POINTS OF NONEXPANSIVE MAPPINGS WITHOUTCOMMUTATIVITY ASSUMPTION YONGHONG YAO, RUDONG CHEN, AND HAIYUN ZHOU Received 11 June 2006; Revised 27 July 2006; Accepted 2 August 2006 We introduc
Trang 1POINTS OF NONEXPANSIVE MAPPINGS WITHOUT
COMMUTATIVITY ASSUMPTION
YONGHONG YAO, RUDONG CHEN, AND HAIYUN ZHOU
Received 11 June 2006; Revised 27 July 2006; Accepted 2 August 2006
We introduce an iteration scheme for nonexpansive mappings in a Hilbert space and prove that the iteration converges strongly to common fixed points of the mappings with-out commutativity assumption
Copyright © 2006 Yonghong Yao et al 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
LetH be a real Hilbert space, and let C be a nonempty closed convex subset of H A
mappingT of C into itself is said to be nonexpansive if
for eachx, y ∈ C For a mapping T of C into itself, we denote by F(T) the set of fixed
points ofT We also denote byNandR +the set of positive integers and nonnegative real numbers, respectively
Baillon [1] proved the first nonlinear ergodic theorem LetC be a nonempty bounded
convex closed subset of a Hilbert spaceH and let T be a nonexpansive mapping of C into
itself Then, for an arbitraryx ∈ C, {(1/(n + 1))n i =0T i x } ∞
n =0converges weakly to a fixed point ofT Wittmann [9] studied the following iteration scheme, which has first been considered by Halpern [3]:
x0= x ∈ C,
x n+1 = α n+1 x +1− α n+1
where a sequence{ α n }in [0,1] is chosen so that limn →∞ α n =0,∞
n =1α n = ∞, and∞
n =1
| α n+1 − α n | < ∞; see also Reich [7] Wittmann proved that for anyx ∈ C, the sequence
Hindawi Publishing Corporation
Fixed Point Theory and Applications
Volume 2006, Article ID 89470, Pages 1 8
DOI 10.1155/FPTA/2006/89470
Trang 2{ x n }defined by (1.2) converges strongly to the unique elementPx ∈ F(T), where P is the
metric projection ofH onto F(T).
Recall that two mappingsS and T of H into itself are called commutative if
for allx, y ∈ H.
Recently, Shimizu and Takahashi [8] have first considered an iteration scheme for two commutative nonexpansive mappingsS and T and proved that the iterations converge
strongly to a common fixed point ofS and T They obtained the following result.
Theorem 1.1 (see [8]) Let H be a Hilbert space, and let C be a nonempty closed convex subset of H Let S and T be nonexpansive mappings of C into itself such that ST = TS and F(S)F(T) is nonempty Suppose that { α n } ∞
n =0⊆ [0, 1] satisfies
(i) limn →∞ α n = 0, and
(ii)∞
n =0α n = ∞
Then, for an arbitrary x ∈ C, the sequence { x n } ∞
n =0generated by x0= x and
x n+1 = α n x +1− α n 2
(n + 1)(n + 2)
n
k =0
i+j = k
S i T j x n, n ≥0, (1.4)
converges strongly to a common fixed point Px of S and T, where P is the metric projection
of H onto F(S)F(T).
Remark 1.2 At this point, we note that the authors have imposed the commutativity on
the mappingsS and T But there are many mappings, that do not satisfy ST = TS For
example, ifX =[−1/2,1/2], and S and T of X into itself are defined by
thenST =sin2x, whereas TS =sinx2
In this paper, we deal with the strong convergence to common fixed points of two nonexpansive mappings in a Hilbert space We consider an iteration scheme for non-expansive mappings without commutativity assumption and prove that the iterations converge strongly to a common fixed point of the mappingsT i,i =1, 2
2 Preliminaries
LetC be a closed convex subset of a Hilbert space H and let S and T be nonexpansive
mappings ofC into itself Then we consider the iteration scheme
x0= x ∈ C,
x n+1 = α n x +1− α n 2
(n + 1)(n + 2)
n
k =0
i+j = k
S i T j y n,
y n = β n x n+
1− β n 2 (n + 1)(n + 2)
n
k =0
i+j = k T i S j x n, n ≥0,
(2.1)
Trang 3where{ α n }and{ β n }are two sequences in [0,1] We know that a Hilbert spaceH satisfies
Opial’s condition [6], that is, if a sequence{ x n }inH converges weakly to an element y of
H and y = z, then
lim inf
n →∞ x n − y< liminf
n →∞ x n − z. (2.2)
In what follows, we will useP Cto denote the metric projection fromH onto C; that is,
for eachx ∈ H, P Cis the only point inC with the property
x − P C x =min
It is known thatP Cis nonexpansive and characterized by the following inequality: given
x ∈ H and v ∈ H, then v = P C x if and only if
Now, we introduce several lemmas for our main result in this paper The first lemma can be found in [4,5,10]
Lemma 2.1 Assume { a n } is a sequence of nonnegative real numbers such that
a n+1 ≤1− γ n
where { γ n } is a sequence in (0, 1) and { δ n } is a sequence such that
(1)∞
n =1γ n = ∞ ;
(2) lim supn →∞ δ n /γ n ≤ 0 or∞
n =1| δ n | < ∞ Then lim n →∞ a n = 0.
Lemma 2.2 Let C be a nonempty bounded closed convex subset of a Hilbert H, and let S,T
be nonexpansive mappings of C into itself For x ∈ C and n ∈ N ∪ {0} , put
G n(x) = 2
(n + 1)(n + 2)
n
k =0
i+j = k S i T j x,
G n(x) = 2
(n + 1)(n + 2)
n
k =0
i+j = k T i S j x.
(2.6)
Then
lim
n → ∞sup
x ∈ C
G n(x) − SG n(x) =0,
lim
n → ∞sup
x ∈ C
G n(x) − TG n(x) =0. (2.7)
Trang 4Proof We first prove lim n →∞supx ∈ C G n(x) − SG n(x) =0.
By an idea in [2], for{ x i,j } ∞
i,j =0,{ x i,j } ∞
i,j =0⊆ C and z n =(1/l n)n
k =0
i+j = k x i,j,z n =
(1/l n)n
k =0
i+j = k x i,j ∈ C, with l n =(n + 1)(n + 2)/2, we have
z n − v 2
= 1
l n
n
k =0
i+j = k
x i,j − v 2
− 1
l n
n
k =0
i+j = k
x i,j − z n 2
(2.8)
for eachv ∈ H For x ∈ C, put x i,j = S i T j x,x i,j = T i S j x and v = Sz n,v = Tz n Then, we have
G n(x) − SG n(x) 2
= l1n
n
k =0
i+j = k
S i T j x − Sz n 2
− l1n
n
k =0
i+j = k
S i T j x − z n 2
= l1n
n
k =0
T k x − Sz n 2
+ 1
l n
n
k =1
i+j = k,i ≥1
S i T j x − Sz n 2
− 1
l n
n
k =0
i+j = k
S i T j x − z n 2
≤ 1
l n
n
k =0
T k x − Sz n 2
+ 1
l n
n
k =1
i+j = k,i ≥1
S i −1T j x − z n 2
− 1
l n
n
k =0
i+j = k
S i T j x − z n 2
= l1n
n
k =0
T k x − Sz n 2
+ 1
l n
n−1
k =0
i+j = k
S i T j x − z n 2
− l1n
n
k =0
i+j = k
S i T j x − z n 2
= l1n
n
k =0
T k x − Sz n 2
− l1n
i+j = n
S i T j x − z n 2
≤ 1
l n
n
k =0
T k x − Sz n 2
n + 2
diam(C) 2
,
(2.9)
where diam(C) is the diameter of C So, we have, for each n ∈ N ∪ {0},
sup
x ∈ C
G n(x) − SG n(x) 2
n + 2
diam(C) 2
and hence
lim
n → ∞sup
x ∈ C
G n(x) − SG n(x) =0. (2.11)
Trang 5Similarly, we have
lim
n → ∞sup
x ∈ C
G n(x) − TG n(x) =0. (2.12)
3 Convergence theorem
Now we can prove a strong convergence theorem in a Hilbert space
Theorem 3.1 Let H be a Hilbert space, and let C be a nonempty closed convex subset of H Let S and T be nonexpansive mappings of C into itself such that F(S)F(T) is nonempty Suppose that { α n } ∞
n =0and { β n } ∞
n =1are two sequences in [0, 1] satisfying the following condi-tions:
(i) limn →∞ α n = 0, and
(ii)∞
n =0α n = ∞
For an arbitrary x ∈ C, the sequence { x n } ∞
n =0is generated by x0= x and
x n+1 = α n x +1− α n 2
(n + 1)(n + 2)
n
k =0
i+j = k
S i T j y n,
y n = β n x n+
1− β n 2 (n + 1)(n + 2)
n
k =0
i+j = k
T i S j x n, n ≥0.
(3.1)
Let
(n + 1)(n + 2)
n
k =0
i+j = k
S i T j y n, z n = 2
(n + 1)(n + 2)
n
k =0
i+j = k
T i S j x n, (3.2)
for each n ∈ N ∪ {0} If there exist subsequences { z n i } ∞
i =0of { z n } ∞
n =0and { z n j } ∞
j =0of { z n } ∞
n =0, respectively, which converge weakly to some common point z in some bounded subset D of C, then the sequence { x n } ∞
n =0defined by ( 3.1 ) converges strongly to P F(S) ∩ F(T) x.
Proof Let x ∈ C and w ∈ F(S)F(T) Putting r = x − w , then the set
D =y ∈ H : y − w ≤ r ∩ C (3.3)
is a nonempty bounded closed convex subset ofC which is S- and T-invariant and
con-tainsx0= x So we may assume, without loss of generality, that S and T are the mappings
ofD into itself Since P is the metric projection of H onto F(S) ∩ F(T), we have
for eachy ∈ F(S)F(T).
Trang 6From (3.4), we have
lim sup
n →∞ z n − Px,x − Px ≤0, lim sup
n →∞ z n − Px,x − Px ≤0. (3.5)
In fact, assume that, there exist two positive real numbersr0andr1such that
lim sup
n →∞ z n − Px,x − Px > r0, lim sup
n →∞ z n − Px,x − Px > r1. (3.6) Since { z n } ∞
n =0 and { z n } ∞
n =0⊆ D are bounded, from (3.6), there exist subsequences
{ z n i } ∞
i =0of{ z n } ∞
n =0and{ z n j } ∞
j =0of{ z n } ∞
n =0, respectively, such that
lim sup
n →∞ z n − Px,x − Px =lim
i →∞ z n i − Px,x − Px > r0, lim sup
n →∞ z n − Px,x − Px =lim
j →∞ z n j − Px,x − Px > r1. (3.7)
By the assumption, we know that{ z n i } ∞
i =0and{ z n j } ∞
j =0converge weakly to some com-mon pointz ∈ D Thus fromLemma 2.2and Opial’s condition, we havez ∈ F(S)F(T).
In fact, ifz = Sz, we have
lim inf
i →∞ z n
i − z< liminf
i →∞ z n
≤lim inf
i →∞ z n i − Sz n i+Sz n
≤lim inf
i →∞ z n
(3.8)
This is a contradiction Therefore, we havez = Sz.
Similarly, we havez = Tz So, we have
On the other hand, since{ z n i }converges weakly toz, we obtain
This is a contradiction Hence, we have
lim sup
n →∞ z n − Px,x − Px ≤0, lim sup
n →∞ z n − Px,x − Px ≤0. (3.11)
Trang 7z n − Px ≤ 2
(n + 1)(n + 2)
n
k =0
i+j = k
T i S j x n − Px 2
≤
2 (n + 1)(n + 2)
n
k =0
i+j = k
x n − Px 2=x n − Px 2
,
y n − Px 2
=β n x n+
1− β n
z n − Px 2
=β n
x n − Px+
1− β n
z n − Px 2
= β2
nx n − Px 2
+ 2β n
1− β n
x n − Px,z n − Px+
1− β n 2 z n − Px 2
≤ β2
nx n − Px 2
+ 2β n
1− β n x n − Px 2
+z n − Px 2
2 +
1− β n 2 z n − Px 2
≤x n − Px 2
.
(3.12) Then, we have
x n+1 − Px 2
=α n x +
1− α n
z n − Px 2
= α2
n x − Px 2+
1− α n 2 z n − Px 2
+ 2α n
1− α n z n − Px,x − Px
≤1− α n 2
2 (n + 1)(n + 2)
n
k =0
i+j = k
S i T j y n − Px 2
+α2
n x − Px 2+ 2α n
1− α n z n − Px,x − Px
≤1− α n 2
2 (n + 1)(n + 2)
n
k =0
i+j = k
y n − Px 2
+α2
n x − Px 2+ 2α n
1− α n z n − Px,x − Px
=1− α n 2 y n − Px 2
+α2
n x − Px 2+ 2α n
1− α n z n − Px,x − Px
≤1− α nx n − Px 2
+α n
α n x − Px 2+ 2
1− α n z n − Px,x − Px .
(3.13) Puttinga n = x n − Px 2, from (3.13), we have
a n+1 ≤1− α n
whereδ n = α n { α n x − Px 2+ 2(1− α n) z n − Px, x − Px }
Trang 8It is easily seen that
lim sup
n →∞ δ n /α n =lim sup
n →∞
α n x − Px 2+ 2
1− α n z n − Px, x − Px ≤0. (3.15)
Now applyingLemma 2.1with (3.15) to (3.14) concludes that x n − Px →0 asn → ∞
References
[1] J.-B Baillon, Un th´eor`eme de type ergodique pour les contractions non lin´eaires dans un espace de
Hilbert, Comptes Rendus de l’Acad´emie des Sciences de Paris, S´erie A-B 280 (1975), no 22,
A1511–A1514.
[2] H Br´ezis and F E Browder, Nonlinear ergodic theorems, Bulletin of the American Mathematical
Society 82 (1976), no 6, 959–961.
[3] B Halpern, Fixed points of nonexpanding maps, Bulletin of the American Mathematical Society
73 (1967), 957–961.
[4] J S Jung, Viscosity approximation methods for a family of finite nonexpansive mappings in Banach
spaces, Nonlinear Analysis 64 (2006), no 11, 2536–2552.
[5] P.-E Maing´e, Viscosity methods for zeroes of accretive operators, Journal of Approximation Theory
140 (2006), no 2, 127–140.
[6] Z Opial, Weak convergence of the sequence of successive approximations for nonexpansive
map-pings, Bulletin of the American Mathematical Society 73 (1967), 591–597.
[7] S Reich, Some problems and results in fixed point theory, Topological Methods in Nonlinear
Func-tional Analysis (Toronto, Ont., 1982), Contemp Math., vol 21, American Mathematical Society, Rhode Island, 1983, pp 179–187.
[8] T Shimizu and W Takahashi, Strong convergence to common fixed points of families of
nonexpan-sive mappings, Journal of Mathematical Analysis and Applications 211 (1997), no 1, 71–83.
[9] R Wittmann, Approximation of fixed points of nonexpansive mappings, Archiv der Mathematik
58 (1992), no 5, 486–491.
[10] H.-K Xu, Viscosity approximation methods for nonexpansive mappings, Journal of Mathematical
Analysis and Applications 298 (2004), no 1, 279–291.
Yonghong Yao: Department of Mathematics, Tianjin Polytechnic University, Tianjin 300160, China
E-mail address:yuyanrong@tjpu.edu.cn
Rudong Chen: Department of Mathematics, Tianjin Polytechnic University, Tianjin 300160, China
E-mail address:chenrd@tjpu.edu.cn
Haiyun Zhou: Department of Mathematics, Shijiazhuang Mechanical Engineering College,
Shijiazhuang 050003, China
E-mail address:witman66@yahoo.com.cn
...[8] T Shimizu and W Takahashi, Strong convergence to common fixed points of families of
nonexpan-sive mappings, Journal of Mathematical Analysis and Applications... zeroes of accretive operators, Journal of Approximation Theory
140 (2006), no 2, 127–140.
[6] Z Opial, Weak convergence of the sequence of. .. (1997), no 1, 71–83.
[9] R Wittmann, Approximation of fixed points of nonexpansive mappings, Archiv der Mathematik
58 (1992), no 5, 486–491.