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NOT NECESSARILY NONEXPANSIVENASEER SHAHZAD AND AMJAD LONE Received 16 October 2004 and in revised form 3 December 2004 LetC be a nonempty closed bounded convex subset of a Banach space X

NOT NECESSARILY NONEXPANSIVE

NASEER SHAHZAD AND AMJAD LONE

Received 16 October 2004 and in revised form 3 December 2004

LetC be a nonempty closed bounded convex subset of a Banach space X whose

char-acteristic of noncompact convexity is less than 1 andT a continuous 1-χ-contractive SL

map (which is not necessarily nonexpansive) fromC to KC(X) satisfying an inwardness

condition, whereKC(X) is the family of all nonempty compact convex subsets of X It

is proved thatT has a fixed point Some fixed points results for noncontinuous maps

are also derived as applications Our result contains, as a special case, a recent result of Benavides and Ram´ırez (2004)

1 Introduction

During the last four decades, various fixed point results for nonexpansive single-valued maps have been extended to multimaps, see, for instance, the works of Benavides and Ram´ırez [2], Kirk and Massa [6], Lami Dozo [7], Lim [8], Markin [10], Xu [12], and the references therein Recently, Benavides and Ram´ırez [3] obtained a fixed point theo-rem for nonexpansive multimaps in a Banach space whose characteristic of noncompact convexity is less than 1 More precisely, they proved the following theorem

Theorem 1.1 (see [3]) Let C be a nonempty closed bounded convex subset of a Banach space X such that
α(X) < 1 and T : C → KC(X) a nonexpansive 1-χ-contractive map If T satisfies

then T has a fixed point.

Benavides and Ram´ırez further remarked that the assumption of nonexpansiveness

in the above theorem can not be avoided In this paper, we prove a fixed point result for multimaps which are not necessarily nonexpansive To establish this, we define a new class

of multimaps which includes nonexpansive maps To show the generality of our result,

we present an example As consequences of our main result, we also derive some fixed point theorems for∗-nonexpansive maps

Copyright©2005 Hindawi Publishing Corporation

Fixed Point Theory and Applications 2005:2 (2005) 169–176

DOI: 10.1155/FPTA.2005.169

2 Preliminaries

LetC be a nonempty closed subset of a Banach space X Let CB(X) denote the family of all

nonempty closed bounded subsets ofX and KC(X) the family of all nonempty compact

convex subsets ofX The Kuratowski and Hausdorff measures of noncompactness of a

nonempty bounded subset ofX are, respectively, defined by

α(B) =inf{ d > 0 : B can be covered by finitely many sets of diameter ≤ d },

χ(B) =inf{ d > 0 : B can be covered by finitely many balls of radius ≤ d } (2.1)

LetH be the Hausdorff metric on CB(X) and T : C → CB(X) a map Then T is called

(1) contraction if there exists a constantk ∈[0, 1) such that

H

T(x),T(y)

≤ k  x − y , ∀ x, y ∈ C; (2.2) (2) nonexpansive if

H

T(x),T(y)

≤  x − y , ∀ x, y ∈ C; (2.3) (3)φ-condensing (resp., 1-φ-contractive), where φ = α( ·) orχ( ·) ifT(C) is bounded

and, for each bounded subsetB of C with φ(B) > 0, the following holds:

φ

T(B)

< φ(B)

resp., φ

T(B)

≤ φ(B)

hereT(B) =x ∈ B T(x);

(4) upper semicontinuous if{ x ∈ C : T(x) ⊂ V }is open in C whenever V ⊂ X is

open;

(5) lower semicontinuous if the set{ x ∈ C : T(x) ∩ V = φ }is open inC whenever

V ⊂ X is open;

(6) continuous (with respect toH) if H(T(x n),T(x)) →0 wheneverx n → x;

(7)∗-nonexpansive (see [5]) if for allx, y ∈ C and u x ∈ T(x) with d(x,u x)=inf{ d(x, z) : z ∈ T(x) }, there exists u y ∈ T(y) with d(y,u y)=inf{ d(y,w) : w ∈ T(y) }

such that

d

u x,u y

A sequence{ x n }is called asymptoticallyT-regular if d(x n,Tx n)→0 asn → ∞

Letφ = α or χ The modulus of noncompactness convexity associated to φ is defined

by

∆X,φ(
)=inf

1− d(0,A) : A ⊂ B Xis convex, withφ(A) ≥
, (2.6) whereB Xis the unit ball ofX The characteristic of noncompact convexity of X associated

with the measure of noncompactnessφ is defined in the following way:

φ(X) =sup

≥0 :∆X,φ(
)=0

Note that

and so

LetC be a nonempty subset of X, Ᏸ a directed set, and { x α:α ∈Ᏸ}a bounded net in

X We use r(C, { x α }) andA(C, { x α }) to denote the asymptotic radius and the asymptotic center of{ x α:α ∈Ᏸ}inC, that is,

r

x,

x α

=lim sup

α

x α − x,

r

C,

x α

=inf

r

x,

x α

:x ∈ C ,

A

C,

x α

=x ∈ C : r

x,

x α

= r

C,

x α

.

(2.10)

It is known thatA(C, { x α }) is a nonempty weakly compact convex set ifC is a nonempty

closed convex subset of a reflexive Banach space For details, we refer the reader to [1,3] LetA be a set and B ⊂ A A net { x α:α ∈Ᏸ}inA is eventually in B if there exists α0 ∈Ᏸ such thatx α ∈ B for all α ≥ α0 A net{ x α:α ∈Ᏸ}in a setA is called an ultranet if either { x α:α ∈Ᏸ}is eventually inB or { x α:α ∈Ᏸ}is eventually inA − B, for each subset B

ofA.

A Banach spaceX is said to satisfy the nonstrict Opial condition if, whenever a

se-quence{ x n }inX converges weakly to x, then for any y ∈ X,

lim sup

n

x n − x ≤lim sup

n

LetC be a nonempty closed convex subset of a Banach space X and x ∈ X Then the

inward setI C(x) is defined by

I C(x) =x + λ(y − x) : y ∈ C, λ ≥0

Note thatC ⊂ I C(x) and I C(x) is convex.

We need the following results in the sequel

Lemma 2.1 (see [9]) Let C be a nonempty closed convex subset of a Banach space X and

T : C → K(X) a contraction If T satisfies

then T has a fixed point.

Lemma 2.2 (see [4]) Let C be a nonempty closed bounded convex subset of a Banach space

X and T : C → KC(X) an upper semicontinuous φ-condensing, where φ( ·)= α( · ) or χ( · ).

If T satisfies

then T has a fixed point.

Lemma 2.3 (see [3]) Let C be a nonempty closed convex subset of a reflexive Banach space

X and { x β:β ∈ D } a bounded ultranet in C Then

r C

A

C,

x β

≤
1−∆X,α
1−)

r

C,

x β

here r C(A(C, { x β }))=inf{sup{ x − y :y ∈ A(C, { x β })}:x ∈ C }

3 Main results

LetC be a nonempty weakly compact convex subset of a Banach space X and T : C → KC(X) a continuous map.

Definition 3.1 The map T is called subsequentially limit-contractive (SL) if for every

asymptoticallyT-regular sequence { x n }inC,

lim sup

n H

T

x n ,T(x)

≤lim sup

n

for allx ∈ A(C, { x n })

Note that ifC is a nonempty closed convex subset of a uniformly convex Banach space

and{ x n }is bounded, thenA(C, { x n }) has a unique asymptotic center, sayx0, and so in the above definition, we have

lim sup

n H

T

x n ,T

x0

≤lim sup

n

x

It is clear that every nonexpansive map is an SL map Several examples of functions can

be constructed which are SL maps but not nonexpansive We include here the follow-ing simple example We further remark thatTheorem 1.1does not apply to the function defined below

Example 3.2 Let C =[0, 3/5] with the usual norm and consider the map T(x) = x2 It is easy to see thatT is an SL map but not nonexpansive Moreover, T is 1-χ-contractive and

has a fixed point

We now prove a result which containsTheorem 1.1, as a special case, and is applicable

to the above example

Theorem 3.3 Let C be a nonempty closed bounded convex subset of a Banach X such that

α(X) < 1 and T : C → KC(X) a continuous, SL, 1-χ-contractive map If T satisfies

then T has a fixed point.

Proof We follow the arguments given in [3] Letx0 ∈ C be fixed Define, for each n ≥1,

a mappingT n:C → KC(X) by

T n(x) : = 1

n x0+



1−1 n



wherex ∈ C Then T nis (1−1/n)-χ-contractive Also T n(x) ⊂ I C(x) for all x ∈ C Now

Lemma 2.1 guarantees that each T n has a fixed point x n ∈ C As a result, we have

limn →∞ d(x n,T(x n))=0 Let{ n α } be an ultranet of the positive integers{ n } SetA = A(C, { x n α }) We claim that

for allx ∈ A To prove our claim, let x ∈ A Since T(x n α) is compact, we can findy n α ∈ T(x n α) such that

x n

α − y n α  = d

x n α,T

x n α



We also havez n α ∈ T(x) such that

y n

α − z n α  = d

y n α,T(x)

We can assume thatz =limα z n α Clearly,z ∈ T(x) We show that z ∈ I A(x) = { x + λ(y − x) : λ ≥0,y ∈ A } SinceT is an SL map and { x n α}is asymptoticallyT-regular, it follows

that

lim sup

α H

T

x n α

 ,T(x)

≤lim sup

α

x n

for allx ∈ A Now

y n

α − z n α  = d

y n α,T(x)

≤ H

T

x n α



and so

lim

α

x

n α − z =lim

α

y

n α − z n α

≤lim sup

α

x n

wherer = r(C, { x n α}) Notice also thatz ∈ T(x) ⊂ I C(x) and so z = x + λ(y − x) for some

λ ≥0 andy ∈ C Without loss of generality, we may assume that λ > 1 Now

y =1

λ z +



1−1 λ



and so

lim

α x n

α − y ≤1

λlimα x n

α − z+1−1

λ

 lim

α x n

α − x ≤ r. (3.12) This implies thaty ∈ A and so z ∈ I A(x) This proves our claim ByLemma 2.3, we have

whereλ : =1−∆X,α(1−)< 1 Now choose x1 ∈ A and for each µ ∈(0, 1), define a mapping

T µ:A → KC(X) by

Then eachT µis aχ-condensing and satisfies

for allx ∈ A NowLemma 2.2guarantees thatT µhas a fixed point As a result, we can get

an asymptoticallyT-regular sequence { x1

n }inA Proceeding as above, we obtain

for allx ∈ A1:= A(C, { x1

n α }) andr C(A1)≤ λr C(A) By induction, for each m ≥1, we can find an asymptoticallyT-regular sequence { x m

n } n ⊆ A m −1 Using the ultranet{ x m

n α } α, we constructA m:= A(C, { x m

n α }) withr C(A m)≤ λ m r C(A) Choose x m ∈ A m Then{ x m } mis a Cauchy sequence Indeed, for eachm ≥1, we have

x m −1− x m  ≤ x m −1− x m n+x m

n − x m

≤diam

A m −1  +x m

for alln ≥1 Now taking lim sup, we see that

x

m −1− x m  ≤diamA m −1+ lim sup

n

x m

n − x m

≤3r C

A m −1 

≤3λ m −1r C(A).

(3.18)

Taking the limit asm → ∞, we get limm →∞  x m −1− x m  =0 This implies that{ x m }is a Cauchy sequence and so is convergent Letx =limm →∞ x m Finally, we show thatx is a

fixed point ofT Since T is an SL map, for m ≥1, we have

lim sup

n H

T

x m n ,T

x m



≤lim sup

n

x m

n − x m. (3.19) Now, we have form ≥1,

d

x m,T

x m

≤x m − x m

n+d

x m n,T

x m n

+H

T

x n m ,T

x m



This implies that

d

x m,T

x m



≤2 lim sup

n

x m − x m

n

Taking the limit asm → ∞, we have limm →∞ d(x m,T(x m))=0 and sox ∈ T(x) This

Theorem 3.3fails if the assumption thatT is an SL map is dropped.

Example 3.4 Let B be the closed unit ball of l2 DefineT : B → B by

T(x) = T

x1,x2, 

= 1−  x 2,x1,x2, 

ThenT is 1-χ-contractive without a fixed point (see [1,2]) We can show that this map is not SL if we consider the following sequence{ x(n) }inB:

x(1)=0,√1

2,

1

√

4,

1

√

8,

1

√

16,

 ,

x(2)=



0,√1

2√

2,

1

√

2√

2,

1

√

2√

4,

1

√

2√

4,

1

√

2√

8,

1

√

2√

8,

 ,

x(3)=



0,√1

3√

2,

1

√

3√

2,

1

√

3√

2,

1

√

3√

4,

1

√

3√

4,

1

√

3√

4,

1

√

3√

8,

1

√

3√

8,

1

√

3√

8,

 , (3.23) and so on

Corollary 3.5 Let C be a nonempty closed bounded convex subset of a Banach space X such that
α(X) < 1 satisfying the nonstrict Opial condition and T : C → KC(X) a nonex-pansive map If T satisfies

then T has a fixed point.

Proof This follows immediately from [2, Theorem 4.5] andTheorem 3.3

Next we present some fixed point results for∗-nonexpansive maps

Theorem 3.6 Let C be a nonempty closed bounded convex subset of a Banach space X such that
α(X) < 1 and T : C → KC(X) a ∗ -nonexpansive, 1-χ-contractive map If T satisfies

then T has a fixed point.

Proof Define

P T(x) =u x ∈ T(x) : d

x,u x

= d

x,T(x)

(3.26) for x ∈ C Since T(x) is compact, P T(x) is nonempty for each x Furthermore, P T is convex and compact valued since T is so Also, P T is nonexpansive because T is ∗ -nonexpansive Let B be a bounded subset of C Then it is easy to see that P T(C) is a

bounded set andχ(P T(B)) ≤ χ(B) Thus P Tis 1-χ-contractive P Talso satisfies

NowTheorem 3.3guarantees thatP Thas a fixed point HenceT has a fixed point.

Similarly, we get the following corollary, which extends [11, Theorem 2] to non-self-multimaps and to spaces satisfying the nonstrict Opial condition

Corollary 3.7 Let C be a nonempty closed bounded convex subset of a Banach space

X such that
α(X) < 1 satisfying the nonstrict Opial condition and T : C → KC(X) a ∗ -nonexpansive map If T satisfies

then T has a fixed point.

Acknowledgment

The authors are indebted to the referees for their valuable comments

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

E-mail address:nshahzad@kau.edu.sa

Amjad Lone: Department of Mathematics, College of Sciences, King Khalid University, P.O Box

9004, Abha, Saudi Arabia

E-mail address:amlone@kku.edu.sa