Abstract
The aim of this paper is to present some coincidence point results for six mappings satisfying the generalized weakly contractive condition in the framework of partially ordered Gmetric spaces. To elucidate our results, we present two examples together with an application of a system of integral equations.
MSC: 47H10, 54H25.
Keywords:
coincidence point; common fixed point; generalized weak contraction; generalized metric space; partially weakly increasing mapping; altering distance function1 Introduction and mathematical preliminaries
Let be a metric space and f be a selfmapping on X. If for some x in X, then x is called a fixed point of f. The set of all fixed points of f is denoted by . If , and for each in a complete metric space X, the sequence , , converges to z, then f is called a Picard operator.
The function is called an altering distance function if φ is continuous and nondecreasing and if and only if [1].
A selfmapping f on X is a weak contraction if the following contractive condition holds:
for all , where φ is an altering distance function.
The concept of a weakly contractive mapping was introduced by Alber and GuerreDelabrere [2] in the setup of Hilbert spaces. Rhoades [3] considered this class of mappings in the setup of metric spaces and proved that a weakly contractive mapping is a Picard operator.
Let f and g be two selfmappings on a nonempty set X. If for some x in X, then x is called a common fixed point of f and g. Sessa [4] defined the concept of weakly commutative maps to obtain common fixed point for a pair of maps. Jungck generalized this idea, first to compatible mappings [5] and then to weakly compatible mappings [6]. There are examples which show that each of these generalizations of commutativity is a proper extension of the previous definition.
Zhang and Song [7] introduced the concept of a generalized φweak contractive mapping as follows.
Selfmappings f and g on a metric space X are called generalized φweak contractions if there exists a lower semicontinuous function with and for all such that for all ,
where
Based on the above definition, they proved the following common fixed point result.
Theorem 1.1[7]
Letbe a complete metric space. Ifare generalizedφweak contractive mappings, then there exists a unique pointsuch that.
For further works in this direction, we refer the reader to [820].
Recently, many researchers have focused on different contractive conditions in complete metric spaces endowed with a partial order and studied fixed point theory in the socalled bistructural spaces. For more details on fixed point results, its applications, comparison of different contractive conditions and related results in ordered metric spaces, we refer the reader to [2140] and the references mentioned therein.
Mustafa and Sims [41] generalized the concept of a metric, in which to every triplet of points of an abstract set, a real number is assigned. Based on the notion of generalized metric spaces, Mustafa et al.[4249] obtained some fixed point theorems for mappings satisfying different contractive conditions. Chugh et al.[50] obtained some fixed point results for maps satisfying property P in Gmetric spaces. Saadati et al.[51] studied fixed point of contractive mappings in partially ordered Gmetric spaces. Shatanawi [52] obtained fixed points of Φmaps in Gmetric spaces. For more details, we refer to [21,5365].
Very recently, Jleli and Samet [66] and Samet et al.[67] noticed that some fixed point theorems in the context of a Gmetric space can be concluded by some existing results in the setting of a (quasi)metric space. In fact, if the contraction condition of the fixed point theorem on a Gmetric space can be reduced to two variables instead of three variables, then one can construct an equivalent fixed point theorem in the setting of a usual metric space. More precisely, in [66,67], the authors noticed that forms a quasimetric. Therefore, if one can transform the contraction condition of existence results in a Gmetric space in such terms, , then the related fixed point results become the known fixed point results in the context of a quasimetric space.
The following definitions and results will be needed in the sequel.
Definition 1.2[41]
Let X be a nonempty set. Suppose that a mapping satisfies:
(G4) (symmetry in all three variables); and
Then G is called a Gmetric on X and is called a Gmetric space.
Definition 1.3[41]
A sequence in a Gmetric space X is:
(i) a Gconvergent sequence if there is such that for any , and , for all , .
(ii) a GCauchy sequence if, for every , there is a natural number such that for all , .
A Gmetric space on X is said to be Gcomplete if every GCauchy sequence in X is Gconvergent in X. It is known that Gconverges to if and only if as .
Lemma 1.4[41]
LetXbe aGmetric space. Then the following are equivalent:
(1) The sequenceisGconvergent tox.
Lemma 1.5[68]
LetXbe aGmetric space. Then the following are equivalent:
For every, there existssuch that for all, ; that is, ifas.
Definition 1.6[41]
Let and be two Gmetric spaces. Then a function is Gcontinuous at a point if and only if it is Gsequentially continuous at x; that is, whenever is Gconvergent to x, is convergent to .
Definition 1.7 A Gmetric on X is said to be symmetric if for all .
Proposition 1.8EveryGmetric onXdefines a metriconXby
For a symmetricGmetric space, one obtains
However, ifGis not symmetric, then the following inequality holds:
Definition 1.9 A partially ordered Gmetric space is said to have the sequential limit comparison property if for every nondecreasing sequence (nonincreasing sequence) in X, implies that ().
Definition 1.10 Let f and g be two selfmaps on a partially ordered set X. A pair is said to be
(i) weakly increasing if and for all [69],
(ii) partially weakly increasing if for all [22].
Let X be a nonempty set and be a given mapping. For every , let .
Definition 1.11 Let be a partially ordered set, and let be mappings such that and . The ordered pair is said to be: (a) weakly increasing with respect to h if and only if for all , for all , and for all [34], (b) partially weakly increasing with respect to h if for all [32].
Remark 1.12 In the above definition: (i) if , we say that f is weakly increasing (partially weakly increasing) with respect to h, (ii) if (the identity mapping on X), then the above definition reduces to a weakly increasing (partially weakly increasing) mapping (see [34,40]).
The following is an example of mappings f, g, h, R, S and T for which all ordered pairs , and are partially weakly increasing with respect to R, S and T but not weakly increasing with respect to them.
Example 1.13 Let . We define functions by
and
Let X be a Gmetric space and . The pair is said to be compatible if and only if , whenever is a sequence in X such that for some .
Definition 1.15 (see, e.g., [67])
A quasimetric on a nonempty set X is a mapping such that (p1) if and only if , (p2) for all . A pair is said to be a quasimetric space.
The study of unique common fixed points of mappings satisfying strict contractive conditions has been at the center of vigorous research activity. The study of common fixed point theorems in generalized metric spaces was initiated by Abbas and Rhoades [56] (see also [21,53,54]). Motivated by the work in [8,13,16,17,22,32] and [40], we prove some coincidence point results for nonlinear generalized weakly contractive mappings in partially ordered Gmetric spaces.
2 Main results
Let be an ordered Gmetric space, and let be six selfmappings. Throughout this paper, unless otherwise stated, for all , let
Let X be any nonempty set and be six mappings such that , and . Let be an arbitrary point of X. Choose such that , such that and such that . This can be done as , and .
Continuing in this way, we construct a sequence defined by: , , and for all . The sequence in X is said to be a Jungcktype iterative sequence with initial guess .
Theorem 2.1Letbe a partially orderedGcompleteGmetric space. Letbe six mappings such that, and. Suppose that for every three comparable elements, we have
whereare altering distance functions. Letf, g, h, R, SandTbe continuous, the pairs, andbe compatible and the pairs, andbe partially weakly increasing with respect toR, SandT, respectively. Then the pairs, andhave a coincidence pointinX. Moreover, if, andare comparable, thenis a coincidence point off, g, h, R, SandT.
Proof Let be a Jungcktype iterative sequence with initial guess in X; that is, , and for all .
As , and , and the pairs , and are partially weakly increasing with respect to R, S and T, so we have
Continuing this process, we obtain for .
We will complete the proof in three steps.
Define . Suppose for some . Then . If , then gives . Indeed,
where
Thus,
which implies that ; that is. Similarly, if , then gives . Also, if , then implies that . Consequently, the sequence becomes constant for .
Suppose that
for each k. We now claim that the following inequality holds:
Let and for , . Then, as , using (2.1), we obtain that
where
Hence, (2.4) implies that
which is possible only if ; that is, , a contradiction to (2.2). Hence, and
Therefore, (2.3) is proved for .
Similarly, it can be shown that
and
Hence, is a nondecreasing sequence of nonnegative real numbers. Therefore, there is such that
Since
taking limit as in (2.8), we obtain
Taking limit as in (2.4), using (2.7), (2.9) and the continuity of ψ and φ, we have . Therefore . Hence,
from our assumptions about φ. Also, from Definition 1.2, part (G3), we have
Step II. We now show that is a GCauchy sequence in X. Because of (2.10), it is sufficient to show that is GCauchy.
We assume on contrary that there exists for which we can find subsequences and of such that and
and is the smallest number such that the above statement holds; i.e.,
From the rectangle inequality and (2.14), we have
Taking limit as in (2.15), from (2.11) and (2.13) we obtain that
Using the rectangle inequality, we have
Taking limit as in (2.17), from (2.16), (2.10), (2.11) and (2.12) we have
Finally,
Taking limit as in (2.19) and using (2.16), (2.10), (2.11) and (2.12), we have
Consider
Taking limit as and using (2.10), (2.11) and (2.12), we have
Therefore,
where
Taking limit as and using (2.11), (2.12), (2.18) and (2.21) in (2.22), we have
a contradiction. Hence, is a GCauchy sequence.
Step III. We will show that f, g, h, R, S and T have a coincidence point.
Since is a GCauchy sequence in the Gcomplete Gmetric space X, there exists such that
and
Hence,
Moreover, from , , and the continuity of T and f, we obtain
By the rectangle inequality, we have
Taking limit as in (2.29), we obtain
which implies that , that is, is a coincidence point of f and T.
Similarly, and . Now, let , and be comparable. By (2.1) we have
where
Hence (2.30) gives
In the following theorem, the continuity assumption on the mappings f, g, h, R, S and T is in fact replaced by the sequential limit comparison property of the space, and the compatibility of the pairs , and is in fact replaced by weak compatibility of the pairs.
Theorem 2.2Letbe a partially orderedGcompleteGmetric space with the sequential limit comparison property, letbe six mappings such that, , and let, RX, SXandTXbeGcomplete subsets of X. Suppose that for comparable elements, we have
whereare altering distance functions. Then the pairs, andhave a coincidence pointinXprovided that the pairs, andare weakly compatible and the pairs, andare partially weakly increasing with respect toR, SandT, respectively. Moreover, if, andare comparable, thenis a coincidence point off, g, h, R, SandT.
Proof Following the proof of Theorem 2.1, there exists such that
Since is Gcomplete and , therefore . Hence, there exists such that and
Similarly, there exists such that and
Now, we prove that w is a coincidence point of f and T. For this purpose, we show that . Suppose opposite . Since as , so .
Therefore, from (2.31), we have
where
Taking limit as in (2.35), as and from (G2) and the fact that , we obtain that
which implies that , a contradiction, so . Again from the above inequality it is easy to see that . So, we have .
As f and T are weakly compatible, we have . Thus is a coincidence point of f and T.
Similarly it can be shown that is a coincidence point of the pairs and .
The rest of the proof can be obtained from the same arguments as those in the proof of Theorem 2.1. □
Remark 2.3 Let be a Gmetric space. Let be mappings. If we define functions in the following way:
and
for all , it is easy to see that both mappings p and q do not satisfy the conditions of Definition 1.15. Hence, Theorem 2.1 and Theorem 2.2 cannot be characterized in the context of quasimetric as it is suggested in [66,67].
Taking in Theorem 2.1, we obtain the following result.
Corollary 2.4Letbe a partially orderedGcompleteGmetric space, and letbe four mappings such that. Suppose that for every three comparable elements, we have
where
andare altering distance functions. Thenf, g, handRhave a coincidence point inXprovided that the pairs, andare partially weakly increasing with respect toRand either
a. fis continuous and the pairis compatible, or
b. gis continuous and the pairis compatible, or
c. his continuous and the pairis compatible.
Taking and in Theorem 2.1, we obtain the following coincidence point result.
Corollary 2.5Letbe a partially orderedGcompleteGmetric space, and letbe two mappings such that. Suppose that for every three comparable elements, we have
where
andare altering distance functions. Then the pairhas a coincidence point inXprovided thatfandRare continuous, the pairis compatible andfis weakly increasing with respect toR.
Example 2.6 Let and G on X be given by for all . We define an ordering ‘⪯’ on X as follows:
Define selfmaps f, g, h, S, T and R on X by
To prove that are partially weakly increasing with respect to R, let be such that ; that is, . By the definition of f and R, we have and . As for all , therefore , or
Therefore, . Hence is partially weakly increasing with respect to R. Similarly, one can show that and are partially weakly increasing with respect to S and T, respectively.
Furthermore, and the pairs , and are compatible. Indeed, if is a sequence in X such that for some , . Therefore, we have
Continuity of sinh^{−1} and sinh implies that
and the uniqueness of the limit gives that . But
So, we have . Since f and T are continuous, we have
Define as and for all , where .
Using the mean value theorem simultaneously for the functions sinh^{−1} and sinh, we have
Thus, (2.1) is satisfied for all . Therefore, all the conditions of Theorem 2.1 are satisfied. Moreover, 0 is a coincidence point of f, g, h, R, S and T.
The following example supports the usability of our results for nonsymmetric Gmetric spaces.
Example 2.7 Let be endowed with the usual order. Let
and
It is easy to see that is a nonsymmetric Gmetric space.
Also, has the sequential limit comparison property. Indeed, for each in X such that for an , there is such that for each , .
Define the selfmaps f, g, h, R, S and T by
We see that
and
Also, RX, SX and TX are Gcomplete. The pairs , and are weakly compatible.
On the other hand, one can easily check that the pairs , and are partially weakly increasing with respect to R, S and T, respectively.
According to the definition of f, g, h and G for each three elements , we see that
But
and
Hence, we have
Therefore, all the conditions of Theorem 2.2 are satisfied. Moreover, 0 is a coincidence point of f, g, h, R, S and T.
Let Λ be the set of all functions satisfying the following conditions:
(I) μ is a positive Lebesgue integrable mapping on each compact subset of .
Remark 2.8 Suppose that there exists such that mappings f, g, h, R, S and T satisfy the following condition:
Then f, g, h, R, S and T have a coincidence point if the other conditions of Theorem 2.1 are satisfied.
For this, define the function . Then (2.38) becomes
Take and . It is easy to verify that and are altering distance functions.
Taking , and in Theorems 2.1 and 2.2, we obtain the following common fixed point result.
Theorem 2.9Letbe a partially orderedGcompleteGmetric space, and letfandgbe two selfmappings onX. Suppose that for every comparable elements,
where
andare altering distance functions. Then the pairhas a common fixed pointzinXprovided that the pairis weakly increasing and either
a. forgis continuous, or
b. Xhas the sequential limit comparison property.
Taking in the above, we obtain the following common fixed point result.
Theorem 2.10Letbe a partially ordered completeGmetric space, and letfbe a selfmapping onX. Suppose that for every comparable elements,
where
andare altering distance functions. Thenfhas a fixed pointzinXprovided thatfis weakly increasing and either
a. fis continuous, or
b. Xhas the sequential limit comparison property.
3 Existence of a common solution for a system of integral equations
Motivated by the work in [21] and [32], we consider the following system of integral equations:
where . The aim of this section is to prove the existence of a solution for (3.1) which belongs to (the set of all continuous realvalued functions defined on ) as an application of Corollary 2.4.
The considered problem can be reformulated as follows.
and
for all and for all . Obviously, the existence of a solution for (3.1) is equivalent to the existence of a common fixed point of f, g and h.
Let
Equip X with the Gmetric given by
for all . Evidently, is a complete Gmetric space. We endow X with the partial ordered ⪯ given by
for all . It is known that has the sequential limit comparison property [37].
Now, we will prove the following result.
Theorem 3.1Suppose that the following hypotheses hold:
(ii) For alland for all, we have
and
(iii) For alland for allwith, we have
whereis a continuous function satisfying
Then system (3.1) has a solution.
Proof From condition (ii), the ordered pairs , and are partially weakly increasing.
Now, let be such that . From condition (iii), for all , we have
Hence,
Similarly, we can show that
and
Therefore, from (3.2), (3.3) and (3.4), we have
Putting, , and in Corollary 2.4, there exists , a common fixed point of f and g and h, which is a solution of (3.1). □
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.
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