Abstract
In this paper, we prove the existence and uniqueness of fixed points of certain cyclic mappings via auxiliary functions in the context of Gmetric spaces, which were introduced by Zead and Sims. In particular, we extend, improve and generalize some earlier results in the literature on this topic.
MSC: 47H10, 54H25.
Keywords:
fixed point; Gmetric space; cyclic maps; cyclic contractions1 Introduction and preliminaries
It is well established that fixed point theory, which mainly concerns the existence and uniqueness of fixed points, is today’s one of the most investigated research areas as a major subfield of nonlinear functional analysis. Historically, the first outstanding result in this field that guaranteed the existence and uniqueness of fixed points was given by Banach [1]. This result, known as the Banach mapping contraction principle, simply states that every contraction mapping has a unique fixed point in a complete metric space. Since the first appearance of the Banach principle, the ever increasing application potential of the fixed point theory in various research fields, such as physics, chemistry, certain engineering branches, economics and many areas of mathematics, has made this topic more crucial than ever. Consequently, after the Banach celebrated principle, many authors have searched for further fixed point results and reported successfully new fixed point theorems conceived by the use of two very effective techniques, combined or separately.
The first one of these techniques is to ‘replace’ the notion of a metric space with a more general space. Quasimetric spaces, partial metric spaces, rectangular metric spaces, fuzzy metric space, bmetric spaces, Dmetric spaces, Gmetric spaces are generalizations of metric spaces and can be considered as examples of ‘replacements’. Amongst all of these spaces, Gmetric spaces, introduced by Zead and Sims [2], are ones of the interesting. Therefore, in the last decade, the notion of a Gmetric space has attracted considerable attention from researchers, especially from fixed point theorists [325].
The second one of these techniques is to modify the conditions on the operator(s). In other words, it entails the examination of certain conditions under which the contraction mapping yields a fixed point. One of the attractive results produced using this approach was given by Kirk et al.[26] in 2003 through the introduction of the concepts of cyclic mappings and best proximity points. After this work, best proximity theorems and, in particular, the fixed point theorems in the context of cyclic mappings have been studied extensively (see, e.g., [2743]).
The two upper mentioned topics, cyclic mappings and Gmetric spaces, have been combined by Aydi in [22] and Karapınar et al. in [36]. In these papers, the existence and uniqueness of fixed points of cyclic mappings are investigated in the framework of Gmetric spaces. In this paper, we aim to improve on certain statements proved on these two topics. For the sake of completeness, we will include basic definitions and crucial results that we need in the rest of this work.
Mustafa and Sims [2] defined the concept of Gmetric spaces as follows.
Definition 1.1 (See [2])
Let X be a nonempty set, be a function satisfying the following properties:
(G4) (symmetry in all three variables),
(G5) (rectangle inequality) for all .
Then the function G is called a generalized metric or, more specifically, a Gmetric on X, and the pair is called a Gmetric space.
Note that every Gmetric on X induces a metric on X defined by
For a better understanding of the subject, we give the following examples of Gmetrics.
Example 1.1 Let be a metric space. The function , defined by
Example 1.2 (See, e.g., [2])
Let . The function , defined by
In their initial paper, Mustafa and Sims [2] also defined the basic topological concepts in Gmetric spaces as follows.
Definition 1.2 (See [2])
Let be a Gmetric space, and let be a sequence of points of X. We say that is Gconvergent to if
that is, for any , there exists such that for all . We call x the limit of the sequence and write or .
Proposition 1.1 (See [2])
Letbe aGmetric space. The following are equivalent:
Definition 1.3 (See [2])
Let be a Gmetric space. A sequence is called a GCauchy sequence if, for any , there exists such that for all , that is, as .
Proposition 1.2 (See [2])
Letbe aGmetric space. Then the following are equivalent:
(2) for any, there existssuch thatfor all.
Definition 1.4 (See [2])
A Gmetric space is called Gcomplete if every GCauchy sequence is Gconvergent in .
Definition 1.5 Let be a Gmetric space. A mapping is said to be continuous if for any three Gconvergent sequences , and converging to x, y and z respectively, is Gconvergent to .
Note that each Gmetric on X generates a topology on X whose base is a family of open Gballs , where for all and . A nonempty set is Gclosed in the Gmetric space if . Observe that
for all . We recall also the following proposition.
Proposition 1.3 (See, e.g., [36])
Letbe aGmetric space andAbe a nonempty subset ofX. The setAisGclosed if for anyGconvergent sequenceinAwith limitx, we have.
Mustafa [5] extended the wellknown Banach contraction principle mapping in the framework of Gmetric spaces as follows.
Theorem 1.1 (See [5])
Letbe a completeGmetric space andbe a mapping satisfying the following condition for all:
where. ThenThas a unique fixed point.
Theorem 1.2 (See [5])
Letbe a completeGmetric space andbe a mapping satisfying the following condition for all:
where. ThenThas a unique fixed point.
Remark 1.1 We notice that the condition (2) implies the condition (3). The converse is true only if . For details, see [5].
Lemma 1.1 ([5])
By the rectangle inequality (G5) together with the symmetry (G4), we have
A map on a metric space is called a weak ϕcontraction if there exists a strictly increasing function with such that
for all . We notice that these types of contractions have also been a subject of extensive research (see, e.g., [4449]). In what follows, we recall the notion of cyclic weak ψcontractions on Gmetric spaces. Let Ψ be the set of continuous functions with and for . In [36], the authors concentrated on two types of cyclic contractions: cyclictype Banach contractions and cyclic weak ϕcontractions.
Theorem 1.3Letbe aGcompleteGmetric space andbe a family of nonemptyGclosed subsets ofXwith. Letbe a map satisfying
Suppose that there exists a functionsuch that the mapTsatisfies the inequality
ThenThas a unique fixed point in.
The following result, which can be considered as a corollary of Theorem 1.3, is stated in [36].
Theorem 1.4 (See [36])
Letbe aGcompleteGmetric space andbe a family of nonemptyGclosed subsets ofX. Letandbe a map satisfying
holds for alland, , thenThas a unique fixed point in.
In this paper, we extend, generalize and enrich the results on the topic in the literature.
2 Main results
We start this section by defining some sets of auxiliary functions. Let ℱ denote all functions such that if and only if . Let Ψ and Φ be the subsets of ℱ such that
Lemma 2.1Letbe aGcompleteGmetric space andbe a sequence inXsuch thatis nonincreasing,
Ifis not a Cauchy sequence, then there existand two sequencesandof positive integers such that the following sequences tend toεwhen:
Proof
Due to Lemma 1.1, we have
Letting regarding the assumption of the lemma, we derive that
If is not GCauchy, then, due to Proposition 1.2, there exist and corresponding subsequences and of ℕ satisfying for which
where is chosen as the smallest integer satisfying (13), that is,
By (13), (14) and the rectangle inequality (G5), we easily derive that
Letting in (15) and using (10), we get
Further,
and
Passing to the limit when and using (10) and (16), we obtain that
In a similar way,
and
Passing to the limit when and using (10) and (16), we obtain that
Furthermore,
and
Passing to the limit when and using (10) and (16), we obtain that
By regarding the assumptions (G3) and (G5) together with the expression (13), we derive the following:
Letting in the inequality above and using (12) and (16), we conclude that
□
Theorem 2.1Letbe aGcompleteGmetric space andbe a family of nonemptyGclosed subsets ofXwith. Letbe a map satisfying
Suppose that there exist functionsandsuch that the mapTsatisfies the inequality
ThenThas a unique fixed point in.
Proof First we show the existence of a fixed point of the map T. For this purpose, we take an arbitrary and define a sequence in the following way:
We have , , , … since T is a cyclic mapping. If for some , then, clearly, the fixed point of the map T is . From now on, assume that for all . Consider the inequality (29) by letting and ,
where
If , then the expression (32) implies that
So, the inequality (34) yields . Thus, we conclude that
This contradicts the assumption for all . So, we derive that
Hence the inequality (32) turns into
Thus, is a nonnegative, nonincreasing sequence that converges to . We will show that . Suppose, on the contrary, that . Taking in (36), we derive that
By the continuity of ψ and the lower semicontinuity of ϕ, we get
Then it follows that . Therefore, we get , that is,
Lemma 1.1 with and implies that
So, we get that
Next, we will show that is a GCauchy sequence in . Suppose, on the contrary, that is not GCauchy. Then, due to Proposition 1.2, there exist and corresponding subsequences and of ℕ satisfying for which
where is chosen as the smallest integer satisfying (42), that is,
By (42), (43) and the rectangle inequality (G5), we easily derive that
Letting in (44) and using (39), we get
Notice that for every there exists satisfying such that
Thus, for large enough values of k, we have , and and lie in the adjacent sets and respectively for some . When we substitute and in the expression (29), we get that
where
By using Lemma 2.1, we obtain that
and
So, we obtain that
So, we have . We deduce that . This contradicts the assumption that is not GCauchy. As a result, the sequence is GCauchy. Since is Gcomplete, it is Gconvergent to a limit, say . It easy to see that . Since , then the subsequence , the subsequence and, continuing in this way, the subsequence . All the m subsequences are Gconvergent in the Gclosed sets and hence they all converge to the same limit . To show that the limit w is the fixed point of T, that is, , we employ (29) with , . This leads to
where
Finally, we prove that the fixed point is unique. Assume that is another fixed point of T such that . Then, since both v and w belong to , we set and in (29), which yields
where
On the other hand, by setting and in (29), we obtain that
where
If , then . Indeed, by definition, we get that . Hence . If , then by (56) and by (55),
and, clearly, . So, we conclude that . Otherwise, . Then by (58), and by (57),
and, clearly, . So, we conclude that . Hence the fixed point of T is unique. □
Remark 2.1 We notice that some fixed point result in the context of Gmetric can be obtained by usual (wellknown) fixed point theorems (see, e.g., [50,51]). In fact, this is not a surprising result due to strong relationship between the usual metric and Gmetric space (see, e.g., [2,3,5]). Note that a Gmetric space tells about the distance of three points instead of two points, which makes it original. We also emphasize that the techniques used in [50,51] are not applicable to our main theorem.
To illustrate Theorem 2.1, we give the following example.
Example 2.1 Let and let be given as . Let and . Define the function as
Clearly, the function G is a Gmetric on X. Define also as and as . Obviously, the map T has a unique fixed point .
It can be easily shown that the map T satisfies the condition (29). Indeed,
which yields
Moreover, we have
We derive from (63) that
On the other hand, we have the following inequality:
By elementary calculation, we conclude from (65) and (64) that
Combining the expressions (62) and (65), we obtain that
Hence, all conditions of Theorem 2.1 are satisfied. Notice that 0 is the unique fixed point of T.
For particular choices of the functions ϕ, ψ, we obtain the following corollaries.
Corollary 2.1Letbe aGcompleteGmetric space andbe a family of nonemptyGclosed subsets ofXwith. Letbe a map satisfying
Suppose that there exists a constantsuch that the mapTsatisfies
ThenThas a unique fixed point in.
Proof The proof is obvious by choosing the functions ϕ, ψ in Theorem 2.1 as and . □
Corollary 2.2Letbe aGcompleteGmetric space andbe a family of nonemptyGclosed subsets ofXwith. Letbe a map satisfying
Suppose that there exist constantsa, b, c, dandewithand there exists a functionsuch that the mapTsatisfies the inequality
for alland, . ThenThas a unique fixed point in.
Proof
Clearly, we have
where
By Corollary 2.1, the map T has a unique fixed point. □
Corollary 2.3Letbe aGcompleteGmetric space andbe a family of nonemptyGclosed subsets ofXwith. Letbe a map satisfying
Suppose that there exist functionsandsuch that the mapTsatisfies the inequality
ThenThas a unique fixed point in.
Proof The expression (75) coincides with the expression (30). Following the lines in the proof of Theorem 2.1, by letting and , we get the desired result. □
Cyclic maps satisfying integral type contractive conditions are amongst common applications of fixed point theorems. In this context, we consider the following applications.
Corollary 2.4Letbe aGcompleteGmetric space andbe a family of nonemptyGclosed subsets ofXwith. Letbe a map satisfying
Suppose also that there exist functionsandsuch that the mapTsatisfies
where
for alland, . ThenThas a unique fixed point in.
Corollary 2.5Letbe aGcompleteGmetric space andbe a family of nonemptyGclosed subsets ofXwith. Letbe a map satisfying
Suppose also that
for alland, . ThenThas a unique fixed point in.
Proof The proof is obvious by choosing the function ϕ, ψ in Corollary 2.4 as and . □
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors read and approved the final manuscript.
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