NONSELF-MAPPINGS IN METRICALLY CONVEX SPACES VIA ALTERING DISTANCES
M. IMDAD AND LADLAY KHAN
Received 13 January 2005 and in revised form 14 September 2005
Some common fixed point theorems for a pair of nonself-mappings in complete met- rically convex metric spaces are proved by altering distances between the points, which generalize earlier results due to M. D. Khan and Bharadwaj (2001), M. S. Khan et al.
(2000), Bianchini (1972), Chatterjea 1972, and others. Some related results are also dis- cussed besides furnishing an illustrative example.
1. Introduction
There exists extensive literature on fixed points of self-mappings in metric and Banach spaces. But in many applications the mappings under examination may not always be self-mappings, therefore fixed point theorems for nonself-mappings form a natural sub- ject for investigation. Assad and Kirk [2] initiated the study of fixed point of nonself- mappings in metrically convex spaces. Indeed while doing so, Assad and Kirk [2] noticed that with some kind of metric convexity, domain and range of the mappings under ex- amination can be considered of more varied type. In recent years, this technique due to Assad and Kirk [2] has been utilized by many researchers of this domain and by now there exists considerable literature on this topic. To mention a few, we cite [1,2,5,6,7,8,11, 12].
Recently, Assad [1] gave sufficient conditions for nonself-mappings defined on a closed subset of complete metrically convex metric spaces satisfying Kannan-type mappings [10] which have been currently generalized by M. S. Khan et al. [12]. For the sake of completeness, we state the main result of M. S. Khan et al. [12].
Theorem1.1. Let(X,d)be a complete metrically convex metric space andK a nonempty closed subset ofX. LetT:K→Xbe a mapping satisfying the inequality
d(Tx,T y)≤amaxd(x,Tx),d(y,T y)+bd(x,T y) +d(y,Tx) (1.1)
Copyright©2005 Hindawi Publishing Corporation
International Journal of Mathematics and Mathematical Sciences 2005:24 (2005) 4029–4039 DOI:10.1155/IJMMS.2005.4029
for everyx,y∈K, whereaandbare nonnegative reals such that max
a+b 1−b, b
1−a−b
=h >0, max
1 +a+b
1−b h, 1 +b 1−a−bh
=h, max{h,h} =h<1.
(1.2)
Further, if for everyx∈δK,Tx∈K,thenThas a unique fixed point inK.
2. Preliminaries
Before proving our results, we collect the relevant definitions and a lemma for our future use.
Definition 2.1[5]. LetKbe a nonempty subset of a metric space (X,d) andF,T:K→X.
The pair (F,T) is said to be weakly commuting if for everyx,y∈K withx=F y and T y∈K,
d(Tx,FT y)≤d(T y,F y). (2.1) Notice that forK=X, this definition reduces to that of Sessa [13].
Definition 2.2[6]. LetKbe a nonempty subset of a metric space (X,d) andF,T:K→X.
The pair (F,T) is said to be compatible if for every sequence{xn} ⊂K and from the relation
nlim→∞dFxn,Txn=0, (2.2)
andTxn∈K(for everyn∈N), it follows that
nlim→∞dT yn,FTxn=0 (2.3)
for every sequenceyn∈Ksuch thatyn=Fxn,n∈N.
Notice that forK=X, this definition reduces to that of Jungck [9].
Definition 2.3. Let (X,d) be a metric space andKa nonempty subset ofX. LetF,T:K→ Xbe a pair of maps which satisfy the condition
φd(Fx,F y)≤amax 1
2φd(Tx,T y),φd(Tx,Fx),φd(T y,F y) +bφd(Tx,F y)+φd(T y,Fx)
(2.4)
for all distinctx,y∈K,a,b≥0 such thata+ 4b <1 and letφ:R+→R+be an increasing
continuous function for which the following properties hold:
φ(t)=0⇐⇒t=0, φ(2t)≤2φ(t). (2.5) ThenFis called generalizedTcontraction mapping ofKintoX.
Definition 2.4[8]. A pair of nonself-mappings (F,T) on a nonempty subsetKof a metric space (X,d) is said to be coincidentally commuting ifTx,Fx∈KandTx=Fximply that FTx=TFx.
Definition 2.5[2]. A metric space (X,d) is said to be metrically convex if for any distinct x,y∈X, there exists a pointz∈Xwithx=z=ysuch that
d(x,z) +d(z,y)=d(x,y). (2.6) Lemma2.6 [2]. LetKbe a nonempty closed subset of a metrically convex metric spaceX. If x∈Kandy /∈K, then there exists a pointz∈δK(the boundary ofK) such that
d(x,z) +d(z,y)=d(x,y). (2.7) 3. Results
Our main result runs as follows.
Theorem3.1. Let(X,d)be a complete metrically convex metric space andK a nonempty closed subset ofX. IfF is a generalizedT contraction mapping ofK intoX satisfying the following:
(i)δK⊆TK,FK∩K⊆TK, (ii)Tx∈δK⇒Fx∈K,
(iii) (F,T)is coincidentally commuting, (iv)TKis closed inX,
thenFandThave a unique common fixed point.
Proof. Firstly, we proceed to construct two sequences{xn}and{yn}in the following way.
Letx∈δK. Then (due toδK⊂TK), there exists a pointx0∈K such thatx=Tx0. SinceTx∈δK⇒Fx∈K, one concludes thatFx0∈FK∩K⊆TK. Letx1∈K be such thaty1=Tx1=Fx0∈K. Lety2=Fx1. Ify2∈K, theny2∈FK∩K⊆TK which implies that there exists a pointx2∈K such that y2=Tx2. Ify2∈/ K, then there exists a point p∈δKsuch that
dTx1,p+dp,y2
=dTx1,y2
. (3.1)
Sincep∈δK⊆TK, there exists a pointx2∈Ksuch thatp=Tx2so that dTx1,Tx2
+dTx2,y2
=dTx1,y2
. (3.2)
Thus, repeating the foregoing arguments, one obtains two sequences{xn}and{yn}such
that
(v) yn+1=Fxn,
(vi)yn∈K⇒yn=Txnoryn∈/ K⇒Txn∈δKand dTxn−1,Txn
+dTxn,yn
=dTxn−1,yn
. (3.3)
We denote
P= Txi∈
Txn:Txi=yi, Q=
Txi∈
Txn:Txi=yi. (3.4)
Obviously, the two consecutive terms cannot lie inQ.
Now, we distinguish the following three cases.
Case 1. IfTxn,Txn+1∈P, then φdTxn,Txn+1
=φdFxn−1,Fxn
≤amax 1
2φdTxn−1,Txn
,φdTxn−1,Fxn−1
,φdTxn,Fxn +bφdTxn−1,Fxn+φdTxn,Fxn−1
=amaxφdyn−1,yn,φdyn,yn+1
+bφdyn−1,yn+1
=amaxφdyn−1,yn,φdyn,yn+1
+bφ2 maxdyn−1,yn
,dyn,yn+1 ,
(3.5) which in turn yields
φdTxn,Txn+1
≤
a+b
1−b
φdTxn−1,Txn
, ifdyn−1,yn
≥dyn+1,yn , 0, ifdyn−1,yn≤dyn+1,yn.
(3.6) Case 2. IfTxn∈PandTxn+1∈Q, then
dTxn,Txn+1
+dTxn+1,yn+1
=dTxn,yn+1
, (3.7)
which in turn yields
dTxn,Txn+1
≤dTxn,yn+1
=dyn,yn+1
, (3.8)
and hence
φdTxn,Txn+1
≤φdTxn,yn+1
=φdyn,yn+1
. (3.9)
Now, as inCase 1, one obtains
φdTxn,Txn+1
≤
a+b
1−b
φdTxn−1,Txn
, ifdyn−1,yn
≥dyn+1,yn
,
0, ifdyn−1,yn
≤dyn+1,yn .
(3.10) Case 3. IfTxn∈QandTxn+1∈P, thenTxn−1∈P. SinceTxnis a convex linear combi- nation ofTxn−1andyn, it follows that
dTxn,Txn+1
≤maxdTxn−1,Txn+1
,dyn,Txn+1
. (3.11)
Now, ifd(Txn−1,Txn+1)≤d(yn,Txn+1), then proceeding as inCase 1, we have
φdTxn,Txn+1
≤
a+b
1−b
φd(Txn−1,Txn), ifdyn−1,yn≥dyn+1,yn,
0, ifdyn−1,yn
≤dyn+1,yn
. (3.12) On the other hand, ifd(Txn,Txn+1)≤d(Txn−1,Txn+1), then
φdTxn,Txn+1
≤φdTxn−1,Txn+1
=φdFxn−2,Fxn
≤amax 1
2φdTxn−2,Txn
,φdTxn−2,Fxn−2
,φdTxn,Fxn
+bφdTxn−2,Fxn+φdTxn,Fxn−2
=amax 1
2φdyn−2,Txn
,φdyn−2,yn−1
,φdTxn,yn+1 +bφdyn−2,yn+1
+φdTxn,yn−1
.
(3.13) Since
1
2φdyn−2,Txn=1 2
φdyn−2,yn−1
+dyn−1,Txn
=maxφdTxn,yn−1
,φdyn−1,yn−2 ,
(3.14)
therefore, one gets φdTxn,Txn+1
=amaxφdTxn,yn−1
,φdyn−1,yn−2
,φdTxn,yn+1
+bφdyn−2,yn+1
+φdTxn,yn−1
.
(3.15)
Note that byCase 2, we haveφ(d(Txn−1,Txn))≤φ(d(Txn−2,Txn−1)), therefore φdTxn,Txn+1
≤amaxφdyn−1,yn−2
,φdTxn,yn+1 +bφdyn−2,yn+1
+bφdyn−2,yn−1
≤amaxφdyn−1,yn−2
,φdTxn,yn+1 +bφ3 maxdyn−2,yn−1
,dyn−1,Txn
,dTxn,yn+1
+bφdyn−2,yn−1
(3.16)
which in turn yields φdTxn,Txn+1
≤
a+ 3b
1−b
φdTxn−2,Txn−1
, ifdyn−2,yn−1
≥dTxn,yn+1
, b
1−a−3b
φdTxn−2,Txn−1
, ifdyn−2,yn−1
≤dTxn,yn+1
. (3.17) Thus in all cases, we have
φdTxn,Txn+1
≤kmaxφdTxn−1,Txn,φdTxn−2,Txn−1
, (3.18)
wherek=max{((a+b)/(1−b)), ((a+ 3b)/(1−b))}.
It can be easily shown by induction that forn≥1, we have φdTxn,Txn+1
≤knmaxφdTx0,Tx1
,φdTx1,Tx2
. (3.19)
Now, for any positive integerp, we have φdTxn,Txn+p
≤φdTxn,Txn+1
+dTxn+1,Txn+2
+···+dTxn+p−1,Txn+p
≤φ1 +k+k2+···+kp−1knmaxdTx0,Tx1
,dTx1,Tx2
≤φ 1
1−k
knmaxdTx0,Tx1
,dTx1,Tx2
,
(3.20) which implies thatφ(d(Txn,Txn+1))→0 asn→ ∞, so that{Txn}is a Cauchy sequence and hence converges to a pointzinX. We assume that a subsequence{Txnk}of{Txn} contained inPandTK is a closed subspace ofX. Since{Txnk}is Cauchy inTK, it con- verges to a pointz∈TK. Letu∈T−1z; thenTu=z. Here, one also needs to note that {Fxnk−1}will also converge toz.
Using (2.4), one can write φdFxnk−1,Fu≤amax
1
2φdTxnk−1,Tu,φdTxnk−1,Fxnk−1
,φd(Tu,Fu) +bφdFxnk−1,Tu+φdFu,Txnk−1
(3.21)
which, on lettingk→ ∞, reduces to φd(Fu,z)≤amax
1
2φd(Tu,z), 0,φd(Tu,Fu)+bφd(Tu,z)+φd(Fu,z), φd(Fu,Tu)≤(a+b)φd(Fu,Tu),
(3.22) yielding therebyTu=Fuwhich shows thatuis a point of coincidence forFandT.
Since the pair (F,T) is coincidentally commuting, therefore
z=Tu=Fu=⇒Fz=FTu=TFu=Tz. (3.23) To prove thatzis the fixed point ofF, consider
φd(Fz,z)=φd(Fz,Fu)
≤amax 1
2φd(Tz,Tu),φd(Tz,Fz),φd(Tu,Fu) +bφd(Tz,Fu)+φd(Tu,Fz)
≤ a
2+ 2b
φd(Fz,z),
(3.24)
which shows thatzis a common fixed point ofFandT. The proof goes on similar lines in case we assume subsequence{Txnk}of{Txn}contained inQ, hence it is omitted. The uniqueness of fixed point follows easily. This completes the proof.
Remark 3.2. By settingT=IK andφ(t)=t, one deduces a result similar to the main results of M. S. Khan et al. [12] and M. D. Khan and Bharadwaj [11].
Remark 3.3. By restricting T =IK and φ(t)=t with b=0, one deduces a result for nonself-mappings satisfying Bianchini-type condition [3].
Remark 3.4. If we chooseT=IK andφ(t)=twitha=0, then, one deduces a result for nonself-mappings satisfying Chatterjea-type condition [4].
Inspired by Imdad [7], we derive a corollary (as an application ofTheorem 3.1) which involves self- as well as nonself-mappings.
Corollary3.5. LetI,J:K→K,F:I(K)→X, andT:J(K)→Xbe four mappings so that FIandTJare nonself-mappings from a closed subsetKtoXwhich satisfy the condition
φd(FIx,FI y)≤amax 1
2φd(TJx,TJ y),φd(TJx,FIx),φd(TJ y,FI y) +bφd(TJx,FI y)+φd(TJ y,FIx),
(3.25) for allx,y∈Kwithx=y,a,b≥0such thata+ 4b <1, and letφ:R+→R+be an increasing continuous function which satisfiesφ(t)=0⇔t=0andφ(2t)≤2φ(t). Suppose that
(vii)δK⊆TJK,FIK∩K⊆TJK, (viii)TJx∈δK⇒FIx∈K,
(ix)TJKis closed inX.
Then(FI,TJ)has a point of coincidencez.Furthermore, if the pair(FI,TJ)is coincidentally commuting, thenzis also the unique common fixed point ofFI andTJ.Moreover, if the pairs(F,I),(FI,I),(T,J),(TJ,J),(FI,T),(FI,J),(TJ,F), and(TJ,I)commute atz, thenz remains a common fixed point ofF,I,T, andJ.
Proof. Since all the conditions ofTheorem 3.1are satisfied, thereforeFI andTJ have a unique common fixed pointz. Now, using the commutativity of various pairs, one can easily show thatzremains the unique common fixed point ofF,I,T, andJ.
In the next theorem, we use “weak commutativity” instead of “coincidentally com- muting property” of (F,T) and alternately replace the “closedness ofTK” by “continuity of the mapForT” to prove the following.
Theorem3.6. Let(X,d)be a complete metrically convex metric space andK a nonempty closed subset ofX. LetF,T:K →X be a pair of maps which satisfy (2.4), (i), and (ii).
Suppose that
(x) (F,T)is a weakly commuting pair, (xi)eitherForTis continuous onK.
ThenFandThave a unique common fixed point.
Proof. On the lines of the proof ofTheorem 3.1, one can show that the sequence{Txn} converges to a pointz inX. Assume that there exists a subsequence {Txnk} of {Txn} which is contained inP. SinceT is continuous,{TTxnk}converges to a point Tz. As Fxnk−1=TxnkandTxnk−1∈K, on using the weak commutativity of (F,T), we have
dTTxnk,FTxnk−1
≤dFxnk−1,Txnk−1
(3.26)
which, on lettingk→ ∞, reduces tod(Tz,FTxnk−1)→0.
Using (2.4), one can write φdFTxnk−1,Fz
≤amax 1
2φdTTxnk−1,Tz,φdTTxnk−1,FTxnk−1
,φd(Tz,Fz) +bφdTTxnk−1,Fz+φdTz,FTxnk−1
,
(3.27)
which, on lettingk→ ∞, reduces to
φd(Tz,Fz)≤amax0, 0,φd(Tz,Fz)+bφd(Fz,Tz)+ 0
≤(a+b)φd(Tz,Fz), (3.28)
yielding therebyFz=Tz.
To prove thatzis the fixed point ofT, consider φdTxnk,Tz=φdFxnk−1,Fz
≤amax 1
2φdTxnk−1,Tz,φdTxnk−1,Fxnk−1
,φd(Tz,Fz) +bφdTxnk−1,Fz+φdTz,Fxnk−1
(3.29) which, on lettingk→ ∞, reduces to
φd(Tz,z)≤ a
2+ 2b
φd(Tz,z), (3.30) implying thereby thatz=Tz. Thusz=Tz=Fz, which shows thatzis a common fixed point ofF andT. In case the subsequence{Txnk}of{Txn}is contained inQ, then the proof goes on similar lines, hence it is omitted. This completes the proof.
Finally, we prove a theorem when “weak commutativity” is replaced by “compatibil- ity.”
Theorem3.7. Let(X,d)be a complete metrically convex metric space andK a nonempty closed subset ofX. LetF,T:K→Xbe a pair of maps satisfying (2.4), (i), and (ii). Suppose that
(xii) (F,T)is a compatible pair, (xiii)FandTare continuous onK.
ThenFandThave a unique common fixed point.
Proof. On the lines of the proof ofTheorem 3.1, one can show that the sequence{Txn} converges to a point z∈K. Again, we assume that a subsequence {Txnk} of {Txn} is contained inP. SinceTxnk=Fxnk−1andTxnk−1∈K, using compatibility of (F,T), we have
nlim→∞dFxnk−1,Txnk−1
=0, lim
n→∞dTTxnk,FTxnk−1
=0. (3.31)
Using continuity ofT, it follows that{FTxnk−1} →Tzask→ ∞. Now, repeating the fore- going arguments (as inTheorem 3.6), one can show thatz=Tz=Fz. This completes the
proof.
Remark 3.8. Corollaries similar toCorollary 3.5can be outlined in the context of Theo- rems3.6and3.7.
4. An illustrative example
Finally, we furnish an example to establish the utility of our results over earlier ones es- pecially those contained in [3,4,11,12] and others.
Example 4.1. Let Rbe the set of reals equipped with usual metric,K= {−2} ∪[0, 1].
DefineF,T:K→Xby Tx=
−2x if 0≤x≤1,
1 ifx= −2, Fx=
−x
2 if 0≤x <1,
0 ifx∈ {−2, 1}, (4.1) whereasφ:R+→R+is defined byφ(t)=t.
SinceδK(the boundary ofK)= {−2, 0, 1}andTK=[−2, 0]∪ {1}which is also closed inR, clearlyδK⊆TK. Furthermore,FK=(−1/2, 0]⇒K∩FK= {0} ⊂TK.
Also
T1= −2∈δK=⇒F1=0∈K, T0=0∈δK=⇒F0=0∈K, T(−2)=1∈δK=⇒F(−2)=0∈K.
(4.2)
Now, ifx,y∈[0, 1), then φd(Fx,F y)=1
2|x−y| =1 2
1
2d(Tx,T y)
≤2 3max
1
2d(Tx,T y),d(Tx,Fx),d(T y,F y) + 1
16
d(Tx,F y) +d(Fx,T y).
(4.3)
Next, ifx∈[0, 1) andy= −2, then φd(Fx,F y)=x
2= 1 3
3x 2
=1
3d(Tx,Fx)
≤2 3max
1
2d(Tx,T y),d(Tx,Fx), (T y,F y)
+ 1 16
d(Tx,F y) +d(Fx,T y).
(4.4)
Finally, ifx=1 andy= −2, then φd(Fx,F y)=0≤2
3max 1
2d(Tx,T y),d(Tx,Fx), (T y,F y) + 1
16
d(Tx,F y) +d(Fx,T y),
(4.5)
which shows that the contraction condition (2.4) is satisfied for every distinctx,y∈K.
Moreover “0” is a point of coincidence asT0=F0. AlsoTF0=0=FT0; hence the pair (F,T) is coincidentally commuting. Thus all the conditions ofTheorem 3.1are satisfied and “0” is the unique common fixed point ofFandT.
However, it is interesting to note thatTheorem 1.1due to M. S. Khan et al. [12] (also Theorem 1 due to M. D. Khan and Bharadwaj [11]) does not work in the context of
mappingT. Otherwise, forx=1,y= −2 (making use of ((a+b)/(1−b))<1 or (b/(1− a−b))<1), contraction condition (1.1) reduces to
3< amax{3, 3}+ 2b <3a+ 21−a
2 <2a+ 1, (4.6)
implying thereby thata >1 which is indeed a contradiction to the fact thata <1. Here, it is worth noting that the mappingsTandFsatisfying (2.4) need not satisfy (1.1) separately (e.g., mappingT) which establishes the utility of our results proved in this paper.
Acknowledgments
The authors are grateful to the learned referees for critical reading of the entire manu- script and suggesting many improvements. The first author is also grateful to the Univer- sity Grants Commission in India for the financial assistance (Project no. F.30-246/2004 (SR)).
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M. Imdad: Department of Mathematics, Aligarh Muslim University, Aligarh 202 002, India E-mail address:[email protected]
Ladlay Khan: Department of Mathematics, Aligarh Muslim University, Aligarh 202 002, India E-mail address:k [email protected]
