Hindawi Publishing Corporation
International Journal of Mathematics and Mathematical Sciences
Volume 2008, Article ID 236269, 9 pages
doi:10.1155/2008/236269
Research Article
Strong Convergence Theorem for Two
Commutative Asymptotically Nonexpansive
Mappings in Hilbert Space
Jianjun Liu, Lili He, and Lei Deng
School of Mathematics and Statistics, Southwest University, Chongqing 400715, China
Correspondence should be addressed to Jianjun Liu, liliu@swu.edu.cn
Received 20 June 2008; Accepted 8 December 2008
Recommended by Enrico Obrecht
C is a bounded closed convex subset of a Hilbert space H, T and S : C → C are two asymptotically
nonexpansive mappings such that ST T S. We establish a strong convergence theorem for S and
T in Hilbert space by hybrid method. The results generalize and unify many corresponding results.
Copyright q 2008 Jianjun Liu 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 C be a bounded closed convex subset of a Hilbert space H. Recall that a mapping T : C →
C is said to be asymptotically nonexpansive mapping if
n
T x − T n y ≤ tn x − y
∀ x, y ∈ C,
1.1
where tn → 1 n → ∞. We may assume that tn ≥ 1 for all n 1, 2, 3, . . . . Denote
by FT the set of fixed points of T . Throughout this paper T and S : C → C are two
commutative asymptotically nonexpansive mappings with asymptotical coefficients {tn } and
{sn }, respectively. Suppose that F : FT ∩FS / ∅ 1, Goebel and Kirk’s theorem makes it
possible. It is well known that FT and FS are convex and closed 1, 2, so is F. PK denotes
the metric projection from H onto a closed convex subset K of H and ωw xn denotes the
weak w-limit set of {xn }. It is well known that a Hilbert space H satisfies Opial’s condition
z, then
3, that is, if a sequence {xn } converges weakly to an element y ∈ H and y /
lim infxn − y < lim infxn − z.
n→∞
n→∞
1.2
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Up to now, fixed points iteration processes for nonexpansive and asymptotically
nonexpansive mappings have been studied extensively by many authors to solve nonlinear
operator equations as well as variational inequalities 4–6. There are many strong
convergence theorems for nonexpansive and asymptotically nonexpansive mappings in
Hilbert space 7, 8.
Especially, Shimizu and Takahashi 7 studied the following iteration process of
nonexpansive mappings for arbitrary x0 ∈ C:
xn1 αn x0 1 − αn
n 2
Si T j xn ,
n 1n 2 k0 ijk
1.3
where {αn } ⊆ 0, 1, limn → ∞ αn 0, ∞
n0 αn ∞. And then they proved that {xn }
converges strongly to PF x0 . This result was extended to two commutative asymptotically
nonexpansive mappings by Shioji and Takahashi 9.
Recently, some attempts to the modified Mann iteration method are made so that
strong convergence is guaranteed. And for hybrid method proposed by Haugazeau 10,
Kim and Xu 8 introduced the following iteration processes for asymptotically nonexpansive
mapping T :
x0 ∈ C chosen arbitrarily,
yn αn xn 1 − αn T n xn ,
2 2
Cn v ∈ C : yn − v ≤ xn − v θn ,
Qn v ∈ C : xn − v, x0 − xn ≥ 0 ,
xn1 PCn Qn x0 ,
1.4
where θn 1 − αn t2n − 1 diam C2 → 0 as n → ∞. Then proved that {xn } converges
strongly to PF x0 . This result was generalized to two asymptotically nonexpansive mappings
by Plubtieng and Ungchittrakool 11.
On the basis of 1.3 and 1.4, we propose a new iteration processes for two
commutative asymptotically nonexpansive mappings S and T :
x0 ∈ C chosen arbitrarily,
yn αn xn 1 − αn
n 2
Si T j xn ,
n 1n 2 k0 ijk
2 2
Cn v ∈ C : yn − v ≤ xn − v θn ,
Qn v ∈ C : xn − v, x0 − xn ≥ 0 ,
xn1 PCn Qn x0 ,
1.5
where θn 1 − αn gn2 − 1 diam C2 , gn 2/n 1n 2 nk0 ijk si tj , for every
n 1, 2, . . . . The purpose of this paper is to prove {xn } converges strongly to PF x0 .
Jianjun Liu 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 7. Letting Ln n 1n 2/2, there holds the identity in a Hilbert space H:
n n yn − v2 1
xi,j − v2 − 1
xi,j − yn 2
Ln k0 ijk
Ln k0 ijk
for {xi,j }∞
⊆ H, yn 1/Ln i,j0
n k0
ijk
2.1
xi,j ∈ H and v ∈ H.
Lemma 2.2. Let C be a bounded closed convex subset of a Hilbert space H, S and T two commutative
asymptotically nonexpansive mappings of C into itself with asymptotical coefficients {sn } and {tn },
respectively. For any x ∈ C, put Fn x 2/n 1n 2 nk0 ijk Si T j x. Then
lim lim sup supFn x − Sl Fn x 0,
l→∞
n→∞
x∈C
lim lim sup supFn x − T l Fn x 0 .
l→∞
n→∞
2.2
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
≤
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
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 2
1 1 Si T j x − Fn x2
s2l Si−l T j x − Fn x −
Ln kl ijk, i≥l
Ln k0 ijk
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International Journal of Mathematics and Mathematical Sciences
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 2
1 1 Si T j x − Fn x2
s2l Si T j x − Fn x −
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
n−l 2
2
1 sl − 1 Si T j x − Fn x .
Ln k0 ijk
2.3
Choose p ∈ F, then there exists a constant M > 0 such that
i j
S T x − p ≤ si tj x − p ≤ M ,
2
n Fn x − p ≤ 1
Si T j x − p ≤ M ,
Ln k0 ijk
2
M
l
S Fn x − p ≤ sl Fn x − p ≤
,
2
2.4
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
s − 1 n 2 − ln 1 − l 2
l 1l
2n 1 − ll
M2 M2 l
M
n 2n 1
n 2n 1
n 2n 1
2.5
−→ 0 n −→ ∞, l −→ ∞.
Similarly, we can prove
lim lim sup supFn x − T l Fn x 0.
l→∞
n→∞
x∈C
2.6
Remark 2.3. Lemma 2.2 extends 7, Lemma 1.
Lemma 2.4. Let S and T be two commutative asymptotically nonexpansive mappings defined on a
bounded closed convex subset C of a Hilbert space H with asymptotical coefficients {sn } and {tn },
respectively. 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 .
Jianjun Liu et al.
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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
xn − Slm x
n n n 1 1 1 i j
i j
lm
i j
≤ xn −
S T xn S T xn − S
S T xn Ln k0 ijk
Ln k0 ijk
Ln k0 ijk
n 1 Slm
Si T j xn − Slm x
Ln k0 ijk
n n n 1 1 1 i j
i j
lm
i j
≤ xn −
S T xn S T xn − S
S T xn Ln k0 ijk
Ln k0 ijk
Ln k0 ijk
n 1 i j
slm S T xn − x
Ln k0 ijk
n n n 1 1
1 ≤ xn −
Si T j xn Si T j xn − Slm
Si T j xn Ln k0 ijk
Ln k0 ijk
Ln k0 ijk
n 1 i j
slm S T xn − xn slm xn − x.
Ln k0 ijk
2.7
By Opial’s condition, for any y ∈ C with y /
x, we have
lim inf xn − x < lim inf xn − y.
n→∞
n→∞
2.8
Let r lim infn → ∞ xn − x and choose a positive number ρ such that
ρ<
r2 ε02
− r.
4
2.9
Then, there exists a subsequence {xnp } of {xn } such that limp → ∞ xnp − x r and xnp − x <
r ρ/4 for all p. By definition of {slm }, there exists a positive integer m0 such that
ρ
slm xnp − x < r ,
4
2.10
n 1 i j
lim xn −
S T xn 0
n → ∞
Ln k0 ijk
2.11
for all m > m0 . Since
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International Journal of Mathematics and Mathematical Sciences
and {slm } is bounded, there exists a positive integer p0 such that
np
1 i j
S T xnp <
xnp −
Lnp k0 ijk
np
1 i j
slm S T xnp − xnp <
Lnp k0 ijk
ρ
,
4
2.12
ρ
4
for all m and p > p0 . By {xnp } ⊂ C is bounded and Lemma 2.2, there exist m1 > m0 and p1 > 0
such that
np
np
ρ
1 1 i j
lm1
i j
S T xnp − S
S T xnp <
4
Lnp k0 ijk
Lnp k0 ijk
2.13
for all p > p1 . By 2.7, 2.10, 2.12, and 2.13, we have
xn − Slm1 x < ρ ρ ρ r ρ r ρ
p
4 4 4
4
2.14
for all p > max{p0 , p1 }. However,
2
lm1
xn − S x x 1 xn − Slm1 x2 p
p
2
2
<
1
xn − x2 −
p
2
r ρ2 r ρ/42 ε02
−
2
2
4
< r ρ2 −
<r
1
Slm1 x − x2
4
2.15
ε02
4
2
for all p > max{p0 , p1 }. This contradicts 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 .
Lemma 2.5 see 12. Let C be a bounded closed convex subset of a Hilbert space H. The set D :
{v ∈ C : y − v2 ≤ x − v2 z, v b} is convex and closed for given x, y, z ∈ C and b ∈ R.
3. Main results
In this section, we prove our main theorem.
Theorem 3.1. Let C be a bounded closed convex subset of a Hilbert H, T and S : C → C be two
commutative asymptotically nonexpansive mappings with asymptotical coefficients {tn } and {sn },
∅, then the
respectively. Suppose that 0 ≤ αn ≤ a for all n, where 0 < a < 1. If F : FT ∩ FS /
sequence generated by 1.5 converges strongly to PF x0 .
Jianjun Liu et al.
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Proof. Note that Cn is convex and closed for all n ≥ 0 by Lemma 2.5. On the other hand, Qn is
convex and closed. So is Cn ∩ Qn .
By definition of {tn } and {sn }, there exists M > 0 such that si tj − 1 ≤ M for all i, j ≥ 0.
On the other hand, for arbitrary ε > 0, there exists N > 0 such that si tj − 1 < ε for all i, j > N.
Hence
n 2
gn − 1 si tj − 1
n 1n 2 k0 ijk
≤
n 2
si tj − 1
n 1n 2 k0 ijk
≤
n
2
n 1n 2 k0
<
2
n 1n 2
si tj − 1 ijk, i≤N
n
n
2
si tj − 1
n 1n 2 k0 ijk, j≤N
si tj − 1
k0 ijk, i≥N1, j≥N1
2N 1M 2N 1M
ε.
n 2
n 2
3.1
Thus limn → ∞ gn 1. Obviously, limn → ∞ θn 0.
Next, we prove that F ⊂ Cn ∩ Qn . Indeed, first of all
2
n 2
2 2
yn − p ≤ αn xn − p 1 − αn Si T j xn − p
n 1n 2 k0 ijk
2 2
≤ αn xn − p 1 − αn gn2 xn − p
2 2 2 xn − p 1 − αn gn2 xn − p − xn − p
2
≤ xn − p θn
3.2
for all p ∈ F. So F ⊂ Cn . It suffices to show that F ⊂ Qn for all n ≥ 0. We prove this by
induction. For n 0, we have F ⊂ C Q0 . Assume that F ⊂ Qn . Since xn1 is the projection of
x0 onto Cn ∩ Qn , we have
xn1 − z, x0 − xn1 ≥ 0 ∀ z ∈ Cn Qn
3.3
As F ⊂ Cn ∩ Qn , 3.3 holds for all z ∈ F, in particular. This together with the definition of
Qn1 implies that F ⊂ Qn1 . Hence, F ⊂ Cn ∩ Qn for all n ≥ 0.
We will show that xn1 −xn → 0 as n → ∞. By the definition of Qn , we have that xn PQn x0 . It follows from xn1 ∈ Cn ∩ Qn ⊂ Qn that xn − x0 ≤ xn1 − x0 . This shows that the
sequence {xn − x0 } is increasing. Since C is bounded, we obtain that limn → ∞ xn − x0 exists.
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International Journal of Mathematics and Mathematical Sciences
Notice again that from xn PQn x0 and xn1 ∈ Qn , we have xn1 − xn , xn − x0 ≥ 0. Hence
xn1 − xn 2 xn1 − x0 − xn − x0 2
2 2
xn1 − x0 xn − x0 − 2 xn1 − x0 , xn − x0
2 2
xn1 − x0 − xn − x0 − 2 xn1 − x0 − xn − x0 , xn − x0
2 2
xn1 − x0 − xn − x0 − 2 xn1 − xn , xn − x0
2 2
≤ xn1 − x0 − xn − x0 −→ 0
3.4
n −→ ∞.
n Now we claim that 2/n 1n 2
the definition of yn , we have
k0
ijk
Si T j xn − xn → 0 as n → ∞. By
n 2
i j
S T xn − xn n 1n 2 k0 ijk
1 yn − xn 1 − αn
1 yn − xn1 xn1 − xn ≤
1 − αn
1 yn − xn1 xn1 − xn .
≤
1−a
3.5
Since xn1 ∈ Cn , yn − xn1 2 ≤ xn − xn1 2 θn → 0 as n → ∞. So yn − xn1 → 0 as
n → ∞. This implies that
n 2
i j
S T xn − xn −→ 0
n 1n 2 k0 ijk
n −→ ∞.
3.6
Since C is bounded closed convex, ωw xn / ∅. It follows from 3.6 and Lemma 2.4
that ωw xn ⊂ F. By the definition of Qn , we have that xn − x0 ≤ PF x0 − x0 for all n ≥ 0.
It follows from the weak lower semi-continuity of the norm that w − x0 ≤ PF x0 − x0 for all w ∈ ωw xn . Since ωw xn ⊂ F, we have w PF x0 for all w ∈ ωw xn . Thus
ωw xn {PF x0 }. Then, {xn } converges to PF x0 weakly. By the fact
xn − PF x0 2 xn − x0 x0 − PF x0 2
2 2
xn − x0 x0 − PF x0 2 xn − x0 , x0 − PF x0
2 ≤ 2 PF x0 − x0 xn − x0 , x0 − PF x0
−→ 0 n −→ ∞,
we have {xn } converges to PF x0 strongly. This completes the proof.
3.7
Jianjun Liu et al.
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The following corollary follows from Theorem 3.1.
Corollary 3.2. Let C be a bounded closed convex subset of a Hilbert H, T and S : C → C be
two commutative nonexpansive mappings. Suppose that 0 ≤ αn ≤ a for all n, where 0 < a < 1. If
F : FT ∩ FS /
∅, then the sequence {xn } generated by
x0 ∈ C chosen arbitrarily,
yn αn xn 1 − αn
n 2
Si T j xn ,
n 1n 2 k0 ijk
2 2 Cn v ∈ C : yn − v ≤ xn − v ,
Qn v ∈ C : xn − v, x0 − xn ≥ 0 ,
xn1 PCn ∩Qn x0 ,
3.8
converges strongly to PF x0 .
Acknowledgment
This work is supported by National Natural Science Foundation of China 10771173.
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