R E S E A R C H
Open Access
Fixed point sets of set-valued mappings
Parunyou Chanthorn
1and Phichet Chaoha
1,2**Correspondence:
phichet.c@chula.ac.th
1Department of Mathematics and
Computer Science, Faculty of Science, Chulalongkorn University, Bangkok, 10330, Thailand
2Centre of Excellence in
Mathematics, CHE, Si Ayutthaya Rd., Bangkok, 10400, Thailand
Abstract
We present new results regarding fixed point sets of various set-valued mappings using the concept of fixed point iteration schemes and the newly defined concept of fixed point resolutions. In particular, we prove that the fixed point sets of certain nonexpansive set-valued mappings are contractible.
MSC: 47H04; 47H09
Keywords: fixed point set; set-valued mapping
1 Introduction
In , Bruck [] gave an intriguing result on the structure of fixed point sets by prov-ing that the fixed point set of a certain nonexpansive mappprov-ing is always a nonexpansive retract of its domain. It took almost thirty years before the result was extended to asymp-totically nonexpansive mappings in []. From the point of view of topological theory, such retraction results enable us to pass various topological properties (for example, connect-edness and contractibility) from the domain of the mapping onto its fixed point set. Re-cently in [], Chaoha introduced the notion of virtually nonexpansive mappings (which includes various nonexpansive-type mappings) on metric spaces and proved that the fixed point set of a virtually nonexpansive mapping is always a retract of a certain subset called the convergence set. Consequently, Chaoha and Atiponrat [] extended the notion of vir-tually nonexpansive mappings on metric spaces to virvir-tually stable mappings on Haus-dorff spaces and presented a retraction result similar to [] for regular spaces. Recently, Chaoha and Chanthorn [] presented the concept of fixed point iteration schemes that unifies well-known iteration processes (for example, Picard, Mann and Ishikawa iteration processes []) and showed that in regular spaces the fixed point set of a certain virtually stable scheme is a retract of its convergence set. Combined with numerous convergence results of iteration processes in the literature, the authors were able to derived some con-tractibility criteria for fixed point sets of mappings in various situations. For set-valued mappings, fewer results on the structure of fixed point sets have been explored. It was recently proved in [] that the fixed point set of a certain quasi-nonexpansive set-valued mapping on a CAT() space is always convex and hence contractible. In this work, we use the concept of virtually stable schemes to acquire retraction results for the fixed point sets of set-valued mappings in appropriate settings. Especially, combined with Nadler’s result in [], we obtain a new contractibility criterion for the fixed point set of a certain set-valuedα-contraction. Then we will construct a sequence of mappings that is not natu-rally a scheme, but surprisingly yields similar retraction and contractibility results for the
fixed point set of a certain nonexpansive set-valued mapping. This immediately calls for the introduction of fixed point resolutions generalizing fixed point iteration schemes at the end of this work.
The paper is organized as follows. In Section , we recall the backgrounds used through-out this work. In particular, we present some useful conditions involving set-valued map-pings as well as the new concept ofα-contractive schemes. In Section , we first con-struct, with the aid of Michael’s selection theorem, virtually stable schemes for certain set-valued mappings to obtain retraction results for fixed point sets. Then we construct anα-contractive scheme for a set-valued α-contraction to obtain a new contractibility criterion for its fixed point set. Finally, we construct a sequence of mappings that is not a fixed point iteration scheme, but still induces retraction and contractibility results for the fixed point set of a certain nonexpansive set-valued mapping.
2 Preliminaries
For a nonempty setX, we letP(X) be the power set ofXand X=P(X) –{∅}. Also, we
denote the fixed point sets of a single-valued mappingf :X→Xand a set-valued map-pingF:X→XbyFix(f) ={x∈X:x=f(x)}andFix(F) ={x∈X:x∈F(x)}, respectively.
Throughout this paper, we always assume that every mapping has nonempty fixed point sets.
For a metric space (X,d),a∈X, andA,B∈X, let • B(a;ε) ={y∈X:d(a,y) <ε},
• D(a;ε) ={y∈X:d(a,y)≤ε}, • S(a;ε) ={y∈X:d(a,y) =ε}, • η(A;ε) =a∈AB(a;ε),
• d(A,B) =inf{d(a,b) :a∈Aandb∈B}, • d(a,B) =d({a},B) =infb∈Bd(a,b), • h(A,B) =supa∈Ad(a,B), and
• CB(X) ={A∈X:Ais closed and bounded}.
TheHausdorff metric[] is the mappingH:CB(X)×CB(X)→Rdefined by
H(A,B) =infε:A⊆η(B;ε) andB⊆η(A;ε)
for eachA,B∈CB(X).
The following facts can be found in [, , ].
Lemma . Let x,y∈X and A,B⊆X.Then we obtain the following:
() d(x,A)≤d(x,y) +d(y,A).
() d(y,A)≤supb∈Bd(b,A) =h(B,A)for eachy∈B. () H(A,B) =max{h(A,B),h(B,A)}for eachA,B∈CB(X).
() For eacha∈Aandε> ,there isb∈Bsuch thatd(a,b)≤H(A,B) +ε.
For a set-valued mappingF:X→X, letP
F :X→P(X) be defined by
PF(x) =
y∈F(x) :d(x,y) =dx,F(x)
• the end-point condition[] ifF(p) ={p}for eachp∈Fix(F), • the proximal conditionifPF(x)=∅for eachx∈X,
• the Chebyshev conditionifPF(x)is a singleton for eachx∈X.
Notice that every single-valued mapping (considered as a set-valued mapping in the trivial way) satisfies all above conditions.
Remark . The definitions of the proximal condition and the Chebyshev condition are motivated by the proximal set and the Chebyshev set, respectively, in []. When a map-pingF:X→Xsatisfies the Chebyshev condition, we will identifyP
F(x) with its element,
i.e.,PFcan be considered as a mapping onXin this case. Note that every set-valued
map-ping with compact values always satisfies the proximal condition.
Recall thatf :X→Yis called aselectionof the set-valued mappingF:X→Yiff(x)∈
F(x) for eachx∈X.
Proposition . If F:X→Xsatisfies the end-point condition and f :X→X is a selection
of F,thenFix(F) =Fix(f).
Lemma . For a metric space(X,d),if F:X→Xsatisfies the proximal condition,then
we have the following:
() Fix(F) =Fix(PF).
() PFsatisfies the end-point condition.
() PF(x) =F(x)∩S(x;d(x,F(x)))for eachx∈X.
() IfFhas closed values,thenPF(x)is closed and bounded for eachx∈X.
Proof () and () are obvious. () is straightforward from the definition ofPF. () follows
directly from () and the fact thatF(x) is closed for eachx∈X.
We now recall some definitions of continuity for set-valued mappings (see [] for more details). For our purpose, letXandY be metric spaces (with no ambiguity, their metrics will be denoted by the same symbol ‘d’). A set-valued mappingF:X→Y is said to be
• upper semi-continuousatxif(xn)is a sequence inXconverging toxandUis an open subset ofYsuch thatF(x)⊆U, then there existsN∈Nsuch thatF(xn)⊆Ufor each
n≥N,
• lower semi-continuousatxif(xn)is a sequence inXconverging toxandy∈F(x), then there exists a sequence(yn)inYsuch thatyn∈F(xn)and(yn)converges toy,
• H-upper semi-continuousatxif for eachε> , there isδ> such thath(F(y),F(x)) <ε
for eachy∈B(x,δ),
• H-lower semi-continuousatxif for eachε> , there isδ> such thath(F(x),F(y)) <ε
for eachy∈B(x,δ),
• (H-) continuousatxifFis (H-) upper and (H-) lower semi-continuous atx.
We say that the set-valued mappingF:X→Yis(upper, H-upper, lower, H-lower semi-,
H-) continuousif it is (upper,H-upper, lower,H-lower semi-,H-) continuous at each point inX. Moreover, forα∈[, ), we say that the mappingF:X→CB(X) is
• anα-contractionifH(F(x),F(y))≤αd(x,y)for eachx,y∈X, • nonexpansiveifH(F(x),F(y))≤d(x,y)for eachx,y∈X,
• ∗-nonexpansive[] if for eachx,y∈Xandux∈PF(x), there isuy∈PF(y)such that
d(ux,uy)≤d(x,y).
It is not difficult to see that every set-valued α-contraction is nonexpansive and H-continuous, while every nonexpansive set-valued mapping is quasi-nonexpansive and continuous. Moreover, when F is single-valued, the above definitions ofα-contraction, nonexpansive mapping and quasi-nonexpansive set-valued mapping coincide with the usual definitions for single-valued mapping.
Lemma .([], Proposition .. and Proposition ..) Let X and Y be metric spaces and F:X→Y be a mapping.
() IfFisH-upper semi-continuous and has compact values,then it is upper semi-continuous.
() IfFisH-lower semi-continuous,then it is lower semi-continuous.
Lemma .([], Proposition .) Let X and Y be metric spaces and F:X→Ybe lower
semi-continuous.If a mapping G:X→Y satisfies G(x) =F(x)for each x∈X,then G is
lower semi-continuous.
Lemma .([], Proposition ..) Let X be a metric space and Y be a Banach space. Assume that F:X→Y and G:X→Y are lower semi-continuous.If G has open convex
values and F(x)∩G(x)=∅for each x∈X,then the mapping:X→Ydefined by
(x) =F(x)∩G(x)
for each x∈X is lower semi-continuous.
Lemma . Every selection of a quasi-nonexpansive set-valued mapping on a metric space satisfying the end-point condition is quasi-nonexpansive.
Proof It is straightforward from the definition.
Lemma . Let X be a metric space.If F:X→Xis∗-nonexpansive with closed values
and satisfies the proximal condition,then the mapping PF:X→CB(X)is nonexpansive.
Proof Letx,y∈Xandε> . For eachz∈PF(x), there isuy∈PF(y) such that
z∈Buy;d(x,y) +ε
⊆
u∈PF(y)
Bu;d(x,y) +ε=ηPF(y);d(x,y) +ε
.
Thus,PF(x)⊆η(PF(y);d(x,y)+ε). Similarly, we havePF(y)⊆η(PF(x);d(x,y)+ε). Therefore,
sinceεis arbitrary,H(PF(x),PF(y))≤d(x,y).
We will see later that the continuity of the mappingPF plays a crucial role in proving
the main result when the set-valued mappingFsatisfies the Chebyshev condition. Also in Theorem .. [], it is shown, by using the compactness of the unit disc of the Euclidean space, that every continuous mappingFfrom the Euclidean spaceRninto Rnsatisfying
the Chebyshev condition induces the continuity ofPF. Therefore, to generalize such a
Lemma . Let X be a compact metric space,p∈X and(xn)be a sequence in X.If every
convergent subsequence of(xn)converges to p,then so does(xn).
Proof Suppose that (xn) does not converge top. Then there exist a neighborhoodU of
pand a subsequence (xnk) of (xn) such thatxnk ∈/ Ufor eachk∈N. By the compactness of X, the sequence (xnk) has a convergent subsequence which is also a subsequence of (xn), say (xmk). Hence, (xmk) converges to a pointpinXby the assumption, and we get a
contradiction.
Theorem . Let X be a compact metric space.If F:X→Xis continuous,satisfies the
Chebyshev condition and has closed values,then PF:X→X is continuous.
Proof Let (xn) be a sequence inXconverging tox∈X. Suppose that (PF(xnk)) is a sub-sequence of (PF(xn)) converging to a pointp∈X. By the upper semi-continuity ofF, for
eachε> , there isN∈Nsuch that
PF(xnk)∈F(xnk)⊆η
F(x);ε
for eachk≥N. Hence,limk→∞d(PF(xnk),F(x)) = and dp,F(x)
≤lim sup
k→∞
dp,PF(xnk)
+dPF(xnk),F(x) = .
Thus,p∈F(x) sinceF(x) is closed. Also, for eacha∈F(xnk) andk∈N,
d(p,x)≤d
p,PF(xnk)
+dPF(xnk),x
≤dp,PF(xnk)
+d(x,xnk) +d
xnk,PF(xnk)
≤dp,PF(xnk)
+d(x,xnk) +d(xnk,a)
≤dp,PF(xnk)
+ d(x,xnk) +d
x,PF(x)
+dPF(x),a
.
SinceFis lower semi-continuous andPF(x)∈F(x), there is a sequence (ynk) inXsuch thatynk∈F(xnk) and (ynk) converges toPF(x). Consequently,
d(p,x)≤lim sup
k→∞
dp,PF(xnk)
+ d(x,xnk) +d
x,PF(x)
+dPF(x),ynk
=dx,PF(x)
=dx,F(x)
= inf
y∈F(x)
d(x,y)≤d(x,p).
It follows thatp∈F(x)∩S(x;d(x,F(x))) =PF(x) by Lemma .(), and hencep=
PF(x) by the Chebyshev condition onF. This proves that each convergent subsequence of
(PF(xn)) converges to the same pointPF(x). Then, by Lemma ., the sequence (PF(xn))
converges toPF(x). Therefore, the mappingPFis continuous.
Lemma . ([]) Let X be a complete and metrically convex metric space. If A is a nonempty closed subset of X,then for each x∈A and y∈/A,there exists z∈∂A(the bound-ary of A)such that
d(x,z) +d(z,y) =d(x,y).
Lemma . Let X be a complete and metrically convex metric space,x,y∈X and s,t∈ [,∞).Then we have H(D(x;s),D(x;t)) =|s–t|.Moreover,if X is a normed space(overR), then H(D(x;t),D(y;t)) =x–y.
Proof WLOG, we assume thats≤t. SinceD(x;s)⊆D(x;t), we have
hD(x;s),D(x;t)= sup
a∈D(x;s)
da,D(x;t)= ,
and henceH(D(x;t),D(x;s)) =h(D(x;t),D(x;s)) by Lemma .(). It is easy to see thatt–s≤ d(y,z) for eachy∈S(x;t) andz∈D(x;s). Also,
t–s≤ inf
z∈D(x;s)d(y,z) =h
y,D(x;s)≤hD(x;t),D(x;s).
On the other hand, for eachy∈S(x;t), there existsz∈S(x;s) such thatd(x,z) +d(z,y) = d(x,y) by Lemma .. Consequently,
hy,D(x;s)≤d(y,z)≤t–s
for eachy∈S(x;t). Thenh(D(x;t),D(x;s))≤t–s.
Now ifXis a normed space andx=y, leta=x+x–ty(x–y)∈S(x;t). Then
x–y+t=
+ t x–y
x–y
=(x–y) + t
x–y(x–y)
=a–y
≤ a–z+z–y ≤ a–z+t
for eachz∈D(y;t). Therefore,x–y ≤ a–zfor eachz∈D(y;t), and hence
x–y ≤ inf
z∈D(y;t)a–z=d
a,D(y;t)≤hD(x;t),D(y;t)≤HD(x;t),D(y;t).
On the other hand, leta∈D(x;t) andε> . Consider the pointb=a+y–x. Thena–b= x–yandb–y=a–x ≤t. Hence, we have
a∈Bb;x–y+ε, b∈D(y;t), and a∈ηD(y;t);x–y+ε.
Consequently,D(x;t)⊆η(D(y;t);x–y+ε). By a similar argument, we can show that D(y;t)⊆η(D(x;t);x–y+ε). It follows thatH(D(x;t),D(y;t))≤ x–y+ε, and sinceεis
The assumption thatXis a normed space in the previous lemma is necessary as shown in the following example.
Example . Consider [, ] with the standard metric. Letx= ,y= , andt= . Then H(D(x;t),D(y;t)) = but|x–y|= .
The following theorem follows directly from the classical Michael’s selection theorem ([], Theorem .).
Theorem . Every lower semi-continuous set-valued mapping from a metric space into a Banach space with closed and convex values admits a continuous selection.
In a (real) Banach space (X, · ), we let
CC(X) =A∈X:Ais convex and CCB(X) =CB(X)∩CC(X), whereCCB(X) is considered as a subspace of the metric space (CB(X),H).
Lemma . Let X be a closed subset of a Banach space E.Assume that f :E→E and ϕ:E→(,∞)are continuous mappings.
() The mapping:E→Edefined by(x) =D(f(x);ϕ(x))for eachx∈Eis lower
semi-continuous.
() The mapping:E→Edefined by(x) =B(f(x);ϕ(x))for eachx∈Eis lower
semi-continuous.
() IfF:X→CC(X)is a lower semi-continuous mapping satisfying(x)∩F(x)=∅for eachx∈X,then there is a continuous mappingg:X→Xsuch that
g(x)∈(x)∩F(x)for eachx∈X.
Proof () Letx∈Eandε> . Sincef andϕare continuous, there isδ> such that
f(x) –f(y)< ε
and ϕ(x) –ϕ(y)< ε
for eachy∈B(x;δ). Thus, by Lemma .,
H(x),(y)≤HDf(x);ϕ(x),Df(x);ϕ(y)
+HDf(x);ϕ(y),Df(y);ϕ(y)
=ϕ(x) –ϕ(y)+f(x) –f(y)<ε
for eachy∈B(x;δ). Therefore,isH-continuous and hence lower semi-continuous by Lemma .().
() It follows directly from () and Lemma ..
() By (), the restriction|X:X→Eis lower semi-continuous. For eachx∈X, since
(x) andF(x) are convex, so is(x)∩F(x). Following Lemma . and Lemma ., the mapping :X→CC(X)⊆CC(E) defined by
for eachx∈Xis lower semi-continuous. By Theorem ., there is a continuous mapping g:X→Esuch that
g(x)∈ (x)⊆(x)∩F(x) =(x)∩F(x)⊆X
for eachx∈X.
We now recall the notion of fixed point iteration schemes and virtually stable schemes defined in []. For a given (nonempty) Hausdorff spaceXand a sequenceS= (sn) of
map-pings onX, letFix(S) =∞n=Fix(sn) and C(S) ={x:limn→∞sn(x) exists}denotethe fixed
point set andthe convergence setofS, respectively. We always haveFix(S)⊆C(S), and
there is a natural mappingr: C(S)→Xdefined by
r(x) = lim
n→∞sn(x)
for eachx∈X. Notice thatrmay not be continuous in general, andFix(S)⊆r(C(S)). For
a sequence (fn) of mappings onX, we write
n
i=jfi for the compositionfn◦fn–◦ · · · ◦fj.
Notice that ifF= (fn) is a sequence of mappings andS= (
n
i=fi), thenFix(S) =Fix(F).
The sequenceS= (ni=fi) is said to be afixed point iteration scheme, or aschemein short,
ifFix(S) =r(C(S)).
Definition . A schemeS= (ni=fi) onXis said to be avirtually stable schemeif for
each common fixed pointpofSand each neighborhoodUofp, there exist a neighborhood V ofpand a subsequence (nk) of (n) such that
nk
i=jfi(V)⊆Ufor eachk∈Nandj≤nk.
Example . From [] it follows that the Picard iteration scheme (fn) for a continuous
quasi-nonexpansive mappingf on a metric space is virtually stable.
Recall that a subsetAof the spaceXis said to be aretractofXif there is a continuous r:X→Asuch thatr(a) =afor eacha∈A. Moreover,ris called aretraction. In addition, a (topological) spaceXis said to becontractibleif there are a pointx∈Xand a
contin-uous mappingf :X×[, ]→Xsuch thatf(x, ) =xandf(x, ) =xfor allx∈X. Note
that a convex subset of a Banach space and a retract of a contractible space are always contractible.
Theorem .([], Theorem .) Let X be a regular space.IfSis a virtually stable scheme having a continuous subsequence,i.e.,there is a subsequence(snk)ofSsuch that each snk is continuous,then r: C(S)→Fix(S)is a retraction.
Next we will introduce the concept of an α-contractive scheme on a metric spaceX which resembles anα-contraction in the sense that it is always virtually stable, and when Xis complete, its convergence set is the whole spaceX.
Definition . Forα∈[, ), the schemeS= (ni=fi) on a metric spaceXis said to be
() For each sequence(tn)∈ {(
n
i=fk+i) :k∈N},
dtn+(x),tn(x)
≤αdtn(x),tn–(x)
for eachn∈N, wheret(x) =xfor eachx∈X.
() The setF={fn:n∈N}is equicontinuous onFix(F).
Example . Iffis anα-contraction on a metric space, then the Picard iteration scheme (fn) isα-contractive.
Theorem . Everyα-contractive schemeSon a metric space X is virtually stable. More-over,if X is complete,thenC(S) =X.
Proof LetS= (ni=fi) be anα-contractive scheme. To show thatSis virtually stable, let
p∈Fix(S) =Fix(F),ε> , andε=min{ε(–α),ε}. SinceFis equicontinuous atp∈Fix(F),
there existsδ> such that
fi
B(p;δ)⊆Bfi(p);ε
for eachfi∈F. WLOG, we may assume thatδ≤ε.
Letn∈N,j≤n,T = (tn) = (
n
i=fj+i), andy∈B(p;δ). Sincep∈Fix(S)⊆Fix(T), we have
t(y) =fj(y)∈B(fj(p);ε) =B(p;ε), and
dy,t(y)
≤d(y,p) +dp,t(y)
<δ+ε≤ε≤
ε .
Consequently,
dt(y),tn–j+(y)
≤
n–j
i=
dti(y),ti–(y)
≤
∞
i=
αi–dt(y),y
≤ε
–α
≤ε . So, d p, n
i=j
fi(y)
≤dp,t(y)
+d
t(y),
n
i=j
fi(y)
=dp,t(y)
+dt(y),tn–j+(y)
<ε.
Hence, for eachp∈Fix(S) andε> , there isδ> such that
n
i=j
fi
B(p;δ)⊆B(p;ε)
for alln∈Nandj≤n. Therefore,Sis virtually stable.
Notice that if a sequence (xn) satisfiesd(xn+,xn)≤αd(xn,xn–) for someα∈[, ) and all
n∈N, then (xn) is a Cauchy sequence. This fact immediately implies that C(S) =Xwhen
Corollary . Let X be a metric space and f :X→X continuous onFix(f).If the(Picard) schemeS= (fn)satisfies
dfn+(x),fn(x)≤αdfn(x),fn–(x)
for someα∈[, )and for all x∈X,thenSisα-contractive and hence virtually stable.
Proof Observe thatF={f}is equicontinuous onFix(F) by the continuity off. Also, since (ni=fk+i) = (fn) =Sfor eachk∈N, thenSisα-contractive and hence virtually stable by
the previous theorem.
3 Main results
We now present some new retraction and contractibility results for fixed point sets of certain set-valued mappings.
Theorem . Let X be a closed subset of a Banach space E. If F :X→CCB(X) is a lower semi-continuous quasi-nonexpansive mapping satisfying the end-point condition, then there is a virtually stable schemeS such thatFix(S) =Fix(F),and henceFix(F)is a retract ofC(S).
Proof Note that CC(X)⊆ CC(E) because X is closed in E. Since F is lower semi-continuous, by Theorem ., F admits a continuous selection, say f : X →X. By Lemma .,f is quasi-nonexpansive, and the scheme S= (fn) is virtually stable by Ex-ample .. Also, by Proposition .,Fix(S) =Fix(f) =Fix(F). Corollary . Let X be a closed subset of a Banach space. If F:X→CC(X) is a ∗ -nonexpansive mapping satisfying the proximal condition,then there is a virtually stable schemeSsuch thatFix(S) =Fix(F),and henceFix(F)is a retract ofC(S).
Proof From Lemma .(), () and Lemma ., the mappingPF :X→CCB(X) is
nonex-pansive satisfying the end-point condition andFix(F) =Fix(PF). Moreover,PF is lower
semi-continuous by Lemma .(). Following Theorem ., the sequence S= (fn) is a
virtually stable scheme with Fix(S) =Fix(f) =Fix(F), wheref is a continuous selection
ofPF.
Theorem . Let X be a compact metric space,and F:X→CB(X)be a set-valued α-contraction satisfying the Chebyshev condition. Then the sequenceS= (sn)of mappings
on X,defined by
s(x) =x and sn(x) =PF
sn–(x)
= (PF)n(x)
for each n∈Nand x∈X,is anα-contractive scheme withFix(S) =Fix(F),and henceFix(F) is a retract of X.
Proof Letx∈Xandn≥. By Lemma .() and (),
dsn–(x),sn(x)
=dsn–(x),F◦sn–(x)
≤hF◦sn–(x),F◦sn–(x)
≤HF◦sn–(x),F◦sn–(x)
≤αdsn–(x),sn–(x)
.
It follows that (sn(x)) is a Cauchy sequence, and hence it converges to a pointp∈X.
Con-sider the following inequality:
dp,F(p)≤dp,F◦sn(x)
+HF◦sn(x),F(p)
≤dp,sn+(x)
+HF◦sn(x),F(p)
.
Since (sn(x)) converges topandFisH-continuous, (F◦sn(x)) converges toF(p).
Conse-quently,d(p,F(p)) = , and sinceF(p) is closed,p∈F(p). This impliesr(C(S))⊆Fix(F). By Lemma .(),
Fix(F) =Fix(PF) =Fix
(PF)n:n∈N
=Fix(S)⊆rC(S).
It follows that the sequenceSis a scheme satisfying the condition in Corollary . with
Fix(S) =Fix(F). Therefore, the schemeSisα-contractive withFix(S) =Fix(F), and hence
Fix(F) is a retract ofXby Lemma . and Theorem .. Remark . The sequence ((PF)n) in the proof of Theorem . is motivated by the
itera-tion process defined in Theorem []. Moreover, in [], Yanagi studied the convergence of an iteration sequence for a sequenceF= (Fn) of set-valued mappings from a complete
metric spaceXintoCB(X) satisfying the relation HFi(x),Fj(y)
≤ad(x,y) +bdx,Fi(x)
+dy,Fj(y)
+cdx,Fj(y)
+dy,Fi(x)
for allx,y∈Xandi,j∈N, wherea,b,c≥ anda+ (a+ )(b+c) < . In this case, we observe that the sequenceS= (ni=PFi) is anα-contractive scheme for someα∈[, ) if eachFi satisfies the Chebyshev condition, and hence, by a similar argument in Theorem ., the common fixed point setFix(F) is a retract ofXwhenXis compact.
Corollary . Let X be a compact contractible metric space.If F:X→CB(X)is an α-contraction satisfying the Chebyshev condition,thenFix(F)is contractible.
Proof It follows directly from Theorem . and the contractibility ofX.
The Chebyshev condition in Corollary . is necessary. For example, in Example [], the mappingF is a contraction that does not satisfy the Chebyshev condition (consider x=), andFix(F) is clearly not contractible. Moreover, the following example shows that the fixed point set of an α-contraction satisfying the Chebyshev condition may not be convex.
Example . Consider the subsetX= [–, ]×[, ] ofR. DefineF:X→CB(X) by
F(x,y) = [–, ]×x
Then, for each (x,y), (x,y), (x,y)∈X,
HF(x,y),F(x,y)
=H
[–, ]×x
, [–, ]×x
=x –x
≤
|x–x| ≤
(x,y) – (x,y),
andPF(x,y) = (x,|x|). Therefore, the mappingFis a -contraction satisfying the
Cheby-shev condition, andFix(F) ={(x,y) :y=|x|}which is not convex.
The following construction, motivated by the iteration process (.) in [], indicates that there is a sequence of mappings that is not naturally a scheme, but still gives rise to a retraction result as well as a contractibility criterion for the fixed point set of a certain nonexpansive set-valued mapping.
LetXbe a closed convex subset of a Banach space,F:X→CCB(X) be anH-continuous mapping, (αn)∞n= be a sequence in [a,b]⊆(, ), and (γn)∞n= be a sequence in (, +∞)
satisfyinglimn→∞γn= .
By Lemma .() and Theorem .,Fadmits a continuous selection, sayg:X→X.
We define mappingss:X→Xands:X→Xby
s(x) =x and s(x) = ( –α)s(x) +αg(x)
for eachx∈X, respectively.
To constructs, consider the mappingϕ:X→(,∞) defined by
ϕ(x) =HF◦s(x),F◦s(x)
+γ
for eachx∈X. Letx∈Xandε> . Sincesandsare continuous andFisH-continuous,
F◦sandF◦sareH-continuous. Then there isδ> such that
HF◦s(x),F◦s(y)
<ε
and H
F◦s(y),F◦s(x)
<ε
for eachy∈B(x,δ). Lety∈B(x;δ), we may assume thatϕ(x) –ϕ(y)≥, and hence
ϕ(x) –ϕ(y)≤HF◦s(x),F◦s(y)
+HF◦s(y),F◦s(y)
+HF◦s(y),F◦s(x)
–HF◦s(y),F◦s(y)
=HF◦s(x),F◦s(y)
+HF◦s(y),F◦s(x)
<ε.
Therefore,ϕis continuous. Also, sinceg(x)∈F(x) =F◦s(x), then by Lemma .(), there
existsa∈F◦s(x) such thata–g(x)<ϕ(x). Thus,
Bg(x);ϕ(x)
Following Lemma ., there exists a continuous mappingg:X→Xsuch that
g(x)∈D
g(x);ϕ(x)
∩F◦s(x)
for eachx∈X. That is,gis a continuous selection ofF◦ssatisfying
g(x) –g(x)≤H
F◦s(x),F◦s(x)
+γ
for eachx∈X. We now defines:X→Xby
s(x) = ( –α)s(x) +αg(x)
for eachx∈X.
Inductively, we obtain a sequenceS= (sn) of continuous mappings such that
sn(x) = ( –αn–)sn–(x) +αn–gn–(x) ()
for eachx∈Xandn≥, wheregn–:X→Xis a continuous selection ofF◦sn–satisfying
gn–(x) –gn–(x)≤H
F◦sn–(x),F◦sn–(x)
+γn–
for eachx∈X.
Recall thatr: C(S)→Xis given by
r(x) = lim
n→∞sn(x)
for eachx∈C(S).
Lemma . Suppose that F:X→CCB(X)satisfies the end-point condition.ThenFix(F) =
Fix(S) =r(C(S)).
Proof It suffices to show thatFix(F)⊆Fix(S) andr(C(S))⊆Fix(F). Supposep∈Fix(F). Sinceg(p)∈F(p) ={p}, we have
s(p) = ( –α)p+αg(p) =p.
Consequently, sinceg(p)∈F◦s(p) ={p},
s(p) = ( –α)s(p) +αg(p) =p.
Inductively, for eachn∈N,sn(p) =p. So,p∈n∈NFix(sn) =Fix(S).
Now, supposep∈r(C(S)). Thenp=r(x) for somex∈C(S).
Sincesn+(x) –sn(x) =αn(gn(x) –sn(x)), (αn) is a sequence in [a,b]⊆(, ) and (sn(x)) is a
convergent sequence, we have
lim sup
n→∞
gn(x) –sn(x)=lim sup n→∞
sn+(x) –sn(x)
αn
Thus,limn→∞gn(x) –sn(x)= . For eachε> , since (sn(x)) converges topandFis
H-continuous, there isN∈Nsuch thatH(F◦sn(x),F(p)) <εfor alln≥N. Consequently,
gn(x)∈F◦sn(x)⊆η(F(p);ε) for alln≥N. That is,limn→∞d(gn(x),F(p)) = . Since, by
Lemma .(),
dp,F(p)≤p–sn(x)+sn(x) –gn(x)+d
gn(x),F(p)
,
it follows that d(p,F(p)) = , and hence p∈ Fix(F) since F(p) is closed. Therefore,
r(C(S))⊆Fix(F).
Lemma . Suppose that F:X→CCB(X)is quasi-nonexpansive and satisfies the end-point condition.Then,for each x∈X,p∈Fix(F)and m≥n,
sm(x) –p≤sn(x) –p,
and r is continuous.
Proof For eachx∈X,p∈Fix(F) andn∈N, we have gn(x) –p≤h
F◦sn(x),{p}
≤HF◦sn(x),F(p)
≤sn(x) –p,
and hence
sn+(x) –p= ( –αn)sn(x) –p+αngn(x) –p
≤( –αn)sn(x) –p+αnsn(x) –p=sn(x) –p.
Inductively, we havesm(x) –p ≤ sn(x) –pwheneverm≥n.
For the continuity of r, let x∈C(S) and ε> . Then there existsN ∈N such that sN(x) –r(x)<ε, and by the continuity ofsN, there existsδ> such thatsN(x) –sN(y)<
ε
wheneverx–y<δ. Hence, for eachy∈C(S) withx–y<δ, there isM≥N such
thatsM(y) –r(y)<ε. Observe that sincer(x)∈Fix(F) and by the above property of (sn),
we have
r(x) –sM(y)≤r(x) –sN(y).
Then it follows that
r(x) –r(y)≤r(x) –sM(y)+sM(y) –r(y)
≤r(x) –sN(y)+sM(y) –r(y)
≤r(x) –sN(x)+sN(x) –sN(y)+sM(y) –r(y)
<ε,
which proves the continuity ofr.
Theorem . Let X be a closed convex subset of a Banach space.If F:X→CCB(X)is H-continuous,quasi-nonexpansive and satisfies the end-point condition,thenFix(F)is a retract ofC(S),where the sequenceSis defined as in().
Corollary . Let X be a compact convex subset of a Banach space and F:X→CCB(X) be a nonexpansive set-valued mapping satisfying the end-point condition.Then the fixed point set of F is contractible.
Proof By Theorem . [], we haveX= C(S), and hence the result follows immediately
from the previous theorem.
The example below shows that, under the assumption of the above corollary, the fixed point set may not be convex.
Example . Consider X= [–, ]×[, ]⊆R with the maximum norm (x,y)=
max{|x|,|y|}. Definef :X→Xby
f(x,y) =x,|x| for each (x,y)∈R.
Thenf is a nonexpansive mapping which is also a nonexpansive set-valued mapping sat-isfying the end-point condition. Observe thatFix(f) ={(x,|x|) :x∈R}is not convex.
Motivated by the last construction, we are ready to extend the notion of fixed point iteration schemes to the notion of fixed point resolutions as follows.
Definition . For a (nonempty) Hausdorff spaceX, a sequenceS= (sn) of mappings of
Xis said to be afixed point resolution, or aresolutionin short, ifr: C(S)→Xgiven by
r(x) = lim
n→∞sn(x)
is continuous andr(C(S))⊆Fix(S).
Hence, the fixed point set of a resolution is always a retract of its convergence set, and whenXis a regular space, every virtually stable scheme onXhaving a continuous subse-quence is always a resolution. The last theorem below summarizes all retraction results in this section in terms of resolutions.
Theorem . Suppose that X is a subset of a Banach space E and F:X→Y⊆Xis a
set-valued mapping.Then there exists a resolutionSwithFix(S) =Fix(F)if one the following conditions holds:
() Xis closed,Y=CCB(X)andFis a quasi-nonexpansive and lower semi-continuous mapping satisfying the end-point condition.
() Xis closed,Y=CC(X)andFis a∗-nonexpansive mapping satisfying the proximal condition.
() Xis compact,Y=CB(X)andFis a set-valuedα-contraction satisfying the Chebyshev condition.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
Acknowledgements
The authors would like to thank A Thamrongthanyalak for useful discussion during the manuscript preparation. The second author is (partially) supported by the Centre of Excellence in Mathematics, the Commission on Higher Education, Thailand.
Received: 23 December 2014 Accepted: 1 April 2015
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