RESULTS IN CERTAIN METRIZABLE TOPOLOGICAL
VECTOR SPACES
NAWAB HUSSAIN AND VASILE BERINDE
Received 27 June 2005; Revised 1 September 2005; Accepted 6 September 2005
We obtain common fixed point results for generalizedI-nonexpansiveR-subweakly com-muting maps on nonstarshaped domain. As applications, we establish noncommutative versions of various best approximation results for this class of maps in certain metrizable topological vector spaces.
Copyright © 2006 N. Hussain and V. Berinde. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, dis-tribution, 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 Hausdorfflocally bounded topological linear space is given by somep-norm, 0< p≤1 (see [9] and references therein). The spaceslpandLp, 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 nonzero x∈X, there exists f ∈X∗such that f(x)=0. In this case the weak topology onX is 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}([12, pages 36 and 37]). However, there are some non-locally convex spacesX (such as the p-normed spaceslp, 0< p <1) whose dualX∗separates the points ofX.
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 approximants tou∈Xout ofM, where dist(u,M)=inf{d(y,u) :y∈M}. Let f :M→Mbe a mapping. A mappingT:M→M
Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2006, Article ID 23582, Pages1–13
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, thenT is called f-nonexpansive. A mappingT:M→M is called condensing if for any bounded subsetBofMwithα(B)>0,α(T(B))< α(B), where α(B)=inf{r >0 :Bcan be covered by a finite number of sets of diameter ≤r}. A map-pingT:M→Mis hemicompact if any sequence{xn}inMhas a convergent subsequence wheneverd(xn,Txn)→0 asn→ ∞. The set of fixed points ofT(resp. f) is denoted by F(T) (resp.F(f)). A pointx∈Mis a common fixed point off andTifx= f x=Tx. The pair{f,T}is called (1) commuting ifT f x= f Txfor allx∈M; (2)R-weakly commut-ing [16] if for allx∈Mthere existsR >0 such thatd(f Tx,T f x)≤Rd(f x,Tx). IfR=1, then the maps are called weakly commuting. The setMis calledq-starshaped withq∈M if the segment [q,x]= {(1−k)q+kx: 0≤k≤1}joiningqtox, is contained inMfor all x∈M. Suppose thatM isq-starshaped withq∈F(f) and is bothT- and f-invariant. ThenTand f are calledR-subweakly commuting onM(see [17]) if for allx∈M, there exists a real numberR >0 such thatd(f Tx,T f x)≤Rdist(f x, [q,Tx]). It is well-known that commuting maps areR-subweakly commuting maps andR-subweakly commuting maps areR-weakly commuting but not conversely in general (see [16,17]).
A setMis said to have property (N) if [7,11] (i)T:M→M,
(ii) (1−kn)q+knTx∈M, for someq∈M and a fixed sequence of real numbers kn(0< kn<1) converging to 1 and for eachx∈M.
A mapping f is said to have property (C) on a setMwith property (N) if f((1−kn)q+ knTx)=(1−kn)f q+knf Txfor eachx∈Mandn∈N.
We extend the concept ofR-subweakly commuting maps to nonstarshaped domain in the following way (see [7]):
Let f andT be self-maps on the setM having property (N) withq∈F(f). Then f andTare calledR-subweakly commuting onM, provided for allx∈M, there exists a real numberR >0 such thatd(f Tx,T f x)≤Rd(f x,Tnx) whereTnx=(1−kn)q+knTx, and the sequence{kn}is as in definition of property (N) ofM. EachT-invariantq-starshaped set has property (N) but not conversely in general. Each affine map on aq-starshaped set Msatisfies condition (C).
Example 1.1[7]. ConsiderX=R2 andM= {(0,y) :y∈[−1, 1]} ∪ {(1−1/(n+ 1), 0) : n∈N} ∪ {(1, 0)}with the metric induced by the norm(a,b) = |a|+|b|, (a,b)∈R2. DefineTonMas follows:
T(0,y)=(0,−y), T
1− 1 n+ 1, 0
=
0, 1− 1 n+ 1
, T(1, 0)=(0, 1). (1.1)
Clearly,M is not starshaped [11] butM has the property (N) for q=(0, 0) andkn= 1−1/(n+ 1). Define I(0,y)=I(1−1/(n+ 1), 0)=(0, 0), I(1, 0)=(1, 0). Then TIx− ITx =0 or 1. Thus for allxinM,TIx−ITx ≤RknTx−Ixwith eachR≥1 and q=(0, 0)∈F(I). ThusIandTareR-subweakly commuting but not commuting onM.
In 1963, Meinardus [10] employed the Schauder fixed point theorem to prove a result regarding invariant approximation. In 1979, Singh [19] proved the following extension of “Meinardus” result.
Theorem1.2. LetTbe a nonexpansive operator on a normed spaceX,Mbe aT-invariant subset of X and u∈F(T). IfPM(u)is nonempty compact and starshaped, then PM(u)∩ F(T)= ∅.
In 1988, Sahab et al. [13] established the following result which containsTheorem 1.2 and many others.
Theorem1.3. LetI andT be selfmaps of a normed spaceXwithu∈F(I)∩F(T),M⊂ X withT(∂M)⊂M, and q∈F(I). IfPM(u)is compact and q-starshaped, I(PM(u))= PM(u),I is continuous and linear onPM(u),I andT are commuting onPM(u)andT is I-nonexpansive onPM(u)∪ {u}, thenPM(u)∩F(T)∩F(I)= ∅.
LetD=PM(u)∩CIM(u), whereCMI (u)= {x∈M:Ix∈PM(u)}.
Theorem1.4 [1, Theorem 3.2]. LetI andT be selfmaps of a Banach spaceX withu∈ F(I)∩F(T),M⊂XwithT(∂M∩M)⊂M. Suppose thatDis closed andq-starshaped with q∈F(I),I(D)=D,Iis linear and continuous onD. IfIandTare commuting onDandT isI-nonexpansive onD∪ {u}withcl(T(D))compact, thenPM(u)∩F(T)∩F(I)= ∅.
Recently, by introducing the concept of non-commuting maps to this area, Shahzad [14–18], Hussain and Khan [6] and Hussain et al. [7], further extended and improved the above mentioned results to non-commuting maps.
The aim of this paper is to prove new results extending and subsuming the above mentioned invariant approximation results. To do this, we establish a general common fixed point theorem forR-subweakly commuting generalizedI-nonexpansive maps on nonstarshaped domain in the setting of locally bounded topological 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], Dotson [3], Guseman and Peters [4], Habiniak [5], Hussain and Khan [6], Hussain et al. [7], Khan and Khan [9], Sahab et al. [13], Shahzad [14–18], and Singh [19].
2. Common fixed point and approximation results
The following common fixed point result is a consequence of Theorem 1 of Berinde [2], which will be needed in the sequel.
Theorem2.1. LetMbe a closed subset of a metric space(X,d)andT and f beR-weakly commuting self-maps of M such thatT(M)⊂ f(M). Suppose there existsk∈(0, 1) such that
d(Tx,T y)≤kmaxd(f x,f y),d(Tx,f x),d(T y,f y),d(Tx,f y),d(T y,f x) (2.1)
We can prove now the following.
Theorem2.2. LetT,Ibe self-maps on a subsetMof ap-normed spaceX. Assume thatM has the property(N)withq∈F(I),I satisfies the condition(C)andM=I(M). Suppose thatTandIareR-subweakly commuting and satisfy
Tx−T yp≤maxIx−I yp, dist(Ix, [Tx,q]), dist(I y, [T y,q]),
dist(Ix, [T y,q]), dist(I y, [Tx,q])
(2.2)
for allx,y∈M. IfT is continuous, thenF(T)∩F(I)= ∅, provided one of the following conditions holds:
(i)Mis closed,cl(T(M))is compact andIis continuous,
(ii)Mis bounded and complete,Tis hemicompact andIis continuous, (iii)Mis bounded and complete,Tis condensing andIis continuous,
(iv)Xis complete with separating dualX∗,M is weakly compact,T is completely con-tinuous andIis continuous.
Proof. DefineTnbyTnx=(1−kn)q+knTxfor allx∈Mand fixed sequence of real num-berskn(0< kn<1) converging to 1. Then, eachTnis a well-defined self-mapping ofMas Mhas property (N) and for eachn,Tn(M)⊂M=I(M). Now the property (C) ofIand theR-subweak commutativity of{T,I}imply that
TnIx−ITnx
p=
knpTIx−ITxp≤knpRdist(Ix, [Tx,q])
≤kn
p
RTnx−Ixp
(2.3)
for allx∈M. This implies that the pair{Tn,I}is (kn)pR-weakly commuting for eachn. Also by (2.2),
Tnx−Tny
p=
knpTx−T yp
≤kn
p
maxIx−I yp, dist(Ix, [Tx,q]), dist(I y, [T y,q]),
dist(Ix, [T y,q]), dist(I y, [Tx,q])
≤kn
pmax Ix−I y
p,Ix−Tnxp,I y−Tnyp,
Ix−Tny
p,I y−Tnxp
(2.4)
for eachx,y∈M.
(ii) 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)= ∅.
(iii) Every condensing map on a complete bounded subset of a metric space is hemi-compact. Hence the result follows from (ii).
(iv) As in (i) there existsxn∈Msuch thatxn=Ixn=Tnxn. SinceMis weakly compact, we can find a subsequence{xm}of {xn}in M converging weakly to y∈M asm→ ∞. Since T is completely continuous, Txm→T y as m→ ∞. Sincekn→1, xm=Tmxm= kmTxm+ (1−km)q→T yasm→ ∞. ThusTxm→T2yasm→ ∞and consequentlyT2y= T yimplies thatTw=w, wherew=T y. Also, sinceIxm=xm→T y=w, using the conti-nuity ofIand the uniqueness of the limit, we haveIw=w. HenceF(T)∩F(I)= ∅. It is clear that eachT-invariantq-starshaped set satisfies the property (N) and ifI is affine, thenIsatisfies the condition (C) andTn(M)⊂I(M) providedT(M)⊂I(M) and q∈F(I).
Corollary2.3. LetMbe a closedq-starshaped subset of ap-normed spaceX, andTand I continuous self-maps ofM. Suppose thatI is affine with q∈F(I), T(M)⊂I(M)and clT(M)is compact. If the pair{T,I} isR-subweakly commuting and satisfy (2.2) for all x,y∈M, thenF(T)∩F(I)= ∅.
Corollary2.4 [18, Theorem 2.2]. Let M be a closedq-starshaped subset of a normed spaceX, andT andI continuous self-maps ofM. Suppose thatI is affine withq∈F(I), T(M)⊂I(M)andclT(M)is compact. If the pair{T,I}isR-subweakly commuting and satisfy, for allx,y∈M,
Tx−T y ≤max
Ix−I y, dist(Ix, [Tx,q]), dist(I y, [T y,q]),
1
2[dist(Ix, [T y,q]) + dist(I y, [Tx,q])]
,
(2.5)
thenF(T)∩F(I)= ∅.
The following corollary improves and generalizes [1, Theorem 2.2].
Corollary2.5. LetMbe a nonempty closed andq-starshaped subset of ap-normed space X and I be continuous self-map ofM. Suppose that I is affine with q∈F(I),T(M)⊂ I(M)and clT(M)is compact. If the pair {T,I}is R-subweakly commuting andT isI -nonexpansive onM, thenF(T)∩F(I)= ∅.
Corollary2.7 [9, Theorem 2]. LetM be a nonempty closed andq-starshaped subset of a p-normed spaceX. If T is nonexpansive self-map ofM and clT(M) is compact, then F(T)= ∅.
We now derive some approximation results.
LetDRM,I(u)=PM(u)∩GRM,I(u), whereGMR,I(u)={x∈M:Ix−up≤(2R+1) dist(u,M)}. The following result extends Theorem 2.3 of Shahzad [16] from theI -nonexpansive-ness ofTto a more general condition.
Theorem2.8. LetMbe subset of ap-normed spaceXandI,T:X→Xbe mappings such thatu∈F(T)∩F(I)for someu∈X andT(∂M∩M)⊂M. IfI(DRM,I(u))=DMR,I(u)and the pair{T,I} isR-subweakly commuting and continuous on DMR,I(u)and satisfy for all x∈DRM,I(u)∪ {u},
Tx−T yp≤
⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩
Ix−Iup if y=u,
maxIx−I yp, dist(Ix, [q,Tx]), dist(I y, [q,T y]),
dist(Ix, [q,T y]), dist(I y, [q,Tx]) ify∈DMR,I(u), (2.6)
thenDMR,I(u)isT-invariant. Suppose thatDMR,I(u)is closed andcl(T(DMR,I(u)))is compact. IfDMR,I(u)has property(N)withq∈F(I), andI satisfies property(C)onDRM,I(u), then PM(u)∩F(I)∩F(T)= ∅.
Proof. Letx∈DRM,I(u). Then,x∈PM(u) and hencex−up=dist(u,M). Note that for anyk∈(0, 1),
ku+ (1−k)x−up=(1−k)px−up<dist(u,M). (2.7)
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) andTandIsatisfy (2.6), we have
Tx−up= Tx−Tup≤ Ix−Iup= Ix−up=dist(u,M). (2.8)
ThusTx∈PM(u). From theR-subweak commutativity of the pair{T,I}and (2.6), it follows that (see also proof of [16, Theorem 2.3]),
ITx−up= ITx−TIx+TIx−Tup≤RTx−Ixp+I2x−Iup
=RTx−u+u−Ixp+I2x−up
≤RTx−up+Ix−up
+I2x−u p
≤(2R+ 1) dist(u,M).
(2.9)
ThusTx∈GRM,I(u). Consequently,T(DMR,I(u))⊂DRM,I(u)=I(DMR,I(u)). NowTheorem 2.2(i)
Remarks 2.9. (1) Ifp=1 andMisq-starshaped withq∈F(I),T(M)⊂I(M) andIis lin-ear onDMR,I(u) inTheorem 2.8, we obtain the conclusion of a recent result [18, Theorem 2.5] for the more general inequality (2.6).
(2) Let CMI (u)= {x∈M :Ix∈PM(u)}. Then I(PM(u))⊂PM(u) implies PM(u)⊂ CI
M(u)⊂GR ,I
M(u) and henceDR ,I
M (u)=PM(u). Consequently,Theorem 2.8remains valid when DRM,I(u)=PM(u). Hence we obtain the following result which contains properly Theorems1.2and1.3and improves and extends Theorem 8 of [5], Theorem 4 in [9], and Theorem 6 in [14,15].
Corollary2.10. LetMbe 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{T,I}isR-subweakly commuting and continuous onPM(u)and satisfy for allx∈PM(u)∪ {u},
Tx−T yp≤
⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩
Ix−Iup ify=u,
maxIx−I yp, dist(Ix, [q,Tx]), dist(I y, [q,T y]),
dist(Ix, [q,T y]), dist(I y, [q,Tx]) ify∈PM(u). (2.10)
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)= ∅.
LetD=PM(u)∩CIM(u), whereCMI (u)= {x∈M:Ix∈PM(u)}. The following result containsTheorem 1.4and many others.
Theorem2.11. LetMbe subset of ap-normed spaceXandI,T:X→Xbe mappings such thatu∈F(T)∩F(I)for someu∈XandT(∂M∩M)⊂M. IfI(D)=Dand the pair{T,I} is commuting and continuous onDand satisfy for allx∈D∪ {u},
Tx−T yp≤
⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩
Ix−Iup ify=u,
maxIx−I yp, dist(Ix, [q,Tx]), dist(I y, [q,T y]),
dist(Ix, [q,T y]), dist(I y, [q,Tx]) ify∈D, (2.11)
thenDisT-invariant. Suppose thatDis closed andcl(T(D))is compact. IfDhas property (N)withq∈F(I), andIsatisfies property(C)onD, thenPM(u)∩F(I)∩F(T)= ∅. Proof. Letx∈D, then proceeding as in the proof ofTheorem 2.8, we obtainTx∈PM(u). Moreover, sinceTcommutes withIonDandTsatisfies (2.11),
ITx−up= TIx−Tup≤I2x−Iup=I2x−up=dist(u,M). (2.12)
Theorem 2.12. Let M be subset of a p-normed spaceX andI,T:X→X be mappings such thatu∈F(T)∩F(I)for someu∈XandT(∂M∩M)⊂M. IfI(D)=Dand the pair {T,I}isR-subweakly commuting and continuous onDand, for allx∈D∪ {u}, satisfies the following inequality,
Tx−T yp≤
⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩
Ix−Iup ify=u,
maxIx−I yp, dist(Ix, [q,Tx]), dist(I y, [q,T y]),
dist(Ix, [q,T y]), dist(I y, [q,Tx]) ify∈D, (2.13)
andIis nonexpansive onPM(u)∪ {u}, thenDisT-invariant. Suppose thatDis closed, has property(N)withq∈F(I),cl(T(D))is compact andI satisfies property(C)onD. Then PM(u)∩F(I)∩F(T)= ∅.
Proof. Letx∈D, then proceeding as in the proof ofTheorem 2.8, we obtainTx∈PM(u). Moreover, sinceIis nonexpansive onPM(u)∪ {u}andTsatisfies (2.13), we obtain
ITx−up≤ Tx−Tup≤ Ix−Iup=dist(u,M). (2.14)
ThusITx∈PM(u) and soTx∈CMI (u). HenceTx∈D. Consequently,T(D)⊂D=I(D). NowTheorem 2.2(i) guarantees thatPM(u)∩F(I)∩F(T)= ∅. Remark 2.13. Notice that approximation results similar to Theorems2.8,2.11, and2.12 can be obtained, usingTheorem 2.2(ii), (iii), and (iv).
Example 2.14. LetX=Rand M= {0, 1, 1−1/(n+ 1) :n∈N}be endowed with usual metric. Define T1=0 and T0=T(1−1/(n+ 1))=1 for alln∈N. Clearly, M is not starshaped butM has the property (N) for q=0 and kn =1−1/(n+ 1), n∈N. Let Ix=x for allx∈M. Now I and T satisfy (2.2) together with all other conditions of Theorem 2.2(i) except the condition thatTis continuous. Note thatF(I)∩F(T)= ∅. Example 2.15. Let X=R2 be endowed with the p-norm ,
p defined by (a,b)p= |a|p+|b|p, (a,b)∈R2.
(1) LetM=A∪B, whereA= {(a,b)∈X: 0≤a≤1, 0≤b≤4}andB= {(a,b)∈X: 2≤a≤3, 0≤b≤4}. DefineT:M→Mby
T(a,b)=
⎧ ⎪ ⎨ ⎪ ⎩
(2,b) if (a,b)∈A,
(1,b) if (a,b)∈B
(2.15)
(2) IfM= {(a,b)∈X: 0≤a <∞, 0≤b≤1}andT:M→Mis defined by
T(a,b)=(a+ 1,b), (a,b)∈M. (2.16)
DefineI(x)=x, for allx∈M. All of the conditions ofTheorem 2.2(i) are satisfied except thatMis compact. NoteF(I)∩F(T)= ∅. Notice thatM, being convex andT-invariant, has the property (N) for any choice ofqand{kn}.
(3) IfM= {(a,b)∈X: 0< a <1, 0< b <1}andT,I:M→Mare defined byT(a,b)= (a/2,b/3), andI(x)=xfor allx∈M. All of the conditions ofTheorem 2.2(i) are satisfied except the fact thatMis closed. HoweverF(I)∩F(T)= ∅.
Example 2.16. LetX=RandM=[0, 1] be endowed with the usual metric. DefineT(x)= 0 andI(x)=1−xfor eachx∈M. All of the conditions ofTheorem 2.2(i) are satisfied except the condition that the pair{I,T}isR-subweakly commuting. NoteF(I)∩F(T)= ∅.
3. Further results
All results of the paper (Theorem 2.2–Remark 2.13) remain valid in the setup of a metriz-able 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 trans-lation invariant and satisfiesdp(αx,αy)≤αpdp(x,y) for any scalarα≥0). Consequently, Theorem 2.2 (i)-(ii) and Theorem 3.3 (i)-(ii) due to Hussain and Khan [6] and Theorem 3.5 (i)-(ii) & (v), (ix)-(x) and Theorem 4.2 (i)-(ii) & (v), (ix)-(x) due to Hussain et al. [7] are extended to a class of maps satisfying a more general inequality.
FromCorollary 2.3, we have the following result which extends [18, Theorem 2.2]; Corollary3.1. LetMbe a closedq-starshaped subset of a metrizable locally convex space (X,d)wheredis translation invariant andd(αx,αy)≤αd(x,y), for eachαwith0< α <1 andx,y∈X. Suppose thatTandIare continuous self-maps ofM,Iis affine withq∈F(I), T(M)⊂I(M)andclT(M)is compact. If the pair{T,I}isR-subweakly commuting and satisfy for allx,y∈M,
d(Tx,T y)≤maxd(Ix,I y), dist(Ix, [Tx,q]), dist(I y, [T y,q]),
dist(Ix, [T y,q]), dist(I y, [Tx,q]), (3.1)
thenF(T)∩F(I)= ∅. We defineCI
M(u)= {x∈M:Ix∈PM(u)}and denote by0the class of closed convex subsets ofXcontaining 0. ForM∈ 0, we defineMu= {x∈M:x ≤2u}. It is clear thatPM(u)⊂Mu∈ 0.
Following result includes [1, Theorem 4.1] and [5, Theorem 8] and provides an ana-logue of [18, Theorem 2.8] in the setting of metrizable locally convex space and contrac-tive condition involved is more general.
satisfies for allx∈M∪ {u},
d(Tx,T y)≤
⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩
d(x,u) ify=u,
maxd(x,y), dist(x, [0,Tx]), dist(y, [0,T y]),
dist(x, [0,T y]), dist(y, [0,Tx]) ify∈M,
(3.2)
then
(i)PM(u)is nonempty, closed, and convex, (ii)T(PM(u))⊂PM(u),
(iii)PM(u)∩F(T)= ∅.
Proof. (i) Letr=dist(u,M). Then there is a minimizing sequence{yn}inM such that limnd(u,yn)=r. As clT(M) is compact so{T yn}has a convergent subsequence{T ym} with limT ym=x0(say) inM. Now by (3.2)
r≤dx0,u=limdT ym,u
≤limdym,u
=limdyn,u
=r. (3.3)
Hencex0∈PM(u). ThusPM(u) is nonempty closed and convex.
(ii) Letz∈PM(u). Thend(Tz,u)=d(Tz,Tu)≤d(z,u)=dist(u,M). This implies that Tz∈PM(u) and soT(PM(u))⊂PM(u).
(iii) As clT(PM(u))⊂clT(M), so clT(PM(u)) is compact. Thus by Corollary 3.1,
PM(u)∩F(T)= ∅.
Theorem3.3. LetX be as inTheorem 3.2andI andT be self-mappings ofX withu∈ F(I)∩F(T)andM∈ 0such thatT(Mu)⊂I(M)⊂M. Suppose thatIis affine and con-tinuous onM, d(Ix,u)≤d(x,u)for allx∈M,clI(M)is compact andI satisfies for all x,y∈M,
d(Ix,I y)≤maxd(x,y), dist(x, [0,Ix]), dist(y, [0,I y]),
dist(x, [0,I y]), dist(y, [0,Ix]).
(3.4)
If the pair{T,I}isR-subweakly commuting andT is continuous onMuand satisfy for all x,y∈Mu∪ {u}, andq∈F(I),
d(Tx,T y)≤
⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩
d(Ix,Iu) ify=u,
maxd(Ix,I y), dist(Ix, [q,Tx]), dist(I y, [q,T y]),
dist(Ix, [q,T y]), dist(I y, [q,Tx]) ify∈Mu,
(3.5)
then
(i)PM(u)is nonempty, closed, and convex, (ii)T(PM(u))⊂I(PM(u))⊂PM(u), (iii)PM(u)∩F(I)∩F(T)= ∅.
thaty=Tz=Ix. By (3.5), we have
d(Ix,u)=d(Tz,Tu)≤d(Iz,Iu)≤d(z,u)=dist(u,M). (3.6)
Hencex∈CI
M(u)=PM(u) and so (ii) holds.
(iii)Theorem 3.2guarantees thatPM(u)∩F(I)= ∅. Thus there existsq∈PM(u) such thatq∈F(I). Hence the conclusion follows fromCorollary 3.1. Following corollary provides the conclusions of [1, Theorem 4.2(a)] and [17, Theorem 2.3], to the setup of metrizable locally convex space.
Corollary3.4. LetXbe as above andI,T be self-mappings ofXwithu∈F(I)∩F(T) andM∈ 0such thatT(Mu)⊂I(M)⊂M. Suppose thatIis affine and continuous onM, d(Ix,u)≤d(x,u)for allx∈M,clI(M)is compact andIis nonexpansive onM. If the pair {T,I}isR-subweakly commuting onMuandTisI-nonexpansive onMu∪ {u}, then
(i)PM(u)is nonempty, closed and convex, (ii)T(PM(u))⊂I(PM(u))⊂PM(u), (iii)PM(u)∩F(I)∩F(T)= ∅.
Let (X,d) be a metric linear space with 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].
Following the arguments of Jungck [8, Theorem 3.2] and using Theorem 2.1 instead of Theorem 3.1 of Jungck [8], we obtain,
Theorem3.5. LetT and f be continuous self-maps of a compact metric space(X,d)with T(X)⊂f(X). IfTandf areR-weakly commuting self-maps ofXsuch that
d(Tx,T y)<maxd(f x,f y),d(Tx,f x),d(T y,f y),d(Tx,f y),d(T y,f x) (3.7)
when right hand side is positive, then there is a unique pointzinXsuch thatTz=f z=z. UsingTheorem 3.5, we establish common fixed point generalization of Theorem 1 of Dotson [3], and Theorem 2 of Guseman and Peters [4].
Theorem3.6. LetT,I be self-maps on a compact subsetMof a metric linear space(X,d) with translation invariant and strictly monotone metricd. Assume thatMhas the property (N)withq∈F(I),Isatisfies the condition(C)andM=I(M). Suppose thatT andI are R-subweakly commuting and satisfy
d(Tx,T y)≤maxd(Ix,I y), dist(Ix, [Tx,q]), dist(I y, [T y,q]),
dist(Ix, [T y,q]), dist(I y, [Tx,q])
(3.8)
for allx,y∈M. IfTandIare continuous, thenF(T)∩F(I)= ∅.
Proof. Proof is similar toTheorem 2.2(i), instead of applying Theorem 2.1, we apply
Similarly, all other results ofSection 2(Corollary 2.3–Theorem 2.12) hold in the set-ting of metric linear space (X,d) with translation invariant and strictly monotone metric dprovided we replace closedness ofM and compactness of clT(M) by compactness of Mand usingTheorem 3.6instead ofTheorem 2.2(i). Consequently, metric linear space versions ofCorollary 2.3–Corollary 2.7improve and extend Theorem 2 and the Corollary in [4].
A metric space (X,d) is said to beS-space [20], if there exists anx0 in X such that for everyt∈(0, 1) there is ad-contractive self-mapping ftofXfor which the inequality d(ft(x),x)≤(1−t)d(x0,x) holds for everyxinX. As an application ofTheorem 3.5and [20, Theorem 1], we obtain the following extension of TheoremsB,K,Z andCin [2] and Theorem 3 of [20] to generalized nonexpansive mappings.
Theorem3.7. Let(X,d)be a compactS-space andT:X→Xsatisfies for allx,y∈X,
d(Tx,T y)≤maxd(x,y),d(x,Tx),d(y,T y),d(x,T y),d(y,Tx). (3.9)
ThenThas a fixed point.
Acknowledgment
The authors are grateful to N. Shahzad for providing preprint of [18].
References
[1] M. A. Al-Thagafi,Common fixed points and best approximation, Journal of Approximation The-ory85(1996), no. 3, 318–323.
[2] V. Berinde, A common fixed point theorem for quasi contractive type mappings, Annales Uni-versitatis Scientiarum Budapestinensis de Rolando E¨otv¨os Nominatae. Sectio Mathematica46 (2003), 101–110 (2004).
[3] W. G. Dotson Jr.,Fixed point theorems for non-expansive mappings on star-shaped subsets of Ba-nach spaces, Journal of the London Mathematical Society. Second Series4(1971/1972), 408– 410.
[4] L. F. Guseman Jr. and B. C. Peters Jr.,Nonexpansive mappings on compact subsets of metric linear spaces, Proceedings of the American Mathematical Society47(1975), 383–386.
[5] L. Habiniak,Fixed point theorems and invariant approximations, Journal of Approximation The-ory56(1989), no. 3, 241–244.
[6] N. Hussain and A. R. Khan,Common fixed-point results in best approximation theory, Applied Mathematics Letters. An International Journal of Rapid Publication16(2003), no. 4, 575–580. [7] N. Hussain, D. O’Regan, and R. P. Agarwal,Common fixed point and invariant approximation
results on non-starshaped domain, Georgian Mathematical Journal12(2005), no. 4, 559–669. [8] G. Jungck,Common fixed points for commuting and compatible maps on compacta, Proceedings
of the American Mathematical Society103(1988), no. 3, 977–983.
[9] L. A. Khan and A. R. Khan,An extension of Brosowski-Meinardus theorem on invariant approxi-mation, Approximation Theory and its Applications. New Series11(1995), no. 4, 1–5. [10] G. Meinardus,Invarianz bei linearen Approximationen, Archive for Rational Mechanics and
Analysis14(1963), 301–303 (German).
[12] W. Rudin,Functional Analysis, 2nd ed., International Series in Pure and Applied Mathematics, McGraw-Hill, New York, 1991.
[13] S. A. Sahab, M. S. Khan, and S. Sessa,A result in best approximation theory, Journal of Approxi-mation Theory55(1988), no. 3, 349–351.
[14] N. Shahzad,A result on best approximation, Tamkang Journal of Mathematics29(1998), no. 3, 223–226.
[15] ,Correction to: “A result on best approximation”, Tamkang Journal of Mathematics30 (1999), no. 2, 165.
[16] ,Noncommuting maps and best approximations, Radovi Matematiˇcki10(2000/2001), no. 1, 77–83.
[17] ,Invariant approximations andR-subweakly commuting maps, Journal of Mathematical Analysis and Applications257(2001), no. 1, 39–45.
[18] ,Invariant approximations, generalizedI-contractions, andR-subweakly commuting maps, Fixed Point Theory and Applications2005(2005), no. 1, 79–86.
[19] S. P. Singh,An application of a fixed-point theorem to approximation theory, Journal of Approxi-mation Theory25(1979), no. 1, 89–90.
[20] L. A. Talman,A fixed point criterion for compactT2-spaces, Proceedings of the American
Mathe-matical Society51(1975), 91–93.
Nawab Hussain: Centre for Advanced Studies in Pure Applied Mathematics, Bahauddin Zakariya University, Multan, Pakistan
Current address: Department of Mathematics, Faculty of Science, King Abdul Aziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia
E-mail address:[email protected]
Vasile Berinde: Department of Mathematics and Computer Science, Faculty of Sciences, North University of Baia Mare, Victoriei Nr. 76, 430122 Baia Mare, Romania
