Common fixed point theorems for non compatible self maps in generalized metric spaces

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R E S E A R C H

Open Access

Common ﬁxed point theorems for

non-compatible self-maps in generalized

metric spaces

Zhongzhi Yang

*

*_{Correspondence:} [email protected] Accounting School, Zhejiang University of Finance and Economics, Hangzhou, 310018, China

Abstract

By usingR-weak commutativity of type (Ag) and non-compatible conditions of

self-mapping pairs in generalized metric space, without the conditions for the completeness of space and the continuity of mappings, we establish some new common ﬁxed point theorems for two self-mappings. Our results diﬀer from other results already known. An example is provided to support our new result.

MSC: 47H10; 54H25; 54E50

Keywords: generalized metric space;R-weakly commuting mappings of type (Ag); non-compatible mapping pairs; common ﬁxed point

1 Introduction and preliminaries

In , Jungck [] proved a common fixed point theorem of commuting mappings in a metric space. In , Sessa [] introduced the concept of weakly commuting mappings, which is a generalization of the concept of commuting mappings, and he has proved some fixed point theorems for weakly commuting mappings. In , Jungck [] introduced more generalized commuting mappings, called compatible mappings, which are more general than commuting and weakly commuting mappings. These concepts have been useful for obtaining more comprehensive fixed point theorems. In , Dhage [] intro-duced the concept ofD-metric space. Recently, Mustafa and Sims [] have shown that most of the results concerning Dhage’sD-metric spaces are invalid. Therefore, they in-troduced an improved version of the generalized metric space structure and called it a

G-metric space [].

Based on the notion of generalized metric spaces, Mustafaet al.[–], Aydiet al.[], Aydi [], Gajié and Stojakovié [], Zhou and Gu [] obtained some fixed point results for mappings satisfying different contractive conditions. Shatanawi [] obtained some fixed point results for-maps inG-metric spaces. Chughet al.[] obtained some fixed point results for maps satisfying propertyPinG-metric spaces. In , Manroet al.[] obtained some fixed point results for expansion mappings inG-metric spaces.

The study of common fixed point problems inG-metric spaces was initiated by Abbas and Rhoades []. Subsequently, many authors obtained many common fixed point the-orems for the mappings satisfying different contractive conditions; see [–] for more details. Recently, some authors have used the (E.A) property in generalized metric space

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to prove common ﬁxed point results, such as Abbaset al.[], Mustafaet al.[], Longet al.[], Gu and Yin [], Gu and Shatanawi [].

Very recently, Jleli and Samet [] and Sametet al.[] noticed that some fixed point theorems in the context of aG-metric space can be concluded by some existing results in the setting of a (quasi-)metric space. In fact, if the contraction condition of the fixed point theorem on aG-metric space can be reduced to two variables instead of three variables, then one can construct an equivalent fixed point theorem in the setting of a usual metric space. More precisely, in [, ], the authors noticed thatd(x,y) =G(x,y,y) forms a quasi-metric. Therefore, if one can transform the contraction condition of existence results in aG-metric space in such terms,G(x,y,y), then the related fixed point results become the well-known fixed point results in the context of a quasi-metric space.

Now we give basic deﬁnitions and some basic results [], which are helpful for improving our main results.

Deﬁnition .[] LetXbe a nonempty set, and letG:X×X×X−→R+be a function satisfying the following axioms:

(G) G(x,y,z) = ifx=y=z;

(G) <G(x,x,y)for allx,y∈Xwithx=y;

(G) G(x,x,y)≤G(x,y,z)for allx,y,z∈Xwithz=y;

(G) G(x,y,z) =G(x,z,y) =G(y,z,x) =· · · (symmetry in all three variables); and (G) G(x,y,z)≤G(x,a,a) +G(a,y,z)for allx,y,z,a∈X(rectangle inequality).

Then the functionGis called a generalized metric, or more speciﬁcally, aG-metric on

Xand the pair (X,G) is called aG-metric space.

It is well known that the functionG(x,y,z) on theG-metric spaceXis jointly continuous in all three of its variables, andG(x,y,z) =  if and only ifx=y=z(see []).

Deﬁnition .[] Let (X,G) be aG-metric space,{xn} ⊂Xbe a sequence. Then the

se-quence{xn}is called:

(i) aG-convergent sequence if, for any> , there is anx∈Xand ann∈N, such that for alln,m≥n,G(xn,xm,x) <;i.e.iflimn,m→∞G(xn,xm,x) = ;

(ii) aG-Cauchy sequence if, for any> , there is ann∈N(the set of natural numbers) such that for alln,m,l≥n,G(xn,xm,xl) <;i.e.ifG(xn,xm,xl)→asn,m,l→ ∞.

AG-metric space (X,G) is said to beG-complete if everyG-Cauchy sequence in (X,G) isG-convergent inX. It is well known that{xn}isG-convergent tox∈X.

Proposition . [] Let (X,G)be a G-metric space, then the following statements are equivalent:

() {xn}isG-convergent tox;

() G(xn,xn,x)→asn→ ∞;

() G(xn,x,x)→asn→ ∞;

() G(xn,xm,x)→asn,m→ ∞.

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Deﬁnition .[] Let (X,G) and (X,G) beG-metric space, andf : (X,G)→(X,G) be a function. Thenf is said to beG-continuous at a pointa∈Xif and only if for every> , there isδ>  such thatx,y∈XandG(a,x,y) <δimpliesG(f(a),f(x),f(y)) <. A function

f isG-continuous atXif and only if it isG-continuous at alla∈X.

Proposition .[] Let(X,G)and(X,G)be a G-metric space.Then f :X→X is G-continuous at x∈X if and only if it is G-sequentially continuous at x,that is,whenever {xn}is G-convergent to x,{f(xn)}is G-convergent to f(x).

Deﬁnition .[] The self-mappingsf andgof aG-metric space (X,G) are said to be compatible iflimn→∞G(fgxn,gfxn,gfxn) =  andlimn→∞G(gfxn,fgxn,fgxn) = , whenever

{xn}is a sequence inXsuch thatlimn→∞fxn=limn→∞gxn=t, for somet∈X.

In , Manroet al.[] introduced the concepts of weakly commuting,R-weakly com-muting mappings, andR-weakly commuting mappings of type (P), (Af), and (Ag) inG

-metric space as follows.

Deﬁnition .[] A pair of self-mappings (f,g) of aG-metric space are said to be weakly commuting if

G(fgx,gfx,gfx)≤G(fx,gx,gx), ∀x∈X.

Deﬁnition .[] A pair of self-mappings (f,g) of aG-metric space are said to beR -weakly commuting if there exists some positive real numberRsuch that

G(fgx,gfx,gfx)≤RG(fx,gx,gx), ∀x∈X.

Remark . IfR≤, thenR-weakly commuting mappings are weakly commuting.

Deﬁnition .[] A pair of self-mappings (f,g) of aG-metric space (X,G) are said to be

(a) R-weakly commuting mappings of type (Af) if there exists some positive real

numberRsuch thatG(fgx,ggx,ggx)≤RG(fx,gx,gx), for allxinX.

(b) R-weakly commuting mappings of type(Ag)if there exists some positive real

numberRsuch thatG(gfx,ﬀx,ﬀx)≤RG(gx,fx,fx), for allxinX.

(c) R-weakly commuting mappings of type(P)if there exists some positive real number Rsuch thatG(ﬀx,ggx,ggx)≤RG(fx,gx,gx), for allxinX.

Proposition .[] Every G-metric on X deﬁnes a metric dGon X by

dG(x,y) =G(x,y,y) +G(y,x,x), ∀x,y∈X.

For a symmetric G-metric space,one obtains

dG(x,y) = G(x,y,y), ∀x,y∈X.

However,if G is not symmetric,then the following inequality holds:



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It is also obvious that

G(x,x,y)≤G(x,y,y).

2 Main results

Theorem . Let(X,G)be a G-metric space and(f,g)be a pair of non-compatible self-mappings with fX⊂gX(here fX denotes the closure of fX).Assume the following conditions are satisﬁed:

G(fx,fy,fz)

≤αmax

G(gx,gy,gz),G(fx,gx,gx)+_G(fy,gy,gy),G(fy,gy,gy)+_G(fz,gz,gz),

G(fz,gz,gz)+G(fx,gx,gx)

 ,

G(fx,gy,gz)+G(gx,fy,gz)

 ,

G(gx,fy,gz)+G(gx,gy,fz) 

(.)

for all x,y,z∈X.Hereα∈[, ).If(f,g)are a pair of R-weakly commuting mappings of type(Ag),then f and g have a unique common ﬁxed point(say t)and both f and g are not

G-continuous at t.

Proof Sincef andgare non-compatible mappings, there exists a sequence{xn} ⊂X, such

that

lim

n→∞fxn=nlim→∞gxn=t, t∈X,

but eitherlimn→∞G(fgxn,gfxn,gfxn) orlimn→∞G(gfxn,fgxn,fgxn) does not exist or exists

and is diﬀerent from zero. Sincet∈fX⊂gX, there must exist au∈Xsatisfyingt=gu. We can assert thatfu=gu. If not, from condition (.), we get

G(fu,fxn,fxn)

≤αmax

G(gu,gxn,gxn),G(fu,gu,gu)+G_(fxn,gxn,gxn),G(fxn,gxn,gxn)+_G(fxn,gxn,gxn), G(fxn,gxn,gxn)+G(fu,gu,gu)

 ,

G(fu,gxn,gxn)+G(gu,fxn,gxn)

 ,

G(gu,fxn,gxn)+G(gu,gxn,fxn) 

.

Lettingn→ ∞at both sides, we obtain

G(fu,gu,gu)

≤αmax

G(gu,gu,gu),G(fu,gu,gu)+_G(gu,gu,gu),G(fu,gu,gu)+_G(fu,gu,gu),

G(fu,gu,gu)+G(fu,gu,gu)

 ,

G(fu,gu,gu)+G(gu,fu,gu)

 ,

G(gu,fu,gu)+G(gu,gu,fu) 

=αG(fu,gu,gu).

Sinceα∈[, ), we getG(fu,gu,gu) = , and sofu=gu.

Since (f,g) are a pair ofR-weakly commuting mappings of type (Ag), we have

G(gfu,ﬀu,ﬀu)≤RG(gu,fu,fu) = .

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Next, we proveffu=fu. In fact, ifffu=fu, from condition (.),fu=guandffu=gfu, we have

G(fu,ﬀu,ﬀu)

≤αmax

G(gu,gfu,gfu),G(fu,gu,gu)+_G(ffu,gfu,gfu),G(ffu,gfu,gfu)+_G(ffu,gfu,gfu),

G(ﬀu,gfu,gfu)+G(fu,gu,gu)

 ,

G(fu,gfu,gfu)+G(gu,ﬀu,gfu)

 ,

G(gu,ﬀu,gfu)+G(gu,gfu,ﬀu) 

=αG(fu,ﬀu,ﬀu).

Fromα∈[, ) we havefu=ffu, which implies thatfu=ffu=gfu, and sot=fuis a common fixed point off andg.

Next we prove that the common ﬁxed pointtis unique.

Actually, supposewis also a common ﬁxed point off andgandw=t, then using the condition (.), we have

G(t,w,w) =G(ft,fw,fw)≤αG(t,w,w),

which implies thatt=w, so that uniqueness is proved.

Now, we prove thatf andgare notG-continuous att. In fact, iff isG-continuous att, we consider the sequence{xn}; then we have

lim

n→∞ﬀxn=ft=t, nlim→∞fgxn=ft=t.

Sincef andgareR-weakly commuting mappings of type (Ag), we get

G(gfxn,ﬀxn,ﬀxn)≤RG(gxn,fxn,fxn),

so that we have

lim

n→∞G(gfxn,ﬀxn,ﬀxn) = ,

it follows that

lim

n→∞gfxn=ft.

Hence, we can get

lim

n→∞G(fgxn,gfxn,gfxn) =G(ft,ft,ft) = 

and

lim

n→∞G(gfxn,fgxn,fgxn) =G(ft,ft,ft) = .

This contradicts withf andgbeing non-compatible, sof is notG-continuous att. IfgisG-continuous att, then we have

lim

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Sincef andgareR-weakly commuting mappings of type (Ag), we get

G(gfxn,ﬀxn,ﬀxn)≤RG(gxn,fxn,fxn),

so that we have

lim

n→∞G(gfxn,ﬀxn,ﬀxn) = ,

and it follows that

lim

n→∞ﬀxn=gt=ft=t.

This contradicts with f being not G-continuous at t, which implies that g is not G

-continuous att. This completes the proof.

Next, we give an example to support Theorem ..

Example . LetX= [, ] be aG-metric space with

G(x,y,z) =|x–y|+|y–z|+|z–x|.

We deﬁne mappingsf andgonXby

fx=

, ifx=  or  <x≤,

, if  <x≤, gx= ⎧ ⎪ ⎨ ⎪ ⎩

, ifx= , , if  <x≤,

x+

 , if  <x≤.

Clearly, from the above functions we know that fX⊂gX, and the pair (f,g) are non-compatible self-maps. To see thatf andgare non-compatible, consider a sequence{xn}=

{ +_n}. We havefxn→,gxn→,fgxn→ andgfxn→. Thus

lim

n→∞G(gfxn,fgxn,fgxn) =G(, , ) = = .

On the other hand, there existsR=  such thatG(gfx,ﬀx,ﬀx)≤RG(gx,fx,fx) for allxinX, that is, the pair (f,g) areR-weakly commuting mappings of type (Ag).

Now we prove that the mappingsf andgsatisfy the condition (.) of Theorem . with

α=_. For this, let

M(x,y,z) =max

G(gx,gy,gz),G(fx,gx,gx)+_G(fy,gy,gy),G(fy,gy,gy)+_G(fz,gz,gz),

G(fz,gz,gz)+G(fx,gx,gx)

 ,

G(fx,gy,gz)+G(gx,fy,gz)

 ,

G(gx,fy,gz)+G(gx,gy,fz) 

.

We consider the following cases:

Case () Ifx,y,z∈ {} ∪(, ], then we haveG(fx,fy,fz) =G(, , ) = , and hence (.) is obviously satisﬁed.

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Case () Ifx=y= ,z∈(, ], then we haveG(fx,fy,fz) =G(, , ) =  andG(gx,gy,gz) =

G(, , ) = . Thus we obtain

G(fx,fy,fz) =  < · =



G(gx,gy,gz)≤ 

M(x,y,z).

Case () Ifx=z= ,y∈(, ], then we haveG(fx,fy,fz) =G(, , ) =  andG(gx,gy,gz) =

G(, , ) = . Thus we obtain

G(fx,fy,fz) =  < · =



G(gx,gy,gz) 

≤M(x,y,z).

Case () Ify=z= ,x∈(, ], then we haveG(fx,fy,fz) =G(, , ) =  andG(gx,gy,gz) =

G(, , ) = . Thus we obtain

G(fx,fy,fz) =  < · =



M(x,y,z).

Case () Ifx= ,y,z∈(, ], then we haveG(fx,fy,fz) =G(, , ) =  andG(gx,gy,gz) =

G(, , ) = . Thus we obtain

G(fx,fy,fz) =  < · =



M(x,y,z).

Case () Ify= ,x,z∈(, ], then we haveG(fx,fy,fz) =G(, , ) =  andG(gx,gy,gz) =

G(, , ) = . Thus we obtain

G(fx,fy,fz) =  < · =



M(x,y,z).

Case () Ifz= ,x,y∈(, ], then we haveG(fx,fy,fz) =G(, , ) =  andG(gx,gy,gz) =

G(, , ) = . Thus we obtain

G(fx,fy,fz) =  < · =



M(x,y,z).

Case () Ifx,y∈(, ],z∈(, ], then we haveG(fx,fy,fz) =G(, , ) =  and

G(fx,gx,gx) +G(fy,gy,gy)

 =

G(, , ) +G(, , )

 = .

Thus we obtain

G(fx,fy,fz) =  < · =

 ·

 ≤



M(x,y,z).

Case () Ifx,z∈(, ],y∈(, ], then we haveG(fx,fy,fz) =G(, , ) =  and

G(fz,gz,gz) +G(fx,gx,gx)

 =

G(, , ) +G(, , )

 = .

Thus we obtain

G(fx,fy,fz) =  < · =

 ·

 ≤



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Case () Ify,z∈(, ],x∈(, ], then we haveG(fx,fy,fz) =G(, , ) =  and

G(fy,gy,gy) +G(fz,gz,gz)

 =

G(, , ) +G(, , )

 = .

Thus we obtain

G(fx,fy,fz) =  < · =

 ·

 ≤



M(x,y,z).

Case () Ifx,y∈(, ],z∈(, ], then we haveG(fx,fy,fz) =G(, , ) =  and

G(fx,gy,gz) +G(gx,fy,gz)

 =

G(,y+_ , ) +G(x+_ , , )

 = .

Thus we obtain

G(fx,fy,fz) =  < · =

 ·

G(fx,gy,gz) +G(gx,fy,gz)

 ≤



M(x,y,z).

Case () Ifx,z∈(, ],y∈(, ], then we haveG(fx,fy,fz) =G(, , ) =  and

 =

G(,x+_ ,x+_ ) +G(, , )

 =  +

x+ 

 ∈(, ].

Thus we obtain

G(fx,fy,fz) =  = · <

 ·

 ≤



M(x,y,z).

Case () Ify,z∈(, ],x∈(, ], then we haveG(fx,fy,fz) =G(, , ) =  and

 =

G(, , ) +G(,y+_ ,y+_ )

 =  +

y+ 

 ∈(, ].

Thus we obtain

G(fx,fy,fz) =  = · <

 ≤



M(x,y,z).

Case () Ifx= ,y∈(, ],z∈(, ], then we haveG(fx,fy,fz) =G(, , ) =  and

 =

G(, , ) +G(, , )

 = .

Thus we obtain

G(fx,fy,fz) =  = ·

 ≤



M(x,y,z).

Case () Ifx= ,y∈(, ],z∈(, ], then we haveG(fx,fy,fz) =G(, , ) =  and

 =

G(, , ) +G(, , )

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Thus we obtain

G(fx,fy,fz) =  = ·

 ≤



M(x,y,z).

Case () Ifx∈(, ],y= ,z∈(, ], then we haveG(fx,fy,fz) =G(, , ) =  and

 =

G(, , ) +G(, , )

 = .

Thus we obtain

G(fx,fy,fz) =  = ·

 ≤



M(x,y,z).

Case () Ifx∈(, ],y= ,z∈(, ], then we haveG(fx,fy,fz) =G(, , ) =  and

 =

G(, , ) +G(, , )

 = .

Thus we obtain

G(fx,fy,fz) =  = ·

 ≤



M(x,y,z).

Case () Ifx∈(, ],y∈(, ],z= , then we haveG(fx,fy,fz) =G(, , ) =  and

 =

G(, , ) +G(, , )

 = .

Thus we obtain

G(fx,fy,fz) =  = ·

 ≤



M(x,y,z).

Case () Ifx∈(, ],y∈(, ],z= , then we haveG(fx,fy,fz) =G(, , ) =  and

 =

G(, , ) +G(, , )

 = .

Thus we obtain

G(fx,fy,fz) =  = ·

 ≤



M(x,y,z).

Then in all the above cases, the mappingsf andgsatisfy the condition (.) of Theorem . withα=_, so that all the conditions of Theorem . are satisﬁed. Moreover,  is the unique common ﬁxed point off andg.

Theorem . Let(X,G)be a G-metric space and let(f,g)be a pair of non-compatible self-mappings such that

G(fx,fy,fz)

≤αmax

G(gx,gy,gz),G(fx,gy,gz),G(fy,gy,gz),G(fz,gx,gy) +G(fy,gx,gz)

 (.)

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(i) for any sequence{xn}that satisﬁes the conditionlimn→∞fxn=limn→∞gxn=t∈X,

we havelimn→∞ﬀxn=ftandlimn→∞gfxn=gt;

(ii) (f,g)are a pair ofR-weakly commuting mappings of type(Ag).

Then f and g have a unique common ﬁxed point in X.

Proof Sincef andgare non-compatible mappings, there exists a sequence{xn} ⊂X, such

that

lim

n→∞fxn=nlim→∞gxn=t, t∈X,

but, eitherlimn→∞G(fgxn,gfxn,gfxn) orlimn→∞G(gfxn,fgxn,fgxn) does not exist or exists

and is diﬀerent from zero. From condition (i), we have

lim

n→∞ﬀxn=ft, nlim→∞gfxn=gt.

Since (f,g) are a pair ofR-weakly commuting mappings of type (Ag), we can get

G(gfxn,ﬀxn,ﬀxn)≤RG(gxn,fxn,fxn).

Lettingn→ ∞, we haveG(gt,ft,ft)≤RG(t,t,t) = . Thus we knowft=gt. From the condition (.), we have

G(ft,fxn,fxn)

≤αmax

G(gt,gxn,gxn),G(ft,gxn,gxn),G(fxn,gxn,gxn),

G(fxn,gt,gxn) +G(fxn,gt,gxn)

 .

Lettingn→ ∞, we get

G(ft,t,t)≤αmax

G(gt,t,t),G(ft,t,t),G(t,t,t),G(t,gt,t) +G(t,gt,t) 

=αG(ft,t,t).

Sinceα∈[, ), therefore,G(ft,t,t) = . Thus, we gett=ft=gt.

Supposewis another ﬁxed point off andgandw=t. Lettingx=t,y=w,z=wunder the condition (.), we obtain

G(ft,fw,fw)

≤αmax

G(gt,gw,gw),G(ft,gw,gw),G(fw,gw,gw),G(fw,gt,gw) +G(fw,gt,gw) 

=αmax

G(t,w,w),G(t,w,w),G(w,w,w),G(w,t,w) +G(w,t,w) 

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Sinceα∈[, ), we ﬁnd thatG(t,w,w) = . Therefore, we havet=w. So, the common ﬁxed point off andgis unique. Thus, we complete the proof.

Competing interests

The author declares that they have no competing interests.

Acknowledgements

The author is grateful to the editor and the reviewer for suggestions, which improved the contents of the article.

Received: 21 February 2014 Accepted: 17 June 2014 Published:23 Jul 2014 References

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10.1186/1029-242X-2014-275