J. Math. Anal. Appl. 341 (2008) 416420 [620033]

J. Math. Anal. Appl. 341 (2008) 416–420
www.elsevier.com/locate/jmaa
Common fixed point results for noncommuting mappings without
continuity in cone metric spaces
M. Abbasa,1, G. Jungckb,∗
aCentre for Advanced Studies in Mathematics and Department of Mathematics, Lahore University of Management Sciences,
54792-Lahore, Pakistan
bDepartment of Mathematics, Bradley University, Peoria, IL 61625, USA
Received 26 March 2007
Available online 11 October 2007
Submitted by B. Sims
Abstract
The existence of coincidence points and common fixed points for mappings satisfying certain contractive conditions, without
appealing to continuity, in a cone metric space is established. These results generalize several well-known comparable results in
the literature.
©2007 Elsevier Inc. All rights reserved.
Keywords: Weakly compatible maps; Common fixed point; Cone metric space
1. Introduction and preliminaries
The study of common fixed points of mappings satisfying certain contractive conditions has been at the centre of
vigorous research activity. In 1976, Jungck [4], proved a common fixed point theorem for commuting maps, gener-alizing the Banach contraction principle. This theorem has many applications but suffers from one drawback—the
results require the continuity of one of the two maps involved. Sessa [11] introduced the notion of weakly commuting
maps. Jungck [5] coined the term compatible mappings in order to generalize the concept of weak commutativity andshowed that weakly commuting maps are compatible but the converse is not true. Pant [9] defined R-weakly commut-
ing maps and proved common fixed point theorems, assuming the continuity of at least one of the mapping. Kannan
[12] proved the existence of a fixed point for a map that can have a discontinuity in a domain, however the maps
involved in every case were continuous at the fixed point. Jungck [7,8] defined a pair of self mappings to be weaklycompatible if they commute at their coincidence points. In recent years, several authors have obtained coincidencepoint results for various classes of mappings on a metric space, utilizing these concepts. For a survey of coincidence
point theory, its applications, comparison of different contractive conditions and related results, we refer to [1,6,9,10]
and references contained therein. Guang and Xian [3] generalized the concept of a metric space, replacing the set of
*Corresponding author.
E-mail addresses: [anonimizat] (M. Abbas), [anonimizat] (G. Jungck).
1Present address: Department of Mathematics, Indiana University, Bloomington, IN 47405-7106, USA.
0022-247X/$ – see front matter ©2007 Elsevier Inc. All rights reserved.
doi:10.1016/j.jmaa.2007.09.070

M. Abbas, G. Jungck / J. Math. Anal. Appl. 341 (2008) 416–420 417
real numbers by an ordered Banach space and obtained some fixed point theorems for mapping satisfying different
contractive conditions. The aim of this paper is to present coincidence point results for two mappings which satisfygeneralized contractive conditions. Common fixed point theorems for a pair of weakly compatible maps, which are
more general than R-weakly commuting, and compatible mappings are obtained in the setting of cone metric spaces
without exploiting the notion of continuity. These theorems generalize the results of Guang and Xian [3], Jungck [4],
Kannan [12] and Pant [9].
Consistent with Guang and Xian [3], the following definitions and results will be needed in the sequel.
LetEbe a real Banach space. A subset PofEis called a cone if and only if:
(a)Pis closed, nonempty and P/negationslash={0};
(b)a,b∈R,a,b/greaterorequalslant0,x,y∈Pimplies ax+by∈P;
(c)P∩(−P)={0}.
Given a cone P⊂E, we define a partial ordering /lessorequalslantwith respect to Pbyx/lessorequalslantyif and only if y−x∈P. A cone P
is called normal if there is a number K> 0 such that for all x,y∈E,
0/lessorequalslantx/lessorequalslantyimplies /bardblx/bardbl/lessorequalslantK/bardbly/bardbl. (1.1)
The least positive number satisfying the above inequality is called the normal constant ofP, while
x/lessmuchystands for
y−x∈intP(interior of P).
Definition 1.1. LetXbe a nonempty set. Suppose that the mapping d:X×X→Esatisfies:
(d1) 0/lessorequalslantd(x,y) for all x,y∈Xandd(x,y) =0 if and only if x=y;
(d2)d(x,y) =d(y,x) for all x,y∈X;
(d3)d(x,y) /lessorequalslantd(x,z) +d(z,y) for all x,y,z ∈X.
Thendis called a cone metric on Xand(X,d) is called a cone metric space. The concept of a cone metric space is
more general than that of a metric space.
Definition 1.2. Let(X,d) be a cone metric space. We say that {xn}is:
(e) a Cauchy sequence if for every cinEwithc/greatermuch0, there is Nsuch that for all n,m > N ,d(xn,xm)/lessmuchc;
(f) a Convergent sequence if for every cinEwithc/greatermuch0, there is Nsuch that for all n>N ,d(xn,x)/lessmuchcfor some
fixedxinX.
A cone metric space Xis said to be complete if every Cauchy sequence in Xis convergent in X. It is known that
{xn}converges to x∈Xif and only if d(xn,x)→0a sn→∞ . The limit of a convergent sequence is unique provided
Pis a normal cone with normal constant K(see Guang and Xian [3] and [2]).
Definition 1.3. Letfandgbe self maps of a set X.I fw=fx=gxfor some xinX, then xis called a coincidence
point offandg, andwis called a point of coincidence offandg.
Proposition 1.4. Letfandgbe weakly compatible self maps of a set X.I ffandghave a unique point of coincidence
w=fx=gx, thenwis the unique common fixed point of fandg.
Proof. Sincew=fx=gxandfandgare weakly compatible, we have fw=fgx=gf x=gw: i.e.,fw=gwis
a point of coincidence of fandg.B u twis the only point of coincidence of fandg,s ow=fw=gw. Moreover
ifz=fz=gz, then zis a point of coincidence of fandg, and therefore z=wby uniqueness. Thus wis a unique
common fixed point of fandg.2

418 M. Abbas, G. Jungck / J. Math. Anal. Appl. 341 (2008) 416–420
2. Common fixed point theorems
In this section we obtain several coincidence and common fixed point theorems for mappings defined on a cone
metric space.
Theorem 2.1. Let(X,d) be a cone metric space, and Pa normal cone with normal constant K. Suppose mappings
f,g:X→Xsatisfy
d(fx,fy) /lessorequalslantkd(gx,gy), for all x,y∈X, (2.1)
where k∈[0,1)is a constant. If the range of gcontains the range of fandg(X) is a complete subspace of X, thenf
andghave a unique point of coincidence in X. Moreover if fandgare weakly compatible, fandghave a unique
common fixed point.
Proof. Letx0be an arbitrary point in X. Choose a point x1inXsuch that f( x 0)=g(x 1). This can be done, since the
range of gcontains the range of f. Continuing this process, having chosen xninX, we obtain xn+1inXsuch that
f( xn)=g(xn+1). Then
d(gx n+1,gxn)=d(fx n,fxn−1)/lessorequalslantkd(gx n,gxn−1)
/lessorequalslantk2d(gx n−1,gxn−2)/lessorequalslant···/lessorequalslantknd(gx 1,gx 0).
Then, for n>m ,
d(gx n,gxm)/lessorequalslantd(gx n,gxn−1)+d(gx n−1,gxn−2)+···+ d(gx m+1,gxm)
/lessorequalslant(kn−1+kn−2+···+ km)d(gx 1,gx 0)
/lessorequalslantkm
1−kd(gx 1,gx 0).
From (1.1),
/vextenddouble/vextenddoubled(gx n,gxm)/vextenddouble/vextenddouble/lessorequalslantkm
1−kK/vextenddouble/vextenddoubled(gx 1,gx 0)/vextenddouble/vextenddouble,
which implies that d(gx n,gxm)→0a sn,m→∞ . Hence {gxn}is a Cauchy sequence. Since g(X) is complete, there
exists a qing(X) such that gxn→qasn→∞ . Consequently, we can find pinXsuch that g(p)=q. Further,
d(gx n,fp)=d(fx n−1,fp)/lessorequalslantkd(gx n−1,g p),
which from (1.1) implies that
/vextenddouble/vextenddoubled(gx n,fp)/vextenddouble/vextenddouble/lessorequalslantKk/vextenddouble/vextenddoubled(gx n−1,g p)/vextenddouble/vextenddouble→0,asn→∞.
Hence d(gx n,fp)→0a sn→∞ .A l s o , d(gx n,g p)→0a sn→∞ . The uniqueness of a limit in a cone metric space
implies that f( p)=g(p) . Now we show that fandghave a unique point of coincidence. For this, assume that there
exists another point qinXsuch that fq=gq.N o w
d(gq,gp) =d(fq,fp) /lessorequalslantkd(gq,gp),
which gives /bardbld(gq,gp) /bardbl=0 andgq=gp. From Proposition 1.4, fandghave a unique common fixed point. 2
Example 2.2. LetE=R2,P={(x,y) ∈E:x,y/greaterorequalslant0}⊂R2,d:R×R→Esuch that d(x,y) =(|x−y|,α|x−y|),
where α>0 is a constant. Define
fx=/braceleftbiggα
β+1x, x /negationslash=0,
γ, x =0,
and
gx=/braceleftbiggαx, x /negationslash=0,
γ, x =0,

M. Abbas, G. Jungck / J. Math. Anal. Appl. 341 (2008) 416–420 419
where β/greaterorequalslant1, and γ/negationslash=0. It may be verified that d(fx,fy) /lessorequalslantkd(gx,gy) , for all x,y∈X, where k=1
β∈(0,1].
Moreover fandghave a coincidence point X.
In above example, fandgdo not commute at the coincidence point 0, and therefore are not weakly compatible.
Andfandgdo not have common fixed point. Thus, this example demonstrates the crucial role of weak compatibility
in our results.
Theorem 2.3. Let(X,d) be a cone metric space and Pa normal cone with normal constant K. Suppose that the
mappings f,g:X→Xsatisfy the contractive condition
d(fx,fy) /lessorequalslantk/parenleftbig
d(fx,gx) +d(fy,gy)/parenrightbig
,for all x,y∈X, (2.2)
where k∈[0,1
2)is a constant. If the range of gcontains the range of fandg(X) is a complete subspace of X, then
fandghave a unique coincidence point in X. Moreover if fandgare weakly compatible, fandghave a unique
common fixed point.
Proof. Letx0be an arbitrary point in X. Choose a point x1inXsuch that f( x 0)=g(x 1). This can be done since the
range of gcontains the range of f. Continuing this process, having chosen xninX, we obtain xn+1inXsuch that
f( xn)=g(xn+1). Then
d(gx n+1,gxn)=d(fx n,fxn−1)/lessorequalslantk/parenleftbig
d(fx n,gxn)+d(fx n−1,gxn−1)/parenrightbig
=k/parenleftbig
d(gx n+1,gxn)+d(gx n,gxn−1)/parenrightbig
.
So
d(gx n+1,gxn)/lessorequalslanthd(gx n,gxn−1),
where h=k
1−k.F o rn>m ,
d(gx n,gxm)/lessorequalslantd(gx n,gxn−1)+d(gx n−1,gxn−2)+···+ d(gx m+1,gxm)
/lessorequalslant/parenleftbig
hn−1+hn−2+···+ hm/parenrightbig
d(gx 1,gx 0)
/lessorequalslanthm
1−hd(gx 1,gx 0),
which from (1.1) implies that /bardbld(gx n,gxm/bardbl/lessorequalslantkm
1−kK/bardbld(gx 1,gx 0)/bardbl. Then d(gx n,gxm)→0a sn,m→∞ , and{gxn}
is a Cauchy sequence. Since g(X) is a complete subspace of X, there exists qing(X) such that gxn→q,a sn→∞ .
Consequently we can find pinXsuch that g(p)=q. Thus,
d(gx n,fp)=d(fx n−1,fp)/lessorequalslantkd(gx n−1,g p),
which implies that/vextenddouble/vextenddoubled(gx n,fp)/vextenddouble/vextenddouble/lessorequalslantKk/vextenddouble/vextenddoubled(gx n−1,g p)/vextenddouble/vextenddouble=0,asn→∞.
Hence d(gx n,fp)→0a sn→∞ .A l s o , d(gx n,g p)→0a sn→∞ . The uniqueness of a limit in a cone metric space
implies that f( p)=g(p) . Now we show that fandghave a unique point of coincidence. For this, assume that there
exists another point qinXsuch that fq=gq.N o w
d(gq,gp) =d(fq,fp)
/lessorequalslantk/parenleftbig
d(fq,gq) +d(fp,gp)/parenrightbig
,
which gives /bardbld(gq,gp) /bardbl=0 andgq=gp. From Proposition 1.4, fandghave a unique common fixed point. 2
Theorem 2.4. Let(X,d) be a cone metric space, and Pa normal cone with normal constant K. Suppose that the
mappings f,g:X→Xsatisfy the contractive condition
d(fx,fy)<k/parenleftbig
d(fx,gy) +d(fy,gx)/parenrightbig
,for all x,y∈X, (2.3)
where k∈[0,1
2)is a constant. If the range of gcontains the range of fandg(X) is a complete subspace of X, then
fandghave a unique coincidence point in X. Moreover if fandgare weakly compatible, fandghave a unique
common fixed point.

420 M. Abbas, G. Jungck / J. Math. Anal. Appl. 341 (2008) 416–420
Proof. Letx0be an arbitrary point in X. Choose a point x1inXsuch that f( x 0)=g(x 1). This can be done, since the
range of gcontains the range of f. Continuing this process, having chosen xninX, we obtain xn+1inXsuch that
f( xn)=g(xn+1). Then
d(gx n+1,gxn)=d(fx n,fxn−1)/lessorequalslantk/parenleftbig
d(fx n,gxn−1)+d(fx n−1,gxn)/parenrightbig
/lessorequalslantk/parenleftbig
d(gx n+1,gxn)+d(gx n,gxn−1)/parenrightbig
.
So
d(gx n+1,gxn)/lessorequalslanthd(gx n,gxn−1),
where h−k
1−k.F o rn>m ,
d(gx n,gxm)/lessorequalslantd(gx n,gxn−1)+d(gx n−1,gxn−2)+···+ d(gx m+1,gxm)
/lessorequalslant/parenleftbig
hn−1+hn−2+···+ hm/parenrightbig
d(gx 1,gx 0)
/lessorequalslanthm
1−hd(gx 1,gx 0).
Following an argument similar to that given in Theorem 2.3, we obtain a point of coincidence of fandg.N o ww e
show that fandghave a unique point of coincidence. For this, assume that there exist pandqinXsuch that fp=gp
andfq=gq.N o w
d(gq,gp) =d(fq,fp)
/lessorequalslantk/parenleftbig
d(fq,gp) +d(fp,gq)/parenrightbig
=2kd(gq,gp)
which implies that d(gq,gp) =0 andgq=gp. From Proposition 1.4, the result follows. 2
The above theorem generalizes Theorem 4 of [3], which itself is a generalization of a result of [12].
Acknowledgment
The first author gratefully acknowledges support provided by Lahore University of Management Sciences (LUMS) during the first author’s stay
at Indiana University Bloomington as a post doctoral fellow.
References
[1] I. Beg, M. Abbas, Coincidence point and invariant approximation for mappings satisfying generalized weak contractive condition, Fixed Point
Theory Appl. 2006 (2006) 1–7, Article ID 74503.
[2] K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, 1985.
[3] L.-G. Huang, X. Zhang, Cone metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl. 332 (2) (2007) 1468–1476.
[4] G. Jungck, Commuting maps and fixed points, Amer. Math. Monthly 83 (1976) 261–263.
[5] G. Jungck, Compatible mappings and common fixed points, Internat. J. Math. Math. Sci. 9 (4) (1986) 771–779.
[6] G. Jungck, Common fixed points for commuting and compatible maps on compacta, Proc. Amer. Math. Soc. 103 (1988) 977–983.
[7] G. Jungck, Common fixed points for noncontinuous nonself maps on non-metric spaces, Far East J. Math. Sci. (FJMS) 4 (1996) 199–215.[8] G. Jungck, B.E. Rhoades, Fixed point for set valued functions without continuity, Indian J. Pure Appl. Math. 29 (3) (1998) 227–238.
[9] R.P. Pant, Common fixed points of noncommuting mappings, J. Math. Anal. Appl. 188 (1994) 436–440.
[10] B.E. Rhoades, A comparison of various definitions of contractive mappings, Trans. Amer. Math. Soc. 26 (1977) 257–290.
[11] S. Sessa, On a weak commutativity condition of mappings in fixed point consideration, Publ. Inst. Math. Soc. 32 (1982) 149–153.
[12] R. Kannan, Some results on fixed points, Bull. Calcutta Math. Soc. 60 (1968) 71–76.