Common Fixed Points, Invariant Approximation and Generalized Weak Contractions

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Volume 2012, Article ID 102980,11pages doi:10.1155/2012/102980

Research Article

Common Fixed Points, Invariant Approximation

and Generalized Weak Contractions

Sumit Chandok

Department of Mathematics, Khalsa College of Engineering & Technology (Punjab Technical University), Ranjit Avenue, Amritsar-143001, India

Correspondence should be addressed to Sumit Chandok,[email protected]

Received 23 March 2012; Revised 23 August 2012; Accepted 31 August 2012

Academic Editor: Harvinder S. Sidhu

Copyrightq2012 Sumit Chandok. 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.

Suﬃcient conditions for the existence of a common fixed point of generalized f-weakly

con-tractive noncommuting mappings are derived. As applications, some results on the set of best approximation for this class of mappings are obtained. The proved results generalize and extend various known results in the literature.

1. Introduction and Preliminaries

It is well known that Banach’s fixed point theorem for contraction mappings is one of the pivotal result of analysis. LetX, dbe a metric space. A mappingT :X → X is said to be

contraction if there exists 0≤k <1 such that for allx, y∈X,

dTx, Ty≤kdx, y. 1.1

If the metric spaceX, dis complete, then the mapping satisfying1.1has a unique fixed point.

A natural question is that whether we can find contractive conditions which will imply existence of fixed point in a complete metric space but will not imply continuity. Kannan1,2

proved the following result, giving an aﬃrmative answer to the above question.

Theorem 1.1. IfT :X → X, whereX, dis a complete metric space, satisfies

dTx, Ty≤kdx, Tx dy, Ty, 1.2

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The mappings satisfying1.2 are called Kannan type mappings. A similar type of contractive condition has been studied by Chatterjea3and he proved the following result.

Theorem 1.2. IfT :X → X, whereX, dis a complete metric space, satisfies

dTx, Ty≤kdx, Tydy, Tx, 1.3

where 0≤k <1/2 andx, y∈X, thenT has a unique fixed point.

In Theorems1.1and1.2there is no requirement of continuity ofT.

A mapT:X → Xis called a weakly contractivesee4–6if for eachx, y∈X,

dTx, Ty≤dx, y−ψdx, y, 1.4

whereψ:0,∞ → 0,∞is continuous and nondecreasing,ψx 0 if and only ifx0 and limψx ∞.

If we takeψx 1−kx, 0 ≤ k < 1, then a weakly contractive mapping is called contraction.

A mapT:X → Xis calledf-weakly contractivesee7if for eachx, y∈X,

dTx, Ty≤dfx, fy−ψdfx, fy, 1.5

wheref :X → X is a self-mapping,ψ :0,∞ → 0,∞is continuous and nondecreasing,

ψx 0 if and only ifx0 and limψx ∞.

If we takeψx 1−kx, 0≤k < 1, then af-weakly contractive mapping is called

f-contraction. Further, iffidentity mapping andψx 1−kx, 0≤k <1, then af-weakly contractive mapping is called contraction.

A mapT :X → Xis called a generalized weakly contractivesee5if for eachx, y∈X,

dTx, Ty≤ 1

2

dx, Tydy, Tx−ψdx, Ty, dy, Tx, 1.6

whereψ:0,∞2 → 0,∞is continuous such thatψx, y 0 if and only ifxy0. If we takeψx, y 1−k/2xy, 0≤k <1/2, then inequality1.6reduces to1.3. Choudhury5shows that generalized weakly contractive mappings are generalizations of contractive mappings given by Chatterjea 1.3, and it constitutes a strictly larger class of mappings than given by Chatterjea.

A mapT:X → Xis called a generalizedf-weakly contractive8if for eachx, y∈X,

dTx, Ty≤ 1

2

dfx, Tydfy, Tx−ψdfx, Ty, dfy, Tx, 1.7

wheref:X → Xis a self-mapping,ψ:0,∞2 → 0,∞is continuous such thatψx, y 0 if and only ifxy0.

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For a nonempty subsetMof a metric spaceX, dandx∈X, an elementy∈Mis said to be a best approximant toxor a bestM-approximant toxifdx, y dx, M≡ inf{dx, k:

k∈M}. The set of all suchy∈Mis denoted byPMx.

A subsetMof a normed linear spaceXis said to be a convex set ifλx 1−λy∈M

for allx, y ∈ Mandλ ∈ 0,1. A setMis said to bep-starshaped, wherep ∈ M, provided

λx 1−λp∈Mfor allx∈Mandλ∈0,1, that is, if the segmentp, x {λx 1−λp: 0 ≤ λ ≤1}joiningptoxis contained inMfor allx ∈M.Mis said to be starshaped if it is

p-starshaped for somep∈M.

Clearly, each convex setMis starshaped but converse is not true.

Suppose thatMis a subset of normed linear spaceX. A mappingT fromMtoX is said to be demiclosed if for every sequence{xn} ⊆Msuch thatxnconverges weakly tox∈M

and{Txn}converges strongly toy ∈Ximplyy Tx.T is said to be demiclosed at 0, if for

every sequence{xn}inMconverging weakly toxand{Txn}converging strongly to 0, then

Tx0.

For a convex subsetMof a normed linear spaceX, a mappingT :M → Mis said to be aﬃne if for allx, y∈M,Tλx 1−λy λTx 1−λTyfor allλ∈0,1.

The ordered pair T, Iof two self-maps of a metric space X, d is called a Banach

operator pair9, if the setFIof fixed points ofI isT-invariant, that is,TFI ⊆ FI. A pointx∈Xis a coincidence pointcommon fixed pointofIandT ifIxTxxIxTx. The set of fixed points resp., coincidence pointsof I andT is denoted byFI, T resp.,

CI, T. The pairI, Tis called commuting ifTIxITxfor allx∈X. Obviously, commuting pair,T, Iis a Banach operator pair but not converselysee9. IfT, Iis a Banach operator pair thenI, Tneed not be a Banach operator pairsee9. If the self-mapsT and I ofX

satisfydITx, Tx ≤ kdIx, x, for allx ∈ X and for some k ≥ 0, ITx TIx whenever

x∈FI, that is,Tx∈FI, thenT, Iis a Banach operator pair. This class of noncommuting mappings is diﬀerent from the known classes of noncommuting mappings namely,R-weakly commuting,R-subweakly commuting, compatible, weakly compatible,Cq-commuting, and

so forth, existing in the literature.

Fixed point theory has gained impetus, due to its wide range of applicability, to resolve diverse problems emanating from the theory of nonlinear diﬀerential equations, theory of nonlinear integral equations, game theory, mathematical economics, control theory, and so forth. For example, in theoretical economics, such as general equilibrium theory, a situation arises where one needs to know whether the solution to a system of equations necessarily exists, or, more specifically, under what conditions will a solution necessarily exist. The mathematical analysis of this question usually relies on fixed point theorems. Hence, finding necessary and suﬃcient conditions for the existence of fixed points is an interesting aspect. Alber and Guerre-Delabriere 4 introduced the concept of weakly contractive mappings and proved the existence of fixed points for single-valued weakly contractive mappings in Hilbert spaces. Thereafter, Rhoades 6 proved a fixed point theorem which is one of the generalizations of Banach’s Fixed Point Theorem in 1922, because the weakly contractions contain many contractions as a special case, and he also showed that some results of 4

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contractive noncommuting mappings are obtained. As applications, we also establish some results on the set of best approximation for this class of mappings. The proved results gener-alize and extend the corresponding results of5,8,9,12–14,18–20and of few others.

2. Main Results

The following result is a consequence of the main theorem of Choudhury5.

Lemma 2.1. Let Mbe a subset of a metric spaceX, dand T is a self-mapping ofMsuch that

clTM⊆M. If clTMis complete andTsatisfies

dTx, Ty≤ 1

2

whereψ :0,∞2 → 0,∞is a continuous mapping such thatψx, y 0 if and only ifxy0,

for allx, y∈X, thenT has a unique fixed point inM.

Corollary 2.2see5. LetT be a self-mapping ofX, whereX, dis a complete metric space. IfT

satisfies

dTx, Ty≤ 1

2

for allx, y∈X, thenT has a unique fixed point.

Ifψx, y 1/2−kxy, 0≤k <1/2, we haveTheorem 1.2.

Theorem 2.3. LetMbe a subset of a metric spaceX, d, andf andT are self-mappings ofMsuch

that clTFf⊆Ff. If clTMis complete,Ffis nonempty, andfandTsatisfy

dTx, Ty≤ 1

2

for allx, y∈X, thenM∩FT∩Ffis a singleton.

Proof. clTFfbeing a subset of clTMis complete and and clTFf⊆ Ff. So for all

x, y∈Ff, we have

dTx, Ty≤ 1

2

dfx, Tydfy, Tx−ψdfx, Ty, dfy, Tx

1

2

dx, Tydy, Tx−ψdx, Ty, dy, Tx.

2.4

Thus, byLemma 2.1,T has a unique fixed pointzinFfand consequently,M∩FT∩Ff

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Corollary 2.4. Let Mbe a subset of a metric spaceX, d, and f andT are self-mappings of M. If clTMis complete,T, fis a Banach operator pair,Ffis nonempty and closed, andf andT satisfy

dTx, Ty≤ 1

2

for allx, y∈X, thenM∩FT∩Ffis a singleton.

Example 2.5. LetX {p, q, r}anddis a metric defined onX. LetT andf be self-mappings ofXsuch thatTp fq,Tq fq,Tr fp,dfp, fq 1,dfq, fr 2,dfr, fp 1.5 and

ψa, b 1/2min{a, b}. ThenT is a generalizedf-weakly contraction, andqis the coin-cidence point ofT andf.

Iffidentity mapping, this example given in5.

Ifψx, y 1/2−kxy, 0≤k <1/2, we have the following result.

Corollary 2.6. LetMbe a subset of a metric spaceX, d, andfandTbe self-mappings ofMsuch

that clTFf⊆Ff. If clTMis complete,Ffis nonempty, andfandTsatisfy

dTx, Ty≤kdfx, Tydfy, Tx, 2.6

where 0≤k <1/2, for allx, y∈M, thenM∩FT∩Ffis a singleton.

Corollary 2.7. LetMbe a subset of a metric spaceX, dandf and T be self-mappings of M. If

clTMis complete,T, f is a Banach operator pair,Ffis nonempty and closed, and f and T satisfy

dTx, Ty≤kdfx, Tydfy, Tx, 2.7

where 0≤k <1/2, for allx, y∈M, thenM∩FT∩Ffis a singleton.

Theorem 2.8. LetMbe a nonempty subset of a normed (resp., Banach) space X, and Tf be

self-mappings ofM. Suppose thatFfisq-starshaped, clTFf⊆Ff(resp.,wclTFf⊆Ff),

clTMis compact (resp.,wclTMis weakly compact, andf−Tis demiclosed at 0), andTsatisfies

Tx−Ty≤ 1

2

distfx,Ty, qdistfy,Tx, q

−ψdistfx,Ty, q,distfy,Tx, q,

2.8

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Proof. For eachn, defineTn : M → MbyTnx 1−knqknTx,x ∈ Mwhereknis a

sequence in0,1such thatkn → 1. SinceFfisq-starshaped and clTFf⊆Ff resp.,

wclTFf⊆Ff, we have

Tnx 1−knqknTx 1−knfqknTx∈F

f, 2.9

for allx∈Ffand so clTnFf⊆Ff resp.,wclTnFf⊆Fffor eachn. Consider

Tnx−TnyknTx−Ty

≤kn

1 2

−ψdistfx,Ty, q,distfy,Tx, q

≤kn

1 2

≤ kn

2 fx−Tnyfy−Tnx,

2.10

for allx, y∈Ff. As clTMis compact, clTnMis compact for eachnand hence complete.

Now byCorollary 2.6, there existsxn∈Msuch thatxnis a common fixed point offandTnfor

eachn. The compactness of clTMimplies that there exists a subsequence{Txni}of{Txn}

such thatTxni → z∈clTM. Since{Txn}is a sequence inTFf,z∈clTFf⊆Ff.

Now, askni → 1, we have

xni Tnixni 1−kniqkniTxni−→z 2.11

andfxni−Txnixni−Txni → 0. Further, we have

Txni−Tz ≤

1 2

distfxni,

Tz, qdistfz,Txni, q

−ψdistfxni,

Tz, q,distfz,Txni, q

1

2

distxni,

Tz, qdistfz,Txni, q

−ψdistxni,

Tz, q,distfz,Txni, q

≤ 1

2

xni−Tzfz−Txni,

2.12

on taking limit, we getzTzand soM∩FT∩Ff/∅.

Next, the weak compactness ofwclTMimplies thatwclTnMis weakly compact

and hence complete. Hence, byCorollary 2.6, for each n ∈ N, there existsxn ∈ Ffsuch

thatxnfxnTnxn. The weak compactness ofwclTMimplies that there is subsequence

{Txni} of {Txn} such that Txni y ∈ wclTM. Since {Txn} is a sequence in TFf,

y∈wclTFf⊆Ff. Also, we havefxni−Txni xni−Txni → 0.

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LetMbe a nonempty subset of a metric spaceX, d. Suppose thatCPMu∩CfMu,

whereCf_Mu {x∈M:fx∈PMu}.

Corollary 2.9. LetX be a normed (resp., Banach) space, andT,f are self-mappings ofX. Ifu∈X,

D⊆C,GD∩Ffisq-starshaped, clTG⊆G(resp.,wclTG⊆G), clTDis compact (resp., wclTDis weakly compact andf−Tis demiclosed at 0), andTsatisfies the inequality2.8for all x, y∈D, thenPMu∩Ff, Tis nonempty.

Corollary 2.10. LetXbe a normed (resp., Banach) space andT, fare self-mappings ofX. Ifu∈X,

D⊆PMu,GD∩Ffisq-starshaped, clTG⊆G(resp.,wclTG⊆G), clTDis compact

(resp.,wclTDis weakly compact andf−Tis demiclosed at 0), andT satisfies the inequality2.8

for allx, y∈D, thenPMu∩Ff, Tis nonempty.

Remark 2.11. Theorem 2.8extends and generalizes the corresponding results of8,9,13,14,

18,19.

LetG0denote the class of closed convex subsets of a normed spaceXcontaining 0. For

M∈G0andp∈X, letMp{x∈M:x ≤2p}. ThenPMp⊂Mp∈G0see12,20.

Theorem 2.12. LetXbe a normed (resp., Banach) space, andT,g are self-mappings ofX. Ifp ∈X

andM∈G0such thatTMp⊆M, clTMpis compact (resp.,wclTMpis weakly compact), and

Tx−p ≤ x−pfor allx∈Mp, thenPMpis nonempty, closed, and convex withTPMp⊆

PMp. If, in addition,Dis a subset ofPMp,G D∩Fgisq-starshaped, clTG ⊆ G(resp.,

wclTG⊆ Gandg−T is demiclosed at 0), andT satisfies inequality2.8for allx, y ∈D, then PMp∩Fg, Tis nonempty.

Proof. Ifp ∈ M, then the results are obvious. So assume that p /∈ M. Ifx ∈ M\Mp, then

x−x0>2p−x0 and sop−x ≥ x − p>p ≥distp, M. Thusαdistp, M≤ p.

Since clTMpis compact, and by the continuity of norm, there existsz ∈clTMpsuch

thatβdistp,clTMp z−p.

On the other hand, if wclTMp is weakly compact, then using Lemma 5.5 of

Singh et al. 22, page 192, we can show that there exists z ∈ wclTMp such that

βdistp, wclTMp z−p.

Hence, in both cases, we have

αdistp, M≤distp,clTMp

β

distp, TMp

≤Tx−p

≤x−p,

2.13

for allx∈Mp. Therefore,αβdistp, M, that is, distp, M distp,clTMp dp, z,

that is,z∈PMpand soPMpis nonempty. The closedness and convexity ofPMpfollow

from that ofM. Now to prove TPMp ⊆ PMp, let y ∈ TPMp. Theny Txforx ∈

PMp. Consider

dp, ydp, Tx≤dp, xdistp, M, 2.14

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The compactness of clTMp resp., weakly compactness ofwclTMpimplies that

clTD is compact resp., wclTD is weakly compact. Hence, the result follows from

Corollary 2.10.

Corollary 2.13. LetXbe a normed (resp., Banach) space, andT,gare self-mappings ofX. Ifp∈X

andM∈G0such thatTMp⊆M, clTMpis compact (resp.,wclTMpis weakly compact), and

Tx−p ≤ x−pfor allx∈Mp, thenPMpis nonempty, closed, and convex withTPMp⊆

PMp. If, in addition,Dis a subset ofPMp,GD∩Fgisq-starshaped and closed (resp., weakly

closed andg−Tis demiclosed at 0),T, gis a Banach operator pair onD, andT satisfies inequality

2.8for allx, y∈D, thenPMp∩Fg, Tis nonempty.

Remark 2.14. Theorem 2.12extends and generalizes the corresponding results of Al-Thagafi

12, Al-Thagafi and Shahzad13, Habiniak18, Narang and Chandok20, and Shahzad

21.

The following result will be used in the sequel.

Lemma 2.15. LetCbe a nonempty subset of a metric spaceX, d,f,gself-maps ofC, clTFf∩

Fg ⊆ Ff∩Fg. Suppose that clTCis complete, andT,f,g satisfy for allx, y ∈ Cand

0≤k <1,

dTx, Ty≤kmax dfx, gy, dTx, fx, dTy, gy, dTx, gy, dTy, fx. 2.15

IfFf∩Fgis nonempty and clTFf∩Fg⊆Ff∩Fg, then there is a common fixed point ofT, fandg.

Proof. clTFf∩Fg, being a closed subset of the complete set clTC, is complete. Further for allx, y∈Ff∩Fg, we have

dTx, Ty≤kmax dfx, gy, dTx, fx, dTy, gy, dTx, gy, dTy, fx

kmax dx, y, dTx, x, dTy, y, dTx, y, dTy, x. 2.16

Hence,T is a generalized contraction onFf∩Fgand clTFf∩Fg ⊆ Ff∩Fg. So by Lemma 3.1 of 13,T has a unique fixed point y in Ff∩Fg and consequently

FT∩Ff∩Fgis a singleton.

Remark 2.16. Iff g, then Theorem 3.2 of Al-Thagafi and Shahzad13is a particular case of

Lemma 2.15.

The following result extends and improves the corresponding results of9,12–14,18,

20.

Theorem 2.17. LetT, g, hbe self-mappings of a Banach spaceX. Ifp ∈ FT∩Fg∩Fhand

M∈G◦such thatTMp⊆gM⊆M⊆hM, clgMpis compact, andTx−p ≤ gx−hp,

gx−u ≤ x−u,hx−ux−ufor allx∈Mp, then

iPMpis nonempty, closed, and convex,

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iiiPMp ∩ FT ∩ Fg ∩ Fh/∅, provided T is continuous, Fg is q-starshaped,

clgFh⊆Fh, and the pairg, hsatisfies the inequality2.8for allx, y ∈PMp,

Fgisq-starshaped withq∈PMp∩Fg∩Fh G, clTFg∩Fh⊆Fg∩Fh

andT, g, hsatisfy for allq∈Fg∩Fhandx, y∈PMp

dTx, Ty≤max dhx, gy,disthx,q, Tx,distgy,q, Ty,disthx,q, Ty,

distgy,q, Tx,

2.17

then there is a common fixed point ofPMp,T,g, andh.

Proof. Proceeding as inTheorem 2.12, we can proveiandii.

By ii, the compactness of clgMp implies that clgPMp and clTPMp are

compact. Hence,Theorem 2.8implies thatFg∩Fh∩PMp/∅.

For eachn∈N, defineTn :X → XbyTnx 1−knqknTx, for eachx∈X, where

{kn} is a sequence in0,1 such thatkn → 1. Then eachTn is a self-mapping of C. Since

clTFg∩Fh⊆Fg∩Fh,Fg∩Fhisq-starshaped withq∈G, so clTnFg∩Fh⊆

Fg∩Fhfor eachn. Consider

_T_n_x₋_T_n_y_k_n_Tx₋_Ty

≤knmax hx−gy,dist

hx,q, Tx,distgy,q, Ty,disthx,q, Ty,

distgy,q, Tx

≤knmax hx−gy,hx−Tnx,gy−Tny,hx−Tny,gy−Tnx,

2.18

for allx, y ∈ PMp. As clTPMpis compact, clTnPMpis compact for eachnand

hence complete. Now byLemma 2.15, there existsxn ∈ Msuch thatxn is a common fixed

point ofg, handTn for eachn. The compactness of clTPMpimplies that there exists a

subsequence{Txni}of{Txn}such thatTxni → z∈clTPMp. Since{Txn}is a sequence in

TFg∩Fh, and clTFg∩Fh⊆Fg∩Fh, thenz∈Fg∩Fh. Now, askni → 1,

we have

xni Tnixni 1−kniqkniTxni −→z, 2.19

Tis continuous, we haveTzzand hencePMp∩FT∩Fg∩Fh/∅.

Remark 2.18. iLetM 0,1. Letdbe defined bydx, y |x−y|. We setTx x/4 and

fxx/2 for allx∈M. Defineψ :0,∞×0,∞ → 0,∞by

ψt, s ts

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Then forx, y∈M, we have

dTx, Tydx

4, y 4 x 4 − y 4 1

4x−y, 2.21

1 2

1 2 x 2 − y 4 y 2 − x 4

−ψx

2 −

y

4

,y

2 − x 4 1 2 x 2 − y 4

y₂ −x

4 −1 8 x 2 − y 4

y₂ −x

4 3 8 x 2 − y 4 y 2 − x 4 . 2.22

From2.21and 2.22, without loss of generality assume thatx ≥ y. Hence, we have two cases:

Case 1. If 2y≥x, from2.22, we have

1 2

3 8 x 2 − y 4 y 2 − x 4 3 8 _x 2 − y 4 y 2 − x 4 3 8 _x 4 y 4 3 32

xy≥ 1

4

x−y.

2.23

Case 2. If 2y < x, from2.22, we have

1 2

3 8 x 2 − y 4 y 2 − x 4 3 8 _x 2 − y 4 −y 2 x 4 3 8 3x 4 3y 4 9 32

x−y≥ 1

4

x−y.

2.24

Thus inequality2.3is satisfied and byTheorem 2.3; 0 is a common fixed point ofT andf.

ii It may be noted that the assumption of linearity or aﬃnity for I is necessary in almost all known results about common fixed points of maps T, I such that T is

I-nonexpansive under the conditions of commuting, weakly commuting, R-subweakly commuting, or compatibilitysee9,12,14,20,21and the literature cited therein, but our results in this paper are independent of the linearity or aﬃnity.

iiiConsider M R2 _{with usual metric}_d_x

1, y1,x2, y2 |x1 −x2||y1 −y2|,

x1, y1,x2, y2 ∈ R2. Define T and I on Mas Tx, y x−2/2,x2 y−4/2and

Ix, y x−2/2, x2_y₋₄_{. Obviously,}_T_is_I_{-nonexpansive,}_I_{-asymptotically nonexpansive,}

butI is not linear or aﬃne. Moreover,FT −2,0,FI {−2, y:y∈R}andCI, T

{x, y:y4−x2_{, x}_∈_R_}_{. Thus, cl}_T_F_I_⊂_F_I_{, which is not a compatible pair}_see₉_,

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Acknowledgment

The author is thankful to the learned referees for the valuable suggestions.

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