FIXED POINT THEOREMS FOR A FAMILY OF HYBRID PAIRS OF MAPPINGS IN METRICALLY CONVEX SPACES

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OF MAPPINGS IN METRICALLY CONVEX SPACES

M. IMDAD AND LADLAY KHAN

Received 30 December 2004 and in revised form 24 March 2005

The present paper establishes some coincidence and common fixed point theorems for a sequence of hybrid-type nonself-mappings defined on a closed subset of a metrically convex metric space. Our results generalize some earlier results due to Khan et al. (2000), Itoh (1977), Khan (1981), Ahmad and Imdad (1992 and 1998), and several others. Some related results are also discussed.

1. Introduction

In recent years several fixed point theorems for hybrid pairs of mappings are proved and by now there exists considerable literature in this direction. To mention a few, one can cite Imdad and Ahmad [10], Pathak [19], Popa [20] and references cited therein. On the other hand Assad and Kirk [4] gave a suﬃcient condition enunciating fixed point of set-valued mappings enjoying specific boundary condition in metrically convex metric spaces. In the current years the work due to Assad and Kirk [4] has inspired extensive activities which includes Itoh [12], Khan [14], Ahmad and Imdad [1,2], Imdad et al. [11] and some others.

Most recently, Huang and Cho [9] and Dhage et al. [6] proved some fixed point theorems for a sequence of set-valued mappings which generalize several results due to Itoh [12], Khan [14], Ahmad and Khan [3] and others. The purpose of this paper is to prove some coincidence and common fixed point theorems for a sequence of hybrid type nonself mappings satisfying certain contraction type condition which is essentially patterned after Khan et al. [15]. Our results either partially or completely generalize earlier results due to Khan et al. [15], Itoh [12], Khan [14], Ahmad and Imdad [1,2], Ahmad and Khan [3] and several others.

2. Preliminaries

Before proving our results, we collect the relevant definitions and results for our future use.

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Let (X,d) be a metric space. Then following Nadler [17], we recall (i)CB(X)={A:Ais nonempty closed and bounded subset ofX}. (ii)C(X)={A:Ais nonempty compact subset ofX}.

(iii) For nonempty subsetsA,BofXandx∈X, d(x,A)=infd(x,a) :a∈A,

H(A,B)=maxsupd(a,B) :a∈A,supd(A,b) :b∈B. (2.1) It is well known (cf. Kuratowski [16]) thatCB(X) is a metric space with the distance Hwhich is known as Hausdorﬀ-Pompeiu metric onX.

The following definitions and lemmas will be frequently used in the sequel.

Definition 2.1. LetK be a nonempty subset of a metric space (X,d),T:K→XandF: K→CB(X). The pair (F,T) is said to be pointwiseR-weakly commuting onKif for given x∈KandTx∈K, there exists someR=R(x)>0 such that

d(T y,FTx)≤R·d(Tx,Fx) for eachy∈K∩Fx. (2.2) Moreover, the pair (F,T) will be calledR-weakly commuting onK if (2.2) holds for eachx∈K,Tx∈Kwith someR >0.

IfR=1, we get the definition of weak commutativity of (F,T) onK due to Hadzic and Gajic [8]. ForK=X Definition 2.1 reduces to “pointwiseR-weak commutativity andR-weak commutativity” for single valued self mappings due to Pant [18].

Definition 2.2[7,8]. LetK be a nonempty subset of a metric space (X,d),T:K→X andF:K→CB(X). The pair (F,T) is said to be weakly commuting (cf. [7]) if for every x,y∈Kwithx∈F yandT y∈K, we have

d(Tx,FT y)≤d(T y,F y), (2.3) whereas the pair (F,T) is said to be compatible (cf. [8]) if for every sequence{xn} ⊂K, from the relation

nlim→∞dFxn,Txn

=0 (2.4)

and Tx_n∈K (for everyn∈N) it follows that lim_n→∞d(T y_n,FTx_n)=0, for every sequence{yn} ⊂Ksuch thatyn∈Fxn,n∈N.

For hybrid pairs of self type mappings these definitions were introduced by Kaneko and Sessa [13].

Definition 2.3[11]. LetKbe a nonempty subset of a metric space (X,d),T:K→Xand F:K→CB(X). The pair (F,T) is said to be quasi-coincidentally commuting if for all coincidence points “x” of (T,F),TFx⊂FTxwheneverFx⊂KandTx∈Kfor allx∈K.

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Definition 2.4[11]. A mappingT:K→X is said to be coincidentally idempotent w.r.t mappingF:K→CB(X), ifTis idempotent at the coincidence points of the pair (F,T).

Definition 2.5[4]. A metric space (X,d) is said to be metrically convex if for anyx,y∈X withx=ythere exists a pointz∈X,x=z=ysuch that

d(x,z) +d(z,y)=d(x,y). (2.5) Lemma 2.6 [4]. Let K be a nonempty closed subset of a metrically convex metric space (X,d). Ifx∈Kandy /∈K then there exists a pointz∈δK (the boundary ofK) such that d(x,z) +d(z,y)=d(x,y).

Lemma2.7 [17]. LetA,B∈CB(X)anda∈A, then for any positive numberq <1 there existsb=b(a)inBsuch thatq·d(a,b)≤H(A,B).

3. Main 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. Let{Fn}^∞_n₌1:K→CB(X)andS,T:K→Xsatisfying

(iv)δK⊆SK∩TK,F_i(K)∩K⊆SK,F_j(K)∩K⊆TK, (v)Tx∈δK⇒Fi(x)⊆K,Sx∈δK⇒Fj(x)⊆K, and

HF_i(x),F_j(y)≤a·max 1

2d(Tx,Sy),dTx,F_i(x),dSy,F_j(y) +bdTx,Fj(y)+dSy,Fi(x),

(3.1)

wherei=2n−1, j=2n,(n∈N),i=j for allx,y∈K withx=y,a,b≥0, and 2b < a,2a+ 3b < q <1,

(vi) (F_i,T)and(F_j,S)are compatible pairs, (vii){Fn},SandTare continuous onK.

Then(F_i,T)as well as(F_j,S)has a point of coincidence.

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. From the implicationTx∈δK which impliesF1(x0)⊆F1(K)∩K⊆SK, letx1∈K be such that y1=Sx1∈F1(x0)⊆K. Sincey1∈F1(x0), there exists a pointy2∈F2(x1) such that

q·dy1,y2

≤HF1

x0

,F2

x1

. (3.2)

Supposey2∈K. Theny2∈F2(K)∩K⊆TKimplies that there exists a pointx2∈Ksuch thaty2=Tx2. Otherwise, ify2∈/ K, then there exists a pointp∈δKsuch that

dSx1,p+dp,y2

=dSx1,y2

. (3.3)

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Sincep∈δK⊆TK, there exists a pointx2∈Kwithp=Tx2so that dSx1,Tx2

+dTx2,y2

=dSx1,y2

. (3.4)

Lety3∈F3(x2) be such thatq·d(y2,y3)≤H(F2(x1),F3(x2)).

Thus, repeating the foregoing arguments, we obtain two sequences{x_n}and{y_n}such that

(viii)y2n∈F2n(x2n−1),y2n+1∈F2n+1(x2n),

(ix)y2n∈K⇒y2n=Tx2nory2n∈/ K⇒Tx2n∈δKand dSx2n−1,Tx2n

+dTx2n,y2n

=dSx2n−1,y2n

, (3.5)

(x)y2n+1∈K⇒y2n+1=Sx2n+1ory2n+1∈/ K⇒Sx2n+1∈δKand dTx2n,Sx2n+1

+dSx2n+1,y2n+1

=dTx2n,y2n+1

. (3.6)

We denote

P◦= Tx2i∈

Tx2n

:Tx2i=y2i , P1=

Tx2i∈ Tx2n

:Tx2i=y2i

, Q◦=

Sx2i+1∈ Sx2n+1

:Sx2i+1=y2i+1

, Q1=

Sx2i+1∈ Sx2n+1

:Sx2i+1=y2i+1 .

(3.7)

One can note that (Tx2n,Sx2n+1)∈P1×Q1and (Sx2n−1,Tx2n)∈Q1×P1. Now, we distinguish the following three cases.

Case 1. If (Tx2n,Sx2n+1)∈P◦×Q◦, then q·dTx2n,Sx2n+1

≤HF2n+1

x2n

,F2n

x2n−1

≤a·max 1

2dTx2n,Sx2n−1

,dTx2n,F2n+1

x2n

,dSx2n−1,F2n

x2n−1

+b·

dTx2n,F2n

x2n−1

+dSx2n−1,F2n+1

x2n

≤a·max 1

2dy2n,y2n−1

,dy2n,y2n+1

,dy2n−1,y2n +b·

dy2n−1,y2n

+dy2n,y2n+1

,

(3.8)

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which in turn yields

dTx2n,Sx2n+1

≤









 a+b

q−b

dSx2n−1,Tx2n

, ifdy2n−1,y2n

≥dy2n+1,y2n

b q−b−a

dSx2n−1,Tx2n

, ifdy2n−1,y2n

≤dy2n+1,y2n

, (3.9) or

dTx2n,Sx2n+1

≤h·dSx2n−1,Tx2n

, (3.10)

whereh=max{((a+b)/(q−b)), (b/(q−b−a))}<1, since 2a+ 3b <1.

Similarly if (Sx2n−1,Tx2n)∈Q◦×P◦, then

dSx2n−1,Tx2n

≤









 a+b

q−b

d(Sx2n−1,Tx2n−2), ifdy2n−2,y2n−1

≥dy2n−1,y2n

b q−b−a

dSx2n−1,Tx2n−2

, ifdy2n−2,y2n−1

≤dy2n−1,y2n , (3.11) or

dSx2n−1,Tx2n

≤h·dSx2n−1,Tx2n−2

, (3.12)

Case 2. If (Tx2n,Sx2n+1)∈P◦×Q1, then dTx2n,Sx2n+1

+dSx2n+1,y2n+1

=dTx2n,y2n+1

, (3.13)

dTx2n,Sx2n+1

≤dTx2n,y2n+1

=dy2n,y2n+1

, (3.14)

and hence

q·dTx2n,Sx2n+1

≤q·dy2n,y2n+1

≤HF2n+1

x2n

,F2n

x2n−1

. (3.15)

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Now, proceeding as inCase 1, we have

dTx2n,Sx2n+1

≤









 a+b

q−b

dSx2n−1,Tx2n), ifdy2n−1,y2n

≥dy2n+1,y2n b

q−b−a

dSx2n−1,Tx2n

, ifdy2n−1,y2n

≤dy2n+1,y2n

, (3.16) or

dTx2n,Sx2n+1

≤h·dSx2n−1,Tx2n

. (3.17)

In case (Sx2n−1,Tx2n)∈Q1×P◦, then as earlier, one also obtains

dSx2n−1,Tx2n

≤









 a+b

q−b

dSx2n−1,Tx2n−2

≥dy2n−1,y2n b

q−b−a

dSx2n−1,Tx2n−2

≤dy2n−1,y2n

, (3.18) or

dSx2n−1,Tx2n

≤h·dSx2n−1,Tx2n−2

, (3.19)

Case 3. If (Tx2n,Sx2n+1)∈P1×Q◦, thenSx2n−1=y2n−1. Proceeding as inCase 1, one gets q·dTx2n,Sx2n+1

=q·dTx2n,y2n+1

≤q·dTx2n,y2n

+q·dy2n,y2n+1

≤q·dSx2n−1,y2n

+HF2n+1(x2n

,F2n

x2n−1

≤q·dSx2n−1,y2n

+a·max 1

2dy2n,y2n−1

,dy2n,y2n+1

,dy2n−1,y2n +bdy2n,y2n

+dy2n−1,y2n+1 ,

(3.20)

dTx2n,Sx2n+1

≤











q+b q−a−b

dSx2n−1,y2n

, ifdy2n−1,y2n

≤dy2n+1,y2n q+a+b

q−b

dSx2n−1,y2n

, ifdy2n−1,y2n

≥dy2n+1,y2n

. (3.21)

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Now, proceeding as earlier, one also obtains

dSx2n−1,y2n

≤









 a+b

q−b

dSx2n−1,Tx2n−2

≥dy2n−1,y2n

b q−a−b

dSx2n−1,Tx2n−2

≤dy2n−1,y2n . (3.22) Therefore combining above inequalities, we have

dTx2n,Sx2n+1

≤k·dSx2n−1,Tx2n−2

, (3.23)

where

k=max

a+b q−b

q+b q−a−b

,

a+b q−b

q+a+b q−b

, b

q−a−b

q+b q−a−b

,

b q−a−b

q+a+b q−b

<1,

(3.24)

since 2a+ 3b <1.

To substantiate that, the inequality 2a+ 3b < q <1 implies all foregoing inequalities, one may note that

2a+ 3b < q=⇒2aq+ 3bq < q², (3.25) or

aq+ab+bq+b²+aq+ 2bq−ab−b²< q², (3.26) or

aq+ab+bq+b²< q²−aq−2bq+ab+b², (3.27) or

a+b q−b

q+b q−a−b

<1, (3.28)

and

2a+ 3b < q=⇒a+ 3b < q, (3.29) or

aq+ 3bq < q²=⇒aq+bq+bq+bq < q², (3.30)

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or

bq+ab+b²< q²−bq−aq+ab−bq+b², (3.31) or

b q−a−b

q+a+b q−b

<1. (3.32)

Similarly one can establish the other inequalities as well. Thus in all the cases, we have dTx2n,Sx2n+1

≤k·maxdSx2n−1,Tx2n

,dTx2n−2,Sx2n−1

(3.33)

whereas

dSx2n+1,Tx2n+2

≤k·maxdSx2n−1,Tx2n

,dTx2n,Sx2n+1

. (3.34)

Now on the lines of Assad and Kirk [4], it can be shown by induction that forn≥1, we have

dTx2n,Sx2n+1

< kⁿ·δ, dSx2n+1,Tx2n+2

< k^n+(1/2)·δ (3.35) whereas

δ=k⁻¹^/²maxdTx0,Sx1

,dSx1,Tx2

. (3.36)

Thus the sequence{Tx0,Sx1,Tx2,Sx3,...,Sx2n−1,Tx2n,Sx2n+1,...}is Cauchy and hence converges to the pointzinX. Then as noted in [7] there exists at least one subsequence {Tx2nk}or{Sx2nk+1}which is contained inP◦ orQ◦ respectively. Suppose that the subsequence{Tx2nk}contained inP◦for eachk∈Nconverges toz. Using compatibility of (F_j,S), we have

klim→∞dSx2nk−1,Fj

x2nk−1

=0 for any even integerj∈N, (3.37)

which implies that lim_k_→∞d(STx2nk,Fj(Sx2nk−1))=0.

Using the continuity ofSandF_j, one obtainsSz∈F_j(z), for any even integer j∈N.

Similarly the continuity ofTandF_iimpliesTz∈F_i(z), for any odd integeri∈N. Now q·d(Tz,Sz)≤HFi(z),Fj(z)

≤a·max 1

2d(Tz,Sz),dTz,Fi(z),dSz,Fj(z) +bdTz,Fj(z)+dSz,Fi(z)

≤a·max 1

2d(Tz,Sz), 0, 0

+bd(Tz,Sz) +d(Tz,Sz)

≤ a

2+ 2b

·d(Tz,Sz),

(3.38)

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yielding therebyTz=Szwhich shows thatzis a common coincidence point of the maps {F_n},SandT.

Remark 3.2. By settingF_i=F(for any odd integeri∈N) andF_j=G(for any even integer j∈N) inTheorem 3.1, one deduces a rectified and sharpened form of a result due to Ahmad and Imdad [2].

Remark 3.3. By settingFi=F (for any odd integeri∈N),Fj=G(for any even integer j∈N) and S=T in Theorem 3.1, one deduces a rectified and improved version of a result due to Ahmad and Imdad [1].

In an attempt to proveTheorem 3.1for pointwiseR-weakly commuting mappings, we have the following.

Theorem3.4. Let(X,d)be a complete metrically convex metric space andK a nonempty closed subset ofX. Let{F_n}^∞_n₌1:K→CB(X)andS,T:K→Xsatisfying (3.1), (iv), (v) and (vii). Suppose that

(xi) (Fi,T)and(Fj,S)are pointwiseR-weakly commuting pairs.

Then(F_i,T)as well as(F_j,S)has a point of coincidence.

Proof. On the lines of the proof ofTheorem 3.1, one can show that the sequence{Tx2n} converges to a point z∈X. Now we assume that there exists a subsequence {Tx2nk} of {Tx2n} which is contained in P◦. Further subsequence {Tx2nk} and{Sx2nk+1}both converge toz∈K as K is a closed subset of the complete metric space (X,d). Since Tx2nk∈Fj(x2nk−1) for any even integer j∈N andSx2nk−1∈K. Using pointwiseR-weak commutativity of (Fj,S), we have

dSF_jx2nk−1

,F_jSx2nk−1

≤R1·dF_jx2nk−1

,Sx2nk−1

(3.39)

for any even integerj∈Nwith someR1>0. Also dSF_jx2nk−1

,F_j(z)≤dSF_jx2nk−1

,F_jSx2nk−1

+HF_jSx2nk−1

,F_j(z). (3.40) Makingk→ ∞in (3.39) and (3.40) and using continuity ofFj as well asS, we getd(Sz, Fj(z))≤0 yielding therebySz∈Fj(z) for any even integerj∈N.

Since y2nk+1∈F_i(x2nk) and{Tx2nk} ∈K, pointwiseR-weak commutativity of (F_i,T) implies

dTF_ix2nk

,F_iTx2nk

≤R2·dF_ix2nk

,Tx2nk

(3.41)

for any odd integeri∈Nwith someR2>0, besides dTF_ix2nk

,F_i(z)≤dTF_ix2nk

,F_iTx2nk

+HF_iTx2nk

,F_i(z). (3.42) Therefore, as earlier the continuity ofFi as well asT impliesd(Tz,Fi(z))≤0 giving therebyTz∈Fi(z) ask→ ∞.

If we assume that there exists a subsequence{Sx2nk+1}contained inQ◦, then analogous arguments establish the earlier conclusions. This concludes the proof.

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In the next theorem, we utilize the closedness ofTKandSKto replace the continuity requirements besides minimizing the commutativity requirements to merely coincidence points.

Theorem3.5. Let(X,d)be a complete metrically convex metric space andK a nonempty closed subset ofX. Let{Fn}^∞_n₌1:K→CB(X)andS,T:K→Xsatisfying (3.1), (iv) and (v).

Suppose that

(xii)TKandSKare closed subspaces ofX. Then () (Fi,T)has a point of coincidence, () (F_j,S)has a point of coincidence.

Moreover,(Fi,T)has a common fixed point ifTis quasi-coincidentally commuting and coincidentally idempotent w.r.tFiwhereas(Fj,S)has a common fixed point providedSis quasi-coincidentally commuting and coincidentally idempotent w.r.tF_j.

Proof. On the lines ofTheorem 3.1, one assumes that there exists a subsequence{Tx2nk} which is contained inP◦andTKas well asSKare closed subspaces ofX. Since{Tx2nk}is Cauchy inTK, it converges to a pointu∈TK. Letv∈T⁻¹u, thenTv=u. Since{Sx2nk+1} is a subsequence of Cauchy sequence,{Sx2nk+1}converges touas well. Using (3.1), one can write

q·dF_i(v),Tx2nk

≤HFi(v),Fj

x2nk−1

≤a·max 1

2dTv,Sx2nk−1

,dSx2nk−1,Fj x2nk−1

,dTv,Fi(v)

+bdTv,Fj x2nk−1

+dSx2nk−1,Fi(v),

(3.43)

which on lettingk→ ∞, reduces to

q·dF_i(v),u≤a·max0,du,F_i(v), 0+b0 +dF_i(v),u

≤(a+b)·du,Fi(v), (3.44)

yielding therebyu∈Fi(v) which implies thatu=Tv∈Fi(v) asFi(v) is closed.

Since Cauchy sequence{Tx2n}converges tou∈Kandu∈F_i(v),u∈F_i(K)∩K⊆SK, there existsw∈Ksuch thatSw=u. Again using (3.1), one gets

q·dSw,Fj(w)=q·dTv,Fj(w)≤HFi(v),Fj(w)

≤a·max 1

2d(Tv,Sw),dTv,Fi(v),dSw,Fj(w) +bdTv,Fj(w)+dSw,Fi(v)

≤(a+b)·dSw,F_j(w),

(3.45)

implying therebySw∈F_j(w), that iswis a coincidence point of (S,F_j).

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If one assumes that there exists a subsequence{Sx2nk+1}contained inQ◦withTK as well asSK are closed subspaces ofX, then noting that{Sx2nk+1}is Cauchy inSK, the foregoing arguments establish thatTv∈Fi(v) andSw∈Fj(w).

Sincevis a coincidence point of (Fi,T) therefore using quasi-coincidentally commut- ing property of (F_i,T) and coincidentally idempotent property ofTw.r.tF_i, one can have Tv∈Fi(v), u=Tv=⇒Tu=TTv=Tv=u, (3.46) thereforeu=Tu=TTv∈TFi(v)⊂Fi(Tv)=Fi(u) which shows thatuis the common fixed point of (Fi,T). Similarly using the quasi-coincidentally commuting property of (F_j,S) and coincidentally idempotent property ofSw.r.tF_j, one can show that (F_j,S) has a common fixed point as well.

By settingS=T=IKinTheorem 3.5, we deduce the following corollary for a sequence of set-valued mappings which is a partially sharpened form of Theorem 2.2 due to Ćirić and Ume [5] as our contraction condition (below) is more general than the condition employed in Ćirić and Ume [5] but Theorem 2.2 due to Ćirić and Ume [5] cannot be derived completely fromTheorem 3.5 as 2a+ 3b <1 does not imply 3a+ 3b+ab <1.

Note that ifa=bandb=cthena+ 2b+ 3c+ac <1 reduces to 3a+ 3b+ab <1.

Corollary3.6. Let(X,d)be a complete metrically convex metric space andKa nonempty closed subset ofX. Let{Fn}^∞_n₌1:K→CB(X)satisfying:

(xiii)x∈δK⇒Fn(x)⊆K, and

HFi(x),Fj(y)≤a·max 1

2d(x,y),dx,Fi(x),dy,Fj(y) +bdx,Fj(y)+dy,Fi(x)

(3.47)

for allx,y∈K withx=y,i=j,a,b≥0such that2a+ 3b <1, then{F_n}has a common fixed point.

Remark 3.7. Theorem 3.5remains true if we substitute closedness of “TKandSK” with closedness of “Fi(K) andFj(K).”

Remark 3.8. By settingS=T=IK inTheorem 3.5, one deduces an extension of a result due to Khan et al. [15] to a sequence of multi-valued mappings.

Remark 3.9. By settingF_n=F(for alln∈N) andS=T=I_KinTheorem 3.5, one deduces a multi-valued version of a result due to Khan et al. [15].

Remark 3.10. By settingFi=F(for any odd integeri∈N),Fj=G(for any even integer j∈N) andS=T=IKinTheorem 3.5, one deduces a sharpened and generalized form of a result due to Khan [14].

Finally, we prove a theorem when “closedness ofK” is replaced by “compactness ofK.”