Common Fixed Point and Approximation Results for Noncommuting Maps on Locally Convex Spaces

Volume 2009, Article ID 207503,14pages doi:10.1155/2009/207503

Research Article

Common Fixed Point and Approximation Results for Noncommuting Maps on Locally Convex Spaces

F. Akbar

and A. R. Khan

1Department of Mathematics, University of Sargodha, Sargodha, Pakistan

2Department of Mathematics and Statistics, King Fahd University of Petroleum & Minerals, Dhahran 31261, Saudi Arabia

Correspondence should be addressed to A. R. Khan,[email protected] Received 21 February 2009; Accepted 14 April 2009

Recommended by Anthony Lau

Common fixed point results for some new classes of nonlinear noncommuting maps on a locally convex space are proved. As applications, related invariant approximation results are obtained.

Our work includes improvements and extension of several recent developments of the existing literature on common fixed points. We also provide illustrative examples to demonstrate the generality of our results over the known ones.

Copyrightq2009 F. Akbar and A. R. Khan. This is an open access article distributed 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

In the sequel, E, τ will be a Hausdorfflocally convex topological vector space. A family {pα :α∈ I}of seminorms defined onEis said to be an associated family of seminorms for τ if the family{γU:γ > 0},whereU _n

i1U_αi andU_αi {x :p_αix<1}, forms a base of neighborhoods of zero forτ. A family{pα : α ∈ I}of seminorms defined onEis called an augmented associated family forτ if{pα :α∈I}is an associated family with property that the seminorm max{pα, p_β} ∈ {pα : α ∈ I}for anyα, β ∈ I. The associated and augmented associated families of seminorms will be denoted byAτandA^∗τ, respectively. It is well known that given a locally convex spaceE, τ,there always exists a family{pα : α∈ I}of seminorms defined onEsuch that{pα:α∈I}A^∗τ see1, page 203.

The following construction will be crucial. Suppose thatMis aτ-bounded subset ofE.

For this setMwe can select a numberλα>0 for eachα∈Isuch thatM⊂λαUα,whereUα {x:p_αx≤1}.Clearly,B

αλ_αU_αisτ-bounded,τ-closed, absolutely convex and contains M. The linear spanE_BofBinEis_∞

n1nB.The Minkowski functional ofBis a norm · _Bon EB. ThusEB, · _Bis a normed space withBas its closed unit ball and sup_αpαx/λα x_B for eachx∈E_Bfor details see1–3.

LetMbe a subset of a locally convex spaceE, τ. LetI, J:M → Mbe mappings. A mappingT :M → Mis calledI, J-Lipschitz if there existsk≥ 0 such thatp_αTx−Ty≤ kp_αIx−Jyfor any x, y ∈ Mand for all p_α ∈ A^∗τ. If k < 1 resp., k 1, then T is called an I, J-contractionresp.,I, J-nonexpansive. A pointx ∈ Mis a common fixed coincidencepoint ofIandTifxIxTxIxTx. The set of coincidence points ofIand T is denoted byCI, T,and the set of fixed points ofT is denoted byFT.The pair{I, T}is called:

1commuting ifTIxITxfor allx∈M;

2R-weakly commuting if for allx ∈ Mand for allp_α ∈ A^∗τ, there existsR > 0 such thatpαITx−TIx≤RpαIx−Tx.IfR1, then the maps are called weakly commuting4;

3compatible 5if for allp_α ∈ A^∗τ, limnp_αTIxn−ITx_n 0 whenever{xn}is a sequence such that limnTxnlimnIxntfor sometinM;

4weakly compatible if they commute at their coincidence points, that is,ITxTIx wheneverIxTx.

Suppose thatMisq-starshaped withq∈FIand is bothT- andI-invariant. ThenT andIare called:

5R-subcommuting onMif for allx ∈Mand for allpα ∈ A^∗τ, there exists a real numberR >0 such thatp_αITx−TIx≤R/kpα1−kqkTx−Ixfor each k∈0,1. IfR1, then the maps are called 1-subcommuting6;

6R-subweakly commuting onMsee7if for allx ∈ Mand for allpα ∈ A^∗τ, there exists a real numberR >0 such thatp_αITx−TIx≤Rd_pαIx,q, Tx, where q, x {1−kqkx: 0≤k≤1}anddpαu, M inf{pαx−u:x∈M};

7Cq-commuting8,9ifITxTIxfor allx∈CqI, T, whereCqI, T ∪{CI, Tk: 0≤k≤1}andT_kx 1−kqkTx.

Ifu ∈ E, M ⊆ E,then we define the set,PMu, of best M-approximations to uas P_Mu {y ∈ M : p_αy−u d_pαu, M, for allp_α ∈ A^∗τ}. A mappingT : M → Eis called demiclosed at 0 if{xα}converges weakly toxand{Txα}converges to 0, then we have Tx0. A locally convex spaceEsatisfies Opial’s condition if for every net{xβ}inEweakly convergent tox∈X,the inequality

lim inf

β→ ∞ p_α x_β−x

<lim inf

β→ ∞ p_α x_β−y

1.1

holds for ally /xandp_α∈A^∗τ}.

In 1963, Meinardus10employed the Schauder fixed point theorem to prove a result regarding invariant approximation. Singh 11, Sahab et al. 12, and Jungck and Sessa 13 proved similar results in best approximation theory. Recently, Hussain and Khan6 have proved more general invariant approximation results for 1-subcommuting maps which extend the work of Jungck and Sessa13and Al-Thagafi14to locally convex spaces. More recently, with the introduction of noncommuting maps to this area, Pant15, Pathak et al.

16, Hussain and Jungck7, and Jungck and Hussain9further extended and improved the above-mentioned results; details on the subject may be found in17,18. For applications of fixed point results of nonlinear mappings in simultaneous best approximation theory and

variational inequalities, we refer the reader to19–21. Fixed point theory of nonexpansive and noncommuting mappings is very rich in Banach spaces and metric spaces 13–17.

However, some partial results have been obtained for these mappings in the setup of locally convex spacessee22and its references. It is remarked that the generalization of a known result in Banach space setting to the case of locally convex spaces is neither trivial nor easy see, e.g.,2,22.

The following general common fixed point result is a consequence ofTheorem 3.1of Jungck5, which will be needed in the sequel.

Theorem 1.1. LetX, dbe a complete metric space, and letT, f, gbe selfmaps ofX. Suppose thatf andgare continuous, the pairs{T, f}and{T, g}are compatible such thatTX⊂fX∩gX. If there existsr∈0,1such that for allx, y∈X,

d

Tx, Ty

≤r max

d fx, gy

, d Tx, fx

, d

Ty, gy ,1

2 d

fx, Ty d

Tx, gy , 1.2

then there is a unique pointzinXsuch thatTzfzgzz.

The aim of this paper is to extend the above well-known result of Jungck to locally convex spaces and establish general common fixed point theorems for generalizedf, g- nonexpansive subcompatible maps in the setting of a locally convex space. We apply our theorems to derive some results on the existence of common fixed points from the set of best approximations. We also establish common fixed point and approximation results for the newly defined class of Banach operator pairs. Our results extend and unify the work of Al- Thagafi14, Chen and Li23, Hussain24, Hussain and Berinde25, Hussain and Jungck 7, Hussain and Khan6, Hussain and Rhoades8, Jungck and Sessa13, Khan and Akbar 19,20, Pathak and Hussain21, Sahab et al.12, Sahney et al.26, Singh11,27, Tarafdar 3, and Taylor28.

2. Subcompatible Maps in Locally Convex Spaces

Recently, Khan et al.29introduced the class of subcompatible mappings as follows:

Definition 2.1. LetMbe aq-starshaped subset of a normed spaceE. For the selfmapsIandT ofMwithq∈FI,we defineSqI, T:∪{SI, Tk: 0≤k≤1},whereTkx 1−kqkTx andSI, Tk {{xn} ⊂M: lim_nIx_nlim_nT_kx_n t∈M}. NowIandT are subcompatible if lim_nITxn−TIx_n0 for all sequences{xn} ∈S_qI, T.

We can extend this definition to a locally convex space by replacing the norm with a family of seminorms.

Clearly, subcompatible maps are compatible but the converse does not hold, in general, as the following example shows.

Example 2.2see29. LetX Rwith usual norm andM 1,∞.LetIx 2x−1 and Tx x²,for allx∈M. Letq1.ThenMisq-starshaped withIqq. Note thatIandTare compatible. For any sequence{xn}inMwith limnxn 2, we have, limnIxn limnT2/3xn 3∈M. However, lim_nITxn−TIx_n/0. ThusIandTare not subcompatible maps.

Note that R-subweakly commuting andR-subcommuting maps are subcompatible.

The following simple example reveals that the converse is not true, in general.

Example 2.3see29. LetXRwith usual norm andM 0,∞.LetIx x/2 if 0≤x <1 andIxxifx≥1, andTx 1/2 if 0≤x <1 andTxx²ifx≥1. ThenMis 1-starshaped withI11 andS_qI, T {{xn}: 1≤x_n <∞}. Note thatIandTare subcompatible but not R-weakly commuting for allR > 0. ThusI andT are neitherR-subweakly commuting nor R-subcommuting maps.

We observe in the following example that the weak commutativity of a pair of selfmaps on a metric space depends on the choice of the metric; this is also true for compatibility,R- weak commutativity, and other variants of commutativity of maps.

Example 2.4see30. LetX Rwith usual metric and M 0,∞.LetIx 1xand Tx 2x². Then|ITx−TIx|2xand|Ix−Tx||x²−x1|. Thus the pairI, Tis not weakly commuting onMwith respect to usual metric. But ifXis endowed with the discrete metric d, thendITx, TIx 1 dIx, Txforx > 1. Thus the pairI, Tis weakly commuting on Mwith respect to discrete metric.

Next we establish a positive result in this direction in the context of linear topologies utilizing Minkowski functional; it extends6, Lemma 2.1.

Lemma 2.5. Let I and T be compatible selfmaps of a τ-bounded subsetM of a Hausdorfflocally convex spaceE, τ. ThenIandTare compatible onMwith respect to · _B.

Proof. By hypothesis, lim_{n→ ∞}p_αITxn − TIx_n 0 for each p_α ∈ A^∗τ whenever lim_{n→ ∞}p_αTxn −t 0 lim_{n→ ∞}p_αIxn−tfor some t ∈ M. Taking supremum on both sides, we get

sup

α

nlim→ ∞pα

ITxn−TIxn

λ_α

sup

α

0 λ_α

, 2.1

whenever

sup

α

nlim→ ∞p_α

Tx_n−t λ_α

sup

α

0 λ_α

sup

α

nlim→ ∞p_α

Ix_n−t λ_α

. 2.2

This implies that

nlim→ ∞sup

α

p_α

ITx_n−TIx_n λα

0, 2.3

whenever

nlim→ ∞sup

α

pα

Tx_n−t λα

0 lim

n→ ∞sup

α

p_α

Ix_n−t λα

. 2.4

Hence limn→ ∞ITxn−TIxn_B 0,whenever lim_{n→ ∞}Txn−t_B 0 limn→ ∞Ixn−t_Bas desired.

There are plenty of spaces which are not normablesee31, page 113. So it is natural and essential to consider fixed point and approximation results in the context of a locally convex space. An application of Lemma 2.5provides the following general common fixed point result.

Theorem 2.6. Let Mbe a nonempty τ-bounded,τ-complete subset of a Hausdorff locally convex spaceE, τand letT, f,andgbe selfmaps ofM.Suppose thatfandgare nonexpansive, the pairs {T, f}and{T, g}are compatible such thatTM⊂fM∩gM. If there existsr∈0,1such that for allx, y∈M,and for allpα∈A^∗τ

p_α

Tx−Ty

≤rmax

p_α

fx−gy , p_α

Tx−fx , p_α

Ty−gy ,1

2 p_α

fx−Ty p_α

Tx−gy , 2.5

then there is a unique pointzinMsuch thatTzfzgzz.

Proof. Since the norm topology on E_B has a base of neighbourhoods of 0 consisting of τ- closed sets andMisτ-sequentially complete, thereforeMis · _B- sequentially complete in EB, · _B; see 3, the proof of Theorem 1.2. ByLemma 2.5, the pairs{T, f}and{T, g}are · _B−compatible maps ofM. From2.5we obtain for anyx, y∈M,

sup

α

p_α

Tx−Ty λ_α

≤r max

sup

α

p_α

fx−gy λ_α

,sup

α

p_α

Tx−fx λ_α

,sup

α

p_α

Ty−gy λ_α

,

1 2

sup

α

p_α

fx−Ty λα

sup

α

p_α

Tx−gy λα

.

2.6

Thus

Tx−Ty

B≤rmaxfx−gy

B,Tx−fx

B,Ty−gy

B, 1

2fx−Ty

BTx−gy

B .

2.7

As f and g are nonexpansive on τ-bounded set M,f, and g are also nonexpansive with respect to · _Band hence continuouscf.6. A comparison of our hypothesis with that of Theorem 1.1tells that we can applyTheorem 1.1toMas a subset ofEB, · _Bto conclude that there exists a uniquezinMsuch thatTzfzgzz.

We now prove the main result of this section.

Theorem 2.7. LetMbe a nonemptyτ-bounded,τ-sequentially complete,q-starshaped subset of a Hausdorfflocally convex spaceE, τand letT, f,andgbe selfmaps ofM.Suppose thatfandgare affine and nonexpansive withq∈Ff∩Fg, andTM⊂fM∩gM. If the pairs{T, f}and

{T, g}are subcompatible and, for allx, y∈Mand for allpα∈A^∗τ,

pα

Tx−Ty

≤max

pα

fx−gy , dpα

fx, Tx, q

, dpα

gy, Ty, q

, 1

2 d_pα

fx, Ty, q

d_pα gy,

Tx, q ,

2.8

thenFT∩Ff∩Fg/∅provided that one of the following conditions holds:

iclTMisτ-sequentially compact, andTis continuous (clstands for closure);

iiMisτ-sequentially compact, andT is continuous;

iiiMis weakly compact inE, τ,andf−T is demiclosed at 0.

Proof. DefineTn:M → Mby

Tnx 1−knqknTx 2.9

for allx∈ Mand a fixed sequence of real numbersk_n 0 < k_n <1converging to 1. Then, eachTnis a selfmap ofMand for eachn≥1,TnM⊂fM∩gMsincefandgare affine andTM ⊂ fM∩gM.Asf is affine and the pair{T, f}is subcompatible, so for any {xm} ⊂Mwith lim_mfx_mlim_mT_nx_mt∈M, we have

limm p_α

T_nfx_m−fT_nx_m

k_nlim

m p_α

Tfx_m−fTx_m 0.

2.10

Thus the pair{Tn, f}is compatible onMfor eachn. Similarly, the pair{Tn, g}is compatible for eachn≥1.

Also by2.8, pα

Tnx−Tny knpα

Tx−Ty

≤knmax

pα

fx−gy , dpα

fx, Tx, q

, dpα

gy, Ty, q

, 1

2 d_pα

fx, Ty, q

d_pα gy,

Tx, q

≤knmax

pα

fx−gy , pα

fx−Tnx , pα

gy−Tny , 1

2 p_α

fx−T_ny p_α

gy−T_nx ,

2.11

for eachx, y ∈ M,pα ∈A^∗τ,and 0 < kn < 1. ByTheorem 2.6, for eachn≥ 1, there exists x_n∈Msuch thatx_nfx_ngx_n T_nx_n.

i The compactness of clTM implies that there exists a subsequence {Txm} of {Txn} and az ∈ clTMsuch thatTx_m → zasm → ∞. Since k_m → 1,x_m T_mx_m 1−k_mqk_mTx_malso converges toz.SinceT,f,andgare continuous, we havez∈FT∩ Ff∩Fg.ThusFT∩Ff∩Fg/∅.

iiProof follows fromi.

iii Since M is weakly compact, there is a subsequence {xm} of {xn} converging weakly to somey ∈ M. But,f andg being affine and continuous are weakly continuous, and the weak topology is Hausdorff, so we havefy y gy. The set Mis bounded, so f−Txm 1−km⁻¹xm−q → 0 asm → ∞.Now the demiclosedness off−T at 0 guarantees thatf−Ty0 and henceFT∩Ff∩Fg/∅.

Theorem 2.7 extends and improves 14, Theorem 2.2, 7, Theorems 2.2-2.3, and Corollaries 2.4–2.7,13, Theorem 6, and the main results of Tarafdar3and Taylor28see also6, Remarks 2.4.

Theorem 2.8. LetMbe a nonemptyτ-bounded,τ-sequentially complete,q-starshaped subset of a Hausdorfflocally convex spaceE, τand letT, f,andgbe selfmaps ofM.Suppose thatfandgare affine and nonexpansive withq∈Ff∩Fg, andTM⊂fM∩gM. If the pairs{T, f}and {T, g}are subcompatible andT isf, g-nonexpansive, thenFT∩Ff∩Fg/∅,provided that one of the following conditions holds

iclTMisτ-sequentially compact;

iiMisτ-sequentially compact;

iiiMis weakly compact inE, τ,f−T is demiclosed at 0.

ivMis weakly compact in an Opial spaceE, τ. Proof. i–iiifollow fromTheorem 2.7.

ivAs iniiiwe havefy y gyandfxm−Txm → 0 asm → ∞.Iffy /Ty, then by the Opial’s condition ofEandf, g-nonexpansiveness ofTwe get,

lim inf

n→ ∞ p_α

fx_m−gy

lim inf

n→ ∞ p_α

fx_m−fy

<lim inf

n→ ∞ p_α

fx_m−Ty

≤lim inf

n→ ∞ pα

fxm−Txm

lim inf

n→ ∞ p_α

Txm−Ty lim inf

n→ ∞ pα

Txm−Ty

≤lim inf

n→ ∞ pα

fxm−gy ,

2.12

which is a contradiction. ThusfyTyand henceFT∩Ff∩Fg/∅.

As 1-subcommuting maps are subcompatible, so by Theorem 2.8, we obtain the following recent result of Hussain and Khan6 without the surjectivity of f. Note that a continuous and affine map is weakly continuous, so the weak continuity offis not required as well.

Corollary 2.9 6, Theorem 2.2. LetMbe a nonempty τ-bounded, τ-sequentially complete, q- starshaped subset of a Hausdorfflocally convex spaceE, τand letT, fbe selfmaps ofM.Suppose that fis affine and nonexpansive withq∈Ff, andTM⊂fM. If the pair{T, f}is 1-subcommuting

andT isf-nonexpansive, thenFT∩Ff/∅,provided that one of the following conditions holds:

iclTMisτ-sequentially compact;

iiMisτ-sequentially compact;

iiiMis weakly compact inE, τ,f−T is demiclosed at 0.

ivMis weakly compact in an Opial spaceE, τ.

The following theorem improves and extends the corresponding approximation results in6–8,11–14,25,27.

Theorem 2.10. LetMbe a nonempty subset of a Hausdorfflocally convex spaceE, τand letf, g, T : E → Ebe mappings such thatu ∈FT∩Ff∩Fgfor someu ∈EandT∂M∩M ⊂ M.

Suppose thatf andg are affine and nonexpansive on P_Muwith q ∈ Ff∩Fg, PMuisτ- bounded,τ-sequentially complete,q-starshaped andfPMu P_Mu gPMu. If the pairs T, fandT, gare subcompatible and, for allx∈PMu∪ {u}andpα∈A^∗τ,

p_α

Tx−Ty

≤

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎩ pα

fx−gu

, if yu, max

p

α

fx−gy , d_pα

fx, q, Tx

, d_pα gy,

q, Ty , 1

2 d_pα

fx, q, Ty

d_pα gy,

q, Tx , ify∈P_Mu,

2.13

thenP_Mu∩Ff∩Fg∩FT/∅, provided that one of the following conditions holds iclTPMuisτ-sequentially compact, andTis continuous;

iiP_Muisτ-sequentially compact, andTis continuous;

iiiP_Muis weakly compact, andf−Tis demiclosed at 0.

Proof. Letx∈ P_Mu. Then for eachp_α,p_αx−u d_pαu, M. Note that for anyk ∈ 0,1, pαku 1−kx−u 1−kpαx−u< dpαu, M.

It follows that the line segment{ku 1−kx: 0< k <1}and the setMare disjoint.

Thusxis not in the interior ofMand sox∈∂M∩M. SinceT∂M∩M⊂M,Txmust be in M. Also sincefx∈PMu,u∈FT∩Ff∩Fg,andT, f, gsatisfy2.13, we have for each p_α,

pαTx−u pαTx−Tu≤pα

fx−gu pα

fx−u

dpαu, M. 2.14

Thus Tx ∈ PMu. Consequently, TPMu ⊂ PMu fPMu gPMu. Now Theorem 2.7guarantees thatPMu∩Ff∩Fg∩FT/∅.

Remark 2.11. One can now easily prove on the lines of the proof of the above theorem that the approximation results are similar to those of Theorems 2.11-2.12 due to Hussain and Jungck 7in the settingof a Hausdorfflocally convex space.

We defineC^I_Mu {x∈M:Ix∈PMu}and denote byI0the class of closed convex subsets ofEcontaining 0. ForM ∈ I0, we defineM_u {x ∈ M : p_αx ≤ 2p_αufor each p_α∈A^∗τ}. It is clear thatP_Mu⊂M_u∈I0.

The following result extends14, Theorem 4.1and7, Theorem 2.14.

Theorem 2.12. Letf, g, T be selfmaps of a Hausdorfflocally convex spaceE, τwithu∈ FT∩ Ff∩FgandM ∈I0such thatTMu ⊂ fM ⊂M gM. Suppose thatp_αfx−u pαx−uandpαgx−u pαx−ufor allx∈Muand for eachpαwhereclfMis compact. Then

iPMuis nonempty, closed, and convex, iiTPMu⊂fPMu⊂P_Mu gPMu,

iiiPMu ∩ Ff ∩ Fg ∩ FT/∅ provided f and g are subcompatible, affine, and nonexpansive onM, and, for someq∈PMuand for allx, y∈PMu,

pα

fx−fy

≤max

pα

gx−gy , dpα

gx, q, fx

, dpα

gy, q, fy

, 1

2 d_pα

gx, q, fy

d_pα gy,

q, fx ,

2.15

T is continuous, the pairs{T, f} and {T, g} are subcompatible on P_Mu and satisfy for all q ∈ Ff∩Fg,

pα

Tx−Ty

≤max

pα

fx−gy , dpα

fx, q, Tx

, dpα

gy, q, Ty

, 1

2 d_pα

fx, q, Ty

d_pα gy,

q, Tx

2.16

for allx, y∈P_Muand for eachp_α∈A^∗τ.

Proof. iWe follow the arguments used in7and8. Letrdpαu, Mfor eachpα.

Then there is a minimizing sequence {yn} in Msuch that lim_n p_αu−y_n r. As clfMis compact so{fyn}has a convergent subsequence{fym}with lim_mfy_mx₀say inM.Now by using

pα

fx−u

≤pαx−u 2.17

we get for eachp_α,

r≤pαx0−u lim

m p_α

fym−u

≤lim

m pα

ym−u lim

n p_α yn−u

r. 2.18

Hencex0∈PMu.ThusPMuis nonempty closed and convex.

iFollows from7, Theorem 2.14.

iiByTheorem 2.7i,P_Mu∩Ff∩Fg/∅,so it follows that there existsq∈P_Mu such thatq∈Ff∩Fg.Henceiiifollows fromTheorem 2.7i.

3. Banach Operator Pair in Locally Convex Spaces

Utilizing similar arguments as above, the following result can be proved which extends recent common fixed point results due to Hussain and Rhoades8, Theorem 2.1and Jungck and Hussain9, Theorem 2.1to the setup of a Hausdorfflocally convex space which is not necessarily metrizable.

Theorem 3.1. LetMbe aτ-bounded subset of a Hausdorfflocally convex spaceE, τ, and letIand letT be weakly compatible self-maps ofM. Assume thatτ −clTM ⊂ IM,τ−clTMis τ-sequentially complete, andTandIsatisfy, for allx, y∈M,pα∈A^∗τand for some 0≤k <1,

p_α

Tx−Ty

≤kmax p_α

Ix−Iy

, p_αIx−Tx, pα

Iy−Ty , p_α

Ix−Ty , p_α

Iy−Tx . 3.1

ThenFI∩FTis a singleton.

As an application ofTheorem 3.1, the analogue of all the results due to Hussain and Berinde25, and Hussain and Rhoades8can be established forC_q-commuting mapsIand T defined on aτ-bounded subsetMof a Hausdorfflocally convex space. We leave details to the reader.

Recently, Chen and Li 23introduced the class of Banach operator pairs, as a new class of noncommuting maps and it has been further studied by Hussain24, Ciric et al.

32, Khan and Akbar 19, 20, and Pathak and Hussain 21. The pair T, f is called a Banach operator pair, if the set Ff is T-invariant, namely, TFf ⊆ Ff. Obviously, commuting pair T, f is a Banach operator pair but converse is not true, in general; see 21,23. IfT, fis a Banach operator pair, thenf, Tneed not be a Banach operator pair cf.23, Example 1.

Chen and Li23proved the following.

Theorem 3.223, Theorems 3.2-3.3. LetMbe aq-starshaped subset of a normed spaceX and let T, I be self-mappings of M. Suppose thatFIis q-starshaped and I is continuous on M. If clTMis compact (resp.,I is weakly continuous,clTMis complete, Mis weakly compact, and eitherI−T is demiclosed at 0 orX satisfies Opial’s condition),T, Iis a Banach operator pair, andT isI-nonexpansive onM, thenM∩FT∩FI/∅.

In this section, we extend and improve the above-mentioned common fixed point results of Chen and Li23in the setup of a Hausdorfflocally convex space.

Lemma 3.3. LetMbe a nonemptyτ-bounded subset of Hausdorfflocally convex spaceE, τ, and let T, f,andgbe self-maps ofM.IfFf∩Fgis nonempty,τ−clTFf∩Fg⊆Ff∩Fg, τ −clTM isτ-sequentially complete, and T, f,and g satisfy for allx, y ∈ M and for some 0≤k <1,

p_α

Tx−Ty

≤k max p_α

fx−gy , p_α

fx−Tx , p_α

gy−Ty , p_α

fx−Ty , p_α

gy−Tx 3.2 thenM∩FT∩Ff∩Fgis singleton.

Proof. Note that τ − clTFf ∩Fg being a subset of τ − clTM is τ-sequentially complete. Further, for allx, y∈Ff∩Fg, we have

pα

Tx−Ty

≤k max pα

fx−gy , pα

fx−Tx , pα

gy−Ty , pα

fx−Ty , pα

gy−Tx

k max pα

x−y

, pαx−Tx, pα

y−Ty , pα

x−Ty , pα

y−Tx .

3.3 HenceT is a generalized contraction onFf∩Fgandτ−clTFf∩Fg⊆Ff∩Fg.

By Theorem 3.1withI identity map,T has a unique fixed point zin Ff∩Fgand consequently,FT∩Ff∩Fgis singleton.

The following result generalizes 19, Theorem 2.3, 24, Theorem 2.11, and 21, Theorem 2.2and improves14, Theorem 2.2and13, Theorem 6.

Theorem 3.4. LetMbe a nonemptyτ-bounded subset of Hausdorfflocally convex (resp., complete) spaceE, τand let T, f,and g be self-maps ofM.Suppose thatFf∩Fgisq-starshaped,τ − clTFf∩Fg⊆Ff∩Fg(resp.,τ−wclTFf∩Fg⊆Ff∩Fg),τ−clTMis compact (resp.,τ−wclTMis weakly compact),Tis continuous onM(resp.,I−Tis demiclosed at 0, whereIstands for identity map) and

p_α

Tx−Ty

≤max p_α

fx−gy , d_pα

fx, q, Tx

, d_pα gy,

q, Ty , d_pα

gy, q, Tx

, d_pα fx,

q, Ty .

3.4

For allx, y∈M,thenM∩FT∩Ff∩Fg/∅.

Proof. DefineTn:Ff∩Fg → Ff∩FgbyTnx 1−knqknTxfor allx∈Ff∩Fg and a fixed sequence of real numbersk_n0 < k_n < 1converging to 1. SinceFf∩Fgis q-starshaped andτ−clTFf∩Fg⊆Ff∩Fg resp.,τ−wclTFf∩Fg⊆Ff∩ Fg, soτ−clTnFf∩Fg⊆Ff∩Fg resp.,τ−wclTnFf∩Fg⊆Ff∩Fg for eachn≥1. Also by3.4,

p_α

T_nx−T_ny

k_np_α

Tx−Ty

≤k_nmax p_α

fx−gy , d_pα

fx, q, Tx

, dpα

gy, q, Ty

, dpα

fx, q, Ty

, dpα

gy,

q, Tx

≤k_n max p_α

fx−gy , p_α

fx−T_nx , p_α

gy−T_ny , p_α

gy−T_nx , p_α

fx−T_ny ,

3.5

for eachx, y∈Ff∩Fgand some 0< kn<1.

Ifτ −clTM is compact, for each n ∈ N,τ −clTnMis τ-compact and hence τ-sequentially complete. ByLemma 3.3, for eachn ≥ 1,there existsx_n ∈ Ff∩Fgsuch thatxn fxn gxn Tnxn.The compactness ofτ −clTMimplies that there exists a subsequence{Txm}of{Txn}such thatTx_m → z ∈clTMasm → ∞. Since{Txm}is a

sequence inTFf∩Fgandτ−clTFf∩Fg⊆Ff∩Fg, thereforez∈Ff∩Fg.

Further,x_mT_mx_m 1−k_mqk_mTx_m → z. By the continuity ofT, we obtainTzz. Thus, M∩FT∩Ff∩Fg/∅proves the first case.

The weak compactness of τ − wclTM implies that τ − wclTnM is weakly compact and henceτ-sequentially complete due to completeness ofX. FromLemma 3.3, for eachn≥1,there existsx_n∈Ff∩Fgsuch thatx_nfx_ngx_nT_nx_n.Moreover, we have pαxn−Txn → 0 asn → ∞. The weak compactness ofτ−wclTMimplies that there is a subsequence{Txm}of{Txn}converging weakly toy ∈τ−wclTMasm → ∞. Since {Txm}is a sequence inTFf∩Fg, thereforey∈τ−wclTFf∩Fg⊆Ff∩Fg.

Also we have,xm−Txm → 0 asm → ∞. IfI −T is demiclosed at 0, theny Ty. Thus M∩FT∩Ff∩Fg/∅.

Corollary 3.5. LetMbe a nonemptyτ-bounded subset of Hausdorfflocally convex (resp., complete) spaceE, τand let T, f,and g be self-maps ofM.Suppose thatFf∩Fgisq-starshaped, and τ-closed (resp.,τ-weakly closed),τ−clTMis compact (resp.,τ−wclTMis weakly compact), Tis continuous onM(resp.,I−T is demiclosed at 0),T, fandT, gare Banach operator pairs and satisfy3.4for allx, y∈M,thenM∩FT∩Ff∩Fg/∅.

LetCPMu∩C^f,g_Mu,whereC^f,g_Mu C^f_Mu∩C^g_MuandC_M^f u {x∈M:fx∈ P_Mu}.It is important to note here thatP_Muis always bounded.

Corollary 3.6. LetEbe a Hausdorfflocally convex (resp., complete) space andT, f,andg be self- maps ofE.Ifu∈ E,D ⊆ C,D0 : D∩Ff∩Fgisq-starshaped,τ−clTD0 ⊆ D0 (resp., τ −wclTD0 ⊆ D₀],τ −clTDis compact (resp.,τ −wclTDis weakly compact),T is continuous onD (resp.,I −T is demiclosed at 0), and3.4holds for allx, y ∈ D,thenP_Mu∩ FT∩Ff∩Fg/∅.

Corollary 3.7. LetEbe a Hausdorfflocally convex (resp., complete) space andT, f,andgbe self-maps ofE.Ifu ∈ E,D ⊆ P_Mu,D₀ : D∩Ff∩Fgisq-starshaped,τ −clTD0 ⊆ D₀ (resp., τ −wclTD0 ⊆ D0),τ −clTDis compact (resp.,τ −wclTDis weakly compact),T is continuous onD (resp.,I −T is demiclosed at 0), and3.4holds for allx, y ∈ D,thenPMu∩ FT∩Ff∩Fg/∅.

Remark 3.8. Khan and Akbar19, Corollaries 2.4–2.8and Chen and Li23, Theorems 4.1 and 4.2are particular cases of Corollaries3.5and3.6.

The following result extends14, Theorem 4.1,7, Theorem 2.14,19, Theorem 2.9, and21, Theorems 2.7–2.11.

Theorem 3.9. Letf, g, Tbe self-maps of a Hausdorfflocally convex spaceE. Ifu∈EandM∈I0

such thatTMu ⊆ M,τ−clTMuis compact andTx−u ≤ x−ufor allx∈ Mu, then P_Mu is nonempty, closed, and convex with TPMu ⊆ P_Mu. If, in addition, D ⊆ P_Mu, D₀ :D∩Ff∩Fgisq-starshaped,τ−clTD0⊆D₀,Tis continuous onD,and3.4holds for allx, y∈D,thenPMu∩FT∩Ff∩Fg/∅.

Proof. We utilizeCorollary 3.5instead ofTheorem 2.7in the proof ofTheorem 2.12.

Remark 3.10. 1The class of Banach operator pairs is different from that of weakly compatible maps; see for example21,23,32.

2 In Example 2.2, the pair T, f is a Banach operator but T and f are not C_q- commuting maps and hence not a subcompatible pair.

Acknowledgments

The author A. R. Khan gratefully acknowledges the support provided by the King Fahd University of Petroleum & Minerals during this research. The authors would like to thank the referees for their valuable suggestions to improve the presentation of the paper.

References

1 G. K ¨othe, Topological Vector Spaces. I, vol. 159 of Die Grundlehren der mathematischen Wissenschaften, Springer, New York, NY, USA, 1969.

2 L. X. Cheng, Y. Zhou, and F. Zhang, “Danes’ drop theorem in locally convex spaces,” Proceedings of the American Mathematical Society, vol. 124, no. 12, pp. 3699–3702, 1996.

3 E. Tarafdar, “Some fixed-point theorems on locally convex linear topological spaces,” Bulletin of the Australian Mathematical Society, vol. 13, no. 2, pp. 241–254, 1975.

4 S. Sessa, “On a weak commutativity condition of mappings in fixed point considerations,” Publications de l’Institut Math´ematique, vol. 3246, pp. 149–153, 1982.

5 G. Jungck, “Common fixed points for commuting and compatible maps on compacta,” Proceedings of the American Mathematical Society, vol. 103, no. 3, pp. 977–983, 1988.

6 N. Hussain and A. R. Khan, “Common fixed-point results in best approximation theory,” Applied Mathematics Letters, vol. 16, no. 4, pp. 575–580, 2003.

7 N. Hussain and G. Jungck, “Common fixed point and invariant approximation results for noncom- muting generalizedf, g-nonexpansive maps,” Journal of Mathematical Analysis and Applications, vol.

321, no. 2, pp. 851–861, 2006.

8 N. Hussain and B. E. Rhoades, “Cq-commuting maps and invariant approximations,” Fixed Point Theory and Applications, vol. 2006, Article ID 24543, 9 pages, 2006.

9 G. Jungck and N. Hussain, “Compatible maps and invariant approximations,” Journal of Mathematical Analysis and Applications, vol. 325, no. 2, pp. 1003–1012, 2007.

10 G. Meinardus, “Invarianz bei linearen Approximationen,” Archive for Rational Mechanics and Analysis, vol. 14, no. 1, pp. 301–303, 1963.

11 S. P. Singh, “An application of a fixed-point theorem to approximation theory,” Journal of Approximation Theory, vol. 25, no. 1, pp. 89–90, 1979.

12 S. A. Sahab, M. S. Khan, and S. Sessa, “A result in best approximation theory,” Journal of Approximation Theory, vol. 55, no. 3, pp. 349–351, 1988.

13 G. Jungck and S. Sessa, “Fixed point theorems in best approximation theory,” Mathematica Japonica, vol. 42, no. 2, pp. 249–252, 1995.

14 M. A. Al-Thagafi, “Common fixed points and best approximation,” Journal of Approximation Theory, vol. 85, no. 3, pp. 318–323, 1996.

15 R. P. Pant, “Common fixed points of noncommuting mappings,” Journal of Mathematical Analysis and Applications, vol. 188, no. 2, pp. 436–440, 1994.

16 H. K. Pathak, Y. J. Cho, and S. M. Kang, “Remarks onR-weakly commuting mappings and common fixed point theorems,” Bulletin of the Korean Mathematical Society, vol. 34, no. 2, pp. 247–257, 1997.

17 M. A. Khamsi and W. A. Kirk, An Introduction to Metric Spaces and Fixed Point Theory, Pure and Applied Mathematics, Wiley-Interscience, New York, NY, USA, 2001.

18 S. Singh, B. Watson, and P. Srivastava, Fixed Point Theory and Best Approximation: The KKM-Map Principle, vol. 424 of Mathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1997.

19 A. R. Khan and F. Akbar, “Best simultaneous approximations, asymptotically nonexpansive mappings and variational inequalities in Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 354, no. 2, pp. 469–477, 2009.

20 A. R. Khan and F. Akbar, “Common fixed points from best simultaneous approximations,” Taiwanese Journal of Mathematics, vol. 13, no. 4, 2009.

21 H. K. Pathak and N. Hussain, “Common fixed points for Banach operator pairs with applications,”

Nonlinear Analysis: Theory, Methods & Applications, vol. 69, no. 9, pp. 2788–2802, 2008.

22 G. L. Cain Jr. and M. Z. Nashed, “Fixed points and stability for a sum of two operators in locally convex spaces,” Pacific Journal of Mathematics, vol. 39, pp. 581–592, 1971.

23 J. Chen and Z. Li, “Common fixed-points for Banach operator pairs in best approximation,” Journal of Mathematical Analysis and Applications, vol. 336, no. 2, pp. 1466–1475, 2007.

24 N. Hussain, “Common fixed points in best approximation for Banach operator pairs with ´Ciri´c type l-contractions,” Journal of Mathematical Analysis and Applications, vol. 338, no. 2, pp. 1351–1363, 2008.

25 N. Hussain and V. Berinde, “Common fixed point and invariant approximation results in certain metrizable topological vector spaces,” Fixed Point Theory and Applications, vol. 2006, Article ID 23582, 13 pages, 2006.

26 B. N. Sahney, K. L. Singh, and J. H. M. Whitfield, “Best approximations in locally convex spaces,”

Journal of Approximation Theory, vol. 38, no. 2, pp. 182–187, 1983.

27 S. P. Singh, “Some results on best approximation in locally convex spaces,” Journal of Approximation Theory, vol. 28, no. 4, pp. 329–332, 1980.

28 W. W. Taylor, “Fixed-point theorems for nonexpansive mappings in linear topological spaces,” Journal of Mathematical Analysis and Applications, vol. 40, no. 1, pp. 164–173, 1972.

29 A. R. Khan, F. Akbar, and N. Sultana, “Random coincidence points of subcompatible multivalued maps with applications,” Carpathian Journal of Mathematics, vol. 24, no. 2, pp. 63–71, 2008.

30 S. L. Singh and A. Tomar, “Weaker forms of commuting maps and existence of fixed points,” Journal of the Korea Society of Mathematical Education. Series B, vol. 10, no. 3, pp. 145–161, 2003.

31 M. Fabian, P. Habala, P. H´ajek, V. Montesinos Santaluc´ıa, J. Pelant, and V. Zizler, Functional Analysis and Infinite-Dimensional Geometry, CMS Books in Mathematics/Ouvrages de Math´ematiques de la SMC, 8, Springer, New York, NY, USA, 2001.

32 L. B. ´Ciri´c, N. Husain, F. Akbar, and J. S. Ume, “Common fixed points for Banach operator pairs from the set of best approximations,” Bulletin of the Belgian Mathematical Society. Simon Stevin, vol. 16, 2009.