R E S E A R C H
Open Access
On new generalizations of Smarzewski’s fixed
point theorem
Wei-Shih Du
**Correspondence:
wsdu@nknucc.nknu.edu.tw Department of Mathematics, National Kaohsiung Normal University, Kaohsiung, 824, Taiwan
Abstract
In this work, we prove some new generalizations of Smarzewski’s fixed point theorem and some new fixed point theorems which are original and quite different from the well-known results in the literature.
MSC: 46B20; 47H09; 54H25
Keywords: strictly convex Banach space; uniformly convex Banach space; uniformly convex in every direction (UCED);
λ
-firmly nonexpansive mapping; asymptotic radius; asymptotic center; (UAC)-property; reactive firmly nonexpansive mapping;Smarzewski’s fixed point theorem
1 Introduction and preliminaries
Let (X, · ) be a normed space with its zero vectorθ. We useB(X) andS(X) to denote respectively theclosed unit ballandunit spherecentered atθ with radius , that is,
B(X) =x∈X:x ≤
and
S(X) =x∈X:x= .
The notion of uniformly convex (UC, for short) Banach space was introduced by Clark-son [], and the research of geometric properties of the Banach space started from . The functionδX: [, ]→[, ], defined by
δX(ε) =inf
–x+y
:x,y∈B(X),x–y ≥ε
forε∈[, ],
is called themodulus of convexity of X. The normed spaceXis calleduniformly convex
ifδX(ε) > for everyε∈(, ]. It is well known that a uniformly convex Banach space is
reflexive and all Hilbert spaces and Banach spacespandLp( <p<∞) all are uniformly
convex; see,e.g., [–] for more details. The normed spaceXis said to bestrictly convex
if x+y< whenever x,y∈S(X) with x–y> . It is obvious that a Banach space
Xis strictly convex if and only ifδX() = . It is well known that the strict convexity of a
normed spaceXcan be characterized by the properties: for any nonzero vectorsx,y∈X, if
x+y=x+y, theny=cxfor some realc> . For eachε> , themodulus of convexity of X in the direction z∈S(X) is defined by
δX(ε,z) =inf
–x+y
:x,y∈B(X),x–y=λz,|λ| ≥ε
.
Clearly,δX(ε) =inf{δX(ε,z) :z∈S(X)}. The Banach spaceXis calleduniformly convex in
every direction(UCED, for short) if for anyz∈S(X) andε> ,δ(ε,z) > . Some character-izations ofUCEDBanach spaces were proved by Dayet al.[]; see also [].
Fact .(see,e.g., [–, ])
(a) EveryUCBanach space isUCED.
(b) EveryUCEDBanach space is strictly convex.
Let (X, · ) be a Banach space andKbe a given nonempty closed subset ofX. Forx∈X
and a bounded sequence{xn} ⊂X, define theasymptotic radius of{xn}at xas the number
rx,{xn}
=lim sup
n→∞ x–xn.
Theasymptotic radius of {xn}with respect to Kis defined by
rK,{xn}
=infrx,{xn}
:x∈K,
and the set
AK,{xn}
=x∈K:rx,{xn}
=rK,{xn}
is called theasymptotic center of {xn}with respect to K. For any bounded sequence{xn}
in X, r(x,{xn}) is easily seen to be a nonnegative, continuous and convex functional of
x∈X. Moreover, ifKis a nonempty convex subset ofX, thenA(K,{xn}) is also convex.
Fact .[, Lemma .] Every bounded sequence in a UCED Banach spaceXhas a unique asymptotic center with respect to any nonempty weakly compact convex subset ofX.
Definition . A normed space (X, · ) is said to have the (UAC)-property if every bounded sequence in Xhas a unique asymptotic center with respect to any nonempty weakly compact convex subset ofX.
According to Facts . and ., it is easy to know that Hilbert spaces,UCBanach spaces andUCEDBanach spaces all have the (UAC)-property.
LetCbe a nonempty subset of a normed space (X, · ) andT:C→Xbe a mapping.
T is said to benonexpansiveif
Tx–Ty ≤ x–y for allx,y∈C.
The concept of firmly nonexpansive mappings was introduced by Bruck []. Letλ∈(, ). The mappingT is said to beλ-firmly nonexpansive[] if
It is obvious that everyλ-firmly nonexpansive mapping is nonexpansive, but the converse is not true. The following example shows that there exists a nonexpansive mapping which is not aλ-firmly nonexpansive mapping for someλ∈(, ).
Example A LetX=Rwith the absolute-value norm| · |andC= [, ]. LetT:C→X
be defined byTx= –x. ThenTis a nonexpansive mapping. Forx= ,y= andλ=, we have
T(x) –T(y) = > = ( –λ)(x–y) +λT(x) –T(y) ,
which deduces thatTis not a -firmly nonexpansive mapping. In fact,Tis notλ-firmly nonexpansive for allλ∈(, ).
In , Browder [], Kirk [] and Göhde [] proved respectively that every nonex-pansive mapping T from a nonempty weakly compact convex subsetK of a uniformly convex Banach spaceXinto itself has a fixed point. It is know that the convexity of sets and mappings plays an important role in fixed point theory and the union of convex sets does not ensure that it is convex. In , Smarzewski [] proved the following interesting theorem.
Theorem . (Smarzewski []) Let X be a uniformly convex Banach space and C = n
k=Ckbe a finite union of nonempty weakly compact convex subsets Ckof X.If T:C→C
is aλ-firmly nonexpansive mapping for someλ∈(, ),then T has a fixed point in C.
Smarzewski’s fixed point theorem (i.e., Theorem .) is not always true ifT is merely nonexpansive, even inX=R.
Example B[] LetX=Rwith the absolute-value norm| · |andC= [–, –]∪[–, –]. Then the mappingT:C→Cdefined byTx= –xis nonexpansive and fixed point free.
In this paper, in order to promote Smarzewski’s fixed point theorem, we first introduce the concept of reactive firmly nonexpansive mappings.
Definition . LetCbe a nonempty subset of a normed space (X, · ) andϕ:C×C→
[, ) be a function. A mappingT:C→Xis said to bereactive firmly nonexpansive with respect toϕif
Tx–Ty ≤ –ϕ(x,y)(x–y) +ϕ(x,y)(Tx–Ty) for allx,y∈C.
Remark .
(a) Every reactive firmly nonexpansive mapping is nonexpansive.
(b) It is obvious that anyλ-firmly nonexpansive mapping is reactive firmly nonexpansive with respect to the functionϕdefined byϕ(s,t) =λfor all(s,t)∈C×C.
Example C LetX=Rwith the absolute-value norm| · |andC= [, ]∪[, ]. Let
T:C→Xbe defined by
Tx:=
Then the following statements hold. (a) Tis a nonexpansive mapping. (b) Tis not -firmly nonexpansive. (c) Defineϕ:C×C→[, )by
ϕ(s,t) :=
, ifs,t∈[, ],
, otherwise.
ThenT is reactive firmly nonexpansive with respect toϕ.
Proof Obviously, statement (a) holds. To see (b), letx= andy= . Since
T(x) –T(y) = > = –
(x–y) +
(Tx–Ty) ,
we show thatTis not -firmly nonexpansive. Finally, we prove (c). We consider the fol-lowing four possible cases to verify
|Tx–Ty| ≤ –ϕ(x,y)(x–y) +ϕ(x,y)(Tx–Ty) (.)
for allx,y∈C.
Case . Ifx,y∈[, ], then
T(x) –T(y) = |x–y|
and
–ϕ(x,y)(x–y) +ϕ(x,y)(Tx–Ty) =
–
(x–y) +
(Tx–Ty)
= |x–y|.
So (.) holds for allx,y∈[, ].
Case . Ifx∈[, ] andy∈[, ], then
T(x) –T(y) = y–
x
and
–ϕ(x,y)(x–y) +ϕ(x,y)(Tx–Ty) =
–
(x–y) +
(Tx–Ty)
= y–
x.
Since
y–
x – y–
x
=
x> ,
Case . Ifx∈[, ] andy∈[, ], then
T(x) –T(y) = x–
y
and
–ϕ(x,y)(x–y) +ϕ(x,y)(Tx–Ty) = x–
y.
Since
x–
y
–
x–
y
=
y> ,
we prove that (.) holds for allx∈[, ] andy∈[, ]. Case . Ifx,y∈[, ], then
T(x) –T(y) = |x–y|
and
–ϕ(x,y)(x–y) +ϕ(x,y)(Tx–Ty) =
–
(x–y) +
(Tx–Ty)
= |x–y|.
So (.) holds for allx,y∈[, ].
By Cases -, we verify that inequality (.) holds for allx,y∈C. HenceT is reactive firmly nonexpansive with respect toϕand (c) is proved.
In this paper, we establish some generalizations of Smarzewski’s fixed point theorem for reactive firmly nonexpansive mappings and some new fixed point theorems which are original and quite different from the well-known results in the literature.
2 New generalizations of Smarzewski’s fixed point theorem and applications to fixed point theory
In this section, we first establish a new fixed point theorem for reactive firmly nonex-pansive mappings which is generalized Smarzewski’s fixed point theorem. We assume <ϕ(s,t) < for all (s,t)∈C×Cin the following main theorem.
Theorem . Let X be a strictly convex Banach space with its zero vector θ and C= n
k=Ck be a finite union of nonempty weakly compact convex subsets Ck of X. Letϕ :
C×C→(, )be a function and T:C→C be a mapping.Suppose that
(a) Xhas the(UAC)-property,
(b) Tis reactive firmly nonexpansive with respect toϕ.
Then T has a fixed point in C.
by
r(x) :=rx,Tjz=lim sup
j→∞
x–Tjz.
For each ≤k≤n, let the number
r(Ck) :=r
Ck,
Tjz=infr(x) :x∈Ck
and the set
A(Ck) :=A
Ck,
Tjz=x∈Ck:r(x) =r(Ck)
be respectively the asymptotic radius and the asymptotic center of the sequence{Tjz}with
respect toCk. SinceXhas the (UAC)-property, letxk∈Ckbe the unique asymptotic center
of{Tjz}with respect toC
kfor ≤k≤n. So
r(xk) =r(Ck) =inf
r(x) :x∈Ck
for each ≤k≤n.
For anykandj, sinceTis nonexpansive, we have
Txk–Tjz≤xk–Tj–z,
which implies
r(Txk) =lim sup j→∞
Txk–Tjz≤lim sup j→∞
xk–Tj–z=r(xk) for allk. (.)
Let
m=minr(xk) : ≤k≤n
.
Clearly,m<∞. For arbitraryx∈C=nk=Ck,x∈Ckx for somekx, ≤kx≤n. Thus we have
r(x)≥r(Ck) =r(xk)≥m. (.)
Taking the infimum forxoverCyieldsr(C,{Tjz})≥m. Conversely, supposer(x
k) =mfor somek, ≤k≤n. Then
m=r(xk)≥inf
x∈Cr(x) =r
C,Tjz. (.)
By (.) and (.), we prove
rC,Tjz=m. (.)
Put
L=xk:r(xk) =m
It is obvious that(L)≤n, where(L) is the cardinal number ofL. We claim that foru∈C
withr(u) =mif and only ifu∈L. Indeed, it suffices to show that ifu∈Cwithr(u) =m, thenu∈L. Letu∈C=nk=Ck. Then there is someku, ≤ku≤n, such thatu∈Cku. So we obtain
r(u) =m≤r(xku)≤r(u), and hence
r(u) =m=r(xku).
By the uniqueness of asymptotic centerxku, we haveu=xku, which means thatu∈L. Next, we will proveTL⊂L(i.e.,LisT-invariant). For anyu∈L, sincer(u) =m, from (.) and (.) it follows that
m=r(u)≥r(Tu)≥inf
x∈Cr(x) =r
C,Tjz=m,
and hencer(Tu) =m. According to our claim, we getTu∈L, which completes the asser-tion. Now, we show thatT has a fixed point inL. Suppose to the contrary thatThas no fixed point inL, that is,Tu=ufor allu∈L. ThenTxk∈/Ckfor allxk∈L. Indeed, ifTxk∈Ck
for somexk∈L, since
m=r(xk)≥r(Txk)≥ inf x∈Ck
r(x) =r(xk) =m,
and by the uniqueness of asymptotic centerxk, we obtainTxk=xk. This means thatxkis a
fixed point forT, contradicting our assumption, henceTxk∈/Ckfor allxk∈L. ForTL⊂L,
there existsw∈Lsuch thatTαw=wfor some ≤α≤(L). Since
=w–Tw=Tαw–Tα+w≤Tα–w–Tαw≤ · · · ≤Tw–Tw≤ w–Tw,
it follows that
<ξ:=w–Tw=Tw–Tw=· · ·=Tα–w–Tαw=Tαw–Tα+w. (.)
SinceTis reactive firmly nonexpansive with respect toϕ, we have
ξ =Tiw–Ti+w
≤ –ϕTi–w,TiwTi–w–Tiw+ϕTi–w,TiwTiw–Ti+w
≤ –ϕTi–w,TiwTi–w–Tiw+ϕTi–w,TiwTiw–Ti+w
=ξ
for ≤i≤α, whereT=I(the identity mapping). Therefore, in view of strict convexity
of the norm, there isti> such that
Ti–w–Tiw=ti
From (.) and (.), we have
ξ=Ti–w–Tiw=tiTiw–Ti+w=tiξ,
which implies thatti= for alli. So, by (.) again, we get
θ=v:=w–Tw=Tw–Tw=· · ·=Tα–w–Tαw.
Since
Tαw=Tα–w–v=Tα–w–v–v=Tα–w– v=· · ·=w–αv,
we obtain
w=Tαw=w–αv,
which impliesv=θ, contradicting the fact thatv=θ. ThereforeTmust have a fixed point
inL⊂Cand this completes the proof.
The following results are immediate consequences of Theorem ..
Corollary . Let X be a strictly convex Banach space with its zero vector θ and C= n
k=Ckbe a finite union of nonempty weakly compact convex subsets Ckof X.Let T:C→
C be a mapping.Suppose that
(a) Xhas the(UAC)-property,
(b) Tisλ-firmly nonexpansive for someλ∈(, ).
Then T has a fixed point in C.
Corollary . Let C=nk=Ckbe a finite union of nonempty weakly compact convex
sub-sets Ckof a UCED Banach space X andϕ:C×C→(, )be a function.If T:C→C is T
is reactive firmly nonexpansive with respect toϕ,then T has a fixed point in C.
Corollary .[, Theorem .] Let C=nk=Ckbe a finite union of nonempty weakly
com-pact convex subsets Ckof a UCED Banach space X.If T:C→C isλ-firmly nonexpansive
for someλ∈(, ),then T has a fixed point in C.
Remark . Theorem . and Corollaries . and . all generalize and improve Smar-zewski’s fixed point theorem, [, Theorem .] and [, Theorems ., .].
In Theorem ., ifC=C=· · ·=Cn:=C, then we obtain the following new fixed point
theorem.
Theorem . Let X be a strictly convex Banach space with its zero vectorθ and C be a nonempty weakly compact convex subset of X.Letϕ:C×C→(, )be a function and T:C→C be a mapping.Suppose that
(a) Xhas the(UAC)-property,
(b) Tis reactive firmly nonexpansive with respect toϕ.
The following results are immediate from Theorem ..
Corollary . Let X be a strictly convex Banach space with its zero vectorθ and C be a nonempty weakly compact convex subset of X.Letϕ:C×C→(, ) be a function and T:C→C be a mapping.Suppose that
(a) Xhas the(UAC)-property,
(b) Tisλ-firmly nonexpansive for someλ∈(, ).
Then T has a fixed point in C.
Corollary . Let C be a nonempty weakly compact convex subset of a UCED Banach space X.Letϕ:C×C→(, )be a function.If T:C→C is reactive firmly nonexpansive with respect toϕ,then T has a fixed point in C.
Corollary . Let C be a nonempty weakly compact convex subset of a UCED Banach space X.If T:C→C isλ-firmly nonexpansive for someλ∈(, ),then T has a fixed point in C.
Definition . LetCbe a nonempty subset of a normed space (X, · ) andα,β∈(, ). A mappingT:C→Xis said to be (α,β)-firmly nonexpansive if
Tx–Ty ≤( –β)( –α)(x–y) +α(Tx–Ty)+β(Tx–Ty) for allx,y∈C.
Finally, by applying Theorem ., we give some new fixed point theorems for (α,β )-firmly nonexpansive mappings.
Theorem . Let X be a strictly convex Banach space with its zero vector θ and C= n
k=Ckbe a finite union of nonempty weakly compact convex subsets Ckof X.Let T:C→
C be a mapping andα,βbe positive real numbers satisfying < ( –α)( –β) < .Suppose that
(a) Xhas the(UAC)-property, (b) Tis(α,β)-firmly nonexpansive.
Then T has a fixed point in C.
Proof Sinceα,β > and < ( –α)( –β) < , we haveα+β–αβ∈(, ). Defineϕ:
C×C→(, ) by
ϕ(s,t) :=α+β–αβ for (s,t)∈C×C.
Due toT is (α,β)-firmly nonexpansive, we obtain
Tx–Ty ≤( –β)( –α)(x–y) +α(Tx–Ty)+β(Tx–Ty)
= –ϕ(x,y)(x–y) +ϕ(x,y)(Tx–Ty)
for all x,y∈C. So, T is reactive firmly nonexpansive with respect toϕ. Therefore the
Corollary . Let C be a nonempty weakly compact convex subset of a UCED Banach space X.Letα,βbe positive real numbers satisfying < ( –α)( –β) < .If T:C→C is
(α,β)-firmly nonexpansive,then T has a fixed point in C.
Corollary . Let C be a nonempty weakly compact convex subset of a uniformly convex Banach space X.Letα,βbe positive real numbers satisfying < ( –α)( –β) < .If T:
C→C is(α,β)-firmly nonexpansive,then T has a fixed point in C.
Competing interests
The author declares that he has no competing interests.
Acknowledgements
The author was supported by Grant No. MOST 103-2115-M-017-001 of the Ministry of Science and Technology of the Republic of China.
Received: 22 September 2014 Accepted: 25 November 2014 Published:12 Dec 2014
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