Abstract
In this paper, we prove the existence and uniqueness of fixed points of certain cyclic mappings via auxiliary functions in the context of G-metric 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.
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1 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. Quasi-metric spaces, partial metric spaces, rectangular metric spaces, fuzzy metric space, b-metric spaces, D-metric spaces, G-metric spaces are generalizations of metric spaces and can be considered as examples of ‘replacements’. Amongst all of these spaces, G-metric spaces, introduced by Zead and Sims [2], are ones of the interesting. Therefore, in the last decade, the notion of a G-metric space has attracted considerable attention from researchers, especially from fixed point theorists [3–25].
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., [27–43]).
The two upper mentioned topics, cyclic mappings and G-metric 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 G-metric 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 G-metric spaces as follows.
Definition 1.1 (See [2])
Let X be a nonempty set, be a function satisfying the following properties:
-
(G1)
if ,
-
(G2)
for all with ,
-
(G3)
for all with ,
-
(G4)
(symmetry in all three variables),
-
(G5)
(rectangle inequality) for all .
Then the function G is called a generalized metric or, more specifically, a G-metric on X, and the pair is called a G-metric space.
Note that every G-metric on X induces a metric on X defined by
For a better understanding of the subject, we give the following examples of G-metrics.
Example 1.1 Let be a metric space. The function , defined by
for all , is a G-metric on X.
Example 1.2 (See, e.g., [2])
Let . The function , defined by
for all , is a G-metric on X.
In their initial paper, Mustafa and Sims [2] also defined the basic topological concepts in G-metric spaces as follows.
Definition 1.2 (See [2])
Let be a G-metric space, and let be a sequence of points of X. We say that is G-convergent 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])
Let be a G-metric space. The following are equivalent:
-
(1)
is G-convergent to x,
-
(2)
as ,
-
(3)
as ,
-
(4)
as .
Definition 1.3 (See [2])
Let be a G-metric space. A sequence is called a G-Cauchy sequence if, for any , there exists such that for all , that is, as .
Proposition 1.2 (See [2])
Let be a G-metric space. Then the following are equivalent:
-
(1)
the sequence is G-Cauchy,
-
(2)
for any , there exists such that for all .
Definition 1.4 (See [2])
A G-metric space is called G-complete if every G-Cauchy sequence is G-convergent in .
Definition 1.5 Let be a G-metric space. A mapping is said to be continuous if for any three G-convergent sequences , and converging to x, y and z respectively, is G-convergent to .
Note that each G-metric on X generates a topology on X whose base is a family of open G-balls , where for all and . A nonempty set is G-closed in the G-metric space if . Observe that
for all . We recall also the following proposition.
Proposition 1.3 (See, e.g., [36])
Let be a G-metric space and A be a nonempty subset of X. The set A is G-closed if for any G-convergent sequence in A with limit x, we have .
Mustafa [5] extended the well-known Banach contraction principle mapping in the framework of G-metric spaces as follows.
Theorem 1.1 (See [5])
Let be a complete G-metric space and be a mapping satisfying the following condition for all :
where . Then T has a unique fixed point.
Theorem 1.2 (See [5])
Let be a complete G-metric space and be a mapping satisfying the following condition for all :
where . Then T has 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 ϕ-contraction 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., [44–49]). In what follows, we recall the notion of cyclic weak ψ-contractions on G-metric spaces. Let Ψ be the set of continuous functions with and for . In [36], the authors concentrated on two types of cyclic contractions: cyclic-type Banach contractions and cyclic weak ϕ-contractions.
Theorem 1.3 Let be a G-complete G-metric space and be a family of nonempty G-closed subsets of X with . Let be a map satisfying
Suppose that there exists a function such that the map T satisfies the inequality
for all and , , where
Then T has 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])
Let be a G-complete G-metric space and be a family of nonempty G-closed subsets of X. Let and be a map satisfying
If there exists such that
holds for all and , , then T has 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 2.1 Let be a G-complete G-metric space and be a sequence in X such that is nonincreasing,
If is not a Cauchy sequence, then there exist and two sequences and of positive integers such that the following sequences tend to ε when :
Proof
Due to Lemma 1.1, we have
Letting regarding the assumption of the lemma, we derive that
If is not G-Cauchy, 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 2.1 Let be a G-complete G-metric space and be a family of nonempty G-closed subsets of X with . Let be a map satisfying
Suppose that there exist functions and such that the map T satisfies the inequality
for all and , , where
Then T has 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 semi-continuity 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 G-Cauchy sequence in . Suppose, on the contrary, that is not G-Cauchy. 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 G-Cauchy. As a result, the sequence is G-Cauchy. Since is G-complete, it is G-convergent 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 G-convergent in the G-closed 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
Passing to limsup as , we get
Thus, and hence , that is, .
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 G-metric can be obtained by usual (well-known) 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 G-metric space (see, e.g., [2, 3, 5]). Note that a G-metric 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 G-metric 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 2.1 Let be a G-complete G-metric space and be a family of nonempty G-closed subsets of X with . Let be a map satisfying
Suppose that there exists a constant such that the map T satisfies
for all and , , where
Then T has a unique fixed point in .
Proof The proof is obvious by choosing the functions ϕ, ψ in Theorem 2.1 as and . □
Corollary 2.2 Let be a G-complete G-metric space and be a family of nonempty G-closed subsets of X with . Let be a map satisfying
Suppose that there exist constants a, b, c, d and e with and there exists a function such that the map T satisfies the inequality
for all and , . Then T has a unique fixed point in .
Proof
Clearly, we have
where
By Corollary 2.1, the map T has a unique fixed point. □
Corollary 2.3 Let be a G-complete G-metric space and be a family of nonempty G-closed subsets of X with . Let be a map satisfying
Suppose that there exist functions and such that the map T satisfies the inequality
for all and , , where
Then T has 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 2.4 Let be a G-complete G-metric space and be a family of nonempty G-closed subsets of X with . Let be a map satisfying
Suppose also that there exist functions and such that the map T satisfies
where
for all and , . Then T has a unique fixed point in .
Corollary 2.5 Let be a G-complete G-metric space and be a family of nonempty G-closed subsets of X with . Let be a map satisfying
Suppose also that
where and
for all and , . Then T has a unique fixed point in .
Proof The proof is obvious by choosing the function ϕ, ψ in Corollary 2.4 as and . □
References
Banach S: Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales. Fundam. Math. 1922, 3: 133–181.
Mustafa Z, Sims B: A new approach to generalized metric spaces. J. Nonlinear Convex Anal. 2006, 7: 289–297.
Mustafa Z, Sims B: Fixed point theorems for contractive mappings in complete G -metric spaces. Fixed Point Theory Appl. 2009., 2009: Article ID 917175 10 pages
Mustafa Z, Obiedat H, Awawdeh F: Some fixed point theorem for mapping on complete G -metric spaces. Fixed Point Theory Appl. 2008., 2008: Article ID 189870 12 pages
Mustafa, Z: A new structure for generalized metric spaces with applications to fixed point theory. Ph.D. Thesis, The University of Newcastle, Australia (2005)
Mustafa Z, Khandaqji M, Shatanawi W: Fixed point results on complete G -metric spaces. Studia Sci. Math. Hung. 2011, 48: 304–319.
Mustafa Z, Aydi H, Karapınar E: Mixed g -monotone property and quadruple fixed point theorems in partially ordered metric space. Fixed Point Theory Appl. 2012., 2012: Article ID 71
Rao KPR, Bhanu Lakshmi K, Mustafa Z: Fixed and related fixed point theorems for three maps in G -metric space. J. Adv. Stud. Topol. 2012, 3(4):12–19.
Mustafa Z: Common fixed points of weakly compatible mappings in G -metric spaces. Appl. Math. Sci. 2012, 6(92):4589–4600.
Shatanawi W, Mustafa Z: On coupled random fixed point results in partially ordered metric spaces. Mat. Vesn. 2012, 64: 139–146.
Mustafa Z: Some new common fixed point theorems under strict contractive conditions in G -metric spaces. J. Appl. Math. 2012., 2012: Article ID 248937 21 pages
Mustafa Z: Mixed g -monotone property and quadruple fixed point theorems in partially ordered G -metric spaces using contractions. Fixed Point Theory Appl. 2012., 2012: Article ID 199
Mustafa Z, Shatanawi W, Bataineh M: Existence of fixed point results in G -metric spaces. Int. J. Math. Math. Sci. 2009., 2009: Article ID 283028 10 pages
Agarwal RP, Karapınar E: Remarks on some coupled fixed point theorems in G -metric spaces. Fixed Point Theory Appl. 2013., 2013: Article ID 2
Aydi H, Postolache M, Shatanawi W: Coupled fixed point results for -weakly contractive mappings in ordered G -metric spaces. Comput. Math. Appl. 2012, 63(1):298–309. 10.1016/j.camwa.2011.11.022
Aydi H, Damjanović B, Samet B, Shatanawi W: Coupled fixed point theorems for nonlinear contractions in partially ordered G -metric spaces. Math. Comput. Model. 2011, 54: 2443–2450. 10.1016/j.mcm.2011.05.059
Luong NV, Thuan NX: Coupled fixed point theorems in partially ordered G -metric spaces. Math. Comput. Model. 2012, 55: 1601–1609. 10.1016/j.mcm.2011.10.058
Aydi H, Karapınar E, Shatanawi W: Tripled fixed point results in generalized metric spaces. J. Appl. Math. 2012., 2012: Article ID 314279
Aydi H, Karapınar E, Mustafa Z: On common fixed points in G -metric spaces using (E.A) property. Comput. Math. Appl. 2012, 64(6):1944–1956. 10.1016/j.camwa.2012.03.051
Tahat N, Aydi H, Karapınar E, Shatanawi W: Common fixed points for single-valued and multi-valued maps satisfying a generalized contraction in G -metric spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 48
Aydi H, Karapınar E, Shatanawi W: Tripled common fixed point results for generalized contractions in ordered generalized metric spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 101
Aydi, H: Generalized cyclic contractions in G-metric spaces. J. Nonlinear Sci. Appl., in press
Karapınar E, Kaymakcalan B, Tas K: On coupled fixed point theorems on partially ordered G -metric spaces. J. Inequal. Appl. 2012., 2012: Article ID 200
Ding HS, Karapınar E: A note on some coupled fixed point theorems on G -metric space. J. Inequal. Appl. 2012., 2012: Article ID 170
Gül U, Karapınar E: On almost contraction in partially ordered metric spaces viz implicit relation. J. Inequal. Appl. 2012., 2012: Article ID 217
Kirk WA, Srinavasan PS, Veeramani P: Fixed points for mapping satisfying cyclical contractive conditions. Fixed Point Theory 2003, 4: 79–89.
Al-Thafai MA, Shahzad N: Convergence and existence for best proximity points. Nonlinear Anal. 2009, 70: 3665–3671. 10.1016/j.na.2008.07.022
Agarwal RP, Alghamdi MA, Shahzad N: Fixed point theory for cyclic generalized contractions in partial metric spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 40. doi:10.1186/1687–1812–2012–40 11 pages
De la Sen M, Agarwal RP: Common fixed points and best proximity points of two cyclic self-mappings. Fixed Point Theory Appl. 2012., 2012: Article ID 136. doi:10.1186/1687–1812–2012–136 17 pages
De la Sen M, Agarwal RP: Fixed point-type results for a class of extended cyclic self-mappings under three general weak contractive conditions of rational type. Fixed Point Theory Appl. 2011., 2011: Article ID 102. doi:10.1186/1687–1812–2011–102 16 pages
Eldered AA, Veeramani P: Convergence and existence for best proximity points. J. Math. Anal. Appl. 2006, 323: 1001–1006. 10.1016/j.jmaa.2005.10.081
Karpagam S, Agrawal S: Best proximity points theorems for cyclic Meir-Keeler contraction maps. Nonlinear Anal., Theory Methods Appl. 2011, 74: 1040–1046. 10.1016/j.na.2010.07.026
Karapınar E: Best proximity points of Kannan type cyclic weak phi-contractions in ordered metric spaces. Analele Stiintifice ale Universitatii Ovidius Constanta 2012, 20(3):51–64.
Karapınar E: Best proximity points of cyclic mappings. Appl. Math. Lett. 2012, 25(11):1761–1766. 10.1016/j.aml.2012.02.008
Karapinar E, Erhan İM: Best proximity points on different type contractions. Appl. Math. Inf. Sci. 2011, 5: 558–569.
Karapınar E, Erhan İM, Yıldiz-Ulus A: Cyclic contractions on G -metric spaces. Abstr. Appl. Anal. 2012., 2012: Article ID 182947. doi:10.1155/2012/182947 15 pages
Alghamdi MA, Petrusel A, Shahzad N: A fixed point theorem for cyclic generalized contractions in metric spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 122. doi:10.1186/1687–1812–2012–122
Pacurar M: Fixed point theory for cyclic Berinde operators. Fixed Point Theory 2011, 12: 419–428.
Păcurar M, Rus IA: Fixed point theory for cyclic φ -contractions. Nonlinear Anal. 2010, 72: 1181–1187. 10.1016/j.na.2009.08.002
Petrusel G: Cyclic representations and periodic points. Stud. Univ. Babeş-Bolyai, Math. 2005, 50: 107–112.
Rezapour Sh, Derafshpour M, Shahzad N: Best proximity point of cyclic φ -contractions in ordered metric spaces. Topol. Methods Nonlinear Anal. 2011, 37: 193–202.
Karapınar E, Erhan IM, Yıldız Ulus A: Fixed point theorem for cyclic maps on partial metric spaces. Appl. Math. Inf. Sci. 2012, 6: 239–244.
Karapınar E, Erhan IM: Cyclic contractions and fixed point theorems. Filomat 2012, 26: 777–782. 10.2298/FIL1204777K
Boyd DW, Wong SW: On nonlinear contractions. Proc. Am. Math. Soc. 1969, 20: 458–464. 10.1090/S0002-9939-1969-0239559-9
Alber YI, Guerre-Delabriere S: Principles of weakly contractive maps in Hilbert spaces.Operator Theory: Advances and Applications. In New Results in Operator Theory and Its Applications. Birkhäuser, Basel; 1997:7–22.
Rhoades BE: Some theorems on weakly contractive maps. Nonlinear Anal. 2001, 47: 851–861.
Zhang Q, Song Y: Fixed point theory for generalized ϕ -weak contractions. Appl. Math. Lett. 2009, 22: 75–78. 10.1016/j.aml.2008.02.007
Karapınar E: Fixed point theory for cyclic ϕ -weak contractions. Appl. Math. Lett. 2011, 24: 822–825. 10.1016/j.aml.2010.12.016
Jachymski J: Equivalent conditions for generalized contractions on (ordered) metric spaces. Nonlinear Anal. 2011, 74: 768–774. 10.1016/j.na.2010.09.025
Samet B, Vetro C, Vetro F: Remarks on G -metric spaces. Int. J. Anal. 2013., 2013: Article ID 917158 6 pages
Jleli M, Samet B: Remarks on G -metric spaces and fixed point theorems. Fixed Point Theory Appl. 2012., 2012: Article ID 210
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Bilgili, N., Karapınar, E. Cyclic contractions via auxiliary functions on G-metric spaces. Fixed Point Theory Appl 2013, 49 (2013). https://doi.org/10.1186/1687-1812-2013-49
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DOI: https://doi.org/10.1186/1687-1812-2013-49