PSEUDOCONTRACTIVE MAPPINGS
YISHENG SONG AND RUDONG CHENReceived 20 March 2006; Revised 24 May 2006; Accepted 28 May 2006
LetKbe a closed convex subset of a real Banach spaceE,T:K→Kis continuous pseudo-contractive mapping, and f :K→Kis a fixedL-Lipschitzian strongly pseudocontractive mapping. For anyt∈(0, 1), letxtbe the unique fixed point oft f+ (1−t)T. We prove that ifThas a fixed point andEhas uniformly Gˆateaux differentiable norm, such that every nonempty closed bounded convex subset ofK has the fixed point property for nonex-pansive self-mappings, then {xt} converges to a fixed point ofT astapproaches to 0. The results presented extend and improve the corresponding results of Morales and Jung (2000) and Hong-Kun Xu (2004).
Copyright © 2006 Y. Song and R. Chen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1. Introduction and preliminaries
LetEbe a real Banach space and letJ denote the normalized duality mapping fromE into 2E∗ given byJ(x)= {f ∈E∗,x,f = xf,x = f}, for allx∈E, whereE∗ denotes the dual space ofEand·,·denotes the generalized duality pairing. IfE∗ is strictly convex, thenJ is single-valued. In the sequel, we will denote the single-valued duality mapping by j, and denoteF(T)= {x∈E; Tx=x}. In Banach spaceE, the fol-lowing result(the subdifferential inequality)is well known [1,5]. For allx,y∈E, for all
j(x+y)∈J(x+y), and for all j(x)∈J(x),
x2+ 2y,j(x)≤ x+y2≤ x2+ 2y,j(x+y). (1.1)
Recall that the norm ofEis said to beGˆateaux differentiable(andEis said to besmooth), if the limit
lim t→0
x+t y − x
t (*)
exists for eachx, y on the unit sphere S(E) ofE. Moreover, if for each y in S(E) the limit defined by (*) is uniformly attained forx in S(E), we say that the norm ofE is uniformly Gˆateaux differentiable. The norm ofEis said to beFr´echet differentiable, if for eachx∈S(E) the limit (*) is attained uniformly fory∈S(E). The norm ofEis said to beuniformly Fr´echet differentiable(andEis said to beuniformly smooth), the limit (*) is attained uniformly for (x,y)∈S(E)×S(E).
The following results which are found in [1,4,5] are well known.
(i) The duality mappingJin smooth Banach spaceEis single-valued and strong-weak∗continuous [5, Lemma 4.3.3].
(ii) IfEis a Banach space with a uniformly Gˆateaux differentiable norm, then the mappingJ:E→E∗is single-valued and norm-to-weak star uniformly continu-ous on bounded sets ofE[5, Theorem 4.3.6].
(iii) In uniformly smooth Banach spaceE, the mappingJ:E→E∗ is single-valued and norm-to-norm uniformly continuous on bounded sets of E[5, Theorem 4.3.6].
(iv) A uniformly convex Banach spaceEis reflexive and strictly convex [5, Theorems 4.1.6 and 4.1.2].
(v) IfKis a nonempty convex subset of a strictly convex Banach spaceEandT:K→ K is a nonexpansive mapping, then fixed point setF(T) ofTis a closed convex subset ofK[5, Theorem 4.5.3].
LetEbe a real Banach space and letT be a mapping with domainD(T) and range R(T) inE.Tis calledpseudocontractive(resp.,strongly) if for anyx,y∈D(T), there exists
j(x−y)∈J(x−y) such that
Tx−T y,j(x−y)≤ x−y2
resp.,Tx−T y,j(x−y)≤βx−y2for some 0< β <1. (1.2)
IfIdenotes the identity operator, then (1.2) implies that
(I−T)x−(I−T)y, j(x−y)≥0, (1.3)
(I−T)x−(I−T)y, j(x−y)≥(1−β)x−y2. (1.4)
LetK be a closed convex subset of a uniformly smooth Banach spaceE,T:K→Ka nonexpansive mapping withF(T)= ∅,f :K→Ka contraction. Then for anyt∈(0, 1), the mapping
Ttf :x−→t f(x) + (1−t)Tx (1.5)
is also contraction. Letxtdenote the unique fixed point ofTtf. In [7], Xu proved that as t↓0,{xt}converges to a fixed pointpofTthat is the unique solution of the variational inequality
LetKbe a nonempty closed convex subset of a Banach spaceE,T:K→Ka continuous pseudocontractive map such thatF(T)= ∅, and f :K→Ka fixed Lipschitzian strongly pseudocontractive map. Then for anyt∈(0, 1),Ttf =t f+ (1−t)T:K→Kis also a con-tinuous strongly pseudocontractive map. Letxtbe the unique fixed point ofTtf (see [1]), that is,
xt=t fxt+ (1−t)Txt. (1.7)
In this paper, our purpose is to prove that{xt}defined by (1.7) strongly converges to a fixed point ofT, which generalizes and improves several recent results. Particularly, it extends and improves [7, Theorems 3.1 and 4.1]. Letμbe a continuous linear functional onl∞satisfyingμ =1=μ(1). Then we know thatμis a mean onNif and only if
infan;n∈N≤μ(a)≤supan;n∈N (1.8)
for everya=(a1,a2,. . .)∈l∞. According to time and circumstances, we useμn(an) instead
ofμ(a). A meanμonNis called aBanach limitif
μn
an
=μn
an+1
(1.9)
for everya=(a1,a2,. . .)∈l∞. Furthermore, we know the following result [6, Lemma 1]
and [5, Lemma 4.5.4].
Lemma1.1 (see [6, Lemma 1]). LetCbe a nonempty closed convex subset of a Banach space Ewith a uniformly Gˆateaux differentiable norm. Let{xn}be a bounded sequence ofEand letμbe a mean onN. letz∈C. Then
μnxn−z
2
=min y∈Cμn
xn−y2
(1.10)
if and only if
μn y−z, j
xn−z
≤0 ∀y∈C, (1.11)
wherejis the duality mapping ofE.
2. Main results
Lemma2.1. LetEbe a Banach space and letK be a nonempty closed convex subset ofE. Suppose thatT:K →K is a pseudocontractive mapping such that for each fixed strongly pseudocontractive map f :K→K, the equation
x=t f(x) + (1−t)Tx (2.1)
has a solutionxtfor eacht∈(0, 1). Suppose thatu∈Kis a fixed point ofT. Then (i){xt}is bounded;
Proof. (i) Asuis a fixed point ofT, we have xt−u2
= tfxt
−u+ (1−t)Txt−u
,jxt−u
=t fxt
−u,jxt−u
+ (1−t) Txt−u,j
xt−u
=t fxt−f(u),jxt−u+tf(u)−u,jxt−u
+ (1−t) Txt−Tu,jxt−u
≤t fxt−f(u),jxt−u+t f(u)−u,jxt−u+ (1−t)xt−u2
≤βtxt−u2+t f(u)−u,jxt−u+ (1−t)xt−u2.
(2.2)
Hence
(1−β)xt−u2≤ f(u)−u,jxt−u≤f(u)−u·xt−u. (2.3)
By (2.3), we get
xt−u≤ 1
1−βf(u)−u, (2.4)
so that{xt: 0< t <1}is bounded.
(ii) Asuis a fixed point ofT, fromxt=t f(xt) + (1−t)Txt, we get
xt−fxt,jxt−u=(1−t) Txt−fxt,jxt−u
= −(1−t) (I−T)xt−(I−T)u,jxt−u
+ (1−t) xt−fxt,jxt−u (using (1.3))
≤(1−t) xt−fxt,jxt−u.
(2.5)
Thereforext−f(xt),j(xt−u) ≤0. The proof is complete. Theorem2.2. LetEbe a reflexive Banach space with a uniformly Gˆateaux differentiable norm. SupposeK is a nonempty closed convex subset of EandT:K→K is a continuous pseudocontractive mapping. Letf :K→Kbe a fixed Lipschitzian strongly pseudocontracitve map fromK toK. Every nonempty closed bounded convex subset ofK has the fixed point property for nonexpansive self-mappings.{xt}(for allt∈(0, 1))is defined by (1.7). Then {xt: 0< t <1}is bounded if and only if, ast→0,xtconverges strongly to a fixed point pof Tsuch thatpis the unique solution inF(T)to the following variational inequality:
Proof. At first, byLemma 2.1(i), the sufficiency is obvious.
Secondly, we show the necessity. Since{xt: 0< t <1}is bounded, f are Lipschitzian mappings, the sets{f(xt) :t∈(0, 1)}are bounded. Byxt=t f(xt) + (1−t)Txt, we have
Txt= 1 1−txt−
t 1−tf
xt
,
Txt≤ 1
1−txt+ t 1−t
fxt.
(2.7)
Therefore, the sets{Txt}are also bounded (usingt→0). This implies that
lim t→0
xt−Txt=lim t→0t
Txt−f
xt=0. (2.8)
We first observe that the mapping 2I−T has a nonexpansive inverse, denoted byA= (2I−T)−1, whereI denotes the identity operator, thenF(T)=F(A) (see [1,5]). By [3,
Theorem 6], we get thatAis a nonexpansive self-mapping onK. UsingA=(2I−T)−1,
we obtain
xt−Txt=(2I−T)xt−xt=A−1xt−xt, xt=AA−1xt, xt−Axt=AA−1x
t−Axt≤A−1xt−xt=xt−Txt.
(2.9)
Since limt→0xt−Txt =0, we have
lim t→0
xt−Axt=0. (2.10)
We claim that the set{xt:t∈(0, 1)}is relatively compact. In fact, let{tn}be a sequence in (0, 1) that converges to 0 (n→ ∞), putxn:=xtn,
g(x)=μnxn−x2 ∀x∈K, (2.11)
whereμis a Banach limit. Define the set
K1=
x∈K:g(x)=inf y∈Kg(y)
. (2.12)
SinceEis a reflexive Banach space,K1is a nonempty bounded closed convex subset ofE
(for more details, see [5]), and since
lim
n→∞xn−Axn=0, (2.13)
for allx∈K1, we get
g(Ax)=μnxn−Ax2=μnAxn−Ax2≤μnxn−x2=g(x). (2.14)
Hence,Ax∈K1, that is,K1is invariant underA. Since every nonempty closed bounded
a fixed pointp∈K1ofA. ByF(T)=F(A),pis also a fixed point ofT. UsingLemma 1.1,
we get
μn x−p,jxn−p≤0 ∀x∈K. (2.15)
By (2.3), and takingx= f(p), we have
μnxn−p2≤ 1
1−βμn f(p)−p,j
xn−p≤0, (2.16)
that is,
μnxn−p2=0. (2.17)
We have proved that for any sequence{xtn}in{xt:t∈(0, 1)}, there exists a subsequence which is still denoted by{xtn}that converges to some fixed pointpofT. To prove that the entire net{xt}converges top, supposed that there exists another sequence{xsk} ⊂ {xt} such thatxsk→q, assk→0, then we also haveq∈F(T) (using limt→0xt−Txt =0). Next we show thatp=qandpis the unique solution inF(T) to the following variational inequality:
(I−f)p,j(p−u)≤0 ∀u∈F(T). (2.18)
Since the sets {xt−u} and{xt−f(xt)} are bounded and the duality mapJ is single-valued and norm to weak∗uniformly continuous on bounded sets of a Banach spaceE with uniformly Gˆateaux differentiable norm, for any u∈F(T), byxsk→q(sk→0), we have
(I−f)xs
k−(I−f)q−→0
sk−→0,
xsk−f
xsk
,jxsk−u
−(I−f)q,j(q−u)
=(I−f)xsk−(I−f)q,j
xsk−u
+(I−f)q,jxsk−u
−j(q−u) ≤(I−f)xsk−(I−f)qxsk−u
+(I−f)q,jxsk−u
−j(q−u)−→0 assk−→0.
(2.19)
Therefore, notingLemma 2.1(ii), for anyu∈F(T), we get
(I−f)q,j(q−u)=lim sk→0 xsk−f
xsk
,jxsk−u
≤0. (2.20)
Similarly, we also can show that
(I−f)p,j(p−u)=lim n→∞ xtn−f
xtn
,jxtn−u
Interchangepanduto obtain
(I−f)q,j(q−p)≤0. (2.22)
Interchangeqanduto obtain
(I−f)p,j(p−q)≤0. (2.23)
This implies that (using (1.4))
(1−β)p−q2≤(I−f)p−(I−f)q,j(p−q)≤0. (2.24)
We must havep=q. The proof is complete.
Since every bounded nonempty closed convex subset with normal structure of the reflexive Banach space has the fixed point property for nonexpansive self-mappings [1,5], and ifF(T)= ∅, byLemma 2.1(i), we have{xt: 0< t <1}is bounded, so that we can obtain the following corollary.
Corollary2.3. LetEbe a reflexive Banach space with a uniformly Gˆateaux differentiable norm. SupposeKis a nonempty closed convex subset ofEwith normal structure andT:K→ Kis a continuous pseudocontractive mapping such thatF(T)= ∅. Let f :K→Kbe a fixed Lipschitzian strongly pseudocontracitve map fromKtoK. Then, ast→0,{xt}(t∈(0, 1)) defined by (1.7) converges strongly to a fixed pointpofTsuch thatpis the unique solution inF(T)to the following variational inequality:
(I−f)p,j(p−u)≤0 ∀u∈F(T). (2.25)
Since every nonempty closed convex subset of a uniformly convex Banach space has normal structure [1,5], we can also obtain the following corollary.
Corollary2.4. LetEbe a uniformly convex Banach space with a uniformly Gˆateaux dif-ferentiable norm. SupposeK is a nonempty closed convex subset of EandT:K→K is a continuous pseudocontractive mapping such thatF(T)= ∅. Let f :K→K be a fixed Lip-schitzian strongly pseudocontracitve map fromK to K. Then, ast→0,{xt} (t∈(0, 1)) defined by (1.7) converges strongly to a fixed pointpofTsuch thatpis the unique solution inF(T)to the following variational inequality:
(I−f)p,j(p−u)≤0 ∀u∈F(T). (2.26)
Every bounded nonempty closed convex subset of uniformly smooth Banach space has normal structure [2, Lemma 8], and every uniformly smooth Banach space is a reflexive Banach space with uniformly Gˆateaux differentiable norm. So that we can also obtain the following corollary.
KtoK. Then, ast→0,{xt}(t∈(0, 1))defined by (1.7) converges strongly to a fixed point pofTsuch thatpis the unique solution inF(T)to the following variational inequality:
(I−f)p,j(p−u)≤0 ∀u∈F(T). (2.27)
Recall the setAofM is aChebyshev set, if for allx∈M, there exactly exists unique elementy∈Asuch thatd(x,y)=d(x,A), where (M,d) is a metric space andd(x,A)= infy∈Ad(x,y). Every nonempty closed convex subsets of a strictly convex and reflexive Banach spaceEis aChebyshev set[4, Corollary 5.1.19].
Theorem 2.6. Let E be a reflexive and strictly convex Banach space with a uniformly Gˆateaux differentiable norm. SupposeKis a nonempty closed convex subset ofEandT:K→ Kis a continuous pseudocontractive mapping such thatF(T)= ∅. Let f :K→Kbe a fixed Lipschitzian strongly pseudocontracitve map fromKtoK. Then, ast→0,{xt}(t∈(0, 1)) defined by (1.7) converges strongly to a fixed pointpofTsuch thatpis the unique solution inF(T)to the following variational inequality:
(I−f)p, j(p−u)≤0 ∀u∈F(T). (2.28)
Proof. ByF(T)= ∅andLemma 2.1(i), we have that{xt: 0< t <1}is bounded. Using the same proof for the necessity ofTheorem 2.2, we can find
K1=
x∈K:g(x)=inf y∈Kg(y)
, (2.29)
K1is a nonempty bounded closed convex subset ofE, andK1is invariant underA. Now
we just need to show that the setK1contains a fixed point ofA. SinceF(T)=F(A)= ∅,
letube one of those. Since every nonempty closed convex subsets of a strictly convex and reflexive Banach spaceEis aChebyshev set, there exists a uniquep∈K1such that
u−p = inf
x∈K1u−x. (2.30)
Next we show thatp=Ap=T p. Byu=AuandAp∈K1,
u−Ap = Au−Ap ≤ u−p. (2.31)
Hencep=Ap. The rest of the proof follows fromTheorem 2.2. The proof is complete. Remark 2.7. We remark thatTheorem 2.6appears to be independent ofTheorem 2.2. On the one hand, it is easy to find examples of spaces which satisfy the fixed point property for nonexpansive self-mappings, which are not strictly convex. On the other hand, it appears to be unknown whether a reflexive and strictly convex Banach space satisfies the fixed point property for nonexpansive self-mappings.
Acknowledgment
References
[1] S.-S. Chang, Y. J. Cho, and H. Zhou,Iterative Methods for Nonlinear Operator Equations in Ba-nach Spaces, Nova Science, New York, 2002.
[2] J. Gao,Modulus of convexity in Banach spaces, Applied Mathematics Letters16(2003), no. 3, 273–278.
[3] R. H. Martin Jr.,Differential equations on closed subsets of a Banach space, Transactions of the American Mathematical Society179(1973), 399–414.
[4] R. E. Megginson,An Introduction to Banach Space Theory, Graduate Texts in Mathematics, vol. 183, Springer, New York, 1998.
[5] W. Takahashi,Nonlinear Functional Analysis. Fixed Point Theory and Its Applications, Yokohama Publishers, Yokohama, 2000.
[6] W. Takahashi and Y. Ueda,On Reich’s strong convergence theorems for resolvents of accretive oper-ators, Journal of Mathematical Analysis and Applications104(1984), no. 2, 546–553.
[7] H.-K. Xu,Viscosity approximation methods for nonexpansive mappings, Journal of Mathematical Analysis and Applications298(2004), no. 1, 279–291.
Yisheng Song: College of Mathematics and Information Science, Henan Normal University, Xinxiang 453007, China
E-mail address:[email protected]
Rudong Chen: Department of Mathematics, Tianjin Polytechnic University, Tianjin 300160, China
