Hindawi Publishing Corporation
International Journal of Mathematics and Mathematical Sciences
Volume 2008, Article ID 649510, 9 pages
doi:10.1155/2008/649510
Research Article
Strong Convergence Theorem of Implicit
Iteration Process for Generalized Asymptotically
Nonexpansive Mappings in Hilbert Space
Lili He, Lei Deng, and Jianjun Liu
School of Mathematics and Statistics, Southwest University, Chongqing 400715, China
Correspondence should be addressed to Lei Deng, denglei math@sina.com
Received 21 May 2008; Accepted 14 August 2008
Recommended by Petru Jebelean
Let C be a nonempty closed and convex subset of a Hilbert space H, let T and S : C → C be
two commutative generalized asymptotically nonexpansive mappings. We introduce
n an implicit
iteration process of S and T defined by xn αn x0 1 − αn 2/n 1n 2 k0 ijk Si T j xn ,
and then prove that {xn } converges strongly to a common fixed point of S and T . The results
generalize and unify the corresponding results.
Copyright q 2008 Lili He 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
Let H be a Hilbert space, C a nonempty closed and convex subset of H. A mapping S is said
to be generalized asymptotically nonexpansive if
n
S x − Sn y ≤ kn x − y rn
1.1
with kn ≥ 0, rn ≥ 0, kn → 1, rn → 0 n → ∞, for each x, y ∈ C, n 0, 1, 2, . . . . If kn 1 and
rn 0, 1.1 reduces to nonexpansive mapping; if rn 0, 1.1 reduces to asymptotically
nonexpansive mapping; if kn 1, 1.1 reduces to asymptotically nonexpansive-type
mapping. So, a generalized asymptotically nonexpansive mapping is much more general
than many other mappings.
Browder 1 introduced the following implicit iteration process of a nonexpansive selfmapping for arbitrary x0 ∈ C:
xn αn x0 1 − αn T xn ,
1.2
and then proved that {xn } converges strongly to the fixed point of T which is nearest to x0 .
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After that, Baillon 2 proved the first nonlinear ergodic theorem: let T be a
nonexpansive self-mapping defined on a nonempty bounded closed and convex set. Then
for arbitrary x ∈ C, {1/n 1 ni0 T i x} converges weakly to a fixed point of T . Those results
have been extended by several authors see, e.g., 3–7.
Recently, Shimizu and Takahashi 8 studied the following implicit iteration process
of asymptotically nonexpansive mappings for arbitrary x0 ∈ C:
n
1 xn αn x0 1 − αn
T i xn ,
n 1 i0
1.3
where αn bn − 1/bn − 1 a, bn 1/n 1 ni0 1 |1 − ki | e−i , {kn } is the asymptotical
coefficient of T . And then proved that {xn } converges strongly to the fixed point of T which
is nearest to x0 .
They also studied the following explicit iteration process of two commutative
nonexpansive self-mappings in 9:
xn1 αn x0 1 − αn
n 2
Si T j xn ,
n 1n 2 k0 ijk
1.4
where {αn } ⊆ 0, 1, limn→∞ αn 0, ∞
n0 αn ∞. And then proved that {xn } converges
strongly to the common fixed point of S and T which is nearest to x0 .
In this paper, we prove the strong convergence theorem of implicit iteration process
for generalized asymptotically nonexpansive mappings which is defined by
xn αn x0 1 − αn
n 2
Si T j xn ,
n 1n 2 k0 ijk
1.5
where {αn } ⊆ 0, 1, limn→∞ αn 0 and lim supn→∞ 1−αn 2/n1n2 nk0 ijk ki tj < 1.
It is well known that a Hilbert space H satisfies Opial’s condition 5, that is, if a
sequence {xn } converges weakly to an element y ∈ H and y / z, then
lim infxn − y < lim infxn − z.
n→∞
n→∞
1.6
Throughout this paper, S and T are two commutative generalized asymptotically
nonexpansive mappings. We denote by FS and FT the set of fixed points of S and T ,
respectively; and suppose that FS FT /∅. S and T satisfy the following conditions:
n
S x − Sn y ≤ kn x − y rn ,
n
T x − T n y ≤ tn x − y bn ,
where kn ≥ 0, rn ≥ 0, tn ≥ 0, bn ≥ 0, and kn → 1, rn → 0, tn → 1, bn → 0 n → ∞.
1.7
Lili He et al.
3
2. Auxiliary lemmas
This section collects some lemmas which will be used to prove the main results in the next
section.
Lemma 2.1 see 9. Letting Ln n 1n 2/2, there holds the identity in a Hilbert space H:
n n xi,j − v2 − 1
xi,j − yn 2
yn − v2 1
Ln k0 ijk
Ln k0 ijk
⊆ H, yn 1/Ln for {xi,j }∞
i,j0
n k0
ijk xi,j
2.1
∈ H, and v ∈ H.
Lemma 2.2. Let C be a nonempty bounded closed convex subset of a Hilbert space H, S and T two
generalized asymptotically nonexpansive mappings of C into itself such that ST T S. For any x ∈ C,
put Fn x 2/n 1n 2 nk0 ijk Si T j x. Then
lim lim supFn x − Sl Fn x 0,
l→∞ n→∞ x∈C lim lim supFn x − T l Fn x 0.
2.2
l→∞ n→∞ x∈C
Proof. Put xi,j Si T j x, v Sl Fn x and Ln n 1n 2/2. It follows from Lemma 2.1
that
Fn x − Sl Fn x2
n n 1 Si T j x − Sl Fn x2 − 1
Si T j x − Fn x2
Ln k0 ijk
Ln k0 ijk
l−1 n
1 Si T j x − Sl Fn x2 1
Si T j x − Sl Fn x2
Ln k0 ijk
Ln kl ijk, i≤l−1
≤
l−1 n
1 Si T j x − Sl Fn x2 1
Si T j x − Sl Fn x2
Ln k0 ijk
Ln kl ijk, i≤l−1
n
n 1 Si T j x − Sl Fn x2 − 1
Si T j x − Fn x2
Ln kl ijk, i≥l
Ln k0 ijk
n
n 2
1 1 Si T j x − Fn x2
kl Si−l T j x − Fn x rl −
Ln kl ijk ,i≥l
Ln k0 ijk
l−1 n
1 Si T j x − Sl Fn x2 1
Si T j x − Sl Fn x2
Ln k0 ijk
Ln kl ijk, i≤l−1
n−l 2
2 i j
1 kl S T x − Fn x 2kl rl Si T j x − Fn x rl2
Ln k0 ijk
−
n 1 Si T j x − Fn x2
Ln k0 ijk
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≤
l−1 n
1 Si T j x − Sl Fn x2 1
Si T j x − Sl Fn x2
Ln k0 ijk
Ln kl ijk, i≤l−1
n−l n−l 2 2kl rl 2
1 Si T j x − Fn x
kl − 1 Si T j x − Fn x Ln k0 ijk
Ln k0 ijk
n 2 − ln 1 − l 2
rl .
n 2n 1
2.3
Choose p ∈ FS
FT , then there exists a constant M > 0 such that
i j
S T x − p ≤ ki T j x − p ri ≤ ki tj x − p ki bj ri ≤ M ,
2
n 1
M
Fn x − p ≤
Si T j x − p ≤
,
Ln k0 ijk
2
2.4
l
S Fn x − p ≤ kl Fn x − p rl ≤ M
2
for all nonnegative integer i, j, l, and n. Hence, Si T j x − Sl Fn x ≤ M, Si T j x − Fn x ≤ M
for all nonnegative integer i, j, l, and n. So
2
supFn x − Sl Fn x
x∈C
2
kl − 1 n 2 − ln 1 − l 2
l 1l
2n 1 − ll
2
2
M M M
≤
n 2n 1
n 2n 1
n 2n 1
2.5
n 2 − ln 1 − l 2
2kl rl n 2 − ln 1 − l
M
rl
n 2n 1
n 2n 1
−→ 0 n −→ ∞, l −→ ∞.
Similarly, we can prove that
lim lim supFn x − T l Fn x 0.
l→∞ n→∞ x∈C
2.6
Remark 2.3. Lemma 2.2 extends 8, Lemma 3 and 9, Lemma 1.
Lemma 2.4. Let S and T be two continuous generalized asymptotically nonexpansive mappings
defined on a nonempty bounded closed convex subset C of a Hilbert space H with ST T S. Let
Ln n 1n 2/2. If {xn } is a sequence in C such that {xn } converges weakly to some x ∈ C
and {xn − 1/Ln nk0 ijk Si T j xn } converges strongly to 0, then x ∈ FS FT .
Lili He et al.
5
Proof. We claim that {Sl x} converges strongly to x as l → ∞. If not, there exist a positive
number ε0 and a subsequence {lm } of {l} such that Slm x − x ≥ ε0 for all m. However, we
have
ni i j
xn − Slm x ≤ xn − 1
S T xni i
i L
ni k0 ijk
ni ni 1 1 i j
lm
i j
S
T
x
−
S
S
T
x
n
n
i
i
L
Lni k0 ijk
ni k0 ijk
ni l
1 i j
lm m
S
S
T
x
x
−
S
ni
Lni k0 ijk
ni 1 i j
≤ xni −
S T xni Lni k0 ijk
ni ni 1 1 i j
lm
i j
S
T
x
−
S
S
T
x
n
n
i
i
L
Lni k0 ijk
ni k0 ijk
2.7
ni 1 i j
rl
klm S
T
x
−
x
ni
m
L
ni k0 ijk
ni 1 i j
≤ xni −
S T xni Lni k0 ijk
ni ni 1 1 i j
lm
i j
S
T
x
−
S
S
T
x
ni
ni L
Lni k0 ijk
ni k0 ijk
ni 1 i j
klm S
T
x
−
x
ni
ni klm xni − x rlm .
L
ni k0 ijk
By Opial’s condition, for any y ∈ C with y / x, we have
lim infxn − x < lim infxn − y.
n→∞
n→∞
Let r lim infn→∞ xn − x and choose a positive number ρ such that
ε2
ρ < r 2 0 − r.
4
2.8
2.9
Then, there exists a subsequence {xni } of {xn } such that limi→∞ xni − x r and xni − x <
r ρ/5 for all i. By definition of {klm } and {rlm }, there exists a positive integer m0 such that
ρ
klm xni − x < r ,
5
rlm <
ρ
5
2.10
for all m > m0 . Since
n 1 i j
x
lim −
S
T
x
n
n 0
n→∞
Ln k0 ijk
2.11
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International Journal of Mathematics and Mathematical Sciences
and {klm } is bounded , there exists a positive integer i0 such that
ni ρ
i j
xn − 1
,
S
T
x
ni <
i L
5
ni k0 ijk
ni 1 ρ
i j
klm S T xni − xni < 5
Lni k0 ijk
2.12
for all m and i > i0 . As {xni } ⊂ C is bounded and by Lemma 2.2, there exist m1 > m0 and i1 > 0
such that
ni ni 1 ρ
1 i j
lm1
i j
S
T
x
−
S
S
T
x
ni
ni <
L
Lni k0 ijk
5
ni k0 ijk
2.13
for all i > i1 . By 2.7, 2.10, 2.12, and 2.13, we have
xn − Slm1 x < ρ ρ ρ r ρ ρ r ρ
i
5 5 5
5 5
2.14
for all i > max{i0 , i1 }. However,
2
lm1
xn − S x x 1 xn − Slm1 x2 i
i
2
2
<
1
xn − x2 −
i
2
1
Slm1 x − x2
4
r ρ2 r ρ/52 ε02
−
2
2
4
< r ρ2 −
2.15
ε02
< r2
4
for all i > max{i0 , i1 }. This contradicts with 2.8. So {Sl x} converges strongly to x and then
x ∈ FS. Similarly, we can get x ∈ FT . Hence, x is a common fixed point of S and T .
3. Main results
In this section, we prove our main theorem.
Theorem 3.1. Let C be a nonempty closed convex subset of a Hilbert space H and let S, T be two
continuous generalized asymptotically nonexpansive mappings of C into itself such that
i Sn x − Sn y ≤ kn x − y rn , kn ≥ 0, rn ≥ 0, kn → 1, rn → 0 (n → ∞),
ii T n x − T n y ≤ tn x − y bn , tn ≥ 0, bn ≥ 0, tn → 1, bn → 0 (n → ∞),
iii ST T S and FS FT /
∅.
For arbitrary x0 ∈ C, the sequence {xn }∞
n0 is defined by
xn αn x0 1 − αn
n 2
Si T j xn ,
n 1n 2 k0 ijk
n ≥ 0,
3.1
Lili He et al.
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with αn ∈ 0, 1 and limn→∞ αn 0. If lim supn→∞ 1 − αn 2/n 1n 2 nk0 ijk ki tj < 1,
to the common fixed point P x0 of S and T , where P is the metric
then {xn }∞
n0 converges strongly
projection of H onto FS FT .
Proof. Firstly, we show that {xn }∞
n0 is bounded. By lim supn→∞ 1 − αn 2/n 1n 2 nk0 ijk ki tj < 1, there exist N > 0 and 0 < a < 1 such that 1 − αn 2/n 1n 2 nk0 ijk ki tj ≤ a for all n > N. Choose p ∈ FS FT , then we have
xn − p αn x0 − p 1 − αn
n i j
2
i j
S T xn − S T p n 1n 2 k0 ijk
≤ αn x0 − p 1 − αn
n 2
ki tj xn − p ki bj ri
n 1n 2 k0 ijk
αn x0 − p 1 − αn
n 2
ki tj xn − p
n 1n 2 k0 ijk
1 − αn
3.2
n n 2
2
ki bj 1 − αn
ri .
n 1n 2 k0 ijk
n 1n 2 k0 ijk
Hence
1 − axn − p ≤ αn x0 − p 1 − αn
1 − αn
n 2
ki bj
n 1n 2 k0 ijk
n 2
ri
n 1n 2 k0 ijk
3.3
for all n > N. So {xn } is bounded.
Letting {xni } be any subsequence of {xn }, there exists a subsequence {xmt } of {xni }
such that {xmt } converges weakly to some u ∈ C. For {xmt } is bounded, so is {2/mt t Si T j xmt }. Thus
1mt 2 m
k0 ijk
mt mt 2
2
Si T j xmt αmt x0 − Si T j xmt
mt 1 mt 2 k0 ijk
mt 1 mt 2 k0 ijk
3.4
xmt − −→ 0t −→ ∞.
It follows from Lemma 2.4 that u ∈ FS
FT . For
mt 2
Si T j P x0 ,
mt 1 mt 2 k0 ijk
αmt P x0 P x0 − 1 − αmt 3.5
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International Journal of Mathematics and Mathematical Sciences
we have
αmt x0 − P x0 , xmt − P x0
xmt − P x0 − 1 − αmt mt i j
2
S T xmt − Si T j P x0 , xmt − P x0
mt 1 mt 2 k0 ijk
xmt
mt i j
2
i j
S T xmt − S T P x0 , xmt − P x0
mt 1 mt 2 k0 ijk
− P x0 − 1 − αmt 2
≥ xmt − P x0 2 − 1 − αmt 2
mt 1 mt 2
×
mt k0 ijk
ki tj xmt − P x0 ki bj ri xmt − P x0 mt 2
ki tj xmt − P x0 2
mt 1 mt 2 k0 ijk
1 − 1 − αmt mt 2
ki bj ri xmt − P x0 .
mt 1 mt 2 k0 ijk
− 1 − αmt 3.6
By hypothesis, there exist M > 0 and 0 < a < 1 such that 1 − αmt 2/mt 1mt t k t ≤ a for all t > M. So
2 m
k0 ijk i j
2
1 − axmt − P x0 mt 2
ki bj ri xmt − P x0 mt 1 mt 2 k0 ijk
≤ αmt x0 − P x0 , xmt − P x0 1 − αmt 3.7
t k b for all t > M. We can easily prove that limn→∞ 1 − αmt 2/mt 1mt 2 m
k0 ijk
i j
ri xmt − P x0 0. Since P is metric projection, x0 − P x0 , x − P x0 ≤ 0 for all x ∈ FS FT .
Hence
x0 − P x0 , xmt − P x0 x0 − P x0 , xmt − u x0 − P x0 , u − P x0
3.8
≤ x0 − P x0 , xmt − u .
Since {xmt } converges weakly to u, lim supt→∞ x0 − P x0 , xmt − P x0 ≤ 0. It follows from 3.7
that {xmt } converges strongly to P x0 . Hence, {xn } converges strongly to P x0 . This completes
the proof.
Acknowledgment
This work is supported by National Natural Science Foundation of China 10771173 and
Natural Science Foundation Project of Henan 2008B110012.
Lili He et al.
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References
1 F. E. Browder, “Convergence of approximants to fixed points of nonexpansive nonlinear mappings in
Banach spaces,” Archive for Rational Mechanics and Analysis, vol. 24, no. 1, pp. 82–90, 1967.
2 J.-B. Baillon, “Un théorème de type ergodique pour les contractions non linéaires dans un espace de
Hilbert,” Comptes Rendus Hebdomadaires des Séances de l’Académie des Sciences. Séries A et B, vol. 280, no.
22, pp. 1511–1514, 1975.
3 N. Hirano and W. Takahashi, “Nonlinear ergodic theorems for nonexpansive mappings in Hilbert
spaces,” Kodai Mathematical Journal, vol. 2, no. 1, pp. 11–25, 1979.
4 S. P. Reich, “Some problems and results in fixed point theory,” in Topological Methods in Nonlinear
Functional Analysis (Toronto, Ont., 1982), vol. 21 of Contemporary Mathematics, pp. 179–187, American
Mathematical Society, Providence, RI, USA, 1983.
5 Z. Opial, “Weak convergence of the sequence of successive approximations for nonexpansive
mappings,” Bulletin of the American Mathematical Society, vol. 73, no. 4, pp. 591–597, 1967.
6 W. Takahashi, Nonlinear Functional Analysis, Kindaikagakusha, Tokyo, Japan, 1988.
7 R. Wittmann, “Approximation of fixed points of nonexpansive mappings,” Archiv der Mathematik, vol.
58, no. 5, pp. 486–491, 1992.
8 T. Shimizu and W. Takahashi, “Strong convergence theorem for asymptotically nonexpansive
mappings,” Nonlinear Analysis: Theory, Methods & Applications, vol. 26, no. 2, pp. 265–272, 1996.
9 T. Shimizu and W. Takahashi, “Strong convergence to common fixed points of families of nonexpansive
mappings,” Journal of Mathematical Analysis and Applications, vol. 211, no. 1, pp. 71–83, 1997.