1. Introduction
Fixed-point theory is a powerful tool for solving pure mathematics and other applied science problems such as computer science, engineering, and so on. Following Banach’s pioneering fixed-point result (also known as the Banach contraction mapping principle) in 1922, many authors have published numerous research papers. The famous Banach fixed-point theorem has been generalized by many researchers from different directions in various structures.
In 1969, one of the essential such generalizations was reported by Meir and Keeler in the article “
A theorem on contraction mappings, [
1]”. More precisely, in this article, the authors introduced the notion of weakly uniformly strict contraction, later called the Meir–Keeler contraction. It is worth mentioning that it is one of the most cited studies in the fixed-point theory literature.
In this study, we conduct a literature review in which we systematically collect studies on Meir–Keeler contraction by starting with the original result of Meir–Keeler [
1]. On the other hand, since there are too many articles and results on this subject, we had to limit ourselves to focus on the Meir–Keeler contraction studies in the framework of the partial metric spaces (and extensions of these spaces;
M- metric and metric-like). This paper will present only the primary studies on this subject.
This paper consists of five sections. In the first section, we shall recollect some fundamental contraction results, including the Meir–Keeler contraction theorem. In
Section 2, we first give the definition of standard partial metric spaces, some critical properties, and examples. Next, we examine Meir–Keeler contraction studies on partial metric space. In addition, we examine Meir–Keeler contraction studies on metric-like space in
Section 3. The Meir–Keeler results on
M-metric space are collected in
Section 4. The last section is reserved for the conclusion.
1.1. Some Definitions and Fundamental Results
In this section, we shall fix some notations and recall some basic definitions and well-known results. Throughout this paper, we denote by the set of all positive integers, that is, . We denote by the set of integers, that is, and and . The symbols the set of all real numbers, denotes the set of all non-negative real numbers, that is, and . Throughout the paper, all considered sets will be presumed nonempty.
Definition 1. Let X denotes a complete metric space with distance function d, and T a function mapping X into itself.
- (i)
(Banach [2]) There exists a number , such that, for each , - (ii)
(Rakotch [3]) There exists a monotone decreasing function such that, for each , - (iii)
(Edelstein [4]) For each , - (iv)
(Kannan [5]) There exists a number , such that, for each , - (v)
(Reich [6]) There exist nonnegative numbers satisfying such that, for each , - (vi)
- (a)
There exists a number , such that, for each - (b)
There exists a number , such that, for each , - (c)
For each ,
- (vii)
(Hardy-Rogers [8]) There exist nonnegative constants, such that, for each , - (viii)
(Ćirić [9]) There exists a constant , such that, for each
Theorem 1 (Fisher [
10] and Khan [
11])
. Let be a metric space and T be a self map on X satisfying the following: if andThen T has a unique fixed point . Also, for each , the sequence converges to .
Theorem 2 (Gupta and Saxena, [
12])
. Let be a complete metric space and let T be a continuous mapping from X into itself satisfying for all , where are constants with and . Then T has a unique fixed point . Also, for all , the sequence converges to Here, we give notations of admissible mappings.
Definition 2. For a nonempty set X, let and .
- (i)
(Samet et al. [13]) We say that T is α-admissible if - (ii)
(Karapınar et al. [14]) A selfmapping T is called triangular α-admissible if - (1)
T is α-admissible and
- (2)
and , for any
- (iii)
(Aydi et al. [15]) Let be a mapping. We say that is a generalized α-admissible pair if for all , we have - (iv)
(Popescu [16]) We say that T is α-orbital admissible if Also, T is called triangular α-orbital admissible if T is α-orbital admissible and
1.2. 1969, Meir and Keeler, a Theorem on Contraction Mappings, [1]
We presume that
is a complete metric space. For a self mapping
T on
X, if there is a real number
such that
for all
, then
T possesses a unique fixed point (i.e., there exists a point
such that
).
This is the initial metric fixed-point result that was stated and proved by Banach [
2]. Meir and Keeler [
1] extended Banach’s metric fixed point theorem by replacing contraction condition with “weakly uniformly strict contraction” as follows:
Theorem 3. We presume that is a complete metric space. For a self mapping T on X, if for each given , there is a so thatthen T has a unique fix point . Moreover, we havefor any initial point . Proof. We first observe that (
1) trivially implies that
T is a strict contraction, i.e.,
Thus, T is continuous and it has at most one fixed point. □
In particular, we have the following known result that can be proved, easily, see e.g., Chu and Diaz [
17].
Lemma 1 ([
17])
. If is a strict contraction and if, for every , is a Cauchy sequence, then T has a unique fixed point and (2) holds. Lemma 2. Condition (1) implies that The proof is by contradiction. Let
. From (
3),
is decreasing with
n. If
, then (
1) fails for
where
is chosen less than
.
Having proved Lemma 2, we now suppose that some sequence is not a Cauchy sequence. Then there exists
such that
. By hypothesis, there exists a
, such that
Formula (
4) will remain true with
replaced by
. From Lemma 2 we can find
M so that
. Pick
so that
. For
j in
,
This implies, since
and
, that there exists
j in
with
However, for all
m and
j,
Therefore, by (
4) and (
5),
which contradicts (
5). This contradiction proves that
must be a Cauchy sequence, and establishes our theorem.
Other authors have extended Banach’s theorem in other ways. We will show that our theorem implies some of their results.
In [
4], Edelstein considers locally contractive mappings and derives as a corollary that any strict contraction of a compact space has a unique fixed point. This result also follows from our theorem, since in a compact space, any strict contraction
is weakly uniformly strict. To prove this, we consider
Since X is compact, this infimum is achieved for some pair of points with . Since T is a strict contraction .
Rakotch [
3] and Boyd and Wong [
18] work with a function
satisfying the following conditions
for all
and
is right upper semicontinuous, assume the inequalities
(as well as other conditions).
The following example shows that (
6) may be violated while the hypothesis (
1) of our theorem is fulfilled.
Example 1. Let with the Euclidean distance, and let be defined as follows: Although T satisfies our condition (1), would have to be 1. On the other hand, Rakotch’s [3] in Corollary monotone), and Boyd and Wong’s [18] in Theorem 1 upper semicontinuous) and Theorem 2 metrically convex) follow easily from our theorem. The following example shows that (
6) may be satisfied in a complete metric space, while the mapping
T has no fixed point. This resolves the question posed by Boyd and Wong.
Example 2. Let , and let . Let for all n. Thenbut there is no fixed point. 1.3. Partial Metric Spaces
First, we give the partial metric space concept.
The concept of metric space is first formulated axiomatically as “
L-space” by Frechét (1906) [
19] and later used as “metric space” (or standard metric) by Hausdorff. Metric space is used effectively in solving problems in many fields, both in natural sciences and applied sciences, beyond mathematics.
To overcome the difficulties of the problems, the concept of metric spaces has been expanded and improved in various ways. Some generalized metric spaces are partial metric, b-metric, metric-like, M-metric, modular metric, quasi metric, etc. In this paper, we focus on very interesting and real generalization metric spaces among them, namely partial metric spaces.
The idea of partial metric spaces, a generalization of metric spaces, is introduced by Matthews [
20] in the early 1990s to handle computer science problems. The most important difference of the partial metric rather than the metric is that in the partial metric, the self-distance,
, need not have to be zero. In other words, partial metric spaces have the existing probability of non-zero self-distance. Furthermore, the topologies of these spaces are quite different from each other. The partial metric space limit is not unique. Recently, some interesting research and survey papers for fixed-point theory on partial metric spaces are published in [
21,
22,
23,
24,
25,
26,
27,
28,
29,
30,
31,
32].
We first recall the definition of partial metric spaces and then some of their critical properties and examples.
Definition 3 ([
20])
. Let X be a nonempty set and let satisfy- (p1)
(symmetry);
- (p2)
if then (equality);
- (p3)
(small self-distances);
- (p4)
(triangularity);
for all . Then the pair is called a partial metric space and p is called a partial metric on X.
Remark 1 ([
20])
. Let a partial metric p on X, the function given by is a metric on X. A partial metric p on X generates a topology on X, having as base the family of open p-balls , where and . Example 3 - (i)
Let and define Then is a partial metric space.
- (ii)
Let and p on X defined by Then is a partial metric space.
- (iii)
Let and given by Then is a partial metric space.
Example 4 (see [
34])
. Let and be a metric space and partial metric space, respectively. Functions given by defined partial metrics on X, where is an arbitrary function and . Definition 4 (see e.g., [
20,
21,
22,
23,
32,
35])
. Let be a partial metric space and be a sequence in .- (i)
A sequence converges to if .
- (ii)
A sequence is called Cauchy if exists (and is finite).
- (iii)
A partial metric spaces is said to be complete if every Cauchy sequence in X converges, with respect to , to a point such that .
- (iv)
A mapping is said to be continuous at if for every , there exists such that )
- (v)
A sequence in is called 0-Cauchy sequence if The space said to be 0-complete if every 0-Cauchy sequence in X converges with respect to to a point such that .
Lemma 3 (see e.g., [
20,
21,
23,
24,
26])
. Let be a partial metric space and be a sequence in .- (a)
If as in with , then we have - (b)
A partial metric spaces is complete if and only if the metric space is complete. Furthermore, - (c)
A sequence is Cauchy in if and only if is Cauchy in .
- (d)
Consider endowed with the partial metric given by for all . Let be a nondecreasing function. If T is continuous with respect to the metric spaces for all , then T is continuous with respect to the partial metric p.
We recall Banach fixed-point theorem in partial metric spaces, as follows;
Theorem 4 ([
20])
. Let T be a mapping of a partial metric space into itself such that: for all , where λ is real number with . Then T has a unique fixed point. 1.4. Metric-like Spaces
The idea of dislocated metric space was first defined by Hitzler and Seda [
36]. In 2012, Amini-Harandi [
37] reintroduced the idea of the dislocated metric space under the name “metric-like”, for more details, see e.g., [
37,
38,
39,
40,
41,
42]. In this study, we prefer the metric-like spaces. The notation of metric-like space is a generalization of that of partial metric space. Now, we recall metric-like (or dislocated metric) notations.
Definition 5 ([
36,
37])
. Let X be a nonempty set. A function is called a metric-like on X if , the following conditions hold:- ()
(nonnegativity),
- ()
(pseudo-indistancy),
- ()
(symmetry),
- ()
(triangularity).
The pair is then called a metric-like space.
Each partial metric space is a metric-like spaces. But the converse is not true. In addition, we give a metric-like which is not a partial metric and so nor metric. Examples that support these remarks are as follows:
Example 5 - (i)
The pair , where is given byis metric-like space. Also, σ is a partial metric. - (ii)
On define the metric-like σ bywhere σ is not a metric. Restricted to becomes , so we return to (i). - (iii)
Let and defined the metric-like given bywhere is not a metric and is not a partial metric.
Definition 6 (See [
37,
42])
. Let be a metric-like space. - (a)
A sequence in X is a Cauchy sequence if exists and is finite.
- (b)
is complete if every Cauchy sequence in X converges with respect to to a point ; that is,
1.5. M-Metric SPACES
In 2014, Asadi et al. [
44] defined the idea of
M-metric spaces.
M-metric spaces are generalizations of standard metric spaces and partial metric spaces.
Definition 7 ([
44])
. Let X be a non-empty set. A function is called a M-metric if the following holds are satisfied:- ()
,
- ()
,
- ()
,
- ()
.
Here we have used the following notations:
- (a)
- (b)
Then the pair is called an M-metric space.
Each partial metric space is an M-metric space, but the converse is not true. Examples that support the remark are as follows:
Example 6 - (i)
Let . Then on X is an M-metric.
- (ii)
Let . Given m on as follows: So, is an M-metric space but it is not a partial metric space.
- (iii)
Let be a metric space. Let be a one to one and nondecreasing or strictly increasing mapping, with given such that Then is an M-metric.
Example 7 ([
45])
. Let ; define the function m on as follows , . Then it is easy to verify that m is an M-metric space but it is not a partial metric space because it does not satisfy the triangle inequality .We said that , for all . Let defined by and define the self-map by for all and . Take and define the corresponding open balls . The open balls are single sets; ∀. So, the M- metric topology on X is the discrete topology and, thus, each map defined on X is lower semicontinuous. Also, , we notice that . So, the function T satisfies the hypotheses of Theorem 20 of [45] and so it has a fixed point. Then, are fixed points. 2. Meir–Keeler Contractions on Partial Metric Spaces
2.1. 2012, Aydi, Karapınar and Rezapour, a Generalized Meir–Keeler-Type Contraction on Partial Metric Spaces, [46]
In [
46], the authors proposed a new notion, namely, a generalization of the Meir–Keeler type contractions on partial metric spaces. First, we give main definition.
Definition 8. Let be a partial metric space. A selfmapping T on is said to be a generalized Meir–Keeler-type contraction if for any there is a so thatwhere . Remark 2. It is evident that for the generalized Meir–Keeler-type contraction T, we always have In case , we have due to (8). Further, if , by (7) we derive the strict inequality . Proposition 1. Let be a partial metric space and a generalized Meir–Keeler-type contraction. Then, for all .
Theorem 5. Let be a 0-complete partial metric space, and let be an orbitally continuous generalized Meir–Keeler-type contraction. Then, T possesses a unique fixed point . In addition, for each and .
Example 8. Let be the set equipped with the partial metric . Clearly, is a 0 -complete partial metric space. Consider defined by . Given , we will show that there exists such that (7) holds for all . Without loss of generality, take . Then, it is easy to show that By letting , we find that (7) holds. Further, due to Lemma 3, we conclude that T is continuous. Consequently, it is orbitally continuous. As a result, all conditions of Theorem 5 are fulfilled. Thus, is the required unique fixed point of T. Example 9. Let be the interval equipped with the partial metric . Consider defined by Take . Given , we have the two following cases.
Case 1 . We have Case 2 and or . We have In each case, it suffices to take in order that (7) holds. Again, by Lemma 3, the mapping T is continuous, and hence it is orbitally continuous. All hypotheses of Theorem 5 are satisfied and is the unique fixed point of T. 2.2. 2012, Aydi and Karapınar, a Meir–Keeler Common Type Fixed-Point Theorem on Partial Metric Spaces, [47]
In this section, we introduce the common fixed-point results of two pairs of weakly compatible self-mappings for the Meir–Keeler type contraction in partial metric space. Now, we give the following results.
Theorem 6. Let , and T be the self maps defined on a complete partial metric space satisfying the following conditions:
- (C1)
and ,
- (C2)
for all , there exists such that for all where - (C3)
- (C4)
for all
If one of the ranges , and is a closed subset of , then
- (I)
A and S have a coincidence point,
- (II)
B and T have a coincidence point.
Moreover, if A and S, as well as, B and T are weakly compatible, then , and T have a unique common fixed point.
Corollary 1. Let , and T be the self maps defined on a partial metric space satisfying the following conditions:
- (C1)
and ,
- (C2)
for all , there exists such that for all where - (C3)
for all with ,
- (C4)
for all x, and .
If one of , or is a complete subspace of X, then
- (I)
A and S have a coincidence point,
- (II)
B and T have a coincidence point.
In addition, if A and S, as well as, B and T are weakly compatible, then , and T have a unique common fixed point.
Denote with the family of nondecreasing functions such that for all , where is the nth iterate of . The following lemma is obvious.
Lemma 4. If , then for all .
Some Equivalence Statements of Meir–Keeler Contraction
We need the following lemma established by Jachymski [
48].
Lemma 5. Let Q be a subset of . Then the following statements are equivalent:
- (J1)
There exists a function such that for any and
- (J1a)
and
- (J1b)
Q and imply .
- (J2)
There exist functions such that, for any , and Q and imply .
- (J3)
There exists an upper semi continuous function such that φ is non-decreasing, for , and implies .
- (J4)
There exists a lower-semi continuous function such that for any δ is non-decreasing for any , and and imply .
- (J5)
There exists a lower-semi continuous function such that for any w is non-decreasing, for and Q implies .
Theorem 7. Let be a partial metric space, and be self-mappings on X. For and for , we define Then the following statements are equivalent.
- (JT1)
There exists a lower-semi continuous function such that, for any and for any and distinct - (JT2)
There exists an upper-semi continuous function such that, φ is non-decreasing, , andfor any and distinct . - (JT3)
There exists a lower-semi continuous function such that, w is non-decreasing, for , andfor any and distinct .
Remark 3. Ćirić et al. [49] assumed that the hypothesis is satisfied for all with and obtained a common fixed-point result. In particular from the assumptions on that holds for and . So, by Theorem 7, (JT1) holds, that is; for all , there exists such that for all x, By Lemma 5 of Jachymski [50], (9) implies (as in metric cases) that the conditions (C2) and (C3) are satisfied, but nothing on the condition (C4). Conversely, in Theorem 6 we have assumed that (C2) and (C3) hold, but we added another condition which is (C4) in order to get a common fixed-point result. Remark 4. Theorem 6 is the analogous of Theorem 1 of Rana et al. [51] on partial metrics, except that the conditions (9) and the fact that , are replaced by the weaker conditions (C2), (C3) and . The condition on a and b is modified due to the fact that may not equal to 0 for . Also, Corollary 1 extends Theorem of Bouhadjera and Djoudi [52] on partial metric cases. Note that Theorem in [52] was improved recently by Akkouchi ([53], Corollary 4.4). Indeed, the Lipschitz constant k is allowed to take values in the interval instead of the case studied in [52], where the constant k belongs to the smaller interval . 2.3. 2012, Erduran and Imdad, Coupled Fixed-Point Theorems for Generalized Meir–Keeler Contractions in Ordered Partial Metric Spaces [54]
Bhaskar and Lakshmikantham [
55] introduced the concept of coupled fixed points and studied some coupled fixed-point theorems.
We achieve this by first considering a function having the mixed monotone property:
Definition 9 ([
55])
. Let be a partially ordered set and . We say that T has the mixed monotone property if is monotone nondecreasing in x and is monotone nonincreasing in y, that is, for any ,and, . This definition coincides with the notion of a mixed monotone function on and ≤ represents the usual total order in .
Definition 10 ([
55])
. We call an element a coupled fixed point of the mapping T ifWe assume that f and T are related by the relation .
Definition 11 ([
55])
. Let be a partially ordered set and d be a metric on X such that is a complete metric space. Further, we endow the product space with the following partial order: Example 10 ([
56])
. Let and be defined byfor all . It is easy to see that T has a unique coupled fixed point . Theorem 8 ([
55])
. Let be a continuous mapping having the mixed monotone property on X. Assume that there exists a withIf there exists such that Then, there exist such that Denote with the family of nondecreasing functions such that for all , where is the nth iterate of . The following lemma is obvious.
Lemma 6. If , then for all .
We recall an important example of coupled fixed point.
Example 11 ([
57])
. Let and be the Euclidean metric. Define a mapping defined byGiven a function as Here, T is mixed monotone, but we said that it does not satisfy Theorem 3.4 condition in [58]. We have such thatequations , . Let, we give in the previous inequality. So, and inequality (10) turns into Such that for any . So, inequality (11) turns intowhich is a contradiction. So, ([58] Theorem 3.4) is not applicable to the operator T in order to prove that is the unique coupled fixed point of T. Now, we present coupled fixed-point results for Meir–Keeler contraction.
Theorem 9 ([
59])
. Let be a partially ordered set and p be a partial metric on X such that is a complete partial metric space. Let be a continuous mapping having the mixed monotone property on X. Assume that for with and , we havewhere . If there exists such that and , then T has coupled fixed point. Furthermore, . Definition 12. Let be a partially ordered partial metric space and be a given mapping. We say that T is a generalized Meir–Keeler type function if for all there exists such that Proposition 2. Let be a partially ordered partial metric space and be a mapping. If (12) is satisfied, then T is a generalized Meir–Keeler type function. Lemma 7. Let be a partially ordered partial metric space and be a given mapping. If T is a generalized Meir–Keeler type function, thenfor all or for all . Lemma 8. Let be a partially ordered partial metric space and be a given mapping. Assume that the following hypotheses hold:
- (i)
T has the mixed strict monotone property,
- (ii)
T is a generalized Meir–Keeler type function,
- (iii)
such that and .
Remark 5. Lemma 8 holds if we replace (iii) by: Theorem 10. Let be a partially ordered set and suppose there is a metric p on X such that is complete partial metric space. Let be mapping satisfying the following hypotheses:
- (i)
T is continuous,
- (ii)
T has the mixed strict monotone property,
- (iii)
T is a generalized Meir–Keeler type function,
- (iv)
such that and .
Then, there exists such that and .
Theorem 11. Let be a partially ordered set. Suppose that there is a metric p on X such that is complete partial metric space. Assume that X has the following properties:
- (a)
if a nondecreasing sequence , then for all n,
- (b)
if a nonincreasing sequence , then for all n.
Let be mapping satisfying the following hypotheses:
- (c)
T is continuous,
- (d)
T has the mixed strict monotone property,
- (e)
T is a generalized Meir–Keeler type function,
- (f)
such that and .
Then, there exists such that and . Furthermore,
2.4. 2012, Erhan, Karapınar and Türkoğlu, Different Types Meir–Keeler Contractions on Partial Metric Spaces, [60]
In this section, we prove the existence fixed point for Meir Keeler type contraction of self-mapping T defined in complete partial metric spaces
Firstly, using Hardy-Rogers [
8], we give the Definition 13.
Definition 13. Suppose that is a self-mapping satisfying the following condition: Given there exists such thatwhere Then T is called Hardy-Rogers type Meir–Keeler contraction.
Proposition 3. Let be a complete partial metric space. Suppose that is a Hardy-Rogers type Meir–Keeler contraction. Then, for we have , as .
The theorem from below generalizes Meir–Keeler contraction.
Theorem 12. Let be a complete partial metric space. Suppose that is a Hardy-Rogers type Meir–Keeler contraction. Then, T has a unique fixed point, say . Moreover, for all .
In Definition 14, we define Kannan [
5] type Meir Keeler contraction, Reich [
6] type Meir Keeler contraction and Chatterjee [
7] type Meir Keeler contraction on partial metric space respectively.
Definition 14. Let be a partial metric space. Let be a self-mapping satisfying the following:
(1) Given there exists such thatwhere Then T is called Kannan type Meir–Keeler contraction.
(2) Given there exists such thatwhere Then T is called Reich type Meir–Keeler contraction.
(3) Given there exists such thatwhere Then T is called Chatterjee type Meir–Keeler contraction.
Theorem 12 is valid also for the notations in Definition 14.
Corollary 2. Let be a complete partial metric space. Suppose that is Meir–Keeler contraction of Kannan type or Reich type or Chattergee type. Then, T has a unique fixed point, say . Moreover, for all .
2.5. 2012, Hussain, Kadelburg, Radenović and Al-Solamy, Comparison Functions and Fixed-Point Results in Partial Metric Spaces, [61]
Contractive conditions with comparison function
of the give form
have been used for obtaining (common) fixed-point theorems of mappings in metric spaces until now the celebrated result of Boyd and Wong [
18]. Also, interesting and different assumptions for the comparison function
can be given as follows.
We consider the following properties of functions . will denote the nth iteration of :
- (I)
for each and for each ,
- (II)
is nondecreasing and for each ,
- (III)
is right-continuous, and for each ,
- (IV)
is nondecreasing and for each .
Lemma 9. - (1)
(II) ⇒ (I).
- (2)
(III) is nondecreasing ⇒ (II).
- (3)
(IV) ⇒ (II).
- (4)
(III) and (IV) are not comparable (even if φ is nondecreasing).
Now, we define Meir–Keeler function via a comparison function in partial metric spaces.
The function
will be called a Meir–Keeler function if
Theorem 13. Let be a complete partial metric space. If satisfies the following conditionfor a Meir–Keeler function φ. Then T has a unique fixed point, say , and for each , the Picard sequence converges to , satisfying . Example 12. Let and d be as in (Example 5.2 in [61]). Consider mapping and function given by Then φ is a Meir–Keeler function. Indeed, for arbitrary choose We will check that T satisfies condition (13) of Theorem 13. In the cases ; and , the left-hand side of (13) is equal to zero. In all other cases ; and , it is Hence, condition (13) always holds true, and mapping T has a unique fixed point . Note again that in the case when standard metric d is used instead of partial metric p, this conclusion cannot be obtained. Indeed, for we have that 2.6. 2013, Chen and Chen, Fixed-Point Results for Meir–Keeler-Type -Contractions on Partial Metric Spaces, [62]
In [
62], Chen and Chen introduce generalized Meir–Keeler-type
-
-contractions on partial metric spaces.
In the section, we denote by the class of functions satisfying the below holds:
is an increasing and continuous function in each coordinate;
for ; and if and only if .
We define generalized Meir–Keeler-type -contractions and new Meir- Keeler-type -contractions in partial metric spaces respectively.
Definition 15. Let be a partial metric space, and . Then T is called a generalized Meir–Keeler-type ϕ-contraction whenever, for each , there exists such that Definition 16. Let be a partial metric space, , and . Then T is called a generalized Meir–Keeler-type -contraction if the following conditions hold:
- (1)
T is α-admissible;
- (2)
for each , there exists such that
Remark 6. Note that if T is a generalized Meir–Keeler-type -contraction, then we have that for all , Also, if , then .
So, if
,
then
.
Theorem 14. Let be a complete partial metric space, and If satisfies the following conditions:
there exists such that ;
if for all , then ;
is a continuous function in each coordinate.
Suppose that is a generalized Meir–Keeler-type -contraction. Then T has a fixed point in X.
Theorem 15. Let be a complete partial metric space and . If X is a generalized Meir–Keeler-type ϕ-contraction, then T has a fixed point in X.
Here, we present Example 13 to support Theorem 15.
Example 13. Let . We define the partial metric p on X by Let be defined aslet be defined asand, let denote Then T is α-admissible.
Without loss of generality, we assume that and verify the inequality (14). For all with , we haveand hence Therefore, all the conditions of Theorem 14 are satisfied, and we obtained that 0 is a fixed point of T. If we letthen it is easy to get the following theorem. 2.7. 2014, Imdad, Sharma and Erduran, Generalized Meir–Keeler Type n-Tupled Fixed-Point Theorems in Ordered Partial Metric Spaces, [63]
In this section, we investigate n-tupled (for even n) fixed-point theorems on ordered partial metric spaces. We use the Meir–Keeler type contraction besides the mixed monotone property to obtain these fixed-point results.
We think n to be an even integer.
Let be a partial metric. We endow times with the partial metric defined for by
Let
be a given mapping. Then for all
and for all
,
, we denote
We give main results, as follows:
Definition 17. Let be a partially ordered set and be a mapping. The mapping T is said to have the mixed strict monotone property if T is nondecreasing in its odd position arguments and nonincreasing in its even position arguments, that is, if,
- (i)
,
- (ii)
,
- (iii)
, ⋮
- (iii)
.
Definition 18. Let be a partially ordered partial metric space and be a given mapping. We say that T is a generalized Meir–Keeler type function if for all 0 there exists such that for with Lemma 10. Let be a partially ordered set and suppose that there is a partial metric p on X such that is a complete partial metric space. Let be a given mapping. If T is a generalized Meir–Keeler type function, then for with or . Lemma 11. Let be a partially ordered set and suppose that there is a partial metric p on X such that is a complete partial metric space. Let be a given mapping. Assume that the following hypotheses hold:
- (1)
T has the mixed strict monotone property,
- (2)
T is a generalized Meir–Keeler type function,
- (3)
there exist with .
Theorem 16. Let be a partially ordered set and suppose that there is a partial metric p on X such that is a complete partial metric space. Let be a given mapping satisfying the following hypotheses:
- (1)
T is continuous,
- (2)
T has the mixed strict monotone property,
- (3)
T is a generalized Meir–Keeler type function,
- (4)
there exist such that
Then there exist such that , .
Theorem 17. Let be a partially ordered set and suppose that there is a partial metric p on X such that is a complete partial metric space. Let be a given mapping. Assume that there exists a function θ from into itself satisfying the following:
- (1)
and for every ,
- (2)
θ is nondecreasing and right continuous,
- (3)
for every , there exists such that
for all . Then T is a generalized Meir–Keeler type function. Example 14. Let . Then is a partially ordered set under the natural ordering of real numbers. Define by . Then is a complete partial metric space.
Now for any fixed even integer , consider the product space times (in short we write ). Define by
Then T has the mixed strict monotone property. Also T is a generalized Meir–Keeler type function. The proof follows in two parts, that is, we prove the following:
For with , The first part is trivial. For second part, we have
Hence all the hypotheses of Theorem 16 are satisfied. Therefore, T has a unique n-tupled fixed point. Here is an n-tupled fixed point of T.
2.8. 2014, Nashine and Kadelburg, Fixed-Point Theorems Using Cyclic Weaker Meir–Keeler Functions in Partial Metric Spaces, [64]
In this section, we give fixed-point results via cyclic weaker Meir Keeler functions in 0-complete partial metric spaces.
First, we recall the definition and fixed-point theorem of the cyclic, as follows;
Definition 19 (Kirk et al. [
65])
. Let X be a nonempty set, and let be a self-mapping. Then is a cyclic representation of X with respect to T if- (a)
, mare non-empty subsets of X;
- (b)
.
Theorem 18 ([
65])
. Let be a complete metric space, and let be a cyclic representation of X with respect to T. Suppose that T satisfies the following conditionwhere and is a function, upper semi-continuous from the right and for . Then, T has a fixed point . Definition 20 (Pǎcurar and Rus [
66])
. Let be a metric space, be closed nonempty subsets of X and . An operator is called a cyclic weaker φ-contraction if- (1)
is a cyclic representation of X with respect to T;
- (2)
there exists a continuous, non-decreasing function with for and such that
for any , where . Theorem 19 ([
66])
. Suppose that T is a cyclic weaker φ-contraction on a complete metric space . Then, T has a fixed point . Here, we introduce fixed-point results via cyclic weaker Meir Keeler conditions in 0-complete partial metric spaces.
Definition 21. As in [67], we assume in this section the following conditions for a weaker Meir–Keeler function ψ: for and ;
for all is decreasing;
for , we have that
- (a)
if , then , and
- (b)
if , then .
Also suppose that is a non-decreasing and continuous function satisfying:
for and ;
φ is subadditive, that is, for every ;
for all if and only if .
The version of Chen [
67]’s definition (in metric space) of cyclic weaker
-contraction in partial metric spaces is as follows.
Definition 22. Let be a partial metric space, be nonempty subsets of X and . An operator is called a cyclic weaker -contraction if
- (1)
is a cyclic representation of X with respect to T;
- (2)
for any where .
Theorem 20. Let be a 0-complete partial metric space, be nonempty closed subsets of and . Suppose that is a cyclic weaker -contraction. Then, T has a unique fixed point . Moreover, .
Now, we give an example that supports Theorem 20.
Example 15. Let and a partial metric be given by If a mapping is given byand , then is a cyclic representation of X with respect to T. Moreover, mapping T is a cyclic weaker -contraction, where and . Indeed, consider the following cases: Case 1 or . Then and relation (15) is trivially satisfied. Case 2 or . Then and . Relation (15) holds as it reduces to . We conclude that T has a unique fixed point (which is ).
Note that, if instead of the given partial metric p its associated metricis used, then for and the respective condition (i.e., condition (ii) of Definition 4 from [67]) is not satisfied since it reduces to Similar conclusion is obtained if the standard Euclidean metric is used.
Hence, this example shows that Theorem 20 is a proper extension of ([
67], Theorem 3).
Later, we introduce fixed-point results via cyclic weaker
Meir Keeler conditions. The results in this part generalized from [
66,
67,
68].
We assume
to be a weaker Meir–Keeler function satisfying conditions
and
. Also consider
to be a non-decreasing and continuous function satisfying
and
are not needed). To complete the results, we need the following notion of a cyclic weaker
-contraction, which is the counterpart of the respective notion from [
67]:
Definition 23. Let be a partial metric space, , and let be nonempty subsets of X such that . An operator is called a cyclic weaker -contraction if
- (1)
is a cyclic representation of X with respect to T;
- (2)
for any where .
Theorem 21. Let be a 0-complete partial metric space, , let be nonempty closed subsets of and . Suppose that is a cyclic weaker -contraction. Then, T has a unique fixed point Moreover, .
We give an example that supports Theorem 21.
Example 16. Let be equipped by the usual partial metric . Let be given by . Then is a cyclic representation of X with respect to T. If and , we will prove that T is a cyclic weaker -contraction. Indeed let, e.g., and , Thenand condition (16) is fulfilled. All other conditions of Theorem 21 are also satisfied and T has a unique fixed point . We also provide another example that supports the main results. The results reveal that it is stronger than from [
67].
Example 17. Let and a partial metric be given by If a mapping is given byand , then is a cyclic representation of X with respect to T. Moreover, mapping T is a cyclic weaker -contraction, where Indeed, consider the following cases:
Case 1 or . Then and relation (16) is trivially satisfied. Case 2 or . Then and . Relation (16) holds as it reduces to . We conclude that T has a unique fixed point (which is ).
Note again that, if instead of the given partial metric p its associated metric is used, then for and the respective condition (i.e., condition (ii) of Definition 5 from [67]) is not satisfied since it reduces to Similar conclusion is obtained if the standard Euclidean metric is used.
So, this example shows that Theorem 21 is a proper extension of ([
67], Theorem 4).
2.9. 2015, Choudhury and Bandyopadhyay, Suzuki Type Common Fixed-Point Theorem in Complete Metric Space and Partial Metric Space, [69]
In this section, we show that a pair of compatible mappings have unique common fixed point in partial metric spaces respectively. First we need the following Suzuki theorem.
Theorem 22 ([
70])
. Define a function θ from onto byLet be a complete metric space. T is a mapping on X. If T satisfy the followingthen T has a fixed point. Definition 24 ([
71])
. Let S and T be mappings from a metric space into itself. Then S and T are said to be compatible if whenever is a sequence in X such that for some z in X. Thus, if as , then S and T are compatible. Definition 25 ([
24])
. Let be a partial metric space and are mappings of X into itself. We say that the pair is partial compatible if the following conditions hold:;
, whenever is a sequence in X such that and for some .
Theorem 23. Let be a complete metric space. Let S be a continuous mappings on X and T be another mapping on X such that is compatible and . Also let for all and for any there exist such thatand Then there exists a unique common fixed point of S and T.
Lemma 12. Let be a partial metric space, T a self map on the constructed metric in [10] and . Thenfor all with . Theorem 24. Let be a complete partial metric space. Let S be a continuous mappings on X, and T be another mapping on X such that is partial compatible and . Also let for all and for any , there exist such thatand Then there exist a unique common fixed point of S and T.
Also, we present an example.
Example 18. Therefore be a complete partial metric space. Define two functions as follows: It is clear that , otherwise for all .
Case I: Let . Then For , there exist such that Hence the result is true for .
Case II: Let . Now, Now for the given , there exist such that Using (18), (19) and the above inequality implies . Hence the result satisfied for . 0 is the unique common fixed point of S and T.
2.10. 2016, Choudhury and Bandyopadhyay, Coupled Meir–Keeler Type Contraction in Metric Spaces with an Application to Partial Metric Spaces, [72]
In this section, we establish a Meir–Keeler type coupled fixed-point results in partial metric spaces.
Definition 26 ([
73])
. Let be a metric space and be a given mapping. T is a generalized Meir–Keeler type function if for all , there exists such that First, we give coupled Meir–Keeler contraction in metric spaces.
Definition 27. Let be a metric space. Let be a given mapping. We say that T is a coupled Meir–Keeler type contraction if it satisfies the following: and for all , there exists such that Definition 28 ([
55])
. Let X be a non-empty set and be a given mapping. An element is said to be a coupled fixed point of the mapping T if and Now, we give coupled Meir–Keeler type contraction in partial metric spaces.
Definition 29. Let be any partial metric space and be a continuous mapping. T is a coupled Meir–Keeler type contraction in partial metric space if T satisfies the following: for all there exists such that Lemma 13. Let be a metric space and satisfy (20), then Lemma 14. Let be a partial metric space, T a self map on the constructed metric in [25,74] and . Thenfor all with . Theorem 25. Let X be any non-empty set and p be a partial metric on X such that is a complete partial metric space. Let be a continuous mapping and T be a coupled Meir–Keeler type contraction in partial metric space, that is, T satisfies (21). Then T has a unique coupled fixed point. Example 19. Let is defined by It is clear that T is a continuous function.
Case 1. For then (22) is trivially satisfied. Case 2. Let for all , there exists such that Hence, from (23)–(25), we get . So, T is a coupled Meir–Keeler type contraction in partial metric space. T has a unique coupled fixed point . 2.11. 2017, Popa and Patriciu, a General Fixed-Point Theorem of Meir–Keeler Type for Mappings Satisfying an Implicit Relation in Partial Metric Spaces, [75]
In this section, Meir–Keeler type for mappings satisfying an implicit relation in partial metric spaces, which generalize (Theorem 2.3 and Corollary 2.4 [
47]) is proved.
Let be self mappings that are defined on a nonempty set X. We say that is a coincidence point of T and S if . We denote by the set of all coincidence points of T and S.
Let
be a metric space. Jungck [
71] defined
T and
S to be compatible if
whenever
is a sequence in
X such that
for some
.
The idea of pointwise
R - weakly commuting mappings is introduced by Pant [
76]. Moreover, Pant [
76] showed that the pointwise
R-weakly commuting is equivalent to commutativity in the coincidence points.
Definition 30 ([
77])
. Two self mappings T and S that are defined on a nonempty set X are said to be weakly compatible if for each . Definition 31. Let be the set of all real continuous mappings satisfying the following conditions:
: F is nonincreasing in variables ,
: implies ,
: implies .
Theorem 26. Let and T be self mappings on a complete partial metric space satisfying the following conditions:
and ,
for all , there exists such that for all ,where , for all with implies ,for all and . If one of is a closed subset of , then
- (a)
,
- (b)
.
Moreover, if A and S, as well B and T are weakly compatible, then and T have a unique common fixed point.
Theorem 27. Let and T be self mappings on a partial metric space satisfying the following conditions: and for all , there exists such that for all ,where , ,for all, for all and . If one of is a complete subspace of , then
- (a)
,
- (b)
.
Moreover, if A and S, as well B and T are weakly compatible, then and T have a unique common fixed point.
2.12. 2018, Gunasekar, Karpagam and Zlatanov, on -Cyclic Orbital MK Contractions in a Partial Metric Space, [78]
A cyclic map with a contractive type of hold named -cyclic orbital Meir–Keeler contraction is established on partial metric space. We provide fixed-point results and best proximity point results in complete partial metric spaces for these maps.
Lim [
79] introduced the concept of
L-functions. We use this function to obtain the main results.
Definition 32 ([
79])
. A function is called an L-function if for every and for every , there exists such that for . Note that every
L-function satisfies the condition
for every
. Suzuki [
80] generalized the
L-function as follows.
Lemma 15 ([
80])
. Let X be a nonempty set, and let . Then, the following are equivalent:- (1)
For each , there exists a , such that .
- (2)
There exists an L-function ϕ (which may be chosen to be a non-decreasing and continuous) such that and .
Suzuki et al. [
81] established the uniform convexity (UC) property. For the investigation of the best proximity points, the notion of the UC of a Banach space plays a crucial role. We introduce UC notion in partial metric space as follows.
Definition 33. Let be a partial metric space and A and B be subsets. The pair is said to satisfy the property UC if the following holds: If are sequences in A and is a sequence in B, such that and for any , there is , so that for all , then for any , there is , so that for .
Now, let us recall the notion of cyclic maps.
Definition 34. Let be a nonempty set and be nonempty subsets of X. A map is called a ρ-cyclic map if , for all , where we use the convention .
Definition 35. Let be a partial metric space and be nonempty subsets of X. A point is said to be a best proximity point of T in , if .
We also give the definition of -cyclic orbital Meir–Keeler contraction.
Definition 36. Let be a partial metric space, , be nonempty subsets of X and be a ρ-cyclic map. The map T is called a ρ-cyclic orbital Meir–Keeler contraction if for some , for each , there exists , such that the following condition:holds for all and for all , where , for . Theorem 28. Let be a complete partial metric space. Let be nonempty and closed subsets of X. Let be a ρ-cyclic orbital Meir–Keeler contraction map with constants equal to zero or .
- (1)
If for all , then is nonempty, and T has a unique fixed point . For any , satisfying (26) with holds. - (2)
If for all and is a partial metric space with property UC, then for every satisfying (26), the sequence converges to a unique point , which is the best proximity point, as well as the unique periodic point of T in . Furthermore, is a best proximity point of T in , which is also a unique periodic point of T in , for each .
Without loss of generality, let us assume that .
Lemma 16. Let be a partial metric space. Let be nonempty subsets of X. Let be a ρ-cyclic orbital Meir–Keeler contraction map with constants equal to zero or . Then, there exists an L-function ϕ such that for an satisfying (26), the following holds: if , then:and if , then , for each , for all and for all . Remark 7. From Lemma 16, it follows that for a ρ-cyclic orbital Meir–Keeler contraction map T, the sequence is non-increasing.
Lemma 17. Let be a partial metric space. Let be nonempty subsets of X. Let be a ρ-cyclic orbital Meir–Keeler contraction map with constants equal to zero or . Then, for any satisfying (26), for all and for each , the sequence converges to . Lemma 18. Let be a complete partial metric space with property UC and be non-empty and closed subsets of X. Let be a ρ-cyclic orbital Meir–Keeler contraction map with constants . Let satisfy (26). Suppose that for each , the sequence converges to . Then: - (a)
;
- (b)
is a best proximity point of T in and , for ;
- (c)
is a periodic point of T with period ρ in .
We use Lemma 19 to check whether a partial metric space is complete.
Lemma 19. Let be a complete metric space and be a partial metric space. Let and . The partial metric space is complete if and only if ω satisfies the condition: if , then .
Corollary 3. Let be a complete metric space and be a continuous function with respect to metric d, and let us consider the partial metric space , where . Then, if is closed in , then it is closed in .
Example 20. Let us consider the metric space , endowed with the metric Let us consider the function Then, is a complete partial metric space, where .
Let . Then, if and only if . By the continuity of ω at any different point from zero, it follows that , provided that . Let be a Cauchy sequence in .
Thus, the limit exists and is finite. This limit is finite if and only if . Consequently, is a Cauchy sequence in if and only if for some and . Consequently, is a complete partial metric space.
Example 21. Let us consider the metric space , endowed with the metric Let us consider the function Then, is a complete partial metric space with if and only if . Let . Then, and , provided that or . Let be a sequence, such that , which is convergent to one with respect to the metric d. Then, it is a Cauchy sequence in , because the limitexists and is finite. Fromit follows that is convergent in if and only if . Example 22. Let us consider the Banach space , where and . Let us endow with the partial metric . From Example 4, is a partial metric. From Lemma 19, it follows that is a complete partial metric space. We consider the sets defined by : . From Corollary 3, it follows that are closed sets in . Let us define a cyclic map by: Let . Let us choose an arbitrary . Let us denote . Then, . Now: Now, .
By solving the equation , we get The function is a continuous function in the interval . From , we get the condition that there exists such that the inequality holds whenever the inequality holds . Consequently, T is a 4-cyclic orbital Meir–Keeler contraction, and x is the unique fixed point.
2.13. 2013, Chen and Karapınar, Fixed-Point Results for the -Meir–Keeler Contraction on Partial Hausdorff Metric Spaces, [82]
We introduce fixed-point results for a multi-valued mapping satisfying the -Meir–Keeler contraction with respect to the partial Hausdorff metric H in complete partial metric spaces.
Let
be a metric space, and let
denote the collection of all nonempty, closed and bounded subsets of
X. For
, we define
where
, and it is well known that
H is called the Hausdorff metric induced by the metric
d. A multi-valued mapping
is called a contraction if
for all
and
. The study of fixed points for multi-valued contractions using the Hausdorff metric was introduced in Nadler [
83].
Theorem 29 ([
83])
. Let be a complete metric space, and let be a multivalued contraction. Then there exists such that . In [
84], authors investigated the partial Hausdorff metric
induced by the partial metric
p. Let
be a partial metric space, and let
be the collection of all nonempty, closed and bounded subset of the partial metric space
. Note that closedness is taken from
, and boundedness is given as follows:
A is a bounded subset in
if there exist
and
such that for all
, we have
For
and
, they define
It is immediate to get that if , then , where A }.
Aydi et al. [
84] also proved the following properties;
Proposition 4 ([
84])
. Let be a partial metric space. For , the following properties hold:- (1)
;
- (2)
;
- (3)
implies that ;
- (4)
.
Proposition 5 ([
84])
. Let be a partial metric space. For , the following properties hold:- (1)
;
- (2)
;
- (3)
;
- (4)
implies that .
Lemma 20 ([
84])
. Let be a partial metric space, and . For any , there exists such that Now, we give the notions of a strictly -admissible and an -Meir Keeler contraction with respect to the partial Hausdorff metric .
Definition 37. Let be a partial metric space, and . We say that T is strictly α-admissible if Definition 38. Let be a partial metric space and . We call T: an α-Meir–Keeler contraction with respect to the partial Hausdorff metric if the following conditions hold:
- ()
T is strictly α-admissible;
- ()
for each , there exists such that
Remark 8. Note that if is a α-Meir–Keeler contraction with respect to the partial Hausdorff metric , then we have that for all Further, if , then .
So, if , then .
The main theorem is as follows;
Theorem 30. Let be a complete partial metric space. Suppose that is an α-Meir–Keeler contraction with respect to the partial Hausdorff metric H and that there exists such that for all . Then T has a fixed point in X (that is, there exists such that .
2.14. 2015, Jen, Chen and Peng, Some New Fixed-Point Theorems for the Meir–Keeler Contractions on Partial Hausdorff Metric Spaces, [85]
In this section, we introduce fixed-point results for a multi-valued mapping concerning with three classes of Meir–Keeler contractions with respect to the partial Hausdorff metric H in partial metric spaces.
Remark 9. It is clear that, if the function is a Reich function (-function), then ψ is also a stronger Meir–Keeler-type function.
Now, we denote by the class of functions satisfying the following conditions:
- (1)
is an increasing and continuous function in each coordinate;
- (2)
for and if and only if .
We now introduce the notion of -Meir–Keeler contraction on partial Hausdorff metric spaces.
Definition 39. Let be a partial metric space, and . We call a -Meir–Keeler contraction with respect to the partial Hausdorff metric , if the following conditions hold:
ψ is a stronger Meir–Keeler-type function;
for all , we have
We state and prove the main fixed-point result for the -Meir–Keeler contraction with respect to the partial Hausdorff metric
Theorem 31. Let be a complete partial metric space. Suppose is a -Meir–Keeler contraction with respect to the partial Hausdorf metric . Then T has a fixed point in X, that is, there exists such that .
Inspired by the Reich function and stronger Meir–Keeler function, we establish the following notion of -Reich’s contraction with respect to the partial Hausdorff metric .
Definition 40. Let be a partial metric space, , and . We call a -Reich’s contraction with respect to the partial Hausdorff metric if the following conditions hold:
- (1)
ψ is a Reich function ( -function);
- (2)
for all , we have
Using Remark 9, Definition
Section 2.14, and Theorem 31, we are easy to get the following theorem.
Theorem 32. Let be a complete partial metric space. Suppose is a -Reich’s contraction with respect to the partial Hausdorff metric . Then T has a fixed point in X, that is, there exists such that .
We let be the class of all non-decreasing function satisfying the following conditions:
- (1)
is a weaker Meir–Keeler-type function;
- (2)
for all is decreasing;
- (3)
for and ,
- (4)
for , if , then , where ;
- (5)
for , if , then .
In [
82], the authors introduce notion of strictly
-admissible.
We now introduce the notion of -Meir–Keeler contraction with respect to the partial Hausdorff metric , as follows:
Definition 41. Let be a partial metric space, , and . We call a -Meir–Keeler contraction with respect to the partial Hausdorff metric if the following conditions hold:
T is strictly α-admissible;
We now state main result for the -Meir–Keeler contraction with respect to the partial Hausdorff metric
Theorem 33. Let be a complete partial metric space. Suppose is an -Meir–Keeler contraction with respect to the partial Hausdorff metric . Suppose also that
- (i)
there exists such that for all ;
- (ii)
if is a sequence in X such that for all n and as , then for all n.
Then T has a fixed point in X (that is, there exists such that .
Note: It is clear that if Meir–Keeler type function then, we have for all ,
We consider the family
such that:
- (1)
for all ;
- (2)
are continuous;
- (3)
if and only if ;
- (4)
is a Meir–Keeler-type function;
- (5)
for all .
We now introduce the notion of -Meir–Keeler contraction on partial Hausdorff metric spaces.
Definition 42. Let be a partial metric space, , and . We call a -Meir–Keeler contraction with respect to the partial Hausdorff metric if the following conditions hold:
- (1)
T is strictly α-admissible;
- (2)
for all , we have
We now state main result for the -Meir–Keeler contraction with respect to the partial Hausdorff metric .
Theorem 34. Let be a complete partial metric space. Suppose is a -Meir–Keeler contraction with respect to the partial Hausdorff metric . Suppose also that
- (i)
there exists such that for all ;
- (ii)
if is a sequence in X such that for all n and as , then for all n.
Then T has a fixed point in X (that is, there exists such that ).
2.15. 2018, Chen, Karapınar and O’Regan, on -Meir–Keeler Contractions on Partial Hausdorff Metric Spaces, [86]
In this section, we present a new -Meir–Keeler contraction for multi-valued mappings and we investigate the existence of fixed-point theorems in partial metric space.
Let be the family of functions satisfying the following conditions:
- (i)
is nondecreasing;
- (ii)
there exist
and
and a convergent series of nonnegative terms
such that
for
and any
.
In the literature such functions are called
-comparison functions (see [
87]).
Lemma 21 (See e.g., [
87])
. If , then the following hold:- (i)
converges to 0 as for all ;
- (ii)
, for any ;
- (iii)
ψ is continuous at 0;
- (iv)
the series converges for any .
We denote by Φ the class of functions satisfying the following conditions:
is an increasing and continuous function in each coordinate;
for each ϕ there exists such that for all ,
We establish the concept of a -Meir–Keeler contraction with respect to the partial Hausdorff metric induced by the partial metric.
Definition 43. Let be a partial metric space, be a mapping and . A multi-valued mapping is called a -Meir–Keeler contraction with respect to the associated partial Hausdorff metric if the following conditions hold:
(C) For each , there exists such thatfor all . A multi-valued mapping
T is called a
-Meir–Keeler-type contraction if
for all
in (
27), that is,
for all
.
Remark 10. Note that if is a -Meir–Keeler contraction with respect to the associated partial Hausdorff metric , then we havefor all . Notice that if , then we have We present main result.
Theorem 35. Let be a complete partial metric space and be a mapping. Suppose that a multi-valued mapping is a -Meir–Keeler contraction with respect to the associated partial Hausdorff metric . Also assume that
- (i)
T is strictly α-admissible;
- (ii)
there exists such that for all ;
- (iii)
if is a sequence in X such that for all n and as , then for all n.
Then T has a fixed point in X, that is, there exists such that .
Example 23. Let be endowed with the partial metric defined by Then is a complete partial metric space, and we have We next define by Then we have
- (1)
if , then ;
- (2)
if and ; then ;
- (3)
if ; then ;
- (4)
,
.
Now, we put . Then all of the hypotheses of Theorem 35 are satisfied. Note is the unique fixed point of T.
2.16. 2020, Afassinou and Narain, Fixed-Point and Endpoint Theorems for -Meir–Keeler Contraction on the Partial Hausdorff Metric, [88]
The purpose of this section is to prove the notion of a multivalued strictly -admissible mappings and a multivalued -Meir–Keeler contractions with respect to the partial Hausdorff metric in partial metric spaces.
Suzuki [
89] established the idea of mappings satisfying condition
which is also known as Suzuki-type generalized nonexpansive mapping.
Definition 44. Let be a metric space. A mapping is said to satisfy condition if for all , Theorem 36. Let be a compact metric space and be a mapping satisfying condition for all . Then T has a unique fixed point.
Definition 45. ([
90]).
Let be a mapping and let be two functions. Then T is called a cyclic -admissible mapping, if- (1)
for some implies that ,
- (2)
for some implies that .
In 2019, Mebawondu et al. [
91] generalized the concept of an
-admissible mapping by introducing the notion of an
-cyclic admissible mapping.
Definition 46 ([
91])
. Let X be a nonempty set, be a mapping and : be two functions. We say that T is an -cyclic admissible mapping, if for all - (1)
,
- (2)
.
Remark 11. It is easy to see that if , we obtain the result from [13]. Inspired by above results, we proved the notion of a multivalued strictly -admissible mappings and a multivalued -Meir–Keeler contraction with respect to partial Hausdorff metric in the partial metric spaces.
Now, we prove the existence theorems for fixed points of these class of mappings.
Definition 47. Let X be a nonempty set, and be three functions. We say that T is strictly -cyclic admissible mapping, if for all and with
- (1)
,
- (2)
.
Remark 12. Clearly if , we have , which is the multivalued version of α-admissible.
Lemma 22. Let X be a nonempty set and be a strictly cyclic admissible mapping. Suppose that there exists such that 1 and , where . Define the sequence by , then implies that and implies that , for all with .
Definition 48. Let be a partial metric space and be two functions. We say that is an -Meir–Keeler contraction with respect to the partial Hausdorff metric , if for each , there exists such thatfor all . Remark 13. If is an -Meir–Keeler contraction with respect to the partial Hausdorff metric , then we havefor all when . On the other hand, observe that if , we clearly have that and using [84] we obtain that . Thus, for all , we get that Remark 14. It is also easy to see that if . Consequently we have obtained the multivalued version of Meir–Keeler type contractions in the framework of partial metric spaces.
Theorem 37. Let be a complete partial metric space and be an -Meir–Keeler contraction with respect to the partial Hausdorff metric . Suppose the following conditions hold:
- (1)
T is strictly -cyclic admissible mapping,
- (2)
there exists and such that and ,
- (3)
If is a sequence such that as and , then for all .
Then T has a fixed point.
Definition 49. Let be a multivalued mapping on a partial metric space .
- (1)
An element is called an endpoint of T if . It is clear that an endpoint of T is also a fixed point of T.
- (2)
T has the approximate endpoint property if there exists a sequence such that or equivaliently if .
Theorem 38. Let be a complete partial metric space and be an -Meir–Keeler contraction with respect to the partial Hausdorff metric . Suppose the following conditions hold:
- (1)
T is strictly -cyclic admissible mapping,
- (2)
there exists and such that and ,
- (3)
If is a sequence such that as and , then for all .
Then T has an endpoint x if and only if T has the approximate endpoint property.
Definition 50. Let be a partial metric space, be a function. We say that is an α-Meir–Keeler contraction with respect to the partial Hausdorff metric , if for each , there exists such thatfor all . Remark 15. If is an α-Meir–Keeler contraction with respect to the partial Hausdorff metric . Then we havefor all when . On the other hand, observe that if , we clearly have that and using [84], we obtain that . Thus, for all , we get Corollary 4. Let be a complete partial metric space and be an α-Meir–Keeler contraction with respect to the partial Hausdorff metric . Suppose the following conditions hold:
- (1)
T is strictly α-admissible mapping,
- (2)
there exists and such that ,
- (3)
If is a sequence such that as and , then , for all .
Then T has a fixed point.
Corollary 5. Let be a complete partial metric space and be an α-Meir–Keeler contraction with respect to the partial Hausdorff metric . Suppose the following conditions hold:
- (1)
T is strictly α-admissible mapping,
- (2)
there exists and such that ,
- (3)
If is a sequence such that as and , then , for all .
Then T has an endpoint x if and only if T has the approximate endpoint property.
2.17. Meir–Keeler Contractions on Various Generalized Partial Metric
2.17.1. 2018, Zhou, Zheng and Zhang, Fixed-Point Theorems in Partial b-Metric Spaces, [92]
We investigate some fixed-point theorem for Meir–Keeler mappings in partial
-metric [
93] space which generalizes and motivate the result of Chatterjea [
7] and Shukla [
93].
Definition 51 ([
94,
95])
. Let X be a (nonempty) set and a real number. A function is a b-metric space on X if following conditions are satisfied: - (i)
if and only if
- (ii)
- (iii)
, for every .
Definition 52. Let X be a nonempty set, be a given real number and let be a mapping such that for all , the following conditions hold:
- (Pb1)
if and only if
- (Pb2)
- (Pb3)
- (Pb4)
Then the pair is called a partial -metric space. The number s is call the coefficient of .
Theorem 39. Let be a complete partial -metric space with coefficient and be a mapping satisfying the following condition:where . Then T has unique fixed point and . Remark 16 ([
7])
. Let be a metric space, a mapping is said to be a C-contraction if there exists such thatholds for all . Taking in Theorem 39, we can get the C-contraction fixed-point theorem in partial metric spaces, taking in Theorem 39. we can get the C-contraction fixed-point theorem in metric spaces.
Theorem 40. Let be a complete partial -metric space with coefficient and be a mapping satisfying the following condition: for each there exists such that Then T has a unique fixed point and .
Remark 17. Let be a complete b-metric space with coefficient be a mapping satisfying the following condition: for each there exists such that Then T has a unique fixed point .
2.17.2. 2019, Vujaković, Aydi, Radenović and Mukheimer, Some Remarks and New Results in Ordered Partial b-Metric Spaces, [96]
We present a new Meir–Keeler type result in partial b-metric spaces.
Theorem 41. Let be a -complete partial b-metric space and let T be a self-mapping on X verifying:
For there is so that Then T has a unique fixed point , and for each .
In [
96], authors announce an open question:
Prove or disprove the following:
Let be a -complete partial b-metric space and let T be a self-mapping on X satisfying:
Given
, there is
such that for all
Then T has a unique fixed point , and for each .
2.17.3. 2020, Hosseinzadeh and Parvaneh, Meir–Keeler Type Contractive Mappings in Modular and Partial Modular Metric Spaces, [97]
In this section, we investigate the new -Meir–Keeler contraction and establish some fixed-point results. Also, we introduce some fixed-point results in modular metric and partial modular metric space.
Definition 53 ([
98,
99,
100])
. A function is called a modular metric on X if the following axioms hold: - (i)
if and only if for all ;
- (ii)
for all and for all ;
- (iii)
for all and for all .
If in the Definition 53, we use the condition
- (i’)
for all and for all ;
instead of (i) then ω is said to be a pseudomodular metric on X. A modular metric ω on X is called regular if the following weaker version of (i) is satisfiedω is called convex if for all and for all the inequality holds: Remark 18. If ω is a pseudomodular metric on a set X, then the function is nonincreasing on for all . Indeed, if , then Definition 54 ([
98,
99,
100])
. Let ω be a pseudomodular on X and fixed. Consider the following two sets:and and are called modular spaces (around ).
Now, we define the notion of -Meir–Keeler contractive mapping as follow.
Definition 55. Let be a modular metric space and let T be a self-mapping on . Also, suppose that and are two functions where is lower ω-semicontinuous in . We say that T is an -Meir–Keeler contraction if for each there exists such thatfor all and for all . Throughout the section, will denotes the set of fixed points of T.
Theorem 42. Let be an ω-complete modular metric space which is ω regular and let be a self-mapping. Assume that there is a function such that the following assertions hold:
- (i)
T is a triangular α-admissible mapping,
- (ii)
T is an Meir–Keeler contraction,
- (iii)
there exists such that - (iv)
T is an ω-continuous mapping.
Then, T has a fixed point such that . Further, if for all , then T has a unique fixed point.
For a self-mapping that is not -continuous we have the following result.
Theorem 43. Let be a ω-complete modular metric space which is ω regular and let be a self-mapping. Assume that there is a function such that the following assertions hold:
- (i)
T is a triangular α-admissible mapping,
- (ii)
T is -Meir–Keeler contraction,
- (iii)
there exists such that - (iv)
if is a sequence in such that for all with as , then .
Then T has a fixed point such that . Further, if for all , then T has a unique fixed point.
Fixed-point results in partial modular metric spaces
Definition 56. A function is called a partial modular metric on X if the following conditions hold:
- (p1)
if and only if for all ;
- (p2)
for all and for all ;
- (p3)
for all and for all ;
- (p4)
for all and for all .
Definition 57. Let be a partial modular metric space and let T be a self-mapping on . Also, suppose that is a function. We say that T is an -Meir–Keeler contractive if for each there exists such thatfor any and for all . Theorem 44. Let be a p-regular p-complete partial modular metric space and let be a self-mapping. Assume that there is a function such that the following assertions hold:
- (i)
T is a triangular α-admissible mapping,
- (ii)
T is -Meir–Keeler contraction,
- (iii)
there exists such that .
- (iv)
if is a sequence in such that for all with as , then .
Then T has a unique fixed point such that for all . Further, if for all Fix , then T has a unique fixed point.
3. Meir–Keeler Contractions on Metric-like Spaces
3.1. 2014, Al-Mezel, Chen, Karapınar and Rakočević, Fixed-Point Results for Various-Admissible Contractive Mappings on Metric-like Spaces, [101]
In this section, we present fixed-point results for -admissible mappings on metric-like space via various auxiliary functions. In particular, we prove the existence of a fixed point of the generalized Meir–Keeler type -contractive self-mapping T defined on a metric-like space X.
Firstly, we present the generalized Meir–Keeler type -contractive mappings. Later, we proved the existence and uniqueness of such mappings in the context of metric-like spaces.
Let
be the class of all function
satisfying the following conditions:
Here, we give main result in metric-like spaces as follows.
Definition 58. Let be a metric-like space and let α: . One says that is called a generalized Meir–Keeler type -contractive mapping if for each there exists such thatfor all and . Remark 19. Note that if T is a generalized Meir–Keeler type -contractive mapping, then we have, for all and , Theorem 45. Let be a complete metric-like space and let be a generalized Meir–Keeler type -contractive mapping where α is transitive. Suppose that
- (i)
T is α-admissible;
- (ii)
there exists such that ;
- (iii)
T is continuous.
Then there exists such that .
Theorem 46. Let be a complete metric-like space and let be a generalized Meir–Keeler type -contractive mapping, where α is transitive. Suppose that
- (i)
T is α-admissible;
- (ii)
there exists such that ;
- (iii)
if is a sequence in X such that for all n and as , then for all n.
Then there exists such that .
For all , we have , where Fix denotes the set of fixed points of T.
Next, we will show that is a unique fixed point of T.
Theorem 47. Adding condition (U) to the hypotheses of Theorem 45 (resp., Theorem 46), one obtains that is the unique fixed point of T.
Fixed-Point Theorems via the Weaker Meir–Keeler Function
Here, we investigate the existence and uniqueness of a fixed point of certain mappings by using the Meir–Keeler function. Now, we recall the notion of the weaker Meir–Keeler function .
Definition 59 (see [
67])
. One calls a weaker Meir–Keeler function if, for each , there exists such that, for with , there exists such that .One denotes by the class of nondecreasing functions satisfying the following conditions:
is a weaker Meir–Keeler function;
for and ;
for all is decreasing;
if , then
And one denotes by Θ the class of functions satisfying the following conditions:
φ is continuous;
for and .
We introduce the generalized weaker Meir–Keeler type - -contractive mappings in metric-like spaces.
Definition 60. Let be a metric-like space, and let α: . One says that is called a generalized weaker Meir–Keeler type -contractive mapping if T is α-admissible and satisfiesfor all , where , and The main result of this section is the following.
Theorem 48. Let be a complete metric-like space and let be a generalized weaker Meir–Keeler type α-contractive mapping where α is transitive. Suppose that
- (i)
T is α-admissible;
- (ii)
there exists such that ;
- (iii)
T is continuous.
Then there exists such that .
Theorem 49. Let be a complete metric-like space and let be a generalized weaker Meir–Keeler type α-contractive mapping where α is transitive. Suppose that
- (i)
T is α-admissible;
- (ii)
there exists such that ;
- (iii)
if is a sequence in X such that for all n and as , then for all n.
Then there exists such that .
Theorem 50. Adding condition (U) to the hypotheses of Theorem 48 (resp., Theorem 49), one obtains that is the unique fixed point of T.
3.2. 2017, Aydi, Felhi, Karapınar and Alshaikh, an Implicit Relation for Meir–Keeler Type Mappings on Metric-like Spaces, [102]
We present an implicit relation for Meir–Keeler type mappings using an auxiliary pair of notations on the metric-like spaces.
Definition 61. Let Γ be the set of all continuous functions such that
- (F1)
: implies that ;
- (F2)
: implies that .
In [
15], Aydi et al. generalized [
13] by introducing a pair of mappings defined generalized
-admissible.
Let be the family of functions satisfying the following conditions:
is nondecreasing;
for all , where is the iterate of .
These functions are called (c)-comparison functions. It is easily proved that if is a (c)-comparison function, then for any .
Now, we establish new generalized -implicit Meir Keeler contractive pair of mappings in the metric-like spaces.
Definition 62. Let be a metric-like space and be given mappings. We say that is a generalized α-implicit Meir–Keeler contractive pair of mappings if there exist and such that
(d1) For all , there exists such thatwhere Remark 20. If we take in (28) and (29),andwhere andthen we say that T is a generalized α-implicit Meir–Keeler contractive mapping. Remark 21. It is obvious that the condition (28) yields Theorem 51. Let T and S be a self-mappings defined on a complete metric-like space and be a generalized α-implicit Meir–Keeler contractive pair of mappings. Suppose that
- (i)
is a generalized α-admissible pair;
- (ii)
there exists such that .
Then there exists a common fixed point of T and S, that is, with .
For the uniqueness of the common fixed point of a generalized contractive pair of mappings, we consider the following hypotheses.
- (H0)
for all .
- (H1)
For all , there exists such that and .
Here, denotes the set of fixed points of T.
Theorem 52. Adding conditions (HO) and (H1) to the hypotheses of Theorem 51 we obtain that is the unique common fixed point of T and S.
Corollary 6. Let T be a self-mapping defined on a complete metric-like space . Suppose that T is an α-implicit Meir–Keeler contractive mapping. Assume that
- (i)
T is an α-admissible mapping;
- (ii)
there exists such that ;
Then there exists a fixed point of T, that is, with .
We omit the proof. It is sufficient to take in Theorem 51.
Corollary 7. Adding conditions (HO) and (H1) to the hypotheses of Corollary 6. we obtain that is the unique fixed point of T.
As it seen above, the implicit relation can be replaced by the continuity of both S and T. Clearly, the continuity hypothesis is heavier than the implicit relation. For the sake of completeness, we put the corresponding results with the proofs.
Definition 63. Let be a metric-like space and be a given mapping. T is said sequentially continuous at if for each sequence in X converging to , we have , that is, .
T is said sequentially continuous on X if T is sequentially continuous at each .
Remark 22. Let be a metric-like space and be a given mapping. If T is continuous on X, then it is sequentially continuous on X.
We introduce new generalized sequentially continuous contractive Meir–Keeler pair of mappings in the metric-like spaces.
Definition 64. Let be a metric-like space and be given mappings. We say that is a generalized sequentially continuous α-contractive Meir–Keeler pair of mappings if there exist and such that
(d1) For all , there exists such thatwhere(d2) are sequentially continuous. Theorem 53. Let T and S be self-mappings defined on a complete metric-like space and be a generalized sequentially continuous α-contractive Meir Keeler pair of mappings. Suppose that
- (i)
is a generalized admissible pair;
- (ii)
there exists such that ;
- (iii)
for all satisfying .
Then there exists a common fixed point of T and S, that is, with .
Theorem 54. Adding conditions (HO) and (H1) to the hypotheses of Theorem 53. we obtain that is the unique common fixed point of T and S.
Corollary 8. Let T be a self-mapping defined on a complete metric-like space . Assume that T is a sequentially continuous α-contractive Meir–Keeler mapping. Suppose that
- (i)
T is α-admissible;
- (ii)
there exists such that ;
- (iii)
for all satisfying .
Then there exists a fixed point of T, that is, with .
We omit the proof since it is sufficient to take in Theorem 53.
Theorem 55. Adding conditions (HO) and (H1) to the hypotheses of Corollary 8, we obtain that is the unique fixed point of T.
3.3. 2019, Karapınar, Chen and Lee, Best Proximity Point Theorems for Two Weak Cyclic Contractions on Metric-like Spaces, [103]
In this section, we the study two best proximity point theorems in the setting of metric-like spaces that are based on cyclic contraction: Meir–Keeler–Kannan type cyclic contractions.
Here, we present the best proximity point results of Meir–Keeler-Kannan type cyclic contractions, as follows;
A mapping
is called Kannan type cyclic contraction, if there exists
such that
for all
, and
where
are nonempty subsets of a metric-like space
.
In [
104], authors introduce the new best proximity point result for a Kannan type cyclic contraction, as follows.
Theorem 56 ([
104])
. Suppose that a mapping is a Kannan type cyclic contraction, where are nonempty, closed subsets of a metric-like space and . If we set for each , for an arbitrary , thenWe have the following:
- (1)
If and has a subsequence which converges to with , then,
- (2)
If and has a subsequence which converges to with , then .
We shall denote set of all Meir–Keeler type function . Note that if , for all we have
Inspired by Meir–Keeler function and Kannan type cyclic contraction, we present the idea of Meir–Keeler-Kannan-type cyclic contraction.
Definition 65. Let and be a cyclic mapping, where A and B be nonempty subsets of a metric-like space . Then, the mapping T is said to be a Meir–Keeler-Kannan type cyclic contraction, iffor all and all . Now, we study the best proximity point results of Meir–Keeler-Kannan type cyclic contraction. The following theorem is generalized Theorem 56.
Lemma 23. Let be a cyclic Meir–Keeler-Kannan type contraction, where A and B be nonempty closed subsets of a metric-like space , and and it is increasing. For , define for each . Then Using Lemma 23 we prove the following best proximity point theorem.
Theorem 57. Let be a cyclic Meir–Keeler-Kannan type contraction, where A and B are nonempty closed subsets of a complete metric-like space . If we set for each , for an arbitrary , then we have the following:
- (1)
If and has a subsequence which converges to with , then .
- (2)
If and has a subsequence which converges to with , then .
We provide an example that satisfies Theorem 57.
Example 24. Let be endowed with the metric-like defined by: Let be defined by: Clearly, is a complete metric-like space, and ϕ is an increasing Meir–Keeler function. Take and , and let be defined byand Then we have and T is a cyclic mapping.
For and , we have that andand Then T is a Meir–Keeler-Kannan type cyclic contraction.
Let . Then for all , we haveand Thus, we get that as So, Lemma 23 holds and we also get that and are the two best proximity points of T.
3.4. 2016, Pasicki, Some Extensions of the Meir–Keeler Theorem, [105]
In this section, we present a new method of proving such Meir–Keeler theorem and give new results.
Lemma 24. Let be a nonnegative sequence such that Then if and only if the following condition is satisfied:
We use the term of dislocated metric following Hitzler and Seda [
36]; dislocated metric differs from metric since
yields
(no equivalence). The topology of a dislocated metric space is generated by balls. If
is a dislocated metric, then the pair
was first defined by Matthews as a metric domain (see [
20]).
In the present section, we put .
Lemma 25. Let be a dislocated metric space, and let T be a self mapping satisfying Then if and only if the following condition holds:for each , there exists such that Definition 66. Let be a dislocated metric space. Then a self mapping T on X is contractive if the following condition is satisfied: If is a contractive mapping, then (32) holds for each . Now, from Lemma 25 we obtain the following: Corollary 9. Let be a dislocated metric space, and let T be a contractive self mapping on X. Then if and only if (33) holds. Then we obtain implies Corollary 10. Let be a dislocated metric space, and let T be a self mapping on X satisfying (35) (or (34) or (32)). Then if and only if (33) holds can be also replaced by in (33). Also, each partial metric is a dislocated metric.
For a mapping
continuous at
and such that
, let us consider
Corollary 11. Let be a partial metric space, and let T be a self mapping on X satisfying (37) (or (35) or (34) or (32)). Then if and only if (33) holds can be also replaced by or by in (33)). Lemma 26. Let be a dislocated metric space, and let T be a self mapping on X satisfying the following conditions:for each , there exists such that Then . In addition, can be replaced by σ in (38) or (39) (so also in both of them). Similarly, if σ is a partial metric, then can be replaced by in (38) or (39). Definition 67. A self mapping T on a dislocated metric space is 0 -continuous at x if implies for each sequence in X; T is 0 -continuous if it is 0 -continuous at each point .
Lemma 27. Let be a dislocated metric space, and let T be a self mapping on X. If is contractive, then T has at most one fixed point; the same holds if and only if satisfies (35) or (37) and σ satisfies if and only if is 0-continuous at x (e.g., if and only if is contractive) and , then and . Theorem 58. Let T be a 0-continuous self mapping on a 0 -complete dislocated metric space . Assume that (35) or (34) holds and the following condition is satisfied:for each , there exists such that implies Then T has a unique fixed point, say x, and .
Theorem 59. Let h be a self mapping on a 0-complete dislocated metric space such that (for some ) satisfies the assumptions of Theorem 58. Then h has a unique fixed point, say x, and .
Theorem 60. Let T be a 0-continuous cyclic self mapping on a 0-complete dislocated metric space , and let the following conditions be satisfied:for each , there exists such that Then T has a unique fixed point, say x, and .
Theorem 61. Let h be a self mapping on a 0-complete dislocated metric space such that (for some ) satisfies the assumptions of Theorem 60. Then has a unique fixed point, say x, and .
Theorem 62. Let be a 0-complete dislocated metric space, and let T be a 0-continuous cyclic self mapping on X such that . Assume that the following conditions hold:for each , there exists such that Then T has a fixed point, say x, such that , and x is unique if σ is a metric. In addition, can be replaced by (or by if σ is a partial metric) in (36) for (40) or (41) (so also for both of them). Theorem 63. Let be a 0-complete dislocated metric space, and let T be a 0-continuous self mapping on X such that . Assume that the following conditions hold:for each , there exists such that Then T has a fixed point, say x, such that , and x is unique if σ is a metric. In addition, can be replaced by (or by if σ is a partial metric) in (36) for (42) or (43) (so also for both of them). 3.5. 2017, Pasicki, Meir and Keeler Were Right, [106]
It Is Shown Here that the Celebrated Fixed-Point Theorem of Meir–Keeler Is Equivalent to the Formally More General Result of Matkowski and Ćirić.
Meir–Keeler condition [
1] was later extended by Matkowski in [
107,
108] and by Ćirić [
9]. They used two conditions. One of them has the following form:
If
is a metric, then the above condition is equivalent to the following one:
and clearly, Meir–Keeler condition implies (
44) (for
).
The second condition of Matkowski and Ćirić is
for each
there exists an
for which
If we assume that (
44) holds, then the above condition is equivalent to the following one:
for each
there exists an
for which
as for
we have
. Obviously, Meir–Keeler condition implies (
44) and (
45).
If is a mapping, then for , the set is an orbit of T.
Lemma 28. Let T be a self mapping on a dislocated metric space , and let be an orbit of T. If the following conditions are satisfied:then . If and (45) holds for and large , then Corollary 12. Let T be a self mapping on a dislocated metric space , and let be an orbit of T. Then (46), () yield . If , (45) holds for and large , and is 0 -complete, then there exists an such that . Lemma 29. Let T be a self mapping on a dislocated metric space , and let be an orbit of T. If (46), () are satisfied, then . If , (45) holds for and large , and is 0-complete, then there exists an such that ; if in addition, T is 0 -continuous at x, then . Theorem 64. Let T be a self mapping on a 0-complete dislocated metric space . If (44), (45) are satisfied for all , then T has a unique fixed point x, and in addition, . Theorem 65. Let T be a self mapping on a 0 -complete dislocated metric space . If Meir–Keeler condition is satisfied for all , then T has a unique fixed point x, and .
Proposition 6. If a mapping satisfies (44) and (45) for all , then Meir–Keeler condition for T replaced , and all Corollary 13. Theorems 64 and 65 are equivalent and the same concerns the classical results of Meir Keeler, and Matkowski, Ćirić.
The next theorem is a tool in proving fixed-point theorems, and it is a consequence of Lemma 29.
Theorem 66. Assume that T is a self mapping on a dislocated metric space . If (46), () are satisfied for an orbit Z of T, then. If is 0 -complete,, and (45) holds for and large , then there exists an x such that ; if in addition, T is 0-continuous at x, then . If is 0 -complete, T and each fulfil the above requirements and x is the unique fixed point of T, then , . Theorem 67. Let T be a self mapping on a 0-complete dislocated metric space , and let (44) hold for all . If (45) is satisfied for each orbit Z of T, then T has a unique fixed point x, and Remark 23. Let us note that in (44) or (45) for Theorem 67 can be replaced byif for used in (44), and if T is 0 -continuous (see the reasoning below). Let us present a more advanced application of Theorem 66.
Also, each partial metric is a dislocated metric.
Let us consider the following conditions
where
Theorem 68. Let T be a self mapping on a 0-complete partial metric space , and let (48) hold. If () is satisfied for each orbit Z of T, and T is 0 -continuous, then T has a unique fixed point x, and . Lemma 30. Let be commuting self mappings on a dislocated metric space , and let the following conditions hold:for each there exists an for which Then satisfies Meir–Keeler conditions for all .
Let us note, that Proposition 65 is a consequence of Lemma 30 for .
Theorem 69. Let be commuting self mappings on a 0-complete dislocated metric space . If (50), (51) hold, then have a unique and common fixed point x, and , Lemma 31. Let be commuting cyclic self mappings on a dislocated metric space , and let the following conditions hold:for each there exists an for which Then for the following condition is satisfied:
for each , there exists an such that 3.6. 2017, Karapınar, a Note on Meir–Keeler Contractions on Dislocated Quasi-b-Metric, [109]
In this section, we show that Meir–Keeler type contractions possess a fixed point in the setting of dislocated quasi-b-metric.
Definition 68 ([
110])
. For a nonempty set X, a dislocated quasi-b-metric is a function such that for all and a fixed constant :if then
Moreover, the pair is called dislocated quasi-b-metric space (DqbMS).
We recall the notion of -comparison function.
For a fixed real number , let be all functions satisfying the conditions
is increasing,
there exist and a convergent series of nonnegative terms such that , for and any .
Any
is called
-comparison function [
111]. For
, in the definition above,
is known as (c)-comparison functions.
We will need the following results.
Lemma 32 ([
87,
111,
112])
. For a comparison function the following hold:- (1)
the series converges for any ;
- (2)
the function defined by is increasing and continuous at 0.
- (3)
each iterate of is also a comparison function;
- (4)
is continuous at 0;
- (5)
for any .
3.6.1. -Meir–Keeler Type Contraction
We introduce the following notion which is an improved version of Meir–Keeler contraction.
Definition 69. Let be a DqbMS. We say that is an -Meir–Keeler type contraction if there exist two functions and satisfying the following condition:
For each , there exists such that Notice that for an -Meir–Keeler type contraction , we have , for any
In what follows we shall state and prove the first main result of this section.
Theorem 70. Suppose that is a complete DqbMS and a self-mapping is a -Meir–Keeler type contraction. Assume also that
- (i)
T is α-orbital admissible;
- (ii)
there exists such that ; and ;
- (iii)
T is continuous.
Then, there exists such that .
Example 25. Let endowed with for all . It is clear that is a complete DqbMS. Define and by: , and We can prove easily T is an -Meir–Keeler type contraction and it is an α-orbital admissible. Moreover, there exists such that . In fact, for , we have Now, we show that T is a continuous. Let in the context of DqbMS , that is, We shall show that T is continuous. Indeed, So all hypotheses of Theorem 70 are satisfied. Consequently, T has a fixed point. Notice that is a fixed point of T.
Theorem 71. Suppose that is a complete DqbMS and a self-mapping is a - Meir–Keeler type contraction. Assume also that
- (i)
T is α-orbital admissible;
- (ii)
there exists such that and ;
- (iii)
if is a sequence in X such that for all n and as , then for all n.
Then, there exists such that .
In the following example, a self-mapping T is not continuous.
Example 26. Let endowed with the dislocated metric for all . Define and by: Clearly T is not continuous at 1 which shows that Theorem 70 is not applicable in this case.
We shall prove that a self-mapping T is an -Meir–Keeler type contraction. Let be given. Take and suppose that we want to show that Suppose that , then and so . Hence Also, T is an α-orbital-admissible. To see this, let , then both . Due to definition of T, we have and . Thus, we get .
Moreover, there exists such that . Indeed, for we have Finally, let be a sequence in X such that for all n and as . Since for all n, by the definition of α, we have for all n and , then .
So, we conclude that all hypotheses of Theorem 71 are fulfilled. So, we proved that T has a fixed point.
3.6.2. Generalized -Meir–Keeler Type Contraction
Definition 70. Suppose that is a DqbMS. A self-mapping is said to be a generalized ( Meir–Keeler type contraction if there exist and such that for each , there exists such thatwhere If a self-mapping is a generalized- -Meir–Keeler type contraction, then we have , for any
The following is the first main result of this section.
Theorem 72. Suppose that is a complete DqbMS, a self-mapping is a generalized-( - Meir Keeler type contraction and the following conditions are fulfilled:
- (i)
T is triangular α-orbital admissible mapping;
- (ii)
there exists such that and ;
- (iii)
T is continuous.
Then, T has a fixed point, that is, there exists such that .
Theorem 73. Suppose that is a complete DqbMS, a self-mapping is a generalized-( α, - Meir Keeler type contraction, where with for a constant . and the following conditions are fulfilled:
- (i)
T is triangular α-orbital admissible mapping;
- (ii)
there exists such that ;
- (iii)
if is a sequence in X such that for all n and as , then for all n.
Then, T has a fixed point, that is, there exists such that .
3.6.3. The Uniqueness of the Fixed Point
Let Fix(T) denotes the set of fixed points of the mapping T.
We, first, recollect the following interesting condition for uniqueness of a fixed point of an -Meir Keeler type contraction.
(H) For all , then there exists such that and , where
Theorem 74. Putting condition (H) to the statements of Theorem 70 (respectively, Theorem 71), we obtain that u is the unique fixed point of T.
The following is an alternative uniqueness condition:
(U) For all Fix , then .
Theorem 75. Putting condition (U) to the statements of Theorem 70(resp. Theorem 71 ), we find that x is the unique fixed point of T.
In what follows, we propose the conditions for the uniqueness of a fixed point of a generalized Meir–Keeler type contraction:
- (H1)
For all Fix , then there exists such that and .
- (H2)
Let
. If there exists a sequence
in
X such that
and
, then
- (H3)
For any , then .
Theorem 76. Putting conditions (H1), (H2) and (H3) to the statements of Theorem 72 (respectively, Theorem 73), we have that x is the unique fixed point of T.
If set for all in Theorem 70, we get the following result:
Theorem 77. Suppose that is a complete DqbMS and a self-mapping is a - Meir–Keeler type contraction. Then, there exists such that .
Notice that -Meir–Keeler type contraction is non-expansive, and hence, it is continuous.
If set for all in Theorem 72 we find the following consequence:
Theorem 78. Suppose that is a complete DqbMS, a self-mapping is a generalized-( α, - Meir Keeler type contraction. If T is continuous T has a fixed point, that is, there exists such that .
3.7. 2016, Gholamian and Khanehgir, Fixed Points of Generalized Meir–Keeler Contraction Mappings in b-Metric-like Spaces, [113]
We recall the following definition.
Definition 71 ([
114])
. Let X be a nonempty set and be a given real number. A function is a b-metric-like if, for all , the following conditions are satisfied: implies ,
,
.
A b-metric-like space is a pair such that X is a nonempty set and is a b-metric like on X. The number s is called the coefficient of .
We establish fixed-point results for the generalized Meir–Keeler type contraction in b metric-like space.
Definition 72. Suppose that is a b-metric-like space with coefficient s. A triangular α-admissible mapping is said to be generalized Meir–Keeler contraction if for every there exists such thatfor all where is a given function. Remark 24. Let T be a generalized Meir–Keeler contractive mapping. Then it is intuitively clear thatfor all when . The definitions of two types (type I and II) of generalized Meir Keeler contractions in b-metric-like spaces are as follows.
Definition 73. Let be a b-metric-like space with coefficient s. A triangular α-admissible mapping is said to be generalized Meir–Keeler contraction of type (I) if for every there exists such thatwherefor all . Definition 74. Let be a b-metric-like space with coefficient s. A triangular α-admissible mapping is said to be generalized Meir–Keeler contraction of type (II) if for every there exists such thatwherefor all . We present two important remark for our new generalized contraction.
Remark 25. Suppose that is a generalized Meir–Keeler contraction of type (I) (respectively, type (II)). Thenfor all when (respectively, . Remark 26. It is readily verified that for all , where and are defined in (54) and (55), respectively. We also present a new theorem for Meir–Keeler type contractions with a rational expression by generalization of idea of Samet et al. [
115].
Theorem 79. Let be a complete b-metric-like space and be a triangular α-admissible mapping. Suppose that the following conditions hold:
- (a)
there exists such that ,
- (b)
if is a sequence in X such that as and for all , then for all ,
- (c)
for each , there exists satisfying the following condition:
Then T has a fixed point in X.
Example 27 shows the validity of Theorem 79.
Example 27. Let . Define as follows: Clearly, is a complete b-metric-like space with . Consider defined by , and . Also, define as follows: It easily can be shown that T is triangular α-admissible. In order to check the condition (56), we choose so thatwhich implies Note that . Now, all conditions of Theorem 79 are satisfied and so T has a fixed point.
Also, let be the b-metric associated to b-metric-like defined by if and , elsewhere. Then condition (56) does not hold in b-metric space . Let , and . Thenfor each . But Theorem 80. Let be a complete b-metric-like space and be an α admissible mapping. Assume that there exists a function satisfying the following conditions:
- (a)
and for every ,
- (b)
θ is nondecreasing and right continuous,
- (c)
for every , there exists such that
for all , then (56) is satisfied. Theorem 81. Let be a complete b-metric-like space with coefficient s and X be a mapping. Suppose that the following conditions hold:
- (a)
T is an orbitally continuous generalized Meir–Keeler contraction of type (I),
- (b)
there exists such that ,
- (c)
if is a sequence in X such that as and for all , then ,
- (d)
or β is a continuous function.
Then T has a fixed point in X.
Theorem 82. Let be a complete b-metric-like space, be a mapping. Suppose that the following conditions hold:
- (a)
T is an orbitally continuous generalized Meir–Keeler contraction of type (II),
- (b)
there exists such that ,
- (c)
if is a sequence in X such that as and for all , then ,
- (d)
or β is a continuous function.
Then T has a fixed point in X.
In fact with the aid of -admissibility of the contraction we will show that orbitally continuity assumption is not required whenever the following condition is satisfied.
(A) If is a sequence in X which converges to z with respect to and satisfies and for all n, then there exists a subsequence of such that and for all k.
Theorem 83. Let be a complete b-metric-like space with coefficient s and satisfies the condition (A). Also, let be a mapping. Suppose that the following conditions hold:
- (a)
is a generalized Meir–Keeler contraction of type (II),
- (b)
there exists such that ,
- (c)
or β is a continuous function.
Then T has a fixed point in X.
Example 28. Let and α be as in Example 27. Consider defined by and . Also, define as follows: In order to check the condition (53), we choose so thatwhich implies . Therefore, the map T is a generalized Meir–Keeler contraction of type (I). Note that T is continuous with respect to and . Now, all conditions of Theorem 81 are satisfied and so T has a fixed point.
Example 29. Let equipped with the b-metric-like defined by It is easy to see that is a complete b-metric-like space, with . Define the self mapping and the functions as follows: Then the mapping T is triangular α-admissible. On the other hand, the condition (A) holds on X. More precisely, if the sequence satisfies , and with respect to , for some , then and, moreover, , which gives us . Hence and .
Next, we prove that T is a generalized Meir–Keeler contraction. We show this in the three following steps.
Step 1. If or .
In this case, and evidently (52) holds. Step 2. Let with .
Let be given and choose . Now if , then Step 3. Let with .
Take . Then the inequalityimplies that Also, notice that and . Moreover, T has fixed points and .
A remarkable fact concerning Example 29 is that the restriction of T to the interval is orbitally continuous and so by the definition of that example fulfills all conditions of Theorem 81, too.
3.8. 2020, Gholamian, Fixed Points of Generalized -Meir–Keeler Type Contractions and Meir–Keeler Contractions through Rational Expression in b-Metric-like Spaces, [116]
In this section, we present some fixed-point results for generalized -Meir–Keeler contractions via rational expression in b-metric-like spaces.
Definition 75. Let be a b-metric-like space with coefficient s. A triangular α admissible mapping is said to be α-admissible Meir–Keeler contraction (or shortly α-Meir–Keeler contraction) if for every , there exists such thatfor all . Applying definition of α-Meir–Keeler contraction, it is clear thatfor all when . Remark 27. Note that our definition of α-Meir–Keeler contraction is different from that of Definition 72. For this, take and defined by , if and 0, otherwise. Then is a b-metric-like space with . Also, consider the mapping defined by and , and functions and defined by It is easily can be checked that T is an α-Meir–Keeler contraction. According to Definition 72, for and we have which does not imply that , Since .
From now on, for convenience, we denote by the set of all functions for a real number .
The definitions of two types (type I and II) of generalized -Meir–Keeler contractions in b-metric-like spaces are as follows.
Definition 76. Let be a b-metric-like space with coefficient s. A triangular α admissible mapping is said to be a generalized α-Meir–Keeler contraction of type (I) if for every there exists such thatwherefor all . Definition 77. Let be a b-metric-like space with coefficient s. A triangular α admissible mapping is said to be a generalized α-Meir–Keeler contraction of type (II) if for every there exists such thatwherefor all . Here, we give fixed-point results for generalized -Meir–Keeler contractions of type (I) in b-metric-like spaces..
Theorem 84. Let be a complete b-metric like space and be a mapping. Suppose that the following conditions hold:
- (a)
T is a continuous generalized α-Meir–Keeler contraction of type (I),
- (b)
there exists such that and ,
- (c)
if is a sequence in X such that as and for all , then .
Then T has a fixed point in X.
Theorem 85. Let be a complete b-metric-like space and be a mapping. Suppose that the following conditions hold:
- (a)
T is a continuous generalized α-Meir–Keeler contraction of type (II),
- (b)
there exists such that and ,
- (c)
if is a sequence in X such that as and for all , then .
Then T has a fixed point in X.
There is an analogous result for -Meir–Keeler contraction.
Proposition 7. Consider a particular case of Theorem 84, whenever T is a generalized α-Meir–Keeler contraction, then T has a fixed point in X.
Example 30. Let equipped with the b-metric-like , where . Then is a complete b-metric-like space with . Consider the mapping and the functions and defined by It easily can be shown that T is triangular α-admissible and continuous. In order to check the condition (57) without loss of generality, we may take . Let be given. Consider the following two cases. Case 1. If , then we have and . We choose so that . It implies that
Case 2. If or , then we have We choose again so that . It follows that Therefore, the map T is a generalized α-Meir–Keeler contraction of type (I). Note that and . Now, all conditions of Theorem 84 are satisfied and so T has a fixed point.
Example 31. Let equipped with the b-metric-like defined by It is easy to see that is a complete b-metric-like space with the coefficient . If we define the mapping and the functions and byandthen the mapping T is triangular α-admissible, which is not continuous. On the other hand, the condition (A) holds. Indeed, if the sequence satisfies or , and , then , and . Hence and . Next, assume that with . Then, for , we choose so that . It implies that Other cases are obvious by the definition of α. Therefore, the mapping T is a generalized α-Meir–Keeler contraction. Also, notice that and . Then, we conclude that all of the assumptions of Proposition 7 are satisfied. Moreover, T has a fixed point .
Now, we introduce fixed-point results via rational expression in b-metric-like space.
Theorem 86. Let be a complete b-metric-like space, be a triangular α-admissible mapping and . Suppose that the following conditions hold:
- (a)
there exists such that and ,
- (b)
if is a sequence in X such that as and for all , then for all ,
- (c)
for each , there exists satisfying the following condition
Then T has a fixed point in X.
Theorem 87. Let be a b-metric-like space, be an α-admissible mapping and . Assume that there exists a function satisfying the following conditions:
- (a)
and for every ,
- (b)
θ is nondecreasing and right continuous,
- (c)
for every , there exists such that
for all . Then (58) is satisfied.