1. Introduction
Classical Banach’s contraction principle [
1] has a very important role in fixed point theory. Despite its considerable importance, it has a weakness that the function needs to be continuous if it satisfies this contraction condition. To remove continuity, many generalizations have been made, such as [
2,
3,
4,
5]. Ćirić [
2] provided some fixed point results for functions satisfying the following contraction condition.
Definition 1. Let be a metric space. A function is called a quasi contraction if there exists a constant satisfyingfor all . Following this, Berinde presented weak contractive mappings, or -weak contractive mappings (later called almost contractive mappings), which are:
Definition 2 ([
6]).
Let be a metric space. A function is called -weak contraction if there exists a constant and some satisfyingfor all . He emphasized that any Kannan, Chatterjee, or Zamfirescu contraction, or any quasi contraction with , is a weak contraction. In addition, he provided a fixed point result for weak contractions.
Theorem 1 ([
6]).
Let be a complete metric space and be a weak contraction. Then(1) is nonempty;
(2) For any the Picard iteration converges to some ;
(3) The following estimates hold for all .
Berinde remarked that this theorem can not guarantee the uniqueness of a fixed point, and he stated that, in this theorem, if the weak contraction
also satisfies
for all
and for some
where
, the uniqueness of the fixed point is attained. He gave more results and properties about weak contractive mappings in [
6,
7,
8,
9]. On the other hand, there is another tendency to extend the Banach contraction principle with partial ordering. To our knowledge, it started with Ran and Reuring [
10] and was followed by many authors, notably Nieto and Lopez [
11,
12]. They presented some fixed point results for nondecreasing, nonincreasing and even not monotone contractions. After that, many fixed point results have been given on partially ordered metric spaces, such as [
13]. While in [
14,
15], the notion of partial metric is combined with partial ordering, in [
16], the notion of M-metric is combined with partial ordering. Moreover, in many works such as [
17,
18,
19,
20], many fixed point results have been given on cone metric spaces. On the other hand, a generalization of metric spaces was made by Cevik and Altun [
21]. They presented
vector metric space(
E-metric space) and gave some fixed point results on this space. According to this work, any metric is a vector metric, but the converse is not true in general. In the last two decades, many extensions of the results in [
21] have been completed, such as [
22,
23,
24,
25]. In [
23], the authors united the concept of partial ordering with vector metric and provided some fixed point theorems on ordered vector metric space; hence, they generalized the results of [
11,
12,
21].
In this paper, we aim to combine the results in [
2,
6,
7,
8] with the notion of vector metric introduced in [
21] and partial ordering. Hence, we extend the results in these works as well as the ones in [
11,
12,
23]. We define ordered vectorial quasi contraction. According to this definition, any ordered quasi contraction (extension of quasi contraction) is an ordered vectorial quasi contraction, but the converse is not true. We also define ordered vectorial almost contraction, which is an extension of
-weak contraction and, we present some fixed point theorems for this kind of family of contractions. In addition, we provide some related examples that show the differences between our results and the ones previously mentioned.
A partially ordered set, whose any two elements contain both supremum and infimum, is called lattice. An ordered vector space
E is a real vector space with an order relation “⩽”, which is compatible with the algebraic structure of
E. In other words,
implies
and
for all
E,
. An ordered vector space is called Riesz space whenever it is a lattice. A Riesz space is labelled Archimedean if
for all
. By the notation
, we mean the sequence
is order-reversing and the infimum of the set
is
a. For other facts and concepts related to Riesz space, we refer [
26,
27]. Now, let us recall some concepts from [
21], especially the definition of vector metric space. A map
is named
vector metric (
E-metric) if it satisfies the conditions
where
X is a nonempty set and
is a Riesz space. Hence,
(briefly
X) is called
vector metric space(
E-metric space). A sequence
of
X vectorially converges (
E-converges) to
if there is a sequence
in
E such that
and
for all
The sequence
is labelled
-Cauchy if there is a sequence
in
E such that
and
for all
n and
p. Additionally, if every
E-Cauchy sequence in
X is
E-convergent, then
X is said to be
E-complete.
Note that a function
is called
vectorial continuous if whenever
where
and
are two vector metric spaces [
28].
2. Ordered Vectorial Quasi and Almost Contractions
Initially, we present the definitions of ordered vectorial quasi contractions and ordered vectorial almost contractions in this part. Unless otherwise stated, we assume that (shortly X) is a vector metric space, and it is equipped with the ordering “⪯”. We also use the notations and , respectively, for the maximum (if exists) and the supremum of the set .
Definition 3. A function on an ordered vector metric space X to itself is named ordered vectorial quasi contraction if there is a number satisfyingfor all where In general, according to a given partial ordering, the concepts of supremum and maximum do not coincide, even on finite sets. In case any two elements of a finite subset of a Riesz space are not comparable, this set has a supremum but may not have a maximum. On the other hand, a finite set whose elements are taken from a Riesz space has a maximum when all the elements are comparable. For instance, if we assume endowed with usual ordering, it can be clearly understood that and coincide. Hence, every ordered quasi contraction is an ordered vectorial quasi contraction, but the reverse is not true (see Example 1). Since any ordered Kannan, Chattarjee and Zamfirescu contraction is an ordered quasi contraction, similar deductions can be made for this type of contractions.
Example 1. Let and It is clear that the relation ⪯ defined asis a partial ordering on X. Let X be equipped with this relation and be equipped with coordinatewise ordering. Then E is a Riesz space and X is a vector metric space with the map defined as . Suppose the function is defined as It is clear that, for all with the function assuresfor . That is is an ordered vectorial quasi contraction. However, is neither a quasi contraction nor an ordered quasi contraction; hence, the results for quasi contractions and ordered quasi contractions can not be applicable to this example. In particular, if we assume and , we see that As a result, does not exist while .
As already mentioned, an ordered quasi contraction is an extension of a quasi contraction. Likewise, an ordered vectorial quasi contraction is an extension of an ordered quasi contraction. However, we do not give fixed point results for this type of contraction because the contraction we focused on, which is ordered vectorial almost contraction, is a generalization of ordered vectorial quasi contraction.
Definition 4. A function on an ordered vector metric space X to itself is named ordered vectorial almost contraction if there is a real number and some satisfyingfor all where The following proposition gives us a relationship between ordered vectorial quasi contractions and ordered vectorial almost contractions.
Proposition 1. Let be an element of the set Then any ordered vectorial quasi contraction with is an ordered vectorial almost contraction.
Since the proof can be made in a similar way as in Proposition 3 in [
6], it is omitted. This proposition is not a necessary condition for an ordered vectorial quasi contraction to be an ordered vectorial almost contraction. In Example 1 for
and
, we have
, which is not a member of the set
and
since
. However, if we take
and
, then we see that
is an ordered vectorial almost contraction.
Now, we give a fixed point theorem for ordered vectorial almost contractions. Throughout the rest of the work, unless otherwise stated, we assume that X is E-complete vector metric space and it is endowed with an order relation “⪯”, and is an Archimedean Riesz space.
Theorem 2. Let is an ordered vectorial almost contraction where and . Suppose that is order-preserving, and one of the following is satisfied
(i) is vectorial continuous;
(ii) For any order-preserving sequence in X, if then for all .
If there exists with , then has a fixed point.
Proof. Let us define a sequence
as
for all
. Since
is order-preserving, we obtain
. Following this process leads us to the result
for all
. It is clear that
is an order-preserving sequence. The function
is an ordered vectorial almost contraction; hence, for a real number
and some
Since
, we have
and we obtain
for all
. Hence, for any
, we deduce that
Due to the fact that and E is an Archimedean Riesz space, it is clear that is E-Cauchy sequence. By E-completeness of X, we deduce that there exists an element x in X such that . That is, there exists a sequence such that for all n and For the rest, we investigate two cases separately.
(i) In case
is vectorial continuous, there exists a sequence
such that
and
for all
n where
is a sequence in
X and
So
for all
. Since
, we have
.
(ii) For any order-preserving sequence
in
X, let
for all
n whenever
. Then, we have
for all
. Since
, so
does. Thus
, that is,
x is a fixed point of
. □
Although this theorem presents the existence of a fixed point, it is not sufficient to say that the fixed point is unique.
Example 2. Let , and it is equipped with an ordering ⪯ defined for any asfor all where “≤” is usual ordering on . Let also be equipped with coordinatewise ordering, and the map is defined asfor all . Suppose be the identity map. That is, for all It is clear that is an ordered vectorial almost contraction with a positive and some . All hypotheses of Theorem 2 are satisfied, but has infinitely many fixed points. Another important role of this example is showing us that any ordered vectorial almost contraction need not to be an ordered vectorial quasi contraction. Indeed, since is the identity map, and for any with , we have Hence, there is not at least one satisfying That is, is not an ordered vectorial quasi contraction.
By applying an additional condition to Theorem 2, we can obtain the uniqueness of the fixed point.
Theorem 3. Let all hypotheses of Theorem 2 be satisfied. Let the function satisfy the propertyfor all where , a and some . If there exists a comparable element for any two elements in X, then T has a unique fixed point. Proof. Let x and y be two fixed points of We have to investigate two cases separately again.
(i) Let
x and
y be comparable elements. Since
we obtain
. That is
.
(ii) Let
x and
y be incomparable elements. Then there exists an element
w comparable with
x and
y. Since
is order-preserving,
is comparable with
w and as a result with
x and
y for any
Hence, we have
for all
Since
and
E is an Archimedean Riesz space,
. Thus, we have
By (i) and (ii), we say that the fixed point of
is unique. □
Example 3. Let and . It is clear that the relation ⪯ defined asis a partial ordering on Let X be equipped with this relation and be equipped with coordinatewise ordering. E is an Archimedean Riesz space, and X is an E-complete vector metric space with the map defined as . Suppose that the function is defined as It is clear that is order-preserving, and for all with satisfiesfor and any . In addition to that, satisfiesfor and any . Additionally, if a sequence in X is order-preserving, then for all . Hence, we see that all hypotheses of Theorem 3 are satisfied, and 0 is the only fixed point of . On the other hand, if we take and , we see that , while and . As a result, we could not find at least one satisfying (1). That is, we can not apply Theorem 1 to this example. Another important aspect of this example is the function is an ordered vectorial quasi contraction for any . However, again if we assume and , we see that there is no such a satisfying (2). So, the results for quasi contraction on metric spaces can not be applicable to this example.