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On Fixed Points of Maximalizing Mappings in Posets
Fixed Point Theory and Applications volume 2010, Article number: 634109 (2009)
Abstract
We use chain methods to prove fixed point results for maximalizing mappings in posets. Concrete examples are also presented.
1. Introduction
According to Bourbaki's fixed point theorem (cf. [1, 2]) a mapping from a partially ordered set
into itself has a fixed point if
is extensive, that is,
for all
, and if every nonempty chain of
has the supremum in
. In [3, Theorem
] the existence of a fixed point is proved for a mapping
which isascending, that is,
implies
. It is easy to verify that every extensive mapping is ascending. In [4] the existence of a fixed point of
is proved if
for some
, and if
issemi-increasing upward, that is,
whenever
and
. This property holds, for instance, if
is ascending orincreasing, that is,
whenever
.
In this paper we prove further generalizations to Bourbaki's fixed point theorem by assuming that a mapping ismaximalizing, that is,
is a maximal element of
for all
. Concrete examples of maximalizing mappings
which have or do not have fixed points are presented. Chain methods introduced in [5, 6] are used in the proofs. These methods are also compared with three other chain methods.
2. Preliminaries
A nonempty set , equipped with a reflexive, antisymmetric, and transitive relation
in
, is called a partially ordered set (poset). An element
of a poset
is called an upper bound of a subset
of
if
for each
. If
, we say that
is thegreatest element of
, and denote
. A lower bound of
and the least element,
, of
are defined similarly, replacing
above by
. If the set of all upper bounds of
has the least element, we call it thesupremum of
and denote it by
. We say that
is amaximal element of
if
, and if
and
imply that
. The infimum of
,
, and a minimal element of
are defined similarly. A subset
of
is called a chain if
or
for all
. We say that
is well ordered if nonempty subsets of
have least elements. Every well-ordered set is a chain.
Let be a nonempty poset. A basis to our considerations is the following chain method (cf. [6, Lemma
]).
Lemma 2.1.
Given and
, there exists a unique well-ordered chain
in
, called a w-o chain of
-iterations, satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F634109/MediaObjects/13663_2009_Article_1317_Equ1_HTML.gif)
If exists in
, then
, and
.
The following result helps to analyze the w-o chain of -iterations.
Lemma 2.2.
Let and
be nonempty subsets of
. If
and
exist, then the equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F634109/MediaObjects/13663_2009_Article_1317_Equ2_HTML.gif)
is valid whenever either of its sides is defined.
Proof.
The sets and
have same upper bounds, which implies the assertion.
A subset of a chain
is called aninitial segment of
if
,
and
imply
. If
is well ordered, then every element
of
which is not the possible maximum of
has a successor:
, in
. The next result gives a characterization of elements of the w-o chain of
-iterations.
Lemma 2.3.
Given and
, let
be the w-o chain of
-iterations. Then the elements of
have the following properties.
.
An element of
has a successor in
if and only if
exists and
, and then
.
If is an initial segment of
and
exists, then
.
If and
is not a successor, then
.
If exists, then
.
Proof.
-
(a)
.
-
(b)
Assume first that
, and that
exists in
. Applying (2.1), Lemma 2.2, and the definition of
we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F634109/MediaObjects/13663_2009_Article_1317_Equ3_HTML.gif)
Moreover, , by definition, whence
.
Assume next that , that
exists, and that
. The previous proof implies the following
-
(i)
There is no element
which satisfies
.
Then , so that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F634109/MediaObjects/13663_2009_Article_1317_Equ4_HTML.gif)
Thus by (2.1). This result and (i) imply that
.
-
(c)
Assume that
is an initial segment of
, and that
exists. If there is
such that
, then
, so that
. Assume next that every element
of
has the successor
in
. Since
by (b), then
. This holds for all
. Since
, then
is an upper bound of
. If
is an upper bound of
, then
for every
. Thus
is an upper bound of
, whence
. But then
, so that
by (2.1).
-
(d)
Assume that
, and that
is not a successor of any element of
. Obviously,
is an upper bound of
. Let
be an upper bound of
. If
, then also
since
is not a successor. Because
by (b), then
. This holds for every
. Since also
, then
is an upper bound of
. Thus
. This holds for every upper bound
of
, whence
.
-
(e)
If
exists, then
by (c) when
, whence
.
In the case when we obtain the following result (cf. [7, Proposition
]).
Lemma 2.4.
Given and
, there exists a unique well-ordered chain
in
, calleda w-o chain of
-iterations of
, satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F634109/MediaObjects/13663_2009_Article_1317_Equ5_HTML.gif)
If , and if
exists, then
, and
.
Lemma 2.4 is in fact a special case of Lemma 2.1, since the assumption implies that
equals to the w-o chain of
-iterations. As for the use of
in fixed point theory and in the theory of discontinuous differential and integral equations, see, for example, [8, 9] and the references therein.
3. Main Results
Let be a nonempty poset. As an application of Lemma 2.1 we will prove our first existence result.
Theorem 3.1.
A mapping has a fixed point if
is maximalizing, that is,
is a maximal element of
for all
, and if
exists in
for some
where
is the w-o chain of
-iterations.
Proof.
If is the w-o chain of
-iterations, and if
exists in
, then
and
by Lemma 2.1. Since
is maximalizing, then
, that is,
is a fixed point of
.
The next result is a consequence of Theorem 3.1. and Lemma 2.3(e).
Proposition 3.2.
Assume that is maximalizing. Given
, let
be the w-o chain of
-iterations. If
exists, it is a fixed point of
if and only if
exists.
Proof.
Assume that exists. It follows from Lemma 2.3(e) that
. If
is a fixed point of
, that is,
, then
, and
.
Assume conversely that exist. Applying (2.1) and Lemma 2.2 we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F634109/MediaObjects/13663_2009_Article_1317_Equ6_HTML.gif)
Thus, by Theorem 3.1, is a fixed point of
.
As a consequence of Proposition 3.2 we obtain the following result.
Corollary 3.3.
If nonempty chains of have supremums, if
is maximalizing, and if
exists for all
, then for each
the maximum of the w-o chain of
-iterations exists and is a fixed point of
.
Proof.
Let be the w-o chain of
-iterations. The given hypotheses imply that both
and
exist. Thus the hypotheses of Proposition 3.2 are valid.
The results of Lemma 2.3 are valid also when is replaced by the w-o chain
of
-iterations of
. As a consequence of these results and Lemma 2.4 we obtain the following generalizations to Bourbaki's fixed point theorem.
Theorem 3.4.
Assume that is maximalizing, and that
for some
, and let
be the w-o chain of
-iterations of
.
-
(a)
If
exists, then
, and
is a fixed point of
.
-
(b)
If
exists, it is a fixed point of
if and only if
exists.
-
(c)
If nonempty chains of
have supremums, and if
exists for all
, then
exists, and
is a fixed point of
The previous results have obvious duals, which imply the following results.
Theorem 3.5.
A mapping has a fixed point if
is minimalizing, that is,
is a minimal element of
for all
, and if
exists in
for some
whenever
is a nonempty chain in
.
Theorem 3.6.
A minimalizing mapping has a fixed point if
exists whenever
is a nonempty chain in
, and if
for some
.
Proposition 3.7.
A minimalizing mapping has a fixed point if every nonempty chain
has the infimum in
, and if
exists for all
.
Remark 3.8.
The hypothesis that is maximalizing can be weakened in Theorems 3.1 and 3.4 and in Proposition 3.2 to the form:
is maximalizing, that is,
is a maximal element of
.
4. Examples and Remarks
We will first present an example of a maximalizing mapping whose fixed point is obtained as the maximum of the w-o chain of -iterations.
Example 4.1.
Let be a closed disc
, ordered coordinate-wise. Let
denote the greatest integer
when
. Define a function
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F634109/MediaObjects/13663_2009_Article_1317_Equ7_HTML.gif)
It is easy to verify that , and that
is maximalizing. To find a fixed point of
, choose
. It follows from Lemma 2.3(b) that the first elements of the w-o chain of
-iterations are successive approximations
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F634109/MediaObjects/13663_2009_Article_1317_Equ8_HTML.gif)
as long as is defined. Denoting
, these successive approximations can be rewritten in the form
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F634109/MediaObjects/13663_2009_Article_1317_Equ9_HTML.gif)
as long as and
, and at least one of these inequalities is strict. Elementary calculations show that
, for every
. Thus (4.3) can be rewritten as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F634109/MediaObjects/13663_2009_Article_1317_Equ10_HTML.gif)
Since the function is increasing
, then
for every
. Thus (4.4) can be reduced to the form
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F634109/MediaObjects/13663_2009_Article_1317_Equ11_HTML.gif)
The sequence is strictly increasing, whence also
is strictly increasing by (4.5). Thus the set
is an initial segment of
. Moreover,
, and if
, then
. Since
is bounded above by
, then
exists, and
. Thus
, and it belongs to
, whence
by Lemma 2.3(c). To determine
, notice that
by (4.5). Thus
, or equivalently,
, so that
. Since
, then
by Lemma 2.3(c). Because
is a maximal element of
, then
. Moreover,
, so that
is a fixed point of
.
The first elements of the w-o chain
of
-iterations can be estimated by the following Maple program (floor
):
(1,1-floor(u) + floor(v)):
(floor
)/2:
:
for to
do
(max
, evalf(max
);
end do;
For instance, .
The verification of the following properties are left to the reader.
-
(i)
If
,
, and
, then the elements of w-o chain
of
-iterations, after two first terms if
, are of the form
,
, where
is increasing and converges to
. Thus
is the maximum of
and a fixed point of
.
-
(ii)
If
,
, or
, then
.
-
(iii)
If
, then
and
,
, where the sequences
and
are bounded and increasing. The limit
of
is the smaller real root of
;
, and the limit
of
is
. Moreover
and
, whence no subsequence of the iteration
converges to a fixed point of
.
-
(iv)
For any choice of
the iterations
and
are not order related when
. The sequence
does not converge, and no subsequence of it converges to a fixed point of
.
-
(v)
Denote
. The function
, defined by (4.1), satisfies
and is maximalizing. The maximum of the w-o chain of
-iterations with
is
, and
is a fixed point of
. If
, then
and
are not comparable.
The following example shows that need not to have a fixed point if either of the hypothesis of Theorem 3.1 is not valid.
Example 4.2.
Denote and
, where
and
are as in Example 4.1. Choose
, and let
be defined by (4.1).
is maximalizing, but
has no fixed points, since
and
. The last hypothesis of Theorem 3.1 is not satisfied.
Denoting , then the set
is a complete join lattice, that is, every nonempty subset of
has the supremum in
. Let
satisfy
and
.
has no fixed points, but
is not maximalizing, since
.
Example 4.3.
The components ,
of the fixed point of
in Example 4.1 form also a solution of the system
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F634109/MediaObjects/13663_2009_Article_1317_Equ12_HTML.gif)
Moreover a Maple program introduced in Example 4.1 serves a method to estimate this solution. When , the estimate is
,
.
Remark 4.4.
The standard "solve" and "fsolve" commands of Maple 12 do not give a solution or its approximation for the system of Example 4.3.
In Example 4.1 the mapping is nonincreasing, nonextensive, nonascending, not semiincreasing upward, and noncontinuous.
Chain is compared in [10] with three other chains which generalize the sequence of ordinary iterations
, and which are used to prove fixed point results for
. These chains are the generalized orbit
defined in [10] (being identical with the set
defined in [11]), the smallest admissible set
containing
(cf. [12–14]), and the smallest complete
-chain
containing
(cf. [10, 15]). If
is extensive, and if nonempty chains of
have supremums, then
, and
is their cofinal subchain (cf. [10, Corollary
]). The common maximum
of these four chains is a fixed point of
. This result implies Bourbaki's Fixed Point Theorem.
On the other hand, if the hypotheses of Theorem 3.4 hold and , then
and
are not necessarily comparable. The successor of such an
in
is
by [14, Proposition
]. In such a case the chains
,
and
attain neither
nor any fixed point of
. For instance when
in Example 4.1, then
, where
is the w-o chain of
-iterations. Since
, then
does not exist,
(see [10]). Thus only
attains a fixed point of
as its maximum. As shown in Example 4.1, the consecutive elements of the iteration sequence
are unordered, and their limits are not fixed points of
. Hence, in these examples also finite combinations of chains
used in [16, Theorem
] to prove a fixed point result are insufficient to attain a fixed point of
.
Neither the above-mentioned four chains nor their duals are available to find fixed points of if
and
are unordered. For instance, they cannot be applied to prove Theorems 3.1 and 3.5 or Propositions 3.2 and 3.7.
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Heikkilä, S. On Fixed Points of Maximalizing Mappings in Posets. Fixed Point Theory Appl 2010, 634109 (2009). https://doi.org/10.1155/2010/634109
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DOI: https://doi.org/10.1155/2010/634109