I -CONTRACTIONS, AND R -SUBWEAKLY COMMUTING MAPS

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I -CONTRACTIONS, AND R -SUBWEAKLY COMMUTING MAPS

NASEER SHAHZAD

Received 11 May 2004 and in revised form 23 August 2004

We present common fixed point theory for generalized contractiveR-subweakly commuting maps and obtain some results on invariant approximation.

1. Introduction and preliminaries

Let S be a subset of a normed spaceX=(X, · ) andT and I self-mappings of X.

ThenT is called (1) nonexpansive on SifTx−T y ≤ x−yfor allx,y∈S; (2)I- nonexpansive on S ifTx−T y ≤ Ix−I y for all x,y∈S; (3) I-contraction on S if there existsk∈[0, 1) such that Tx−T y ≤kIx−I yfor allx,y∈S. The set of fixed points ofT (resp.,I) is denoted byF(T) (resp.,F(I)). The setSis called (4) p- starshaped with p∈Sif for allx∈S, the segment [x,p] joiningx top is contained in S(i.e.,kx+ (1−k)p∈Sfor allx∈Sand all realkwith 0≤k≤1); (5) convex ifSisp- starshaped for all p∈S. The convex hull co(S) ofSis the smallest convex set inX that containsS, and the closed convex hull cl co(S) ofSis the closure of its convex hull. The mappingT is called (6) compact if clT(D) is compact for every bounded subsetD of S. The mappingsT andI are said to be (7) commuting onSifITx=TIxfor allx∈S;

(8)R-weakly commuting on S[7] if there exists R∈(0,∞) such thatTIx−ITx ≤ RTx−Ix for allx∈S. Suppose S⊂X is p-starshaped with p∈F(I) and is both T- andI-invariant. ThenT andI are called (8) R-subweakly commuting onS[11] if there existsR∈(0,∞) such thatTIx−ITx ≤Rdist(Ix, [Tx,p]) for all x∈S, where dist(Ix, [Tx,p])=inf{Ix−z:z∈[Tx,p]}. Clearly commutativity impliesR-subweak commutativity, but the converse may not be true (see [11]).

The setPS(x) = {y∈S:y−x =dist(x,S) }is called the set of best approximants to

x∈X out ofS, where dist(x,S)=inf{y−x:y∈S}. We defineC_S^I(x)= {x∈S:Ix∈ P_S(x)}and denote by0the class of closed convex subsets ofXcontaining 0. ForS∈ 0, we defineSx= {x∈S:x ≤2x}. It is clear thatPS(x)⊂Sx∈ 0.

In 1963, Meinardus [6] employed the Schauder fixed point theorem to establish the existence of invariant approximations. Afterwards, Brosowski [2] obtained the following extension of the Meinardus result.

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Theorem 1.1. Let T be a linear and nonexpansive self-mapping of a normed spaceX, S⊂Xsuch thatT(S)⊂S, andx∈F(T). IfP_S(x) is nonempty, compact, and convex, then PS(x)∩F(T)= ∅.

Singh [15] observed thatTheorem 1.1is still true if the linearity ofTis dropped and PS(x) is only starshaped. He further remarked, in [16], that Brosowski’s theorem remains valid ifTis nonexpansive only onP_S(x) ∪{x}. Then Hicks and Humphries [5] improved Singh’s result by weakening the assumptionT(S)⊂StoT(∂S)⊂S; here∂Sdenotes the boundary ofS.

On the other hand, Subrahmanyam [18] generalized the Meinardus result as follows.

Theorem 1.2. Let T be a nonexpansive self-mapping of X, S a finite-dimensional T-invariant subspace ofX, andx∈F(T). ThenPS(x)∩F(T)= ∅.

In 1981, Smoluk [17] noted that the finite dimensionality ofSinTheorem 1.2can be replaced by the linearity and compactness ofT. Subsequently, Habiniak [4] observed that the linearity ofTin Smoluk’s result is superfluous.

In 1988, Sahab et al. [8] established the following result which contains Singh’s result as a special case.

Theorem1.3. LetTandIbe self-mappings of a normed spaceX,S⊂Xsuch thatT(∂S)⊂ S, andx∈F(T)∩F(I). SupposeTisI-nonexpansive onP_S(x)∪{x},Iis linear and contin- uous onPS(x), andT andI are commuting onPS(x). If PS(x) is nonempty, compact, and p-starshaped withp∈F(I), and ifI(PS(x)) =PS(x), thenPS(x)∩F(T)∩F(I)= ∅.

Recently, Al-Thagafi [1] generalizedTheorem 1.3and proved some results on invariant approximations for commuting mappings. More recently, with the introduction of noncommuting maps to this area, Shahzad [9,10,11,12,13,14] further extended Al-Thagafi’s results and obtained a number of results regarding best approximations. The purpose of this paper is to present common fixed point theory for generalizedI-contraction andR- subweakly commuting maps. As applications, some invariant approximation results are also obtained. Our results extend, generalize, and complement those of Al-Thagafi [1], Brosowski [2], Dotson Jr. [3], Habiniak [4], Hicks and Humphries [5], Meinardus [6], Sahab et al. [8], Shahzad [9,10,11,12], Singh [15,16], Smoluk [17], and Subrahmanyam [18].

2. Main results

Theorem 2.1. Let Sbe a closed subset of a metric space(X,d), and T andI R-weakly commuting self-mappings ofSsuch that T(S)⊂I(S). Suppose there exists k∈[0, 1)such that

d(Tx,T y)≤kmax

d(Ix,I y),d(Ix,Tx),d(I y,T y),1 2

d(Ix,T y) +d(I y,Tx) (2.1)

for allx,y∈S. If cl(T(S))is complete andTis continuous, thenS∩F(T)∩F(I)is singleton.

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Proof. Letx0∈S and letx1∈S be such thatIx1=Tx0. Inductively, choosex_nso that Ix_n=Tx_n−1. This is possible sinceT(S)⊂I(S). Notice

dIx_n+1,Ix_n=dTx_n,Tx_n−1

≤kmax

dIxn,Ixn−1

,dIxn,Txn

,dIxn−1,Txn−1

, 1

2

dIxn,Txn−1

+dIxn−1,Txn

=kmax

dIxn,Ixn−1

,dIxn,Txn , dIx_n−1,Tx_n−1

,1

2dIx_n−1,Tx_n

≤kmax

dIxn,Ixn−1

,dIxn,Txn , 1

2

dIx_n−1,Ix_n+dIx_n,Tx_n

≤kdIxn,Ixn−1

(2.2)

for all n. This shows that {Ixn} is a Cauchy sequence inS. Consequently, {Txn} is a Cauchy sequence. The completeness of cl(T(S)) further implies thatTxn→y∈S and soIx_n→yasn→ ∞. SinceTandIareR-weakly commuting, we have

dTIxn,ITxn

≤RdTxn,Ixn

. (2.3)

This implies thatITxn→T yasn→ ∞. Now dTx_n,TTx_n≤kmax

dIx_n,ITx_n,dIx_n,Tx_n,dITx_n,TTx_n, 1

2

dIxn,TTxn

+dITxn,Txn

.

(2.4)

Taking the limit asn→ ∞, we obtain dy,T y≤kmax

d(y,T y),d(y,y),d(T y,T y), 1

2

d(y,T y) +d(T y,y)

=kd(y,T y),

(2.5)

which implies y=T y. Since T(S)⊂I(S), we can choosez∈Ssuch that y=T y=Iz.

Since

dTTxn,Tz≤kmax

dITxn,Iz,dITxn,TTxn

,dIz,Tz, 1

2

dITx_n,Tz+dIz,TTx_n,

(2.6)

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taking the limit asn→ ∞yields

d(T y,Tz)≤kd(T y,Tz). (2.7)

This implies thatT y=Tz. Therefore,y=T y=Tz=Iz. Using theR-weak commutativ- ity ofTandI, we obtain

d(T y,I y)=d(TIz,ITz)≤Rd(Tz,Iz)=0. (2.8) Thusy=T y=I y. Clearlyyis a unique common fixed point ofTandI. HenceS∩F(T)∩

F(I) is singleton.

Theorem2.2. LetSbe a closed subset of a normed spaceX, andT andIcontinuous self- mappings ofSsuch thatT(S)⊂I(S). SupposeI is linear,p∈F(I),Sis p-starshaped, and cl(T(S))is compact. IfTandIareR-subweakly commuting and satisfy

Tx−T y ≤max

Ix−I y, distIx, [Tx,p], distI y, [T y,p], 1

2

distIx, [T y,p]+ distI y, [Tx,p]

(2.9)

for allx,y∈S, thenS∩F(T)∩F(I)= ∅.

Proof. Choose a sequence{k_n} ⊂[0, 1) such thatk_n→1 asn→ ∞. Define, for eachn, a mapT_nbyT_n(x)=k_nTx+ (1−k_n)pfor eachx∈S. Then eachT_nis a self-mapping ofS.

Furthermore,Tn(S)⊂I(S) for eachnsinceIis linear andT(S)⊂I(S). Now the linearity ofIand theR-subweak commutativity ofTandIimply that

TnIx−ITnx=knTIx−ITx ≤knRdistIx, [Tx,p]

≤knRTnx−Ix (2.10)

for allx∈S. This shows thatTnandIareknR-weakly commuting for eachn. Also T_nx−T_ny=k_nTx−T y

≤k_nmax

2

distIx, [T y,p]+ distI y, [Tx,p]

≤k_nmax

Ix−I y,Ix−T_nx,I y−T_ny, 1

2Ix−Tny+I y−Tnx

(2.11)

for allx,y∈S. Now Theorem 2.1 guarantees thatF(T_n)∩F(I)= {x_n}for somex_n∈S.

The compactness of cl(T(S)) implies that there exists a subsequence{x_m}of{x_n}such

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thatx_m→y∈Sasm→ ∞. By the continuity ofTandI, we havey∈F(T)∩F(I). Hence

S∩F(T)∩F(I)= ∅.

The following corollaries extend and generalize [3, Theorem 1] and [4, Theorem 4].

Corollary2.3. LetSbe a closed subset of a normed spaceX, andTandIcontinuous self- mappings ofSsuch thatT(S)⊂I(S). SupposeI is linear,p∈F(I),Sis p-starshaped, and cl(T(S))is compact. IfTandIareR-subweakly commuting andT isI-nonexpansive onS, thenS∩F(T)∩F(I)= ∅.

Corollary2.4. Let Sbe a closed subset of a normed spaceX, andT andI continuous self-mappings ofSsuch thatT(S)⊂I(S). SupposeI is linear, p∈F(I),Sis p-starshaped, andcl(T(S))is compact. IfT andI are commuting and satisfy (2.9) for allx,y∈S, then S∩F(T)∩F(I)= ∅.

LetD^R_S^,^I(x) =PS(x)∩G^R_S^,^I(x), where

G^R_S^,Î(x) = x∈S:Ix−x ≤(2R+ 1) dist(x,S). (2.12) Theorem2.5. LetTandIbe self-mappings of a normed spaceXwithx∈F(T)∩F(I)and S⊂X such thatT(∂S∩S)⊂S. SupposeI is linear onD^R_S^,Î(x), p∈F(I),D^R_S^,Î(x) is closed and p-starshaped, clT(D_S^R^,Î(x)) is compact, andI(D_S^R^,Î(x)) =D_S^R^,Î(x). IfT andI areR- subweakly commuting and continuous onD^R_S^,Î(x)and satisfy, for allx∈D^R_S^,Î(x)∪{x},

Tx−T y ≤











Ix−Ix ify=x, max

2

distIx, [T y,p]+ distI y, [Tx,p] ify∈D_S^R^,^I(x), (2.13) thenP_S(x)∩F(T)∩F(I)= ∅.

Proof. Letx∈D^R,I_S (x). Then x∈∂S∩S(see [1]) and soTx∈SsinceT(∂S∩S)⊂S. Now Tx−x = Tx−Tx ≤ Ix−Ix = Ix−x =dist(x,S). (2.14) This shows thatTx∈PS(x). From the R-subweak commutativity ofTandI, it follows that ITx−x = ITx−Tx ≤RTx−Ix+I²x−Ix≤(2R+ 1) dist(x,S). (2.15) This implies thatTx∈G^R_S^,Î(x). Consequently, Tx∈D_S^R^,Î(x) and soT(D_S^R^,Î(x))⊂D_S^R^,Î(x) = I(D^R_S^,Î(x)). Now Theorem 2.2guarantees thatP_S(x) ∩F(T)∩F(I)= ∅. Theorem2.6. LetTandIbe self-mappings of a normed spaceXwithx∈F(T)∩F(I)and S⊂X such thatT(∂S∩S)⊂I(S)⊂S. SupposeI is linear onD_S^R^,Î(x), p∈F(I),D^R_S^,Î(x) is closed andp-starshaped,clT(D^R_S^,Î(x)) is compact, andI(G^R_S^,Î(x)) ∩D^R_S^,Î(x) ⊂I(D^R_S^,Î(x)) ⊂ D^R_S^,Î(x). If TandI areR-subweakly commuting and continuous onD^R_S^,Î(x) and satisfy, for allx∈D^R_S^,Î(x)∪{x}, (2.13), thenPS(x)∩F(T)∩F(I)= ∅.

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Proof. Let x∈D^R_S^,Î(x). Then, as in Theorem 2.5, Tx∈D_S^R^,Î(x), that is, T(D^R_S^,Î(x)) ⊂ D^R_S^,Î(x). Also (1−k)x+kx−x<dist(x,S) for allk∈(0, 1). This implies thatx∈∂S∩S (see [1]) and soT(D_S^R^,Î(x)) ⊂T(∂S∩S)⊂I(S). Thus we can choosey∈Ssuch thatTx= I y. Since I y=Tx∈PS(x), it follows that y∈G^R_S^,Î(x). Consequently, T(D_S^R^,Î(x))⊂ I(G^R_S^,Î(x)) ⊂PS(x). Therefore, T(D^R_S^,Î(x)) ⊂I(G^R_S^,Î(x))∩D_S^R^,Î(x)⊂I(D_S^R^,Î(x)) ⊂D^R_S^,Î(x).

NowTheorem 2.2guarantees thatP_S(x)∩F(T)∩F(I)= ∅. Remark 2.7. Theorems2.5 and 2.6 remain valid when D^R_S^,^I(x) =PS(x). If I(PS(x)) ⊂ P_S(x), then P_S(x) ⊂C^I_S(x) ⊂G^R,I_S (x) (see [1]) and so D_S^R,I(x) =P_S(x). Consequently,Theo- rem 2.5containsTheorem 1.3as a special case.

The following result includes [1, Theorem 4.1] and [4, Theorem 8]. It also contains the well-known results due to Smoluk [17] and Subrahmanyam [18].

Theorem2.8. LetT be a self-mapping of a normed spaceX withx∈F(T)andS∈ 0

such thatT(S_x)⊂S. IfclT(S_x)is compact andT is continuous onS_x and satisfies for all x∈Sx∪{x}

Tx−T y ≤











x−x ify=x,

max

x−y, distx, [Tx, 0], disty, [T y, 0], 1

2

distx, [T y, 0]+ disty, [Tx, 0] ify∈Sx,

(2.16)

then

(i)PS(x)is nonempty, closed, and convex, (ii)T(PS(x))⊂PS(x),

(iii)P_S(x)∩F(T)= ∅.

Proof. (i) We may assume thatx∈S. Ifx∈S\Sx, thenx>2x. Notice that x−x ≥ x − x>x ≥distx,S x

. (2.17)

Consequently, dist(x,S x)=dist(x,S) ≤ x. Alsoz−x =dist(x, cl T(Sx)) for somez∈ clT(Sx). Thus

distx,S x

≤distx, cl TSx

≤distx,T Sx

≤ Tx−x = Tx−Tx

≤ x−x

(2.18)

for allx∈S_x. This implies thatz−x =dist(x,S) and so P_S(x) is nonempty. Further- more, it is closed and convex.

(ii) Lety∈PS(x). Then

T y−x = T y−Tx ≤ y−x =dist(x,S). (2.19) This implies thatT y∈P_S(x) and so T(P_S(x))⊂P_S(x).

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(iii)Theorem 2.2 guarantees thatP_S(x) ∩F(T)= ∅ since clT(P_S(x)) ⊂clT(S_x) and

clT(Sx) is compact.

Theorem2.9. LetIandTbe self-mappings of a normed spaceXwithx∈F(I)∩F(T)and S∈ 0such thatT(Sx)⊂I(S)⊂S. Suppose thatIis linear,Ix−x = x−xfor allx∈S, clI(Sx)is compact andIsatisfies, for allx,y∈Sx,

Ix−I y ≤max

x−y, distx, [Ix, 0], disty, [I y, 0], 1

2

distx, [I y, 0]+ disty, [Ix, 0].

(2.20)

IfIandTareR-subweakly commuting and continuous onSxand satisfy, for allx∈Sx∪{x}, andp∈F(I),

Tx−T y ≤











Ix−Ix ify=x, max

2

distIx, [T y,p]+ distI y, [Tx,p] ify∈Sx, (2.21)

then

(i)P_S(x)is nonempty, closed, and convex, (ii)T(PS(x))⊂I(PS(x)) ⊂PS(x), (iii)P_S(x)∩F(I)∩F(T)= ∅.

Proof. FromTheorem 2.8, (i) follows immediately. Also, we haveI(PS(x)) ⊂PS(x). Let y∈T(P_S(x)). Since T(Sx)⊂I(S) andP_S(x) ⊂Sx, there existz∈P_S(x) and x1∈Ssuch thaty=Tz=Ix1. Furthermore, we have

Ix1−x= Tz−Tx ≤ Iz−Ix ≤ z−x =d(x,S). (2.22)

Thusx1∈C^I_S(x) =PS(x) and so (ii) holds.

Since, byTheorem 2.8,P_S(x)∩F(I)= ∅, it follows that there existsp∈P_S(x) such that p∈F(I). Hence (iii) follows fromTheorem 2.2.

The following corollary extends [1, Theorem 4.2(a)] to a class of noncommuting maps.

Corollary2.10. LetIandTbe self-mappings of a normed spaceXwithx∈F(I)∩F(T) andS∈ 0such thatT(Sx)⊂I(S)⊂S. Suppose thatIis linear,Ix−x = x−xfor all x∈S,clI(S_x)is compact, andIis nonexpansive onS_x. IfIandTareR-subweakly commut- ing onSxandTisI-nonexpansive onSx∪{x}, then

(i)PS(x)is nonempty, closed and convex, (ii)T(P_S(x))⊂I(P_S(x)) ⊂P_S(x), and (iii)P_S(x)∩F(I)∩F(T)= ∅.

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Acknowledgment

The author would like to thank the referee for his suggestions.

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Naseer Shahzad: Department of Mathematics, King Abdul Aziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

E-mail address:[email protected]

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