We prove that ifT has a fixed point and E has uniformly Gˆateaux differentiable norm, such that every nonempty closed bounded convex subset ofK has the fixed point property for nonex-pan
Trang 1PSEUDOCONTRACTIVE MAPPINGS
YISHENG SONG AND RUDONG CHEN
Received 20 March 2006; Revised 24 May 2006; Accepted 28 May 2006
LetK be a closed convex subset of a real Banach space E, T : K → K is continuous
pseudo-contractive mapping, and f : K → K is a fixed L-Lipschitzian strongly pseudocontractive
mapping For anyt ∈(0, 1), letx tbe the unique fixed point oft f + (1 − t)T We prove that
ifT has a fixed point and E has uniformly Gˆateaux differentiable norm, such that every nonempty closed bounded convex subset ofK has the fixed point property for
nonex-pansive self-mappings, then {x t } converges to a fixed point ofT as t approaches 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
LetE be a real Banach space and let J denote the normalized duality mapping from E
into 2E ∗ given byJ(x) = { f ∈ E ∗,x, f = x f ,x = f }, for allx ∈ E, where E ∗
denotes the dual space ofE and ·,·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 denote F(T) = {x ∈ E; Tx = x} In Banach spaceE, the fol-lowing result (the subdi fferential 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+ 2
y, j(x)
≤ x + y2≤ x2+ 2
y, j(x + y)
Recall that the norm ofE is said to be Gˆateaux di fferentiable (and E is said to be smooth),
if the limit
lim
t →0
x + t y − x
Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2006, Article ID 28950, Pages 1 9
DOI 10.1155/JIA/2006/28950
Trang 2exists for eachx, y on the unit sphere S(E) of E Moreover, if for each y in S(E) the
limit defined by (*) is uniformly attained forx in S(E), we say that the norm of E is uniformly Gˆateaux differentiable The norm of E is said to be Fr´echet differentiable, if for
eachx ∈ S(E) the limit (*) is attained uniformly fory ∈ S(E) The norm of E is said to
be uniformly Fr´echet di fferentiable (and E is said to be uniformly 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 mappingJ in smooth Banach space E is single-valued and
strong-weak∗continuous [5, Lemma 4.3.3]
(ii) IfE is 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 mapping J : 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 spaceE is reflexive and strictly convex [5, Theorems 4.1.6 and 4.1.2]
(v) IfK is a nonempty convex subset of a strictly convex Banach space E and T : K →
K is a nonexpansive mapping, then fixed point set F(T) of T is a closed convex
subset ofK [5, Theorem 4.5.3]
LetE be a real Banach space and let T be a mapping with domain D(T) and range R(T) in E T is called pseudocontractive (resp., strongly) if for any x, 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
IfI denotes the identity operator, then (1.2) implies that
(I − T)x −(I − T)y, j(x − y)
(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 space E, T : K → K a
nonexpansive mapping withF(T) ,f : K → K a contraction Then for any t ∈(0, 1), the mapping
is also contraction Letx tdenote the unique fixed point ofT t f In [7], Xu proved that as
t ↓0,{x t }converges to a fixed pointp of T that is the unique solution of the variational
inequality
(I − f )u, j(u − p)
Trang 3LetK be a nonempty closed convex subset of a Banach space E, T : K → K a continuous
pseudocontractive map such thatF(T) , and f : K → K a fixed Lipschitzian strongly
pseudocontractive map Then for anyt ∈(0, 1),T t f = t f + (1 − t)T : K → K is also a
con-tinuous strongly pseudocontractive map Letx tbe the unique fixed point ofT t f (see [1]), that is,
x t = t f
x t
In this paper, our purpose is to prove that{x t }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] Letμ be a continuous linear functional
onl ∞satisfyingμ =1= μ(1) Then we know that μ is a mean onNif and only if
inf
a n;n ∈ N≤ μ(a) ≤sup
for everya =(a1,a2, .) ∈ l ∞ According to time and circumstances, we useμ n(a n) instead
ofμ(a) A mean μ onNis called a Banach limit if
μ n
a n
= μ n
a n+1
(1.9) for everya =(a1,a2, .) ∈ l ∞ Furthermore, we know the following result [6, Lemma 1] and [5, Lemma 4.5.4]
Lemma 1.1 (see [6, Lemma 1]) Let C be a nonempty closed convex subset of a Banach space
E with a uniformly Gˆateaux differentiable norm Let {x n } be a bounded sequence of E and let μ be a mean onN letz ∈ C Then
μ nx n − z 2
=min
y ∈ C μ nx n − y 2
(1.10)
if and only if
μ n
y − z, j
where j is the duality mapping of E.
2 Main results
Lemma 2.1 Let E be a Banach space and let K be a nonempty closed convex subset of E Suppose that T : K → K is a pseudocontractive mapping such that for each fixed strongly pseudocontractive map f : K → K, the equation
has a solution x t for each t ∈ (0, 1) Suppose that u ∈ K is a fixed point of T Then
(i){x t } is bounded;
(ii)x t − f (x t),j(x t − u) ≤ 0.
Trang 4Proof (i) As u is a fixed point of T, we have
x t − u 2
= t
f
x t
− u + (1− t)
Tx t − u
,
x t − u
= t
f
x t
− u, j
x t − u + (1− t)
Tx t − u, j
x t − u
= t
f
x t
− f (u), j
x t − u +t
f (u) − u, j
x t − u
+ (1− t)
Tx t − Tu, j
x t − u
≤ t
f
x t
− f (u), j
x t − u +t
f (u) − u, j
x t − u + (1− t)x t − u 2
≤ βtx t − u 2
+t
f (u) − u, j
x t − u + (1− t)x t − u 2
.
(2.2) Hence
(1− β)x t − u 2
≤ f (u) − u, j
x t − u ≤f (u) − u·x t − u. (2.3)
By (2.3), we get
x t − u ≤ 1
so that{x t: 0< t < 1}is bounded
(ii) Asu is a fixed point of T, from x t = t f (x t) + (1− t)Tx t, we get
x t − f
x t
,
x t − u =(1− t)
Tx t − f
x t ,
x t − u
= −(1− t)
(I − T)x t −(I − T)u, j
x t − u
+ (1− t)
x t − f
x t ,
x t − u (using (1.3))
≤(1− t)
x t − f
x t
,
x t − u
(2.5)
Thereforex t − f (x t),j(x t − u) ≤0 The proof is complete
Theorem 2.2 Let E be a reflexive Banach space with a uniformly Gˆateaux differentiable norm Suppose K is a nonempty closed convex subset of E and T : K → K is a continuous pseudocontractive mapping Let f : K → K be a fixed Lipschitzian strongly pseudocontracitve map from K to K Every nonempty closed bounded convex subset of K has the fixed point property for nonexpansive self-mappings {x t } (for all t ∈ (0, 1)) is defined by (1.7 ) Then {x t: 0< t < 1} is bounded if and only if, as t → 0, x t converges strongly to a fixed point p of
T such that p is the unique solution in F(T) to the following variational inequality:
(I − f )p, j(p − u)
Trang 5Proof At first, byLemma 2.1(i), the sufficiency is obvious.
Secondly, we show the necessity Since{x t: 0< t < 1}is bounded, f are Lipschitzian
mappings, the sets{ f (x t) :t ∈(0, 1)}are bounded Byx t = t f (x t) + (1− t)Tx t, we have
Tx t = 1
1− t x t − t
1− t f
x t ,
Tx t ≤ 1
1− tx t+ t
1− t
f
Therefore, the sets{Tx t }are also bounded (usingt →0) This implies that
lim
t →0
x t − Tx t =lim
t →0tTx t − f
We first observe that the mapping 2I − T has a nonexpansive inverse, denoted by A =
(2I − T) −1, whereI denotes the identity operator, then F(T) = F(A) (see [1,5]) By [3, Theorem 6], we get thatA is a nonexpansive self-mapping on K Using A =(2I − T) −1,
we obtain
x t − Tx t =(2I − T)x t − x t = A −1x t − x t, x t = AA −1x t,
x t − Ax t = AA −1x t − Ax t ≤ A −1x t − x t = x t − Tx t. (2.9) Since limt →0x t − Tx t =0, we have
lim
t →0
We claim that the set{x t:t ∈(0, 1)}is relatively compact In fact, let{t n }be a sequence
in (0, 1) that converges to 0 (n → ∞), putx n:= x t n,
g(x) = μ nx n − x 2
whereμ is a Banach limit Define the set
K1=
x ∈ K : g(x) =inf
y ∈ K g(y)
SinceE is a reflexive Banach space, K1is a nonempty bounded closed convex subset ofE
(for more details, see [5]), and since
lim
for allx ∈ K1, we get
g(Ax) = μ nx n − Ax 2
= μ nAx n − Ax 2
≤ μ nx n − x 2
= g(x). (2.14) Hence,Ax ∈ K1, that is,K1is invariant underA Since every nonempty closed bounded
convex subset ofK has the fixed point property for nonexpansive self-mappings, there is
Trang 6a fixed pointp ∈ K1ofA By F(T) = F(A), p is also a fixed point of T UsingLemma 1.1,
we get
μ n
x − p, j
By (2.3), and takingx = f (p), we have
μ nx n − p 2
≤ 1
1− β μ n
f (p) − p, j
that is,
μ nx n − p 2
We have proved that for any sequence{x t n }in{x t:t ∈(0, 1)}, there exists a subsequence which is still denoted by{x t n }that converges to some fixed pointp of T To prove that the
entire net{x t }converges top, supposed that there exists another sequence {x s k } ⊂ {x t }
such thatx s k → q, as s k →0, then we also haveq ∈ F(T) (using lim t →0x t − Tx t =0) Next we show thatp = q and p is the unique solution in F(T) to the following variational
inequality:
(I − f )p, j(p − u)
Since the sets {x t − u} and{x t − f (x t)} 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), by x s k → q(s k →0), we have
(I − f )x s
k −(I − f )q −→0
s k −→0
,
x s k − f
x s k
,
x s k − u
−(I − f )q, j(q − u)
= (I − f )x s k −(I − f )q, j
x s k − u
+ (I − f )q, j
x s k − u
− j(q − u)
≤(I − f )x s
k −(I − f )qx s
k − u
+
(I − f )q, j
x s k − u
− j(q − u) 0 ass k −→0.
(2.19) Therefore, notingLemma 2.1(ii), for anyu ∈ F(T), we get
(I − f )q, j(q − u)
=lim
s k →0
x s k − f
x s k ,
x s k − u ≤0. (2.20) Similarly, we also can show that
(I − f )p, j(p − u)
=lim
n →∞
x t n − f
x t n
,
x t n − u ≤0. (2.21)
Trang 7Interchangep and u to obtain
(I − f )q, j(q − p)
Interchangeq and u to obtain
(I − f )p, j(p − q)
This implies that (using (1.4))
(1− β)p − q2≤(I − f )p −(I − f )q, j(p − q)
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{x t: 0< t < 1}is bounded, so that we can obtain the following corollary
Corollary 2.3 Let E be a reflexive Banach space with a uniformly Gˆateaux differentiable norm Suppose K is a nonempty closed convex subset of E with normal structure and T : K →
K is a continuous pseudocontractive mapping such that F(T) Let f : K → K be a fixed Lipschitzian strongly pseudocontracitve map from K to K Then, as t → 0, {x t }(t ∈(0, 1))
defined by ( 1.7 ) converges strongly to a fixed point p of T such that p is the unique solution
in F(T) to the following variational inequality:
(I − f )p, j(p − u)
Since every nonempty closed convex subset of a uniformly convex Banach space has normal structure [1,5], we can also obtain the following corollary
Corollary 2.4 Let E be a uniformly convex Banach space with a uniformly Gˆateaux dif-ferentiable norm Suppose K is a nonempty closed convex subset of E and T : K → K is a continuous pseudocontractive mapping such that F(T) Let f : K → K be a fixed Lip-schitzian strongly pseudocontracitve map from K to K Then, as t → 0, {x t } (t ∈(0, 1))
defined by ( 1.7 ) converges strongly to a fixed point p of T such that p is the unique solution
in F(T) to the following variational inequality:
(I − f )p, j(p − u)
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
Corollary 2.5 Let E be a uniformly smooth Banach space Suppose K is a nonempty closed convex subset of E and T : K → K is a continuous pseudocontractive mapping such that F(T) Let f : K → K be a fixed Lipschitzian strongly pseudocontracitve map from
Trang 8K to K Then, as t → 0, {x t }(t ∈ (0, 1)) defined by (1.7 ) converges strongly to a fixed point
p of T such that p is the unique solution in F(T) to the following variational inequality:
(I − f )p, j(p − u)
Recall the setA of M is a Chebyshev set, if for all x ∈ M, there exactly exists unique
elementy ∈ A such that d(x, y) = d(x, A), where (M, d) is a metric space and d(x, A) =
infy ∈ A d(x, y) Every nonempty closed convex subsets of a strictly convex and reflexive
Banach spaceE is a Chebyshev 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 Suppose K is a nonempty closed convex subset of E and T : K →
K is a continuous pseudocontractive mapping such that F(T) Let f : K → K be a fixed Lipschitzian strongly pseudocontracitve map from K to K Then, as t → 0, {x t }(t ∈(0, 1))
defined by ( 1.7 ) converges strongly to a fixed point p of T such that p is the unique solution
in F(T) to the following variational inequality:
(I − f )p, j(p − u)
Proof By F(T) andLemma 2.1(i), we have that{x t: 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 ∈ K g(y)
K1is a nonempty bounded closed convex subset ofE, and K1is invariant underA Now
we just need to show that the setK1contains a fixed point ofA Since F(T) = F(A) , letu be one of those Since every nonempty closed convex subsets of a strictly convex and
reflexive Banach spaceE is a Chebyshev set, there exists a unique p ∈ K1such that
u − p = inf
x ∈ K1
Next we show thatp = Ap = T p By u = Au and Ap ∈ K1,
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
This work is supported by the National Natural Science Foundation of China Grant no 10471033
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Yisheng Song: College of Mathematics and Information Science, Henan Normal University,
Xinxiang 453007, China
E-mail address:songyisheng123@yahoo.com.cn
Rudong Chen: Department of Mathematics, Tianjin Polytechnic University, Tianjin 300160, China
E-mail address:chenrd@tjpu.edu.cn
... and norm to weak∗uniformly continuous on bounded sets of a Banach spaceEwith uniformly Gˆateaux differentiable norm, for any u ∈ F(T), by x s... space with a uniformly Gˆateaux differentiable norm Suppose K is a nonempty closed convex subset of E with normal structure and T : K →
K is a continuous pseudocontractive. .. subset of a uniformly convex Banach space has normal structure [1,5], we can also obtain the following corollary
Corollary 2.4 Let E be a uniformly convex Banach space with a uniformly Gˆateaux