Abstract
In this paper we prove a coincidence point result in a space which does not have to satisfy any of the classical axioms that define a metric space. Furthermore, the ambient space need not be ordered and does not have to be complete. Then, this result may be applied in a wide range of different settings (metric spaces, quasi-metric spaces, pseudo-metric spaces, semi-metric spaces, pseudo-quasi-metric spaces, partial metric spaces, G-spaces, etc.). Finally, we illustrate how this result clarifies and improves some well-known, recent results on this topic.
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1 Introduction
Fixed point theory plays a crucial role in nonlinear functional analysis since, among other reasons, fixed point results are used to prove the existence (and also uniqueness) of solution when solving various types of equations. The Banach contraction principle is considered to be the pioneering result of the fixed point theory, and it is the most celebrated result in this field. The simplicity of its proof and the possibility of attaining the fixed point by using successive approximations let this theorem become a very useful tool in analysis and in applied mathematics. The great significance of Banach’s principle, and the reason it is possibly one of the most frequently cited fixed point theorems in all of analysis, lies in the fact that its proof contains elements of fundamental importance to the theoretical and practical treatment of mathematical equations. After the appearance of this result in Banach’s thesis in 1922, a great number of extensions (in many occasions, as well-known as the original result, such as those by Krasnoselskii and Zabreiko, Edelstein, Browder, Schauder, Göhde, Kirk, and Caristi; a comprehensive study can be found in [1]) have been proved in various different frameworks (see [2, 3] in partial metric spaces, [4–7] in G-metric spaces, [8, 9] in fuzzy metric spaces, [10, 11] in intuitionistic fuzzy metric spaces, [12, 13] in probabilistic metric spaces and [14, 15] in Menger spaces).
In recent times, one of the most attractive research topics in fixed point theory is to prove the existence of a fixed point in metric spaces endowed with partial orders. An initial result in this direction was given by Turinici [16] in 1986. Following this line of research, Ran and Reurings [17] (and later Nieto and Rodríguez-López [18]) used a partial order on the ambient metric space to introduce a slightly different contractivity condition, which must be only verified by comparable points. Thus, they reported two versions of the Banach contraction principle in partially ordered sets and applied them to the study of some applications to matrix equations. Their proofs involved combining the ideas of the iterative technique in the contractive mapping principle with those in the monotone technique. This approach led to a very recent branch of this field, with applications to matrix equations and ordinary differential equations. The literature on this topic has exponentially risen in recent years. To mention some advances on this topic, we highlight the following ones. Firstly, in order to guarantee the existence and uniqueness of a solution of periodic boundary value problems, Gnana-Bhaskar and Lakshmikantham [19] (and, subsequently, Lakshmikantham and Ćirić [20]) proved, in 2006, the existence and uniqueness of a coupled fixed point (a notion introduced by Guo and Lakshmikantham in [21]) in the setting of partially ordered metric spaces by introducing the notion of mixed monotone property. Later, the notions of tripled fixed point, quadruple fixed point and multidimensional fixed point were introduced by Berinde and Borcut [22], by Karapinar and Luong [23] and by Berzig and Samet [24] (see also [25]), respectively.
But the two main ingredients of all extensions are, basically, the same that we can find in the Banach contraction principle: a complete metric space and a self-mapping verifying a contractive condition. Although modern versions use, in many cases, different kind of mappings, the more intensively studied condition is based on the idea that the distance between the images of any two points (comparable or not) is upper bound by the product of a constant (small enough) and the distance between those points. The main aim of this manuscript is to provide a result powerful enough to guarantee that a nonlinear operator T has, at least, a fixed point, even when we consider that a measure mapping does not have to be an underlying metric structure on the ambient space X and the binary relationship is not necessarily a partial order on X. To do this, we present a result which can be applied in the following adverse conditions: the framework is a set X provided with a preorder and a measure mapping that does not necessarily verify any of the four classical properties of a metric space (in fact, it need not be one of the following metric structures: a metric, a pseudo-metric, a quasi-metric, a pseudo-quasi-metric or a semi-metric). Furthermore, d has not to be symmetric and the triangular inequality must only be verified by a kind of comparable points. Even if d would verify some of the classical properties of a metric, would not be a complete space. In this setting, none of the theorems proved until now can be applied to guarantee that a nonlinear operator (even if it is a contractive mapping) has, at least, a fixed point. We illustrate our results with a particular example. Finally, we show that they extend and improve some well-known fixed point theorems.
2 Preliminaries
Preliminaries and notation about coincidence points can also be found in [25]. Let n be a positive integer. Henceforth, X will denote a nonempty set and will denote the product space . Throughout this manuscript, m and k will denote nonnegative integers and . Unless otherwise stated, ‘for all m’ will mean ‘for all ’ and ‘for all i’ will mean ‘for all ’. In the sequel, let and be three mappings. For brevity, will be denoted by Tx.
Definition 2.1 A binary relation on X is a nonempty subset ℛ of . For simplicity, we will write if , and we will say that ≼ is the binary relation. We will write when and , and we will write when . We will say that x and y are ≼-comparable if or .
A binary relation ≼ on X is transitive if for all such that and . A preorder (or a quasi-order) ≼ on X is a binary relation on X that is reflexive (i.e., for all ) and transitive. In such a case, we say that is a preordered space (or a preordered set). If a preorder ≼ is also antisymmetric ( and implies ), then ≼ is called a partial order, and is a partially ordered space.
All partial orders and equivalence relations are preorders, but preorders are more general. From now on, will always denote a preordered space.
Definition 2.2 A metric on X is a mapping satisfying
for all . If d is a metric on X, we say that is a metric space.
The function d is a premetric if it satisfies (); a pseudo-metric if it satisfies (), () and (); a quasi-metric (or a nonsymmetric metric) if it satisfies (), () and (); a quasi-pseudo-metric if it satisfies () and (); and a semi-metric if it satisfies (), () and ().
Remark 2.1 We point out that there exist different notions of premetric that are not universally accepted. For instance, Kasahara [26] used the term premetric to refer to a quasi-pseudo-metric defined on a subset of . However, Kim [27], in the same issue as Kasahara, preferred using the term quasi-pseudo-metric (see also Reilly et al. [28]). For our purposes and for the sake of clarity, we prefer using the previous definitions because we consider that it is a more modern nomenclature.
Definition 2.3 A fixed point of a self-mapping is a point such that . A coincidence point between two mappings is a point such that . A common fixed point of is a point such that .
Remark 2.2 If are commuting and is a coincidence point of T and g, then is also a coincidence point of T and g.
Definition 2.4 If is a preordered space and are two mappings, we will say that T is a -nondecreasing mapping if for all such that . If g is the identity mapping on X, T is nondecreasing (w.r.t. ≼).
3 An illustrative example
Let be the real interval and let provided with the following binary relation:
Define , for all , by
Let g the identity mapping on . Define by
Then the following statements hold. Proofs can be found in Appendix 2.
-
1.
The binary relation ≼ is a preorder on , but it is not a partial order on .
-
2.
The measure mapping d does not hold any of the four classical properties ()-() that define a metric space. Indeed, it is not a metric on , neither a premetric nor any of the following: a pseudo-metric, a quasi-metric, a pseudo-quasi-metric, a semi-metric or a partial metric.
-
3.
Even if d would verify some of the metric properties of Definition 2.2, would not be a complete space.
-
4.
T is not a d-contraction (that is, there is no such that for all ) because , but .
Therefore, none of the theorems proved until now can be applied to the quadruple in order to guarantee that T has a fixed point.
4 Test functions
One of the most important ingredients of a contractivity condition is the kind of involved functions. Recently, many classes of families have been introduced, like altering distance functions, comparison functions, -comparison functions, Geraghty functions, etc. In this section, we present the kind of functions we will use and we show how other classes can be seen as particular cases.
Definition 4.1 (Agarwal et al. [29])
We will denote by ℱ the family of all pairs , where are functions, verifying the following three conditions.
-
() ψ is nondecreasing.
-
() If there exists such that , then and .
-
() If are sequences such that , and verifying and for all k, then .
Example 4.1 If and we define and for all , then . Furthermore, for all .
Notice that axiom () does not necessarily imply the well known condition . Furthermore, we do not impose any continuity condition neither on ψ nor on φ. In order to prove that the family ℱ is very general, next we will show a variety of pairs of functions in ℱ that have been previously considered by other authors in the past.
A function is lower semi-continuous if for all sequence such that . Similarly, ϕ is upper semi-continuous if, in the same conditions, .
Definition 4.2 (Khan et al. [30])
An altering distance function is a continuous, nondecreasing function such that if and only if .
Proposition 4.1 If ϕ is an altering distance function and verifies , then .
The following lemma shows some examples of pairs in ℱ.
Lemma 4.1 (see [29])
Let be two functions such that ψ is an altering distance function.
-
1.
If φ is lower semi-continuous and , then .
-
2.
If φ is continuous and verifies , then .
-
3.
If ψ and φ are altering distance functions, then .
Notice that the condition is not necessary.
Proof We prove item (1). Conditions () and () are obvious. Next, assume that are sequences such that , and verify and for all k. Therefore, . Hence for all k. Letting and taking into account that ψ is continuous, we deduce that . As and φ is lower semi-continuous, we deduce that . Hence . The other two items immediately follow from item 1. □
Example 4.2 (see [29])
-
1.
If and we define and for all , then . The case is usually included in other papers, but the case is new.
-
2.
If for all , then . Notice that, in this case, () holds because it is impossible to find such kind of sequences since . In this case, the condition does not hold.
Corollary 4.1 Let be two functions such that ψ is an altering distance function and ϕ is upper semi-continuous verifying and for all . Then , where .
Proof It follows from item 1 of Lemma 4.1 because is lower semi-continuous and verifies . □
A function is a Geraghty function if the condition implies that .
Lemma 4.2 (Agarwal et al. [29])
If α is a Geraghty function and we define and for all , then .
In [31], the authors used a contractivity condition as follows:
where ψ is an altering distance function and β is a Geraghty function.
Lemma 4.3 If ψ is an altering distance function and β is a Geraghty function, then .
Proof Let , that is, for all . Notice that the image of is contained in the image of β, which is in . Therefore, for all (if , both members are equal, and if , then since ).
() Since ψ is an altering distance function, then it is nondecreasing.
() Assume that there exists such that . Then . Since , then , which means that . In such a case, because it is an altering distance function.
() Let be sequences such that , and verify and for all k. Since ψ is continuous, . Moreover,
Let us show that reasoning by contradiction. Suppose that . Since ψ is nondecreasing,
In particular, and
Letting , we deduce that . Since β is a Geraghty function, , which contradicts that because and ψ is an altering distance function. □
In [32], the authors used a contractivity condition as follows:
ψ is continuous and φ verifies that implies that .
Lemma 4.4 Let be two functions such that ψ is an altering distance function and φ verifies the following condition:
Then .
Proof () Since ψ is an altering distance function, then it is nondecreasing.
() Assume that there exists such that . Letting for all and applying (2), we deduce that . In such a case, because it is an altering distance function.
() Let be sequences such that , and verify and for all k. Hence for all k. Letting and taking into account that ψ is continuous, we deduce that . By condition (2), . □
A comparison function is a nondecreasing function such that for all .
Lemma 4.5 If ϕ is a continuous comparison function, and we define and for all , then .
Proof It is clear that every comparison function ϕ verifies for all . In such a case, if ϕ is continuous, then , so for all . Moreover, if , then .
() Since ψ is an altering distance function, then it is nondecreasing.
() Assume that there exists such that . Then , so . In such a case, because it is an altering distance function.
() Let be sequences such that , and verify and for all k. Therefore
Letting , we deduce that . Therefore, as ϕ is continuous, , which is only possible when . □
In [33], Berzig et al. introduced the notion of pair of generalized altering distance functions, which is a pair , where , verifying the following conditions:
-
(a1)
ψ is continuous;
-
(a2)
ψ is nondecreasing;
-
(a3)
(a3) .
The condition (a3) was introduced by Popescu in [34] and Moradi and Farajzadeh in [35]. Notice that the above conditions do not determine the values and . If in , then . This is the case of Lemma 4.4.
After we have shown many different contexts in which some pairs of ℱ appear, we present some of their useful properties.
Lemma 4.6 (Agarwal et al. [29])
Let .
-
1.
If and , then either or . In any case, .
-
2.
If and , then .
-
3.
If are such that for all k and , then .
-
4.
If and for all k, then .
Remark 4.1 In [36], Berzig introduced the class of shifting distance functions, which are pairs of functions verifying the following two conditions:
-
(i)
for , if , then ;
-
(ii)
for with , if for all , then .
Pairs of functions in ℱ are intimately related with the class of pairs of shifting distance functions, but they are different. On the one hand, pairs in can take values in ℝ, but pairs in ℱ take values in . On the other hand, if a pair verifies () and (), then the pair satisfies (i). Furthermore, if satisfies (ii), then satisfies ().
5 A fixed point theorem without an underlying metric structure
The main aim of this section is to show sufficient conditions in order to ensure that T and g (given in Section 3) have a coincidence point. To set the framework, throughout this section, let be a preordered space, and let and be three mappings. The following definitions are usually considered when X has a metric structure. However, we do not suppose, a priori, any condition on the mapping d. Indeed, we will only be able to prove that d takes nonnegative values as a consequence of a particular version of the triangular inequality. However, in general, we do not consider necessary to assume this sign constraint.
Definition 5.1 We will say that a sequence :
-
d-converges to (and we will write or simply ) if for all there exists such that for all ;
-
is d-Cauchy if for all there exists such that for all .
We will say that is complete if every d-Cauchy sequence in X is d-convergent in X.
With respect to the previous notions, the following remarks must be done.
Remark 5.1
-
When the distance measure d is not symmetric (that is, it does not verify axiom ()), the definition of convergence or Cauchyness of sequences usually depends on the side, because and can be different. This is the case, for instance, of quasi-metric spaces. In such cases, the previous definitions correspond to the idea of right-convergence and right-Cauchyness (see Jleli and Samet [37]) because the more advanced term (which is nearer to the limit) is placed at the right argument of d. Similarly, it can be defined the notions of left-convergence (using ) and left-Cauchyness (using ) of sequences. Notice that some of the concepts we will present can also be introduced by the right side or by the left side. However, in order not to complicate the notation, we prefer avoiding the term right- in all definitions and theorems.
-
In [[38], Section 3], the authors did a complete study (completion, topology and powerdomains) of spaces verifying axioms () and (), which they called generalized metric spaces. They solved the previous discussion using the terms forward convergent sequences (for right-convergent sequences) and backward convergent sequences (for left-convergent sequences). However, as we will not assume () nor (), we also prefer avoiding these prefixes.
-
Notice that if d does not verify (), then the limit of a sequence, if there exists, might not be unique.
-
And if d only takes nonpositive values, then all sequences converge to all points.
Definition 5.2 We will say that a subset is -nondecreasing-closed if any d-limit of any ≼-nondecreasing sequence of points of A is also in A.
Similarly can be defined the concepts of -nonincreasing-closed set and -monotone-closed set, and, more generally, a d-closed set, when any d-limit of any convergent sequence of points of A is also in A.
Definition 5.3 A mapping is -nondecreasing-continuous at if we have that d-converges to for all ≼-nondecreasing sequence d-convergent to .
In a similar way, the concepts of -nonincreasing-continuous mapping and -monotone-continuous mapping may be considered and, more generally, a d-continuous mapping, when d-converges to for all sequence d-convergent to .
Definition 5.4 We will say that a point is a d-precoincidence point of T and g if .
In the following results, we will assume some of the following conditions.
-
(a)
.
-
(b)
T is -nondecreasing.
-
(c)
There exists such that .
-
(d)
There exists such that
(3) -
(e)
for all such that .
-
(f)
for all such that .
-
(g)
Every ≼-nondecreasing, d-Cauchy sequence in X is d-convergent in X.
-
(h)
If is a nondecreasing sequence and d-converges to , then for all m.
-
(i)
for all such that and .
-
(j)
Every d-precoincidence point of T and g is a coincidence point of T and g (that is, if , then ).
As we have shown in Section 3, notice that a mapping d verifying (a)-(i) does not have to verify any of the conditions that define a metric on X.
Remark 5.2 A priori, d can take negative values in ℝ. However, condition (f) lets us prove some constraints about the sign of d. Indeed, if we take in condition (f), we deduce that for all . Furthermore, if in (f), it follows that for all such that . This does not mean that d is nonnegative because d could take negative values when or x and z are not ≼-comparable.
Remark 5.3 As we shall show in the proofs, the mapping d could only be considered on the set , that is, we will only use . In this case, the previous remark shows that, as usual, .
Theorem 5.1 Let be a preordered space and let be two mappings verifying (a)-(c). Then there exists a sequence such that for all . In particular,
Proof Since , there exists such that . Then . Since T is -nondecreasing, . Now , so there exists such that . Then . Since T is -nondecreasing, . Repeating this argument, there exists a sequence such that for all . □
Theorem 5.2 Let be a preordered space and let and be three mappings verifying (a)-(e). Then any sequence such that for all , is d-Cauchy ( is given as in Theorem 5.1).
Proof Since for all , it follows from (d) that, for all ,
By item 4 of Lemma 4.6, the sequence d-converges to zero. Using the same reasoning, since for all , it follows that also d-converges to zero. Therefore
Let us show that is d-Cauchy reasoning by contradiction. Suppose that is not d-Cauchy. Then there exist and partial subsequences and verifying
( is the least integer number, greater than , such that ). Since , we have . By (e),
Therefore, the sequence satisfies
Now, let us apply condition (e) and (5) to , and we deduce, for all k,
By (d), we also have, for all k,
By item 1 of Lemma 4.6, it follows that
Joining this inequality and (7), we deduce that, for all k,
Letting and using (4) and (6), we deduce that the sequence also verifies , and by (8), we have that for all k. Since , axiom () guarantees that , which is a contradiction with the fact that . This contradiction shows that is a d-Cauchy sequence. □
After the previous technical results, we give the main results of this manuscript.
Theorem 5.3 Let be a preordered space and let and be three mappings which fulfil conditions (a)-(h). Assume that the following condition holds.
(p) is -nondecreasing-closed.
Then there exists such that the sequence (defined in Theorem 5.2) d-converges to gz and to Tz. Furthermore, if (i) holds, then z is a d-precoincidence point of T and g.
Notice that, in the previous result, g and T need not be continuous.
Proof Theorem 5.2 guarantees that is d-Cauchy. Since it is ≼-nondecreasing, condition (g) implies that there exists such that d-converges to y (that is, ). Moreover, since is -nondecreasing-closed, , so there exists such that . Applying (h), for all m and, hence,
Since , item 3 of Lemma 4.6 guarantees that , that is, d-converges to Tz.
Now suppose that (i) holds. By (h), for all m and (i) implies that
Therefore . □
Theorem 5.4 Let be a preordered space and let and be three mappings which fulfil conditions (a)-(h). Suppose also:
-
(p′)
T and g are -nondecreasing-continuous and commuting and, at least, one of the following conditions holds:
-
() T is a ≼-nondecreasing mapping.
-
() g is a ≼-nondecreasing mapping.
-
() If and is a ≼-nondecreasing sequence such that d-converges to z and to ω at the same time, then .
-
Then there exists such that the sequence (defined in Theorem 5.2) d-converges to gy and to Ty at the same time. Furthermore, if (i) holds, then y is a d-precoincidence point of T and g.
In any case, T and g have, at least, a d-precoincidence point.
Proof Theorem 5.2 guarantees that is d-Cauchy. Since it is ≼-nondecreasing, condition (g) implies that there exists such that d-converges to y. Thus, taking into account that g and T are -nondecreasing-continuous, d-converges to gy and d-converges to Ty. Furthermore, since T and g are commuting, for all m, which means that d-converges, at the same time, to gy and to Ty.
Next, assume that (i) holds, and we claim that y is a d-precoincidence point of T and g. Firstly, if T (or g) is a ≼-nondecreasing mapping, then the sequence is ≼-nondecreasing. Since it d-converges to gy and to Ty, property (h) implies that and for all m. Reasoning as in (9), we conclude that y is a d-precoincidence point of T and g. Secondly, if T and g are not necessarily ≼-nondecreasing mappings, we could apply () to the sequence in order to deduce that , that is, y is a d-precoincidence point of T and g. □
We summarize and improve all previous results in the following theorem.
Theorem 5.5 Let be a preordered space and let and be three mappings verifying the following properties.
-
(a)
.
-
(b)
T is -nondecreasing.
-
(c)
There exists such that .
-
(d)
There exist such that
-
(e)
for all such that .
-
(f)
for all such that .
-
(g)
Every ≼-nondecreasing, d-Cauchy sequence in X is d-convergent in X.
-
(h)
If is a ≼-nondecreasing sequence and d-converges to , then for all m.
-
(i)
for all such that and .
-
(j)
Every d-precoincidence point of T and g is a coincidence point of T and g.
Assume also either
-
(p)
is -nondecreasing-closed, or
(p′) T and g are -nondecreasing-continuous and commuting and, at least, one of the following conditions holds:
-
() T is a ≼-nondecreasing mapping.
-
() g is a ≼-nondecreasing mapping.
-
() If and is a ≼-nondecreasing sequence such that d-converges to z and to ω at the same time, then .
Then T and g have, at least, a coincidence point.
Notice that condition (j) does not mean that implies .
Remark 5.4 If g is the identity mapping on the ambient space, then the quadruple introduced in Section 3 verifies all conditions (a)-(j) and (p)-() (see more details in Appendix 2).
Remark 5.5 Obviously, similar results can be stated changing the following hypothesis.
() T is -nonincreasing.
() There exists such that .
() Every nonincreasing, d-Cauchy sequence in X is d-convergent in X.
() If is a nonincreasing sequence and d-converges to , then for all m.
() is -nonincreasing-closed.
() T and g are -nonincreasing-continuous and commuting and, at least, one of the following conditions holds:
() T is a ≼-nonincreasing mapping.
() g is a ≼-nonincreasing mapping.
() If and is a ≼-nonincreasing sequence such that d-converges to z and to ω at the same time, then .
The unicity of the coincidence point cannot be guaranteed unless additional conditions are imposed. A result in this direction is the following.
Theorem 5.6 Under the hypothesis of Theorem 5.5, let be two coincidence points of T and g verifying that there exists such that and . Then .
Proof Define . Since , there exists such that . Repeating this process, there exists a sequence such that for all . We claim that and . Firstly, we reason using x, but the same argument is valid for y.
Indeed, notice that . As T is -nondecreasing, then , which means that . Again, implies , which means that . By induction, it is possible to prove that for all . Using condition (d), it follows that for all . Thus, by item 4 of Lemma 4.6, . The same argument proves that for all and . As a consequence, by (i), for all m, which lets us conclude that . □
Corollary 5.1 Under the hypothesis of Theorem 5.5, assume the following conditions.
-
For all coincidence points of T and g, there exists such that and .
-
g is injective on the set of all coincidence points of T and g.
-
If verify , then .
Then T and g have a unique coincidence point. Furthermore, if T and g are commuting, it is a common fixed point of T and g.
Proof Let be two coincidence points of T and g. Then and . By Theorem 5.6, . Therefore . As g is injective on the set of all coincidence points of T and g, we conclude that .
Now let be a coincidence point of T and g, and let . By Remark 2.2, z is also a coincidence point of T and g. Then , so x is a common fixed point of T and g. □
Taking and for all in the previous results, we obtain the following particular case.
Corollary 5.2 Theorems 5.1, 5.2, 5.3, 5.4, 5.5, 5.6 and Corollary 5.1 also hold if we replace condition (d) by the following one.
() There exists such that for all for which .
6 Consequences
This section is devoted to show how to apply Theorem 5.5 in many different contexts, and how to deduce unidimensional, coupled, tripled, quadruple and multidimensional fixed point theorems (for completeness, they are included in Appendix 1).
6.1 Fixed/coincidence point theorems in partially ordered metric spaces
In this subsection we show that different results in partially ordered metric spaces, including unidimensional, coupled, tripled, quadruple and multidimensional fixed point theorems, can be seen as simple consequences of Theorem 5.5.
Corollary 6.1 Theorems A.1, A.2 and A.3 follow from Theorem 5.5.
However, Theorems A.1 and A.2 cannot be applied to the example of Section 3.
Corollary 6.2 Theorem A.4 follows from Theorem 5.5.
Proof Let provided with the partial order if and only if and , and the metric given by for all . Define by for all , and let G be the identity mapping on Y. Let . Then the hypothesis of Theorem A.4 implies the hypothesis of Theorem 5.5 (for instance, is -nondecreasing because F has the mixed ≼-monotone property). The contractivity condition holds since, if ,
Theorem 5.5 assures us that and G have a coincidence point, that is, F has a coupled fixed point. □
Tripled, quadruple and multidimensional theorems can be proved similarly using , and , obtaining the following result.
Corollary 6.3 Theorems A.5, A.6 and A.7 follow from Theorem 5.5.
We remark that the techniques used in this paper might be applied in order to prove other coupled, tripled, quadruple, n-tupled fixed point theorems in the framework of various abstract spaces, e.g., partial metric spaces, cone metric spaces, fuzzy metric spaces, b-metric spaces, etc.
6.2 Fixed/coincidence point theorems in quasi-metric spaces
Recall that a mapping is a quasi-metric on X if it satisfies (), () and (), that is, if it verifies, for all :
-
() if and only if, ,
-
() .
In such a case, the pair is called a quasi-metric space. Some preliminaries about convergence, Cauchy sequences and completeness in quasi-metric spaces can be found in [29, 37].
Theorem 6.1 Let be a complete quasi-metric space and let be given mappings. Suppose that and that there exists such that
Then T and g have a unique coincidence point.
Proof Assume that ≼ is the preorder in X given by for all , that is, all points are ≼-comparable. Then all conditions of Theorem 5.5 hold. Moreover, as is -nondecreasing-closed, we deduce that T and g have, at least, a coincidence point. Furthermore, if u and v are two distinct coincidence points of T and g, then
which is a contradiction. □
If g is the identity mapping on X, we have the following statement.
Corollary 6.4 Let be a complete quasi-metric space and let be a given mapping. Suppose that there exists such that
Then T has a unique fixed point.
If for all , we have the following particular case.
Corollary 6.5 (Jleli and Samet [37], Theorem 3.2)
Let be a complete quasi-metric space and let be a mapping satisfying
where is continuous with . Then T has a unique fixed point.
6.3 Fixed/coincidence point theorems in G-metric spaces
Following [7, 39], recall that a generalized metric on X (or, more specifically, a G-metric on X) is a mapping verifying the following properties.
Definition 6.1
() if ;
() for all with ;
() for all with ;
() (symmetry in all three variables);
() (rectangle inequality) for all .
In such a case, the pair is called a G-metric space. Some preliminaries about convergence, Cauchy sequences and completeness in quasi-metric spaces can be found in [7, 29, 37, 39].
Lemma 6.1 (see, e.g., [29, 37])
Let be a G-metric space and let us define by
Then the following properties hold.
-
1.
and are quasi-metrics on X. Moreover,
(10) -
2.
In and in , a sequence is right-convergent (respectively, left-convergent) if and only if it is convergent. In such a case, its right-limit, its left-limit and its limit coincide.
-
3.
In and in , a sequence is right-Cauchy (respectively, left-Cauchy) if and only if it is Cauchy.
-
4.
In and in , every right-convergent (respectively, left-convergent) sequence has a unique right-limit (respectively, left-limit).
-
5.
If and , then .
-
6.
If , then is G-Cauchy ⟺ is -Cauchy ⟺ is -Cauchy.
-
7.
is complete ⟺ is complete ⟺ is complete.
Theorem 6.2 Let be a complete G-metric space and let be given mappings. Suppose that and that there exists such that
Then T and g have a unique coincidence point.
Proof It follows from Theorem 6.1 applied to the quasi-metric for all (as in Lemma 6.1). □
To conclude the paper, we include two appendices: in the first one, we recall some celebrated theorems that can be seem as particular cases of our main results; in the second one, we prove the statements announced in Section 3 and why our results can be applied.
Appendix 1: Some recent results we generalize
The following statements are well-known fixed point theorems in partially ordered metric spaces.
Theorem A.1 (Ran and Reurings [17])
Let be an ordered set endowed with a metric d and be a given mapping. Suppose that the following conditions hold:
-
(a)
is complete.
-
(b)
T is nondecreasing (w.r.t. ≼).
-
(c)
T is continuous.
-
(d)
There exists such that .
-
(e)
There exists a constant such that for all with .
Then T has a fixed point. Moreover, if for all there exists such that and , we obtain uniqueness of the fixed point.
Nieto and Rodríguez-López [18] slightly modified the hypothesis of the previous result obtaining the following theorem.
Theorem A.2 (Nieto and Rodríguez-López [18])
Let be an ordered set endowed with a metric d and be a given mapping. Suppose that the following conditions hold:
-
(a)
is complete.
-
(b)
T is nondecreasing (w.r.t. ≼).
-
(c)
If a nondecreasing sequence in X converges to some point , then for all m.
-
(d)
There exists such that .
-
(e)
There exists a constant such that for all with .
Then T has a fixed point. Moreover, if for all there exists such that and , we obtain uniqueness of the fixed point.
Theorem A.3 (Harjani and Sadarangani [40])
Let be a partially ordered set and suppose that there exists a metric d in X such that is a complete metric space. Let be a nondecreasing mapping such that
where ψ and ϕ are altering distance functions. Also assume that, at least, one of the following conditions holds.
-
(i)
T is continuous, or
-
(ii)
if a nondecreasing sequence in X converges to some point , then for all n.
If there exists with , then T has a fixed point.
Theorem A.4 (Bhaskar and Lakshmikantham [19])
Let be a partially ordered set endowed with a metric d. Let be a given mapping. Suppose that the following conditions hold:
-
(i)
is complete;
-
(ii)
F has the mixed monotone property;
-
(iii)
F is continuous or X has the following properties:
-
() if a nondecreasing sequence in X converges to some point , then for all n,
-
() if a decreasing sequence in X converges to some point , then for all n;
-
-
(iv)
there exist such that and ;
-
(v)
there exists a constant such that for all with and ,
Then F has a coupled fixed point . Moreover, if for all there exists such that and , we have uniqueness of the coupled fixed point and .
Theorem A.5 (Berinde and Borcut [22])
Let be a partially ordered set and suppose that there is a metric d on X such that is a complete metric space. Let be a mapping having the mixed g-monotone property. Assume that there exist constants with such that
for all with , , . Suppose that either F is continuous or has the following properties:
-
(a)
if a nondecreasing sequence , then for all m;
-
(b)
if a nondecreasing sequence , then for all m.
If there exist such that
then there exist such that
A quadruple version was obtained by Karapınar and Luong in [23].
Theorem A.6 (Karapınar and Luong [23])
Let be a partially ordered set and be a complete metric space. Let be a mapping having the mixed monotone property. Assume that there exists a constant such that
for all with , , and . Suppose that there exist such that
Suppose that either F is continuous or has the following properties:
-
(a)
if a nondecreasing sequence , then for all m;
-
(b)
if a nondecreasing sequence , then for all m.
Then there exist such that
Later, Berzig and Samet extended the previous result to the multidimensional case in the following way.
Theorem A.7 (Berzig and Samet [24])
Let be a partially ordered set and suppose that there is a metric d on X such that is a complete metric space. For N, m positive integers, , , let be a continuous mapping having the m-mixed monotone property. Assume that there exist the constants with for which
for all such that
If there exists such that
where , , , and , then there exists satisfying
Theorem A.8 (Dutta and Choudhury [41])
Let be a complete metric space and let be a self-mapping satisfying the inequality
for all , where are both continuous and monotone nondecreasing functions with if and only if . Then T has a unique fixed point.
Theorem A.9 (Luong and Thuan [42])
Let be a partially ordered set and suppose that there is a metric d on X such that is a complete metric space. Let be a mapping having the mixed monotone property on X such that there exist two elements with and . Suppose that there exist and such that
for all with and . Suppose either
-
(a)
F is continuous, or
-
(b)
X has the following properties:
-
(i)
if a nondecreasing sequence , then for all m,
-
(ii)
if a nonincreasing sequence , then for all m.
-
(i)
Then there exist such that and , that is, F has a coupled fixed point in X.
Appendix 2: Proof of statements of Section 3 and Remark 5.4
Recall that , ,
and and are defined by
We firstly prove some properties of this space.
() The binary relation ≼ is a preorder on (reflexive and transitive), but it is not a partial order since , and . Furthermore, if and , then
If , then either or or . The same is true if and .
() d does not verify () because .
() d does not verify () since but .
() d does not verify () since and .
() d does not verify (): if , and , then and .
() is not complete. Since and is the Euclidean metric on , then the sequence is d-Cauchy but it is not d-convergent in .
() If is a ≼-nondecreasing sequence in , then one, and only one, of the following cases holds.
⊳ for all m.
⊳ for all m.
⊳ There exists such that for all m (that is, is a constant sequence).
It follows from that fact that points of (respectively, , ) are only ≼-related with points of (respectively, , themselves), and for all .
() If is a ≼-nondecreasing, d-convergent sequence in , then one, and only one, of the following cases holds.
⊳ for all m. In this case, its d-limit is also in .
⊳ for all m. In this case, its d-limit is also in .
⊳ There exists such that for all m. In this case, its d-limit is .
It follows from (). Notice that since .
() T is not a d-contraction (that is, there is no such that for all ) because but .
Now we prove assertions (a)-(j), (p) and (p′) taking into account that g is the identity mapping on .
-
(a)
. It is obvious.
-
(b)
T is ≼-nondecreasing. Let be such that and . By (), if , then and . Then , being , so . If , then , so . The case is impossible since .
-
(c)
There exists such that . If , then .
-
(d)
There exists such that for all for which . Let and assume that are such that . By (), if , then and . If , then and . If , then and . Therefore .
-
(e)
for all such that . By (), there are only three cases. If , the triangular inequality holds because it is true in ℝ provided with the Euclidean metric. If , then all distances are zero. Finally, if , all distances are either zero (if ) or 0.5 (if ).
-
(f)
for all such that . It is similar to the previous one using ().
-
(g)
Every d-Cauchy, ≼-nondecreasing sequence in is d-convergent. It follows from ().
-
(h)
If is a ≼-nondecreasing sequence and d-converges to , then for all m. It also follows from ().
-
(i)
for all such that and . It is similar to (e) using ().
-
(j)
Every d-precoincidence point of T and g is a coincidence point of T and g. Assume that . Since , it is impossible or (the only cases, far from , in which the distance between them can be zero). Then implies that and , so .
-
(p)
is ≼-nondecreasing d-closed. It is immediate: indeed, is d-closed.
(p′) T and g are -nondecreasing-continuous and commuting, and g is ≼-nondecreasing. It is only necessary to prove that T is -nondecreasing-continuous. Actually, it follows from () since there are only three cases: if for all m, its d-limit is also in , and d-converges to ; if for all m, then and for all m; finally, if there exists such that for all m, then for all m.
We point out that T is not d-continuous: if and , then but does not d-converge to .
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Acknowledgements
This article was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. The second author acknowledges with thanks DSR for financial support. The first author has been partially supported by Junta de Andalucía by project FQM-268 of the Andalusian CICYE.
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Roldán-López-de-Hierro, AF., Shahzad, N. Some fixed/coincidence point theorems under -contractivity conditions without an underlying metric structure. Fixed Point Theory Appl 2014, 218 (2014). https://doi.org/10.1186/1687-1812-2014-218
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DOI: https://doi.org/10.1186/1687-1812-2014-218