Λ (I) we
n
mean the set I{1,2,...,n}. So, the elements of Λ(I) are written as infinite words
α = α α ...α α ... and the elements of Λ (I) are written as finite words
1 2 m m+1 n
α = α α ...α (n, which is the length of ω, is denoted by |ω|).
1 2 n
By Λ∗(I) we denote the set of all finite words, i.e. Λ∗(I) d=ef ∪ Λ (I)∪
n
n∈N∗
{λ}, where λ is the empty word.
For α = α α ...α α ... ∈ Λ(I) and n ∈ N, we shall use the following
1 2 m m+1
not
notation: [α] = α α ...α if n ≥ 1 and λ if n = 0.
n 1 2 n
For two words α ∈ Λ (B)and β ∈ Λ (B) or β ∈ Λ(B), by αβ we mean
n m
the concatenation of the words α and β, i.e. αβ = α α ...α β β ...β and
1 2 n 1 2 m
respectively αβ = α α ...α β β ...β β ....
1 2 n 1 2 m m+1
On Λ(I) we consider the metric given by d (α,β) = ∞ 1−δβαkk, where
Λ P 3k
k=1
1, if x = y
δy = { .
x 0, if x 6= y
Remark 2.6. The function τ : Λ(I) → Λ(I), given by τ (α) = iα for
i i
every α ∈ Λ(I), is continuous.
Remark 2.7.
i) The convergence in the compact metric space (Λ(I),d ) is the conver-
Λ
gence on components.
ii) If I is finite, then (Λ(I),d ) is compact.
Λ
Given the functions f : X → X, where X is a given set and i ∈ I, we
i
shall use the following notations:
i) f = Id ;
λ X
not
ii) f = f ◦f ◦...◦f for every α ,α ,...,α ∈ I;
α1α2...αm α1 α2 αm 1 2 m
iii) Y n=ot f (Y) for every α ∈ Λ∗(I) and every Y ⊆ X.
α α
Definition 2.8 (topological self-similar set, topological self-similar sys-
tem). A compact Hausdorff topological space K is called a topological self-
similar set if there exist continuous functions f , f , ..., f : K → K, where
1 2 N
N ∈ N∗, and a continuous surjection π : Λ({1,2,...,N}) → K such that the
diagram
Λ({1,2,...,N}) →τi Λ({1,2,...,N})
π ↓ ↓ π
K → K
fi
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commutes for all i ∈ {1,2,...,N}.
We say that (K,(f ) ), a topological self-similar set together with
i i∈{1,2,...,N}
the set of continuous maps as above, is a topological self-similar system.
The above definition is Definition 0.3 from [14].
Theorem 2.9 (see Theorem 3.1 from [19]). For every topological self-
similar system (K,(f ) ) there exist a metric δ on K which is com-
i i∈{1,2,...,N}
patible with the original topology and a comparison function ϕ : [0,∞) →
[0,∞) such that δ(f (x),f (y)) ≤ ϕ(δ(x,y)) for each i ∈ {1,2,...,N} and
i i
each x,y ∈ K.
Definition 2.10 (iterated function system). Given a complete metric
space (X,d), an iterated function system is a pair S = ((X,d),(f ) ),
i i∈{1,2,...,N}
where f : X → X is a continuous function for each i ∈ {1,2,...,N},N ∈ N∗.
i
3. THE RESULTS
Some considerations on iterated function systems consisting of
ϕ-contractions
We start with a result that emphasizes some properties of iterated func-
tion systems consisting of ϕ-contractions.
Proposition 3.1. Let us consider an iterated function system S =
((X,d),(f ) ) consisting of ϕ-contractions, where ϕ is a comparison func-
i i∈I
tion and the metric space (X,d) is complete and bounded. Then:
a) For every α ∈ Λ(I), the set ∩ X has a unique element which is
n∈N∗ [α]n
denoted by a .
α
b) If a 6= a , where α,β ∈ Λ(I), then there exists n ∈ N∗ such that
α β 0
X ∩X = ∅.
[α]n0 [β]n0
Proof.
a) Let us consider α = α α ...α ... ∈ Λ(I) and n ∈ N∗. As for ev-
1 2 m
ery x,y ∈ X there exist u,v ∈ X such that x = f (u) and y =
[α]n α1α2...αn
fi areϕ-contractions
f (v), we have d(x,y) = d(f (u),f (v)) ≤
α1α2...αn α1α2...αn α1α2...αn
ϕisincreasing
ϕ[n](d(u,v)) ≤ ϕ[n](d(X)),sod(X ) ≤ ϕ[n](d(X)),henced(X ) ≤
[α]n [α]n
ϕ[n](d(X)) for every n ∈ N∗. As (X,d) is complete, making use of Remark
5
2.2, i) and the fact that X ⊆ X for every n ∈ N∗, we conclude that
[α]n+1 [α]n
the set ∩ X has one element denoted by a , i.e.
n∈N [α]n α
∩ X = {a }. (1)
n∈N [α]n α
Remark2.4
Letusnotethatf (a ) ∈ f ( ∩ X ) ⊆ ∩ f (X ) ⊆ ∩ f (X ) =
i α i n∈N∗ [α]n n∈N∗ i [α]n n∈N∗ i [α]n
(1)
∩ X = {a }, so
n∈N∗ [iα]n iα
f (a ) = a , (2)
i α iα
for every i ∈ I and every α ∈ Λ(I). For α = α α ...α ... ∈ Λ(I) and
1 2 n
n ∈ N∗, with the notation β = α α ...α ... ∈ Λ(I), we have a =
n n+1 n+2 m α
(2)
a = f (a ) ∈ X . Hence {a } ⊆ ∩ X ⊆ ∩ X = {a }, so
[α]nβn [α]n βn [α]n α n∈N∗ [α]n n∈N∗ [α]n α
{a } = ∩ X .
α n∈N∗ [α]n
b) Let us consider α,β ∈ Λ(I) such that a 6= a . Then Remark 2.2,
α β
i) assures the existence of a n ∈ N∗ such that ϕ[n0](d(X)) < d(aα,aβ).
0 3
Consequently, since (as we have seen above) d(X ) ≤ ϕ[n0](d(X)) and
[α]n0
d(X ) ≤ ϕ[n0](d(X)), we get d(X ) < d(aα,aβ) and d(X ) < d(aα,aβ).
[β]n0 [α]n0 3 [β]n0 3
If, by reductio ad absurdum, X ∩X 6= ∅, then choosing x ∈ X ∩
[α]n0 [β]n0 [α]n0
X , we get the following contradiction: d(a ,a ) ≤ d(a ,x)+d(x,a ) ≤
[β]n0 α β α β
d(X )+d(X ) < 2d(aα,aβ). (cid:3)
[α]n0 [β]n0 3
Remark 3.2.
i) With the notation A = {a | α ∈ Λ(I)}, the function π : Λ(I) → A,
α
given by π(α) = a for every α ∈ Λ(I), is continuous.
α
Indeed, given a fixed α ∈ Λ(I), as limd(X ) = 0, for every ε > 0 there
[α]n
n→∞
exists m ∈ N∗ such that X ⊆ B(a ,ε), so B(α, 1 ) ⊆ {ω ∈ Λ(I) | [ω] =
[α]m α 3m m
[α] } ⊆ π−1(X ) ⊆ π−1(B(a ,ε)), i.e. π(B(α, 1 )) ⊆ B(π(α),ε).
m [α]m α 3m
ii) Considering the function F : K(X) → K(X) given by F (C) =
S S
∪f (C) for every C ∈ K(X), using (2) from the proof of Proposition 3.1,
i
i∈I
we infer that F (A) = A, i.e., taking into account the uniqueness of the
S
fixed point of F (see Theorem 2.5 from [7]), A is the attractor of the
S
iterated function system S. Moreover, the same result guarantees that
[n]
limh(F (B),A) = 0foreveryB ∈ K(X),wherehdesignatestheHausdorff-
S
n→∞
Pompeiu metric.
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The notion of family of functions having attractor
As X = X, the above considerations suggest the following:
[α]0
Definition 3.3. We say that a family of functions F = (f ) , where
i i∈I
f : X → X and I is finite, has attractor if the following two properties are
i
valid:
a) For every α ∈ Λ(I), the set ∩ X has a unique element which
n∈N [α]n
is denoted by a .
α
b) If a 6= a , where α,β ∈ Λ(I), then there exists n ∈ N such that
α β 0
X ∩X = ∅.
[α]n0 [β]n0
def
The set A = {a | α ∈ Λ(I)} is called the attractor of F.
α
A metric on the attractor which makes ϕ-contractions all the
functions of a family having attractor
Theorem 3.4. If F = (f ) is a family of functions having attractor
i i∈I
A, then there exist a metric d on A and a comparison function ϕ such that
d(f (x),f (y)) ≤ ϕ(d(x,y)) for every i ∈ I and every x,y ∈ A.
i i
Proof. Considering the function π : Λ(I) → A, given by π(α) = a for
α
every α ∈ Λ(I), the binary relation on Λ(I), given by α ∼ β if and only
if π(α) = π(β), turns out to be an equivalence relation. We transport the
quotient topology on Λ(I)(cid:30) ∼ on the topology τ on A via the bijection
A
g : Λ(I)(cid:30) ∼→ A given by g([α]) = π(α) for every [α] ∈ Λ(I)(cid:30) ∼.
Note that:
i) g is a homeomorphism;
ii) the function p : Λ(I) → Λ(I)(cid:30) ∼, given by p(α) = [α] for every
α ∈ Λ(I), is continuous;
iii) π = g ◦p is continuous.
Claim 1. f ◦π = π ◦τ for every i ∈ I.
i i
Justification of claim 1. We have (f ◦ π)(α) = f (a ) ∈ f ( ∩ X ) ⊆
i i α i n∈N [α]n
∩ f (X ) = ∩ X = {a } = {(π ◦ τ )(α)} for every i ∈ I and every
n∈N i [α]n n∈N [iα]n iα i
α ∈ Λ(I).
Note that Claim 1 implies that A = ∪f (A).
i
i∈I
Claim 2. f : (A,τ ) → (A,τ ) is continuous for every i ∈ I.
i A A
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Justification of claim 2. Taking into account i), it suffices to prove that
f ◦g : Λ(I)(cid:30) ∼→ A, given by
i
Claim1
(f ◦g)([α]) = (π ◦τ )(α), (1)
i i
Claim1
is continuous. Since α ∼ β ⇔ π(α) = π(β) ⇒ (f ◦π)(α) = (f ◦π)(β) ⇔
i i
(1)
(π ◦ τ )(α) = (π ◦ τ )(β) ⇔ (f ◦ g)([α]) = (f ◦ g)([β]) and the function
i i i i
h = π ◦τ : Λ(I) → A, described by h(α) = (f ◦g)([α]) for every α ∈ Λ(I),
i i
is continuous (as a composition of continuous functions; see Remark 2.6 and
iii)), relying on Theorem 4.3, page 126, from [6], we get the conclusion.
Claim 3. (A,τ ) is compact.
A
Justification of claim 3. From ii) and Remark 2.7, ii), we conclude that
Λ(I)(cid:30) ∼ is compact. Using i), we get the conclusion.
Claim 4. The set R = {(α,β) ∈ Λ(I)×Λ(I) | α ∼ β} is closed.
Justification of claim 4. Let us consider (α,β) ∈ R. Then there ex-
ists ((α ,β )) ⊆ R such that lim(α ,β ) = (α,β) and consequently
n n n∈N n n
n→∞
limα = α and limβ = β. If a 6= a , then, according to the property
n n α β
n→∞ n→∞
b) from the definition of a family of functions having attractor, there exists
n ∈ N such that X ∩ X = ∅. As limα = α and limβ = β,
0 [α]n0 [β]n0 n→∞ n n→∞ n
there exists n ∈ N such that [α ] = [α] and [β ] = [β] for every
1 n n0 n0 n n0 n0
n ∈ N, n ≥ n (see Remark 2.7, i)). But α ∼ β (because (α ,β ) ∈ R),
1 n n n n
i.e. a = a , and therefore we get the following contradiction: a = a ∈
αn βn αn βn
X ∩ X = X ∩ X = ∅. Hence a = a , i.e. α ∼ β, so
[αn]n0 [βn]n0 [α]n0 [β]n0 α β
(α,β) ∈ R. Therefore R is closed.
Claim 5. (A,τ ) is Hausdorff.
A
Justification of claim 5. From the compactness of Λ(I) (see Remark 2.7,
ii)) and Claim 4, we infer that Λ(I)(cid:30) ∼ is Hausdorff. Using i) we get the
conclusion.
Claims 1, 2, 3 and 5 assure us that (A,(f ) ) is a topological self-similar
i i∈I
system and, based on Theorem 2.9, there exist a metric d on A compat-
ible with τ and a comparison function ϕ : [0,∞) → [0,∞) such that
A
d(f (x),f (y)) ≤ ϕ(d(x),d(y)) for every i ∈ I and every x,y ∈ A. (cid:3)
i i
Let us consider the function n : X → N∪{∞} given by n(x) = sup{m ∈
N | x ∈ F[m](X)} for every x ∈ X, where F : P(X) → P(X) is described by
def
F(C) = ∪f (C) for every C ∈ P(X) = {Y | Y ⊆ X}.
i
i∈I
8
The following result provides an alternative characterization of the at-
tractor A via the function n.
Proposition 3.5. In the framework of the above theorem, we have A =
{x ∈ X | n(x) = ∞}.
Proof.
”⊆” If x ∈ A, then there exists α ∈ Λ(I) such that x = a , hence
α
x ∈ X ⊆ F[m](X) for every m ∈ N. So n(x) = sup{m ∈ N | x ∈
[α]m
F[m](X)} = supN = ∞.
”⊇” Since n(x) = sup{m ∈ N | x ∈ F[m](X)} = ∞, for every m ∈ N
there exists α ∈ Λ (I) such that x ∈ X . There exists i ∈ I such
m m αm 1
that {ω ∈ Λ∗(I) | x ∈ X } is infinite. Indeed, if this is not the case,
i1ω
then the set M d=ef {ω ∈ Λ∗(I) | x ∈ X } is finite for every i ∈ I. If
i iω
def
m = max{|iω| | ω ∈ M }, then we get the contradiction that there exists no
i i
α ∈ Λ (I) such that x ∈ X , where m = 1+max{m | i ∈ I}. Repeating
m+1 α i
this procedure we get α = α α ...α ... ∈ Λ(I) such that x ∈ ∩ X , i.e.
1 2 n n∈N [α]n
x = a ∈ A. (cid:3)
α
∼
The family of sets {X | α ∈ Λ∗(I)} associated to a family of
α
functions having attractor
Given a family of functions F = (f ) having attractor A, in the sequel,
i i∈I
for α ∈ Λ∗(I) we shall use the following notations:
∼
not not
Y = {a | X ∩X 6= ∅ for every n ∈ N} and X = X ∪Y .
α β α [β]n α α α
Proposition 3.6 (The properties of the sets X and Y ). In the above
α α
framework, we have:
a) A ⊆ Y ⊆ A for every α ∈ Λ∗(I);
α α
∼
b) X ⊆ X ⊆ X ∪A for every α ∈ Λ∗(I);
α α α
c) Y ⊆ Y for every α ∈ Λ(I) and every n ∈ N;
[α]n+1 [α]n
d) ∩ (X ∪Y ) = ( ∩ X )∪( ∩ Y ) for every α ∈ Λ(I);
n∈N [α]n [α]n n∈N [α]n n∈N [α]n
e) ∩ Y = {a } for every α ∈ Λ(I);
n∈N [α]n α
f) A∩X ⊆ Y for every α ∈ Λ∗(I);
α α
g) f (Y ) ⊆ Y for every α ∈ Λ∗(I) and every i ∈ I.
i α iα
Proof.
9
a) If z ∈ A , then there exists γ ∈ Λ(I) such that z = f (a ), so z ∈ X .
α α γ α
Moreover, z = f (a ) ∈ f ( ∩ X ) ⊆ ∩ f (X ) ⊆ ∩ X = {a },
α γ α n∈N [γ]n n∈N α [γ]n n∈N [αγ]n αγ
hence z = a ∈ X ∩X for every n ∈ N, i.e. z ∈ Y .
αγ α [αγ]n α
b) It results immediately from a).
c) If z ∈ Y , then there exists β ∈ Λ(I) such that z = a and
[α]n+1 β
X ∩ X 6= ∅ for every k ∈ N. As X ∩ X ⊆ X ∩ X ,
[α]n+1 [β]k [α]n+1 [β]k [α]n [β]k
we deduce that X ∩X 6= ∅ for every k ∈ N, i.e. z = a ∈ Y .
[α]n [β]k β [α]n
d)
”⊇” It is clear.
”⊆” Let ussuppose that there exists x ∈ X ∪Y for every n ∈ N such
[α]n [α]n
thatx ∈/ ( ∩ X )∪( ∩ Y ), i.e. thereexistn ,n ∈ Nsuchthatx ∈/ X
n∈N [α]n n∈N [α]n 1 2 [α]n1
and x ∈/ Y . Then, in view of c), we have x ∈/ X and x ∈/ Y which
[α]n2 [α]m [α]m
leads to the contradiction x ∈/ X ∪Y , where m = max{n ,n }.
[α]m [α]m 1 2
e)
Claim1fromtheproofofTheorem3.4 a)
”⊇” We have a ∈ ∩ A ⊆ ∩ Y .
α n∈N [α]n n∈N [α]n
”⊆” If c ∈ ∩ Y , then there exists (β ) ⊆ π−1({c}) ⊆ Λ(I) such
n∈N [α]n n n∈N
that
X ∩X 6= ∅, (1)
[α]n [βn]k
for every n,k ∈ N. The compactness of Λ(I) (see Remark 2.7, ii)) assures the
existence of a subsequence (β ) of (β ) and of an element β ∈ Λ(I)
nl l∈N n n∈N
such that limβ = β. As π(β ) = c, i.e. a = c, and π is continuous (see
l→∞ nl nl βnl
Remark 3.2, i)), we infer that π(β) = c, i.e. a = c. By replacing β with
β j
β for all j ∈ {n +1,...,n −1}, we can suppose that limβ = β. Hence
nl l−1 l n→∞ n
for every l ∈ N there exists n ∈ N, n > l such that
l l
[β ] = [β] , (2)
n l l
(1)
for all n ∈ N, n ≥ n . Hence X ∩ X 6= ∅, i.e., in view of (2),
l [α]nl [βnl]l
X ∩X 6= ∅ and since X ⊆ X , we infer that X ∩X 6= ∅ for
[α]nl [β]l [α]nl [α]l [α]l [β]l
every l ∈ N. Therefore, taking into account the property b) of a family of
functions having attractor, we conclude that a = a = c.
α β
f) If z ∈ A∩X , then there exists γ ∈ Λ(I) such that z = a ∈ X , so
α γ [γ]n
z ∈ X ∩X and therefore X ∩X 6= ∅ for every n ∈ N. Consequently
α [γ]n α [γ]n
z = a ∈ Y .
γ α
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