COMMUTING MAPPINGS IN D -METRIC SPACES

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ON COMMON FIXED POINTS OF PAIRS OF A SINGLE AND A MULTIVALUED COINCIDENTALLY

COMMUTING MAPPINGS IN D -METRIC SPACES

B. C. DHAGE, A. JENNIFER ASHA, and S. M. KANG Received 10 December 2002 and in revised form 15 February 2003

The present paper studies some common fixed-point theorems for pairs of a single-valued and a multivalued coincidentally commuting mappings inD-metric spaces satisfying a certain generalized contraction condition. Our result general- izes more than a dozen known fixed-point theorems inD-metric spaces including those of Dhage (2000) and Rhoades (1996).

2000 Mathematics Subject Classification: 47H10, 54H25.

1. Introduction. The concept of aD-metric space introduced by the first author in [1] is as follows. A nonempty set, together with a functionρ:X×X× X→[0,∞), is called aD-metric spaceand denoted by(X,ρ)if the functionρ, called aD-metriconX, satisfies the following properties:

(i) ρ(x,y,z)=0x=y=z(coincidence),

(ii) ρ(x,y,z)=0=ρ(p{x,y,z})(symmetry), wherepis a permutation, (iii) ρ(x,y,z)≤ρ(x,y,a)+ρ(x,a,z)+ρ(a,y,z)for allx,y,z,a∈X(tetra-

hedral inequality).

It is known that theD-metricρin a continuous function onX³in the topol- ogy ofD-metric convergence is Hausdorff. The details of aD-metric space and its topological properties appear in Dhage [8]. Some specific examples of a D-metric space are presented in Dhage [2].

A sequence {x_n} ⊂X is calledconvergent and converges to a point x if lim_m,nρ(x_m,x_n,x) =0. Again a sequence {x_n} ⊂X is called D-Cauchy if limm,n,pρ(xm,xn,xp)=0. A completeD-metric spaceXis one in which every D-Cauchy sequence converges to a point inX. A subsetSof aD-metric space Xis calledboundedif there exists a constantk >0 such thatρ(x,y,z)≤kfor allx,y,z∈Xand the constantkis called aD-boundofS. The smallest among all suchD-boundskofSis called thediameterofXand it is denoted byδ(S).

Let 2^X and CB(X)denote the classes of nonempty closed and nonempty, closed, bounded subsets ofX, respectively. A correspondenceF:X→2^X is called a multivalued mappingon a D-metric space X, and a point u∈X is called afixed pointofFifu∈Fu.

In [3], the first author has defined a notion of the generalized or Kasusai D-metric onX. Letκ:(CB(X))³→[0,∞)be a function defined by

κ(A,B,C)=inf

>0|A∪B⊂N(c,), B∪C⊂N(A,), C∪A⊂N(B,) , (1.1) whereN(A,)= ∪a∈AN(a,), N(a,)= {x∈N^∗(a,)|ρ(a,x,y) < for all y∈N^∗(a,)}, andN^∗(a,)= {x∈X|ρ(a,x,x) < }.

The definition (1.1) is equivalent to κ(A,B,C)=max

sup

a∈A, b∈BD(a,b,c), sup

b∈B, c∈CD(b,c,A), sup

c∈C, a∈AD(c,a,B)

, (1.2) whereD(a,b,c)=inf{ρ(a,b,c)|c∈C}.

Define

D(A,B,C)=inf

ρ(a,b,c)|a∈A, b∈B, c∈C ,

δ(A,B,C)=supρ(a,b,c)|a∈A, b∈B, c∈C. (1.3) Notice thatDandδare continuous functions on(CB(X))³and satisfy

D(A,B,C)≤κ(A,B,C)≤δ(A,B,C). (1.4) A multivalued mappingF:X→CB(X)is calledcontinuousif

limm,nρx_m,x_n,x

=0 ⇒κFx_m,Fx_n,Fx

=0. (1.5)

In [3], the first author has proved some fixed-point theorem for multivalued contraction mappings inD-metric spaces, and in [5] he has proved some com- mon fixed-point theorems for coincidentally commuting single-valued map- pings inD-metric spaces satisfying a condition of generalized contraction.

In this paper, we prove some common fixed-point theorems for a pair of singlevalued and multivalued mappings in aD-metric space satisfying a con- traction condition more general than that given in Dhage [1,2,3,4,5,7] and Rhoades [12]. The results of this paper are new to the fixed-point theory in D-metric spaces and include nearly a dozen of known fixed-point theorems as special cases (see [1,2,3,4,5,6,7,8,9,10,12]).

2. Preliminaries. Before going to the main results of this paper, we give some preliminaries needed in the sequel.

LetF:X→2^X. Then by an orbit ofF at a pointx∈Xwe mean a setOF(x) inXdefined by

OF(x)=

x0=x, x_n+1∈Fxn, n≥0

. (2.1)

An orbitOF(x)is calledboundedifδ(OF(x)) <∞, and aD-metric spaceX is calledF-orbitally bounded ifOF(x)is bounded for eachx∈X. Again anF- orbitO_F(x)is calledcompleteif everyD-Cauchy sequence inO_F(x)converges to a point inX. AD-metric spaceXis said to beF-orbitally completeifOF(x) is complete for eachx∈X. Finally,Fis calledF-orbitally continuousif for any sequence{x_n} ⊆O_F(x), we have

limm,nρ

xm,xn,x^∗

=0 ⇒lim

m,nκ

Fxm,Fxn,Fx^∗

=0 (2.2)

for eachx∈X.

Let Φ denote the class of all functionsφ :[0,∞)→[0,∞) satisfying the following properties:

(i) φis continuous, (ii) φis nondecreasing, (iii) φ(t) < t,t >0, (iv) _∞

n=1φⁿ(t) <∞for allt∈[0,∞).

The functionφis called aLipschitz control functionorLipschitz growth func- tionand the usual growth function isφ(t)=αt, 0≤t <1. The following lemma concerning the functionφappears in [7].

Lemma2.1. Ifφ∈Φ, thenφⁿ(t)=0for eachn∈Nandlimnφⁿ(t)=0for eacht∈[0,∞).

We need the followingD-Cauchy principle of Dhage [7] in the sequel.

Lemma2.2(D-Cauchy principle). Let{x_n}be a bounded sequence in aD- metric spaceXwithD-boundksatisfying, for some positive real numberr,

ρx_n,x_n+1,x_m

≤φⁿk^{r 1/r} (2.3)

for allm > n∈N, whereφ:[0,∞)→[0,∞)satisfies_∞

n=1φⁿ(t) <∞for each t∈[0,∞). Then{xn}is aD-Cauchy sequence inX.

Proof. The proof appears in [7], but for the sake of completeness we give the details. Letp,t∈Nbe arbitrary but fixed. Then from (2.3) it follows that

ρ

xn,x_n+1,x_n+p

≤ φⁿ

k^{r 1/r}, ρ

xn,x_n+1,x_n+p+t

≤ φⁿ

k^{r 1/r}, (2.4)

for alln∈N.

Now by repeated application of the tetrahedral inequality, we obtain ρx_n,x_n+p,x_n+p+t

≤ρ

xn,x_n+1,x_n+p +ρ

xn,x_n+1,x_n+p+t +ρ

x_n+1,x_n+p,x_n+p+t

≤ρ

xn,x_n+1,x_n+p +ρ

xn,x_n+1,x_n+p+t +ρ

x_n+1,x_n+2,x_n+p +ρ

x_n+1,x_n+2,x_n+p+t +ρ

x_n+2,x_n+p,x_n+p+t

≤2 φⁿ

k^{r 1/r}+2 φⁿ⁺¹

k^{r 1/r}+ρ

x_n+2,x_n+p,x_n+p+t

≤2

φⁿk^{r 1/r}+···+φ^n+p−2k^{r 1/r}

+ρx_n+p−1,x_n+p,x_n+p+t

≤2

n+p−1

j=n

φ^j k^{r 1/r}.

(2.5) Since_∞

n=1φⁿ(t) <∞for eacht∈[0,∞), we have_∞

j=1[φ^j(k^r)]^1/r<∞and so lim_nn+p−1

j=n [φ^j(k^r)]^1/r=0. Now from (2.5) it follows that

n→∞limρ

xn,x_n+p,x_n+p+t

=0. (2.6)

This proves that{x_n}is aD-Cauchy sequence inXand the proof of the lemma is complete.

As a direct application ofLemma 2.2, we obtain the following result proved in [5].

Lemma2.3. Let{x_n}be a bounded sequence in a D-metric space X with D-boundksatisfying

ρ

xn,x_n+1,xm

≤λⁿk (2.7)

for allm > n∈N, where0≤λ <1. Then{x_n}isD-Cauchy.

We use contractive conditions of the form a^r≤φ

b^r

(2.8) for some positive real numberr, whereaandbare nonnegative real numbers and φ∈Φ, because sometimes inequality (2.8) holds, but for the same real numbersaandb, the inequality

a≤φ(b) (2.9)

does not hold. To see this, letφ:R⁺→R⁺be a function defined by φ(t)= αt

1+t, 0≤α <1. (2.10) Obviously the functionφis continuous, nondecreasing and satisfiesφ(t)= αt/(1+t) < tfort >0. Again since

∞ n=1

φⁿ(t)= ∞ n=1

αⁿt

1+t+···+αⁿ⁻¹t<

∞ n=1

αⁿ<∞, (2.11)

we have thatφ∈Φ.

Now fora=1/2 andb=2/3, we have, by (2.9), 1

2≤φ2 3

=(2/3)α 1+2/3=2

5α, (2.12)

which is not true since 0≤α <1. But for the same values ofaandb, we have a positive real numberr=2 such that

1 2

2

=1 4≤4α

13=φ 2

3 2

(2.13) for 13/16≤α <1. Hence inequality (2.8) holds. Thus inequality (2.9) does not imply inequality (2.8). Actually, inequalities (2.8) and (2.9) are independent. To show that inequality (2.8) does not imply inequality (2.9), leta=1/4,b=4/9, andr=1/2. Clearly, inequality (2.8) does not hold, but for the same values of a,b, andr, one has

1 4≤4α

13=α(4/9) 1+4/9=φ4

9

(2.14) forα≥13/16, and so inequality (2.9) holds. Thus inequalities (2.8) and (2.9) are independent.

In the following sections, we will prove the main results of this paper.

3. Weak commuting mappings in D-metric spaces. Let F : X →2^X and g:X→X. Then the pair{F,g}of maps is calledlimit coincidentif limnFxn= {limngxn}for some sequence{xn}inX, andcoincidentif there exists a point u∈Xsuch thatFu= {gu}. Again two mapsFandgare calledlimit commut- ingif lim_nFgx_n= {lim_ngFx_n}, where{x_n}is a sequence inX, andcommuting ifFgx= {gFx}for allx∈X. Two mapsFandgare calledlimit coincidentally commuting if their limit coincidence implies the limit commutativity onX.

Similarly, they are calledcoincidentally commuting if they are commuting at the coincidence points. Again two mapsFandgare said to belimit pseudocom- mutingif limnFgx_n∩limngFx_n=φ, that is, lim_nD(Fgx_n,gFx_n,gFx_n)=0, where{x_n}is a sequence inX, andpseudocommutingifFgx∩gFx≠∅for eachx∈X. Finally, the pair{F,g}is calledlimit coincidentally pseudocommut- ingif its limit coincidence implies the limit pseudocommutativity onX, and coincidentally pseudocommutingif it is pseudocommuting at the coincidence points. It is known that a coincidentally commuting pair is limit coincidentally commuting and a coincidentally pseudocommuting pair is limit coincidentally pseudocommuting, but the converse implications need not hold. A pair of maps{F,g}isweak commutingif it is either limit commuting, coincidentally commuting, limit pseudocommuting, or coincidentally pseudocommuting on X. Below, we will prove some common fixed-point theorems for each of these weak commuting mappings onD-metric spaces.

3.1. Limit coincidentally commuting maps inD-metric spaces. LetF:X→ 2^Xandg:X→X. By an(F/g)-orbit of the pair{F,g}of maps at a pointx∈X, we mean a setO_F(gx)inXdefined by

OF(gx)=

yn|y0=gx0, yn=gxn∈Fx_n−1, n∈N,wherex0=x (3.1) for some sequence{x_n}inX. The orbitO_F(gx)is well defined for eachx∈X ifF(X)⊆g(X). ByO_F(gx)we denote the closure of the setO_F(gx)inX.

AD-metric spaceXis called(F/g)-orbitally bounded ifδ(OF(gx)) <∞for eachx ∈X. FurtherX is called (F/g)-orbitally completeif every D-Cauchy sequence{x_n} ⊂O_F(gx)converges to a point inXfor eachx∈X. Finally, a mappingT:X→CB(X)is called(F/g)-orbitally continuousif for any{xn} ⊂ OF(gx),xn→x^∗implies thatT xn→T x^∗for eachx∈X.

Theorem3.1. LetF:X→CB(X)andg:X→Xbe two mappings satisfying, for some positive real numberr,

δ^r(Fx,Fy,Fz)

≤φ max

ρ^r(gx,gy,gz),δ^r(Fx,Fy,gz),δ^r(gx,Fx,gz),

δ^r(gy,Fy,gz),δ^r(gx,Fy,gz),δ^r(gy,Fx,gz) (3.2) for allx,y,z∈X, whereφ∈Φ. Suppose that

(a) F(X)⊆g(X)andg(X)is bounded, (b) {F,g}is limit coincidentally commuting,

(c) F orgis(F/g)-orbitally continuous.

Further ifXis(F/g)-orbitally completeD-metric space, thenFandghave a unique common fixed pointu∈Xsuch thatFu= {u} =gu. Moreover, ifgis continuous atu, thenFis also continuous atuin the KasubaiD-metric onX.

Proof. Letx∈Xbe arbitrary and define a sequence{yn}inXas follows.

Takex0=xandy0=gx0. Choose a pointy1∈Fx0=X1. SinceF(X)⊆g(X), there is a pointx1∈Xsuch thaty1=gx1. Again choose a pointy2∈Fx1=X2. By hypothesis (a), there is a pointx2∈Xsuch thaty2=gx2. Proceeding in this way, by induction there is a sequence{x_n}of points inXsuch that

y0=gx0, y_n+1=gx_n+1∈X_n+1=Fx_n, n=0,1,2,.... (3.3) From hypothesis (a), it follows that

δX_m,X_n,X_p

≤δg(X)

=k <∞ (3.4) for allm,n,p∈N.

Now there are two cases.

Case 1. Suppose that y_r = y_r+1 for some r ∈N. Then we havegx_r = gx_r+1=ufor someu∈X.

We will show thatFxr= {u}. Suppose not. Then by (3.2), δ^r

Fx_r,Fx_r,u

=δ^r

Fxr,Fxr,gx_r+1

≤δ^r

Fxr,Fxr,Fxr

≤φ

maxρ^rgx_r,gx_r,gx_r,δ^rgx_r,Fx_r,gx_r,δ^rFx_r,Fx_r,gx_r

≤φ

max0,δ^rgx_r,Fx_r,gx_r,δ^rFx_r,Fx_r,u

=φ max

δ^r

u,Fx_r,u ,δ^r

Fx_r,Fx_r,u

=φ δ^r

u,Fxr,u

(3.5)

becauseδ^r(Fxr,Fxr,u)≤φ(δ^r(Fxr,Fxr,u))is not possible in view ofφ∈Φ. Again by (3.2),

δ^r

Fxr,u,u

=δ^r

Fxr,gx_r+1,gx_r+1

≤δ^r

Fxr,Fxr,Fxr

≤φ max

δ^r

u,Fxr,u ,δ^r

Fxr,Fxr,u

=φδ^rFx_r,Fx_r,u.

(3.6)

Substituting (3.6) in (3.5), we obtain δ^r

Fxr,Fxr,u

≤φ² δ^r

Fxr,Fxr,u

, (3.7)

which is a contradiction sinceφ∈Φ. HenceFx_r =u. SinceF andgare limit coincidentally commuting, one hasFgx_r= {gFx_r}.

We will show thatuis a common fixed point ofF and g such thatFu= {u} =gu.

Now,

δ^r(Fu,gu,u)=δ^r

FFxr,Fgxr,Fxr

≤φ max

ρ^r

gFxr,ggxr,gxr ,δ^r

FFxr,Fgxr,gxr , δ^rgFx_r,FFx_r,gx_r,δ^rggx_r,Fgx_r,gx_r, δ^rgFx_r,Fgx_r,gx_r,δ^rggx_r,FFx_r,gx_r

=φ max

ρ^r

gFx_r,ggx_r,gx_r ,δ^r

ggx_r,FFx_r,gx_r

=φ

δ^r(Fu,gu,u) ,

(3.8)

which is possible only whenFu= {u} =gusinceφ∈Φ.

Case2. Assume thatyn=y_n+1for eachn∈N. We will show that{yn}is aD-Cauchy sequence inX. Letx=x0,y=x1, andz=x_m−1,m≥1. Then by (3.2),

ρ^r

y1,y2,ym

≤δ^r

Fx0,Fx1,Fx_m−1

≤φ

maxρ^rgx0,gx1,gx_m−1,δ^rFx0,Fx1,gx_m−1,δ^rgx0,Fx0,gx_m−1, δ^rgx1,Fx1,gx_m−1,δ^rgx0,Fx1,gx_m−1,δ^rgx1,Fx0,gx_m−1

≤φ max

δ^r

X0,X1,X_m−1 ,δ^r

X1,X2,X_m−1 ,δ^r

X0,X1,X_m−1 , δ^r

X1,X2,X_m−1 ,δ^r

X0,X2,X_m−1 ,δ^r

X1,X1,X_m−1

≤φ

0≤a≤1,max1≤b≤2δ^rX_a,X_b,X_m−1

≤φ k^r

,

(3.9) that is,

ρ

y1,y2,y_m

≤ φ

k^{r 1/r}. (3.10)

Similarly, lettingx=x1,y=x2, andz=z_m−1,m≥2 in (3.2), we obtain ρ^ry2,y3,y_m

≤δ^r

Fx1,Fx2,Fx_m−1

≤φ max

ρ^r

gx1,gx2,gx_m−1 ,δ^r

Fx1,Fx2,gx_m−1 , δ^r

gx1,Fx1,gx_m−1 ,δ^r

gx2,Fx2,gx_m−1 , δ^rgx1,Fx2,gx_m−1,δ^rgx2,Fx1,gx_m−1

≤φ

maxδ^rFx0,Fx1,Fx_m−2,δ^rFx1,Fx2,Fx_m−2, δ^r

Fx0,Fx1,Fx_m−2

,δ^r

Fx1,Fx2,Fx_m−2

, δ^r

Fx0,Fx2,Fx_m−2 ,δ^r

Fx1,Fx1,Fx_m−2

≤φ

φ

0≤a≤2,1≤b≤3max δ^r

Xa,Xb,X_m−2

≤φ φ

k^r

=φ²k^r,

(3.11)

that is,

ρ

y2,y3,y_m

≤ φ²

k^{r 1/r}. (3.12)

In general, by induction,

ρy_n,y_n+1,y_m

≤φⁿk^{r 1/r} (3.13)

for allm > n∈N.

Hence, the application ofLemma 2.2yields that{yn}is aD-Cauchy sequence inX. TheD-metric spaceX being complete, there is a pointu∈Xsuch that lim_ny_n=u. The definition of{y_n}implies that lim_ngx_n=u. We will show that limnFxn= {u}.

Now, limn δ^r

Fxn,Fxn,u

=lim

n δ^r

Fxn,Fxn,y_n+1

≤lim

n δ^rFx_n,Fx_n,Fx_n

≤lim

n φ max

ρ^r

gx_n,gx_n,gx_n ,δ^r

Fx_n,Fx_n,gx_n ,δ^r

gx_n,Fx_n,gx_n

=lim

n φ max

δ^r

Fxn,Fxn,u ,0

=φ limn δ^r

Fx_n,Fx_n,u ,

(3.14) which implies that limnFxn=u. Thus we have

limn Fxn= {u} =lim

n gxn. (3.15)

SinceF andgare limit coincidentally commuting, one has limn Fgxn=

limn gFxn

. (3.16)

Suppose thatgis(F/g)-orbitally continuous onX. Then we have limn Fgxn=lim

n gFxn=lim

n ggxn=gu. (3.17)

First, we will show thatuis a common fixed point ofFandg. Suppose not.

Then we have δ^r(u,u,gu)=lim

n δ^r

Fxn,Fxn,gFxn

=lim

n δ^rFx_n,Fx_n,Fgx_n

≤lim

n φ

maxρ^rgx_n,gx_n,ggx_n, δ^r

Fxn,Fxn,ggxn ,δ^r

gxn,Fxn,ggxn

=φ max

limn δ^rgx_n,gx_n,ggx_n,lim

n δ^rFx_n,Fx_n,ggx_n

=φ

δ^r(u,u,gu) ,

(3.18) which is a contradiction and hencegu=u.

Again δ^r(Fu,gu,u)

=lim

n δ^r

Fu,Fx_n,Fgx_n

≤lim

n φ max

ρ^r

gu,gxn,ggxn ,δ^r

Fu,Fxn,ggxn ,δ^r

gu,Fu,ggxn , δ^r

gxn,Fxn,ggxn ,δ^r

gu,Fxn,ggxn ,δ^r

gxn,Fu,ggxn

=φ max

ρ^r(gu,u,gu),δ^r(Fu,u,gu),δ^r(gu,Fu,gu), δ^r(u,u,gu),δ^r(gu,u,gu),δ^r(u,Fu,gu)

=φδ^r(Fu,gu,u),

(3.19) which is possible only whenFu= {u} =gusinceφ∈Φ. Thusuis a common fixed point ofF andg.

Next, suppose thatFis(F/g)-orbitally continuous onX. Then we have limn Fgx_n=lim

n gFx_n=lim

n FFx_n=Fu= {z}. (3.20) We will show thatzis a common fixed point ofF andg. SinceF(X)⊆g(X), there is a pointv∈Xsuch thatFu=gv=z. We will show thatFv=gv= {z}. By (3.2),

δ^r(Fv,gv,Fv)

=lim

n δ^rFv,Fv,FFx_n

≤lim

n φ

maxρ^rgv,gv,gFx_n,δ^rFv,gv,gFx_n,δ^rgv,Fv,gFx_n

=φ max

δ^r(gv,gv,gv),δ^r(Fv,gv,z) ,

(3.21) that is,

δ^r(Fv,gv,z)≤φ

δ^r(Fv,gv,z)

, (3.22)

which implies thatFv=gv= {z}sinceφ∈Φ.

SinceF andg are limit coincidentally commuting, they are coincidentally commuting onX. Therefore, we haveFgv=gFv. Now, proceeding with the arguments as inCase 1, it is proved thatzis a common fixed point ofFandg.

To prove the uniqueness, letz^∗(=z)be another common fixed point ofF andg. Then by (3.2),

ρ^r z,z,z^∗

=δ^r

Fz,Fz,Fz^∗

≤φ

maxρ^rgz,gz,gz^∗,δ^rFz,Fz,gz^∗, δ^rgz,Fz,gz^∗,δ^rgz,Fz,gz^∗

=φ ρ^r

z,z,z^∗ ,

(3.23)

which is a contradiction. Hencez=z^∗. ThenF andghave a unique common fixed pointz∈XwithFz= {z} =gz.

Finally, suppose thatgis continuous at the common fixed pointzofF and g. Then we will prove thatF is also continuous atz. Let{zn}be any sequence

inXconverging to the common fixed pointz. Sincegis continuous onX, we have

limm,nρz_m,z_n,z

=0 ⇒lim

m,nρgz_m,gz_n,gz

=0. (3.24)

From (1.2), it follows that κ

Fzm,Fzn,Fz

≤δ

Fzm,Fzn,Fz

. (3.25)

Now,

δ^rFz_m,Fz_n,Fz

≤φ max

ρ^r

gz_m,gz_n,gz ,δ^r

Fz_m,Fz_n,gz ,δ^r

gz_m,Fz_m,gz , δ^r

gzn,Fzn,gz ,δ^r

gzm,Fzn,gz ,δ^r

gzn,Fzm,gz .

(3.26)

Therefore,

limm,nδ^rFz_m,Fz_n,Fz

≤lim

m,nφ

maxρ^rgz_m,gz_n,gz,δ^rFz_m,Fz_n,Fz,δ^rgz_m,Fz_m,z, δ^rgz_n,Fz_n,z,δ^rgz_m,Fz_n,z,δ^rgz_n,Fz_m,z

=φ max

0,lim

m,nδ^r

Fzm,Fzn,Fz ,limm δ^r

z,Fzm,z ,limn δ^r

z,Fzn,z

=φ max

limm δ^r

z,Fzm,z ,limn δ^r

z,Fzn,z .

(3.27) But

limm δ^rz,Fz_m,z

=lim

m δ^r

Fz,Fz,Fzm

≤lim

m φ max

ρ^r

gz,gz,gzm ,δ^r

Fz,Fz,gzm ,δ^r

gz,Fz,gzm

=φ

max{0,0,0}

=0.

(3.28)

Similarly, limnδ^r(z,Fzn,z)=0. Substituting these estimates in (3.27) yields that

limm,nδ^r

Fzm,Fzn,Fz

=0 (3.29)

or

limm,nδ

Fzm,Fzn,Fz

=0. (3.30)

Now from (3.25), it follows that

limm,nκFz_m,Fz_n,Fz

=0, (3.31)

and soFis continuous at the common fixed pointzofFandg. This completes the proof.

Lettingg=I, the identity map onXandr=1, inTheorem 3.1, we obtain the following corollary.

Corollary3.2. LetF:X→CB(X)be a multivalued mapping satisfying δ(Fx,Fy,Fz)≤φ

ρ(x,y,z),δ(Fx,Fy,z),δ(x,Fx,z),

δ(y,Fy,z),δ(x,Fy,z),δ(y,Fx,z) (3.32) for all x,y,z∈X, where φ∈Φ. Further if X is F-orbitally bounded and F- orbitally completeD-metric space, thenF has a unique fixed pointu∈Xsuch thatFu= {u}andF is continuous atu.

Corollary3.3. LetF:X→CB(X)be a multivalued mapping satisfying δ(Fx,Fy,Fz)≤λmax

ρ(x,y,z),δ(Fx,Fy,z),δ(x,Fx,z),

δ(y,Fy,z),δ(x,Fy,z),δ(y,Fx,z) (3.33) for allx,y,z∈X, where0≤λ <1. Further ifX is F-orbitally bounded and F-orbitally completeD-metric space, thenFhas a unique fixed pointu∈Xsuch thatFu= {u}andF is continuous atu.

Corollary 3.3 includes the following fixed point of Dhage [3] as a special case.

Corollary 3.4 (see [3]). Let X be a bounded and complete D-metric space and letF:X→CB(X)be a multivalued mapping satisfying

δ(Fx,Fy,Fz)≤λρ(x,y,z) (3.34) for allx,y,z∈X, where0≤λ <1. ThenFhas a unique fixed pointu∈Xsuch thatFu= {u}andF is continuous atu.

Corollary3.5. Letf ,g:X→Xbe two mappings satisfying ρ^r(f x,f y,f z)

≤φ max

ρ^r(gx,gy,gz),ρ^r(f x,f y,gz),ρ^r(gx,f x,gz),

ρ^r(gy,f y,gz),ρ^r(gx,f y,gz),ρ^r(gy,f x,gz) (3.35) for allx,y,z∈X, whereφ∈Φ. Suppose that

(a) f (X)⊆g(X),

(b) {f ,g}is limit coincidentally commuting, (c) forgis continuous.

Further ifXis(f /g)-orbitally bounded and(f /g)-orbitally completeD-metric space, thenfandghave a unique common fixed pointu∈X. Moreover, ifgis continuous atu, thenfis also continuous atu.

Remark 3.6. Note that Corollary 3.5includes the class of pairs of fixed- point mappings of Dhage [7] characterized by the inequality

ρ^r(f x,f y,f z)

≤φ

maxρ^r(gx,gy,gz),ρ^r(gx,f x,gz),

ρ^r(gy,f y,gz),ρ^r(gx,f y,gz),ρ^r(gy,f x,gz) (3.36) for allx,y,z∈Xandφ∈Φ.

Corollary3.7. Letf ,g:X→Xbe two mappings satisfying for some posi- tive real numbersp,q, andr,

ρ^r

f^px,f^py,f^pz

≤φ

maxρ^rg^qx,g^qy,g^qz,ρ^rf^px,f^py,g^qz, ρ^rg^qx,f^px,g^qz,ρ^rg^qy,f^py,g^qz, ρ^r

g^qx,f^py,g^qz ,ρ^r

g^qy,f^px,g^qz

(3.37)

for allx,y,z∈X, whereφ∈Φ. Suppose that (a) f^p(X)⊆g^q(X),

(b) {f ,g}is commuting, (c) forgis continuous.

Further ifXis an(f^p/g^q)-orbitally bounded and(f^p/g^q)-orbitally complete D-metric space, thenf andghave a unique common fixed pointu∈X. More- over, ifgis continuous atu, thenf^pis also continuous atu.

Proof. LetS=f^pandT=g^q. Then byCorollary 3.5,SandThave a unique common fixed pointu∈X, that is,Su=f^pu=u=g^qu=T u. Now by com- mutativity off andg, we obtain

f u=ff^pu

=f^p(f u), f u=fg^qu

=g^q(f u). (3.38) This shows thatf uis again a common fixed point of f^p and g^q. By the uniqueness ofu, we havef u=u. Similarly it is proved thatgu=u. Thusf andghave a unique common fixed pointu∈X. Further ifgis continuous on X,g^qis continuous onXand by application ofCorollary 3.5yields thatf^pis continuous atu. This completes the proof.

Corollary 3.7includes the class of pairs of fixed-point mappings of Dhage [7] characterized by the inequality

ρ^r

f^px,f^py,f^pz

≤φ

maxρ^rg^qx,g^qy,g^qy,ρ^rg^qx,f^px,g^qz,

ρ^rg^qy,f^py,g^qz,ρ^rg^qx,f^py,g^qz,ρ^rg^qy,f^px,g^qz (3.39) for allx,y,z∈Xandφ∈Φ.

Corollary3.8. Letf be a self-map of aD-metric spaceXsatisfying ρ(f x,f y,f z)≤λmax

ρ(x,y,z),ρ(f x,f y,z),ρ(x,f x,z),

ρ(y,f y,z),ρ(x,f y,z),ρ(y,f x,z) (3.40) for allx,y,z∈X, where0≤λ <1. Further if X is f-orbitally bounded and f-orbitally complete, thenfhas a unique fixed pointu∈Xandfis continuous atu.

Corollary3.9. Letf be a self-map of aD-metric spaceXsatisfying, for some positive real numberp,

ρf^px,f^py,f^pz

≤λmax

ρ(x,y,z),ρ

f^px,f^py,z ,ρ

x,f^px,z , ρ

y,f^py,z ,ρ

x,f^py,z ,ρ

y,f^px,z (3.41) for allx,y,z∈X, where0≤λ <1. Further if X is f-orbitally bounded and f-orbitally complete, thenf has a unique fixed pointu∈X,f^pis continuous, andfisf-orbitally continuous atu.

Note that Corollaries3.8and3.9include the fixed-point theorems of Rhoades [12] and Dhage [9] for the mappings characterized by the inequalities

ρ(f x,f y,f z)≤λmaxρ(x,y,z),ρ(x,f x,z),

ρ(y,f y,z),ρ(x,f y,z),ρ(y,f x,z)

, (3.42) ρ

f^px,f^py,f^pz

≤λmax

ρ(x,y,z),ρ

x,f^px,z , ρ

y,f^py,z ,ρ

x,f^py,z ,ρ

y,f^px,z , (3.43) for allx,y,z∈Xand 0≤λ <1.

3.2. Coincidentally commuting mappings. The coincidentally commuting mappings require some stronger condition than limit coincidentally commut- ing mappings and a good number of mathematicians have studied them on metric andD-metric spaces for the existence of their common fixed point. See [5,11] and the references therein. The novelty of the fixed-point theorems for these coincidentally commuting mappings lies in the fact that here we do not require any of the maps under consideration to be continuous. Below, we prove a result in this direction and derive some interesting corollaries.

Theorem3.10. LetXbe aD-metric space and letF:X→CB(X)andg:X→ Xbe two mappings satisfying (3.2). Further suppose that

(a) F(X)⊆g(X),

(b) g(X)is bounded and complete, (c) {F,g}is coincidentally commuting.

ThenF andghave a unique common fixed pointu∈Xsuch thatFu= {u} = gu. Moreover, if g is continuous at u, thenF is also continuous at u in the KasubaiD-metric onX.

Proof. Letx∈X be arbitrary and define a sequence{y_n} ⊂X by (3.3).

Clearly the sequence{yn}is well defined sinceF(X)⊆g(X). Further we note that{yn} ⊆g(X). We prove the conclusion of the theorem in two cases.

Case 1. Suppose thaty_r =y_r+1 for somer ∈N. Then proceeding with the arguments similar toCase 1of the proof ofTheorem 3.1, it is proved that yr=uis a common fixed point ofF andgsuch thatFu= {u} =gu.

Case2. Assume thaty_n=y_n+1for eachn∈N. Then followingCase 2of the proof ofTheorem 3.1, it is shown that{yn}is aD-Cauchy sequence. Since g(X)is complete, there is a pointz∈g(X)such that limnyn=z=limngxn. We will show that lim_nFx_n= {z}.

Now, limn δ^r

Fxn,Fxn,z

=lim

n δ^rFx_n,Fx_n,y_n+1

≤lim

n δ^r

Fxn,Fxn,Fxn

≤lim

n φ max

ρ^r

gxn,gxn,gxn ,δ^r

Fxn,Fxn,gxn ,δ^r

gxn,Fxn,gxn

=φ max

0,lim

n δ^r

Fx_n,Fx_n,z

=φ

limn δ^rFx_n,Fx_n,z,

(3.44) which gives that limnFx_n= {z}.

Sincez∈g(X), there is a pointu∈Xsuch thatgu=u. We will show that Fu= {z} =gu. Now,

δ^r(Fu,z,z)

=lim

n δ^rFu,Fx_n,Fx_n

=lim

n δ^r

Fx_n,Fx_n,Fu

≤lim

n φ max

ρ^r

gu,gxn,gxn ,δ^r

Fxn,Fxn,gu ,δ^r

gxn,Fxn,gu

=φ

max{0,0,0}

=φ(0)

=0

(3.45) and soFu=gu= {z}. Thusuis a coincidence point ofFandg. The rest of the proof is similar toCase 2of the proof ofTheorem 3.1. We omitted the details.

As a consequence ofTheorem 3.10, we obtain the following corollaries.

Corollary3.11. Letf ,g:X→X be two mappings satisfying (3.35). Sup- pose that

(a) f (X)⊆g(X),

(b) g(X)is bounded and complete, (c) {f ,g}is coincidentally commuting.

Thenfandghave a unique common fixed pointuand ifgis continuous atu, thenf is also continuous atu.

Corollary 3.12. Let X be aD-metric space and let f ,g:X→X be two mappings satisfying

ρ(f x,f y,f z)

≤λmaxρ(gx,gy,gz),ρ(f x,f y,gz),ρ(gx,f x,gz),

ρ(gy,f y,gz),ρ(gx,f y,gz),ρ(gy,f x,gz) (3.46) for allx,y,z∈X, where0≤λ <1. Further suppose that hypotheses (a), (b), and (c) ofCorollary 3.11hold. Thenfandghave a unique common fixed point u∈Xand ifgis continuous atu, thenf is also continuous atu.

Corollary 3.12includes a common fixed-point theorem of Dhage [5] for the mappingsfandgon aD-metric space characterized by the inequality

ρ(f x,f y,gz)

≤λmax

ρ(gx,gy,gz),ρ(gx,f x,gz),

ρ(gy,f y,gz),ρ(gx,f y,gz),ρ(gy,f x,gz) (3.47) for allx,y,z∈Xand 0≤λ <1.

Corollary 3.13. Let X be aD-metric space and let f ,g:X→X be two mappings satisfying (3.37). Further suppose that

(a) f^p(X)⊆g^q(X),

(b) g^p(X)is bounded and complete, (c) {f ,g}is commuting.

Thenf andghave a unique common fixed pointuand ifg^qis continuous at u, thenf^pis also continuous atu.

Notice thatCorollary 3.13includes a class of common fixed-point mappings fandgon aD-metric spaceXcharacterized by the inequality

ρf^px,f^py,f^pz

≤λmaxρg^qx,g^qy,g^qz,ρg^qx,f^px,g^qz, ρ

g^qy,f^py,g^qz ,ρ

g^qx,f^py,g^qz ,ρ

g^qy,f^px,g^qz (3.48) for allx,y,z∈Xand 0≤λ <1. See [5].

4. Weak commuting mappings in compact D-metric spaces. In this sec- tion, we prove some common fixed-point theorems for the pairs of singleval- ued and multivalued coincidentally commuting mappings on aD-metric space satisfying a contraction condition more general than (4.3). But in this case the D-metric space under consideration is required to satisfy a stronger condition of compactness and the mappings under consideration are required to satisfy the continuity condition on theD-metric spaces. Our results of this section generalize some earlier known fixed-point theorems such as those of Dhage [9] and Rhoades [12] for single maps as well as for a pair of maps onD-metric spaces.

Theorem4.1. LetXbe a compactD-metric space and letF:X→CB(X)and g:X→Xbe two continuous mappings satisfying, for some positive real number r,

δ^r(Fx,Fy,Fz)

<maxρ^r(gx,gy,gz),δ^r(Fx,Fy,gz),δ^r(gx,Fx,gz),

δ^r(gy,Fy,gz),δ^r(gx,Fy,gz),δ^r(gy,Fx,gz) (4.1) for allx,y,z∈Xfor which the right-hand side is not zero. Further suppose that

(a) F(X)⊆g(X),

(b) {F,g}is limit coincidentally commuting.

ThenF andghave a unique common fixed pointu∈Xsuch thatFu= {u} = gu.

Proof. From inequality (4.3), it follows that ifFandghave a common fixed pointu∈X, then it is unique andFu= {u} =gu. SinceX is compact andδ is continuous, both sides of inequality (4.1) are bounded onX. Now, there are two cases.

Case1. Suppose that the right-hand side of (4.1) is zero for somex,y,z∈ X. Then, we have

Fx=gx=gz, Fy=gy=gz. (4.2)

Now, proceeding with the arguments similar to Case 1 of the proof of Theorem 3.1, it is proved that u=Fx= gx is a common fixed point of F andgand so it is unique.

Case2. Suppose that the right-hand side of inequality (4.1) is not zero for allx,y,z∈X. Define a mappingT:X×X×X→(0,∞)by

T (x,y,z)=δ^r(Fx,Fy,Fz)

M(x,y,z) , (4.3)

where

M(x,y,z)=max

ρ^r(gx,gy,gz),δ^r(Fx,Fy,gz),δ^r(gx,Fx,gz), δ^r(gy,Fy,gz),δ^r(gx,Fy,gz),δ^r(gy,Fx,gz)

. (4.4)

Clearly, the functionTis well defined sinceM(x,y,z)=0 for allx,y,z∈X.

SinceF andgare continuous, from the compactness ofX it follows that the function T attains its maximum on X³ at some point u,v,w ∈X. Call the valuec. It is clear from (4.1) that 0< c <1. By the definition ofc, we have T (x,y,z)≤cfor allx,y,z∈X. This further, in view of (4.3), implies that

δ^r(Fx,Fy,Fz)

≤cM(x,y,z)

=cmax

ρ^r(gx,gy,gz),δ^r(Fx,Fy,Fz),δ^r(gx,Fx,gx), δ^r(gy,Fy,gz),δ^r(gx,Fy,gz),δ^r(gy,Fx,gz)

(4.5)

for allx,y,z∈X.

AsX is compact, it is complete and g(X)is bounded in view of the con- tinuity ofgon X. Now, the desired conclusion follows by an application of Theorem 3.1. This completes the proof.

Now we derive some interesting corollaries.

Corollary4.2. LetXbe a compactD-metric space and letF:X→CB(X) be a continuous mapping satisfying

δ(Fx,Fy,Fz) <max

ρ(x,y,z),δ(Fx,Fy,z),δ(x,Fx,z),

δ(y,Fy,z),δ(x,Fy,z),δ(y,Fx,z) (4.6)

for allx,y,z∈Xfor which the right-hand side is not zero. ThenFhas a unique fixed pointu∈Xsuch thatFu= {u}.

Proof. The proof follows by lettingg=I inTheorem 4.1, whereI is the identity map onX.

Corollary4.3(see [3]). LetXbe a compactD-metric space and letF:X→ CB(X)be a continuous mapping satisfying

δ(Fx,Fy,Fz) < ρ(x,y,z) (4.7)

for allx,y,z∈X for whichρ(x,y,z)=0. ThenF has a unique fixed point u∈Xsuch thatFu= {u}.

Corollary4.4. LetXbe a compactD-metric space and letf ,g:X→Xbe two continuous mappings satisfying

ρ(f x,f y,f z) <max

ρ(gx,gy,gz),ρ(f x,f y,gz),ρ(gx,f x,gz),

ρ(gy,f y,gz),ρ(gx,f y,gz),ρ(gy,f x,gz) (4.8)

for allx,y,z∈Xfor which the right-hand side is not zero. Further suppose that (a) f (X)⊆g(X),

(b) {f ,g}is limit coincidentally commuting.

Thenf andghave a unique common fixed point.

Proof. The proof follows by lettingF = {f}, a single-valued mapping in Theorem 4.1.

Corollary4.5. LetXbe a compactD-metric space and letf:X→Xbe a continuous mapping satisfying

ρ(f x,f y,f z) <max

ρ(x,y,z),ρ(f x,f y,z),ρ(x,f x,z),

ρ(y,f y,z),ρ(x,f y,z),ρ(y,f x,z) (4.9)

for allx,y,z∈Xfor which the right-hand side is not zero. Thenfhas a unique fixed point.

Proof. The conclusion follows by lettingg=IinCorollary 4.4, whereI is the identity map onX.

Note that Corollaries4.4and4.5include the fixed-point theorems of Dhage [5] and Rhoades [12] for the mappingsfandgon aD-metric spaceXcharac- terized by the inequalities

ρ(f x,f y,f z) <maxρ(gx,gy,gz),ρ(gx,f x,gz),

ρ(gy,f y,gz),ρ(gx,f y,gz),ρ(gy,f x,gz)

, (4.10) ρ(f x,f y,f z) <max

ρ(x,y,z),ρ(x,f x,z),

ρ(y,f y,z),ρ(x,f y,z),ρ(y,f x,z)

, (4.11)

respectively.

Theorem4.6. LetXbe aD-metric space and letF:X→CB(X),g:X→X be two continuous mappings satisfying (4.1). Suppose further that

(a) F(X)⊆g(X), (b) g(X)is compact,

(c) {f ,g}is coincidentally commuting.

ThenF andghave a unique common fixed pointu∈Xsuch thatFu= {u} = gu.

Proof. LetA=g(X). ThenA is a compactD-metric space andF and g define the maps F :A→CB(A) and g:A→A. Now, the desired conclusion follows by an application ofTheorem 4.1.

Corollary 4.7. Let X be a D-metric space and letf ,g : X →X be two continuous mappings satisfying (4.8). Further suppose that