Common fixed points of weakly reciprocally continuous maps using a gauge function

Mathematical Theory and Modeling www.iiste.org
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.2, No.3, 2012

Common Fixed Points of Weakly Reciprocally Continuous
Maps using a Gauge Function
R. K. Bisht*, Jay Singh, AjayGairola, R. U. Joshi
Department of Mathematics, D. S. B. Campus, Kumaun University, Nainital-263002, India
* E-mail of the corresponding author: ravindra.bisht@yahoo.com

Abstract
The aim of the present paper is to obtain a common fixed point theorem by employing the recently
introduced notion of weak reciprocal continuity. We demonstrate that weak reciprocal continuity ensures
the existence of fixed points under contractive conditions which otherwise do not ensure the existence of
fixed points. Our result generalize and extend several well-known fixed point theorems due to Boyd and
Wong (1969), Jungck(1976), Pant (1994) and Pathak et al (1997).

Keywords: Fixed point theorems, compatible maps, A-compatible maps, T-compatible maps, reciprocal
continuity, weak reciprocal continuity

1. Introduction
The question of continuity of contractive maps in general and of continuity at fixed points in particular
emerged with the publication of two research papers by Kannan (1968, 1969) in 1968 and 1969 respectively.
These two papers generated unprecedented interest in the fixed point theory of contractive maps which, in
turn, resulted in vigorous research activity on the existence of fixed points of contractive maps and the
question of continuity of contractive maps at their fixed points turned into an open question.

In (1998), 30 years after Kannan’s celebrated papers, Pant (1998) introduced the notion of reciprocal
continuity for a pair of mappings and as an application of this concept obtained the first fixed point theorem,
in which the common fixed point was a point of discontinuity.

Definition 1.1
Two self mappings A and T of a metric space (X, d) are defined to be reciprocally continuous iff
limnATxn=At and limnTAxn= Tt, whenever {xn} is a sequence in X such that limnAxn= limnTxn= t for some t
in X.

The notion of reciprocal continuity has been employed by many researchers in diverse settings to establish
fixed point theorems which admit discontinuity at the fixed point. Imdad et al (2009) used this concept in
the setting of non-self mappings. Singh and Mishra (2002) have used reciprocal continuity to establish
general fixed point theorems for hybrid pairs of single valued and multi-valued maps. P.Balasubramaniam
et al (2002) extended the study of reciprocal continuity to fuzzy metric spaces. Suneel Kumar et al (2008)
studied this concept in the setting of probabilistic metric space. S. Murlishankar et al (2009) established a
common fixed point theorem in an intuitionistic fuzzy metric space using contractive condition of integral
type. Chugh et al (2003) and Kumar et al (2002) have, in the setting of metric spaces, obtained interesting
fixed point theorems which do not force the map to be continuous at the fixed point.

The notion of reciprocal continuity is mainly applicable to compatible mapping satisfying contractive
conditions. To widen the scope of the study of fixed points from the class of compatible mappings

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Vol.2, No.3, 2012
satisfying contractive conditions to a wider class including compatible as well as noncompatible
mappings satisfying contractive, nonexpansive or Lipschitz type condition Pant et al(2011) generalized the
notion of reciprocal continuity by introducing the new concept of weak reciprocal continuity as follows:

Definition 1.2
Two self mappings A and T of a metric space (X, d) are defined to be weak reciprocally continuous iff
limnATxn=At or limnTAxn= Tt, whenever {xn} is a sequence in X such that limnAxn= limnTxn= t for some t in
X.
Jungck(1986) generalized the notion of weakly commuting maps by introducing the concept of compatible
maps.
Definition 1.3
Two self mappings A and T of a metric space (X, d) are compatible iff limn d(ATxn, TAxn)= 0, whenever
{xn} is a sequence in X such that limnAxn= limnTxn= t for some t in X.
In (1993), Jungck et al (1993) further generalized the concept of weakly commuting mappings by
introducing the notion of compatible of type (A).
Definition 1.4
Two self mappings A and T of a metric space (X, d) are compatible of type (A) iff limn d(AAxn, TAxn)= 0
and limn d(ATxn, TTxn)= 0, whenever {xn} is a sequence in X such that limnAxn= limnTxn= t for some t in
X.
In (1998), Pathak et al (1998) generalized the notion of compatibility of type (A) by introducing the two
analogous notions of compatibility, i.e., A-compatible and T-compatible.
Definition1.5
Two self mappings A and T of a metric space (X, d) are A-compatible iff limn d(ATxn, TTxn)= 0, whenever
Definition 1.6
Two self mappings A and T of a metric space (X, d) are T-compatible iff limnd(AAxn, TAxn)= 0, whenever
It may be noted that the notions compatible and A-comaptible (or T-compatible, compatible mappings of
type (A)) are independent to each other. If both A and T are continuous, then all the analogous notions of
compatibility including compatibility are equivalent to each other.
As an application of weak reciprocal continuity we prove a common fixed point theorem for a
contractive condition that extend the scope of the study of common fixed point theorems from the class
of compatible or analogous compatible continuous mappings to a wider class of mappings which also
includes discontinuous mappings.

1. Main Results
Theorem 2.1
Let A and T be weakly reciprocally continuous self mappings of a complete metric space (X, d) satisfying
(i) AX TX
(ii) d(Ax, Ay) (max{ d(Tx, Ty), d(Ax, Tx), d(Ay,Ty), [d(Ax, Ty) + d(Ay, Tx)]/2 }),
where : R+ R+ denotes an upper semi continuous function such that (t) < t for each t > 0.
If A and T are either compatible or A-compatible or T-compatible then A and T have a unique common
fixed point.

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Vol.2, No.3, 2012

Proof.
Let x0 be any point in X. Define sequences {xn} and {yn} in X given by the rule
yn = Axn = Txn+1.
This can be done since AX TX. Then using (ii) we obtain
d(yn, yn+1) = d(Axn, Axn+1) (max{d(Txn, Txn+1), d(Axn, Txn), d(Axn+1, Txn+1),
[d(Axn, Txn+1) + d(Axn+1, Txn)]/2}).
= (d(yn-1,yn) < d(yn-1,yn).
Thus d(yn, yn+1) (d(yn-1, yn) < d(yn-1,yn).
(2.1)
Similarly, d (yn-1, yn) (d(yn-2, yn-1) < d(yn-2, yn-1).
(2.2)

Thus we see that {d(yn,yn+1)} isstrictly decreasing sequence of positive numbers and hence tends to a limit
r 0.
Suppose r > 0, then relation (2.1) on making n → ∞ and in view of upper semi continuity of yields
r (r) < r, a contradiction. Hence r = limn →∞ d (yn, yn+1) = 0. We claim {yn} is a Cauchy sequence.
Suppose it is not. Then there exists an ε> 0 and a subsequence {yni} of {yn} such that d(yni, yni + 1) > 2ε.
Since limnd(yn, yn+1) = 0,there exists integers mi satisfying ni< mi< ni+1 such that
d(yni, ymi) . If not then
d(yni, yni +1) d(yni, yni + 1 – 1) + d(yni + 1 – 1, yni + 1) <ε + d(yni + 1 – 1, yni + 1) < 2ε,
a contradiction. If mi be the smallest integer such that d(yni, ymi) >ε, then
ε d(yni, ymi) < d(yni, ymi - 2) + d(ymi – 2 , ymi – 1) + d(ymi – 1, ymi)
<ε + d(ymi – 2 , ymi – 1) + d(ymi – 1, ymi),
that is, there exists integer mi satisfying ni< mi<ni + 1 such that
d(yni, ymi) ε and limn d(yni, ymi) = ε,
(2.3)
without loss of generality we can assume that ni is odd and mi is even. Now by virtue of (1), we have
d(yni + 1,ymi + 1) ((d(yni , ymi) + d(yni, yni + 1)).
Now on letting ni→ ∞ and in view of (2.3) and upper semi continuity of , the above relation yields
( ) <ε , a contradiction. Hence {yn} is a Cauchy sequence. Since X is complete, there exists a point t
in X such that yn→ u as n → ∞. Moreover,
yn = Axn = Txn+1→ u.
(2.4)

Suppose that A and T are compatible mappings. Now, weak reciprocal continuity of A and T implies that
limnATxn=Au or limnTAxn= Tu. Let limnTAxn= Tu. Then compatibility of A and T yields limnd(ATxn, TAxn)=0,
i.e., limnATxn=Tu. By virtue of (2.4) this yields limn ATxn+1= limnAAxn=Tu. If Au≠Tu then using (ii) we
get d(Au, AAxn) (max{d(Tu, TAxn), d(Au, Tu), d(AAxn, TAxn), [d(Au, TAxn) + d(AAxn, Tu)]/2}. On
letting n → ∞ we get d(Au, Tu) (d(Au, Tu)) < d(Au, Tu), a contradiction. Hence Au=Tu. Again
compatibility of A and T implies commutativity at coincidence points. Hence ATu=TAu=AAu=TTu. Further,
if Au≠AAu then in view of (ii), we get d(Au, AAu) (max{d(Tu, TAu), d(Au, Tu), d(AAu, TAu), [d(Au,
TAu) + d(AAu, Tu)]/2} = (d(Au, AAu)) < d(Au, AAu), a contradiction. Hence Au = AAu =TAu. Therefore,
Au = Tu is a common fixed point of A and T.
Next suppose that limnATxn=Au. Then AX TX implies that Au=Tv for some v in X and limnATxn=Au =Tv.
Compatibility of A and T implies, limnTAxn=Tv. By virtue of (2.4) this also yields limn ATxn+1=
limnAAxn=Tv. If Av≠Tv then using (ii) we get d(Av, AAxn) (max{d(Tv, TAxn), d(Av, Tv), d(AAxn, TAxn),
[d(Av, TAxn) + d(AAxn, Tv)]/2}. On letting n → ∞ we get d(Av, Tv) (d(Av, Tv)) < d(Av, Tv), a
contradiction. Hence Av=Tv. Compatibility of A and T implies commutativity at coincidence points.
Hence ATv=TAv=AAv=TTv. Further, if Av≠AAv then in view of (ii), we get d(Av, AAv) (d(Av, AAv)) <
d(Av, AAv), a contradiction. Hence Av = AAv =TAv. Therefore, Av = Tv is a common fixed point of A and T.

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Vol.2, No.3, 2012
Now, suppose that A and T are A-compatible mappings. Weak reciprocal continuity of A and T
implies that limnATxn=Au or limnTAxn= Tu. Let limnTAxn= Tu. Then A-compatibility of A and T yields
limnd(ATxn, TTxn)=0. By virtue of limn TTxn+1= limnTAxn=Tu this yields limnATxn=Tu. If Au≠Tu then using
(ii) we get d(Au, ATxn) (max{d(Tu, TTxn), d(Au, Tu), d(ATxn, TTxn), [d(Au, TTxn) + d(ATxn,
Tu)]/2}. On letting n → ∞ we get d(Au, Tu) (d(Au, Tu)) < d(Au, Tu), a contradiction. Hence Au=Tu.
Again A- compatibility implies commutativity at coincidence points. Hence ATu=TAu=AAu=TTu. Further,
if Au≠AAu then in view of (ii), we get d(Au, AAu) (max{d(Tu, TAu), d(Au, Tu), d(AAu, TAu), [d(Au,
TAu) + d(AAu, Tu)]/2}= (d(Au, AAu)) < d(Au, AAu), a contradiction. Hence Au = AAu =TAu. Therefore,
Au = Tu is a common fixed point of A and T.

Next suppose that limnATxn=Au. Then AX TX implies that Au=Tv for some v in X and limnATxn=Au =Tv.
A-Compatibility of A and T implies, limnTTxn=Tv. If Av≠Tv then using (ii) we get d(Av, ATxn)
(max{d(Tv, TTxn), d(Av, Tv), d(ATxn, TTxn), [d(Av, TTxn) + d(ATxn, Tv)]/2}. On letting n → ∞ we get
d(Av, Tv) (d(Av, Tv)) < d(Av, Tv), a contradiction. Hence Av=Tv. A-Compatibility implies
commutativity at coincidence points. Hence ATv=TAv=AAv=TTv. Further, if Av≠AAv then in view of (ii),
we get d(Av, AAv) (d(Av, AAv)) < d(Av, AAv), a contradiction. Hence Av = AAv =TAv. Therefore, Av =
Tv is a common fixed point of A and T.

Finally, suppose that A and T are T-compatible mappings. Now, weak reciprocal continuity of A and T
implies that limnATxn=Au or limnTAxn= Tu. Let limnTAxn= Tu. Then T-compatibility of A and T yields
limn d(TAxn, AAxn)=0, i.e., limnAAxn= Tu. If Au≠Tu then using (ii) we get d(Au, AAxn) (max{d(Tu,
TAxn), d(Au, Tu), d(AAxn, TAxn), [d(Au, TAxn) + d(AAxn, Tu)]/2}. On letting n → ∞ we get d(Au, Tu)
(d(Au, Tu)) < d(Au, Tu), a contradiction. Hence Au=Tu. Again T- compatibility implies commutativity at
coincidence points. Hence ATu=TAu=AAu=TTu. Further, if Au≠AAu then in view of (ii), we get d(Au,
AAu) (max{d(Tu, TAu), d(Au, Tu), d(AAu, TA), [d(Au, TAu) + d(AAu, Tu)]/2} = (d(Au, AAu)) <
d(Au, AAu), a contradiction. Hence Au = AAu =TAu. Therefore, Au = Tu is a common fixed point of A and
T.

Next suppose that limnATxn=Au. Then AX TX implies that Au=Tv for some v in X and limnATxn=Au
=Tv. By virtue of (2.4) this yields limn ATxn+1= limnAAxn=Au=Tv. T-Compatibility of A and T implies,
limnTAxn=Tv. If Av≠Tv then using (ii) we get d(Av, AAxn) (max{d(Tv, TAxn), d(Av, Tv), d(AAxn, TAxn),
[d(Av, TAxn) + d(AAxn, Tv)]/2}. On letting n → ∞ we get d(Av, Tv) (d(Av, Tv)) < d(Av, Tv), a
contradiction. Hence Av=Tv. T-Compatibility implies commutativity at coincidence points. Hence
ATv=TAv=AAv=TTv. Further, if Av≠AAv then in view of (ii), we get d(Av, AAv) (d(Av, AAv)) < d(Av,
AAv), a contradiction. Hence Av = AAv =TAv. Therefore, Av = Tv is a common fixed point of A and T.
Uniqueness of the common fixed point theorem follows easily in each of the three cases.
We now furnish an example to illustrate Theorems 2.1.

Example 2.1
Let X = [2, 20] and d be the usual metric on X. Define A and T: X X as follows
Ax = 2 if x = 2 or x > 5, Ax = 6 if 2 < x 5,
Tx = 2, Tx = 12 if 2 < x 5, Tx = x- 3 if x > 5.
Then A and T satisfy all the conditions of Theorem 2.1 and have a common fixed point at x = 2. It can be
verified in this example that the mappings A and T are T-compatible. It can also be noted that A and T are
weakly reciprocally continuous. To see this, let {xn} be a sequence in X such that fxn→t, gxn→t for some t.
Then t=2 and either {xn}=2 for each n or {xn}=5+ εn where ε→ 0 as n→ ∞. If xn=2 for each n, ATxn→2
=A2 and TAxn→ 2=T2. If xn=5+ εn, then Axn→2, Txn→2, ATxn→6≠A2 and TAxn→ 2=T2. Thus, limn
TAxn=T2 but limn ATxn ≠A2. Hence A and T are weakly reciprocally continuous. It is also obvious that A
and T are not reciprocally continuous mappings.

Remark 2.1
Theorem 2.1 contains proper generalizations of many important fixed point theorems, we mention only
those due to Boyd and Wong (1969), Jungck (1976), Pant (1994) and Pathak et al (1997).

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Vol.2, No.3, 2012

As a direct consequence of the above theorem we get the following corollary.

Corollary 2.1
Let A and T be reciprocally continuous self mappings of a complete metric space (X, d) satisfying
(i) AX TX
(ii) d(Ax, Ay) (max{ d(Tx, Ty), d(Ax, Tx), d(Ay,Ty), [d(Ax, Ty) + d(Ay, Tx)]/2 }),
where, : R+ R+ denotes an upper semi continuous function such that (t) < t for each t > 0.
fixed point.

If we let (t) = kt, o k < 1, then we get the following corollaries:

Corollary 2.2
Let A and T be weakly reciprocally continuous self mappings of a complete metric space (X, d) satisfying
(i) AX TX
(ii) d(Ax, Ay) k(max{ d(Tx, Ty), d(Ax, Tx), d(Ay,Ty), [d(Ax, Ty) + d(Ay, Tx)]/2 }), o k < 1,
fixed point.

Corollary 2.3
Let A and T be reciprocally continuous self mappings of a complete metric space (X, d) satisfying
(i) AX TX
(ii) d(Ax, Ay) k(max{ d(Tx, Ty), d(Ax, Tx), d(Ay,Ty), [d(Ax, Ty) + d(Ay, Tx)]/2 }), o k < 1,
fixed point.

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Common fixed points of weakly reciprocally continuous maps using a gauge function

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