Endpoints of Multi-Valued Weak Contractions on the Metric Space of Partially Ordered Groups ()
1. Introduction
Let
be a complete metric space. Denote by
the class of all nonempty closed and bounded subsets of X. Denote by
the Hausdorff metric of A and B with respect to d, that is,
for all
, where
. Further let
be a multi-valued/set-valued map. A point
is called a fixed point of T if
. Define
. A point
is called an endpoint/a stationary point of a multi-valued map
if
. We denote the set of all endpoints of T by
.
The investigation of endpoint of multi-valued mappings is an important extending of the study of fixed point, which was made as early as 30 years ago, and has received great attention in recent years, see e.g. ref. [1] and ref. [2], and the references therein. In particular, Amini-Harandi [1] (2010) proved the Theorem 1.1 below.
Theorem 1.1 (Theorem 2.1 of [1]). Let
be a complete metric space and
be a set-valued map that satisfies
(1.1)
for each
, where
is upper semicontinuous (u.s.c.),
for each
and satisfies
. Then T has a unique endpoint if and only if T has the approximate endpoint property. (i.e.
.)
Huang and Zhang ref. [3] (2007) introduced the concept of cone metric space, and established some fixed point theorems for contractive type maps in a normal cone metric space. Subsequently, some other authors gave many results about the fixed point theory in cone metric spaces. For example, Rezapour and Hamlbarani ref. [4] (2008) generalized some results of [3]. Raja and Vaezpour ref. [5] (2008) presented some extensions of Banach’s Contraction Principle in complete cone metric spaces. Aage and Salunke ref. [6] (2011) proved some fixed point theorems for the expansion onto mappings on complete cone metric spaces. Also, many common fixed point theorems were proved for maps on cone metric spaces in some literatures, for example, see Ilić and Rakočević ref. [7] (2008); Arshad, Azam and Vetro ref. [8] (2009), whose results generalized and unified many fixed point theorems. Rezapour and Haghi ref. [9] (2009), as well as Haghi and Rezapour ref. [10] (2010) studied fixed points of multifunctions (i.e. multi-valued mappings) on normal cone metric spaces and on regular cone metric spaces, respectively. Moreover, Wardowski ref. [11] (2009) introduced a kind of set-valued contractions in cone metric spaces and established endpoint and fixed point theorems for his contractions.
In addition, Rezapour and Haghi [9] (2009) introduced the concept of cone topology on cone metric space. Lakshmikantham and Ćirić ref. [12] (2009) introduced the concept of a mixed g-monotone mapping and prove coupled coincidence and coupled common fixed point theorems for such nonlinear contractive mappings in partially ordered complete metric spaces. Harjani and Sadarangani ref. [13] (2009) present some fixed point theorems for weakly contractive maps in a complete metric space endowed with a partial order. And Zhang ref. [14] (2010) proved some new fixed point and coupled fixed point theorems for multivalued monotone mappings in ordered metric spaces. Finally, Amini-Harandi ref. [15] (2011) studied fixed point theorems for a kind of generalized quasicontraction maps in so called the vector modular spaces.
Motivated by the contributions stated above, the present work studies the endpoint in the abstract metric space. The remainder of the paper is organized as follows. In Section 2, it introduces the metric space valued in a partially ordered group endowed with a topological structure and the metric space valued in a partially ordered module endowed with a topological structure, and establishes some fundamental concepts of analysis on the introduced spaces, such as the convergence of sequences, which extends the theory of cone metric space; the multi-valued weak contractions, and so on. In Section 3, it focuses on addressing the endpoint theory in the metric space of partially ordered group. And finally in Section 4, it focuses on addressing the endpoint theory in the metric space of partially ordered module.
2. Preliminaries
This section provides necessary preliminaries for our discussions.
We first make the following explanations. For a partial order
of a set, we write
to indicate that
but
, where
and
are elements of the set. And for a group
with partial order
, we write
and
to indicate respectively the sets
and
, where
indicates the identity element of
.
Definition 2.1. Let G be an abelian/a commutative group with partial order
. We call G a
-partially ordered group, a partially ordered group for simplicity, if
satisfies the law (g1)
. Let further G be an R-module and the integral ring R be a ≤-partially ordered group. Assume that the partial order < satisfies the law (r1):
, where 1 and 0 are the unit element and the identity element of R, respectively. Assume also that the partial orders < and
satisfy the law (m1):
and
(i.e.
). Then we call G an
-partially ordered module, a partially ordered module for simplicity.
Remark 2.2. 1) For convenience, we focus our attention to study under the assumption that there exist non-identity elements in group G below. 2) Note that each element of a group has an inverse element. From (g1), we can easily obtain the order relation: (g1)'
. In addition, from (m1), we can easily obtain the order relation: (m1)'
and
. 3) From (m1), we can also obtain the order relations: (m2)
and
; and (m2)'
and
. In fact, let
, then, by (g1), we have
. Let also
, i.e.
. Then, by (m1), we have
. From (g1), this leads to
. So we have (m2). Finally, from (m2), it is obvious that we have (m2)'. 4) It is obvious that the partially ordered module is a special kind of the partially ordered group.
Example 2.3. Let E be a Banach space over the real field
and P be a subset of E. P is called a cone if and only if: 1) P is closed, nonempty, and
; 2)
; 3)
and
. Here
denotes all the non-negative real numbers. For a given cone P of E, define the partial order
on E by
if and only if
, see [3]. Then it can be easily verified that E is an
-partially ordered module, and therefore, of course, is a
-partially ordered group. Here ≤ is the usual order of
.
In the following part of this section G is supposed either of a
-partially ordered group and an
-partially ordered module unless otherwise specified.
Definition 2.4. Let
be a non-empty relation of G.
is called an analytic topological structure of partially ordered group G if it satisfies: (t1)
; (t2)
; (t3)
; (t4)
; and (t5)
, there exists
such that
.
is called an analytic topological structure of partially ordered module G if it also satisfies: (t6)
and
.
Remark 2.5. In the definition above, for
is non-empty, there are actually infinite elements
such that
in
. In fact, since
is a non-empty relation, there exist at least two elements
and
such that
. By (t3), we have
. Thus, according to (t5), the result holds.
Example 2.6. For the partially ordered module E of Example 2.3, define the relation
by
if and only if
, where
denotes the interior of P, see [3] and [4]. Then we can verify that
is an analytic topological structure of E. In fact, it is obvious that
satisfies (t1), (t3), (t5) and (t6). To prove (t2), let
and
. Then, from
, we have
. So there is an
such that
, where
and
indicates the norm of
. Consider
. Let
. Then
. This implies
. On the other hand,
for
. So
. Namely
. Hence
, e.g.
, that is, (t2) holds. To prove (t4), assume
. Let
. Then
. By regarding
as
, we have
for all
, where
represents all the natural numbers. This leads to
because
(in norm) and P is
closed. So, by
, we have
. That is, (t4) holds. Therefore,
is an analytic topological structure of E.
Definition 2.7. Let
be an analytic topological structure of G and
. A sequence
of
is said to be convergent to
(in
) if
, there is a natural number N such that
for all
, denoted by
or
.
Remark 2.8. 1) Let
be an analytic topological structure of G, which is different from the
. Suppose
is also an analytic topological structure of G, and sequence
of
converges to
in
. Then we can easily know that
converges to
in
. (In fact, let
. Then, from (t5), there exists
such that
. For the
, since
in
, there is a natural number N such that
for all
. By (t2) and
, this leads to
for all
. Hence
in
.) That is, the convergence in
is stronger than in
. So, in the case, the convergence in
can be regarded as a kind of weak convergence. 2) For the analytic topological structure
of the partially ordered module E in Example 2.6, it can be easily verified that
is different from the
if E is a two-dimensional Euclidean space and
. 3) It can be easily verified that for an analytic topological structure
of G,
, there is a natural number N such that
for all
. In fact, assume
.
, let
. Then, by (t3), we have
. So, there is a natural number N such that
for all
. From (g1)' and (t3), this leads to
for all
. Conversely,
, let
. Then
. So, there is a natural number N such that
for all
. Note that
. This shows
.
Remark 2.9. For the E and the analytic topological structure
of Example 2.6, let
be a sequence in
. Assume
in norm. Then,
, there exists
such that
. Due to
in norm, there exists also a natural number N such that
for all
. Therefore,
, that is,
, for all
. This implies that
in
if
in norm.
G always associates with an analytic topological structure
and the convergence of the sequences of
is in
are assumed below.
Lemma 2.10. Let
and
be two sequences of
. We have the three conclusions as follows. 1) If
, then
is unique. 2) If
and
, then
. 3) If
for all
and
, then
.
Proof. Proving 1). Let
. Then there is a natural number
such that
for all
. On the other hand,
, since
, there is a natural number
such that
for all
. Let
. Then,
and
. From (t2), this leads to
. By virtue of (t4), we have
. Hence 1) holds.
Proving 2). Let
. By (t5), there exists
such that
. For
and
, there are natural numbers
and
such that
and
. Put
. We have:
. Hence,
. That is 2) holds.
Proving 3). Arguing by contradiction, assume
. Then there exists a
and a subsequence
such that
does not hold for all
. From
, by (g1)', we have
. This implies that
does not hold for all
. (In fact, if for some
,
, then, from (t2) and
, we have
, which contradicts that
does not hold.) Hence
. The contradiction shows 3) holds.
Definition 2.11. G is called regular if every decreasing sequence
of
is convergent. That is, if a sequence
of
satisfies
for all
, then exists a
such that
converges to
.
Remark 2.12. 1) Let
and
.
is called the infimum of A if and only if
is a lower bound of A, and
for each lower bound c of A, denoted by
. 2) It is obvious that there is at most one infimum for each subset of
. In fact, for any
, let
and
be two infimums of A. Then both
and
hold. Hence
. This shows that A has at most one infimum. 3) In particular, G is regular if for each non-empty subset A of
,
exists and there exists a sequence
of A such that
converges to
. Actually, let
be a decreasing sequence of
. Then, in the case,
exists and there exists a subsequence
of
such that
converges to
. Since
converges to
,
, there is a natural number I such that
for all
. For
decreasing, this leads to
for all
. That is,
converges to
. Hence G is regular.
Definition 2.13. Let X be a non-empty set. Suppose the mapping
satisfies
(d1)
for all
and
if and only if
,
(d2)
for all
,
(d3)
for all
.
Then d is called a metric (on X) valued in partially ordered group G, and
is called a metric space valued in partially ordered group G, when G is a partially ordered group; a metric of group and a metric space of group for simplicity, respectively. (Then d is called a metric valued in partially ordered module G, and
is called a metric space valued in partially ordered module G, when G is a partially ordered module.)
In the rest of this section, we always assume that
is either of a metric space valued in partially ordered group G and a metric space valued in partially ordered module G.
Definition 2.14. For given
, let
and
be a sequence in X.
1) We call that
converges to
if and only if
, denoted by
or
.
2)
is a Cauchy sequence if and only if
, that is,
, there is a natural number N such that
for all
.
3)
is complete if and only if every Cauchy sequence is convergent.
4)
is regular if and only if G is regular.
Remark 2.15. The relation between the regular space and the complete space is an interesting question for further research.
Definition 2.16. Given
, let
be a multi-valued mapping and
be a mapping with
for all
. T is called a multi-valued (
-)weak contraction on
if, for all different
,
, there exists
such that
(2.1)
T is called a global multi-valued (
-)weak contraction on
if, for all different
, we have
(2.2)
The weak contraction T is called to satisfy C-condition (convergence condition) if
, then
, where
and
are two sequences of X.
Remark 2.17. 1) It is obvious that a global multi-valued weak contraction is a multi-valued weak contraction. 2) The weak contraction T is called to satisfy C'-condition if
, that is,
, there is a N such that
for all
and
, then
. 3) If T satisfies C-condition, then it also satisfies C'-condition. In fact, for the set
is countable, it can be rewritten as the sequence
. Assume
. Let
. Then there exists a natural number N such that
whenever
and
. Because the set
is finite, there is a natural number I such that if
and
, then
. This implies
, we have
. Hence
. For T satisfies C-condition, we have
. Further, due to
,
, there is a natural number
such that
whenever
. Since the set
is finite, there is a natural number
such that if
,
and
, then
. That is,
,
, we have
. Note that
when
. This leads to
.
Definition 2.18. A map
on
is said to have approximate endpoint property if there exist a sequence
of X and a sequence
of
with
such that
(2.3)
for all
.
Remark 2.19. When
is the usual complete metric space, it can be easily verified that T has the approximate endpoint property in Theorem 1.1 and T has the approximate endpoint property defined in Definition 2.18 are equivalent. (In fact, if T has the approximate endpoint property in Theorem 1.1, that is,
, then there is a sequence
of X such that
. Let
. Then
and
. This shows that T has the approximate endpoint property defined in Definition 2.18. On the other hand, if T has the approximate endpoint property defined in Definition 2.18, that is, there exist a sequence
of X and a sequence
of
with
such that
, then
for all
. This implies that
, namely,
T has the approximate endpoint property in Theorem 1.1. Hence we have the equivalence stated above.)
Lemma 2.20. Let T be a multi-valued weak contraction on
. Then we have the following two conclusions. 1) T has approximate endpoint property if T has endpoints. 2) T has one endpoint at most. (i.e.
. Here
denotes the cardinal number of
Proof. 1) is obvious. In fact, let
be an endpoint of T. Put
and
for all
. Then
and (2.3) holds for all
. Hence T has the approximate endpoint property. To prove 2), assume
. Then, there exist
such that
. From (2.1), we have
. Note that
for any
. This implies
. Hence, from (d1), we have
. This contradicts
. So
, that is, 2) holds.
3. Main Results
In this section, we always assume that
is a metric space valued in partially ordered group G.
Now we are ready to prove our main results. We first present the following Theorem 3.1, which extends Theorem 1.1 (Theorem 2.1 of Amini-Harandi [1]) to the case of the metric space of group.
Theorem 3.1. Let T be a multi-valued weak contraction on complete
and satisfy C-condition. Then T has a unique endpoint if and only if it has the approximate endpoint property.
Proof. The necessity is clear from the 1) of Lemma 2.20. Next we prove the sufficiency.
Since T has the approximate endpoint property, there exist sequences
of X and
of
satisfying (2.3) and
. If there exists a subsequence
such that
being the same point
of X for all
, then we can easily know that
is an endpoint of T from (2.3). (In fact, for any given
, we have
for all
. Since
, we have
. So we have
for all
. This implies
from (t4), that is,
. Hence
is an endpoint of T.) Otherwise, without loss of generality, we can assume
whenever
and continue to prove as follows.
For any different
, let
. Then
(3.1)
Since
, according to (2.1), there exists
such that
Using this and (3.1), we further obtain
(3.2)
By (2.3), we have
and
. From (3.2), this leads to
(3.3)
For
, we have
. So,
. On the other hand, noting that
, following the proof on the 2) of Lemma 2.10, we can easily know that
. Thus we can obtain
from (3.3). This implies
for T satisfies the C-condition, which leads to
satisfies the C'-condition, see the 2) and 3) of Remark 2.17. Hence,
is a Cauchy sequence. Since
is complete, there is a
such that
.
We show
is an endpoint of T below.
Since
whenever
, without loss of generality, we can assume
for any
. Let
. Then, for all
, we have
(3.4)
For
, by (2.1), there exists
such that
(3.5)
In terms of (3.4), (3.5) and (2.3), we obtain
Since
and
, by the 2) of Lemma 2.10, we obtain
. So,
for all
. Note also that
. From (t4), we have
. Hence
. That is,
.
Finally, the uniqueness is directly obtained from the 2) of Lemma 2.20. The proof completes.
Remark 3.2. Here we make a simple explanation for Theorem 3.1 extending Theorem 1.1. Firstly, it is obvious that for the usual order ≤ of the real field
,
is a ≤-partially ordered group with analytic topological structure >. Further, due that
in Theorem 1.1 is a complete metric space, it is a complete metric space of the group
with analytic topological structure >. That is,
satisfies the requirement of Theorem 3.1. Secondly, for the
in Theorem 1.1, let
, then
is a mapping from
to
and
for all
. For the mapping T in Theorem 1.1 and the
defined above, we have
for all different
, that is,
This leads to
. Thus,
for
is closed and bounded,
, there is a
such that
. This shows that T is a multi-valued weak contraction on the space
. Thirdly, if
, then
, that is, T satisfies the C-condition. In fact, if
does not converge to 0, then there exist a
and a subsequence
such that
for all
. We show the fact is true as follows. Let
and
do not converge to 0. If
is unbounded, without loss of generality, we can assume that
increases and converges to
. For
and
,
we have
Hence there exist a
and a subsequence
such that
for all
, that is, the fact is true. If
is bounded, without loss of generality, we assume that
increases and converges to a point
. Then, for
is u.s.c. at
, i.e.
, and
, we have
.
Note that
increases and
.
We have
Hence the fact is also true. That is,
for all
. This contradicts
converges to 0. Hence T satisfies the C-condition. Finally, for the T has the approximate endpoint property of Theorem 1.1, from Remark 2.19, it has the approximate endpoint property (defined in Definition 2.18). Hence we can directly obtain Theorem 1.1 from Theorem 3.1.
Next we further present the following Theorem 3.3, which shows, in the setting that
is complete and regular, if the global multi-valued weak contraction satisfies C-condition, then it has the approximate endpoint property, so has a unique endpoint from Theorem 3.1.
Theorem 3.3. Let
be complete and regular, T be a global multi-valued weak contraction on
and satisfy C-condition. Then T has a unique endpoint.
Proof. We first prove the existence of endpoints.
Arguing by contradiction, assume T has no endpoint. Then for any
, there is at least one
such that
. Hence there must be a sequence
of X such that
and
for all
. Note T is a global multi-valued weak contraction. In terms of
,
, (2.2) and
for
, we have
(3.6)
for all
. Hence the sequence
is decreasing. So, for G is regular, there exists
such that
. Hence, from (3.6), we have
. Further, according to the 3) of Lemma 2.10, we obtain
. For T satisfies C-condition, this leads to
. And by (3.6) we further have
. Now let
and
. Then we have
and
. That is, T has the approximate endpoint property. Thus, by Theorem 3.1, T has endpoints. This contradicts our assumption. Hence the existence of endpoints is true.
Finally, the uniqueness follows directly from the 2) of Lemma 2.20. This ends the proof.
For the single-valued weak contraction can be regarded as a kind of specific global multi-valued weak contraction, from Theorem 3.3, we can immediately derive the Corollary 3.4 below, which generalizes Lemma 2.4 and Corollary 2.5 of [1].
Corollary 3.4. Let
be complete and regular,
be a single-valued weak contraction on
and satisfy C-condition. Then T has a unique fixed point.
Remark 3.5. The multi-valued weak contraction cannot have the approximate endpoint property, even in the usual metric space, for instance, see the Example 2.3 of [1].
4. Endpoint Theory for the Metric Space of Module
Note that a metric space of module is a special metric space of group. As applications of the results proved above, this section discusses the endpoint theory for the metric space of module. We always assume
is a metric space of module G in this section.
By regarding
as
, we can easily derive the next Theorem 4.1 from Theorem 3.1 and Theorem 3.3.
Theorem 4.1. Let
be complete,
be a multi-valued mapping. Let also
be a mapping, which satisfies: for any two sequences
and
of X with
, there is an
such that
for all
, as well as
has multiplicative inverse
and
. Then we have the following two conclusions.
1) Suppose for all different
,
, there exists
such that
. Then T has a unique endpoint if and only if T has the approximate endpoint property.
2) Let
be regular. Suppose for all different
, we have
,
. Then T has a unique endpoint.
Proof. Let
. Then
is a mapping. Since
, by (m2), see the 3) of remark 2.2, we have
for all
. We prove the statement that if
, then
below.
Let
and
be two sequences of X. Then
.
So, if
, then
(4.1)
Hence there exists an
such that
. By (g1)', this further leads to
. Since also
, by (m2)', we have
(4.2)
In terms of (4.1) and (4.2), we obtain
. For
, we have
. Let
. By (t6), we have
. Hence, there exists a natural N such that
for all
. For
, from (t6), we obtain also
for all
. This implies
.
For conclusion 1), by
for all
and the statement proved above, it is obvious that T is a multi-valued weak contraction on the space
and satisfies C-condition. Hence we can immediately know that the conclusion is true from Theorem 3.1. For conclusion 2), T is clearly a global multi-valued weak contraction and satisfies C-condition. Note that
is regular. We can immediately know that the conclusion is true from Theorem 3.3. This completes the proof.
Replacing, in Theorem 4.1,
by
, we directly obtain the following Corollary 4.2.
Corollary 4.2. Let
be complete,
be a multi-valued mapping. Let also
with
and
. 1) Suppose T satisfies for all different
,
, there exists
such that
. Then T has a unique endpoint if and only if T has the approximate endpoint property. 2) Let
be regular. Suppose T satisfies for all different
,
. Then T has a unique endpoint.
Proof. Let
. And then applying Theorem 4.1, we obtain the Corollary instantly.
Finally, in the 2) of Corollary 4.2, replacing also multi-valued mapping by single-valued mapping, we obtain Corollary 4.3 below.
Corollary 4.3. Let
be complete and regular,
be a single-valued mapping,
with
and
. Suppose T satisfies for all different
,
. Then T has a unique fixed point.
Remark 4.4. In particular, when
is the usual complete metric space, Corollary 4.3 is just the famous Banach fixed point theorem.
Acknowledgements
The author cordially thanks the anonymous referees for their valuable comments which lead to the improvement of this paper.