NOT NECESSARILY NONEXPANSIVE
NASEER SHAHZAD AND AMJAD LONEReceived 16 October 2004 and in revised form 3 December 2004
LetCbe a nonempty closed bounded convex subset of a Banach spaceX whose char-acteristic of noncompact convexity is less than 1 andTa continuous 1-χ-contractive SL map (which is not necessarily nonexpansive) fromCtoKC(X) satisfying an inwardness condition, whereKC(X) is the family of all nonempty compact convex subsets ofX. 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.
Theorem1.1 (see [3]). LetCbe a nonempty closed bounded convex subset of a Banach spaceXsuch thatα(X)<1andT:C→KC(X)a nonexpansive1-χ-contractive map. IfT satisfies
T(x)⊂IC(x) ∀x∈C, (1.1)
thenThas 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.
2. Preliminaries
LetCbe a nonempty closed subset of a Banach spaceX. LetCB(X) denote the family of all nonempty closed bounded subsets ofXandKC(X) the family of all nonempty compact convex subsets ofX. The Kuratowski and Hausdorffmeasures of noncompactness of a nonempty bounded subset ofXare, respectively, defined by
α(B)=inf{d >0 :Bcan be covered by finitely many sets of diameter≤d},
χ(B)=inf{d >0 :Bcan be covered by finitely many balls of radius≤d}. (2.1)
LetHbe the Hausdorffmetric onCB(X) andT:C→CB(X) a map. ThenTis called (1) contraction if there exists a constantk∈[0, 1) such that
HT(x),T(y)≤kx−y, ∀x,y∈C; (2.2)
(2) nonexpansive if
HT(x),T(y)≤ x−y, ∀x,y∈C; (2.3)
(3)φ-condensing (resp., 1-φ-contractive), whereφ=α(·) orχ(·) ifT(C) is bounded and, for each bounded subsetBofCwithφ(B)>0, the following holds:
φT(B)< φ(B) resp.,φT(B)≤φ(B); (2.4)
hereT(B)=x∈BT(x);
(4) upper semicontinuous if{x∈C:T(x)⊂V}is open in CwheneverV ⊂X is open;
(5) lower semicontinuous if the set{x∈C:T(x)∩V =φ}is open inCwhenever V⊂Xis open;
(6) continuous (with respect toH) ifH(T(xn),T(x))→0 wheneverxn→x;
(7)∗-nonexpansive (see [5]) if for allx,y∈Candux∈T(x) withd(x,ux)=inf{d(x, z) :z∈T(x)}, there exists uy∈T(y) with d(y,uy)=inf{d(y,w) :w∈T(y)} such that
dux,uy
≤d(x,y). (2.5)
A sequence{xn}is called asymptoticallyT-regular ifd(xn,Txn)→0 asn→ ∞.
Letφ=αorχ. The modulus of noncompactness convexity associated toφis defined by
∆X,φ()=inf1−d(0,A) :A⊂BXis convex, withφ(A)≥, (2.6)
whereBXis the unit ball ofX. The characteristic of noncompact convexity ofXassociated with the measure of noncompactnessφis defined in the following way:
φ(X)=sup
≥0 :∆X,φ()=0
Note that
∆X,α()≤∆X,χ() (2.8)
and so
α(X)≥χ(X). (2.9)
LetCbe a nonempty subset ofX,Ᏸa directed set, and{xα:α∈Ᏸ}a bounded net in X. We user(C,{xα}) andA(C,{xα}) to denote the asymptotic radius and the asymptotic center of{xα:α∈Ᏸ}inC, that is,
rx,xα=lim sup α
xα−x,
rC,xα=infrx,xα:x∈C,
AC,xα
=x∈C:rx,xα
=rC,xα
.
(2.10)
It is known thatA(C,{xα}) is a nonempty weakly compact convex set ifCis a nonempty closed convex subset of a reflexive Banach space. For details, we refer the reader to [1,3]. LetAbe a set andB⊂A. A net{xα:α∈Ᏸ}inAis eventually inBif there existsα0∈Ᏸ such thatxα∈Bfor allα≥α0. A net{xα:α∈Ᏸ}in a setAis called an ultranet if either {xα:α∈Ᏸ}is eventually inBor{xα:α∈Ᏸ}is eventually inA−B, for each subsetB ofA.
A Banach spaceX is said to satisfy the nonstrict Opial condition if, whenever a se-quence{xn}inXconverges weakly tox, then for anyy∈X,
lim sup n
xn−x≤lim sup
n
xn−y. (2.11)
LetCbe a nonempty closed convex subset of a Banach spaceXandx∈X. Then the inward setIC(x) is defined by
IC(x)=
x+λ(y−x) :y∈C,λ≥0. (2.12)
Note thatC⊂IC(x) andIC(x) is convex. We need the following results in the sequel.
Lemma2.1 (see [9]). LetCbe a nonempty closed convex subset of a Banach spaceXand T:C→K(X)a contraction. IfTsatisfies
T(x)⊂IC(x) ∀x∈C, (2.13)
thenThas a fixed point.
Lemma2.2 (see [4]). LetCbe a nonempty closed bounded convex subset of a Banach space XandT:C→KC(X)an upper semicontinuousφ-condensing, whereφ(·)=α(·)orχ(·). IfTsatisfies
T(x)∩IC(x) = ∅ ∀x∈C, (2.14)
Lemma2.3 (see [3]). LetCbe a nonempty closed convex subset of a reflexive Banach space Xand{xβ:β∈D}a bounded ultranet inC. Then
rC
AC,xβ
≤1−∆X,α
1−)rC,xβ
; (2.15)
hererC(A(C,{xβ}))=inf{sup{x−y:y∈A(C,{xβ})}:x∈C}.
3. Main results
LetCbe a nonempty weakly compact convex subset of a Banach spaceX andT:C→ KC(X) a continuous map.
Definition 3.1. The mapT is called subsequentially limit-contractive (SL) if for every asymptoticallyT-regular sequence{xn}inC,
lim sup
n H
Txn
,T(x)≤lim sup n
xn−x (3.1)
for allx∈A(C,{xn}).
Note that ifCis a nonempty closed convex subset of a uniformly convex Banach space and{xn}is bounded, thenA(C,{xn}) has a unique asymptotic center, sayx0, and so in the above definition, we have
lim sup
n H
Txn
,Tx0
≤lim sup n
x
n−x0. (3.2)
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. LetC=[0, 3/5] with the usual norm and consider the mapT(x)=x2. It is easy to see thatTis an SL map but not nonexpansive. Moreover,Tis 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.
Theorem3.3. LetCbe a nonempty closed bounded convex subset of a BanachXsuch that α(X)<1andT:C→KC(X)a continuous,SL,1-χ-contractive map. IfTsatisfies
T(x)⊂IC(x) ∀x∈C, (3.3)
thenThas a fixed point.
Proof. We follow the arguments given in [3]. Letx0∈Cbe fixed. Define, for eachn≥1, a mappingTn:C→KC(X) by
Tn(x) :=n1x0+
1−1 n
wherex∈C. ThenTnis (1−1/n)-χ-contractive. AlsoTn(x)⊂IC(x) for allx∈C. Now Lemma 2.1 guarantees that each Tn has a fixed point xn∈C. As a result, we have limn→∞d(xn,T(xn))=0. Let{nα} be an ultranet of the positive integers{n}. SetA=
A(C,{xnα}).We claim that
T(x)∩IA(x) = ∅ (3.5)
for allx∈A. To prove our claim, letx∈A. SinceT(xnα) is compact, we can findynα∈ T(xnα) such that
xnα−ynα=d
xnα,Txnα. (3.6)
We also haveznα∈T(x) such that
ynα−znα=d
ynα,T(x). (3.7)
We can assume thatz=limαznα. Clearly,z∈T(x). We show thatz∈IA(x)= {x+λ(y− x) :λ≥0,y∈A}. SinceTis an SL map and{xnα}is asymptoticallyT-regular, it follows that
lim sup
α H
Txnα,T(x)≤lim sup α
xnα−x (3.8)
for allx∈A. Now
ynα−znα=d
ynα,T(x)
≤HTxnα
,T(x) (3.9)
and so
lim α
x
nα−z=limα ynα−znα ≤lim sup
α
xnα−x=r, (3.10)
wherer=r(C,{xnα}). Notice also thatz∈T(x)⊂IC(x) and soz=x+λ(y−x) for some λ≥0 andy∈C. Without loss of generality, we may assume thatλ >1. Now
y=1 λz+
1−1 λ
x (3.11)
and so
lim
α xnα−y≤ 1
λlimα xnα−z+
1−1 λ
lim
α xnα−x≤r. (3.12)
This implies thaty∈Aand soz∈IA(x). This proves our claim. ByLemma 2.3, we have
whereλ:=1−∆X,α(1−)<1. Now choosex1∈Aand for eachµ∈(0, 1), define a mapping Tµ:A→KC(X) by
Tµ(x)=µx1+ (1−µ)T(x). (3.14)
Then eachTµis aχ-condensing and satisfies
Tµ(x)∩IA(x) = ∅ (3.15)
for allx∈A. NowLemma 2.2guarantees thatTµhas a fixed point. As a result, we can get an asymptoticallyT-regular sequence{x1
n}inA. Proceeding as above, we obtain
T(x)∩IA1(x) = ∅ (3.16)
for allx∈A1:=A(C,{x1
nα}) andrC(A1)≤λrC(A). By induction, for eachm≥1, we can find an asymptoticallyT-regular sequence{xm
n}n⊆Am−1. Using the ultranet{xmnα}α, we constructAm:=A(C,{xm
nα}) withrC(Am)≤λmrC(A). Choosexm∈Am. Then{xm}mis a Cauchy sequence. Indeed, for eachm≥1, we have
xm−1−xm≤xm−1−xm
n+xnm−xm ≤diamAm−1+xm
n −xm,
(3.17)
for alln≥1. Now taking lim sup, we see that
x
m−1−xm≤diamAm−1+ lim sup n
xm n −xm
≤3rC
Am−1
≤3λm−1rC(A).
(3.18)
Taking the limit asm→ ∞, we get limm→∞xm−1−xm =0. This implies that{xm}is a Cauchy sequence and so is convergent. Letx=limm→∞xm. Finally, we show thatxis a fixed point ofT. SinceTis an SL map, form≥1, we have
lim sup
n H
Txmn
,Txm≤lim sup n
xm
n−xm. (3.19)
Now, we have form≥1,
dxm,Txm≤xm−xnm+dxmn,Txmn
+HTxnm,Txm. (3.20)
This implies that
dxm,Txm≤2 lim sup n
xm−xm n
≤2λm−1rC(A). (3.21)
Taking the limit asm→ ∞, we have limm→∞d(xm,T(xm))=0 and sox∈T(x). This
Theorem 3.3fails if the assumption thatTis an SL map is dropped. Example 3.4. LetBbe the closed unit ball ofl2. DefineT:B→Bby
T(x)=Tx1,x2,...
= 1− x2,x 1,x2,...
. (3.22)
ThenTis 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.
Corollary3.5. LetCbe a nonempty closed bounded convex subset of a Banach spaceX such thatα(X)<1satisfying the nonstrict Opial condition andT:C→KC(X)a nonex-pansive map. IfTsatisfies
T(x)⊂IC(x) ∀x∈C, (3.24)
thenThas 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.
Theorem3.6. LetCbe a nonempty closed bounded convex subset of a Banach spaceXsuch thatα(X)<1andT:C→KC(X)a∗-nonexpansive,1-χ-contractive map. IfTsatisfies
T(x)⊂IC(x) ∀x∈C, (3.25)
thenThas a fixed point. Proof. Define
PT(x)=
ux∈T(x) :d
x,ux
=dx,T(x) (3.26)
for x∈C. Since T(x) is compact, PT(x) is nonempty for each x. Furthermore, PT is convex and compact valued since T is so. Also, PT is nonexpansive because T is ∗-nonexpansive. Let Bbe a bounded subset ofC. Then it is easy to see thatPT(C) is a bounded set andχ(PT(B))≤χ(B). ThusPTis 1-χ-contractive.PTalso satisfies
PT(x)⊂IC(x) ∀x∈C. (3.27)
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 andT:C→KC(X)a ∗ -nonexpansive map. IfTsatisfies
T(x)⊂IC(x) ∀x∈C, (3.28)
thenThas 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