N. HUSSAIN AND B. E. RHOADES
Received 20 December 2005; Revised 29 March 2006; Accepted 4 April 2006
We obtain common fixed point results for generalizedI-nonexpansive Cq-commuting maps. As applications, various best approximation results for this class of maps are de- rived in the setup of certain metrizable topological vector spaces.
Copyright © 2006 N. Hussain and B. E. Rhoades. This is an open access article distrib- uted under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1. Introduction and preliminaries
LetXbe a linear space. A p-norm onX is a real-valued function onX with 0< p≤1, satisfying the following conditions:
(i)xp≥0 andxp=0⇔x=0, (ii)αxp= |α|pxp,
(iii)x+yp≤ xp+yp,
for allx,y∈X and all scalarsα. The pair (X, · p) is called a p-normed space. It is a metric linear space with a translation invariant metricdpdefined bydp(x,y)= x−yp
for allx,y∈X. If p=1, we obtain the concept of the usual normed space. It is well known that the topology of every Hausdorff locally bounded topological linear space is given by some p-norm, 0< p≤1 (see [7,13] and references therein). The spaceslp
andLp, 0< p≤1, are p-normed spaces. A p-normed space is not necessarily a locally convex space. Recall that dual spaceX∗(the dual ofX) separates points ofXif for each nonzerox∈X, there exists f ∈X∗such that f(x)=0. In this case the weak topology on Xis well defined and is Hausdorff. Notice that ifXis not locally convex space, thenX∗ need not separate the points ofX. For example, ifX=Lp[0, 1], 0< p <1, thenX∗= {0} [17, pages 36–37]. However, there are some nonlocally convex spacesX(such as the p- normed spaceslp, 0< p <1) whose dualX∗separates the points ofX. In the sequel, we will assume thatX∗separates points of ap-normed spaceXwhenever weak topology is under consideration.
Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2006, Article ID 24543, Pages1–9 DOI10.1155/FPTA/2006/24543
LetXbe a metric linear space andMa nonempty subset ofX. The setPM(u)= x∈ M:d(x,u)=dist(u,M)is called the set of best approximations tou∈Xout ofM, where dist(u,M)=infd(y,u) :y∈M. Let f :M→Mbe a mapping. A mappingT:M→M is called an f-contraction if there exists 0≤k <1 such thatd(Tx,T y)≤k d(f x,f y) for anyx,y∈M. Ifk=1, thenTis calledf-nonexpansive. The set of fixed points ofT(resp., f) is denoted byF(T) (resp.,F(f)). A pointx∈Mis a common fixed (coincidence) point of f andTifx= f x=Tx(f x=Tx). The set of coincidence points of f andTis denoted byC(f,T). A mappingT:M→Mis called
(1) hemicompact if any sequence{xn}inMhas a convergent subsequence whenever d(xn,Txn)→0 asn→ ∞;
(2) completely continuous if{xn}converges weakly toxwhich implies that{Txn} converges strongly toTx;
(3) demiclosed at 0 if for every sequence{xn} ∈Msuch that{xn}converges weakly toxand{Txn}converges strongly to 0, we haveTx=0.
The pair{f,T}is called
(4) commuting ifT f x= f Txfor allx∈M;
(5)R-weakly commuting if for allx∈Mthere existsR >0 such thatd(f Tx,T f x)≤ R d(f x,Tx). IfR=1, then the maps are called weakly commuting;
(6) compatible [10] if limnd(T f xn,f Txn)=0 whenever{xn}is a sequence such that limnTxn=limnf xn=tfor sometinM;
(7) weakly compatible [2,11] if they commute at their coincidence points, that is, if f Tx=T f xwhenever f x=Tx. The setMis calledq-starshaped withq∈Mif the segment [q,x]= {(1−k)q+kx: 0≤k≤1}joiningqtoxis contained inM for allx∈M. Suppose thatMisq-starshaped withq∈F(f) and is bothT- and
f-invariant. ThenTand f are called
(8)R-subcommuting onM(see [19,20]) if for allx∈M, there exists a real number R >0 such thatd(f Tx,T f x)≤(R/k)d((1−k)q+kTx,f x) for eachk∈(0, 1];
(9)R-subweakly commuting onM (see [7,21]) if for allx∈M, there exists a real numberR >0 such thatd(f Tx,T f x)≤Rdist(f x, [q,Tx]);
(10)Cq-commuting [2] if f Tx=T f xfor allx∈Cq(f,T), whereCq(f,T)= ∪{C(f, Tk) : 0≤k≤1} and Tkx=(1−k)q+kTx. Clearly, Cq-commuting maps are weakly compatible but not conversely in general.R-subcommuting andR-sub- weakly commuting maps areCq-commuting but the converse does not hold in general [2].
Meinardus [14] employed the Schauder fixed point theorem to prove a result regarding invariant approximation. Singh [22] proved the following extension of “Meinardus’s”
result.
Theorem 1.1. LetT be a nonexpansive operator on a normed spaceX,M aT-invariant subset ofX, andu∈F(T). IfPM(u) is nonempty compact and starshaped, thenPM(u)∩ F(T)= ∅.
Sahab et al. [18] established an invariant approximation result which containsTheo- rem 1.1. Further generalizations of the result of Meinardus are obtained by Al-Thagafi [1],
Shahzad [19–21], Hussain and Berinde [7], Rhoades and Saliga [16], and O’Regan and Shahzad [15].
The aim of this paper is to establish a general common fixed point theorem forCq- commuting generalizedI-nonexpansive maps in the setting of locally bounded topolog- ical vector spaces, locally convex topological vector spaces, and metric linear spaces. We apply a new theorem to derive some results on the existence of best approximations. Our results unify and extend the results of Al-Thagafi [1], Al-Thagafi and Shahzad [2], Dot- son [3], Guseman and Peters [4], Habiniak [5], Hussain [6], Hussain and Berinde [7], Hussain and Khan [8], Hussain et al. [9], Jungck and Sessa [12], Khan and Khan [13], O’Regan and Shahzad [15], Rhoades and Saliga [16], Sahab et al. [18], Shahzad [19–21], and Singh [22].
2. Common fixed point and approximation results
The following result extends and improves [2, Theorem 2.1], [21, Theorem 2.1], and [15, Lemma 2.1].
Theorem 2.1. LetMbe a subset of a metric space (X,d), and letI andTbe weakly com- patible self-maps ofM. Assume that cl(T(M))⊂I(M), cl(T(M)) is complete, andTandI satisfy for allx,y∈Mand 0≤h <1,
dTx,T y≤hmaxdIx,I y,dIx,Tx,dI y,T y,dIx,T y,dI y,Tx. (2.1) ThenF(I)∩F(T) is a singleton.
Proof. AsT(M)⊂I(M), one can choosexninMforn∈N, such thatTxn=Ixn+1. Then following the arguments in [15, Lemma 2.1], we infer that{Txn}is a Cauchy sequence.
It follows from the completeness of cl(T(M)) thatTxn→wfor somew∈Mand hence Ixn→wasn→ ∞. Consequently, limnIxn=limnTxn=w∈cl(T(M))⊂I(M). Thusw= I yfor somey∈M. Notice that for alln≥1, we have
dw,T y≤dw,Txn+dTxn,T y≤dw,Txn +hmaxdIxn,I y,dTxn,Ixn
,dT y,I y,dT y,Ixn
,dTxn,I y. (2.2) Lettingn→ ∞, we obtainI y=w=T y. We now show thatT yis a common fixed point of IandT. SinceIandTare weakly compatible andI y=T y, we obtain by the definition of weak compatibility thatIT y=TI y. Thus we haveT2y=TI y=IT yand so by inequality (2.1),
d(TT y,T y)≤hmaxd(IT y,I y),d(IT y,TT y),d(I y,T y),d(IT y,T y),d(I y,TT y)
≤hd(IT y,T y).
(2.3) HenceTT y=T yash∈(0, 1) and soT y=TT y=IT y. This implies thatT yis a com- mon fixed point ofT andI. Inequality (2.1) further implies the uniqueness of the com-
mon fixed pointT y. HenceF(I)∩F(T) is a singleton.
We can prove now the following.
Theorem 2.2. LetI andT be self-maps on aq-starshaped subsetMof a p-normed space X. Assume that cl(T(M))⊂I(M),q∈F(I), andI is affine. Suppose thatTandI areCq- commuting and satisfy
Tx−T yp≤max
⎧⎨
⎩
Ix−I yp, distIx, [Tx,q], distI y, [T y,q], distIx, [T y,q], distI y, [Tx,q]
⎫⎬
⎭ (2.4)
for allx,y∈M. IfT is continuous, thenF(T)∩F(I)= ∅, provided one of the following conditions holds:
(i) cl(T(M)) is compact andIis continuous;
(ii)Mis complete,F(I) is bounded, andTis a compact map;
(iii)Mis bounded, and complete,Tis hemicompact andIis continuous;
(iv)Xis complete,Mis weakly compact,Iis weakly continuous, andI−Tis demiclosed at 0;
(v)Xis complete,Mis weakly compact,Tis completely continuous, andIis continuous.
Proof. DefineTn:M→Mby
Tnx=
1−knq+knTx (2.5)
for someqand allx∈Mand a fixed sequence of real numberskn(0< kn<1) converging to 1. Then, for eachn, cl(Tn(M))⊂I(M) asM isq-starshaped, cl(T(M))⊂I(M),I is affine, andIq=q. AsIandTareCq-commuting andIis affine withIq=q, then for each x∈Cq(I,T),
ITnx= 1−kn
q+knITx= 1−kn
q+knTIx=TnIx. (2.6) ThusITnx=TnIxfor eachx∈C(I,Tn)⊂Cq(I,T). HenceIandTnare weakly compatible for alln. Also by (2.4),
Tnx−Tnyp=
knpTx−T yp
≤ knp
maxIx−I yp, distIx, [Tx,q], distI y, [T y,q], distIx, [T y,q], distI y, [Tx,q]
≤ kn
p
maxIx−I yp,Ix−Tnxp,I y−Tnyp, Ix−Tnyp,I y−Tnxp,
(2.7)
for eachx,y∈M.
(i) Since cl(T(M)) is compact, cl(Tn(M)) is also compact. ByTheorem 2.1, for each n≥1, there existsxn∈M such thatxn=Ixn= Tnxn. The compactness of cl(T(M)) implies that there exists a subsequence{Txm}of{Txn}such thatTxm→yasm→ ∞. Then the definition ofTmxm impliesxm→y, so by the continuity ofT andI, we have y∈F(T)∩F(I). ThusF(T)∩F(I)= ∅.
(ii) As in (i), there is a uniquexn∈Msuch thatxn=Tnxn=Ixn. AsTis compact and {xn}being inF(I) is bounded, so{Txn}has a subsequence{Txm}such that{Txm} →y asm→ ∞. Then the definition ofTmxmimpliesxm→y, so by the continuity ofTandI, we havey∈F(T)∩F(I). ThusF(T)∩F(I)= ∅.
(iii) As in (i), there existsxn∈M such thatxn=Ixn=Tnxn, andM is bounded, so xn−Txn=(1−(kn)−1)(xn−q)→0 asn→ ∞and hencedp(xn,Txn)→0 asn→ ∞. The hemicompactness ofTimplies that{xn}has a subsequence{xj}which converges to some z∈M. By the continuity ofTandIwe havez∈F(T)∩F(I). ThusF(T)∩F(I)= ∅.
(iv) As in (i), there existsxn∈Msuch thatxn=Ixn=Tnxn. SinceMis weakly com- pact, we can find a subsequence{xm}of{xn}inMconverging weakly toy∈Masm→ ∞ and asI is weakly continuous soI y=y. By (iii)Ixm−Txm→0 asm→ ∞. The demi- closedness ofI−Tat 0 implies thatI y=T y. ThusF(T)∩F(I)= ∅.
(v) As in (iv), we can find a subsequence{xm}of {xn} in M converging weakly to y∈M asm→ ∞. SinceT is completely continuous,Txm →T y asm→ ∞. Sincekn→ 1,xm=Tmxm=kmTxm+ (1−km)q→T y asm→ ∞. ThusTxm →T2y asm→ ∞and consequentlyT2y=T y implies thatTw=w, where w=T y. Also, since Ixm=xm→ T y=w, using the continuity ofIand the uniqueness of the limit, we haveIw=w. Hence
F(T)∩F(I)= ∅.
The following corollary improves and generalizes [2, Theorem 2.2] and [7, Theorem 2.2].
Corollary 2.3. LetMbe aq-starshaped subset of ap-normed spaceX, andIandTcontin- uous self-maps ofM. Suppose thatIis affine withq∈F(I), cl(T(M))⊂I(M), and cl(T(M)) is compact. If the pair{I,T}isR-subweakly commuting and satisfies (2.4) for allx,y∈M, thenF(T)∩F(I)= ∅.
Remark 2.4. Theorem 2.2extends and improves Al-Thagafi’s [1, Theorem 2.2], Dotson’s [3, Theorem 1], Habiniak’s [5, Theorem 4], Hussain and Berinde’s [7, Theorem 2.2], O’Regan and Shahzad’s [15, Theorem 2.2], Shahzad’s [21, Theorem 2.2], and the main result of Rhoades and Saliga [16].
The following provides the conclusion of [13, Theorem 2] without the closedness of M.
Corollary 2.5. LetMbe a nonemptyq-starshaped subset of ap-normed spaceX. IfTis nonexpansive self-map ofMand cl(T(M)) is compact, thenF(T)= ∅.
The following result contains properlyTheorem 1.1, [18, Theorem 3], and improves and extends [2, Theorem 3.1], [5, Theorem 8], [13, Theorem 4], and [19, Theorem 6].
Theorem 2.6. LetM be a subset of ap-normed spaceXand letI,T:X→Xbe mappings such thatu∈F(T)∩F(I) for someu∈XandT(∂M∩M)⊂M. Assume thatI(PM(u))= PM(u) and the pair{I,T}isCq-commuting and continuous onPM(u) and satisfies for all x∈PM(u)∪ {u},
Tx−T yp≤
⎧⎪
⎪⎪
⎨
⎪⎪
⎪⎩
Ix−Iup ify=u,
maxIx−I yp, distIx, [q,Tx], distI y, [q,T y],
distIx, [q,T y], distI y, [q,Tx] ify∈PM(u).
(2.8) Suppose thatPM(u) is closed,q-starshaped withq∈F(I),I is affine, and cl(T(PM(u))) is compact. ThenPM(u)∩F(I)∩F(T)= ∅.
Proof. Letx∈PM(u). Thenx−up=dist(u,M). Note that for anyk∈(0, 1),ku+ (1− k)x−up=(1−k)px−up<dist(u,M).
It follows that the line segment{ku+ (1−k)x: 0< k <1}and the setMare disjoint.
Thusxis not in the interior ofMand sox∈∂M∩M. SinceT(∂M∩M)⊂M,Txmust be inM. Also sinceIx∈PM(u),u∈F(T)∩F(I) andT, andIsatisfy (2.8), we have
Tx−up= Tx−Tup≤ Ix−Iup= Ix−up=dist(u,M). (2.9) ThusTx∈PM(u).Theorem 2.2(i) further guarantees thatPM(u)∩F(I)∩F(T)= ∅.
LetD=PM(u)∩CIM(u), whereCMI (u)=
x∈M:Ix∈PM(u).
The following result contains [1, Theorem 3.2], extends [2, Theorem 3.2], and pro- vides a nonlocally convex space analogue of [8, Theorem 3.3] for more general class of maps.
Theorem 2.7. LetM be a subset of ap-normed spaceX, andI andT:X→Xmappings such thatu∈F(T)∩F(I) for someu∈XandT(∂M∩M)⊂M. Suppose thatDis closed q-starshaped withq∈F(I),Iis affine, cl(T(D)) is compact,I(D)=D, and the pair{T,I} isCq-commuting and continuous onDand, for allx∈D∪ {u}, satisfies the following in- equality:
Tx−T yp≤
⎧⎪
⎪⎪
⎨
⎪⎪
⎪⎩
Ix−Iup ify=u,
maxIx−I yp, distIx, [q,Tx], distI y, [q,T y],
distIx, [q,T y], distI y, [q,Tx] ify∈D.
(2.10) IfIis nonexpansive onPM(u)∪ {u}, thenPM(u)∩F(I)∩F(T)= ∅.
Proof. Letx∈D, then proceeding as in the proof ofTheorem 2.6, we obtainTx∈PM(u).
Moreover, sinceIis nonexpansive onPM(u)∪ {u}andTsatisfies (2.10), we obtain ITx−up≤ Tx−Tup≤ Ix−Iup=dist(u,M). (2.11) ThusITx∈PM(u) and soTx∈CIM(u). HenceTx∈D. Consequently, cl(T(D))⊂D= I(D). NowTheorem 2.2(i) guarantees thatPM(u)∩F(I)∩F(T)= ∅.
Remark 2.8. Notice that approximation results similar to Theorems2.6–2.7can be ob- tained, usingTheorem 2.2(ii)–(v).
3. Further remarks
(1) All results of the paper (Theorem 2.2–Remark 2.8) remain valid in the setup of a metrizable locally convex topological vector space (TVS) (X,d), where dis translation invariant andd(αx,αy)≤αd(x,y), for eachαwith 0< α <1 andx,y∈X(recall thatdp is translation invariant and satisfiesdp(αx,αy)≤αpdp(x,y) for any scalarα≥0).
Consequently, Hussain and Khan’s [8, Theorems 2.2–3.3] are improved and extended.
(2) Following the arguments as above, we can obtain all of the recent best approxi- mation results due to Hussain and Berinde’s [7, Theorem 3.2–Corollary 3.4] for more general class ofCq-commuting mapsIandT.
(3) A subsetMof a linear spaceXis said to have property (N) with respect toT[7,9]
if
(i)T:M→M,
(ii) (1−kn)q+knTx∈M, for someq∈Mand a fixed sequence of real numberskn (0< kn<1) converging to 1 and for eachx∈M.
A mappingI is said to have property (C) on a setM with property (N) ifI((1−kn)q+ knTx)=(1−kn)Iq+knITxfor eachx∈Mandn∈N.
All of the results of the paper (Theorem 2.2–Remark 2.8) remain valid, providedI is assumed to be surjective and theq-starshapedness of the setM and affineness ofI are replaced by the property (N) and property (C), respectively, in the setup of p-normed spaces and metrizable locally convex topological vector spaces (TVS) (X,d) whered is translation invariant andd(αx,αy)≤αd(x,y), for eachαwith 0< α <1 andx,y∈X.
Consequently, recent results due to Hussain [6], Hussain and Berinde [7], and Hussain et al. [9] are extended to a more general class ofCq-commuting maps.
(4) Let (X,d) be a metric linear space with a translation invariant metricd. We say that the metricdis strictly monotone [4] ifx=0 and 0< t <1 implyd(0,tx)< d(0,x). Each p-norm generates a translation invariant metric, which is strictly monotone [4,7].
Using [10, Theorem 3.2], we establish the following generalization of Al-Thagafi and Shahzad’s [2, Theorem 2.2 ], Dotson’s [3, Theorem 1], Guseman and Peters’s [4, Theorem 2], and Hussain and Berinde’s [7, Theorem 3.6].
Theorem 3.1. LetTandIbe self-maps on a compact subsetMof a metric linear space (X,d) with translation invariant and strictly monotone metricd. Assume thatMisq-starshaped, cl(T(M))⊂I(M),q∈F(I), andIis affine (orMhas the property (N) withq∈F(I),Isatis- fies the condition (C), andM=I(M)). Suppose thatTandIare continuous,Cq-commuting and satisfy
dTx,T y≤max
⎧⎪
⎨
⎪⎩
dIx,I y, distIx, [Tx,q], distI y, [T y,q], 1
2
distIx, [T y,q]+ distI y, [Tx,q]
⎫⎪
⎬
⎪⎭ (3.1)
for allx,y∈M. ThenF(T)∩F(I)= ∅.
Proof. Two continuous maps defined on a compact domain are compatible if and only if they are weakly compatible (cf. [10, Corollary 2.3]). To obtain the result, use an argument similar to that inTheorem 2.2(i) and apply [10, Theorem 3.2] instead ofTheorem 2.1.
(5) Similarly, all other results ofSection 2 (Corollary 2.3–Theorem 2.7) hold in the setting of metric linear space (X,d) with translation invariant and strictly monotone metricdprovided we replace compactness of cl(T(M)) by compactness ofMand using Theorem 3.1instead ofTheorem 2.2(i).
Acknowledgment
The authors would like to thank the referee for his valuable suggestions to improve the presentation of the paper.
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N. Hussain: Department of Mathematics, King Abdul Aziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia
E-mail address:[email protected]
B. E. Rhoades: Department of Mathematics, Indiana University, Bloomington, IN 47405-7106, USA
E-mail address:[email protected]
