Generalized Asymptotic Pointwise Contractions and Nonexpansive Mappings Involving Orbits

Volume 2010, Article ID 458265,19pages doi:10.1155/2010/458265

Research Article

Generalized Asymptotic Pointwise Contractions and Nonexpansive Mappings Involving Orbits

Adriana Nicolae

Department of Applied Mathematics, Babes¸-Bolyai University, Kog˘alniceanu 1, 400084 Cluj-Napoca, Romania

Correspondence should be addressed to Adriana Nicolae,[email protected] Received 30 September 2009; Accepted 25 November 2009

Academic Editor: Mohamed A. Khamsi

Copyrightq2010 Adriana Nicolae. 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 give fixed point results for classes of mappings that generalize pointwise contractions, asymptotic contractions, asymptotic pointwise contractions, and nonexpansive and asymptotic nonexpansive mappings. We consider the case of metric spaces and, in particular, CAT0spaces.

We also study the well-posedness of these fixed point problems.

1. Introduction

Four recent papers1–4present simple and elegant proofs of fixed point results for pointwise contractions, asymptotic pointwise contractions, and asymptotic nonexpansive mappings.

Kirk and Xu 1 study these mappings in the context of weakly compact convex subsets of Banach spaces, respectively, in uniformly convex Banach spaces. Hussain and Khamsi 2 consider these problems in the framework of metric spaces and CAT0 spaces. In3, the authors prove coincidence results for asymptotic pointwise nonexpansive mappings.

Esp´ınola et al. 4 examine the existence of fixed points and convergence of iterates for asymptotic pointwise contractions in uniformly convex metric spaces.

In this paper we do not consider more general spaces, but instead we formulate less restrictive conditions for the mappings and show that the conclusions of the theorems still stand even in such weaker settings.

2. Preliminaries

LetX, dbe a metric space. Forz∈Xandr >0 we denote the closed ball centered atzwith radiusrbyBz, r :{x∈X:dx, z≤r}.

LetK ⊆Xand letT :K → K. Throughout this paper we will denote the fixed point set ofT by FixT. The mappingT is called a Picard operator if it has a unique fixed pointz andTⁿx_n∈Nconverges tozfor eachx∈K.

A sequencexn_n∈N ⊆ K is said to be an approximate fixed point sequence for the mappingTif limn→ ∞dxn, Txn 0.

The fixed point problem forT is well-posedsee5,6ifT has a unique fixed point and every approximate fixed point sequence converges to the unique fixed point ofT.

A mapping T : X → X is called a pointwise contraction if there exists a function α:X → 0,1such that

d

Tx, T y

≤αxd x, y

for everyx, y∈X. 2.1

LetT :X → Xand forn∈Nletαn:X → Rsuch that d

Tⁿx, Tⁿ y

≤αnxd x, y

for everyx, y∈X. 2.2

If the sequenceαn_n∈Nconverges pointwise to the functionα:X → 0,1, thenT is called an asymptotic pointwise contraction.

If for every x ∈ X, lim sup_{n→ ∞}αnx ≤ 1, then T is called an asymptotic pointwise nonexpansive mapping.

If there exists 0 < k < 1 such that for everyx ∈X, lim sup_{n→ ∞}αnx ≤k, thenT is called a strongly asymptotic pointwise contraction.

For a mappingT :X → Xandx∈Xwe define the orbit starting atxby OTx

x, Tx, T²x, . . . , Tⁿx, . . .

, 2.3

whereTⁿ¹x TTⁿxforn≥0 andT⁰x x. Denote alsoOTx, y OTx∪OTy.

GivenD ⊆Xandx∈X, the numberrxD sup_y∈Ddx, yis called the radius ofD relative tox. The diameter ofDis diamD sup_x,y∈Ddx, yand the cover ofDis defined

as covD

{B:Bis a closed ball andD⊆B}.

As in2, we say that a familyFof subsets ofXdefines a convexity structure onXif it contains the closed balls and is stable by intersection. A subset ofX is admissible if it is a nonempty intersection of closed balls. The class of admissible subsets ofXdenoted byAX defines a convexity structure onX. A convexity structureFis called compact if any family Aα_α∈Γof elements ofFhas nonempty intersection provided

α∈FAα/∅for any finite subset F⊆Γ.

According to2, for a convexity structure F, a functionϕ : X → R is called F- convex if{x : ϕx ≤ r} ∈ F for any r ≥ 0. A type is defined as ϕ : X → R, ϕu lim sup_{n→ ∞}du, xnwhere xn_n∈Nis a bounded sequence inX. A convexity structureFis T-stable if all types areF-convex.

The following lemma is mentioned in2.

Lemma 2.1. LetX be a metric space andFa compact convexity structure onX which isT-stable.

Then for any typeϕthere isx0∈Xsuch that ϕx0 inf

x∈Xϕx. 2.4

A metric spaceX, dis a geodesic space if every two pointsx, y ∈ X can be joined by a geodesic. A geodesic fromxtoyis a mappingc : 0, l → X, where0, l ⊆ R, such thatc0 x, cl y,anddct, ct |t−t|for every t, t ∈ 0, l. The imagec0, l ofcforms a geodesic segment which joinsxandy. A geodesic triangleΔx1, x2, x3consists of three pointsx1, x2,andx3inXthe vertices of the triangleand three geodesic segments corresponding to each pair of points the edges of the triangle. For the geodesic traingle Δ Δx1, x2, x3, a comparison triangle is the triangleΔ Δx1, x2, x3 in the Euclidean spaceE²such thatdxi, xj d_E²xi, xjfori, j ∈ {1,2,3}. A geodesic triangleΔsatisfies the CAT0inequality if for every comparison triangleΔofΔand for everyx, y∈Δwe have

d x, y

≤dE² x, y

, 2.5

wherex, y ∈ Δare the comparison points ofxandy. A geodesic metric space is a CAT0 space if every geodesic traingle satisfies the CAT0inequality. In a similar way we can define CATkspaces fork >0 ork <0 using the model spacesM_k².

A geodesic space is a CAT0space if and only if it satisfies the following inequality known as theCNinequality of Bruhat and Tits7. Letx, y1, y2be points of a CAT0space and letmbe the midpoint ofy1, y2. Then

dx, m²≤ 1 2d

x, y1

₂ 1

2d x, y2

₂

−1 4d

y1, y2

₂

. 2.6

It is also knownsee8that in a complete CAT0space, respectively, in a closed convex subset of a complete CAT0space every type attains its infimum at a single point. For more details about CATkspaces one can consult, for instance, the papers9,10.

In2, the authors prove the following fixed point theorems.

Theorem 2.2. Let X be a bounded metric space. Assume that the convexity structure AX is compact. LetT :X → Xbe a pointwise contraction. ThenTis a Picard operator.

Theorem 2.3. Let X be a bounded metric space. Assume that the convexity structure AX is compact. LetT :X → Xbe a strongly asymptotic pointwise contraction. ThenTis a Picard operator.

Theorem 2.4. LetXbe a bounded metric space. Assume that there exists a convexity structureFthat is compact andT-stable. LetT :X → Xbe an asymptotic pointwise contraction. ThenT is a Picard operator.

Theorem 2.5. Let X be a complete CAT0 space and let K be a nonempty, bounded, closed and convex subset ofX. Then any mappingT :K → Kthat is asymptotic pointwise nonexpansive has a fixed point. Moreover, FixTis closed and convex.

The purpose of this paper is to present fixed point theorems for mappings that satisfy more general conditions than the ones which appear in the classical definitions of pointwise contractions, asymptotic contractions, asymptotic pointwise contractions and asymptotic nonexpansive mappings. Besides this, we show that the fixed point problems are well-posed.

Some generalizations of nonexpansive mappings are also considered. We work in the context of metric spaces and CAT0spaces.

3. Generalizations Using the Radius of the Orbit

In the sequel we extend the results obtained by Hussain and Khamsi 2using the radius of the orbit. We also study the well-posedness of the fixed point problem. We start by introducing a property for a mappingT : X → X, whereX is a metric space. Namely, we will say thatTsatisfies propertySif

Sfor every approximate fixed point sequence xn_n∈N and for every m ∈ N, the sequencedxn, T^mxn_n∈Nconverges to 0 uniformly with respect to m.

For instance, if for everyx∈X,dx, T²x≤dx, Txthen propertySis fulfilled.

Proposition 3.1. LetX be a metric space and letT :X → X be a mapping which satisfiesS. If xn_n∈Nis an approximate fixed point sequence, then for everym∈Nand everyx∈X,

lim sup

n→ ∞ dx, T^mxn lim sup

n→ ∞ dx, xn, 3.1

lim sup

n→ ∞ rxOTxn lim sup

n→ ∞ dx, xn, 3.2

nlim→ ∞diam O_Txn 0. 3.3

Proof. Since T satisfies S and xn_n∈N is an approximate fixed point sequence, it easily follows that3.1holds. To prove3.2, let >0. Then there existsm∈Nsuch that

rxOTxn≤dx, T^mxn ≤dx, xn dxn, T^mxn . 3.4 Taking the superior limit,

lim sup

n→ ∞ rxOTxn≤lim sup

n→ ∞ dx, xn . 3.5

Hence,3.2holds. Now let again >0. Then there existm1, m2 ∈Nsuch that

diamOTxn≤dT^m1xn, T^m2xn ≤dxn, T^m1xn dxn, T^m2xn . 3.6 We only need to letn → ∞in the above relation to prove3.3.

Theorem 3.2. LetXbe a bounded metric space such thatAXis compact. Also letT :X → Xfor which there existsα:X → 0,1such that

d

Tx, T y

≤αxrx

OT

y

for everyx, y∈X. 3.7

ThenT is a Picard operator. Moreover, if additionallyT satisfiesS, then the fixed point problem is well-posed.

Proof. BecauseAXis compact, there exists a nonempty minimalT-invariantK ∈ AXfor which covTK K. Ifx, y∈KthenrxOTy≤rxK.In a similar way as in the proof of Theorem 3.1 of2we show now thatT has a fixed point. Letx∈K. Then,

d

Tx, T y

≤αxrx

OT

y

≤αxrxK for everyy∈X. 3.8

This means that TK ⊆ BT x, αxrxK, so K covTK ⊆ BTx, αxr xK.

Therefore,

rTxK≤αxrxK. 3.9

Denote

Kx

y∈K:ryK≤rxK . 3.10

Kx∈ AXsince it is nonempty andKx

y∈KBy, r xK∩K.

Lety ∈ Kx. As above we haveK ⊆ BTy, αyr yK ⊆ BT y, αyrxKand hence Ty ∈ Kx. BecauseK is minimal T-invariant it follows that Kx K. This yields ryK rxKfor everyx, y ∈K. In particular,r_TxK rxKand using3.9we obtain rxK 0 which implies thatKconsists of exactly one point which will be fixed underT.

Now supposex, y∈X, x /yare fixed points ofT. Then d

x, y

≤αxrx

OT

y

αxd x, y

. 3.11

This means thatαx≥1 which is impossible.

Letzdenote the unique fixed point ofT, letx ∈ X andlx lim sup_{n→ ∞}dz, Tⁿx.

Observe that the sequencerzOTTⁿx_n∈Nis decreasing and bounded below by 0 so its limit exists and is preciselylx. Then

lx≤αzlim

n→ ∞rz

OT

Tⁿ⁻¹x

αzlx. 3.12

This implies thatlx0 and hence limn→ ∞Tⁿx z.

Next we prove that the problem is well-posed. Letxn_n∈N be an approximate fixed point sequence. We know that

dz, xn≤dxn, Txn dTxn, Tz≤dxn, Txn αzrzOTxn. 3.13

Taking the superior limit and applying3.2ofProposition 3.1forz, lim sup

n→ ∞ dz, xn≤αzlim sup

n→ ∞ dz, xn, 3.14

which implies lim_{n→ ∞}dz, xn 0.

We remark that if in the above resultT is, in particular, a pointwise contraction then the fixed point problem is well-posed without additional assumptions forT.

Next we give an example of a mapping which is not a pointwise contraction, but fulfills 3.7.

Example 3.3. LetT :0,1 → 0,1,

Tx

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎩ 1−x

2 , ifx≥ 1 2, 3

4x, ifx < 1

2, 3.15

and letα:0,1 → 0,1,

αx

⎧⎪

⎪⎨

⎪⎪

⎩ 1

2, if x≥ 1 2, 3

4x², if x < 1 2.

3.16

ThenT is not a pointwise contraction, but3.7is verified.

Proof. T is not continuous, so it is not nonexpansive and hence it cannot be a pointwise contraction. Ifx, y ≥ 1/2 orx, y < 1/2 the conclusion is immediate. Supposex ≥ 1/2 and y <1/2. Then

rx

OT

y

x, ryOTx max

x−y, y . 3.17

iIfTx−Ty≥0, then 1−x

2 − 3 4y≤ x

2 αxrx

OT

y , 1−x

2 −3 4y≤

3

4 y² x−y

≤α y

ryOTx.

3.18

The above is true because 1/2−5/4x <0≤y²x−y.

iiIfTx−Ty<0, then 3

4y−1−x 2 ≤ −1

8 x 2 < x

2 αxrx

OT

y , 3

4y−1−x

2 ≤

3 4 y²

y≤α

y

ryOTx.

3.19

Theorem 3.4. LetXbe a bounded metric space,T : X → X,and suppose there exists a convexity structureFwhich is compact andT-stable. Assume

d

Tⁿx, Tⁿ y

≤αnxrx

OT

y

for everyx, y∈X, 3.20

where for eachn ∈ N, αn : X → R,and the sequenceαn_n∈Nconverges pointwise to a function α:X → 0,1. ThenT is a Picard operator. Moreover, if additionallyT satisfiesS, then the fixed point problem is well-posed.

Proof. AssumeT has two fixed pointsx, y∈X, x /y. Then for eachn∈N, d

x, y

≤αnxd x, y

. 3.21

Whenn → ∞we obtainαx≥1 which is false. Hence,Thas at most one fixed point.

Letx∈X. We considerϕ:X → R, ϕu lim sup

n→ ∞ du, Tⁿx foru∈X. 3.22

BecauseFis compact andT-stable there existsz∈Xsuch that ϕz inf

u∈Xϕu. 3.23

Forp∈N,

ϕz≤ϕT^pz≤αpzlim

n→ ∞rzOTTⁿx αpzϕz. 3.24

Letting p → ∞in the above relation yieldsϕz 0 so Tⁿx_n∈N converges toz which will be the unique fixed point of T because dTz, Tⁿ¹x ≤ α1zrzOTTⁿx and limn→ ∞rzOTTⁿx 0.Thus, all the Picard iterates will converge toz.

Letxn_n∈Nbe an approximate fixed point sequence and letm∈N. Then

dz, xn≤dxn, T^mxn dT^mxn, T^mz≤dxn, T^mxn αmzrzOTxn. 3.25

Taking the superior limit and applying3.2ofProposition 3.1, lim sup

n→ ∞ dz, xn≤αmzlim sup

n→ ∞ dz, xn. 3.26

Lettingm → ∞we have lim_{n→ ∞}dz, xn 0.

Theorem 3.5. LetXbe a complete CAT0space and letK⊆Xbe nonempty, bounded, closed, and convex. LetT :K → Kand forn∈N, letαn :K → Rbe such that lim sup_{n→ ∞}αnx≤1 for all x∈K. If for alln∈N,

d

Tⁿx, Tⁿ y

≤αnxrx

OT

y

for everyx, y∈K, 3.27

thenT has a fixed point. Moreover, Fix(T) is closed and convex.

Proof. The idea of the proof follows to a certain extend the proof of Theorem 5.1 in2. Let x∈K. Denoteϕ:K → R,

ϕu lim sup

n→ ∞ du, Tⁿx foru∈K. 3.28

SinceKis a nonempty, closed, and convex subset of a complete CAT0space there exists a uniquez∈Ksuch that

ϕz inf

u∈Kϕu. 3.29

Forp∈N,

ϕT^pz≤αpzlim

n→ ∞rzOTTⁿx αpzϕz. 3.30

Letp, q ∈ Nand letmdenote the midpoint of the segmentT^pz, T^qz. Using the CN inequality, we have

dm, Tⁿx²≤ 1

2dT^pz, Tⁿx²1

2dT^qz, Tⁿx²−1

4dT^pz, T^qz². 3.31 Lettingn → ∞and consideringϕz≤ϕm, we have

ϕz² ≤ 1

2ϕT^pz²1

2ϕT^qz²−1

4dT^pz, T^qz²

≤ 1

2αpz²ϕz²1

2αqz²ϕz²−1

4dT^pz, T^qz².

3.32

Lettingp, q → ∞we obtain thatTⁿz_n∈Nis a Cauchy sequence which converges toω∈K.

As in the proof ofTheorem 3.4we can show thatωis a fixed point forT. To prove that FixT is closed takexn_n∈Na sequence of fixed points which converges tox^∗∈K. Then

dTx^∗, Txn≤α1x^∗dx^∗, xn, 3.33

which shows thatx^∗is a fixed point ofT.

The fact that FixTis convex follows from theCNinequality. Letx, y∈FixTand letmbe the midpoint ofx, y. Forn∈Nwe have

dm, Tⁿm²≤ 1

2dx, Tⁿm²1 2d

y, Tⁿm₂

−1 4d

x, y₂

≤ 1

2αnm²rmOTx²1

2αnm²rm

OT

y2−1 4d

x, y2

1

2αnm²

dm, x²d

m, y₂

−1 4d

x, y₂ 1

4

αnm²−1 d

x, y₂ .

3.34

Lettingn → ∞we obtain lim_{n→ ∞}Tⁿm m. This yieldsmwhich is a fixed point since lim sup

n→ ∞ d

Tm, Tⁿ¹m

≤α1mlim sup

n→ ∞ dm, Tⁿm. 3.35

Hence, FixTis convex.

We conclude this section by proving a demi-closed principle similarly to 2, Proposition 1. To this end, for K ⊆ X, K closed and convex andϕ : K → R, ϕx lim sup_{n→ ∞}dx, xn, as in2, we introduce the following notation:

xn

ωϕ iffϕω inf

x∈Kϕx, 3.36

where the bounded sequencexn_n∈Nis contained inK.

Theorem 3.6. LetXbe a CAT0space and letK⊆X,Kbounded, closed, and convex. LetT :K → KsatisfySand forn∈ N, letαn : K → R be such that lim sup_{n→ ∞}αnx ≤ 1 for allx ∈K. Suppose that forn∈N,

d

Tⁿx, Tⁿ y

≤αnxrx

OT

y

for everyx, y∈K. 3.37

Let alsoxn_n∈N⊆Kbe an approximate fixed point sequence such thatxn

ω.ϕ Thenω∈Fix(T).

Proof. Using3.1ofProposition 3.1we obtain that for everyx∈Kand everyp∈N, ϕx lim sup

n→ ∞ dx, T^pxn. 3.38

Applying3.2ofProposition 3.1forω, we have ϕT^pω lim sup

n→ ∞ dT^pω, T^pxn≤αpωlim sup

n→ ∞ rωOTxn αpωϕω. 3.39

Letp ∈ Nand letmbe the midpoint ofω, T^pω. As in the above proof, using the CN inequality we have

ϕm²≤ 1

2ϕω² 1

2ϕT^pω²−1

4dω, T^pω². 3.40

Sinceϕω≤ϕm,

ϕω² ≤ 1

2ϕω²1

2αpω²ϕω²−1

4dω, T^pω². 3.41

Lettingp → ∞, we have limp→ ∞T^pω ω. This meansω∈FixTbecause lim sup

p→ ∞ d

Tω, T^p1ω

≤α1ωlim sup

p→ ∞ dω, T^pω. 3.42

4. Generalized Strongly Asymptotic Pointwise Contractions

In this section we generalize the strongly asymptotic pointwise contraction condition, by using the diameter of the orbit. We begin with a fixed point result that holds in a complete metric space.

Theorem 4.1. LetX be a complete metric space and letT : X → X be a mapping with bounded orbits that is orbitally continuous. Also, forn∈N, letαn:X → Rfor which there exists 0< k <1 such that for everyx∈X, lim sup_{n→ ∞}αnx≤k. If for eachn∈N,

d

Tⁿx, Tⁿ y

≤αnxdiam OT

x,y

for every x,y∈X, 4.1

thenT is a Picard operator. Moreover, if additionallyT satisfiesS, then the fixed point problem is well-posed.

Proof. First, suppose thatT has two fixed pointsx, y∈X, x /y. Then for eachn∈N, d

x, y

≤αnxd x, y

. 4.2

Lettingn → ∞we obtain that k ≥ 1 which is impossible. Hence, T has at most one fixed point. Letx ∈ X. Notice that the sequencediam OTTⁿx_n∈Nis decreasing and bounded below by 0 so it converges tolx≥0. Forn, p1, p2∈N, p1≤p2we have

dT^np1x, T^np2x≤αnp1xdiamOTx. 4.3 Taking the supremum with respect top1andp2and then lettingn → ∞we obtain

lx≤kdiamOTx. 4.4

Forp∈N,

lx lim

n→ ∞diamOTTⁿT^px≤kdiamOTT^px. 4.5 Lettingp → ∞in the above relation we havelx ≤klxwhich implies thatlx 0. This means that the sequenceTⁿx_n∈Nis Cauchy so it converges to a pointz∈X. BecauseT is orbitally continuous it follows that zis a fixed point, which is unique. Therefore, all Picard iterates converge toz.

Next we prove that the problem is well-posed. Letxn_n∈N be an approximate fixed point sequence. Taking into account3.2applied forzand3.3ofProposition 3.1,

lim sup

n→ ∞ diamOTz, xn lim sup

n→ ∞ diam{z} ∪OTxn lim sup

n→ ∞ dz, xn. 4.6 Knowing that

dz, xn≤dxn, T^mxn dT^mxn, T^mz≤dxn, T^mxn αmzdiamOTz, xn, 4.7 and taking the superior limit we obtain

lim sup

n→ ∞ dz, xn≤αmzlim sup

n→ ∞ dz, xn. 4.8

If we let herem → ∞it is clear thatxn_n∈Nconverges toz.

A similar result can be given in a bounded metric space where the convexity structure defined by the class of admissible subsets is compact.

Theorem 4.2. LetX be a bounded metric space such thatAXis compact and letT :X → Xbe an orbitally continuous mapping. Also, forn∈N, letαn :X → Rfor which there exists 0< k <1 such that for everyx∈X, lim sup_{n→ ∞}αnx≤k. If for eachn∈N,

d

Tⁿx, Tⁿ y

≤αnxdiam OT

x,y

for every x,y∈X, 4.9 thenT is a Picard operator. Moreover, if additionallyT satisfiesS, then the fixed point problem is well-posed.

Proof. Letx∈X. Denoteϕ:X → R, ϕu lim sup

n→ ∞ du, Tⁿx foru∈X. 4.10

As in the proof ofTheorem 4.1one can show thatT has at most one fixed point and for each x ∈ X, the sequenceTⁿx_n∈N is Cauchy. This means that limn→ ∞ϕTⁿx 0 for each x∈X. BecauseAXis compact we can choose

ω∈

n≥1

cov

T^kx:k≥n

. 4.11

Following the argument of 2, Theorem 4.1 we can show that ϕω 0. For the sake of completeness we also include this part of the proof. The definition ofϕyields that foru∈X and every >0 there existsn0∈Nsuch that for anyn≥n0,

du, Tⁿx≤ϕu . 4.12

Hence,Tⁿx∈Bu, ϕu for everyn≥n0and so cov{Tⁿx:n≥n0}⊆B

u, ϕu

. 4.13

Therefore,ω∈Bu, ϕu for each >0. This impliesdω, u≤ϕuwhich holds for every u∈X. Thus,

ϕω lim sup

n→ ∞ dω, Tⁿx≤lim sup

n→ ∞ ϕTⁿx 0. 4.14

Now it is clear thatTⁿx_n∈Nconverges toω. BecauseT is orbitally continuous,ωwill be the unique fixed point and all the Picard iterates will converge to this unique fixed point.

The fact that every approximate fixed point sequencexn_n∈Nconverges toω can be proved identically as inTheorem 4.1.

In connection with the use of the diameter of the orbit, Walter11obtained a fixed point theorem that may be stated as follows.

Theorem 4.3Walter11. LetX, dbe a complete metric space and letT :X → Xbe a mapping with bounded orbits. If there exists a continuous, increasing functionϕ:R → Rfor whichϕr< r for everyr >0 and

d

Tx, T y

≤ϕ diam

O_T x,y

for everyx, y∈X, 4.15

thenT is a Picard operator.

We conclude this section by proving an asymptotic version of this result. In this way we extend the notion of asymptotic contraction introduced by Kirk in12.

Theorem 4.4. LetX, dbe a complete metric space and letT :X → Xbe an orbitally continuous mapping with bounded orbits. Suppose there exist a continuous functionϕ : R → R satisfying ϕt < tfor allt > 0 and the functionsϕn : R → R such that the sequenceϕn_n∈Nconverges pointwise toϕand for eachn∈N,

d

Tⁿx, Tⁿ y

≤ϕn

diamO_T x,y

for allx, y∈X, 4.16

thenT is a Picard operator. Moreover, if additionallyT satisfiesSand ϕn is continuous for each n∈N, then the fixed point problem is well-posed.

Proof. The proof follows closely the ideas presented in the proof ofTheorem 4.1.

We begin by supposing thatThas two fixed pointsx, y∈X, x /y. Then for eachn∈N, d

x, y

≤ϕn

d x, y

. 4.17

Lettingn → ∞we obtain thatdx, y≤ϕdx, ywhich is impossible. Hence,Thas at most one fixed point.

Notice that forx ∈ X the sequencediamOTTⁿx_n∈Nis decreasing and bounded below by 0 so it converges tolx≥0. Forn, p1, p2∈N, p1≤p2we have

dT^np1x, T^np2x≤ϕnp1diamOTx. 4.18 Thus,lx≤ϕdiamOTx.

Forp∈N,

lx lim

n→ ∞diamOTTⁿT^px≤ϕdiamOTT^px. 4.19 Hence,lx≤ϕlxwhich implies thatlx 0 and the proof may be continued as inTheorem 4.1 in order to conclude thatT is a Picard operator.

Letz∈Xbe the unique fixed point ofTand letxn_n∈Nbe an approximate fixed point sequence. To show that the problem is well-posed, takexnp_p∈Na subsequence of xn_n∈N such that

lim sup

n→ ∞ dz, xn lim

p→ ∞d z, xnp

. 4.20

Because every subsequence of xn_n∈N is also an approximate fixed point sequence, the conclusions ofProposition 3.1still stand forxnp_p∈N. This yields

lim sup

p→ ∞ diamOT

z, xnp

lim sup

p→ ∞ diam

{z} ∪OT

xnp

lim

p→ ∞d z, xnp

. 4.21

But since

d z, xnp

≤diamOT

z, xnp

, 4.22

by passing to the inferior limit follows limp→ ∞diamOTz, xnp limp→ ∞dz, xnp. Form∈N,

d z, xnp

≤d

xnp, T^m xnp

d T^m

xnp

, T^mz

≤d

xnp, T^m xnp

ϕm

diamOT

z, xnp

.

4.23

If we let herep → ∞, we have limp→ ∞dz, xnp ≤ ϕmlimp→ ∞dz, xnp.Passing here to the limit with respect tomimplies limp→ ∞dz, xnp ≤ ϕlimp→ ∞dz, xnpand this means lim_{p→ ∞}dz, xnp 0.Because of4.20it follows thatxn_n∈Nconverges toz.

5. Some Generalized Nonexpansive Mappings in CAT(0) Spaces

In this section we give fixed point results in CAT0spaces for two classes of mappings which are more general than the nonexpansive ones.

Theorem 5.1. LetXbe a bounded complete CAT0space and letT :X → Xbe such that for every x, y∈X,

d

Tx, T y

≤rx

OT

y

. 5.1

ThenT has a fixed point. Moreover, Fix(T) is closed and convex.

Proof. Letx∈X. Denoteϕ:X → R, ϕu lim sup

n→ ∞ du, Tⁿx foru∈X. 5.2

SinceXis a complete CAT0space there exists a uniquez∈Xsuch that ϕz inf

u∈Xϕu. 5.3

Supposingzis not a fixed point ofT, we have ϕz< ϕTz≤ lim

n→ ∞rz

OT

Tⁿ⁻¹x

ϕz. 5.4

This is a contradiction and thusz∈FixT.

Letxn_n∈Nbe a sequence of fixed points which converges tox^∗∈X. Then,

dTx^∗, Txn≤dx^∗, xn 5.5

which proves thatx^∗is a fixed point ofT so FixTis closed.

Now takex, y ∈FixT. We show that the midpoint ofx, ydenoted bymis a fixed point ofTusing theCNinequality. More precisely we have

dm, Tm²≤ 1

2dTm, Tx²1 2d

Tm, T y₂

−1 4d

x, y₂

≤ 1

2dm, x²1 2d

m, y₂

− 1 4d

x, y₂ 0.

5.6

Hence, FixTis convex.

A simple example of a mapping which is not nonexpansive, but satisfies5.1, is the following.

Example 5.2. LetT :0,1−→0,1,

Tx

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎩ x

2, ifx≥ 1 2, x

4, ifx < 1

2. 5.7

ThenT is not nonexpansive but5.1is verified.

Proof. T is not continuous, so it cannot be nonexpansive. To show that5.1holds, we only consider the situation whenx≥1/2 andy <1/2 because in all other the condition is clearly satisfied. Then|Tx−Ty|x/2−y/4.We can easily observe that

rx

OT

y

x≥ x 2 −y

4, ryOTx max

x−y, y .

5.8

If3/4y≤x/2 thenx/2−y/4≤x−y.Otherwise,x/2−y/4≤3/4y−y/4y/2≤y.In this way we have shown that5.1is accomplished.

Theorem 5.3. LetXbe a bounded complete CAT0space and letT :X → Xbe such that for every x, y∈X,

d

Tx, T y

≤diam

{x} ∪OT

y

, 5.9

d

Tx, T y

≤rx

OT

y sup

k,p∈N

diam

T^kx

∪OT

T^kp

y

−diamOT

T^kp

y

. 5.10

ThenT has a fixed point. Moreover, Fix(T) is closed and convex.

Proof. Letx∈X. Denoteϕ:X → R, ϕu lim sup

n→ ∞ du, Tⁿx foru∈X. 5.11

SinceXis a complete CAT0space there exists a uniquez∈Xsuch that ϕz inf

u∈Xϕu. 5.12

Let lx limn→ ∞diamOTTⁿx. This limit exists since the sequence is decreasing and bounded below by 0.

Supposezis not a fixed point ofT. Then

lim sup

n→ ∞ dz, Tⁿx ϕz< ϕTz≤ lim

n→ ∞diam{z} ∪OTTⁿx. 5.13

This means that

n→ ∞limdiam{z} ∪OTTⁿx lx, 5.14 lim sup

n→ ∞ dTz, Tⁿx≤lx,

nlim→ ∞diam{Tz} ∪OTTⁿx lx.

5.15

Inductively, it follows that fork≥0,

lim sup

n→ ∞ d

T^kz, Tⁿx

≤lx. 5.16

Letk, p, n∈Nand letdk,ndiam{T^kz} ∪OTT^nkx. Obviously,

diam T^kz

∪OT

T^npkx

≤dk,n, 5.17

sinceOTT^npkx⊆OTT^nkx.

Because of5.9we have

rT^kz

OT

T^nkx

≤diam

T^k−1z

∪OT

T^nk−1x

. 5.18

Since diamOTT^nkx≤diamOTT^nk−1x, it is clear thatdk,n≤d_k−1,n. Hence,

sup

k∈Ndk,ndiam{z} ∪OTTⁿx. 5.19

Letsnsup_k,p∈Ndiam{T^kz} ∪OTT^npkx−diamOTT^npkx.

Then,

sn≤sup

k∈Ndk,n− inf

k,p∈NdiamOT

T^npkx

≤diam{z} ∪OTTⁿx−lx. 5.20

Taking into account5.14, limn→ ∞sn0.Now, ϕTz lim sup

n→ ∞ dTz, Tⁿx≤ lim

n→ ∞rz

OT

Tⁿ⁻¹x lim

n→ ∞sn−1

lim sup

n→ ∞ d

z, Tⁿ⁻¹x

ϕz, 5.21

which is a contradiction. Hence,Tz z.

The fact that FixTis closed and convex follows as in the previous proof.

Remark 5.4. It is clear that nonexpansive mappings and mappings for which 5.1 holds satisfy 5.9 and 5.10. However, there are mappings which satisfy these two conditions without verifying5.1as the following example shows.

Example 5.5. The set0,1with the usual metric is a CAT0space. Let us takeT :0,1 → 0,1,

Tx

⎧⎪

⎪⎨

⎪⎪

⎩ 2

3x, ifx≥ 1 2, x

4, ifx < 1 2.

5.22

ThenT does not satisfy5.1but conditions5.9,5.10hold.

Proof. To prove thatT does not verify5.1we takex1/2 andy1/4. Then|Tx−Ty|

1/3−1/1613/48.However,

r1/4

OT

1 2

1

4 < 13

48. 5.23

Next we show that5.9and5.10hold. We only need to consider the case whenx≥1/2 and y <1/2 because in all the other situations this is evident. Then|Tx−Ty| 2/3x−y/4.

Since

diam

{x} ∪OT

y

diam

y ∪OTx

x≥ 2 3x−y

4, 5.24

relation5.9is satisfied.

Also,

rx

OT

y

≥x−y 4 ≥ 2

3x−y 4, ryOTx≥x−y.

5.25

Since sup_p∈Ndiam{y} ∪OTT^px−diamOTT^px≥3/4y,we obtainx−y 3/4y≥ 2/3x−y/4.Hence, relation5.10is also accomplished.

Remark 5.6. If we replace condition5.9ofTheorem 5.3with d

Tx, T y

≤αxdiam

{x} ∪OT

y

for everyx, y∈X, 5.26 whereα:X → 0,1, then we may conclude thatThas s unique fixed point.

It is also clear that a pointwise contraction satisfies these conditions so we can apply this result to prove that it has a unique fixed point.

We next prove a demi-closed principle. We will use the notations introduced at the end ofSection 3.

Theorem 5.7. LetXbe a CAT0space,K⊆X,Kbounded, closed, and convex. LetT:K → Kbe a mapping that safisfiesSand5.9for eachx, y∈Kand letxn_n∈N⊆Kbe an approximate fixed point sequence such thatxn

ω.ϕ Thenω∈Fix(T).

Proof. Using3.1ofProposition 3.1we haveϕx lim sup_{n→ ∞}dx, Txn.Applying3.2 and3.3ofProposition 3.1forω,

lim sup

n→ ∞ diam{ω} ∪OTxn lim sup

n→ ∞ dω, xn. 5.27

Then,

ϕTω lim sup

n→ ∞ dTω, Txn≤lim sup

n→ ∞ diam{ω} ∪OTxn ϕω. 5.28 Letmdenote the midpoint ofω, Tω. TheCNinequality yields

dm, xn²≤ 1

2dω, xn²1

2dTω, xn²−1

4dω, Tω². 5.29

Taking the superior limit, we have ϕm² ≤ 1

2ϕω²1

2ϕTω²−1

4dω, Tω². 5.30

But sincexn

ω,ϕ

1

4dω, Tω² ≤ 1

2ϕω²1

2ϕω²−ϕω²0. 5.31

Hence,ω∈FixT.

We conclude this paper with the following remarks.

Remark 5.8. All the above results obtained in the context of CAT0spaces also hold in the more general setting used in4of uniformly convex metric spaces with monotone modulus of convexity.

Remark 5.9. In a similar way as for nonexpansive mappings, one can develop a theory for the classes of mappings introduced in this section. An interesting idea would be to study the approximate fixed point property of such mappings. A nice synthesis in the case of nonexpansive mappings can be found in the recent paper of Kirk13.

Acknowledgment

The author wishes to thank the financial support provided from programs cofinanced by The Sectoral Operational Programme Human Resources Development, Contract POS DRU 6/1.5/S/3—“Doctoral studies: through science towards society.”

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