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

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Volume 2012, Article ID 248937,21pages doi:10.1155/2012/248937

Research Article

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

Zead Mustafa

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

Correspondence should be addressed to Zead Mustafa,[email protected] Received 7 May 2012; Revised 24 June 2012; Accepted 2 August 2012 Academic Editor: Ya Ping Fang

Copyrightq2012 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.

We introduce some new types of pairs of mappingsf, g on G-metric space called G-weakly commuting of typeAfand G-R-weakly commuting of typeAf. 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 Jungck1proved a common fixed point theorem for commuting maps, but his results required the continuity of one of the maps.

Sessa2in 1982 first introduced a weaker version of commutativity for a pair of self- maps, and it is shown in Sessa 2that weakly commuting pair of maps in metric pace is commuting, but the converse may not be true.

Later, Jungck3introduced 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, Jungck4defined 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 ⇒ weakly commuting maps⇒compatible maps⇒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 see5–7.

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However, the study of common fixed point of noncompatible mappings has recently been initiated by Pantsee8,9.

In 2002 Amari and El Moutawakil10defined a new property called E.A. property which generalizes the concept of noncompatible mappings, and they proved some common fixed point theorem.

Definition 1.1see10. LetSandT be two self-mappings of a metric spaceX, d. We say thatTandSsatisfy the E.A. property if there exists a sequencexnsuch that

nlim→ ∞Txn lim

n→ ∞Sxnt, for somet∈X. 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.11–15obtained some fixed point results for mapping satisfying diﬀerent contractive conditions on complete G-metric space, while in16the completeness property was omitted and replaced by suﬃcient conditions, where these conditions do not imply the completeness property.

Chugh et al.17obtained some fixed point results for maps satisfying property P in G-metric spaces. Saadati et al.18studied fixed point of contractive mappings in partially ordered G-metric spaces. Shatanawi obtained fixed points ofφ-maps in G-metric spaces19 and a number of fixed point results for the two weakly increasing mappings with respect to partial ordering in G-metric spaces20. In21,22authors 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 2.3–2.6 generalize Theorems 2.1–2.4 of11.

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 in13 Theorem 2.1and Corollary 3.3 is G-version of Theorem 2 from10in the case of two self-mappings. Also this corollary is in relation with Theorem 2.5 of23.

In25the 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 thatTheorem 2.13 is an extension and generalization of1Theorem 2.2, Corollary 2.3, Theorem 2.6, Corollaries 2.7 and 2.8 in26and2Theorem 2.4 and Corollary 2.5 in27.

Aydi et al.28established some common fixed point results for two mappingsfand gon G-metric spaces with assumption thatfis a generalized weakly G-contraction mappings of type A and B with respect tog.

In this paper, we define new types of self-maps f and g on G-metric space called G-weakly commuting of typeAf andG-R-weakly commuting of typeAf. Also we obtain several common fixed point results for these mappings under certain contractive condition inG-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.

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Definition 1.2see29. AG-metric space is a pairX, G, whereXis a nonempty set, andG is a nonnegative real-valued function defined onX×X×Xsuch that for allx, y, z, a ∈Xwe have

G1Gx, y, z 0 ifxyz,

G20< Gx, x, y; for allx, y∈X, with x /y,

G3Gx, x, y≤Gx, y, z, for allx, y, z∈X, with z /y,

G4Gx, y, z Gx, z, y Gy, z, x · · ·,symmetry in all three variables, G5Gx, y, z≤Gx, a, a Ga, y, z, for allx, y, z, a ∈X,rectangle inequality.

The functionGis calledG-metric onX.

EveryG-metric onXdefines a metricd_GonXby dG

x, y G

x, y, y G

y, x, x

∀x, y∈X. 1.2

Example 1.3see29. LetX, dbe a metric space, and defineGsandGmonX×X×XtoR by

G_s x, y, z

d x, y

d y, z

dx, z, Gm

x, y, z

max d

x, y , d

y, z

, dx, z

, 1.3

for allx, y, z∈X. ThenX, GsandX, GmareG-metric spaces.

Example 1.4see29. LetXR, and defineG:X×X×X → R , by

G x, y, z

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎩

x−y y−z |x−z|, if all x, y,and zare strictly positive or they are all strictly negative or allx, y,andzare zero, 1 x−y y−z |x−z|, otherwise,

1.4

thenX, Gis G-metric space.

Definition 1.5see29. A sequencexnin a G-metric spaceX is said to converge if there existsx ∈ X such that lim_{n,m→ ∞}Gx, xn, x_m 0, and one says that the sequencexnisG- convergent tox. We callxthe limit of the sequencexnand writexn → xor limn→ ∞xnx through this paper we mean byN the set of all natural numbers.

Proposition 1.6see29. LetXbeG-metric space. Then the following statements are equivalent:

1 xnisG-convergent tox, 2Gxn, x_n, x → 0, asn → ∞, 3Gxn, x, x → 0, asn → ∞, 4Gxm, x_n, x → 0, asm, n → ∞.

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Definition 1.7see29. In aG-metric spaceX, a sequencexnis said to beG-Cauchy if given ε >0, there isN ∈N such thatGxn, x_m, x_l< ε, for alln, m, l≥N. That isGxn, x_m, x_l → 0 asn, m, l → ∞.

Proposition 1.8see29. In aG-metric spaceX, the following statements are equivalent:

1the sequencexnisG-Cauchy;

2for everyε >0, there exists N∈N such thatGxn, x_m, x_m< ε, for alln, m≥N.

Definition 1.9 see 29. A G-metric space X, G is called symmetric G-metric space if Gx, y, y Gy, x, xfor allx, y∈Xand called nonsymmetric if it is not symmetric.

Example 1.10. LetXN be the set of all natural numbers, and define G:X×X×X → R such that for allx, y, z∈X:

Gx, y, z 0 ifxyz, Gx, y, y x y, ifx < y, Gx, y, y x y 1/2, ifx > y,

Gx, y, z x y zifx /y /zand symmetry in all three variables.

Then,X, Gis G-metric space and nonsymmetric since ifx < y, we haveGx, y, y x y /x y 1/2Gy, x, x.

Proposition 1.11 see 29. LetX be a G-metric space; then the function Gx, y, z is jointly continuous in all three of its variables.

Definition 1.12 see 29. A G-metric space X is said to be complete if every G-Cauchy sequence inXisG-convergent inX.

Definition 1.13see23. Letfandgbe self-maps of a setX. Ifwfxgxfor somex∈X, thenxis called a coincidence point offandg, andwis called a point of coincidence offand g.

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

Proposition 1.14see23. Letfandgbe weakly compatible self-maps of a setX. Iffandghave a unique point of coincidencewfxgx, thenwis the unique common fixed point offandg.

In 2001, Abbas et al. 30 introduce a new type of pairs of mappings f, g called R-weakly commuting and they proved a unique common fixed point of four R-weakly commuting, maps satisfying generalized contractive condition.

Definition 1.15see30. LetXbe a G-metric space, and letfandgbe two self-mappings of X; thenfandgare called R-weakly commuting if there exists a positive real numberRsuch that

G f

gx , f

gx , g

fx

≤RG

fx, fx, gx

hold for eachx∈X. 1.5

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Very recently, Mustafa et al.31introduce some new types of pairs of mappingsf, g on G-metric space called G-weakly commuting of typeG_f and G-R-weakly commuting of typeG_f, and they obtained several common fixed point results by using E.A. property.

Definition 1.16see31. A pair of self-mappingsf, gof a G-metric spaceX, Gis said to beG-weakly commuting of typeG_f if

G

fgx, gfx, ffx

≤G

fx, gx, fx

, ∀x∈X. 1.6

Definition 1.17see31. A pair of self-mappingsf, gof aG-metric spaceX, Gis said to beG-R-weakly commuting of typeGf if there exists some positive real numberRsuch that

G

fgx, gfx, ffx

≤R G

fx, gx, fx

, ∀x∈X. 1.7

Remark 1.18. TheG-R-weakly commuting maps of typeGf are R-weakly commuting since Gfgx, fgx, gfx≤Gfgx, gfx, ffx≤Gfx, gx, fx, 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 Af for pairs of mappingf, gand comparison between this concept and Definitions1.15,1.16, and1.17is 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-mappingsf, gof a G-metric spaceX, Gis said to be G-weakly commuting of typeA_f if

G

fgx, ggx, ffx

≤G

fx, gx, fx

, ∀x∈X. 2.1

Definition 2.2. A pair of self-mappingsf,gof a G-metric spaceX, Gis said to be G-R-weakly commuting of typeAf if there exists some positive real numberRsuch that

G

fgx, ggx, ffx

≤R G

fx, gx, fx

, ∀x∈X. 2.2

Remark 2.3. TheG-weakly commuting maps of typeA_f areG-R-weakly commuting of type A_f. Reciprocally, if R ≤ 1, then G-R-weakly commuting maps of type A_f are G-weakly commuting of typeAf.

If we interchangef andgin2.1and2.2, then the pair of mappingsf, gis called G-weakly commuting of typeA_gandG-R-weakly commuting of typeA_g, respectively.

Example 2.4. LetX 0,3/4, with the G-metricGx, y, z |x−y| |y−z| |x−z|, for all x, y, z ∈X. Definef, g:X → X by,fx 1/4x², gx x²; then as an easy calculation one can show that Gfgx, ggx, ffx 126/64x⁴ ≤ Gfx, gx, fx 6/4x², for allx ∈ X. Then the pairf, gis G-Weakly commuting of typeAf and G-R-Weakly commuting of typeA_f.

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Example 2.5. LetX 2,∞, with the G-metricGx, y, z |x−y| |y−z| |x−z|, for all x, y, z ∈X. Definef, g :X → Xby,fx x 1, gx 2x 1, then forx2 we see that Gfgx, ggx, ffx Ggfx, ggx, ffx 20 andGfx, gx, fx Ggx, fx, gx 6. Therefore the pairf, gis notG-weakly commuting of typeAf orAg, but it is G-R-weakly commuting of typeAf andAgforR≥4.

The following examples show a pair of mappingsf, gthatG-weakly commuting of typeGf need not beG-weakly commuting of typeAf.

Example 2.6. LetX 0,89/100, with the G-metricGx, y, z max{|x−y|,|y−z|,|x−z|}, for allx, y, z ∈ X. Definefx 1/4x², gx x²; then we see thatGfgx, gfx, ffx 15/16x⁴ and Gfx, gx, fx 3/4x², while as an easy calculation one can show that for x 88/100 we have Gfgx, ggx, ffx 59/100 Ggx, fx, gx 58/100. Therefore the pairf, gis notG-weakly commuting of typeA_f, but it is G-weakly commuting of typeGf.

The following example shows that

1a pair of mappingsf, gthat isG-weakly commuting of typeAf need not beG- weakly commuting of typeA_g;

2a pair of mappingsf, gthat isG-weakly commuting of typeA_f need not beG- weakly commuting of typeG_f;

3a pair of mappings f, gthat isG-R-weakly commuting of type A_f need not be R-weakly commuting;

Example 2.7. LetX 2,9and Gx, y, z max{|x−y|,|y−z|,|x−z|}for all x, y, z ∈ X.

Define the mappingsf, g:X → Xby

fx

⎧⎪

⎪⎨

⎪⎪

⎩

2 ifx2, 6 if 2< x≤5, 5 ifx >5,

gx

⎧⎪

⎪⎨

⎪⎪

⎩

2 ifx2, 9 if 2< x≤5, 6 ifx >5.

2.3

Then,

f

gx

2 ifx2,

5 ifx >2, g fx

⎧⎪

⎪⎨

⎪⎪

⎩

2 ifx2, 6 if 2< x≤5, 9 ifx >5,

g

gx

2 ifx2, 6 ifx >2, f

fx

⎧⎪

⎪⎨

⎪⎪

⎩

2 ifx2, 5 if 2< x≤5, 6 ifx >5.

2.4

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Moreover,

fx−gx

⎧⎪

⎪⎨

⎪⎪

⎩

0 ifx2, 3 if 2< x≤5, 1 ifx >5.

2.5

Ifx2, we haveGfgx, ggx, ffx 0Gfx, gx, fx.

If 2< x≤5, we have

G f

gx

, f

fx

, g

gx

max{0,1,1} ≤3G

fx, gx, fx

. 2.6

Ifx >5, then

G f

gx

, f

fx

, g

gx

max{1,1,0} ≤1G

gx, fx, gx

. 2.7

Thus,fandgareGweakly commuting of typeA_f, but forx6, we have

G g

f6

, f f6

, g

g6

max{3,3,0}1G

g6, f6, g6

. 2.8

Therefore, the pairf, gis not G-weakly commuting of type A_g, but it is G-weakly commuting of typeA_f.

Also forx7, we have

G f

g7

, g

f7

, f

f7

41G

g6, f6, g6

. 2.9

Therefore, the pairf, gis not G-weakly commuting of typeG_f.

AS an easy calculation one can see thatf, gareG-R-weakly commuting of typeAg

forR 3; but forx6 we haveGfg6, fg6, gf6 4 3Gf6, f6, g6 3, hencef, gis NOTR-weakly commuting forR3.

Lemma 2.8. IffandgareG-weakly commuting of typeAf orG-R-weakly commuting of typeAf, thenfandgare weakly compatible.

Proof. Letxbe a coincidence point off andg, that is, fx gx; then if the pairf, gis G-weakly commuting of typeAf, we have

G f

gx

, g

fx

, f

gx

G f

gx

, g gx

, f fx

≤G

fx, gx, fx 0.

2.10

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It followsfgx gfx; then they commute at their coincidence point.

Similarly, if the pairf, gisG-R-weakly commuting of typeG_f, we have G

f gx

, g fx

, f gx

G f

gx

, g

gx

, f

fx

≤RG

fx, gx, fx 0.

2.11

Thusfgx gfx; then the pairf, gis weakly compatible.

The following example shows that

1the converse ofLemma 2.8failsfor the case ofG-weakly commutativity,

2a pair of mappings f, g that is R-weakly commuting need not be G-R-weakly commuting of typeA_f,

3a pair of mappings f, g that is R-weakly commuting need not be G-R-weakly commuting of typeGf.

Example 2.9. LetX 1, ∞andGx, y, z max{|x−y|,|y−z|,|x−z|}. Definef, g:X → X byfx 2x−1 andgx x². We see thatx1 is the only coincidence point andfg1 f1 1 andgf1 g1 1, sofandgare weakly compatible.

But, by an easy calculation, one can see that forx3 we have, G

f

gx

, g

gx

, f

fx

724G

fx, gx, fx

. 2.12

Therefore,fandgare notG-weakly commuting of typeA_f.

Also, we see thatGfgx, fgx, gfx 2x²−4x 2≤2Gfx, fx, gx 2x² −2x 1; therefore the mappings f, gare R-weakly commuting forR 2, but for x 4 we haveGfg4, gg4, ff4 243 2Gf4, g4, f4 18; hencef, g are notG-R-weakly commuting of typeAf forR2 andGfg4, gf4, ff4 49 2Gf4, g4, f4 18; hencef, gare notG-R-weakly commuting of typeGf forR2.

Now, we rewriteDefinition 1.1onG-metric spaces setting.

Definition 2.10. LetSandT be two self-mappings of aG-metric spaceX, G. We say thatT andSsatisfy the E.A. property if there exists a sequencexnsuch thatTxnandSxnG- converge totfor somet∈X; that is, thanks toProposition 1.6,

n−→∞limGTxn, Tx_n, t lim

n−→∞GSxn, Sx_n, t 0. 2.13

Remark 2.11. In view of1.2andExample 1.3,Definition 1.1is equivalent toDefinition 2.10.

In the following example, we show that iffandg satisfy the E.A. property, then the pairf, gneed not beG-weakly commuting of typeAf.

Example 2.12. We return to Example 2.9. Let x_n 1 1/3n. We have lim_n−→∞fxn

lim_n−→∞1 2/3n 1, and lim_n−→∞gxn lim_n−→∞1 1/3n² 1, therefore, lim_n−→∞fxn lim_n−→∞gxn 1 ∈ 1,∞. Then f and g satisfy the E.A. property, but we know that the pairf, gis notG-weakly commuting of typeA_f.

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Following Matkowski see 32, let Φ be the set of all functions φ such that φ : 0,∞ → 0,∞be a nondecreasing function with lim_n_{→ ∞}φⁿt 0 for all t ∈ 0, ∞. If φ∈Φ, thenφis calledΦ-map. IfφisΦ-map, then it is easy to show that

1φt< tfor allt∈0, ∞, 2φ0 0.

2.2. Some Common Fixed Point Results We start this section with the following theorem.

Theorem 2.13. LetX, Gbe aG-metric space; suppose mappingsf, g:X → Xsatisfy the following condition:

1fandgbeG-weakly commuting of typeAf, 2fX⊆gX,

3gXisG-complete subspace ofX,

4Gfx, fy, fz≤φMx, y, z, for all x, y, z∈X, where

M x, y, z

max

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩ G

gx, g y

, gz , G

gx, fx, fx ,1

2G

gx, f y

, f y

, 1

2G

gx, fz, fz , G

g y

, f y

, G g

y

, fx, fx , G

g y

, fz, fz , G

gz, fz, fz , G

gz, fx, fx , G

gz, f y

, f y

⎫⎪

⎪⎪

⎬

⎪⎪

⎪⎭ .

2.14

Thenfandghave a unique common fixed point.

Proof. Let x0 ∈ X, and then choose x1 ∈ X such that fx0 gx1 and x2 ∈ X where fx1 gx2; then by induction we can define a sequenceyn∈Xas follows:

yn fxn gx_{n 1}, n∈N∪ {0} 2.15

We will show that the sequenceynisG-cauchy sequence:

G

y_n, y_{n 1}, y_{n 1} G

fxn, fxn 1, fxn 1

≤φMxn, x_{n 1}, x_{n 1}, 2.16 where

Mxn, x_{n 1}, x_{n 1} max

⎧⎪

⎪⎨

⎪⎪

⎩ G

gxn, gxn 1, gxn 1 , G

gxn, fxn, fxn , 1

2G

gxn, fx_{n 1}, fx_{n 1} , G

gx_{n 1}, fxn, fxn , G

gxn 1, fxn 1, fxn 1 ,

⎫⎪

⎪⎬

⎪⎪

⎭ max

⎧⎨

⎩G

y_n−1, y_n, y_n ,1

2G

y_n−1, y_{n 1}, y_{n 1} , G

yn, yn, yn

, G

yn, y_{n 1}, y_{n 1} ,

⎫⎬

⎭.

2.17

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We will have diﬀerent cases.

Case1: if Mxn, x_{n 1}, x_{n 1} Gyn, y_{n 1}, y_{n 1}, thenGyn, y_{n 1}, y_{n 1} ≤ φGyn, y_{n 1}, y_{n 1} < Gyn, y_{n 1}, y_{n 1}, which is contradiction.

Case 2: if Mxn, x_{n 1}, x_{n 1} 1/2Gyn−1, y_{n 1}, y_{n 1}, then in this case we have max{Gy_n−1, y_n, y_n, Gyn, y_{n 1}, y_{n 1}} < 1/2Gy_n−1, y_{n 1}, y_{n 1}, which implies that

G

y_n−1, yn, yn

G

yn, y_{n 1}, y_{n 1}

< G

y_n−1, y_{n 1}, y_{n 1}

, 2.18

but from G-metric propertyG5we have G

y_n−1, y_{n 1}, y_{n 1}

≤G

y_n−1, yn, yn

G

yn, y_{n 1}, y_{n 1}

. 2.19

Thus, from2.18and2.19we see that case2is impossible.

Then, we must have the case

Mxn, x_{n 1}, x_{n 1} G

y_n−1, y_n, y_n

. 2.20

Thus, forn∈N∪ {0}and from2.16we have, G

y_n, y_{n 1}, y_{n 1}

≤φ G

y_n−1, y_n, y_n

≤φ² G

y_n−2, y_n−1, y_n−1 ...

≤φⁿ G

y₀, y₁, y₁ . . .1.

2.21

Given >0, since lim_n−→∞φⁿGy0, y1, y1 0, andφ< , there is an integerno ∈ N, such that

φⁿ G

y0, y1, y1

< −φ, ∀n≥n0. 2.22

Hence, we have G

yn, y_{n 1}, y_{n 1}

≤φⁿ G

y0, y1, y1

< −φ. 2.23

Now form, n∈N;m > n, we claim that G

yn, ym, ym

< , ∀m≥n≥n0. 2.24

We will prove2.24by induction onm.

Inequality2.24holds formn 1, by using2.23and the fact that−φ< .

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Assume2.24holds formk. Formk 1, we have G

yn, y_{k 1}, y_{k 1}

≤G

yn, y_{n 1}, y_{n 1} G

y_{n 1}, y_{k 1}, y_{k 1}

< −φ φ G

y_n, y_k, y_k

< −φ φ .

2.25

By induction onm, we conclude that2.24holds for allm≥n≥n0.

Hence, the sequenceyn gx_{n 1}isG-cauchy sequence ingX; sincegX isG- complete, then there existst∈gXsuch that lim_n_{→ ∞}gxn tlim_n_{→ ∞}fxn.

Thus, there existsp∈Xsuch thatgp t, also limn→ ∞fxn gp.

We will show thatfp gp. Supposing thatfp/gp, then condition4implies that,Gfp, fp, fxn≤φMp, p, xn,where

M

p, p, x_n

max

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩ G

g p

, g p

, gxn , G

g p

, f p

,1 2G

g p

, f p

, 1

2G g

p

, fxn, fxn , G

g p

, f p

, G g

p , f

p , G

g p

, fxn, fxn , G

gxn, fxn, fxn , G

gxn, f p

, f p

, G

gxn, f p

, f p

⎫⎪

⎪⎪

⎬

⎪⎪

⎪⎭ . 2.26

Taking the limit asn → ∞and using the fact that the functionGis continuous we get

G f

p , f

p , g

p

≤φ

max

G g

p , f

p ,1

2G g

p , f

p

φ G

g p

, f p

, f

p

.

2.27

Therefore, G

f p

, f p

, g p

≤φ G

g p

, f p

, f

p

< G g

p , f

p

, 2.28

which is contradiction; hencefpgp.

SincefandgareG-weakly commuting of typeA_f, thenGfgp, ggp, ffp

≤Gfp, gp, fp 0.

Thus,ffp fgp gfp ggp; it follows thatft fgp gfp gt.

Finally, we will show thatt:fpis common fixed point offandg.

Supposing thatft /t, then G

ft, t, t G

ft, f p

, f p

≤φ M

t, p, p

, 2.29

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where

M t, p, p

max

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩ G

gt, g p

, g p

, G

gt, ft, ft ,1

2G gt, f

p , f

p , 1

2G gt, f

p , f

p , G

g p

, f p

, G g

p

, ft, ft , G

g p

, f p

, G g

p , f

p , G

g p

, ft, ft , G

g p

, f p

⎫⎪

⎪⎪

⎬

⎪⎪

⎪⎭ .

2.30

Sincegt ft, andgp fp, therefore2.30implies that

M t, p, p

max

G

ft, t, t ,1

2G

ft, t, t , G

t, ft, ft

. 2.31

Hence,2.29becomes G

ft, t, t

≤φ max

G

ft, t, t , G

t, ft, ft φ

G

t, ft, ft

< G

t, ft, ft

. 2.32

Similarly we get,

G

t, ft, ft

< G

ft, t, t

. 2.33

So,

G

ft, t, t

< G

ft, t, t

, 2.34

a contradiction which implies thattftgt. Thentis a common fixed point.

To prove uniqueness suppose we have u and vsuch that u /v,fu gu uand fvgvv; then condition4implies that

Gu, v, v≤φGv, u, u. 2.35

Therefore,

Gu, v, v≤φGv, u, u< Gv, u, u. 2.36

Similarly,Gv, u, u < Gu, v, v; thusGu, v, v < Gu, v, v, a contradiction which implies thatuv. Thentis a unique common fixed point offandg.

Now we give an example to support ourresult.

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Example 2.14. LetX 0,4/3, and defineG:X×X×X → 0,∞byGx, y, z max{|x− y|,|y−z|,|x−z|}andf, g:X → Xbyfx x³/8, gx x³/2 andφt 2/3t. Then,

agXisG-complete subspace ofX, bfX⊂gX,

cfandgareG-weakly commuting of typeAf, dfandgsatisfy condition4ofTheorem 2.13.

It is clear thataandbare satisfied.

To show c, as an easy calculation one can show that ∀x ∈ X; we have Gfgx, ggx, ffx max{3/64x⁹,63/4096x⁹,255/4096x⁹} ≤ 3/8x³ Gfx, gx, fx. Thenfandgare G-weakly commuting of typeAf.

To showd, forx, y, z∈Xwe have

G

fx, f y

, fz 1

8maxx³−y³,y³−z³,x³−z³

≤ 1

3maxx³−y³,y³−z³,x³−z³ 2

3 1

2maxx³−y³,y³−z³,x³−z³ φ

gx, g y

, gz

≤φ M

x, y, z .

2.37

Therefore, all hypotheses ofTheorem 2.13are satisfied andx0 unique common fixed point offandg.

Corollary 2.15. LetX, Gbe aG-metric space, and suppose mappingsf, g : X → X satisfy the following conditions:

1fandgbe G -weakly commuting of typeA_f, 2fX⊆gX,

4Gfx, fy, fz≤kMx, y, z, where

M x, y, z

max

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩ G

gx, g y

, gz , G

gx, fx, fx ,1

2G

gx, f y

, f y

, 1

2G

gx, fz, fz , G

g y

, f y

), G g

y

, fx, fx , G

g y

, fz, fz , G

gz, fz, fz , G

gz, fx, fx , G

gz, f y

, f y

⎫⎪

⎪⎪

⎬

⎪⎪

⎪⎭ ,

2.38 for allx, y, z∈X, wherek∈0,1; thenfandghave a unique common fixed point.

Proof. It suﬃces to takeφt ktinTheorem 2.13.

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Theorem 2.16. LetX, Gbe aG-metric space. Suppose the mappingsf, g:X → XareG-weakly commuting of typeA_gand satisfy the following condition:

1fandgsatisfy E.A. property, 2gXis closed subspace ofX,

3Gfx, fy, fz≤kMx, y, z, where

M x, y, z

max

⎧⎪

⎪⎨

⎪⎪

⎩ G

gx, fx, fx G

g y

, f y

G

gz, fz, fz ,

G

gx, f y

, f y

G g

y

, fx, fx G

gz, f y

, f

y

, G

gx, fz, fz G

g y

, fz, fz G

gz, fx, fx

⎫⎪

⎪⎬

⎪⎪

⎭ 2.39

for allx, y, z∈X, wherek∈0,1/3; thenfandghave a unique common fixed point.

Proof. Since f and g satisfy E.A. property, there exists in X a sequence xn satisfying limn→ ∞fxn limn→ ∞gxn tfor somet∈X.

Since gXis closed subspace ofXand lim_{n→ ∞}gxn t, there existsp∈Xsuch that gp t, also lim_n_{→ ∞}fxn gp.

We will show thatfp gpsupposing thatfp/gp, then condition3implies that

G f

p , f

p , fxn

≤kM p, p, xn

, 2.40

where, M

p, p, xn

max

⎧⎪

⎪⎨

⎪⎪

⎩ G

g p

, f p

G g

p , f

p G

gxn, fxn, fxn ,

G g

p , f

p G

g p

, f p

G

gxn, f p

, f

p

, G

g p

, fxn, fxn G

g p

, fxn, fxn G

gxn, f p

, f

p

⎫⎪

⎪⎬

⎪⎪

⎭. 2.41

Taking the limit asn → ∞and using the fact that the functionGis continuous, we get G

f p

, f p

, g p

≤3kG g

p , f

p

, 2.42

which is contradiction sincek∈0,1/3, sofpgp. SincefandgareG-weakly commuting of typeAg, then

G gf

p , gg

p , ff

p

≤G g

p , f

p , g

p

0. 2.43 Therefore,fgp ffp gfp ggp; then

ft:fg p

gf p

gt. 2.44

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Finally, we will show thattfpis common fixed point offandg.

Supposing thatft /t, then G

ft, t, t G

ft, f p

, f p

≤kM t, p, p

, 2.45

where

M t, p, p

max

⎧⎪

⎪⎨

⎪⎪

⎩ G

gt, ft, ft G

g p

, f p

G g

p , f

p

, G

gt, f p

, f p

G g

p

, ft, ft G

g p

, f p

, f

p

, G

gt, f p

, f p

G g

p , f

p G

g p

, ft, ft

⎫⎪

⎪⎬

⎪⎪

⎭. 2.46

Butft gtandfp gp. Thus, G

ft, t, t

≤kmax G

t, ft, ft ,

G

ft, t, t G

t, ft, ft

< k G

ft, t, t G

t, ft, ft

. 2.47

Hence,

G

ft, t, t

≤ k

1−k

G

t, ft, ft

. 2.48

Adjusting similarly, we get

G

t, ft, ft

≤ k 1−k

G

ft, t, t

. 2.49

Therefore,

G ft, t, t

≤ k

1−k 2

G ft, t, t

, 2.50

a contradiction which implies thatfttfp, butgtftt. Thentis a common fixed point offandg.

To prove uniqueness, suppose we have uand vsuch that u /v,fu gu uand fvgvv; then

Gu, v, v G

fu, fv, fv

≤k{Gu, v, v Gv, u, u}. 2.51

Hence,Gu, v, v≤k/1−kGv, u, u.

Similarly,Gv, u, u≤k/1−kGu, v, v.

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Therefore,Gu, v, v≤k/1−k²Gu, v, va contradiction which implies thatu v. Thentis a unique common fixed point offandg.

Now we give an example to support our result.

Example 2.17. LetX 0,3/4, defineG:X×X×X → 0,∞by

Gx, y, z max{|x−y|,|y−z|,|x−z|}and letf, g:X → Xbyfx x²/5,gx x². Then,

agXis closed subspace ofX,

bfandgare G-weakly commuting of typeA_g, cfandgsatisfy E.A. property.

dfandgsatisfy condition4fork 1/4.

Proof. ais obvious.

To show b, as an easy calculation one can show that for allx ∈ X; we have Ggfx, ggx, ffx max{4/125x⁴,24/25x⁴,124/125x⁴} ≤ 4/5x² Gfx, gx, fx. ThenfandgareG-weakly commuting of typeA_g.

To showc, if we consider the sequence{xn}{1/2n}, thenfxn → 0 andgxn → 0 asn → ∞. Thus,fandgsatisfy the E.A. property.

To showd, forx, y, z∈Xwe have

x² 5 −y²

5 ≤ x²

5 y²

5 ,

y²

5 − z² 5

≤ y²

5 z²

5 ,

x²

5 − z² 5

≤ x²

5 z²

5 .

2.52

Then

G

fx, f y

, fz

max

x² 5 −y²

5 ,

y²

5 −z² 5

,

x²

5 −z² 5

≤ x² 5

y² 5

z² 5 1

4 4x²

5 4y²

5 4z²

5

1 4

G

gx, fx, fx G

g y

, f y

, f y G

gz, fz, fz

≤kM x, y, z

.

2.53

Therefore, all hypotheses ofTheorem 2.16are satisfied fork1/4 andx0, a unique common fixed point offandg.

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Theorem 2.18. LetX, Gbe aG-metric space, and suppose mappingsf, g:X → XbeG-R-weakly commuting of typeA_f. Suppose that there exists a mappingψ :X → 0,∞such that

1fX⊂gX,

3Ggx, fx, fx< ψgx−ψfx, for allx∈X,

G

fx, f y

, fz

<max G

gx, g y

, gz , G

gx, fx, g y

, G

gz, fz, fx , G

g y

, f y

, fz

, 2.54

for allx, y, z∈X; thenfandghave a unique common fixed point.

Proof. Letx₀∈X, and then choosex₁∈Xsuch thatfx0 gx1andx₂ ∈Xwherefx1

gx2; then by induction we can define a sequenceyn∈Xas follows:

ynfxn gxn 1, n∈N∪ {0}. 2.55

We will show that the sequenceynisG-cauchy sequence:

G

gxn, gx_{n 1}, gx_{n 1} G

gxn, fxn, fxn

< ψ gxn

−ψ fxn ψ

gxn

−ψ

gxn 1 .

2.56

Consideranψgxn, n1,2,3,4, . . ., then 0≤G

gxn, gxn 1, gxn 1

< an−a_{n 1}. 2.57 Thus, the sequence an is nonincreasing and bounded below by 0; hence anis convergent sequence.

On the other hand we have, fromG5and2.57, that form, n∈N;m > n

G

gxn, gxn m, gxn m

≤^{n m−1}

jn

G g

x_j , g

x_{j 1} , g

x_{j 1}

<

n m−1

jn

a_j−a_{j 1}

Telescoping sum a_n−a_{n m}.

2.58

Therefore, the sequencegxnis G-cauchy sequence ingX.

SincegXisG-complete subspace, then there existst∈gXsuch that limn→ ∞gxn

t; havingt∈gXthere existsp∈Xsuch thatgp t, also lim_n_{→ ∞}fxn gp t.

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We will show thatfp gp; supposing thatfp/gp, then condition4implies that

G f

p , f

p

, fxn

<max G

g p

, g p

, gxn , G

g p

, f p

, g p

, G

g p

, f p

, fxn , G

gxn, fxn, f p

. 2.59

Taking the limit asn → ∞, we get

G f

p , f

p , g

p

<max

⎧⎪

⎪⎨

⎪⎪

⎩ G

g p

, g p

, G

g p

, f p

, g p

, G

g p

, g p

, f p

⎫⎪

⎪⎬

⎪⎪

⎭, 2.60

hence,

G f

p , f

p , g

p

< G g

p , g

p , f

p

. 2.61

Adjusting similarly, we get G

g p

, g p

, f p

< G f

p , f

p , g

p

2.62

Therefore, G

f p

, f p

, g p

< G g

p , f

p , g

p

< G f

p , f

p , g

p

. 2.63

Thus, a contradiction impliesfpgp.

SincefandgareG-weakly commuting of typeAf, then G

f g

p , g

g p

, f f

p

≤G f

p , g

p , f

p

0. 2.64 Thus,ffp fgp gfp ggp, thenft fgp gfp gt.

Finally, we will show thattfpis common fixed point offandg.

Suppose thatft /t, so

G

ft, t, t G

ft, f p

, f p

<max

⎧⎪

⎪⎨

⎪⎪

⎩ G

gt, g p

, g p

, G

gt, ft, g p

, G

g p

, f p

⎫⎪

⎪⎬

⎪⎪

⎭. 2.65

Sincegp fpandgt ft, therefore2.65implies that G

ft, t, t

< G

ft, ft, t

. 2.66

Similarly, we haveGft, ft, t< Gft, t, t.

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A contradiction implies thatftfpt. Thentis a common fixed point.

To prove uniqueness suppose we haveuandvsuch thatu /vwherefuguuand fvgvv; then as an easy calculation one can get

Gu, v, v< Gv, u, u. 2.67

Similarly,Gv, u, u< Gu, v, v, a contradiction which implies thatuv. Then,tis a unique common fixed point offandg.

Now we give an example to support our result.

Example 2.19. LetX 1,∞,ψ : X −→ 0,∞such thatψt 3t, t ∈ X and Gx, y, z max{|x−y|,|y−z|,|z−x|}. Definef, g:X−→Xbyfx 2x−1 andgx 3x−2.

Then,

afX⊂gX,

bgXisG-complete subspace ofX,

cGgx, fx, fx< ψgx−ψfx, for allx∈X, dfandgsatisfy condition4ofTheorem 2.18.

Then as an easy calculation one can see thatGfgx, ggx, ffx max{2x− 2,5x−5,3x−3} 5x−5 ≤ Rx−1 RGfx, gx, fx, forR ≥ 5, thenf andg are G-R-weakly commuting of typeAf.

Also we see thatfX⊂gXandgXis G-complete subspace ofX.

To provec, for allx∈Xwe see that G

gx, fx, fx

x−1≤3x−3ψ

gx

−ψ

fx

. 2.68

To proved, for allx, y, z∈Xwe have G

fx, fy, fz

2 maxx−y,y−z,|x−z|

≤3 maxx−y,y−z,|x−z|

G

gx, gy, gz

≤max G

gx, g y

, gz , G

gx, fx, g y

, G

gz, fz, fx , G

g y

, f y

, fz .

2.69

Therefore, all hypotheses of the previous theorem are satisfied and x 1 a unique common fixed point offandg.

Note that the main result of Mustafa33is not applicable in this case. Indeed, for yz1 andx3,

G

f3, f1, f1

4>2kkG3,1,1 ∀k∈0,1. 2.70

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Also, the Banach principle34is not applicable. Indeed, fordx, y |x−y|for all x, y∈Xwe have forx /y

d

fx, f y

2x−y> kx−y ∀k∈0,1. 2.71 Corollary 2.20. Theorems 2.13,2.16, and 2.18 remain true if we replace, respectively, G-weakly commuting of type A_f, G-weakly commuting of type A_g, weakly compatible and G-R-weakly commuting of typeAf by any one of them (retaining the rest of hypothesis).

Corollary 2.21. Some corollaries could be derived from Theorems2.13,2.16, and2.18by takingzy orgId_X.

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