R E S E A R C H
Open Access
The structure of fixed-point sets of
Lipschitzian type semigroups
DR Sahu
1*, RP Agarwal
2and Donal O’Regan
3*Correspondence: [email protected] 1Department of Mathematics, Banaras Hindu University, Varanasi, 221005, India
Full list of author information is available at the end of the article
Abstract
The purpose of this paper is to establish some results on the structure of fixed point sets for one-parameter semigroups of nonlinear mappings which are not necessarily Lipschitzian in Banach spaces. Our results improve several known existence and convergence fixed point theorems for semigroups which are not necessarily Lipschitzian.
MSC: Primary 47H09; 47H10; secondary 47B20; 54C15
Keywords: asymptotic center; normal structure coefficient; pseudo-contractive semigroup; sunny nonexpansive retraction; uniformly convex Banach space; uniformly Lipschitzian semigroup; variational inequality
1 Introduction
LetCbe a nonempty subset of a Banach spaceXandT:C→Ca mapping. We useFix(T) to denote the set of all fixed points ofT. A nonempty closed convex subsetDofCis said to satisfy property (ω) with respect to mappingT[] if
(ω) ωT(x)⊂D for everyx∈D,
whereωT(x) denotes the set of all weak subsequential limits of{Tnx:n∈N}. Moreover,T is said to satisfy the (ω)-fixed point propertyifThas a fixed point in every nonempty closed convex subsetDofCwhich satisfies property (ω). For a Lipschitzian mappingS:C→C, we use the symbolσ(S) to denote the exact Lipschitz constant ofS,i.e.,
σ(S) =infk∈[,∞] :Sx–Sy ≤kx–yfor allx,y∈C.
A mappingT:C→Cis said to be
() nonexpansiveifσ(T) = ,
() asymptotically nonexpansive[] ifσ(Tn)≥for alln∈Nandlimn→∞σ(Tn) = ,
() uniformlyL-Lipschitzianifσ(Tn) =Lfor alln∈Nand for someL∈(,∞).
In general, the fixed-point set of a nonexpansive mapping need not be convex and can be extremely irregular. Suppose thatCis a nonempty closed convex bounded subset of a Banach spaceXandT :C→Cis a nonexpansive mapping withFix(T)=∅. Obviously,
Fix(T) is a closed set.Fix(T) is convex ifXis strictly convex (see [, ]).
Nonexpansive retracts have been studied in several contexts (for example, convex geom-etry [], extension problems [], fixed point theory [], optimal sets []). It is well known
that ifCis a nonempty closed convex bounded subset of a Banach space and if a nonex-pansive mappingT:C→Chas a fixed point in every nonempty closed convex subset of Cwhich is invariant underT, thenFix(T) is a nonexpansive retract ofC(that is, there exists a nonexpansive mappingR:C→Fix(T) such thatR|Fix(T)=I) (see [, Theorem ]). The Bruck result was extended by Benavides and Ramirez [] to the case of asymptotically nonexpansive mappings if the spaceXwas sufficiently regular.
The class of asymptotically nonexpansive mappings was introduced by Goebel and Kirk [] in and they proved that ifCis a nonempty closed convex bounded subset of a uniformly convex Banach space, then every asymptotically nonexpansive self-mapping of Chas a fixed point. Several authors have studied the existence of fixed points of asymp-totically nonexpansive mappings in Banach spaces having rich geometric structure, see [, , ].
There is a class of mappings which lies strictly between the class of contraction mappings and the class of nonexpansive mappings. The class of pointwise contractions was intro-duced in Belluce and Kirk [], and later it was called ‘generalized contractions’ in []. Banach’s celebrated contraction principle was extended to this larger class of mappings as follows:
Theorem .[, ] Let C be a nonempty weakly compact convex subset of a Banach space and T :C→C a pointwise contraction.Then T has a unique fixed point x*,and
{Tnx}converges strongly to x*for each x∈C.
Kirk [] combined the ideas of pointwise contraction [] and asymptotic contraction [] and introduced the concept of an asymptotic pointwise contraction. He announced that an asymptotic pointwise contraction defined on a closed convex bounded subset of a super-reflexive Banach space has a fixed point. Recently, Kirk and Xu [] gave a simple and elementary proof of the fact that an asymptotic pointwise contraction defined on a weakly compact convex set always has a unique fixed point (with convergence of Picard iterates). They also introduced the concept of pointwise asymptotically nonexpansive mapping and proved that every pointwise asymptotically nonexpansive mapping defined on a closed convex bounded subset of a uniformly convex Banach space has a fixed point.
Every asymptotically nonexpansive mapping is uniformlyL-Lipschitzian, and the (ω )-fixed point property of uniformlyL-Lipschitzian mappings is closely related to the class of nonexpansive and asymptotically nonexpansive mappings. In this connection, a deep result of Casini and Maluta [] was generalized by Lim and Xu [] as follows:
Theorem LX(Lim and Xu [, Theorem ]) Let X be a Banach space with a uniform normal structure and let N(X)be the normal structure coefficient of X.Let C be a nonempty bounded subset of X and T:C→C a uniformly L-Lipschitzian mapping with L<√N(X). Then T satisfies the(ω)-fixed point property.
The (ω)-fixed point property plays a key role in the existence and approximation of so-lutions of fixed point problems and variational inequality problems, see [–].
unique global solutions to time evolution equations of the formu=Au, whereAis a com-bination of derivatives. In many applications, semigroups are not necessarily Lipschitzian. It is an interesting problem to extend fixed point existence results, namely Theorem LX, for semigroups of nonlinear mappings which are not necessarily Lipschitzian.
Motivated by the results above, in this paper we establish some results on the structure of fixed point sets for one-parameter semigroups of nonlinear mappings which are not nec-essarily Lipschitzian in Banach spaces. Our theorems significantly extend Theorem LX to more general Banach spaces and to a more general class of operators. We obtain a general convergence theorem for semigroups of non-Lipschitzian pseudo-contractive mappings. Our results improve several known fixed point problems and variational inequality prob-lems for semigroups which are not necessarily Lipschitzian.
2 Preliminaries
LetR+denote the set of nonnegative real numbers, and letN
denote the set of nonneg-ative integers. Throughout this paper,Gdenotes an unbounded set ofR+:= [,∞) such thats+t∈Gfor alls,t∈Gands–t∈Gfor alls,t∈Gwiths≥t(oftenG=NorR+).
2.1 Lipschitzian type mappings
LetCbe a nonempty subset of a Banach spaceXandT:C→Ca mapping. ThenT is called
(i) pointwise contractive[] if there exists a functionα:C→[, )such that Tx–Ty ≤α(x)x–yfor allx,y∈C;
(ii) asymptotic pointwise contractive[] if for eachn∈N, there exists a function αn:C→[, )such thatTnx–Tny ≤αn(x)x–yfor allx,y∈C, whereαn→α
pointwise onC;
(iii) pointwise asymptotically nonexpansive[] if for each integern∈N, Tnx–Tny ≤α
n(x)x–yfor allx,y∈C, whereαn→pointwise;
(iv) asymptotically nonexpansive in the intermediate sense[] providedTis uniformly continuous and
lim sup
n→∞
sup
x,y∈C
Tnx–Tny–x–y≤; (.)
(v) mapping of asymptotically nonexpansive type[] if
lim sup
n→∞ supy∈C
Tnx–Tny–x–y≤ for allx∈C.
Fix a sequence{an}in [,∞) with an→. A mappingT:C→Cis said to benearly Lipschitzianwith respect to{an}[] if for eachn∈N, there exists a constantkn> such that
Tnx–Tny≤knx–y+an (.)
for allx,y∈C. The infimum of constantsknin (.) is callednearly Lipschitz constantand is denoted byη(Tn). A nearly Lipschitzian mappingT with the sequence{(a
(i) nearly contractiveifη(Tn) < for alln∈N,
(ii) nearly uniformlyL-Lipschitzianifη(Tn)≤Lfor alln∈N,
(iii) nearly uniformlyk-contractiveifη(Tn)≤k< for alln∈N,
(iv) nearly nonexpansiveifη(Tn) = for alln∈N,
(v) nearly asymptotically nonexpansiveifη(Tn)≥for alln∈Nwith
limn→∞η(Tn) = .
The mappingTis said to bedemicontinuousif, whenever a sequence{xn}inCconverges strongly tox∈C, then{Txn}converges weakly toTx. The mappingTis said to beweakly contractiveif
Tx–Ty ≤ x–y–ψx–y for allx,y∈C,
whereψ: [,∞)→[,∞) is a continuous and nondecreasing function such thatψ() = ,
ψ(t) > fort> andlimt→∞ψ(t) =∞.
LetC be a convex subset of a Banach spaceXandDa nonempty subset ofC. Then a continuous mappingPfromContoDis called aretractionifPx=xfor allx∈D,i.e., P=P. A retractionPis said to besunnyifP(Px+t(x–Px)) =Pxfor eachx∈Candt≥ withPx+t(x–Px)∈C. If the sunny retractionPis also nonexpansive, thenDis said to be asunny nonexpansive retractofC.
In what follows, we shall make use of the following lemmas:
Lemma .[] Let C be a nonempty closed convex subset of a Banach space X and T:C→ C a continuous strongly pseudo-contractive mapping.Then T has a unique fixed point in C.
Lemma .(Goebel and Reich [, Lemma .]) Let C be a convex subset of a smooth Banach space X,D a nonempty subset of C and P a retraction from C onto D.Then the following are equivalent:
(a) Pis sunny and nonexpansive.
(b) x–Px,J(z–Px) ≤for allx∈C,z∈D. (c) x–y,J(Px–Py) ≥ Px–Pyfor allx,y∈C.
2.2 Semigroups
LetCbe a nonempty subset of a Banach spaceX. The one-parameter familyF :={T(t) : t∈G}is said to be astrongly continuous semigroup of mappingsfromCinto itself if
(I) T()x=xfor allx∈C;
(II) T(s+t) =T(s)T(t)for alls,t∈G;
(III) for eachx∈C, the mappingT(·)xfromGintoCis continuous.
We denote by Fix(F) the set of all common fixed points of F, i.e., Fix(F) :=
t∈GFix(T(t)). For a Lipschitzian semigroupF, we write
σ(F) :=lim inf
t→∞ σ
T(t).
IfF satisfies (I)-(III) and
lim
thenF is calledasymptotically regular on C. IfF satisfies (I)-(III) and
lim
t→∞
sup
x∈C
T(t)x–T(s)T(t)x= for alls> and boundedC⊆C, (uar)
thenF is calleduniformly asymptotically regular on C. A Lipschitzian semigroupF is called a
(i) uniformlyL-Lipschitzian semigroupifsupt∈Gσ(T(t)) =L<∞; (ii) nonexpansive semigroup ifσ(T(t)) = for allt∈G;
(iii) asymptotically nonexpansive semigroupifσ(T(t))≥for allt∈Gand limt→∞σ(T(t)) = .
2.3 Asymptotic center
Throughout the paper, (X, · ) is a Banach space which is assumed not to be Schur. That is,Xhas weakly convergent sequences that are not norm convergent. LetCbe a nonempty closed convex subset of a Banach spaceXand{xt}t∈G a bounded set inX. Consider the functionalra(·,{xt}t∈G) :X→R+defined by
rax,{xt}t∈G
=lim sup
Gt→∞xt–x, x∈X.
The infimum ofra(·,{xt}t∈G) overC is said to be theasymptotic radiusof {xt}t∈G with respect toCand is denoted byra(C,{xt}t∈G). A pointz∈C is said to be anasymptotic centerof{xt}t∈Gwith respect toCif
raz,{xt}t∈G
=infrax,{xt}t∈G
:x∈C.
The set of all asymptotic centers of{xt}t∈Gwith respect toCis denoted byZa(C,{xt}t∈G). A numberdiama({xt}t∈G) =lim supGk→∞(sup{xs–xt:s,t≥k}) is called an asymptotic diameter of{xt}t∈G. It is well known that ifXis reflexive, thenZa(C,{xt}t∈G) is nonempty
closed convex and bounded, and ifXis uniformly convex, thenZa(C,{xt}t∈G) consists only of a single point,{z}=Za(C,{xt}t∈G),i.e.,z∈Cis the unique point which minimizes the functional
lim sup
Gt→∞xt–x
overx∈C.
2.4 Normal structure
Normal structure plays a key role in some problems of metric fixed point theory. LetCbe a nonempty bounded subset of a Banach spaceX. We denote by
diam(C) = sup
x,y∈C x–y,
the diameter ofC. Put
rC(C) =inf
x∈C
sup
y∈C x–y
This nonnegative real number is called the Chebyshev radius ofCrelative to itself. The normal structure coefficientN(X) of a Banach spaceXis defined [] by
N(X) =inf
diam(C)
rC(C) :Cis nonempty bounded convex subset
ofXwith diam(C) >
.
The spaceXis said to have theuniformly normal structureifN(X) > . It is well known that, for every uniformly convex Banach spaceX,N(X) > . A weakly convergent sequence coefficient ofXis defined (see []) by
WCS(X) =sup
k:klim sup
n→∞
xn<diama
{xn}for all{xn}inXwithxn
.
It is proved in [, Theorem ] that
WCS(X) =β(X) :=infD{xn}:xn,xn →,
whereD[{xn}] :=lim supm→∞(lim supn→∞xm–xn). It is readily seen that
≤N(X)≤WCS(X)≤.
The spaceXis said to have theweak uniformly normal structureifWCS(X) > . IfXis a reflexive Banach space with modulus of convexityδX, then
–δX()≤N(X)≤WCS(X).
Thus, ifXis a uniformly convex Banach space, thenWCS(X) > and also the equation
α WCS(X)δ
– X
–
α
=
has a unique solutionα> . A general formula forWCS(X) in an arbitrary Banach space is not known. In particular, it has been calculated that for a Hilbert spaceH,
WCS(H) =√;
forp(≤p<∞),
WCS(p) = /p
and for∞,
Remark . pis an example of a reflexive Banach space such thatN(X) andWCS(X) are different. Indeed,
N(p) = (p–)/p< /p=WCS(p), for <p< .
A Banach spaceXis said to satisfy the Opial condition, if whenever a sequence{xn}inX converges weakly tox, then
lim inf
n→∞ xn–x<lim infn→∞ xn–y for ally∈X\ {x}.
The Opial modulusrXofXis defined by
rX(c) =inf
lim inf
n→∞ xn+x–
,
wherec> and the infimum is taken over allx∈Xwithx ≥cand all sequences{xn} inXsuch thatw–limn→∞xn= andlim infn→∞xn ≥. For any Banach spaceX, we
have the following inequality:
+rX()≤WCS(X).
3 Nonemptiness of common fixed-point sets First, we introduce some wider classes of semigroups.
Definition . LetCbe a nonempty subset of a normed spaceXandF :={T(t) :t∈G}a strongly continuous semigroup of mappings fromCinto itself. The semigroupF is said to benearly Lipschitzianif there exist a functiona(·) :G→[,∞) withlimt→∞a(t) =
and a functionη(·) :G→(,∞) such that
T(t)x–T(t)y≤ηT(t)x–y+a(t) for allx,y∈Candt∈G.
For a nearly Lipschitzian semigroupF, we write
η(F) :=lim inf
t→∞ η
T(t).
We sayF is
(a) pointwise nearly Lipschitzianif for eacht∈G, there exist a function
a(·) :G→[,∞)withlimt→∞a(t) = and a functionαt(·) :C→(,∞)such that
T(t)x–T(t)y≤αt(x)
x–y+a(t) for allx,y∈C;
(b) pointwise nearly uniformlyα(·)-Lipschitzianif there exist a function
a(·) :G→[,∞)withlimt→∞a(t) = and a functionα(·) :C→(,∞)such that
(c) asymptotic pointwise nearly Lipschitzianif for eacht∈G, there exist a function
a(·) :G→[,∞)withlimt→∞a(t) = and two functionsαt(·),α(·) :C→(,∞)
withlimt→∞αt=αpointwise such that T(t)x–T(t)y≤αt(x)
x–y+a(t) for allx,y∈C.
We say that an asymptotic pointwise nearly Lipschitzian semigroup F is pointwise nearly asymptotically nonexpansive (pointwise asymptotically nonexpansive) ifαt(x)≥ for allt∈Gandlimt→∞αt= pointwise (a(t) = andαt(x)≥ for allt∈Gandαt(x)→ pointwise). Further, we say that an asymptotic pointwise nearly Lipschitzian semigroup F is asymptotic pointwise nearly contractive ifαt→α pointwise and α(x)≤k< for allx∈C. The semigroupF is said to be nearly uniformlyL-Lipschitzian if there exist a constantL∈[,∞) and a functiona(·) :G→[,∞) withlimt→∞a(t) = such that
T(t)x–T(t)y≤Lx–y+a(t) for allx,y∈C.
The nearly uniformlyL-Lipschitzian semigroup will be called nearly nonexpansive semi-group.
Before presenting the main result of this section, we give another definition:
Definition . LetC be a nonempty weakly compact convex subset of Banach space X,F :={T(t) :t∈G}a strongly continuous semigroup of mappings from Cinto itself. A nonempty closed convex subsetDofCis said to satisfy property (ω) with respect to semigroupF if
(ω) ωF(x)⊂Dfor everyx∈D,
whereωF(x) denotes the set of all weak limits of{T(tn)x:n∈N}astn→ ∞.
The semigroupF is said to satisfy the (ω)-fixed point propertyifF has a common fixed point in every nonempty closed convex subsetDofCwhich satisfies property (ω).
We now establish that a semigroupF of a certain class of Lipschitzian type mappings satisfies the (ω)-fixed point property.
Theorem . Let X be a Banach space with weak uniformly normal structure, C a nonempty weakly compact convex subset of X andF ={T(t) :t∈G}a strongly contin-uous semigroup of demicontincontin-uous mappings from C into itself. Suppose that for each t∈G,there exist a function a(·) :G→[,∞)withlimt→∞a(t) = and two functions αt(·),α(·) :C→(,∞)withlimt→∞αt=α pointwise andsupx∈Cα(x) <
√
WCS(X)such that
T(t)x–T(t)y≤αt(x)x–y+a(t) for all x,y∈C.
Also suppose that there exists a nonempty closed convex subset M of C which satisfies prop-erty(ω)with respect toF.Then:
(a) For arbitraryx∈M,there exist a sequence{tn}inGwithlimn→∞tn=∞and an
iterative sequence{xm}inMdefined by
xm=w– lim
(b) IfF is asymptotically regular onC,then there exists an elementx*∈M∩Fix(F)
such that{xm}converges strongly tox*. Proof
(a) Since one can easily construct a nonempty closed convex separable subsetCof C which is invariant under eachT(t) (i.e.,T(t)(C)⊂Cfort∈G), we may assume thatC itself is separable.
The separability ofCmakes it possible to select a sequence{T(tn)x}of{T(t)x}t∈Gsuch that
T(tn)x
is weakly convergent for everyx∈C
and
lim
n→∞αtn(x) =α(x) for everyx∈C.
For anyx∈M⊂C, consider a sequence{T(tn)x}inC. Supposew–limn→∞T(tn)x= x∈C. Using property (ω), we obtain thatx∈M. Now we can construct a sequence{xm} inMin the following way:
⎧ ⎨ ⎩
x∈M arbitrary,
xm=w–limn→∞T(tn)xm– for allm∈N.
(b) The weak asymptotic regularity ofF ensures thatxm=w–limn→∞T(tn+tr)xm–, tr∈G. We now show that{xm}converges strongly to a common fixed point ofF. Set
L:=sup
x∈C
α(x), Dm:=lim sup
n→∞
xm–T(tn)xm
and
Rm:=lim sup
n→∞
xm+–T(tn)xm
for allm= , , , . . . . By the property ofWCS(X), we have
Rm=lim sup
n→∞
xm+–T(tn)xm≤
WCS(X)D
T(tn)xm. (.)
By the asymptotic regularity ofF and thew-l.s.c. of the norm · , we have
Dm =lim sup r→∞
xm–T(tr)xm
≤lim sup
r→∞
lim sup
s→∞
T(ts)xm––T(tr)xm
≤lim sup
r→∞
lim sup
s→∞
T(ts)xm––T(tr+ts)xm–+T(tr+ts)xm––T(tr)xm
≤lim sup
r→∞
lim sup
s→∞
=α(xm)lim sup
s→∞
T(ts)xm––xm
≤LRm–.
On the other hand, by the asymptotic regularity ofF, we have
DT(tn)xm=lim sup
n→∞
lim sup
r→∞
T(tn)xm–T(tr)xm
≤lim sup
n→∞
lim sup
r→∞
T(tn)xm–T(tn+tr)xm
+T(tn+tr)xm–T(tr)xm
≤lim sup
n→∞
lim sup
r→∞
αtn(xm)xm–T(tr)xm+a(tn)
=LDm
≤LRm–.
Setλ:=WCSL(X)< . From (.), we obtain
Rm≤λRm–≤λRm–≤ · · · ≤λmR→ asm→ ∞.
For anym∈N, one can see that
xm+–xm ≤lim sup n→∞
xm+–T(tn)xm+T(tn)xm–xm
≤Rm+Dm
=Rm+LRm–
≤(λ+L)Rm– ..
.
≤(λ+L)λm–R, (.)
so it follows that{xm}is a Cauchy sequence inM. Letlimm→∞xm=v∈M. Observe that
v–T(tn)v≤ v–xm++xm+–T(tn)xm+T(tn)xm–T(tn)v
≤ v–xm++xm+–T(tn)xm+αtn(xm)
xm–v+a(tn).
Taking the limit superior asn→ ∞on both sides, we get
lim sup
n→∞
v–T(tn)v≤ v–xm++Rm+α(xm)xm–v
≤ v–xm++Rm+Lxm–v → asm→ ∞.
By the uniqueness of the weak limit of {T(t)T(tn)v}n∈N, we have T(t)v=v. Therefore,
v∈M∩Fix(F).
Theorem . generalizes the result due to Górnicki [] in the context of the (ω)-fixed point property for a wider class of mappings. Theorem . also extends corresponding results of Sahu, Agarwal and O’Regan [], Sahu, Liu and Kang [] and Sahu, Petruşel and Yao [] for asymptotic pointwise nearly Lipschitzian semigroups. AsN(X)≤WCS(X), and there are Banach spaces for whichN(X) = whileWCS(X) > , the following result is an improvement on Casini and Maluta [] and Lim and Xu [, Theorem ].
Corollary . Let X be a Banach space with weak uniformly normal structure, C a nonempty weakly compact convex subset of X andF ={T(t) :t∈G}a strongly continuous semigroup of demicontinuous nearly uniformly L-Lipschitzian of mappings from C into it-self.Suppose that L<√WCS(X)and that there exists a nonempty closed convex subset M of C which satisfies property(ω)with respect toF.Then:
(a) For arbitraryx∈M,there exist a sequence{tn}inGwithlimn→∞tn=∞and an
iterative sequence{xm}inMdefined by
xm=w– lim
n→∞T(tn)xm– for allm∈N.
(b) IfF is asymptotically regular onC,then there exists an elementx*∈M∩Fix(F)
such that{xm}converges strongly tox*.
Corollary . Let X be a Banach space with weak uniformly normal structure, C a nonempty weakly compact convex subset of X andF ={T(t) :t∈G}a strongly continu-ous semigroup of demicontinucontinu-ous nearly uniformly L-Lipschitzian asymptotically regular mappings from C into itself such that L<√WCS(X).ThenF has a common fixed point in C.
4 Common fixed-point sets as Lipschitzian retracts
Theorem . Let X be a uniformly Banach space with the Opial condition,C a nonempty closed convex bounded subset of X andF ={T(t) :t∈G}a strongly continuous semigroup of demicontinuous mappings from C into itself.Suppose that there exists a function a(·) : G→[,∞)withlimt→∞a(t) = such that
T(t)x–T(t)y≤ηT(t)x–y+a(t) for all x,y∈C,
whereη(T(t))∈(,∞)withη(F) :=lim inft→∞η(T(t)) <√WCS(X).Also suppose thatF
is asymptotically regular on C.ThenFix(F)=∅andFix(F)isη(F)-Lipschitzian retract of C.
Proof Using similar arguments as in the proof of Theorem .(a), we may select a sequence
{T(tn)x}of{T(t)x}t∈Gsuch thatlimn→∞η(T(tn)) =η(F) and
Let A:C→C denote a mapping which associates with a givenx∈C a unique z∈
Za(C,{T(t)x}t∈G), that is,z=Ax. SinceAx=w–limn→∞T(tn)xfor allx∈C, it follows from the lower weak semi-continuity of the norm that
Ax–Ay ≤η(F)x–y for allx,y∈C,
i.e.,Aisη(F)-Lipschitzian mapping. It follows thatAis uniformly continuous.
For anyx=x∈C, consider a sequence{T(tn)x}inC. Supposew–limn→∞T(tn)x= x∈C. Now we can construct a sequence{xm}inCin the following way:
⎧ ⎨ ⎩
x=x∈C arbitrary,
xm=w–limn→∞T(tn)xm– for allm∈N.
(.)
From (.), we have
xm+=Axm for allm∈N.
Set L := η(F), λ := WCSL(X) < , Dm := lim supn→∞xm – T(tn)xm and Rm :=
lim supn→∞xm+–T(tn)xmfor allm= , , , . . . . From (.), we have
Am+x–Amx=xm+–xm ≤(λ+L)λm–R
≤(λ+L)λm–diam(C)
forx∈Candm∈N. It follows that
∞
m=
sup
x∈C
Am+x–Amx<∞.
Thus, the sequence{Amx}converges uniformly to a functionQdefined by
Qx= lim
m→∞A
mx for allx∈C.
Forx=x, we have
Qx–T(tn)Qx≤ Qx–xm+xm–T(tn)xm+T(tn)xm–T(tn)Qx
≤ Qx–xm+xm–T(tn)xm+ηT(tn)xm–Qx+a(tn).
Taking the limit superior asn→ ∞, we get
lim sup
n→∞
Qx–T(tn)Qx≤ +η(F)xm–Qx+Dm→ asm→ ∞.
Corollary . Let X be a uniformly Banach space with the Opial condition,C a nonempty closed convex bounded subset of X and T:C→C a demicontinuous asymptotically reg-ular nearly Lipschitzian mapping such thatη∞(T) :=lim infn→∞η(Tn) <
√
WCS(X).Then
Fix(T)=∅andFix(T)is aη∞(T)-Lipschitzian retract of C.
One sees from Theorem . that if η(F) = , then Fix(F) is a nonexpansive retract ofC. In the next section, we show thatFix(F) is a sunny nonexpansive retract ofCwhen F ={T(t) :t∈R+}a strongly continuous semigroup of asymptotically pseudo-contractive mappings (see Theorem .).
5 Common fixed-point sets as sunny nonexpansive retracts
LetCbe a nonempty subset of a Banach spaceXandF ={T(t) :t∈G}a semigroup of mappings fromCinto itself. A sequence{xn}inCis said to anapproximating fixed point sequence of F iflimn→∞(xn–T(t)xn) = for allt∈G. The family{I–T(t) :t∈G}is demiclosed at zero if{yn}is a sequence inCweakly converging toz∈Candlimn→∞(xn– T(t)yn) = for allt∈Gimplyz=T(t)zfor allt∈G. Following [], we say thatF has property (A) if for every bounded set{xt}t∈GinC, we have
lim
t→∞xt–T(t)xt= implies tlim→∞xt–T(s)xt= for alls∈G.
In [], Schu introduced the concept of asymptotically pseudo-contractive mapping as follows:
LetHbe a real Hilbert space whose inner product and norm are denoted by·,·and
· respectively. LetCbe a nonempty subset ofH andT:C→Ca mapping. ThenT is called an asymptotically pseudo-contractive mapping if there exists a sequence{kn}in [,∞) withlimn→∞kn= such that
Tnx–Tny,x–y≤knx–y for allx,y∈Candn∈N.
The class of asymptotically pseudo-contractive mappings contain properly the class of asymptotically nonexpansive mappings. The following example shows that a continuous asymptotically pseudo-contractive mapping is not necessarily asymptotically nonexpan-sive.
Example . LetX=RandC= [, ]. DefineT:C→Cby
Tx= –x//, x∈C.
Note thatTis a pseudo-contractive mapping which is not Lipschitzian (see []). Since T is not Lipschitzian, it is not asymptotically nonexpansive. It is shown in [] thatT is an asymptotically pseudo-contractive mapping with sequence{}.
LetF ={T(t) :t∈G}be a semigroup of mappings fromCinto itself. ThenF is said to be pseudo-contractive if
Remark .
(i) The semigroupF is pseudo-contractive if and only if the following holds:
T(t)x–T(t)y≤ x–y+x–T(t)x–y–T(t)y for allx,y∈Candt∈G.
(ii) Every nonexpansive semigroup must be a continuously pseudo-contractive semigroup.
We sayF is asymptotically pseudo-contractive if there exists a functionk(·) :G→[,∞) withlimt→∞k(t) = such that
T(t)x–T(t)y,x–y≤k(t)x–y for allx,y∈Candt∈G.
Example . LetX=R,b∈(, ),C= [, ] andG= [,∞). Fort> , defineT(t) :C→C by
T(t)x= ⎧ ⎨ ⎩
btx, ifx∈[, /];
, ifx∈(/, ]
and define
T()x=x, x∈C.
SetC:= [, /] andC:= (/, ]. Note that
T(t)x–T(t)y,x–y=bt|x–y|≤ |x–y| for allx,y∈Candt>
and
T(t)x–T(t)y,x–y= ≤ |x–y| for allx,y∈Candt> .
Forx∈Candy∈C, we havex–y≤, and hence
T(t)x–T(t)y,x–y=btx– (x–y)≤≤ |x–y| for allt> .
Thus,
T(t)x–T(t)y,x–y≤ |x–y| for allx,y∈Candt≥.
Therefore, F ={T(t) :t≥} is an asymptotically pseudo-contractive semigroup with functionk= . Moreover, for eacht> ,T(t) is discontinuous atx= / and henceF is not a Lipschitzian semigroup.
We begin with the following:
uniformly continuous nearly uniformly L-Lipschitzian asymptotically pseudo-contractive mappings from C into itself.Then the family{I–T(t) :t∈G}is demiclosed at zero.
Proof Assume that {yn} is a sequence in C weakly converging to z and limn→∞(yn– T(s)yn) = for alls∈G. Lett∈Gwitht= and let{tm}be a sequence inGdefined bytm=mtfor allm∈N. Notice thatz∈C. Fixα∈(, /( +L)) and define
zm= ( –α)z+αT(tm)z for allm∈N.
SinceTis uniformly continuous, we haveyn–T(tm)z → asn→ ∞for fixedm∈N. Indeed, for fixedm∈N, we have
yn–T(tm)yn≤yn–T(t)yn+T(t)yn–T(t)yn+· · ·
+T(m– )tyn–T(mt)yn
for alln∈N. SinceF is a uniformly continuous semigroup, it follows thatlimn→∞yn– T(tm)yn = for each fixed m ∈ N. Noticing that F is an asymptotically pseudo-contractive semigroup, for fixedm∈N, we have
z–zm,I–T(tm)zm=z–yn,I–T(tm)zm+yn–zm,I–T(tm)zm
=z–yn,I–T(tm)zm
+yn–zm,I–T(tm)zm–I–T(tm)yn
+yn–zm,I–T(tm)yn
≤z–yn,I–T(tm)zm+k(tn) – yn–zm
+yn–zm,
I–T(tm)
yn
. (.)
Sinceynzandlimn→∞yn–T(tm)yn= , it follows from (.) that
z–zm,I–T(tm)zm≤k(tn) – diam(C).
Note that
z–T(tm)z =z–T(tm)z,z–T(tm)z
=
α
z–zm,z–T(tm)z
=
α
z–zm,I–T(tm)z–I–T(tm)zm+I–T(tm)zm
=
α
z–zm,I–T(tm)z–I–T(tm)zm+z–zm,I–T(tm)zm
≤ α
z–zm+z–zmT(tm)z–T(tm)zm+
k(tm) –
diam(C)
≤ α
z–zm+Lz–zmz–zm+a(tm)+k(tm) – diam(C)
≤ α
which implies that
α –α( +L)z–T(tm)z
≤a(tm)L+k(tm) –
diam(C) for allm∈N. (.)
Lettingm→ ∞in (.), we obtain thatT(tm)z→z. It follows from the continuity of T(t) that
z= lim
m→∞T(t+tm)z=mlim→∞T(t)T(tm)z=T(t)z.
Therefore,z=T(t)zfor allt∈G.
The following result extends the celebrated convergence theorem of Browder [] and many results concerning Browder’s convergence theorem to a semigroup of uniformly continuous nearly uniformlyL-Lipschitzian asymptotically pseudo-contractive mappings.
Theorem . Let C be a nonempty closed convex bounded subset of a real Hilbert space H andF ={T(t) :t∈R+}a strongly continuous semigroup of uniformly continuous nearly uniformly L-Lipschitzian asymptotically pseudo-contractive mappings from C into itself. Let{bn}be a sequence in(, )and{tn}a sequence in(,∞)such that k(tn) – <bnfor all n∈N,limn→∞bn=limn→∞k(tbnn)–= andlimn→∞tn=∞.Then:
(a) There exists a sequence{yn}inCdefined by
yn=bnu+ ( –bn)T(tn)yn, n∈N. (.)
(b) IfF has property(A),thenFix(F)=∅and{yn}converges strongly toy*∈Fix(F)
such that
y*–u,y*–v≤ for allv∈Fix(F). (.)
Proof
(a) LetF ={T(t) :t∈R+}be a strongly continuous semigroup of asymptotically pseudo-contractive mappings with a net{k(t) :t∈(,∞)}. Setn:=k(tbnn)–. Notek(tn) – <bnfor alln∈N, it follows thatn< ≤k(tn) and hence ( –bn)k(tn) < for alln∈N. Then, for eachn∈N, the mappingGn:C→Cdefined by
Gny:=bnu+ ( –bn)T(tn)y, y∈C
is continuous and strongly pseudo-contractive. Indeed, forx,yinC, we have
Gnx–Gny,x–y = ( –bn)T(tn)x–T(tn)y,x–y
≤( –bn)k(tn)x–y.
Therefore, by Lemma ., there exists a sequence{yn}inCdescribed by (.).
bounded, we can assume that a subsequence{yni}of{yn}such thatynizfor somez∈C.
By Theorem ., we havez∈Fix(F). Forv∈Fix(F), we have
yn–T(tn)yn,yn–v
=yn–v+T(tn)v–T(tn)yn,yn–v
≥–k(tn) –
yn–v.
From (.), we have
yn–u,yn–v= ( –bn)T(tn)yn–u,yn–v
= ( –bn)T(tn)yn–yn+yn–u,yn–v,
so it follows that
yn–u,yn–v ≤ –bn bn
T(tn)yn–yn,yn–v
≤( –bn)nyn–v. (.)
Sincelimn→∞n= andCis bounded, it follows from (.) that
lim sup
n→∞
yn–u,yn–v ≤ for allv∈Fix(F). (.)
We claim that the set{yn}is sequentially compact. Forv∈Fix(F), we have
yn–v =
bn(u–v) + ( –bn)
T(tn)yn–v
,yn–v
≤bnu–v,yn–v+ ( –bn)k(tn)yn–v,
which implies that
yn–v≤ bn
– ( –bn)k(tn)u–v,yn–v. (.)
By the weak compactness ofC, there exists a weakly convergent subsequence{yni} ⊆ {yn}.
Suppose thatyniy*∈Casi→ ∞. Since{yn}is an approximating fixed point sequence
ofF, we infer from Theorem . thaty*∈Fix(F). In (.), interchangevandy*to obtain that
yni–y*≤ bni
– ( –bni)k(tni)
u–y*,yni–y*.
Sinceyniy*, we get thatyn
i→y
*. Hence the set{yn}is sequentially compact.
Next, we show that yn→y*. Suppose, for contradiction, that {ynj} is another
subse-quence of{yn}such thatynj→z
*=y*. It is easy to see thatz*∈Fix(F). Observe that
yn–u,yn–z*–y*–u,y*–z*≤yn–u,yn–z*–y*–u,yn–z*
≤yn–u–y*–uyn–z*
+y*–u,yn–y* for alln∈N.
Sinceyni→y *, we get
yni–u,yni–z
*→y*–u,y*–z*.
From (.), we obtain
y*–u,y*–z*≤. (.)
Similarly, we have
z*–u,z*–y*≤. (.)
Adding inequalities (.) and (.) yields
y*–z*,y*–z*≤,
a contradiction. In a similar way it can be shown that each cluster point of the sequence
{yn}is equal toy*. Therefore, the entire sequence{yn}converges strongly toy*. It is easy
to see, from (.), that the inequality (.) holds.
Theorem . Let C be a nonempty closed convex bounded subset of a real Hilbert space H andF ={T(t) :t∈R+}a strongly continuous semigroup of uniformly continuous nearly uniformly L-Lipschitzian asymptotically pseudo-contractive mappings from C into itself. Suppose thatF has property(A).ThenFix(F)=∅andFix(F)is a sunny nonexpansive retract of C.
Proof Assume thatF ={T(t) :t∈R+}is a semigroup of asymptotically pseudo-contrac-tive mappings fromCinto itself with a functionk(·) :R+→[,∞) withlim
t→∞k(t) = .
Without loss of generality, we may assume that{bn}in (, ) and{tn}in (,∞) such that k(tn) – <bnfor alln∈N,limn→∞bn=limn→∞k(tbnn)–= andlimn→∞tn=∞. Then, for an arbitrarily fixed elementu∈C, there exists a sequence{yn}inCdefined by (.). By Theorem .(b),Fix(F)=∅.
By Theorem .(b),{yn}converges strongly to an elementy*∈Fix(F) such that the in-equality (.) holds. Define a mappingQ:C→Fix(F) by
Qu= lim
n→∞yn, u∈C.
In view of (.), we have
Qu–u,Qu–v ≤ for allu∈Candv∈Fix(F).
6 Application
LetCbe a nonempty convex subset of a real Hilbert spaceHandDa nonempty subset ofC. For a nonlinear mappingF:C→H, the variational inequality problemVIP(F,C) overDis to find a pointx*∈Dsuch that
Fx*,v–x*≥ for allv∈D.
It is important to note that the theory of variational inequalities has played an important role in the study of many diverse disciplines, for example, partial differential equations, optimal control, optimization, mathematical programming, mechanics, finance,etc.; see, for example, [, ] and references therein.
We now turn our attention to dealing with the problem of the existence of solutions of
VIP(C,F) by sunny nonexpansive retractions.
Following Wong, Sahu and Yao [, Proposition .], one can show that the variational inequality problem VIP(C,F) withF =I–f is equivalent to the fixed point problem. Indeed,
Proposition . Let C be a nonempty convex subset of a smooth Banach space X and F ={T(t) :t∈R+}a strongly continuous semigroup of mappings from C into itself with
Fix(F)=∅.Let f :C→C be a mapping withF=I–f and let Q be the sunny nonexpan-sive retraction from C ontoFix(F).Then x*is a solution of variational inequality problem
VIP(C,F)overFix(F)if and only if x*is a fixed point of Qf.
The following result improves the so-called viscosity approximation method which was first introduced by Moudafi [] from nonexpansive mappings to a semigroup of pseudo-contractive mappings.
Theorem . Let C be a nonempty closed convex bounded subset of a real Hilbert space H, f :C→C a weakly contractive mapping with functionψ andF ={T(t) :t∈R+}a strongly continuous semigroup of uniformly continuous pseudo-contractive mappings from C into itself.Suppose thatF has property(A)andF is nearly nonexpansive with function a: [,∞)→[,∞).Let{bn}be a sequence in(, )and{tn}a sequence in(,∞)such that
limn→∞bn=limn→∞ab(tnn)= andlimn→∞tn=∞.Then,we have the following:
(a) The variational inequality problemVIP(C,I–f)overFix(F)has a unique solution inFix(F).
(b) There exists a sequence{yn}inCdefined by
yn=bnf(yn) + ( –bn)T(tn)yn, n∈N (.)
such that{yn}converges strongly to the unique solution of the variational inequality problemVIP(C,I–f).
Proof
an element ofFix(F). It follows from Proposition . thatx*is the unique solution of the variational inequality problemVIP(C,I–f) overFix(F).
(b) For eachn∈N, the mappingFn:C→Cdefined by
Fny:=bnfy+ ( –bn)T(tn)y, y∈C
is continuous and strongly pseudo-contractive. In fact, for allx,y∈Candn∈N, we have
Fnx–Fny,x–y=bnfx–fy,x–y+ ( –bn)T(tn)x–T(tn)y,x–y
≤bnfx–fyx–y+ ( –bn)x–y
≤bn
x–y–ψx–yx–y+ ( –bn)x–y
=x–y–bnψ
x–yx–y.
Hence eachFn is continuous bnψ-strongly pseudo-contractive. Therefore, by [, ], there exists a sequence{yn}inC described by (.). As in Theorem .(a), we may de-fine a sequence{zn}inCby
zn=bnfx*+ ( –bn)T(tn)zn for alln∈N.
By Theorem ., we have thatzn→x*=Qfx*. Observe that
yn–zn ≤bnfyn–fx*+ ( –bn)T(tn)yn–T(tn)zn
≤bn
fyn–fzn+fzn–fx*+ ( –bn)T(tn)yn–T(tn)zn
≤bnyn–zn–ψyn–zn+zn–x*–ψzn–x*
+ ( –bn)yn–zn+a(tn)
≤ yn–zn–bnψyn–zn+bnzn–x*+a(tn).
It follows that
ψyn–zn≤zn–x*+a(tn) bn
for alln∈N.
Thus,yn–zn →. Therefore,xn→x*=Qfx*.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
DRS designed of the study, performed the nonlinear analysis and also wrote the article. DO participated in the design of the study, carried out the materials and helped to check the manuscript. RA conceived of the study, participated in its design and also helped to draft the manuscript. All authors read and approved the final manuscript.
Author details
1Department of Mathematics, Banaras Hindu University, Varanasi, 221005, India.2Department of Mathematics, Texas A&M University, Kingsville, TX, USA.3School of Mathematics, Statistics and Applied Mathematics, National University of Ireland Galway, Galway, Ireland.
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doi:10.1186/1687-1812-2012-163
