Abstract
The aim of this paper is to prove some common fixed-point theorems for weakly compatible mappings in Menger spaces satisfying common property (E.A). Some examples are also given which demonstrate the validity of our results. As an application of our main result, we present a common fixed-point theorem for four finite families of self-mappings in Menger spaces. Our result is an improved probabilistic version of the result of Sedghi et al. [Gen. Math. 18:3-12, 2010].
MSC:54H25, 47H10, 54E70.
Similar content being viewed by others
1 Introduction
In 1922, Banach proved the principal contraction result [1]. As we know, there have been published many works about fixed-point theory for different kinds of contractions on some spaces such as quasi-metric spaces [2], cone metric spaces [3], convex metric spaces [4], partially ordered metric spaces [5–9], G-metric spaces [10–14], partial metric spaces [15, 16], quasi-partial metric spaces [17], fuzzy metric spaces [18], and Menger spaces [19]. Also, studies either on approximate fixed point or on qualitative aspects of numerical procedures for approximating fixed points are available in the literature; see [4, 20, 21].
Jungck and Rhoades [22] weakened the notion of compatibility by introducing the notion of weakly compatible mappings (extended by Singh and Jain [23] to probabilistic metric space) and proved common fixed-point theorems without assuming continuity of the involved mappings in metric spaces. In 2002, Aamri and Moutawakil [24] introduced the notion of property (E.A) (extended by Kubiaczyk and Sharma [25] to probabilistic metric space) for self-mappings which contained the class of noncompatible mappings due to Pant [26]. Further, Liu et al. [27] defined the notion of common property (E.A) (extended by Ali et al. [28] to probabilistic metric space) which contains the property (E.A) and proved several fixed-point theorems under hybrid contractive conditions. Since then, there has been continuous and intense research activity in fixed-point theory and by now there exists an extensive literature (e.g. [29–33] and the references therein).
Many mathematicians proved several common fixed-point theorems for contraction mappings in Menger spaces by using different notions viz. compatible mappings, weakly compatible mappings, property (E.A), common property (E.A) (see [28, 34–51]).
In the present paper, we prove some common fixed-point theorems for weakly compatible mappings in Menger space using the common property (E.A). Some examples are also derived which demonstrate the validity of our results. As an application of our main result, we extend the related results to four finite families of self-mappings in Menger spaces.
2 Preliminaries
In the sequel, ℝ, , and ℕ denote the set of real numbers, the set of nonnegative real numbers, and the set of positive integers, respectively.
Definition 2.1 [52]
A triangular norm ∗ (shortly t-norm) is a binary operation on the unit interval such that for all the following conditions are satisfied:
-
(1)
,
-
(2)
,
-
(3)
whenever and ,
-
(4)
.
Examples of t-norms are , , and .
Definition 2.2 [52]
A mapping is called a distribution function if it is nondecreasing and left continuous with and . We shall denote the set of all distribution functions on by ℑ, while H will always denotes the specific distribution function defined by
If X is a nonempty set, is called a probabilistic distance on X and is usually denoted by .
Definition 2.3 [52]
The ordered pair is called a probabilistic metric space (shortly, PM-space) if X is a nonempty set and ℱ is a probabilistic distance satisfying the following conditions:
-
(1)
for all if and only if ,
-
(2)
for all ,
-
(3)
for all and for all ,
-
(4)
, for and .
Every metric space can always be realized as a probabilistic metric space defined by for all and . So probabilistic metric spaces offer a wider framework (than that of the metric spaces) and are general enough to cover even wider statistical situations.
Definition 2.4 [19]
A Menger space is a triplet where is a probabilistic metric space and ∗ is a t-norm satisfying the following condition:
for all and .
Throughout this paper, is considered to be a Menger space with condition for all . Every fuzzy metric space may be a Menger space by considering defined by for all .
Definition 2.5 [52]
Let be a Menger space and ∗ be a t-norm. Then
-
(1)
a sequence in X is said to converge to a point x in X if and only if for every and , there exists an integer such that for all ;
-
(2)
a sequence in X is said to be Cauchy if for every and , there exists an integer such that for all .
A Menger space in which every Cauchy sequence is convergent is said to be complete.
Definition 2.6 [53]
A pair of self-mappings of a Menger space is said to be compatible if for all , whenever is a sequence in X such that for some .
Definition 2.7 [28]
A pair of self-mappings of a Menger space is said to be noncompatible if there exists at least one sequence in X such that for some , but, for some , either or the limit does not exist.
Definition 2.8 [25]
A pair of self-mappings of a Menger space is said to satisfy property (E.A) if there exists a sequence in X such that
for some .
Remark 2.1 From Definition 2.8, it is easy to see that any two noncompatible self-mappings of satisfy property (E.A) but the reverse need not be true (see [[40], Example 1]).
Definition 2.9 [34]
Two pairs and of self-mappings of a Menger space are said to satisfy the common property (E.A) if there exist two sequences , in X such that
for some .
Definition 2.10 [22]
A pair of self-mappings of a nonempty set X is said to be weakly compatible (or coincidentally commuting) if they commute at their coincidence points, i.e. if for some , then .
Remark 2.2 If self-mappings A and S of a Menger space are compatible then they are weakly compatible but the reverse need not be true (see [[23], Example 1]).
Remark 2.3 It is noticed that the notion of weak compatibility and the (E.A) property are independent to each other (see [[54], Example 2.2]).
Definition 2.11 [41]
Two families of self-mappings and are said to be pairwise commuting if:
-
(1)
, ,
-
(2)
, ,
-
(3)
, , .
3 Main results
Let Φ is a set of all increasing and continuous functions , such that for every .
Example 3.1 Let defined by . It is easy to see that .
Before proving our main theorems, we begin with the following lemma.
Lemma 3.1 Let A, B, S and T be self-mappings of a Menger space , where ∗ is a continuous t-norm. Suppose that
-
(1)
or ,
-
(2)
the pair or satisfies property (E.A),
-
(3)
converges for every sequence in X whenever converges or converges for every sequence in X whenever converges,
-
(4)
there exist and such that
(3.1)
holds for all , . Then the pairs and share the common property (E.A).
Proof Suppose the pair satisfies property (E.A), then there exists a sequence in X such that
for some . Since , hence for each there corresponds a sequence such that . Therefore,
Thus in all, we have , and . By (3), the sequence converges and in all we need to show that as . Let for as . Then, it is enough to show that . Suppose that , then there exists such that
In order to establish the claim embodied in (3.4), let us assume that (3.4) does not hold. Then we have for all . Repeatedly using this equality, we obtain
as . This shows that for all , which contradicts , and hence (3.4) is proved. Using inequality (3.1), with , , we get
for all . As , it follows that
as , we have
which contradicts (3.4). Therefore, . Hence the pairs and share the common property (E.A). □
Remark 3.1 In general, the converse of Lemma 3.1 is not true (see [[28], Example 3.1]).
Now we prove a common fixed-point theorem for two pairs of mappings in Menger space which is an extension of the main result of Sedghi et al. [55] in a version of Menger space.
Theorem 3.1 Let A, B, S and T be self-mappings of a Menger space , where ∗ is a continuous t-norm satisfying inequality (3.1) of Lemma 3.1. Suppose that
-
(1)
the pairs and share the common property (E.A),
-
(2)
and are closed subsets of X.
Then the pairs and have a coincidence point each. Moreover, A, B, S, and T have a unique common fixed point provided both pairs and are weakly compatible.
Proof Since the pairs and share the common property (E.A), there exist two sequences and in X such that
for some . Since is a closed subset of X, hence . Therefore, there exists a point such that . Now we assert that . Suppose that , then there exists such that
In order to establish the claim embodied in (3.6), let us assume that (3.6) does not hold. Then we have for all . Repeatedly using this equality, we obtain
as . This shows that for all , which contradicts and hence (3.6) is proved. Using inequality (3.1), with , , we get
for all . As , it follows that
as , we have
which contradicts (3.6). Therefore and hence u is a coincidence point of .
If is a closed subset of X. Therefore there exists a point such that . Now we assert that . Let, on the contrary, . As earlier, there exists such that
To support the claim, let it be untrue. Then we have for all . Repeatedly using this equality, we obtain
as . This shows that for all , which contradicts and hence (3.7) is proved. Using inequality (3.1), with , , we get
for all . As , it follows that
as , we have
which contradicts (3.7). Therefore , which shows that v is a coincidence point of the pair .
Since the pair is weakly compatible, therefore . Now we assert that z is a common fixed point of . If , then on using (3.1) with , , we get, for some ,
for all . As , we have
which is a contradiction. Hence , i.e. z is a common fixed point of . Also the pair is weakly compatible, therefore . Now we show that z is also a common fixed point of . If , then on using (3.1) with , , we get, for some ,
for all . As , we have
which is a contradiction. Therefore , which shows that z is a common fixed point of the pair . Therefore z is a common fixed point of both pairs and . The uniqueness of common fixed point is an easy consequence of inequality (3.1). □
Remark 3.2 Theorem 3.1 is an improved probabilistic version of the result of Sedghi et al. [[55], Theorem 1] for two pairs of self-mappings without any requirement on containment of ranges amongst the involved mappings.
The following example illustrates Theorem 3.1.
Example 3.2 Let be a Menger space, where , with continuous t-norm ∗ is defined by for all and
for all . The function ϕ is defined as in Example 3.1. Define the self-mappings A, B, S, and T by
We take , or , . We have
Therefore, both pairs and satisfy the common property (E.A).
It is noted that and . On the other hand, and are closed subsets of X. Thus, all the conditions of Theorem 3.1 are satisfied and 2 is a unique common fixed point of the pairs and , which also remains a point of coincidence as well. Also, all the involved mappings are even discontinuous at their unique common fixed point 2.
Remark 3.3 In fact, the mapping ℱ in Example 3.2 is also a fuzzy metric. However, the result of Sedghi et al. [[55], Theorem 1] cannot be used for this case since and .
Theorem 3.2 The conclusion of Theorem 3.1 remains true if the condition (2) of Theorem 3.1 is replaced by the following:
(2)′ and , where is the closure range of A and is the closure range of B.
Proof Since the pairs and satisfy the common property (E.A), there exist two sequences and in X such that
for some . Then since and there exists a point such that . By the proof of Theorem 3.1, we can show that the pair has a coincidence point, call it v, i.e. . Since and there exists a point such that . Similarly we can also prove that the pair has a coincidence point, call it u, i.e. . The rest of the proof is on the lines of the proof of Theorem 3.1, hence it is omitted. □
Corollary 3.1 The conclusions of Theorems 3.1-3.2 remain true if condition (2) of Theorem 3.1 and condition (2)′ of Theorem 3.2 are replaced by the following:
(2)″ and are closed subsets of X provided and .
Theorem 3.3 Let be a Menger space, where ∗ is a continuous t-norm. Let A, B, S and T be mappings from X into itself and satisfying the conditions (1)-(4) of Lemma 3.1. Suppose that
-
(5)
(or ) is a closed subset of X.
Then the pairs and have a coincidence point each. Moreover, A, B, S and T have a unique common fixed point provided both pairs and are weakly compatible.
Proof In view of Lemma 3.1, the pairs and share the common property (E.A), i.e. there exist two sequences and in X such that
for some .
If is a closed subset of X, then on the lines of Theorem 3.1, we can show that the pair has coincidence point, say u, i.e. . Since and , there exists such that . The rest of the proof runs along the lines of the proof of Theorem 3.1, therefore details are omitted. □
Remark 3.4 Theorem 3.3 is also a partial improvement of Theorem 3.1 besides relaxing the closedness of one of the subspaces.
Example 3.3 In setting of Example 3.2, replace the self-mappings A, B, S and T by
It is noted that and . Also the pairs and commute at 2 which is their common coincidence point. Thus all the conditions of Theorems 3.2-3.3 and Corollary 3.1 are satisfied and 2 is a unique common fixed point of A, B, S and T. Here, it may be pointed out that Theorem 3.1 is not applicable to this example as is not a closed subset of X. Also, notice that some mappings in this example are even discontinuous at their unique common fixed point 2.
By choosing A, B, S, and T suitably, we can drive a multitude of common fixed-point theorems for a pair or triod of self-mappings. If we take and in Theorem 3.1 then we get the following natural result which is an improved probabilistic version of the result of Sedghi et al. [[55], Theorem 1].
Corollary 3.2 Let be a Menger space, where ∗ is a continuous t-norm. Let A and S be mappings from X into itself and satisfying the following conditions:
-
(1)
The pair shares property (E.A),
-
(2)
is a closed subset of X,
-
(3)
there exist and such that
(3.8)
holds for all and . Then the pair has a coincidence point. Moreover, A and S have a unique common fixed point provided the pair is weakly compatible.
Our next theorem is proved for six self-mappings in Menger space, which extends earlier proved Theorem 3.1.
Theorem 3.4 Let be a Menger space, where ∗ is a continuous t-norm. Let A, B, R, S, H and T be mappings from X into itself and satisfying the following conditions:
-
(1)
The pairs and share the common property (E.A),
-
(2)
and are closed subsets of X,
-
(3)
there exist and such that
(3.9)
holds for all and . Then the pairs and have a coincidence point each. Moreover, A, B, R, S, H, and T have a unique common fixed point provided the pairs and commute pairwise (i.e. , , , , , and ).
Proof By Theorem 3.1, A, B, TH and SR have a unique common fixed point z in X. We show that z is a unique common fixed point of the self-mappings A, R and S. If , then on using (3.9) with , , we get, for some ,
for all . As , we have
which is a contradiction. Therefore, and so . Similarly, we get . Hence z is a unique common fixed point of self-mappings A, B, R, S, H and T in X. □
Corollary 3.3 Let be a Menger space, where ∗ is a continuous t-norm. Let , , and be four finite families from X into itself such that , , and , which satisfy the inequality (3.1). If the pairs and share the common property (E.A) along with closedness of and , then and have a point of coincidence each.
Moreover, , , and have a unique common fixed point provided the pairs of families and are commute pairwise, where , , and .
Proof The proof of this theorem is similar to that of Theorem 3.1 contained in Imdad et al. [41], hence details are omitted. □
Remark 3.5 Corollary 3.3 extends the result of Sedghi et al. [[55], Theorem 2] to four finite families of self-mappings.
By setting , , , and in Corollary 3.3, we deduce the following.
Corollary 3.4 Let be a Menger space, where ∗ is a continuous t-norm. Let A, B, S and T be mappings from X into itself such that the pairs and share the common property (E.A). Then there exist , and such that
holds for all and . If and are closed subsets of X, then the pairs and have a point of coincidence each. Further, A, B, S, and T have a unique common fixed point provided both pairs and commute pairwise.
Conclusion
Theorem 3.1 is proved for two pairs of weakly compatible mappings in Menger spaces using common property (E.A). Theorem 3.1 is an improved probabilistic version of the result of Sedghi et al. [[55], Theorem 1] for two pairs of mappings without any requirement on containment of ranges amongst the involved mappings. Several results (Theorem 3.2, Theorem 3.3 and Corollary 3.1) are also defined for the existence of fixed points in Menger spaces. Example 3.2 and Example 3.3 are furnished in support of our results. As an extension of our main result, Theorem 3.4 is proved for six self-mappings using the notion of pairwise commuting whereas Corollary 3.3 extends Theorem 3.1 to four finite families of self-mappings.
References
Banach S: Sur les operations dans les ensembles abstraits et leur application aux equations integrales. Fundam. Math. 1922, 3: 133–181.
Hicks TL: Fixed point theorems for quasi-metric spaces. Math. Jpn. 1988,33(2):231–236.
Choudhury BS, Metiya N: Coincidence point and fixed point theorems in ordered cone metric spaces. J. Adv. Math. Stud. 2012,5(2):20–31.
Olatinwo MO, Postolache M: Stability results for Jungck-type iterative processes in convex metric spaces. Appl. Math. Comput. 2012,218(12):6727–6732. 10.1016/j.amc.2011.12.038
Aydi H, Karapınar E, Postolache M: Tripled coincidence point theorems for weak φ -contractions in partially ordered metric spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 44
Aydi H, Shatanawi W, Postolache M, Mustafa Z, Tahat N: Theorems for Boyd-Wong type contractions in ordered metric spaces. Abstr. Appl. Anal. 2012., 2012: Article ID 359054
Chandok S, Postolache M: Fixed point theorem for weakly Chatterjea-type cyclic contractions. Fixed Point Theory Appl. 2013., 2013: Article ID 28
Choudhury BS, Metiya N, Postolache M: A generalized weak contraction principle with applications to coupled coincidence point problems. Fixed Point Theory Appl. 2013., 2013: Article ID 152
Shatanawi W, Postolache M: Common fixed point results of mappings for nonlinear contractions of cyclic form in ordered metric spaces. Fixed Point Theory Appl. 2013., 2013: Article ID 60
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
Chandok S, Mustafa Z, Postolache M: Coupled common fixed point theorems for mixed g -monotone mappings in partially ordered G -metric spaces. U. Politeh. Buch. Ser. A 2013,75(4):11–24.
Shatanawi W, Pitea A: Fixed and coupled fixed point theorems of omega-distance for nonlinear contraction. Fixed Point Theory Appl. 2013., 2013: Article ID 275
Shatanawi W, Postolache M: Some fixed point results for a G -weak contraction in G -metric spaces. Abstr. Appl. Anal. 2012., 2012: Article ID 815870
Shatanawi W, Pitea A: Omega-distance and coupled fixed point in G -metric spaces. Fixed Point Theory Appl. 2013., 2013: Article ID 208
Aydi H: Fixed point results for weakly contractive mappings in ordered partial metric spaces. J. Adv. Math. Stud. 2011,4(2):1–12.
Shatanawi W, Postolache M: Coincidence and fixed point results for generalized weak contractions in the sense of Berinde on partial metric spaces. Fixed Point Theory Appl. 2013., 2013: Article ID 54
Shatanawi W, Pitea A: Some coupled fixed point theorems in quasi-partial metric spaces. Fixed Point Theory Appl. 2013., 2013: Article ID 153
Grabiec M: Fixed points in fuzzy metric spaces. Fuzzy Sets Syst. 1988, 27: 385–389. 10.1016/0165-0114(88)90064-4
Menger K: Statistical metrics. Proc. Natl. Acad. Sci. USA 1942, 28: 535–537. 10.1073/pnas.28.12.535
Haghi RH, Postolache M, Rezapour S: On T -stability of the Picard iteration for generalized φ -contraction mappings. Abstr. Appl. Anal. 2012., 2012: Article ID 658971
Miandaragh MA, Postolache M, Rezapour S: Some approximate fixed point results for generalized α -contractive mappings. U. Politeh. Buch. Ser. A 2013,75(2):3–10.
Jungck G, Rhoades BE: Fixed points for set valued functions without continuity. Indian J. Pure Appl. Math. 1998,29(3):227–238. MR1617919
Singh B, Jain S: A fixed point theorem in Menger space through weak compatibility. J. Math. Anal. Appl. 2005,301(2):439–448. 10.1016/j.jmaa.2004.07.036
Aamri M, Moutawakil DEl: Some new common fixed point theorems under strict contractive conditions. J. Math. Anal. Appl. 2002,270(1):181–188. 10.1016/S0022-247X(02)00059-8
Kubiaczyk I, Sharma S: Some common fixed point theorems in Menger space under strict contractive conditions. Southeast Asian Bull. Math. 2008,32(1):117–124. MR2385106 Zbl 1199.54223
Pant RP: Common fixed point theorems for contractive maps. J. Math. Anal. Appl. 1998, 226: 251–258. MR1646430 10.1006/jmaa.1998.6029
Liu Y, Wu J, Li Z: Common fixed points of single-valued and multi-valued maps. Int. J. Math. Math. Sci. 2005, 19: 3045–3055.
Ali J, Imdad M, Bahuguna D: Common fixed point theorems in Menger spaces with common property (E.A). Comput. Math. Appl. 2010,60(12):3152–3159. MR2739482 (2011g:47124) Zbl 1207.54050 10.1016/j.camwa.2010.10.020
Abbas M, Nazir T, Radenović S: Common fixed point of power contraction mappings satisfying (E.A) property in generalized metric spaces. Appl. Math. Comput. 2013, 219: 7663–7670. 10.1016/j.amc.2012.12.090
Cakić N, Kadelburg Z, Radenović S, Razani A: Common fixed point results in cone metric spaces for a family of weakly compatible maps. Adv. Appl. Math. Sci. 2009,1(1):183–201.
Janković S, Golubović Z, Radenović S: Compatible and weakly compatible mappings in cone metric spaces. Math. Comput. Model. 2010, 52: 1728–1738. 10.1016/j.mcm.2010.06.043
Kadelburg Z, Radenović S, Rosić B: Strict contractive conditions and common fixed point theorems in cone metric spaces. Fixed Point Theory Appl. 2009., 2009: Article ID 173838
Long W, Abbas M, Nazir T, Radenović S: Common fixed point for two pairs of mappings satisfying (E.A) property in generalized metric spaces. Abstr. Appl. Anal. 2012., 2012: Article ID 394830
Ali J, Imdad M, Miheţ D, Tanveer M: Common fixed points of strict contractions in Menger spaces. Acta Math. Hung. 2011,132(4):367–386. 10.1007/s10474-011-0105-3
Altun I, Tanveer M, Miheţ D, Imdad M: Common fixed point theorems of integral type in Menger PM spaces. J. Nonlinear Anal. Optim., Theory Appl. 2012,3(1):55–66.
Beg I, Abbas M: Common fixed points of weakly compatible and noncommuting mappings in Menger spaces. Int. J. Mod. Math. 2008,3(3):261–269.
Chauhan S, Pant BD: Common fixed point theorem for weakly compatible mappings in Menger space. J. Adv. Res. Pure Math. 2011,3(2):107–119. 10.5373/jarpm.585.100210
Cho YJ, Park KS, Park WT, Kim JK: Coincidence point theorems in probabilistic metric spaces. Kobe J. Math. 1991,8(2):119–131.
Fang JX: Common fixed point theorems of compatible and weakly compatible maps in Menger spaces. Nonlinear Anal. 2009,71(5–6):1833–1843. 10.1016/j.na.2009.01.018
Fang JX, Gao Y: Common fixed point theorems under strict contractive conditions in Menger spaces. Nonlinear Anal. 2009,70(1):184–193. 10.1016/j.na.2007.11.045
Imdad M, Ali J, Tanveer M: Coincidence and common fixed point theorems for nonlinear contractions in Menger PM spaces. Chaos Solitons Fractals 2009,42(5):3121–3129. MR2562820 (2010j:54064) Zbl 1198.54076 10.1016/j.chaos.2009.04.017
Imdad M, Tanveer M, Hassan M: Some common fixed point theorems in Menger PM spaces. Fixed Point Theory Appl. 2010., 2010: Article ID 819269
Kumar, S, Chauhan, S, Pant, BD: Common fixed point theorem for noncompatible maps in probabilistic metric space. Surv. Math. Appl. (in press)
Kumar S, Pant BD: Common fixed point theorems in probabilistic metric spaces using implicit relation and property (E.A). Bull. Allahabad Math. Soc. 2010,25(2):223–235.
Kutukcu S: A fixed point theorem in Menger spaces. Int. Math. Forum 2006,1(32):1543–1554.
Miheţ D: A note on a common fixed point theorem in probabilistic metric spaces. Acta Math. Hung. 2009,125(1–2):127–130. 10.1007/s10474-009-8238-3
O’Regan D, Saadati R: Nonlinear contraction theorems in probabilistic spaces. Appl. Math. Comput. 2008,195(1):86–93. MR2379198 Zbl 1135.54315 10.1016/j.amc.2007.04.070
Pant BD, Abbas M, Chauhan S: Coincidences and common fixed points of weakly compatible mappings in Menger spaces. J. Indian Math. Soc. 2013,80(1–2):127–139.
Pant BD, Chauhan S: A contraction theorem in Menger space. Tamkang J. Math. 2011,42(1):59–68. MR2815806 Zbl 1217.54053
Pant BD, Chauhan S: Common fixed point theorems for two pairs of weakly compatible mappings in Menger spaces and fuzzy metric spaces. Sci. Stud. Res. Ser. Math. Inform. 2011,21(2):81–96.
Saadati R, O’Regan D, Vaezpour SM, Kim JK: Generalized distance and common fixed point theorems in Menger probabilistic metric spaces. Bull. Iran. Math. Soc. 2009,35(2):97–117.
Schweizer B, Sklar A: Statistical metric spaces. Pac. J. Math. 1960, 10: 313–334. 10.2140/pjm.1960.10.313
Mishra SN: Common fixed points of compatible mappings in PM-spaces. Math. Jpn. 1991,36(2):283–289.
Pathak HK, López RR, Verma RK: A common fixed point theorem using implicit relation and property (E.A) in metric spaces. Filomat 2007,21(2):211–234. 10.2298/FIL0702211P
Sedghi S, Shobe N, Aliouche A: A common fixed point theorem for weakly compatible mappings in fuzzy metric spaces. Gen. Math. 2010,18(3):3–12.
Acknowledgements
The authors would like to thank the referees for their useful comments on the manuscript.
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors read and approved the final manuscript.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Chauhan, S., Dalal, S., Sintunavarat, W. et al. Common property (E.A) and existence of fixed points in Menger spaces. J Inequal Appl 2014, 56 (2014). https://doi.org/10.1186/1029-242X-2014-56
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/1029-242X-2014-56