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Journal of Applied Mathematics
Volumeย 2012ย (2012), Article IDย 248937, 21 pages
http://dx.doi.org/10.1155/2012/248937
Research Article

Some New Common Fixed Point Theorems under Strict Contractive Conditions in G-Metric Spaces

Department of Mathematics, The Hashemite University, P.O. Box 330127, Zarqa 13115, Jordan

Received 7 May 2012; Revised 24 June 2012; Accepted 2 August 2012

Academic Editor: Ya Pingย Fang

Copyright ยฉ 2012 Zead Mustafa. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We introduce some new types of pairs of mappings (๐‘“,๐‘”) on G-metric space called G-weakly commuting of type (๐ด๐‘“) and G-R-weakly commuting of type (๐ด๐‘“). We obtain also several common fixed point results for these mappings under certain contractive condition in G-metric space. Also some examples illustrated to support our results, and comparison between different types of pairs of mappings are studied.

1. Introduction and Preliminaries

The study of common fixed points of mappings satisfying certain contractive conditions has been at the center of strong research activity and, being the area of the fixed point theory, has very important application in applied mathematics and sciences. In 1976 Jungck [1] proved a common fixed point theorem for commuting maps, but his results required the continuity of one of the maps.

Sessa [2] in 1982 first introduced a weaker version of commutativity for a pair of self-maps, and it is shown in Sessa [2] that weakly commuting pair of maps in metric pace is commuting, but the converse may not be true.

Later, Jungck [3] introduced the notion of compatible mappings in order to generalize the concepts of weak commutativity and showed that weak commuting map is compatible, but the reverse implication may not hold.

In 1996, Jungck [4] defined a pair of self-mappings to be weakly compatible if they commute at their coincidence points.

Therefore, we have one-way implication, namely, commuting maps โ‡’ weakly commuting maps โ‡’ compatible maps โ‡’ weakly Compatible maps. Recently various authors have introduced coincidence points results for various classes of mappings on metric spaces for more detail of coincidence point theory and related results see [5โ€“7].

However, the study of common fixed point of noncompatible mappings has recently been initiated by Pant (see [8, 9]).

In 2002 Amari and El Moutawakil [10] defined a new property called E.A. property which generalizes the concept of noncompatible mappings, and they proved some common fixed point theorem.

Definition 1.1 (see [10]). Let ๐‘† and ๐‘‡ be two self-mappings of a metric space (๐‘‹,๐‘‘). We say that ๐‘‡ and ๐‘† satisfy the E.A. property if there exists a sequence (๐‘ฅ๐‘›) such that lim๐‘›โ†’โˆž๐‘‡๐‘ฅ๐‘›=lim๐‘›โ†’โˆž๐‘†๐‘ฅ๐‘›=๐‘ก,forsome๐‘กโˆˆ๐‘‹.(1.1)

In 2005 Zead Mustafa and Brailey Sims introduced the notion of G-metric spaces as generalization of the concept of ordinary metric spaces. Based on the notion of G-metric space Mustafa et al. [11โ€“15] obtained some fixed point results for mapping satisfying different contractive conditions on complete G-metric space, while in [16] the completeness property was omitted and replaced by sufficient conditions, where these conditions do not imply the completeness property.

Chugh et al. [17] obtained some fixed point results for maps satisfying property P in G-metric spaces. Saadati et al. [18] studied fixed point of contractive mappings in partially ordered G-metric spaces. Shatanawi obtained fixed points of ๐œ™-maps in G-metric spaces [19] and a number of fixed point results for the two weakly increasing mappings with respect to partial ordering in G-metric spaces [20]. In [21, 22] authors established coupled fixed point theorems in a partially ordered G-metric spaces.

Abbas and rhoades [23] proved several common fixed points for noncommuting mappings without continuity in G-metric space, and they show that the results 2.3โ€“2.6 generalize Theorems 2.1โ€“2.4 of [11].

In [24] Abbas et al. proved several unique common fixed points for mappings satisfying E.A. property under generalized contraction condition and show that Corollary 3.1 extends the main result in [13] (Theorem 2.1) and Corollary 3.3 is G-version of Theorem 2 from [10] in the case of two self-mappings. Also this corollary is in relation with Theorem 2.5 of [23].

In [25] the authors proved some coupled coincidence and common coupled fixed point results for mappings defined on a set equipped with two G-metric spaces and these results do not rely on continuity of mappings involved therein as well as they show that Theorem 2.13 is an extension and generalization of (1) Theorem 2.2, Corollary 2.3, Theorem 2.6, Corollaries 2.7 and 2.8 in [26] and (2) Theorem 2.4 and Corollary 2.5 in [27].

Aydi et al. [28] established some common fixed point results for two mappings ๐‘“ and ๐‘” on G-metric spaces with assumption that ๐‘“ is a generalized weakly G-contraction mappings of type A and B with respect to ๐‘”.

In this paper, we define new types of self-maps ๐‘“ and ๐‘” on ๐บ-metric space called ๐บ-weakly commuting of type ๐ด๐‘“ and ๐บ-๐‘…-weakly commuting of type ๐ด๐‘“. Also we obtain several common fixed point results for these mappings under certain contractive condition in ๐บ-metric space, and some examples are illustrated to support our results, and a comparison between different types of pairs of mappings are stated.

The following definitions and results will be needed in the sequel.

Definition 1.2 (see [29]). A ๐บ-metric space is a pair (๐‘‹,๐บ), where ๐‘‹ is a nonempty set, and ๐บ is a nonnegative real-valued function defined on ๐‘‹ร—๐‘‹ร—๐‘‹ such that for all ๐‘ฅ,๐‘ฆ,๐‘ง,๐‘Žโˆˆ๐‘‹ we have(G1)๐บ(๐‘ฅ,๐‘ฆ,๐‘ง)=0 if ๐‘ฅ=๐‘ฆ=๐‘ง,(G2)0<๐บ(๐‘ฅ,๐‘ฅ,๐‘ฆ);forall๐‘ฅ,๐‘ฆโˆˆ๐‘‹,๐‘ค๐‘–๐‘กโ„Ž๐‘ฅโ‰ ๐‘ฆ,(G3)๐บ(๐‘ฅ,๐‘ฅ,๐‘ฆ)โ‰ค๐บ(๐‘ฅ,๐‘ฆ,๐‘ง),forall๐‘ฅ,๐‘ฆ,๐‘งโˆˆ๐‘‹,๐‘ค๐‘–๐‘กโ„Ž๐‘งโ‰ ๐‘ฆ,(G4)๐บ(๐‘ฅ,๐‘ฆ,๐‘ง)=๐บ(๐‘ฅ,๐‘ง,๐‘ฆ)=๐บ(๐‘ฆ,๐‘ง,๐‘ฅ)=โ‹ฏ, (symmetry in all three variables),(G5)๐บ(๐‘ฅ,๐‘ฆ,๐‘ง)โ‰ค๐บ(๐‘ฅ,๐‘Ž,๐‘Ž)+๐บ(๐‘Ž,๐‘ฆ,๐‘ง), for all ๐‘ฅ,๐‘ฆ,๐‘ง,๐‘Žโˆˆ๐‘‹, (rectangle inequality).
The function ๐บ is called ๐บ-metric on ๐‘‹.

Every ๐บ-metric on ๐‘‹ defines a metric ๐‘‘๐บ on ๐‘‹ by ๐‘‘๐บ(๐‘ฅ,๐‘ฆ)=๐บ(๐‘ฅ,๐‘ฆ,๐‘ฆ)+๐บ(๐‘ฆ,๐‘ฅ,๐‘ฅ)โˆ€๐‘ฅ,๐‘ฆโˆˆ๐‘‹.(1.2)

Example 1.3 (see [29]). Let (๐‘‹,๐‘‘) be a metric space, and define ๐บ๐‘  and ๐บ๐‘š on ๐‘‹ร—๐‘‹ร—๐‘‹ to ๐‘+ by ๐บ๐‘ ๐บ(๐‘ฅ,๐‘ฆ,๐‘ง)=๐‘‘(๐‘ฅ,๐‘ฆ)+๐‘‘(๐‘ฆ,๐‘ง)+๐‘‘(๐‘ฅ,๐‘ง),๐‘š(๐‘ฅ,๐‘ฆ,๐‘ง)=max{๐‘‘(๐‘ฅ,๐‘ฆ),๐‘‘(๐‘ฆ,๐‘ง),๐‘‘(๐‘ฅ,๐‘ง)},(1.3) for all ๐‘ฅ,๐‘ฆ,๐‘งโˆˆ๐‘‹. Then (๐‘‹,๐บ๐‘ ) and (๐‘‹,๐บ๐‘š) are ๐บ-metric spaces.

Example 1.4 (see [29]). Let ๐‘‹=๐‘, and define ๐บโˆถ๐‘‹ร—๐‘‹ร—๐‘‹โ†’๐‘+, by โŽงโŽชโŽจโŽชโŽฉ||||+||||||||+||||๐บ(๐‘ฅ,๐‘ฆ,๐‘ง)=๐‘ฅโˆ’๐‘ฆ๐‘ฆโˆ’๐‘ง+|๐‘ฅโˆ’๐‘ง|,ifall๐‘ฅ,๐‘ฆ,and๐‘งarestrictlypositiveortheyareallstrictlynegativeorall๐‘ฅ,๐‘ฆ,and๐‘งarezero,1+๐‘ฅโˆ’๐‘ฆ๐‘ฆโˆ’๐‘ง+|๐‘ฅโˆ’๐‘ง|,otherwise,(1.4) then (๐‘‹,๐บ) is G-metric space.

Definition 1.5 (see [29]). A sequence (๐‘ฅ๐‘›) in a ๐บ-metric space ๐‘‹ is said to converge if there exists ๐‘ฅโˆˆ๐‘‹ such that lim๐‘›,๐‘šโ†’โˆž๐บ(๐‘ฅ,๐‘ฅ๐‘›,๐‘ฅ๐‘š)=0, and one says that the sequence (๐‘ฅ๐‘›) is ๐บ-convergent to ๐‘ฅ. We call ๐‘ฅ the limit of the sequence (๐‘ฅ๐‘›) and write ๐‘ฅ๐‘›โ†’๐‘ฅ or lim๐‘›โ†’โˆž๐‘ฅ๐‘›=๐‘ฅ (through this paper we mean by ๐ the set of all natural numbers).

Proposition 1.6 (see [29]). Let ๐‘‹ be ๐บ-metric space. Then the following statements are equivalent:(1)(๐‘ฅ๐‘›) is ๐บ-convergent to ๐‘ฅ,(2)๐บ(๐‘ฅ๐‘›,๐‘ฅ๐‘›,๐‘ฅ)โ†’0, as ๐‘›โ†’โˆž,(3)๐บ(๐‘ฅ๐‘›,๐‘ฅ,๐‘ฅ)โ†’0, as ๐‘›โ†’โˆž,(4)๐บ(๐‘ฅ๐‘š,๐‘ฅ๐‘›,๐‘ฅ)โ†’0, as ๐‘š,๐‘›โ†’โˆž.

Definition 1.7 (see [29]). In a ๐บ-metric space ๐‘‹, a sequence (๐‘ฅ๐‘›) is said to be ๐บ-Cauchy if given ๐œ€>0, there is ๐‘โˆˆ๐ such that ๐บ(๐‘ฅ๐‘›,๐‘ฅ๐‘š,๐‘ฅ๐‘™)<๐œ€, for all ๐‘›,๐‘š,๐‘™โ‰ฅ๐‘. That is ๐บ(๐‘ฅ๐‘›,๐‘ฅ๐‘š,๐‘ฅ๐‘™)โ†’0 as ๐‘›,๐‘š,๐‘™โ†’โˆž.

Proposition 1.8 (see [29]). In a ๐บ-metric space ๐‘‹, the following statements are equivalent:(1)the sequence (๐‘ฅ๐‘›) is ๐บ-Cauchy;(2)for every ๐œ€>0,๐‘กโ„Ž๐‘’๐‘Ÿ๐‘’๐‘’๐‘ฅ๐‘–๐‘ ๐‘ก๐‘ ๐‘โˆˆ๐ such that ๐บ(๐‘ฅ๐‘›,๐‘ฅ๐‘š,๐‘ฅ๐‘š)<๐œ€, for all ๐‘›,๐‘šโ‰ฅ๐‘.

Definition 1.9 (see [29]). A ๐บ-metric space (๐‘‹,๐บ) is called symmetric ๐บ-metric space if ๐บ(๐‘ฅ,๐‘ฆ,๐‘ฆ)=๐บ(๐‘ฆ,๐‘ฅ,๐‘ฅ) for all ๐‘ฅ,๐‘ฆโˆˆ๐‘‹ and called nonsymmetric if it is not symmetric.

Example 1.10. Let ๐‘‹=๐ be the set of all natural numbers, and define โ€‰๐บโˆถ๐‘‹ร—๐‘‹ร—๐‘‹โ†’๐‘ such that for all ๐‘ฅ,๐‘ฆ,๐‘งโˆˆ๐‘‹: โ€‰๐บ(๐‘ฅ,๐‘ฆ,๐‘ง)=0 if ๐‘ฅ=๐‘ฆ=๐‘ง,โ€‰๐บ(๐‘ฅ,๐‘ฆ,๐‘ฆ)=๐‘ฅ+๐‘ฆ, if ๐‘ฅ<๐‘ฆ,โ€‰๐บ(๐‘ฅ,๐‘ฆ,๐‘ฆ)=๐‘ฅ+๐‘ฆ+1/2, if ๐‘ฅ>๐‘ฆ,โ€‰๐บ(๐‘ฅ,๐‘ฆ,๐‘ง)=๐‘ฅ+๐‘ฆ+๐‘ง if ๐‘ฅโ‰ ๐‘ฆโ‰ ๐‘ง and symmetry in all three variables.
Then, (๐‘‹,๐บ) is G-metric space and nonsymmetric since if ๐‘ฅ<๐‘ฆ, we have ๐บ(๐‘ฅ,๐‘ฆ,๐‘ฆ)=๐‘ฅ+๐‘ฆโ‰ ๐‘ฅ+๐‘ฆ+1/2=๐บ(๐‘ฆ,๐‘ฅ,๐‘ฅ).

Proposition 1.11 (see [29]). Let ๐‘‹ be a ๐บ-metric space; then the function ๐บ(๐‘ฅ,๐‘ฆ,๐‘ง) is jointly continuous in all three of its variables.

Definition 1.12 (see [29]). A ๐บ-metric space ๐‘‹ is said to be complete if every ๐บ-Cauchy sequence in ๐‘‹ is ๐บ-convergent in ๐‘‹.

Definition 1.13 (see [23]). Let ๐‘“ and ๐‘” be self-maps of a set ๐‘‹. If ๐‘ค=๐‘“๐‘ฅ=๐‘”๐‘ฅ for some ๐‘ฅโˆˆ๐‘‹, then ๐‘ฅ is called a coincidence point of ๐‘“ and ๐‘”, and ๐‘ค is called a point of coincidence of ๐‘“ and ๐‘”.

Recall that a pair of self-mappings are called weakly compatible if they commute at their coincidence points.

Proposition 1.14 (see [23]). Let ๐‘“ and ๐‘” be weakly compatible self-maps of a set ๐‘‹. If ๐‘“ and ๐‘” have a unique point of coincidence ๐‘ค=๐‘“๐‘ฅ=๐‘”๐‘ฅ, then ๐‘ค is the unique common fixed point of ๐‘“ and ๐‘”.

In 2001, Abbas et al. [30] introduce a new type of pairs of mappings (๐‘“,๐‘”) called R-weakly commuting and they proved a unique common fixed point of four R-weakly commuting, maps satisfying generalized contractive condition.

Definition 1.15 (see [30]). Let ๐‘‹ be a G-metric space, and let ๐‘“ and ๐‘” be two self-mappings of ๐‘‹; then ๐‘“ and ๐‘” are called R-weakly commuting if there exists a positive real number ๐‘… such that ๐บ(๐‘“(๐‘”(๐‘ฅ)),๐‘“(๐‘”(๐‘ฅ)),๐‘”(๐‘“(๐‘ฅ)))โ‰ค๐‘…๐บ(๐‘“(๐‘ฅ),๐‘“(๐‘ฅ),๐‘”(๐‘ฅ))holdforeach๐‘ฅโˆˆ๐‘‹.(1.5)

Very recently, Mustafa et al. [31] introduce some new types of pairs of mappings (๐‘“,๐‘”) on G-metric space called G-weakly commuting of type ๐บ๐‘“ and G-R-weakly commuting of type ๐บ๐‘“, and they obtained several common fixed point results by using E.A. property.

Definition 1.16 (see [31]). A pair of self-mappings (๐‘“,๐‘”) of a G-metric space (๐‘‹,๐บ) is said to be ๐บ-weakly commuting of type ๐บ๐‘“ if ๐บ(๐‘“๐‘”๐‘ฅ,๐‘”๐‘“๐‘ฅ,๐‘“๐‘“๐‘ฅ)โ‰ค๐บ(๐‘“๐‘ฅ,๐‘”๐‘ฅ,๐‘“๐‘ฅ),โˆ€๐‘ฅโˆˆ๐‘‹.(1.6)

Definition 1.17 (see [31]). A pair of self-mappings (๐‘“,๐‘”) of a ๐บ-metric space (๐‘‹,๐บ) is said to be ๐บ-๐‘…-weakly commuting of type ๐บ๐‘“ if there exists some positive real number ๐‘… such that ๐บ(๐‘“๐‘”๐‘ฅ,๐‘”๐‘“๐‘ฅ,๐‘“๐‘“๐‘ฅ)โ‰ค๐‘…๐บ(๐‘“๐‘ฅ,๐‘”๐‘ฅ,๐‘“๐‘ฅ),โˆ€๐‘ฅโˆˆ๐‘‹.(1.7)

Remark 1.18. The ๐บ-๐‘…-weakly commuting maps of type ๐บ๐‘“ are ๐‘…-weakly commuting since ๐บ(๐‘“๐‘”๐‘ฅ,๐‘“๐‘”๐‘ฅ,๐‘”๐‘“๐‘ฅ)โ‰ค๐บ(๐‘“๐‘”๐‘ฅ,๐‘”๐‘“๐‘ฅ,๐‘“๐‘“๐‘ฅ)โ‰ค๐บ(๐‘“๐‘ฅ,๐‘”๐‘ฅ,๐‘“๐‘ฅ), but the converse need not be true.

2. Main Results

2.1. New Concepts and Some Properties

In this section we introduce the concept of G-weakly commuting of type ๐ด๐‘“ for pairs of mapping (๐‘“,๐‘”) and comparison between this concept and Definitions 1.15, 1.16, and 1.17 is studied as well as examples illustrated to show that these types of mappings are different.

First, we introduce the following concepts as follows.

Definition 2.1. A pair of self-mappings (๐‘“,๐‘”) of a G-metric space (๐‘‹,๐บ) is said to be G-weakly commuting of type ๐ด๐‘“ if ๐บ(๐‘“๐‘”๐‘ฅ,๐‘”๐‘”๐‘ฅ,๐‘“๐‘“๐‘ฅ)โ‰ค๐บ(๐‘“๐‘ฅ,๐‘”๐‘ฅ,๐‘“๐‘ฅ),โˆ€๐‘ฅโˆˆ๐‘‹.(2.1)

Definition 2.2. A pair of self-mappings (f,g) of a G-metric space (๐‘‹,๐บ) is said to be G-R-weakly commuting of type ๐ด๐‘“ if there exists some positive real number ๐‘… such that ๐บ(๐‘“๐‘”๐‘ฅ,๐‘”๐‘”๐‘ฅ,๐‘“๐‘“๐‘ฅ)โ‰ค๐‘…๐บ(๐‘“๐‘ฅ,๐‘”๐‘ฅ,๐‘“๐‘ฅ),โˆ€๐‘ฅโˆˆ๐‘‹.(2.2)

Remark 2.3. The ๐บ-weakly commuting maps of type ๐ด๐‘“ are ๐บ-๐‘…-weakly commuting of type ๐ด๐‘“. Reciprocally, if ๐‘…โ‰ค1, then ๐บ-๐‘…-weakly commuting maps of type ๐ด๐‘“ are ๐บ-weakly commuting of type ๐ด๐‘“.
If we interchange ๐‘“ and ๐‘” in (2.1) and (2.2), then the pair of mappings (๐‘“,๐‘”) is called ๐บ-weakly commuting of type ๐ด๐‘” and ๐บ-๐‘…-weakly commuting of type ๐ด๐‘”, respectively.

Example 2.4. Let ๐‘‹=[0,3/4], with the G-metric ๐บ(๐‘ฅ,๐‘ฆ,๐‘ง)=|๐‘ฅโˆ’๐‘ฆ|+|๐‘ฆโˆ’๐‘ง|+|๐‘ฅโˆ’๐‘ง|, for all ๐‘ฅ,๐‘ฆ,๐‘งโˆˆ๐‘‹. Define ๐‘“,๐‘”โˆถ๐‘‹โ†’๐‘‹ by, ๐‘“(๐‘ฅ)=(1/4)๐‘ฅ2,๐‘”(๐‘ฅ)=๐‘ฅ2; then as an easy calculation one can show that ๐บ(๐‘“๐‘”๐‘ฅ,๐‘”๐‘”๐‘ฅ,๐‘“๐‘“๐‘ฅ)=(126/64)๐‘ฅ4โ‰ค๐บ(๐‘“๐‘ฅ,๐‘”๐‘ฅ,๐‘“๐‘ฅ)=(6/4)๐‘ฅ2,forall๐‘ฅโˆˆ๐‘‹. Then the pair (๐‘“,๐‘”) is G-Weakly commuting of type ๐ด๐‘“ and G-R-Weakly commuting of type ๐ด๐‘“.

Example 2.5. Let ๐‘‹=[2,โˆž], with the G-metric ๐บ(๐‘ฅ,๐‘ฆ,๐‘ง)=|๐‘ฅโˆ’๐‘ฆ|+|๐‘ฆโˆ’๐‘ง|+|๐‘ฅโˆ’๐‘ง|, for all ๐‘ฅ,๐‘ฆ,๐‘งโˆˆ๐‘‹. Define ๐‘“,๐‘”โˆถ๐‘‹โ†’๐‘‹ by, ๐‘“(๐‘ฅ)=๐‘ฅ+1,๐‘”(๐‘ฅ)=2๐‘ฅ+1, then for ๐‘ฅ=2 we see that ๐บ(๐‘“๐‘”๐‘ฅ,๐‘”๐‘”๐‘ฅ,๐‘“๐‘“๐‘ฅ)=๐บ(๐‘”๐‘“๐‘ฅ,๐‘”๐‘”๐‘ฅ,๐‘“๐‘“๐‘ฅ)=20 and ๐บ(๐‘“๐‘ฅ,๐‘”๐‘ฅ,๐‘“๐‘ฅ)=๐บ(๐‘”๐‘ฅ,๐‘“๐‘ฅ,๐‘”๐‘ฅ)=6. Therefore the pair (๐‘“,๐‘”) is not ๐บ-weakly commuting of type ๐ด๐‘“ or ๐ด๐‘”, but it is G-R-weakly commuting of type ๐ด๐‘“ (and ๐ด๐‘”) for ๐‘…โ‰ฅ4.

The following examples show a pair of mappings (๐‘“,๐‘”) that ๐บ-weakly commuting of type ๐บ๐‘“ need not be ๐บ-weakly commuting of type ๐ด๐‘“.

Example 2.6. Let ๐‘‹=[0,89/100], with the G-metric ๐บ(๐‘ฅ,๐‘ฆ,๐‘ง)=max{|๐‘ฅโˆ’๐‘ฆ|,|๐‘ฆโˆ’๐‘ง|,|๐‘ฅโˆ’๐‘ง|}, for all ๐‘ฅ,๐‘ฆ,๐‘งโˆˆ๐‘‹. Define ๐‘“(๐‘ฅ)=(1/4)๐‘ฅ2,๐‘”(๐‘ฅ)=๐‘ฅ2; then we see that ๐บ(๐‘“๐‘”๐‘ฅ,๐‘”๐‘“๐‘ฅ,๐‘“๐‘“๐‘ฅ)=(15/16)๐‘ฅ4 and ๐บ(๐‘“๐‘ฅ,๐‘”๐‘ฅ,๐‘“๐‘ฅ)=(3/4)๐‘ฅ2, while as an easy calculation one can show that for ๐‘ฅ=88/100 we have ๐บ(๐‘“(๐‘”(๐‘ฅ),๐‘”๐‘”(๐‘ฅ),๐‘“(๐‘“(๐‘ฅ)))=(59/100)โ‰ฐ๐บ(๐‘”(๐‘ฅ),๐‘“(๐‘ฅ),๐‘”(๐‘ฅ))=(58/100). Therefore the pair (๐‘“,๐‘”) is not ๐บ-weakly commuting of type ๐ด๐‘“, but it is G-weakly commuting of type ๐บ๐‘“.

The following example shows that(1)a pair of mappings (๐‘“,๐‘”) that is ๐บ-weakly commuting of type ๐ด๐‘“ need not be ๐บ-weakly commuting of type ๐ด๐‘”;(2)a pair of mappings (๐‘“,๐‘”) that is ๐บ-weakly commuting of type ๐ด๐‘“ need not be ๐บ-weakly commuting of type ๐บ๐‘“;(3)a pair of mappings (๐‘“,๐‘”) that is ๐บ-๐‘…-weakly commuting of type ๐ด๐‘“ need not be ๐‘…-weakly commuting;

Example 2.7. Let ๐‘‹=[2,9] and ๐บ(๐‘ฅ,๐‘ฆ,๐‘ง)=๐‘š๐‘Ž๐‘ฅ{|๐‘ฅโˆ’๐‘ฆ|,|๐‘ฆโˆ’๐‘ง|,|๐‘ฅโˆ’๐‘ง|} for all ๐‘ฅ,๐‘ฆ,๐‘งโˆˆ๐‘‹. Define the mappings ๐‘“,๐‘”โˆถ๐‘‹โ†’๐‘‹ by โŽงโŽชโŽจโŽชโŽฉโŽงโŽชโŽจโŽชโŽฉ๐‘“(๐‘ฅ)=2if๐‘ฅ=2,6if2<๐‘ฅโ‰ค5,5if๐‘ฅ>5,๐‘”(๐‘ฅ)=2if๐‘ฅ=2,9if2<๐‘ฅโ‰ค5,6if๐‘ฅ>5.(2.3)
Then, ๎‚ปโŽงโŽชโŽจโŽชโŽฉ๎‚ปโŽงโŽชโŽจโŽชโŽฉ๐‘“(๐‘”(๐‘ฅ))=2if๐‘ฅ=2,5if๐‘ฅ>2,๐‘”(๐‘“(๐‘ฅ))=2if๐‘ฅ=2,6if2<๐‘ฅโ‰ค5,9if๐‘ฅ>5,๐‘”(๐‘”(๐‘ฅ))=2if๐‘ฅ=2,6if๐‘ฅ>2,๐‘“(๐‘“(๐‘ฅ))=2if๐‘ฅ=2,5if2<๐‘ฅโ‰ค5,6if๐‘ฅ>5.(2.4)
Moreover, ||||=โŽงโŽชโŽจโŽชโŽฉ๐‘“(๐‘ฅ)โˆ’๐‘”(๐‘ฅ)0if๐‘ฅ=2,3if2<๐‘ฅโ‰ค5,1if๐‘ฅ>5.(2.5)
If ๐‘ฅ=2, we have ๐บ(๐‘“(๐‘”(๐‘ฅ)),๐‘”(๐‘”(๐‘ฅ)),๐‘“(๐‘“(๐‘ฅ)))=0=๐บ(๐‘“(๐‘ฅ),๐‘”(๐‘ฅ),๐‘“(๐‘ฅ)).
If 2<๐‘ฅโ‰ค5, we have ๐บ(๐‘“(๐‘”(๐‘ฅ)),๐‘“(๐‘“(๐‘ฅ)),๐‘”(๐‘”(๐‘ฅ)))=max{0,1,1}โ‰ค3=๐บ(๐‘“(๐‘ฅ),๐‘”(๐‘ฅ),๐‘“(๐‘ฅ)).(2.6)
If ๐‘ฅ>5, then ๐บ(๐‘“(๐‘”(๐‘ฅ)),๐‘“(๐‘“(๐‘ฅ)),๐‘”(๐‘”(๐‘ฅ)))=max{1,1,0}โ‰ค1=๐บ(๐‘”(๐‘ฅ),๐‘“(๐‘ฅ),๐‘”(๐‘ฅ)).(2.7)
Thus, ๐‘“ and ๐‘” are ๐บ weakly commuting of type ๐ด๐‘“, but for ๐‘ฅ=6, we have ๐บ(๐‘”(๐‘“(6)),๐‘“(๐‘“(6)),๐‘”(๐‘”(6)))=max{3,3,0}โ‰ฐ1=๐บ(๐‘”(6),๐‘“(6),๐‘”(6)).(2.8)
Therefore, the pair (๐‘“,๐‘”) is not G-weakly commuting of type ๐ด๐‘”, but it is G-weakly commuting of type ๐ด๐‘“.
Also for ๐‘ฅ=7, we have ๐บ(๐‘“(๐‘”(7)),๐‘”(๐‘“(7)),๐‘“(๐‘“(7)))=4โ‰ฐ1=๐บ(๐‘”(6),๐‘“(6),๐‘”(6)).(2.9)
Therefore, the pair (๐‘“,๐‘”) is not G-weakly commuting of type ๐บ๐‘“.
AS an easy calculation one can see that (๐‘“,๐‘”) are ๐บ-๐‘…-weakly commuting of type ๐ด๐‘” for ๐‘…=3; but for ๐‘ฅ=6 we have ๐บ(๐‘“(๐‘”(6)),๐‘“(๐‘”(6)),๐‘”(๐‘“(6)))=4โ‰ฐ3๐บ(๐‘“(6),๐‘“(6),๐‘”(6))=3, hence (๐‘“,๐‘”) is NOT ๐‘…-weakly commuting for ๐‘…=3.

Lemma 2.8. If ๐‘“ and ๐‘” are ๐บ-weakly commuting of type ๐ด๐‘“ or ๐บ-๐‘…-weakly commuting of type ๐ด๐‘“, then ๐‘“ and ๐‘” are weakly compatible.

Proof. Let ๐‘ฅ be a coincidence point of ๐‘“ and ๐‘”, that is, ๐‘“(๐‘ฅ)=๐‘”(๐‘ฅ); then if the pair (๐‘“,๐‘”) is ๐บ-weakly commuting of type ๐ด๐‘“, we have ๐บ(๐‘“(๐‘”(๐‘ฅ)),๐‘”(๐‘“(๐‘ฅ)),๐‘“(๐‘”(๐‘ฅ)))=๐บ(๐‘“(๐‘”(๐‘ฅ)),๐‘”(๐‘”(๐‘ฅ)),๐‘“(๐‘“(๐‘ฅ)))โ‰ค๐บ(๐‘“(๐‘ฅ),๐‘”(๐‘ฅ),๐‘“(๐‘ฅ))=0.(2.10)
It follows ๐‘“(๐‘”(๐‘ฅ))=๐‘”(๐‘“(๐‘ฅ)); then they commute at their coincidence point.
Similarly, if the pair (๐‘“,๐‘”) is ๐บ-๐‘…-weakly commuting of type ๐บ๐‘“, we have ๐บ(๐‘“(๐‘”(๐‘ฅ)),๐‘”(๐‘“(๐‘ฅ)),๐‘“(๐‘”(๐‘ฅ)))=๐บ(๐‘“(๐‘”(๐‘ฅ)),๐‘”(๐‘”(๐‘ฅ)),๐‘“(๐‘“(๐‘ฅ)))โ‰ค๐‘…๐บ(๐‘“(๐‘ฅ),๐‘”(๐‘ฅ),๐‘“(๐‘ฅ))=0.(2.11)
Thus ๐‘“(๐‘”(๐‘ฅ))=๐‘”(๐‘“(๐‘ฅ)); then the pair (๐‘“,๐‘”) is weakly compatible.

The following example shows that(1)the converse of Lemma 2.8 fails (for the case of ๐บ-weakly commutativity),(2)a pair of mappings (๐‘“,๐‘”) that is ๐‘…-weakly commuting need not be ๐บ-๐‘…-weakly commuting of type ๐ด๐‘“,(3)a pair of mappings (๐‘“,๐‘”) that is ๐‘…-weakly commuting need not be ๐บ-๐‘…-weakly commuting of type ๐บ๐‘“.

Example 2.9. Let ๐‘‹=[1,+โˆž) and ๐บ(๐‘ฅ,๐‘ฆ,๐‘ง)=max{|๐‘ฅโˆ’๐‘ฆ|,|๐‘ฆโˆ’๐‘ง|,|๐‘ฅโˆ’๐‘ง|}. Define ๐‘“,๐‘”โˆถ๐‘‹โ†’๐‘‹ by ๐‘“(๐‘ฅ)=2๐‘ฅโˆ’1 and ๐‘”(๐‘ฅ)=๐‘ฅ2. We see that ๐‘ฅ=1 is the only coincidence point and ๐‘“(๐‘”(1))=๐‘“(1)=1 and ๐‘”(๐‘“(1))=๐‘”(1)=1, so ๐‘“ and ๐‘” are weakly compatible.
But, by an easy calculation, one can see that for ๐‘ฅ=3 we have, ๐บ(๐‘“(๐‘”(๐‘ฅ)),๐‘”(๐‘”(๐‘ฅ)),๐‘“(๐‘“(๐‘ฅ)))=72โ‰ฐ4=๐บ(๐‘“(๐‘ฅ),๐‘”(๐‘ฅ),๐‘“(๐‘ฅ)).(2.12) Therefore, ๐‘“ and ๐‘” are not ๐บ-weakly commuting of type ๐ด๐‘“.
Also, we see that ๐บ(๐‘“(๐‘”(๐‘ฅ)),๐‘“(๐‘”(๐‘ฅ)),๐‘”(๐‘“(๐‘ฅ)))=2๐‘ฅ2โˆ’4๐‘ฅ+2โ‰ค2๐บ(๐‘“(๐‘ฅ),๐‘“(๐‘ฅ),๐‘”(๐‘ฅ))=2(๐‘ฅ2โˆ’2๐‘ฅ+1); therefore the mappings (๐‘“,๐‘”) are ๐‘…-weakly commuting for ๐‘…=2, but for ๐‘ฅ=4 we have ๐บ(๐‘“(๐‘”(4)),๐‘”(๐‘”(4)),๐‘“(๐‘“(4)))=243โ‰ฐ2๐บ(๐‘“(4),๐‘”(4),๐‘“(4))=18; hence (๐‘“,๐‘”) are not ๐บ-๐‘…-weakly commuting of type ๐ด๐‘“ for ๐‘…=2 and ๐บ(๐‘“(๐‘”(4)),๐‘”(๐‘“(4)),๐‘“(๐‘“(4)))=49โ‰ฐ2๐บ(๐‘“(4),๐‘”(4),๐‘“(4))=18; hence (๐‘“,๐‘”) are not ๐บ-๐‘…-weakly commuting of type ๐บ๐‘“ for ๐‘…=2.

Now, we rewrite Definition 1.1 on ๐บ-metric spaces setting.

Definition 2.10. Let ๐‘† and ๐‘‡ be two self-mappings of a ๐บ-metric space (๐‘‹,๐บ). We say that ๐‘‡ and ๐‘† satisfy the E.A. property if there exists a sequence (๐‘ฅ๐‘›) such that (๐‘‡๐‘ฅ๐‘›) and (๐‘†๐‘ฅ๐‘›)๐บ-converge to ๐‘ก for some ๐‘กโˆˆ๐‘‹; that is, thanks to Proposition 1.6, lim๐‘›โŸถโˆž๐บ๎€ท๐‘‡๐‘ฅ๐‘›,๐‘‡๐‘ฅ๐‘›๎€ธ,๐‘ก=lim๐‘›โŸถโˆž๐บ๎€ท๐‘†๐‘ฅ๐‘›,๐‘†๐‘ฅ๐‘›๎€ธ,๐‘ก=0.(2.13)

Remark 2.11. In view of (1.2) and Example 1.3, Definition 1.1 is equivalent to Definition 2.10.

In the following example, we show that if ๐‘“ and ๐‘” satisfy the E.A. property, then the pair (๐‘“,๐‘”) need not be ๐บ-weakly commuting of type ๐ด๐‘“.

Example 2.12. We return to Example 2.9. Let ๐‘ฅ๐‘›=1+(1/3๐‘›). We have lim๐‘›โ†’โˆž๐‘“(๐‘ฅ๐‘›)=lim๐‘›โ†’โˆž(1+(2/3๐‘›))=1, and lim๐‘›โ†’โˆž๐‘”(๐‘ฅ๐‘›)=lim๐‘›โ†’โˆž(1+(1/3๐‘›))2=1, therefore, lim๐‘›โ†’โˆž๐‘“(๐‘ฅ๐‘›)=lim๐‘›โ†’โˆž๐‘”(๐‘ฅ๐‘›)=1โˆˆ[1,โˆž). Then ๐‘“ and ๐‘” satisfy the E.A. property, but we know that the pair (๐‘“,๐‘”) is not ๐บ-weakly commuting of type ๐ด๐‘“.

Following Matkowski (see [32]), let ฮฆ be the set of all functions ๐œ™ such that ๐œ™โˆถ[0,โˆž)โ†’[0,โˆž) be a nondecreasing function with lim๐‘›โ†’โˆž๐œ™๐‘›(๐‘ก)=0 for all ๐‘กโˆˆ(0,+โˆž). If ๐œ™โˆˆฮฆ, then ๐œ™ is called ฮฆ-map. If ๐œ™ is ฮฆ-map, then it is easy to show that(1)๐œ™(๐‘ก)<๐‘ก for all ๐‘กโˆˆ(0,+โˆž),(2)๐œ™(0)=0.

2.2. Some Common Fixed Point Results

We start this section with the following theorem.

Theorem 2.13. Let (๐‘‹,๐บ) be a ๐บ-metric space; suppose mappings ๐‘“,๐‘”โˆถ๐‘‹โ†’๐‘‹ satisfy the following condition: (1)๐‘“ and ๐‘” be ๐บ-weakly commuting of type ๐ด๐‘“, (2)๐‘“(๐‘‹)โŠ†๐‘”(๐‘‹),(3)๐‘”(๐‘‹) is ๐บ-complete subspace of ๐‘‹, (4)๐บ(๐‘“(๐‘ฅ),๐‘“(๐‘ฆ),๐‘“(๐‘ง))โ‰ค๐œ™(๐‘€(๐‘ฅ,๐‘ฆ,๐‘ง)),๐‘“๐‘œ๐‘Ÿ๐‘Ž๐‘™๐‘™๐‘ฅ,๐‘ฆ,๐‘งโˆˆ๐‘‹,๐‘คโ„Ž๐‘’๐‘Ÿ๐‘’โŽงโŽชโŽชโŽชโŽจโŽชโŽชโŽชโŽฉ1๐‘€(๐‘ฅ,๐‘ฆ,๐‘ง)=max๐บ(๐‘”(๐‘ฅ),๐‘”(๐‘ฆ),๐‘”(๐‘ง)),๐บ(๐‘”(๐‘ฅ),๐‘“(๐‘ฅ),๐‘“(๐‘ฅ)),21๐บ(๐‘”(๐‘ฅ),๐‘“(๐‘ฆ),๐‘“(๐‘ฆ)),2โŽซโŽชโŽชโŽชโŽฌโŽชโŽชโŽชโŽญ๐บ(๐‘”(๐‘ฅ),๐‘“(๐‘ง),๐‘“(๐‘ง)),๐บ(๐‘”(๐‘ฆ),๐‘“(๐‘ฆ),๐‘“(๐‘ฆ)),๐บ(๐‘”(๐‘ฆ),๐‘“(๐‘ฅ),๐‘“(๐‘ฅ)),๐บ(๐‘”(๐‘ฆ),๐‘“(๐‘ง),๐‘“(๐‘ง)),๐บ(๐‘”(๐‘ง),๐‘“(๐‘ง),๐‘“(๐‘ง)),๐บ(๐‘”(๐‘ง),๐‘“(๐‘ฅ),๐‘“(๐‘ฅ)),๐บ(๐‘”(๐‘ง),๐‘“(๐‘ฆ),๐‘“(๐‘ฆ)).(2.14) Then ๐‘“ and ๐‘” have a unique common fixed point.

Proof. Let ๐‘ฅ0โˆˆ๐‘‹, and then choose ๐‘ฅ1โˆˆ๐‘‹ such that ๐‘“(๐‘ฅ0)=๐‘”(๐‘ฅ1) and ๐‘ฅ2โˆˆ๐‘‹ where ๐‘“(๐‘ฅ1)=๐‘”(๐‘ฅ2); then by induction we can define a sequence (๐‘ฆ๐‘›)โˆˆ๐‘‹ as follows: ๐‘ฆ๐‘›๎€ท๐‘ฅ=๐‘“๐‘›๎€ธ๎€ท๐‘ฅ=๐‘”๐‘›+1๎€ธ,๐‘›โˆˆ๐‘โˆช{0}(2.15) We will show that the sequence (๐‘ฆ๐‘›) is ๐บ-cauchy sequence: ๐บ๎€ท๐‘ฆ๐‘›,๐‘ฆ๐‘›+1,๐‘ฆ๐‘›+1๎€ธ๎€ท๐‘“๎€ท๐‘ฅ=๐บ๐‘›๎€ธ๎€ท๐‘ฅ,๐‘“๐‘›+1๎€ธ๎€ท๐‘ฅ,๐‘“๐‘›+1๎€ท๐‘€๎€ท๐‘ฅ๎€ธ๎€ธโ‰ค๐œ™๐‘›,๐‘ฅ๐‘›+1,๐‘ฅ๐‘›+1๎€ธ๎€ธ,(2.16) where ๐‘€๎€ท๐‘ฅ๐‘›,๐‘ฅ๐‘›+1,๐‘ฅ๐‘›+1๎€ธโŽงโŽชโŽชโŽจโŽชโŽชโŽฉ๐บ๎€ท๐‘”๎€ท๐‘ฅ=max๐‘›๎€ธ๎€ท๐‘ฅ,๐‘”๐‘›+1๎€ธ๎€ท๐‘ฅ,๐‘”๐‘›+1๎€ท๐‘”๎€ท๐‘ฅ๎€ธ๎€ธ,๐บ๐‘›๎€ธ๎€ท๐‘ฅ,๐‘“๐‘›๎€ธ๎€ท๐‘ฅ,๐‘“๐‘›,1๎€ธ๎€ธ2๐บ๎€ท๐‘”๎€ท๐‘ฅ๐‘›๎€ธ๎€ท๐‘ฅ,๐‘“๐‘›+1๎€ธ๎€ท๐‘ฅ,๐‘“๐‘›+1๎€ท๐‘”๎€ท๐‘ฅ๎€ธ๎€ธ,๐บ๐‘›+1๎€ธ๎€ท๐‘ฅ,๐‘“๐‘›๎€ธ๎€ท๐‘ฅ,๐‘“๐‘›,๐บ๎€ท๐‘”๎€ท๐‘ฅ๎€ธ๎€ธ๐‘›+1๎€ธ๎€ท๐‘ฅ,๐‘“๐‘›+1๎€ธ๎€ท๐‘ฅ,๐‘“๐‘›+1,โŽซโŽชโŽชโŽฌโŽชโŽชโŽญโŽงโŽชโŽจโŽชโŽฉ๐บ๎€ท๐‘ฆ๎€ธ๎€ธ=max๐‘›โˆ’1,๐‘ฆ๐‘›,๐‘ฆ๐‘›๎€ธ,12๐บ๎€ท๐‘ฆ๐‘›โˆ’1,๐‘ฆ๐‘›+1,๐‘ฆ๐‘›+1๎€ธ,๐บ๎€ท๐‘ฆ๐‘›,๐‘ฆ๐‘›,๐‘ฆ๐‘›๎€ธ๎€ท๐‘ฆ,๐บ๐‘›,๐‘ฆ๐‘›+1,๐‘ฆ๐‘›+1๎€ธ,โŽซโŽชโŽฌโŽชโŽญ.(2.17)
We will have different cases.โ€‰Case (1): if ๐‘€(๐‘ฅ๐‘›,๐‘ฅ๐‘›+1,๐‘ฅ๐‘›+1)=๐บ(๐‘ฆ๐‘›,๐‘ฆ๐‘›+1,๐‘ฆ๐‘›+1), then ๐บ(๐‘ฆ๐‘›,๐‘ฆ๐‘›+1,๐‘ฆ๐‘›+1)โ‰ค๐œ™(๐บ(๐‘ฆ๐‘›,๐‘ฆ๐‘›+1,๐‘ฆ๐‘›+1))โ€‰<๐บ(๐‘ฆ๐‘›,๐‘ฆ๐‘›+1,๐‘ฆ๐‘›+1), which is contradiction.โ€‰Case (2): if ๐‘€(๐‘ฅ๐‘›,๐‘ฅ๐‘›+1,๐‘ฅ๐‘›+1)=(1/2)๐บ(๐‘ฆ๐‘›โˆ’1,๐‘ฆ๐‘›+1,๐‘ฆ๐‘›+1), then in this case we havemax{๐บ(๐‘ฆ๐‘›โˆ’1,๐‘ฆ๐‘›,๐‘ฆ๐‘›),๐บ(๐‘ฆ๐‘›,๐‘ฆ๐‘›+1,๐‘ฆ๐‘›+1)}<(1/2)๐บ(๐‘ฆ๐‘›โˆ’1,๐‘ฆ๐‘›+1,๐‘ฆ๐‘›+1), which implies that ๐บ๎€ท๐‘ฆ๐‘›โˆ’1,๐‘ฆ๐‘›,๐‘ฆ๐‘›๎€ธ๎€ท๐‘ฆ+๐บ๐‘›,๐‘ฆ๐‘›+1,๐‘ฆ๐‘›+1๎€ธ๎€ท๐‘ฆ<๐บ๐‘›โˆ’1,๐‘ฆ๐‘›+1,๐‘ฆ๐‘›+1๎€ธ,(2.18)
but from G-metric property (G5) we have ๐บ๎€ท๐‘ฆ๐‘›โˆ’1,๐‘ฆ๐‘›+1,๐‘ฆ๐‘›+1๎€ธ๎€ท๐‘ฆโ‰ค๐บ๐‘›โˆ’1,๐‘ฆ๐‘›,๐‘ฆ๐‘›๎€ธ๎€ท๐‘ฆ+๐บ๐‘›,๐‘ฆ๐‘›+1,๐‘ฆ๐‘›+1๎€ธ.(2.19)
Thus, from (2.18) and (2.19) we see that caseโ€‰(2) is impossible.
Then, we must have the case ๐‘€๎€ท๐‘ฅ๐‘›,๐‘ฅ๐‘›+1,๐‘ฅ๐‘›+1๎€ธ๎€ท๐‘ฆ=๐บ๐‘›โˆ’1,๐‘ฆ๐‘›,๐‘ฆ๐‘›๎€ธ.(2.20)
Thus, for ๐‘›โˆˆ๐โˆช{0} and from (2.16) we have, ๐บ๎€ท๐‘ฆ๐‘›,๐‘ฆ๐‘›+1,๐‘ฆ๐‘›+1๎€ธ๎€ท๐บ๎€ท๐‘ฆโ‰ค๐œ™๐‘›โˆ’1,๐‘ฆ๐‘›,๐‘ฆ๐‘›๎€ธ๎€ธโ‰ค๐œ™2๎€ท๐บ๎€ท๐‘ฆ๐‘›โˆ’2,๐‘ฆ๐‘›โˆ’1,๐‘ฆ๐‘›โˆ’1โ‹ฎ๎€ธ๎€ธโ‰ค๐œ™๐‘›๎€ท๐บ๎€ท๐‘ฆ0,๐‘ฆ1,๐‘ฆ1โ€ฆ๎€ธ๎€ธ(1).(2.21)
Given๐œ–>0, since lim๐‘›โ†’โˆž๐œ™๐‘›(๐บ(๐‘ฆ0,๐‘ฆ1,๐‘ฆ1))=0, and ๐œ™(๐œ–)<๐œ–, there is an integer ๐‘›๐‘œโˆˆ๐, such that ๐œ™๐‘›๎€ท๐บ๎€ท๐‘ฆ0,๐‘ฆ1,๐‘ฆ1๎€ธ๎€ธ<๐œ–โˆ’๐œ™(๐œ–),โˆ€๐‘›โ‰ฅ๐‘›0.(2.22)
Hence, we have ๐บ๎€ท๐‘ฆ๐‘›,๐‘ฆ๐‘›+1,๐‘ฆ๐‘›+1๎€ธโ‰ค๐œ™๐‘›๎€ท๐บ๎€ท๐‘ฆ0,๐‘ฆ1,๐‘ฆ1๎€ธ๎€ธ<๐œ–โˆ’๐œ™(๐œ–).(2.23)
Now for ๐‘š,๐‘›โˆˆ๐; ๐‘š>๐‘›, we claim that ๐บ๎€ท๐‘ฆ๐‘›,๐‘ฆ๐‘š,๐‘ฆ๐‘š๎€ธ<๐œ–,โˆ€๐‘šโ‰ฅ๐‘›โ‰ฅ๐‘›0.(2.24)
We will prove (2.24) by induction on ๐‘š.
Inequality (2.24) holds for ๐‘š=๐‘›+1, by using (2.23) and the fact that ๐œ–โˆ’๐œ™(๐œ–)<๐œ–.
Assume (2.24) holds for ๐‘š=๐‘˜. For ๐‘š=๐‘˜+1, we have ๐บ๎€ท๐‘ฆ๐‘›,๐‘ฆ๐‘˜+1,๐‘ฆ๐‘˜+1๎€ธ๎€ท๐‘ฆโ‰ค๐บ๐‘›,๐‘ฆ๐‘›+1,๐‘ฆ๐‘›+1๎€ธ๎€ท๐‘ฆ+๐บ๐‘›+1,๐‘ฆ๐‘˜+1,๐‘ฆ๐‘˜+1๎€ธ๎€ท๐บ๎€ท๐‘ฆ<๐œ–โˆ’๐œ™(๐œ–)+๐œ™๐‘›,๐‘ฆ๐‘˜,๐‘ฆ๐‘˜๎€ธ๎€ธ<๐œ–โˆ’๐œ™(๐œ–)+๐œ™(๐œ–)=๐œ–.(2.25)
By induction on ๐‘š, we conclude that (2.24) holds for all ๐‘šโ‰ฅ๐‘›โ‰ฅ๐‘›0.
Hence, the sequence (๐‘ฆ๐‘›)=๐‘”(๐‘ฅ๐‘›+1) is ๐บ-cauchy sequence in ๐‘”(๐‘‹); since ๐‘”(๐‘‹) is ๐บ-complete, then there exists ๐‘กโˆˆ๐‘”(๐‘‹) such that lim๐‘›โ†’โˆž๐‘”(๐‘ฅ๐‘›)=๐‘ก=lim๐‘›โ†’โˆž๐‘“(๐‘ฅ๐‘›).
Thus, there exists ๐‘โˆˆ๐‘‹ such that ๐‘”(๐‘)=๐‘ก, also lim๐‘›โ†’โˆž๐‘“(๐‘ฅ๐‘›)=๐‘”(๐‘).
We will show that ๐‘“(๐‘)=๐‘”(๐‘). Supposing that ๐‘“(๐‘)โ‰ ๐‘”(๐‘), then condition (4) implies that, ๐บ(๐‘“(๐‘),๐‘“(๐‘),๐‘“(๐‘ฅ๐‘›))โ‰ค๐œ™(๐‘€(๐‘,๐‘,๐‘ฅ๐‘›)), where ๐‘€๎€ท๐‘,๐‘,๐‘ฅ๐‘›๎€ธโŽงโŽชโŽชโŽชโŽจโŽชโŽชโŽชโŽฉ๐บ๎€ท๎€ท๐‘ฅ=max๐‘”(๐‘),๐‘”(๐‘),๐‘”๐‘›1๎€ธ๎€ธ,๐บ(๐‘”(๐‘),๐‘“(๐‘),๐‘“(๐‘)),21๐บ(๐‘”(๐‘),๐‘“(๐‘),๐‘“(๐‘)),2๐บ๎€ท๐‘”๎€ท๐‘ฅ(๐‘),๐‘“๐‘›๎€ธ๎€ท๐‘ฅ,๐‘“๐‘›๐บ๎€ท๎€ท๐‘ฅ๎€ธ๎€ธ,๐บ(๐‘”(๐‘),๐‘“(๐‘),๐‘“(๐‘)),๐บ(๐‘”(๐‘),๐‘“(๐‘),๐‘“(๐‘)),๐‘”(๐‘),๐‘“๐‘›๎€ธ๎€ท๐‘ฅ,๐‘“๐‘›๎€ท๐‘”๎€ท๐‘ฅ๎€ธ๎€ธ,๐บ๐‘›๎€ธ๎€ท๐‘ฅ,๐‘“๐‘›๎€ธ๎€ท๐‘ฅ,๐‘“๐‘›๎€ท๐‘”๎€ท๐‘ฅ๎€ธ๎€ธ,๐บ๐‘›๎€ธ๎€ธ,๐บ๎€ท๐‘”๎€ท๐‘ฅ,๐‘“(๐‘),๐‘“(๐‘)๐‘›๎€ธ๎€ธโŽซโŽชโŽชโŽชโŽฌโŽชโŽชโŽชโŽญ.,๐‘“(๐‘),๐‘“(๐‘)(2.26)
Taking the limit as ๐‘›โ†’โˆž and using the fact that the function ๐บ is continuous we get๎‚€๎‚†1๐บ(๐‘“(๐‘),๐‘“(๐‘),๐‘”(๐‘))โ‰ค๐œ™max๐บ(๐‘”(๐‘),๐‘“(๐‘),๐‘“(๐‘)),2๐บ(๐‘”(๐‘),๐‘“(๐‘),๐‘“(๐‘))๎‚‡๎‚=๐œ™(๐บ(๐‘”(๐‘),๐‘“(๐‘),๐‘“(๐‘))).(2.27) Therefore, ๐บ(๐‘“(๐‘),๐‘“(๐‘),๐‘”(๐‘))โ‰ค๐œ™(๐บ(๐‘”(๐‘),๐‘“(๐‘),๐‘“(๐‘)))<๐บ(๐‘”(๐‘),๐‘“(๐‘),๐‘“(๐‘)),(2.28) which is contradiction; hence ๐‘“๐‘=๐‘”๐‘.
Since ๐‘“ and ๐‘” are ๐บ-weakly commuting of type ๐ด๐‘“, then (๐บ(๐‘“(๐‘”(๐‘)),๐‘”(๐‘”(๐‘)),๐‘“(๐‘“(๐‘)))โ€‰ โ‰ค๐บ(๐‘“(๐‘),๐‘”(๐‘),๐‘“(๐‘))=0).
Thus, ๐‘“๐‘“(๐‘)=๐‘“๐‘”(๐‘)=๐‘”๐‘“(๐‘)=๐‘”๐‘”(๐‘); it follows that ๐‘“(๐‘ก)=๐‘“๐‘”(๐‘)=๐‘”๐‘“(๐‘)=๐‘”(๐‘ก).
Finally, we will show that ๐‘กโˆถ=๐‘“(๐‘) is common fixed point of ๐‘“ and ๐‘”.
Supposing that ๐‘“๐‘กโ‰ ๐‘ก, then ๐บ(๐‘“(๐‘ก),๐‘ก,๐‘ก)=๐บ(๐‘“(๐‘ก),๐‘“(๐‘),๐‘“(๐‘))โ‰ค๐œ™(๐‘€(๐‘ก,๐‘,๐‘)),(2.29) where โŽงโŽชโŽชโŽชโŽจโŽชโŽชโŽชโŽฉ1๐‘€(๐‘ก,๐‘,๐‘)=max๐บ(๐‘”(๐‘ก),๐‘”(๐‘),๐‘”(๐‘)),๐บ(๐‘”(๐‘ก),๐‘“(๐‘ก),๐‘“(๐‘ก)),21๐บ(๐‘”(๐‘ก),๐‘“(๐‘),๐‘“(๐‘)),2โŽซโŽชโŽชโŽชโŽฌโŽชโŽชโŽชโŽญ๐บ(๐‘”(๐‘ก),๐‘“(๐‘),๐‘“(๐‘)),๐บ(๐‘”(๐‘),๐‘“(๐‘),๐‘“(๐‘)),๐บ(๐‘”(๐‘),๐‘“(๐‘ก),๐‘“(๐‘ก)),๐บ(๐‘”(๐‘),๐‘“(๐‘),๐‘“(๐‘)),๐บ(๐‘”(๐‘),๐‘“(๐‘),๐‘“(๐‘)),๐บ(๐‘”(๐‘),๐‘“(๐‘ก),๐‘“(๐‘ก)),๐บ(๐‘”(๐‘),๐‘“(๐‘),๐‘“(๐‘)).(2.30) Since ๐‘”(๐‘ก)=๐‘“(๐‘ก), and ๐‘”(๐‘)=๐‘“(๐‘), therefore (2.30) implies that ๎‚†1๐‘€(๐‘ก,๐‘,๐‘)=max๐บ(๐‘“(๐‘ก),๐‘ก,๐‘ก),2๎‚‡.๐บ(๐‘“(๐‘ก),๐‘ก,๐‘ก),๐บ(๐‘ก,๐‘“(๐‘ก),๐‘“(๐‘ก))(2.31) Hence, (2.29) becomes ๐บ(๐‘“(๐‘ก),๐‘ก,๐‘ก)โ‰ค๐œ™(max{๐บ(๐‘“(๐‘ก),๐‘ก,๐‘ก),๐บ(๐‘ก,๐‘“(๐‘ก),๐‘“(๐‘ก))})=๐œ™(๐บ(๐‘ก,๐‘“(๐‘ก),๐‘“(๐‘ก)))<๐บ(๐‘ก,๐‘“(๐‘ก),๐‘“(๐‘ก)).(2.32) Similarly we get, ๐บ(๐‘ก,๐‘“(๐‘ก),๐‘“(๐‘ก))<๐บ(๐‘“(๐‘ก),๐‘ก,๐‘ก).(2.33) So, ๐บ(๐‘“(๐‘ก),๐‘ก,๐‘ก)<๐บ(๐‘“(๐‘ก),๐‘ก,๐‘ก),(2.34) a contradiction which implies that ๐‘ก=๐‘“๐‘ก=๐‘”๐‘ก. Then ๐‘ก is a common fixed point.
To prove uniqueness suppose we have ๐‘ข and ๐‘ฃ such that ๐‘ขโ‰ ๐‘ฃ, ๐‘“๐‘ข=๐‘”๐‘ข=๐‘ข and ๐‘“๐‘ฃ=๐‘”๐‘ฃ=๐‘ฃ; then condition (4) implies that ๐บ(๐‘ข,๐‘ฃ,๐‘ฃ)โ‰ค๐œ™(๐บ(๐‘ฃ,๐‘ข,๐‘ข)).(2.35) Therefore, ๐บ(๐‘ข,๐‘ฃ,๐‘ฃ)โ‰ค๐œ™(๐บ(๐‘ฃ,๐‘ข,๐‘ข))<๐บ(๐‘ฃ,๐‘ข,๐‘ข).(2.36) Similarly, ๐บ(๐‘ฃ,๐‘ข,๐‘ข)<๐บ(๐‘ข,๐‘ฃ,๐‘ฃ); thus ๐บ(๐‘ข,๐‘ฃ,๐‘ฃ)<๐บ(๐‘ข,๐‘ฃ,๐‘ฃ), a contradiction which implies that ๐‘ข=๐‘ฃ. Then ๐‘ก is a unique common fixed point of ๐‘“ and ๐‘”.

Now we give an example to support our result.

Example 2.14. Let ๐‘‹=[0,4/3], and define ๐บโˆถ๐‘‹ร—๐‘‹ร—๐‘‹โ†’[0,โˆž) by ๐บ(๐‘ฅ,๐‘ฆ,๐‘ง)=max{|๐‘ฅโˆ’๐‘ฆ|,|๐‘ฆโˆ’๐‘ง|,|๐‘ฅโˆ’๐‘ง|} and ๐‘“,๐‘”โˆถ๐‘‹โ†’๐‘‹ by ๐‘“(๐‘ฅ)=๐‘ฅ3/8,๐‘”(๐‘ฅ)=๐‘ฅ3/2 and ๐œ™(๐‘ก)=(2/3)๐‘ก. Then,(a)๐‘”(๐‘‹) is ๐บ-complete subspace of ๐‘‹,(b)๐‘“(๐‘‹)โŠ‚๐‘”(๐‘‹),(c)๐‘“ and ๐‘” are ๐บ-weakly commuting of type ๐ด๐‘“, (d)๐‘“ and ๐‘” satisfy condition (4) of Theorem 2.13.
It is clear that (a) and (b) are satisfied.
To show (c), as an easy calculation one can show that โˆ€๐‘ฅโˆˆ๐‘‹; we have ๐บ(๐‘“(๐‘”(๐‘ฅ)),๐‘”(๐‘”(๐‘ฅ)),๐‘“(๐‘“(๐‘ฅ)))=max{(3/64)๐‘ฅ9,(63/4096)๐‘ฅ9,(255/4096)๐‘ฅ9}โ‰ค(3/8)๐‘ฅ3=๐บ(๐‘“(๐‘ฅ),๐‘”(๐‘ฅ),๐‘“(๐‘ฅ)). Then ๐‘“ and ๐‘” are G-weakly commuting of type ๐ด๐‘“.
To show (d), for ๐‘ฅ,๐‘ฆ,๐‘งโˆˆ๐‘‹ we have 1๐บ(๐‘“(๐‘ฅ),๐‘“(๐‘ฆ),๐‘“(๐‘ง))=8๎€ฝ||๐‘ฅmax3โˆ’๐‘ฆ3||,||๐‘ฆ3โˆ’๐‘ง3||,||๐‘ฅ3โˆ’๐‘ง3||๎€พโ‰ค13๎€ฝ||๐‘ฅmax3โˆ’๐‘ฆ3||,||๐‘ฆ3โˆ’๐‘ง3||,||๐‘ฅ3โˆ’๐‘ง3||๎€พ=23๎‚€12๎€ฝ||๐‘ฅmax3โˆ’๐‘ฆ3||,||๐‘ฆ3โˆ’๐‘ง3||,||๐‘ฅ3โˆ’๐‘ง3||๎€พ๎‚=๐œ™(๐‘”(๐‘ฅ),๐‘”(๐‘ฆ),๐‘”(๐‘ง))โ‰ค๐œ™(๐‘€(๐‘ฅ,๐‘ฆ,๐‘ง)).(2.37)
Therefore, all hypotheses of Theorem 2.13 are satisfied and ๐‘ฅ=0 unique common fixed point of ๐‘“ and ๐‘”.

Corollary 2.15. Let (๐‘‹,๐บ) be a ๐บ-metric space, and suppose mappings ๐‘“,๐‘”โˆถ๐‘‹โ†’๐‘‹ satisfy the following conditions: (1)๐‘“ and ๐‘” be G -weakly commuting of type ๐ด๐‘“, (2)๐‘“(๐‘‹)โŠ†๐‘”(๐‘‹), (3)๐‘”(๐‘‹) is ๐บ-complete subspace of ๐‘‹, (4)๐บ(๐‘“(๐‘ฅ),๐‘“(๐‘ฆ),๐‘“(๐‘ง))โ‰ค๐‘˜๐‘€(๐‘ฅ,๐‘ฆ,๐‘ง),๐‘คโ„Ž๐‘’๐‘Ÿ๐‘’โŽงโŽชโŽชโŽชโŽจโŽชโŽชโŽชโŽฉ1๐‘€(๐‘ฅ,๐‘ฆ,๐‘ง)=max๐บ(๐‘”(๐‘ฅ),๐‘”(๐‘ฆ),๐‘”(๐‘ง)),๐บ(๐‘”(๐‘ฅ),๐‘“(๐‘ฅ),๐‘“(๐‘ฅ)),21๐บ(๐‘”(๐‘ฅ),๐‘“(๐‘ฆ),๐‘“(๐‘ฆ)),2โŽซโŽชโŽชโŽชโŽฌโŽชโŽชโŽชโŽญ๐บ(๐‘”(๐‘ฅ),๐‘“(๐‘ง),๐‘“(๐‘ง)),๐บ(๐‘”(๐‘ฆ),๐‘“(๐‘ฆ),๐‘“(๐‘ฆ))),๐บ(๐‘”(๐‘ฆ),๐‘“(๐‘ฅ),๐‘“(๐‘ฅ)),๐บ(๐‘”(๐‘ฆ),๐‘“(๐‘ง),๐‘“(๐‘ง)),๐บ(๐‘”(๐‘ง),๐‘“(๐‘ง),๐‘“(๐‘ง)),๐บ(๐‘”(๐‘ง),๐‘“(๐‘ฅ),๐‘“(๐‘ฅ)),๐บ(๐‘”(๐‘ง),๐‘“(๐‘ฆ),๐‘“(๐‘ฆ)),(2.38) for all ๐‘ฅ,๐‘ฆ,๐‘งโˆˆ๐‘‹, where ๐‘˜โˆˆ[0,1); then ๐‘“ and ๐‘” have a unique common fixed point.

Proof. It suffices to take ๐œ™(๐‘ก)=๐‘˜๐‘ก in Theorem 2.13.

Theorem 2.16. Let (๐‘‹,๐บ) be a ๐บ-metric space. Suppose the mappings ๐‘“,๐‘”โˆถ๐‘‹โ†’๐‘‹ are ๐บ-weakly commuting of type ๐ด๐‘” and satisfy the following condition: (1)๐‘“ and ๐‘” satisfy E.A. property, (2)๐‘”(๐‘‹) is closed subspace of ๐‘‹, (3)๐บ(๐‘“(๐‘ฅ),๐‘“(๐‘ฆ),๐‘“(๐‘ง))โ‰ค๐‘˜๐‘€(๐‘ฅ,๐‘ฆ,๐‘ง),๐‘คโ„Ž๐‘’๐‘Ÿ๐‘’โŽงโŽชโŽชโŽจโŽชโŽชโŽฉ[],[],[]โŽซโŽชโŽชโŽฌโŽชโŽชโŽญ๐‘€(๐‘ฅ,๐‘ฆ,๐‘ง)=max๐บ(๐‘”(๐‘ฅ),๐‘“(๐‘ฅ),๐‘“(๐‘ฅ))+๐บ(๐‘”(๐‘ฆ),๐‘“(๐‘ฆ),๐‘“(๐‘ฆ))+๐บ(๐‘”(๐‘ง),๐‘“(๐‘ง),๐‘“(๐‘ง))๐บ(๐‘”(๐‘ฅ),๐‘“(๐‘ฆ),๐‘“(๐‘ฆ))+๐บ(๐‘”(๐‘ฆ),๐‘“(๐‘ฅ),๐‘“(๐‘ฅ))+๐บ(๐‘”(๐‘ง),๐‘“(๐‘ฆ),๐‘“(๐‘ฆ))๐บ(๐‘”(๐‘ฅ),๐‘“(๐‘ง),๐‘“(๐‘ง))+๐บ(๐‘”(๐‘ฆ),๐‘“(๐‘ง),๐‘“(๐‘ง))+๐บ(๐‘”(๐‘ง),๐‘“(๐‘ฅ),๐‘“(๐‘ฅ))(2.39) for all ๐‘ฅ,๐‘ฆ,๐‘งโˆˆ๐‘‹, where ๐‘˜โˆˆ[0,1/3); then ๐‘“ and ๐‘” have a unique common fixed point.

Proof. Since ๐‘“ and ๐‘” satisfy E.A. property, there exists in ๐‘‹ a sequence (๐‘ฅ๐‘›) satisfying lim๐‘›โ†’โˆž๐‘“(๐‘ฅ๐‘›)=lim๐‘›โ†’โˆž๐‘”(๐‘ฅ๐‘›)=๐‘ก for some ๐‘กโˆˆ๐‘‹.
Since ๐‘”(๐‘‹) is closed subspace of ๐‘‹ and lim๐‘›โ†’โˆž๐‘”(๐‘ฅ๐‘›)=๐‘ก, there exists ๐‘โˆˆ๐‘‹ such that ๐‘”(๐‘)=๐‘ก, alsolim๐‘›โ†’โˆž๐‘“(๐‘ฅ๐‘›)=๐‘”(๐‘).
We will show that ๐‘“(๐‘)=๐‘”(๐‘) supposing that ๐‘“(๐‘)โ‰ ๐‘”(๐‘), then condition (3) implies that ๐บ๎€ท๐‘“๎€ท๐‘ฅ(๐‘),๐‘“(๐‘),๐‘“๐‘›๎€ท๎€ธ๎€ธโ‰ค๐‘˜๐‘€๐‘,๐‘,๐‘ฅ๐‘›๎€ธ,(2.40) where, ๐‘€๎€ท๐‘,๐‘,๐‘ฅ๐‘›๎€ธโŽงโŽชโŽชโŽจโŽชโŽชโŽฉ๎€บ๎€ท๐‘”๎€ท๐‘ฅ=max๐บ(๐‘”(๐‘),๐‘“(๐‘),๐‘“(๐‘))+๐บ(๐‘”(๐‘),๐‘“(๐‘),๐‘“(๐‘))+๐บ๐‘›๎€ธ๎€ท๐‘ฅ,๐‘“๐‘›๎€ธ๎€ท๐‘ฅ,๐‘“๐‘›,๎€บ๐บ๎€ท๐‘”๎€ท๐‘ฅ๎€ธ๎€ธ๎€ป(๐‘”(๐‘),๐‘“(๐‘),๐‘“(๐‘))+๐บ(๐‘”(๐‘),๐‘“(๐‘),๐‘“(๐‘))+๐บ๐‘›๎€ธ,๎€บ๐บ๎€ท๎€ท๐‘ฅ,๐‘“(๐‘),๐‘“(๐‘)๎€ธ๎€ป๐‘”(๐‘),๐‘“๐‘›๎€ธ๎€ท๐‘ฅ,๐‘“๐‘›๎€ท๎€ท๐‘ฅ๎€ธ๎€ธ+๐บ๐‘”(๐‘),๐‘“๐‘›๎€ธ๎€ท๐‘ฅ,๐‘“๐‘›๎€ท๐‘”๎€ท๐‘ฅ๎€ธ๎€ธ+๐บ๐‘›๎€ธโŽซโŽชโŽชโŽฌโŽชโŽชโŽญ.,๐‘“(๐‘),๐‘“(๐‘)๎€ธ๎€ป(2.41) Taking the limit as ๐‘›โ†’โˆž and using the fact that the function ๐บ is continuous, we get ๐บ(๐‘“(๐‘),๐‘“(๐‘),๐‘”(๐‘))โ‰ค3๐‘˜๐บ(๐‘”(๐‘),๐‘“(๐‘),๐‘“(๐‘)),(2.42) which is contradiction since ๐‘˜โˆˆ[0,1/3), so ๐‘“๐‘=๐‘”๐‘. Since ๐‘“ and ๐‘” are ๐บ-weakly commuting of type ๐ด๐‘”, then ๐บ(๐‘”๐‘“(๐‘),๐‘”๐‘”(๐‘),๐‘“๐‘“(๐‘))โ‰ค๐บ(๐‘”(๐‘),๐‘“(๐‘),๐‘”(๐‘))=0.(2.43) Therefore, ๐‘“(๐‘”(๐‘))=๐‘“๐‘“(๐‘)=๐‘”๐‘“(๐‘)=๐‘”๐‘”(๐‘); then ๐‘“(๐‘ก)โˆถ=๐‘“๐‘”(๐‘)=๐‘”๐‘“(๐‘)=๐‘”(๐‘ก).(2.44) Finally, we will show that ๐‘ก=๐‘“(๐‘) is common fixed point of ๐‘“ and ๐‘”.
Supposing that ๐‘“๐‘กโ‰ ๐‘ก, then ๐บ(๐‘“(๐‘ก),๐‘ก,๐‘ก)=๐บ(๐‘“(๐‘ก),๐‘“(๐‘),๐‘“(๐‘))โ‰ค๐‘˜๐‘€(๐‘ก,๐‘,๐‘),(2.45) where โŽงโŽชโŽชโŽจโŽชโŽชโŽฉ[],[],[]โŽซโŽชโŽชโŽฌโŽชโŽชโŽญ๐‘€(๐‘ก,๐‘,๐‘)=max๐บ(๐‘”(๐‘ก),๐‘“(๐‘ก),๐‘“(๐‘ก))+๐บ(๐‘”(๐‘),๐‘“(๐‘),๐‘“(๐‘))+๐บ(๐‘”(๐‘),๐‘“(๐‘),๐‘“(๐‘))๐บ(๐‘”(๐‘ก),๐‘“(๐‘),๐‘“(๐‘))+๐บ(๐‘”(๐‘),๐‘“(๐‘ก),๐‘“(๐‘ก))+๐บ(๐‘”(๐‘),๐‘“(๐‘),๐‘“(๐‘))๐บ(๐‘”(๐‘ก),๐‘“(๐‘),๐‘“(๐‘))+๐บ(๐‘”(๐‘),๐‘“(๐‘),๐‘“(๐‘))+๐บ(๐‘”(๐‘),๐‘“(๐‘ก),๐‘“(๐‘ก)).(2.46) But ๐‘“(๐‘ก)=๐‘”(๐‘ก) and ๐‘“(๐‘)=๐‘”(๐‘). Thus, []}๐บ(๐‘“(๐‘ก),๐‘ก,๐‘ก)โ‰ค๐‘˜max{๐บ(๐‘ก,๐‘“(๐‘ก),๐‘“(๐‘ก)),๐บ(๐‘“(๐‘ก),๐‘ก,๐‘ก)+๐บ(๐‘ก,๐‘“(๐‘ก),๐‘“(๐‘ก))<๐‘˜{๐บ(๐‘“(๐‘ก),๐‘ก,๐‘ก)+๐บ(๐‘ก,๐‘“(๐‘ก),๐‘“(๐‘ก))}.(2.47) Hence, ๎‚€๐‘˜๐บ(๐‘“(๐‘ก),๐‘ก,๐‘ก)โ‰ค๎‚1โˆ’๐‘˜๐บ(๐‘ก,๐‘“(๐‘ก),๐‘“(๐‘ก)).(2.48)
Adjusting similarly, we get ๎‚€๐‘˜๐บ(๐‘ก,๐‘“(๐‘ก),๐‘“(๐‘ก))โ‰ค๎‚1โˆ’๐‘˜๐บ(๐‘“๐‘ก,๐‘ก,๐‘ก).(2.49) Therefore, ๎‚€๐‘˜๐บ(๐‘“๐‘ก,๐‘ก,๐‘ก)โ‰ค๎‚1โˆ’๐‘˜2๐บ(๐‘“๐‘ก,๐‘ก,๐‘ก),(2.50) a contradiction which implies that ๐‘“๐‘ก=๐‘ก=๐‘“๐‘, but ๐‘”๐‘ก=๐‘“๐‘ก=๐‘ก. Then ๐‘ก is a common fixed point of ๐‘“ and ๐‘”.
To prove uniqueness, suppose we have ๐‘ข and ๐‘ฃ such that ๐‘ขโ‰ ๐‘ฃ, ๐‘“๐‘ข=๐‘”๐‘ข=๐‘ข and ๐‘“๐‘ฃ=๐‘”๐‘ฃ=๐‘ฃ; then ๐บ(๐‘ข,๐‘ฃ,๐‘ฃ)=๐บ(๐‘“(๐‘ข),๐‘“(๐‘ฃ),๐‘“(๐‘ฃ))โ‰ค๐‘˜{๐บ(๐‘ข,๐‘ฃ,๐‘ฃ)+๐บ(๐‘ฃ,๐‘ข,๐‘ข)}.(2.51)โ€‰Hence, ๐บ(๐‘ข,๐‘ฃ,๐‘ฃ)โ‰ค(๐‘˜/(1โˆ’๐‘˜))๐บ(๐‘ฃ,๐‘ข,๐‘ข).โ€‰Similarly, ๐บ(๐‘ฃ,๐‘ข,๐‘ข)โ‰ค(๐‘˜/(1โˆ’๐‘˜))๐บ(๐‘ข,๐‘ฃ,๐‘ฃ).โ€‰Therefore, ๐บ(๐‘ข,๐‘ฃ,๐‘ฃ)โ‰ค(๐‘˜/(1โˆ’๐‘˜))2๐บ(๐‘ข,๐‘ฃ,๐‘ฃ)a contradiction which implies that ๐‘ข=๐‘ฃ. Then ๐‘ก is a unique common fixed point of ๐‘“ and ๐‘”.
Now we give an example to support our result.

Example 2.17. Let ๐‘‹=[0,3/4], define ๐บโˆถ๐‘‹ร—๐‘‹ร—๐‘‹โ†’[0,โˆž) by
๐บ(๐‘ฅ,๐‘ฆ,๐‘ง)=max{|๐‘ฅโˆ’๐‘ฆ|,|๐‘ฆโˆ’๐‘ง|,|๐‘ฅโˆ’๐‘ง|} and let ๐‘“,๐‘”โˆถ๐‘‹โ†’๐‘‹ by ๐‘“(๐‘ฅ)=๐‘ฅ2/5, ๐‘”(๐‘ฅ)=๐‘ฅ2.
Then,(a)๐‘”(๐‘‹) is closed subspace of ๐‘‹,(b)๐‘“ and ๐‘” are G-weakly commuting of type ๐ด๐‘”,(c)๐‘“ and ๐‘” satisfy E.A. property. (d)๐‘“ and ๐‘” satisfy condition (4) for ๐‘˜=(1/4).

Proof. (a) is obvious.
To show (b), as an easy calculation one can show that forall๐‘ฅโˆˆ๐‘‹; we have ๐บ(๐‘”(๐‘“(๐‘ฅ)),๐‘”(๐‘”(๐‘ฅ)),๐‘“(๐‘“(๐‘ฅ)))=max{(4/125)๐‘ฅ4,(24/25)๐‘ฅ4,(124/125)๐‘ฅ4}โ‰ค(4/5)๐‘ฅ2=๐บ(๐‘“(๐‘ฅ),๐‘”(๐‘ฅ),๐‘“(๐‘ฅ)). Then ๐‘“ and ๐‘” are ๐บ-weakly commuting of type ๐ด๐‘”.
To show (c), if we consider the sequence {๐‘ฅ๐‘›}={1/2๐‘›}, then ๐‘“๐‘ฅ๐‘›โ†’0 and ๐‘”๐‘ฅ๐‘›โ†’0 as ๐‘›โ†’โˆž. Thus, ๐‘“ and ๐‘” satisfy the E.A. property.
To show (d), for ๐‘ฅ,๐‘ฆ,๐‘งโˆˆ๐‘‹ we have ||||๐‘ฅ25โˆ’๐‘ฆ25||||โ‰ค๐‘ฅ25+๐‘ฆ25,||||๐‘ฆ25โˆ’๐‘ง25||||โ‰ค๐‘ฆ25+๐‘ง25,||||๐‘ฅ25โˆ’๐‘ง25||||โ‰ค๐‘ฅ25+๐‘ง25.(2.52)
Then๎ƒฏ||||๐‘ฅ๐บ(๐‘“(๐‘ฅ),๐‘“(๐‘ฆ),๐‘“(๐‘ง))=max25โˆ’๐‘ฆ25||||,||||๐‘ฆ25โˆ’๐‘ง25||||,||||๐‘ฅ25โˆ’๐‘ง25||||๎ƒฐโ‰ค๐‘ฅ25+๐‘ฆ25+๐‘ง25=14๎‚ต4๐‘ฅ25+4๐‘ฆ25+4๐‘ง25๎‚ถ=14(๐บ(๐‘”(๐‘ฅ),๐‘“(๐‘ฅ),๐‘“(๐‘ฅ))+๐บ(๐‘”(๐‘ฆ),๐‘“(๐‘ฆ),๐‘“(๐‘ฆ))+๐บ(๐‘”(๐‘ง),๐‘“(๐‘ง),๐‘“(๐‘ง)))โ‰ค๐‘˜๐‘€(๐‘ฅ,๐‘ฆ,๐‘ง).(2.53)
Therefore, all hypotheses of Theorem 2.16 are satisfied for ๐‘˜=1/4 and ๐‘ฅ=0, a unique common fixed point of ๐‘“ and ๐‘”.

Theorem 2.18. Let (๐‘‹,๐บ) be a ๐บ-metric space, and suppose mappings ๐‘“,๐‘”โˆถ๐‘‹โ†’๐‘‹ be ๐บ-๐‘…-weakly commuting of type ๐ด๐‘“. Suppose that there exists a mapping ๐œ“โˆถ๐‘‹โ†’[0,โˆž) such that (1)๐‘“(๐‘‹)โŠ‚๐‘”(๐‘‹), (2)๐‘”(๐‘‹) is ๐บ-complete subspace of ๐‘‹, (3)๐บ(๐‘”๐‘ฅ,๐‘“๐‘ฅ,๐‘“๐‘ฅ)<๐œ“(๐‘”(๐‘ฅ))โˆ’๐œ“(๐‘“(๐‘ฅ)),forall๐‘ฅโˆˆ๐‘‹, โŽงโŽชโŽจโŽชโŽฉโŽซโŽชโŽฌโŽชโŽญ๐บ(๐‘“(๐‘ฅ),๐‘“(๐‘ฆ),๐‘“(๐‘ง))<max๐บ(๐‘”(๐‘ฅ),๐‘”(๐‘ฆ),๐‘”(๐‘ง)),๐บ(๐‘”(๐‘ฅ),๐‘“(๐‘ฅ),๐‘”(๐‘ฆ)),๐บ(๐‘”(๐‘ง),๐‘“(๐‘ง),๐‘“(๐‘ฅ)),๐บ(๐‘”(๐‘ฆ),๐‘“(๐‘ฆ),๐‘“(๐‘ง)),(2.54) for all ๐‘ฅ,๐‘ฆ,๐‘งโˆˆ๐‘‹; then ๐‘“ and ๐‘” have a unique common fixed point.

Proof. Let ๐‘ฅ0โˆˆ๐‘‹, and then choose ๐‘ฅ1โˆˆ๐‘‹ such that ๐‘“(๐‘ฅ0)=๐‘”(๐‘ฅ1) and ๐‘ฅ2โˆˆ๐‘‹ where ๐‘“(๐‘ฅ1)=๐‘”(๐‘ฅ2); then by induction we can define a sequence (๐‘ฆ๐‘›)โˆˆ๐‘‹ as follows: ๐‘ฆ๐‘›๎€ท๐‘ฅ=๐‘“๐‘›๎€ธ๎€ท๐‘ฅ=๐‘”๐‘›+1๎€ธ,๐‘›โˆˆ๐‘โˆช{0}.(2.55) We will show that the sequence (๐‘ฆ๐‘›) is ๐บ-cauchy sequence: ๐บ๎€ท๐‘”๎€ท๐‘ฅ๐‘›๎€ธ๎€ท๐‘ฅ,๐‘”๐‘›+1๎€ธ๎€ท๐‘ฅ,๐‘”๐‘›+1๎€ท๐‘”๎€ท๐‘ฅ๎€ธ๎€ธ=๐บ๐‘›๎€ธ๎€ท๐‘ฅ,๐‘“๐‘›๎€ธ๎€ท๐‘ฅ,๐‘“๐‘›๎€ท๐‘”๎€ท๐‘ฅ๎€ธ๎€ธ<๐œ“๐‘›๎€ท๐‘“๎€ท๐‘ฅ๎€ธ๎€ธโˆ’๐œ“๐‘›๎€ท๐‘”๎€ท๐‘ฅ๎€ธ๎€ธ=๐œ“๐‘›๎€ท๐‘”๎€ท๐‘ฅ๎€ธ๎€ธโˆ’๐œ“๐‘›+1.๎€ธ๎€ธ(2.56)
Consider ๐‘Ž๐‘›=๐œ“(๐‘”(๐‘ฅ๐‘›)),๐‘›=1,2,3,4,โ€ฆ, then ๎€ท๐‘”๎€ท๐‘ฅ0โ‰ค๐บ๐‘›๎€ธ๎€ท๐‘ฅ,๐‘”๐‘›+1๎€ธ๎€ท๐‘ฅ,๐‘”๐‘›+1๎€ธ๎€ธ<๐‘Ž๐‘›โˆ’๐‘Ž๐‘›+1.(2.57)
Thus, the sequence (๐‘Ž๐‘›) is nonincreasing and bounded below by 0; hence (๐‘Ž๐‘›)is convergent sequence.
On the other hand we have, from (G5) and (2.57), that for ๐‘š,๐‘›โˆˆ๐;๐‘š>๐‘›๐บ๎€ท๐‘”๎€ท๐‘ฅ๐‘›๎€ธ๎€ท๐‘ฅ,๐‘”๐‘›+๐‘š๎€ธ๎€ท๐‘ฅ,๐‘”๐‘›+๐‘šโ‰ค๎€ธ๎€ธ๐‘›+๐‘šโˆ’1๎“๐‘—=๐‘›๐บ๎€ท๐‘”๎€ท๐‘ฅ๐‘—๎€ธ๎€ท๐‘ฅ,๐‘”๐‘—+1๎€ธ๎€ท๐‘ฅ,๐‘”๐‘—+1<๎€ธ๎€ธ๐‘›+๐‘šโˆ’1๎“๐‘—=๐‘›๐‘Ž๐‘—โˆ’๐‘Ž๐‘—+1(Telescopingsum)=๐‘Ž๐‘›โˆ’๐‘Ž๐‘›+๐‘š.(2.58)
Therefore, the sequence (๐‘”(๐‘ฅ๐‘›)) is G-cauchy sequence in ๐‘”(๐‘‹).
Since ๐‘”(๐‘‹) is ๐บ-complete subspace, then there exists ๐‘กโˆˆ๐‘”(๐‘‹) such that lim๐‘›โ†’โˆž๐‘”(๐‘ฅ๐‘›)=๐‘ก; having ๐‘กโˆˆ๐‘”(๐‘‹) there exists ๐‘โˆˆ๐‘‹ such that ๐‘”(๐‘)=๐‘ก, also lim๐‘›โ†’โˆž๐‘“(๐‘ฅ๐‘›)=๐‘”(๐‘)=๐‘ก.
We will show that ๐‘“(๐‘)=๐‘”(๐‘); supposing that ๐‘“(๐‘)โ‰ ๐‘”(๐‘), then condition (4) implies that ๐บ๎€ท๎€ท๐‘ฅ๐‘“(๐‘),๐‘“(๐‘),๐‘“๐‘›โŽงโŽชโŽจโŽชโŽฉ๐บ๎€ท๎€ท๐‘ฅ๎€ธ๎€ธ<max๐‘”(๐‘),๐‘”(๐‘),๐‘”๐‘›๐บ๎€ท๎€ท๐‘ฅ๎€ธ๎€ธ,๐บ(๐‘”(๐‘),๐‘“(๐‘),๐‘”(๐‘)),๐‘”(๐‘),๐‘“(๐‘),๐‘“๐‘›๎€ท๐‘”๎€ท๐‘ฅ๎€ธ๎€ธ,๐บ๐‘›๎€ธ๎€ท๐‘ฅ,๐‘“๐‘›๎€ธ๎€ธโŽซโŽชโŽฌโŽชโŽญ,๐‘“(๐‘).(2.59) Taking the limit as ๐‘›โ†’โˆž, we get โŽงโŽชโŽชโŽจโŽชโŽชโŽฉโŽซโŽชโŽชโŽฌโŽชโŽชโŽญ๐บ(๐‘“(๐‘),๐‘“(๐‘),๐‘”(๐‘))<max๐บ(๐‘”(๐‘),๐‘”(๐‘),๐‘”(๐‘)),๐บ(๐‘”(๐‘),๐‘“(๐‘),๐‘”(๐‘)),๐บ(๐‘”(๐‘),๐‘”(๐‘),๐‘“(๐‘)),(2.60) hence, ๐บ(๐‘“(๐‘),๐‘“(๐‘),๐‘”(๐‘))<๐บ(๐‘”(๐‘),๐‘”(๐‘),๐‘“(๐‘)).(2.61) Adjusting similarly, we get ๐บ(๐‘”(๐‘),๐‘”(๐‘),๐‘“(๐‘))<๐บ(๐‘“(๐‘),๐‘“(๐‘),๐‘”(๐‘))(2.62)
Therefore, ๐บ(๐‘“(๐‘),๐‘“(๐‘),๐‘”(๐‘))<๐บ(๐‘”(๐‘),๐‘“(๐‘),๐‘”(๐‘))<๐บ(๐‘“(๐‘),๐‘“(๐‘),๐‘”(๐‘)).(2.63)
Thus, a contradiction implies ๐‘“๐‘=๐‘”๐‘.
Since ๐‘“ and ๐‘” are ๐บ-weakly commuting of type ๐ด๐‘“, then ๐บ(๐‘“(๐‘”(๐‘)),๐‘”(๐‘”(๐‘)),๐‘“(๐‘“(๐‘)))โ‰ค๐บ(๐‘“(๐‘),๐‘”(๐‘),๐‘“(๐‘))=0.(2.64)
Thus, ๐‘“๐‘“(๐‘)=๐‘“๐‘”(๐‘)=๐‘”๐‘“(๐‘)=๐‘”๐‘”(๐‘), then ๐‘“(๐‘ก)=๐‘“๐‘”(๐‘)=๐‘”๐‘“(๐‘)=๐‘”(๐‘ก).
Finally, we will show that ๐‘ก=๐‘“(๐‘) is common fixed point of ๐‘“ and ๐‘”.
Suppose that ๐‘“๐‘กโ‰ ๐‘ก, so โŽงโŽชโŽชโŽจโŽชโŽชโŽฉโŽซโŽชโŽชโŽฌโŽชโŽชโŽญ๐บ(๐‘“(๐‘ก),๐‘ก,๐‘ก)=๐บ(๐‘“(๐‘ก),๐‘“(๐‘),๐‘“(๐‘))<max๐บ(๐‘”(๐‘ก),๐‘”(๐‘),๐‘”(๐‘)),๐บ(๐‘”(๐‘ก),๐‘“(๐‘ก),๐‘”(๐‘)),๐บ(๐‘”(๐‘),๐‘“(๐‘),๐‘“(๐‘)).(2.65)
Since ๐‘”(๐‘)=๐‘“(๐‘) and ๐‘”(๐‘ก)=๐‘“(๐‘ก), therefore (2.65) implies that ๐บ(๐‘“(๐‘ก),๐‘ก,๐‘ก)<๐บ(๐‘“(๐‘ก),๐‘“(๐‘ก),๐‘ก).(2.66)
Similarly, we have ๐บ(๐‘“(๐‘ก),๐‘“(๐‘ก),๐‘ก)<๐บ(๐‘“(๐‘ก),๐‘ก,๐‘ก).
A contradiction implies that ๐‘“๐‘ก=๐‘“๐‘=๐‘ก. Then ๐‘ก is a common fixed point.
To prove uniqueness suppose we have ๐‘ข and ๐‘ฃ such that ๐‘ขโ‰ ๐‘ฃ where ๐‘“๐‘ข=๐‘”๐‘ข=๐‘ข and ๐‘“๐‘ฃ=๐‘”๐‘ฃ=๐‘ฃ; then as an easy calculation one can get ๐บ(๐‘ข,๐‘ฃ,๐‘ฃ)<๐บ(๐‘ฃ,๐‘ข,๐‘ข).(2.67) Similarly, ๐บ(๐‘ฃ,๐‘ข,๐‘ข)<๐บ(๐‘ข,๐‘ฃ,๐‘ฃ), a contradiction which implies that ๐‘ข=๐‘ฃ. Then, ๐‘ก is a unique common fixed point of ๐‘“ and ๐‘”.

Now we give an example to support our result.

Example 2.19. Let ๐‘‹=[1,โˆž), ๐œ“โˆถ๐‘‹โ†’[0,โˆž) such that ๐œ“(๐‘ก)=3๐‘ก,๐‘กโˆˆ๐‘‹ and ๐บ(๐‘ฅ,๐‘ฆ,๐‘ง)=max{|๐‘ฅโˆ’๐‘ฆ|,|๐‘ฆโˆ’๐‘ง|,|๐‘งโˆ’๐‘ฅ|}. Define ๐‘“,๐‘”โˆถ๐‘‹โ†’๐‘‹ by ๐‘“(๐‘ฅ)=2๐‘ฅโˆ’1 and ๐‘”(๐‘ฅ)=3๐‘ฅโˆ’2.
Then, (a)๐‘“(๐‘‹)โŠ‚๐‘”(๐‘‹), (b)๐‘”(๐‘‹) is ๐บ-complete subspace of ๐‘‹, (c)๐บ(๐‘”๐‘ฅ,๐‘“๐‘ฅ,๐‘“๐‘ฅ)<๐œ“(๐‘”(๐‘ฅ))โˆ’๐œ“(๐‘“(๐‘ฅ)),forall๐‘ฅโˆˆ๐‘‹,(d)๐‘“ and ๐‘” satisfy condition (4) of Theorem 2.18.
Then as an easy calculation one can see that ๐บ