Invariant Points and

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Volume 2011, Article ID 579819,10pages doi:10.1155/2011/579819

Research Article

Invariant Points and

ε

-Simultaneous

Approximation

Sumit Chandok and T. D. Narang

Department of Mathematics, Guru Nanak Dev University, Amritsar-143005, India

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

Received 10 December 2010; Revised 18 March 2011; Accepted 19 April 2011

Academic Editor: S. M. Gusein-Zade

Copyrightq2011 S. Chandok and T. D. Narang. 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.

We generalize and extend Brosowski-Meinardus type results on invariant points from the set of

best approximation to the set ofε-simultaneous approximation. As a consequence some results

onε-approximation and best approximation are also deduced. The results proved in this paper

generalize and extend some of the known results on the subject.

1. Introduction and Preliminaries

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.

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Theorem 1.1. LetT be a linear and nonexpansive operator on a normed linear spaceE. LetCbe a

T-invariant subset of EandxaT-invariant point. If the setPCxof best C-approximants toxis

nonempty, compact, and convex, then it contains aT-invariant point.

Subsequently, various generalizations of Brosowski’s results appeared in the literature. Singh 3 observed that the linearity of the operator T and convexity of the set PCx in

Theorem 1.1can be relaxed and proved the following.

Theorem 1.2. LetT :E → Ebe a nonexpansive self-mapping on a normed linear spaceE. LetCbe

aT-invariant subset ofEandxaT-invariant point. If the setPCxis nonempty, compact, and star

shaped, then it contains aT-invariant point.

Singh 4 further showed that Theorem 1.2 remains valid if T is assumed to be nonexpansive only on PCx∪ {x}. Since then, many results have been obtained in this directionsee Chandok and Narang5,6, Mukherjee and Som7, Mukherjee and Verma

8, Narang and Chandok9–11, Rao and Mariadoss12, and references cited therein. In this paper we prove some similar types of results onT-invariant points for the set ofε-simultaneous approximation in a metric spaceX, d. Some results onT-invariant points for the set ofε-approximation and best approximation are also deduced. The results proved in the paper generalize and extend some of the results of6,8–13and of few others.

LetGbe a nonempty subset of a metric spaceX, dand letFbe a nonempty bounded subset ofX. Forx ∈X, letdFx sup{dy, x :y ∈F},DF, G inf{dFx :x∈G}and

PGF {g0∈G:dFg0 DF, G}. An elementg0 ∈PGFis said to be a best simultaneous

approximation ofFwith respect toG.

For ε > 0, we define PGεF {g0 ∈ G : dFg0 ≤ DF,G ε} {g0 ∈ G :

sup_y∈Fdy, g0 ≤ infg∈Gsupy∈Fdy, g ε}. An element g0 ∈ PGεF is said to be a

ε-simultaneous approximation ofFwith respect toG.

It can be easily seen that forε >0, the setPGεFis always a nonempty bounded set

and is closed ifGis closed.

In caseF {p},p∈X, then elements ofPGpare called best approximations topinG

and ofPGεpare calledε-approximation topinG.

A sequenceyninG is called aε-minimizing sequence forF, if lim sup_x∈Fdx, yn ≤ DF, G ε. The setGis said to beε-simultaneous approximatively compact with respect toFif for everyx∈F, eachε-minimizing sequenceyninGhas a subsequenceyniconverging to an

element ofG.

LetX, dbe a metric space. A continuous mappingW:X×X×0,1 → Xis said to be a convex structure onXif for allx, y∈Xandλ∈0,1,

du, Wx, y, λ≤λdu, x 1−λdu, y 1.1

holds for allu∈X. The metric spaceX, dtogether with a convex structure is called a convex

metric space14.

A convex metric spaceX, dis said to satisfy Property (I)15if for allx, y, p∈Xand

λ∈0,1,

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A normed linear space and each of its convex subset are simple examples of convex metric spaces withW given byWx, y, λ λx 1−λyforx, y ∈Xand 0 ≤λ≤1. There are many convex metric spaces which are not normed linear spacessee14. PropertyIis always satisfied in a normed linear space.

A subsetKof a convex metric spaceX, dis said to be

ia convex set14ifWx, y, λ∈Kfor allx, y∈Kandλ∈0,1;

iip-star shaped16wherep∈K, providedWx, p, λ∈Kfor allx∈Kandλ∈0,1;

iiistar shaped if it isp-star shaped for somep∈K.

Clearly, each convex set is star shaped but not conversely. A self-mapT on a metric spaceX, dis said to be

icontraction if there existsk, 0≤k <1 such thatdTx, Ty≤kdx, yfor allx, y∈X;

iinonexpansive ifdTx, Ty≤dx, yfor allx, y∈X;

iiiquasi-nonexpansive if the setFTof fixed points ofT is nonempty anddTx, p ≤

dx, pfor allx∈Xandp∈FT.

A nonexpansive mapping T on X with FT/∅ is quasi-nonexpansive, but not conversely. A linear quasi-nonexpansive mapping on a Banach space is nonexpansive. But there exist continuous and discontinuous nonlinear quasi-nonexpansive mappings that are not nonexpansive.

2. Main Results

To start with, we prove the following proposition onε-simultaneous approximation which will be used in the sequel.

Proposition 2.1. LetFbe a nonempty bounded subset of a metric spaceX, d, and letCbe a

non-empty subset ofX. If Cisε-simultaneous approximatively compact with respect toF, then the set

PCεFis a nonempty compact subset ofC.

Proof. Sinceε >0,PCεFis nonempty. We now show thatPCεFis compact. Letynbe a

sequence inPCεF. Then lim sup_x∈Fdx, yn≤DF, C ε, that is,ynis anε-minimizing

sequence forC. SinceCisε-simultaneous approximatively compact with respect toF, there is a subsequenceynisuch thatyni → y∈C. Consider

sup

x∈F d

x, ysup

x∈F d

x,limyni

lim sup

x∈F d

x, yni

≤DF, C ε.

2.1

This implies thaty ∈ PCεF. Thus we get a subsequenceyni ofyn converging to an

elementy∈PCεF. HencePCεFis compact.

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Corollary 2.2see 9. If Cis an ε-approximatively compact set in a metric space X, d then

PCεxis a nonempty compact set.

ForF{x}andε0, we have the following result on the set of best approximation.

Corollary 2.3see10. LetCbe a nonempty approximatively compact subset of a metric space

X, d,x∈X, andPCbe the metric projection ofXontoCdefined byPCx {y∈C:dx, y} ≡

dx, C. ThenPCxis a nonempty compact subset ofC.

We will be using the following result of Hardy and Rogers17in proving our first theorem.

Lemma 2.4. LetT be a mapping from a complete metric spaceX, dinto itself satisfying

dTx, Ty≤adx, Tx dy, Ty bdy, Tx dx, Ty cdx, y, 2.2

for anyx, y∈X, wherea, b, andcare nonnegative numbers such that 2a 2b c≤1. ThenThas a

unique fixed pointuinX. In fact for anyx∈X, the sequence{Tn_x}_{converges to}_u.

Theorem 2.5. LetTbe a continuous self-map on a complete convex metric spaceX, dwith Property

(I) and satisfying inequality2.2, letCbe aT-invariant subset ofX, and letFbe a nonempty bounded

subset ofXsuch thatTxxfor allx∈F. IfPCεFis compact, and star shaped, then it contains a

T-invariant point.

Proof. Letz∈PCεFbe arbitrary. Then by2.2, we have for allx∈F

dx, Tz dTx, Tz

≤adx, Tx dz, Tz bdz, Tx dx, Tz cdx, z

adz, Tz bdz, Tx dx, Tz cdx, z

≤adz, x dx, Tz bdz, x dx, Tx dx, Tz cdx, z

a b cdx, z a bdx, Tz.

2.3

This gives

1−a−bdx, Tz≤a b cdx, z

dx, Tz≤dx, z 2.4

since 2a 2b c≤1. Therefore, using definition ofPCεF, we get

sup

x∈F{dx, Tz} ≤supx∈F{dx, z} ≤DF, C ε. 2.5

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Let q be the star-center of PCεF. Define Tn : PCεF → PCεF as Tnx

WTx, q, λn,x∈PCεFwhereynis a sequence in0,1such thatλn → 1. Consider

dTnx, TnydWTx, q, λn, WTy, q, λn

≤λnd

Tx, Ty

≤λnadx, Tx dy, Ty bdy, Tx dx, Ty cdx, y

≤adx, Tx dy, Ty bdy, Tx dx, Ty cdx, y,

2.6

where2a 2b c≤1. Therefore byLemma 2.4, eachTnhas a unique fixed pointzninPCεF.

SincePCεFis compact, there is a subsequencezniofznsuch thatzni → z◦∈PCεF.

We claim thatTz◦ z◦. Considerzni Tnizni WTzni, q, λni → Tz◦, asT is continuous.

Thuszni → Tz◦and consequently,Tz◦z◦, that is,z◦∈PCεFis aT-invariant point.

Since for anε-simultaneous approximatively compact subsetCof a metric spaceX, d

the set of ε-simultaneous C-approximant is nonempty and compact Proposition 2.1, we have the following result.

Corollary 2.6. LetTbe a continuous self-map on a complete convex metric spaceX, dwith Property

(I) and satisfying inequality2.2, letFbe a nonempty bounded subset ofXsuch thatTxxfor all

x∈F, and letCbe aT-invariant subset ofX. IfCisε-simultaneous approximatively compact with

respect toFandPCεFis star shaped, then it contains aT-invariant point.

Corollary 2.7see8. LetT be a continuous self-map on a Banach spaceXsatisfying2.2, letC

be an approximatively compact andT-invariant subset ofX. LetTxi xii 1,2for somex1, x2

not in clC. If the set of best simultaneousC-approximants tox1, x2is star shaped, then it contains a

T-invariant point.

Corollary 2.8see11. LetT be a mapping on a metric spaceX, d, letCbe aT-invariant subset

ofXandxaT-invariant point. IfPCxis a nonempty, compact set for which there exists a contractive

jointly continuous family of functions andT is nonexpansive onPCx∪ {x}thenPCxcontains

aT-invariant point.

Corollary 2.9. Let T be a mapping on a convex metric spaceX, dwith Property (I),letCbe an

approximatively compact,p-star shaped,T-invariant subset ofXand letxbe aT-invariant point. If

Tis nonexpansive onPCx∪ {x}, thenPCxcontains aT-invariant point.

Corollary 2.10see10, Theorem 4. LetT be a quasi-nonexpansive mapping on a convex metric

spaceX, dwith Property (I), letCbe aT-invariant subset ofX, and letxbe aT-invariant point. If

PCxis nonempty, compact, and star shaped, andTis nonexpansive onPCx, thenPCxcontains

aT-invariant point.

Corollary 2.11see10, Theorem 5. LetT be a quasi-nonexpansive mapping on a convex metric

spaceX, dwith Property (I), letCbe an approximatively compact,T-invariant subset ofX, and letx

be aT-invariant point. IfPCxis star shaped andT is nonexpansive onPCx, thenPCxcontains

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Remark 2.12. Theorem 2.5improves and generalizes Theorem 1 of Narang and Chandok9

and of Rao and Mariadoss12.

Definition 2.13. A subsetK of a metric spaceX, dis said to be contractive if there exists a

sequencefnof contraction mappings ofKinto itself such thatfny → yfor eachy∈K.

Theorem 2.14. LetTbe a nonexpansive self-mapping on a metric spaceX, d, letCbe aT-invariant

subset ofX, and letFbe a nonempty bounded subset ofXsuch thatTx xfor allx∈F. If the set

PCεFis compact and contractive, then the setPCεFcontains aT-invariant point.

Proof. Proceeding as in Theorem 2.5, we can prove that T is a self-map of PCεF. Since

PCεFis contractive, there exists a sequencefnof contraction mapping ofPCεFinto

itself such thatfnz → zfor everyz∈PCεF.

Clearly,fnT is a contraction on the compact setPCεFfor eachnand so by Banach contraction principle, each fnT has a unique fixed point, say zn in PCεF. Now the

compactness ofPCεFimplies that the sequenceznhas a subsequencezni → z◦ ∈D.

We claim thatz◦ is a fixed point ofT. Letε > 0 be given. Sincezni → z◦andfnTz◦ → Tz◦,

there exist a positive integermsuch that for allni≥m

dzni, z◦<

ε

2, d

fniTz◦, Tz◦

< ε

2. 2.7

Again,

dfniTzni, fniTz◦≤dzni, z◦< ε₂. 2.8

Hence

dfniTzni, Tz◦≤dfniTzni, fniTz◦ dfniTz◦, Tz◦

< ε

2

ε

2,

2.9

that is,dfniTzni, Tz◦< εfor allni ≥mand sofniTzni → Tz◦. ButfniTzni zni → z◦and

thereforeTz◦z◦.

UsingProposition 2.1we have the following result.

Corollary 2.15. LetTbe a nonexpansive self-mapping on a metric spaceX, d, letCbe aT-invariant

subset ofX, and letFbe a nonempty bounded subset of Xsuch thatTx xfor allx ∈ F. IfCis

ε-simultaneous approximatively compact with respect toF and the set PCεFis contractive, and

T-invariant, thenPCεFcontains aT-invariant point.

Corollary 2.16see9. LetT be a self-mapping on a metric spaceX, d, letGbe aT-invariant

subset ofX, and letxbe aT-invariant point. If the setDofε-approximant toxis compact, contractive

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Remark 2.17. Theorem 2.14 also improves and generalizes the corresponding results of Brosowski2, Mukherjee and Verma8,13, Chandok and Narang9, Rao and Mariadoss

12, and of Singh3.

Definition 2.18. For each bounded subsetGof a metric spaceX, d, the Kuratowski’s measure

of noncompactness ofG,αGis defined as

αG infε >0 :Gis covered by a finite number of closed

balls centered at points ofX of radius≤ε. 2.10

A mappingT :X → Xis called condensing if for all bounded setsG⊂X,αTG≤αG.

We will be using the following result of 18 on fixed points of nonexpansive condensing maps.

Lemma 2.19. LetXbe a complete contractive metric space with contractions{fn}. LetCbe a closed

bounded subsets ofXandT :C → Cis nonexpansive and condensing, thenThas a fixed point inC.

Using the above lemma andTheorem 2.5, we now prove the following result.

Theorem 2.20. LetX, dbe a complete, contractive metric space with contractionsfn. Let Gbe a

closed and bounded subset ofXand letFbe a nonempty bounded subset ofX. IfT is a nonexpansive

and condensing self-map onXsuch thatTxxfor allx∈F, thenP_GεFhas aT-invariant point.

Proof. As G is closed and bounded, PGεF is nonempty, closed and bounded. Using

Theorem 2.5, we can prove that T is a self-map of PGεF. Now a direct application of

Lemma 2.19, gives aT-invariant point inPGεF.

Corollary 2.21 see 8, Theorem 3.1. Let X be a complete, contractive metric space with

contractionsfn. LetGbe a closed and bounded subset ofX. IfTis a nonexpansive and condensing

self-map onXsuch thatTx1 x1andTx2 x2for somex1, x2 ∈X, andD PGx1, x2is nonempty,

then it has aT-invariant point.

Corollary 2.22 see 12, Theorem 4. Let X be a complete, contractive metric space with

contractionsfn. LetGbe a closed and bounded subset ofX. IfT is a nonexpansive and condensing

self-map onX such thatTx xfor somex∈ X, andPGxis nonempty, then it has aT-invariant

point.

Definition 2.23. A mappingT on a metric spaceX, dis called a Kannan mapping19if there

existsα∈0,1/2such that

dTx, Ty≤αdx, Tx dy, Ty 2.11

for allx, y∈X.

Kannan19proved that ifX is complete, then every Kannan mapping has a unique fixed point.

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Theorem 2.24. LetGbe a nonempty subset of a complete metric spaceX, dand letFbe a nonempty

bounded subset ofX. LetTbe a self-map onXwithTxxfor allx∈FandTm_satisfies,

dTmy, Tmz≤αdy, Tmy dz, Tmz, 2.12

for some positive integerm, ally, z∈ Xand some fixed 0 < α < 1/2. IfD PGεFis compact,

then it has a unique fixed point ofT.

Proof. AsTxx,Tn_x_x_{for all positive integers}_n_{. Let}_y0_∈_D_{. Then, for 0}_{< α <}₁_/_2,

dx, Tmy0

dTmx, Tmy0

≤αdx, Tmx dy0, Tmy0

αdy0, Tmy0

≤αdy0, x dx, Tmy0,

2.13

which implies that

dx, Tmy0

≤ α

1−αd

y0, x

. 2.14

Further, we have

sup

x∈F d

x, Tmy0≤sup

x∈F

dy0, x≤DF, G ε. _2.15

Therefore,Tm_y0_∈_D_,_Tm_D_⊂_D_{. Since}_Tm_{satisfies the conditions of Kannan map,}_Tm_{has a}

unique fixed pointx0inD. Now,TmTx0 TTmx0 Tx0, implies thatTx0is a fixed point ofTm_{. But the fixed point of}_Tm_{is unique and equals}_x

0. ThereforeTx0x0and hencex0is a

unique fixed point ofTinD.

Corollary 2.25. LetFbe a nonempty bounded subset of a complete metric spaceX, dandGa subset

ofX. LetGbeε-simultaneous approximatively compact with respect toF, andTa self map onXwith

Txxfor allx∈F, andTm_satisfies

dTmy, Tmz≤αdy, Tmy dz, Tmz, 2.16

for some positive integerm, ally, z∈Xand some fixed 0< α <1/2 thenDPGεFhas a unique

fixed point ofT.

Remarks 2.26. Theorem 2.24extends and generalizes Theorem 3.2 of Mukherjee and Verma8

and Theorem 5 of Rao and Mariadoss12from the set of best simultaneous approximation and best approximation, respectively, toε-simultaneous approximation.

Forε >0, we defineRGεF {g0 ∈G: sup_g∈Gdg, g0 ε≤infg∈Gsup_y∈Fdy, g}.

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A mappingT : X → X satisfies condition (A)see13ifdTx, y ≤ dx, yfor all

x, y∈X.

We now prove a result forT-invariant points from the set ofε-simultaneous coapproxi-mations.

Theorem 2.27. LetT be a self-map satisfying condition (A) and inequality2.2on a convex metric

spaceX, dsatisfying Property (I), letGbe a subset ofX, and letFbe a nonempty bounded subset of

Xsuch thatRGεFis compact and star shaped. ThenRGεFcontains aT-invariant point.

Proof. Letg◦∈RGεF. Consider

dTg◦, g ε≤dg◦, g ε≤_g∈Ginfsup

y∈F d

y, g, _2.17

and soTg◦ ∈ RGεF, that is,T : RGεF → RGεF. SinceRGεFis star shaped, there

existsp∈RGεFsuch thatWz, p, λ∈RGεFfor allz∈RGεF,λ∈0,1. Letkn,0≤ kn < 1, be a sequence of real numbers such that kn → 1 asn → ∞. DefineTn asTnz

WTz, p, kn,z∈RGεF. SinceT is a self-map onRGεFandRGεFis star shaped, each

Tnis a well defined and mapsRGεFintoRGεF. Moreover,

dTny, Tnz

dWTy, p, kn

, WTz, p, kn

≤kndTy, Tz

≤kn

ady, Ty dz, Tz bdz, Ty dy, Tz cdy, z,

2.18

wherekn2a 2b c≤1. So byLemma 2.4eachTn has a unique fixed pointxn ∈RGεF, that is,Tnxn xnfor eachn. SinceRGεFis compact,xnhas a subsequencexni → x ∈ RGεF. Now, we claim thatTx x. Asdxni, Tx≤ dxni, x. On lettingn → ∞, we have xni → Tx. ThereforeTxx, that is,xisT-invariant, hence the result.

ForF{x},x∈X, we have the following result on the set ofε-coapproximation.

Corollary 2.28 see9, Theorem 4. Let T be a self-map satisfying condition (A) on a convex

metric spaceX, dsatisfying Property (I), let Gbe a subset ofX such that RGεx is nonempty

compact, star shaped, and letT be nonexpansive onRGεx. Then there exists ag◦∈RGεxsuch

thatTg◦g◦.

Remarks 2.29. iTheorem 2.27also improves and generalizes Theorem 4.1 of Mukherjee and

Verma13from the set of best approximation toε-simultaneous approximation.

iiBy taking F {x1, x2}, x1, x2 ∈ X, the setPGεF respectively,RGεF) is the

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Acknowledgments

The authors are thankful to the learned referee for careful reading and very valuable suggestions leading to an improvement of the paper. The research work of the second author has been supported by the UGC India under the Emeritus Fellowship.

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