Hindawi Publishing Corporation
Fixed Point Theory and Applications
Volume 2011, Article ID 706206, 13 pages
doi:10.1155/2011/706206
Research Article
Strong Convergence Theorems of the Ishikawa
Process with Errors for Strictly Pseudocontractive
Mapping of Browder-Petryshyn Type in
Banach Spaces
Yu-Chao Tang,1, 2 Yong Cai,1 and Li-Wei Liu1
1
2
Department of Mathematics, NanChang University, Nanchang 330031, China
Department of Mathematics, Xi’an Jiaotong University, Xi’an 710049, China
Correspondence should be addressed to Yu-Chao Tang, hhaaoo1331@yahoo.com.cn
Received 18 December 2010; Accepted 23 February 2011
Academic Editor: M. de la Sen
Copyright q 2011 Yu-Chao Tang 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.
We prove several strong convergence theorems for the Ishikawa iterative sequence with errors to
a fixed point of strictly pseudocontractive mapping of Browder-Petryshyn type in Banach spaces
and give sufficient and necessary conditions for the convergence of the scheme to a fixed point of
the mapping. The results presented in this work give an affirmative answer to the open question
raised by Zeng et al. 2006, and generalize the corresponding result of Zeng et al. 2006, Osilike and
Udomene 2001, and others.
1. Introduction and Preliminaries
Let E be a real Banach space and E∗ its dual. ·, · denotes the generalized duality pairing
∗
between E and E∗ . Let J : E → 2E be the normalized duality mapping defined by the
following:
2 Jx f ∈ E∗ : x, f x2 f ,
∀x ∈ E.
1.1
It is well known that if E is smooth, then J is single-valued. In this paper, we denote a singlevalued selection of the normalized duality mapping by j. I denotes the identity operator.
FT is the fixed point set of T , that is, FT {x : T x x}.
2
Fixed Point Theory and Applications
Definition 1.1 see 1. A mapping T : DT ⊂ E → E is said to be strictly pseudocontractive
if there exists λ > 0 and jx − y ∈ Jx − y, such that
2
2
T x − T y, j x − y ≤ x − y − λx − y − T x − T y ,
∀x, y ∈ DT .
1.2
Remark 1.2. i Without loss of generality, we may assume λ ∈ 0, 1. Inequality 1.2 can be
written in the form
2
I − T x − I − T y, j x − y ≥ λI − T x − I − T y .
1.3
ii If E is a Hilbert space, then inequality 1.2 is equivalent to the following inequality:
T x − T y2 ≤ x − y2 kI − T x − I − T y2 ,
k 1 − 2λ < 1.
1.4
iii T is a Lipschitz continuous mapping, that is, ∃L > 0, s.t, T x − T y ≤ Lx − y. In
fact, by 1.3, we have
x − y ≥ λx − y − T x − T y ≥ λT x − T y − λx − y,
1.5
so that,
T x − T ≤ Lx − y,
∀x, y ∈ DT ,
1.6
where L λ 1/λ.
Definition 1.3. A mapping T : DT ⊂ E → E is said to be
i compact, if for any bounded sequence {xn } in DT , there exists a strongly
convergent subsequence of {T xn }, or
ii demicompact, if for any bounded sequence {xn } in DT , whenever {xn − T xn } is
strongly convergent, there exists a strongly convergent subsequence of {xn }.
Let us recall some important iterative processes.
Definition 1.4 Ishikawa iterative process with errors in the sense of Liu 2. Let K be a
nonempty convex subset of E with K K ⊆ K. For any x1 ∈ K, the sequence {xn } is defined
as follows:
xn1 1 − αn xn αn T yn un ,
yn 1 − βn xn βn T xn vn ,
n ≥ 1,
1.7
where {αn } and {βn } are appropriate sequences in 0, 1, and {un }, {vn }, are appropriate
sequences in K.
Fixed Point Theory and Applications
3
If βn vn 0 for all n, then 1.7 reduces to Mann iterative process with errors as
follows:
xn1 1 − αn xn αn T xn un .
1.8
Definition 1.5 Ishikawa iterative process with errors in the sense of Xu 3. Let K be a
nonempty convex subset of E. For any x1 ∈ K, the sequence {xn } is defined as follows:
xn1 1 − αn − γn xn αn T yn γn un ,
yn 1 − βn − δn xn βn T xn δn vn ,
n ≥ 1,
1.9
where {un } and {vn } are bounded sequences in K, and {αn }, {γn }, {βn }, {δn } are real
sequences in 0, 1 satisfying αn γn ≤ 1, βn δn ≤ 1, for all n ≥ 1.
If βn δn 0 for all n, then 1.9 reduces to Mann iterative process with errors as
follows:
xn1 1 − αn − γn xn αn T xn γn un .
1.10
Remark 1.6. i If un vn 0 in 1.7 or γn δn 0 in 1.9, then 1.7 and 1.9 reduce to
Ishikawa iterative process 4,
xn1 1 − αn xn αn T yn ,
yn 1 − βn xn βn T xn ,
n ≥ 1.
1.11
ii If un 0 in 1.8 or γn 0 in 1.10, then 1.8 and 1.10 reduce to Mann iterative
process 5,
xn1 1 − αn xn αn T xn ,
n ≥ 1.
1.12
In 1974, Rhoades 6 proved strong convergence theorem by the Mann iterative
process to a fixed point of strictly pseudocontractive mapping defined on a nonempty
compact convex subset of a Hilbert space. In 2001, Osilike and Udomene 7 proved weak and
strong convergence theorems for strictly pseudocontractive mapping in a real q-uniformly
smooth Banach space E which is also uniform convex.
In 2006, Zeng et al. 8 established the sufficient and necessary conditions on the strong
convergence to a fixed point of strictly pseudocontractive mapping in a real q-uniformly
smooth Banach space. They got the following main results.
Theorem 1.7. Let q > 1 and E be a real q-uniformly smooth Banach space, let K be a nonempty closed
convex subset of E with K K ⊆ K, and let T : K → K be a strictly pseudocontractive mapping
with FT /
∅. Let {un } be a bounded sequence in K. Let {αn } and {βn } be real sequences in 0, 1
4
Fixed Point Theory and Applications
satisfying the following conditions:
i
∞
n1
un < ∞;
ii αn ≤ λq/cq 1/q−1 , and
depending on q.
∞
n1
βnτ < ∞, where τ min{1, q − 1} and cq is a constant
From an arbitrary x1 ∈ K, let {xn } be defined by the following:
xn1 1 − αn xn αn T yn un ,
yn 1 − βn xn βn T xn , n ≥ 1.
1.13
Then {xn } converges strongly to a fixed point of T if and only if {xn } is bounded and
lim infn → ∞ dxn , FT 0, where dx, FT infp∈FT x − p.
In the end of Zeng et al. 8, they raised an open question.
Open Question 1. Can the Ishikawa iterative process with errors 1.7 be extended to Theorem
1.7?
At the same year, Zeng et al. 9 proved the following strong convergence theorem for
strictly pseudocontractive mappings.
Theorem 1.8. Let q > 1 and E be a real q-uniformly smooth Banach space. Let K be a nonempty closed
convex subset of E, and let T : K → K be compact or demicompact, and strictly pseudocontractive
with FT /
∅. Let {un } be a bounded sequence in K. Let {αn }, {βn }, and {γn } be real sequences in
0, 1 satisfying the following conditions:
i αn γn ≤ 1, for all n ≥ 1;
ii limn → ∞ αn < λq/cq 1/q−1 , limn → ∞ βn < 1/L and
iii
∞
n1 γn
< ∞ and
∞
n1
∞
n1
αn ∞;
αn βnτ < ∞, where τ min{1, q − 1}.
From an arbitrary x1 ∈ K, let {xn } be defined by the following:
xn1 1 − αn − γn xn αn T yn γn un ,
yn 1 − βn xn βn T xn , n ≥ 1.
1.14
If {xn } is the bounded sequence, then {xn } converges strongly to a fixed point of T .
They raised another open question.
Open Question 2. Can the Ishikawa iterative process with errors 1.9 be extended to Theorem
1.8?
Fixed Point Theory and Applications
5
We have answered the Open Question 1 in 10. The purpose of this paper is to
answer the Open Question 2, and we prove some strong convergence theorems for strictly
pseudocontractive mapping in Banach spaces, which improve Theorem 1.8 in the following:
i q-uniformly smooth Banach spaces can be replaced by general Banach spaces.
ii Remove the boundedness assumption of {xn }.
iii Iterative process 1.14 can be replaced by Ishikawa iterative process with errors
1.9.
Respectively, our results improve and generalize the corresponding results of Zeng el al. 8,
Osilike and Udomene 7, and others.
In the sequel, we will need the following lemmas.
Lemma 1.9 see 11. Let {an }, {bn }, {cn } be sequences of nonnegative real numbers satisfying the
inequality
an1 ≤ 1 cn an bn ,
n ≥ 1.
1.15
∞
If ∞
n1 cn < ∞,
n1 bn < ∞, we have (i) limn → ∞ an exists. (ii) In particular, if lim infn → ∞ an 0, then limn → ∞ an 0.
∗
Lemma 1.10 see 12. Let E be a Banach space and J : E → 2E be the normalized duality
mapping, then for any x, y ∈ E, the following conclusions hold
i x y2 ≤ x2 2y, jx y, for all jx y ∈ Jx y;
ii x y2 ≥ x2 2y, jx, for all jx ∈ Jx.
2. Main Results
In the rest of paper, we denote by L the Lipschitz constant.
Lemma 2.1. Let K be a nonempty closed convex subset of a real Banach space E. Let T : K → K be a
strictly pseudocontractive mapping with FT / ∅. Let x1 ∈ K; {xn } is defined by 1.9 and satisfying
the following conditions:
i βn ≤ αn , δn ≤ γn , ∞
n1 γn < ∞;
∞ 2
∞
ii n1 αn < ∞, n1 αn ∞.
Then
1 there exist two sequences {rn }, {sn } in 0, ∞, such that
∞, and
xn1 − q ≤ 1 rn xn − q sn ,
Furthermore, limn → ∞ xn − q exists.
∞
n1 rn
∀q ∈ FT , n ≥ 1.
< ∞,
∞
n1 sn
<
2.1
6
Fixed Point Theory and Applications
2 For any integer n, m ≥ 1, there exists a constant M1 > 0, such that
xnm − q ≤ M1 xn − q M1
nm−1
sk ,
∀q ∈ FT .
2.2
kn
Proof. 1 Let q ∈ FT . Since {un } and {vn } are bounded sequences in K, we have
0 < M : max supun − q, supvn − q
n≥1
< ∞.
2.3
n≥1
Since T is a strictly pseudocontractive mapping, by Remark 1.2i,
2
I − T x − I − T y, j x − y ≥ λI − T x − I − T y ≥ 0.
2.4
By Kato 13, the above inequality is equivalent to
x − y ≤ x − y γ I − T x − I − T y ,
∀x, y ∈ K, γ > 0.
2.5
Let an αn γn and from 1.9, we have
xn1 1 − an xn an T yn γn un − T yn .
2.6
xn 1 an xn1 an I − T xn1 − an xn 2a2n xn − T yn
an T xn1 − T yn γn 1 2an T yn − un .
2.7
q 1 an q an I − T q − an q.
2.8
xn − q 1 an xn1 − q an I − T xn1 − I − T q − an xn − q
2a2n xn − T yn an T xn1 − T yn γn 1 2an T yn − un .
2.9
It follows that
Observe that
From 2.7 and 2.8, we have
Fixed Point Theory and Applications
7
By inequality 2.5, we get
xn − q ≥ 1 an xn1 − q an I − T xn1 − I − T q 1 an
− an xn − q − 2a2n xn − T yn − an T xn1 − T yn − γn 1 2an T yn − un 2.10
≥ 1 an xn1 − q − an xn − q − 2a2n xn − T yn − an T xn1 − T yn − γn 1 2an T yn − un .
So
xn1 − q ≤ xn − q 2a2n xn − T yn an T xn1 − T yn γn 1 2an T yn − un .
2.11
Furthermore, set bn βn δn ≤ 1, then
yn 1 − bn xn bn T xn δn vn − T xn .
2.12
We make the following estimations.
yn − q 1 − bn xn − q bn T xn − q δn vn − T xn ≤ 1 L − 1bn xn − q δn vn − T xn ≤ Lxn − q δn Lxn − q δn M
2.13
L1 δn xn − q δn M,
xn − T yn ≤ xn − q q − T yn ≤ xn − q Lyn − q
≤ xn − q L2 1 δn xn − q Lδn M
2.14
1 L2 1 δn xn − q Lδn M,
T yn − un ≤ Lyn − q un − q
≤ L2 1 δn xn − q 1 Lδn M,
2.15
8
Fixed Point Theory and Applications
T xn1 − T yn ≤ Lxn1 − yn Lxn − yn an T yn − xn γn un − T yn ≤ Lxn − yn Lan T yn − xn Lγn T yn − un Lbn xn −T xn δn T xn − vn Lan T yn − xn Lγn T yn − un 2.16
≤ Lbn xn −T xn Lδn T xn − vn Lan T yn − xn Lγn T yn − un ≤ L1 Lbn xn − q L2 δn xn − q Lan 1 L2 1 δn xn − q
L3 γn 1 δn xn − q Lδn M L2 an δn M Lγn 1 Lδn M.
Substituting 2.14, 2.15, and 2.16 in 2.11, we obtain
xn1 − q ≤ xn − q 2a2n 1 L2 1 δn xn − q L1 Lan bn xn − q
an L2 δn xn − q La2n 1 L2 1 δn xn − q
an L3 γn 1 δn xn − q 2a2n Lδn M an Lδn M L2 a2n δn M
2.17
Lan γn 1 Lδn M
1 rn xn − q sn ,
where
rn 2a2n 1 L2 1 δn L1 Lan bn an L2 δn
La2n 1 L2 1 δn an L3 γn 1 δn ,
2.18
sn 2a2n Lδn M an Lδn M L2 a2n δn M Lan γn 1 Lδn M.
∞
By conditions i and ii, we have ∞
n1 rn < ∞,
n1 sn < ∞. It follows from Lemma 1.9
that limn → ∞ xn − q exists. This completes the proof of part 1.
2 If x ≥ 0, then 1 x ≤ ex . For any integer n, m ≥ 1 and from part 1, we have
xnm − q ≤ 1 rnm−1 xnm−1 − q snm−1
≤ ernm−1 ernm−2 xnm−2 − q ernm−1 snm−2 snm−1
···
≤e
nm−1
kn
rk
xn − q e
nm−1
kn
rk
nm−1
sk
kn
≤ M1 xn − q M1
nm−1
sk ,
kn
where M1 e
∞
k1 rk
. This completes the proof of part 2.
2.19
Fixed Point Theory and Applications
9
Lemma 2.2. Let K be a nonempty closed convex subset of a real Banach space E. Let T : K → K be
a strictly pseudocontractive mapping with FT /
∅. Let {xn } be defined as in Lemma 2.1. Then there
exists a subsequence xnj of {xn }, such that
lim xnj − T xnj 0.
2.20
j→∞
Proof. Let q ∈ FT . It follows from 1.3, 1.9, and Lemma 1.10i that
xn1 − q2
2
xn − q αn T yn − xn γn un − xn 2
≤ xn − q 2 αn T yn − xn γn un − xn , j xn1 − q
2
xn − q − 2αn xn1 − T xn1 , j xn1 − q 2αn xn1 − xn , j xn1 − q
2αn T yn − T xn1 , j xn1 − q 2γn un − xn , j xn1 − q
2
≤ xn − q − 2αn λxn1 − T xn1 2 2α2n T yn − xn , j xn1 − q
2αn T yn − T xn1 , j xn1 − q 2αn γn 2γn un − xn , j xn1 − q
2
≤ xn − q − 2αn λxn1 − T xn1 2 2α2n T yn − xn xn1 − q
2αn Lyn − xn1 xn1 − q 2αn γn 2γn un − xn xn1 − q.
2.21
Let
kn 2α2n T yn − xn xn1 − q 2αn Lyn − xn1 xn1 − q
2αn γn 2γn un − xn xn1 − q.
2.22
Then 2.21 becomes
xn1 − q2 ≤ xn − q2 − 2αn λxn1 − T xn1 2 kn .
2.23
From Lemma 2.11, limn → ∞ xn − q exists. So {xn − q} is bounded. By inequalities
2.14, 2.15, and 2.16, the sequences {T yn −xn }, {yn −xn1 }, {un −xn } are all bounded.
∞
∞
2
Notice the conditions of ∞
n1 αn < ∞ and
n1 γn < ∞, then
n1 kn < ∞. It follows from
2.23 that
2 2
2αn λxn1 − T xn1 2 ≤ xn − q − xn1 − q kn ,
2.24
so
n
n
2 2λ αi xi1 − T xi1 2 ≤ x1 − q ki .
i1
Hence,
∞
n1
αn xn1 − T xn1 2 < ∞. Since
2.25
i1
∞
n1
αn ∞, so we have limn → ∞ xn1 −T xn1 0.
10
Fixed Point Theory and Applications
By virtue of Lemma 1.10ii, we obtain
xn1 − T xn1 2
2
1 − αn − γn xn αn T yn γn un − T xn1 2
xn − T xn T xn − T xn1 αn T yn − xn γn un − xn ≥ xn − T xn 2 2 T xn −T xn1 αn T yn − xn γn un − xn , jxn − Txn ,
2.26
therefore,
xn − T xn 2 ≤ xn1 − T xn1 2 2 T xn1 − T xn , jxn − T xn 2αn xn − T yn , jxn − T xn 2γn xn − un , jxn − T xn ≤ xn1 − T xn1 2 2T xn1 − T xn xn − T xn 2.27
2αn xn − T yn xn − T xn 2γn xn − un xn − T xn .
Observe the right side of the above inequality, since
T xn1 − T xn ≤ Lxn1 − xn ≤ αn T yn − xn γn un − xn −→ 0 n −→ ∞,
2.28
and {xn − T xn }, {xn − T yn }, {xn − un } are all bounded. Together with limn → ∞ xn1 −
T xn1 0, then limn → ∞ xn − T xn 0, that is, there exists a subsequence xnj of {xn }, such
that
lim xnj − T xnj 0.
j →∞
2.29
Theorem 2.3. Let K be a nonempty closed convex subset of a real Banach space E. Let T : K → K be
a strictly pseudocontractive mapping with FT / ∅. Let {xn } be defined as in Lemma 2.1, then {xn }
converges strongly to a fixed point of T if and only if lim infn → ∞ dxn , FT 0, where dx, FT infp∈FT x − p.
Proof. The necessity is obvious. So, we will prove the sufficiency. From Lemma 2.11, we
have
xn1 − q ≤ 1 rn xn − q sn ,
∀q ∈ FT , n ≥ 1.
2.30
Therefore,
dxn1 , FT ≤ 1 rn dxn , FT sn .
2.31
Fixed Point Theory and Applications
11
∞
Note that ∞
n1 rn < ∞,
n1 sn < ∞. By Lemma 1.9 and lim infn → ∞ dxn , FT 0, we get
limn → ∞ dxn , FT 0.
Next, we prove {xn } is a cauchy sequence. For each ε > 0, there exists a natural number
n1 , such that
dxn , FT ≤
ε
,
12M1
∀n ≥ n1 ,
2.32
where M1 is the constant in Lemma 2.1 2. Hence, there exists p1 ∈ FT and a natural
number n2 > n1 , such that
xn2 − p1 ≤
ε
,
4M1
∞
ε
.
4M1
sk <
kn2
2.33
From Lemma 2.12 and 2.33, for all n ≥ n2 , we have
xnm − xn ≤ xnm − p1 p1 − xn ≤ 2M1 xn2 − p1 M1
≤ 2M1
nm−1
n−1
kn2
kn2
sk M1
sk
2.34
ε
ε
2M1
ε.
4M1
4M1
Hence, {xn } is a cauchy sequence. Since K is a closed subset of E, so {xn } converges strongly
to a p ∈ K.
Finally, we prove p ∈ FT . In fact, since dp, FT 0. So, for any ε1 > 0, there exists
p ∈ FT , such that p − p < ε1 . Then we have
T p − p ≤ T p − p p − p
≤ 1 Lε1 .
2.35
By the arbitrary of ε1 , we know that T p − p 0. Therefore, p ∈ FT .
A mapping T : K → K is said to satisfy ConditionA 14, if there exists a
nondecreasing function f : 0, ∞ → 0, ∞ with f0 0, fr > 0, for all r ∈ 0, ∞
such that x − T x ≥ fdx, FT for all x ∈ K.
Theorem 2.4. Let K be a nonempty closed convex subset of a real Banach space E. Let T : K → K
be a strictly pseudocontractive mapping with FT /
∅, and satisfy Condition(A). Let {xn } be defined
as in Lemma 2.1. Then {xn } converges strongly to a fixed point of T .
12
Fixed Point Theory and Applications
Proof. By Lemma 2.2, there exists a subsequence xnj of {xn }, such that
lim xnj − T xnj 0.
2.36
j →∞
By ConditionA, limj → ∞ fdxnj , FT 0. Since f is a nondecreasing function and f0 0, therefore limj → ∞ dxnj , FT 0. The rest of the proof is the same to Theorem 2.3.
Theorem 2.5. Let K be a nonempty closed convex subset of a real Banach space E. Let T : K → K be
a compact and strictly pseudocontractive mapping with FT /
∅. Let {xn } be defined as in Lemma 2.1.
Then {xn } converges strongly to a fixed point of T .
Proof. From Lemma 2.1 1, it follows that limn → ∞ xn − q exists, for any q ∈ FT . By
Lemma 2.2, there exists a subsequence {xnj } of {xn } such that limj → ∞ xnj − T xnj 0. Since
{xnj } is bounded and T is compact, {T xnj } has a strongly convergent subsequence. Without
loss of generality, we may assume that {T xnj } converges strongly to p ∈ K. Next, we prove
p ∈ FT .
j −→ ∞ ,
xnj − p ≤ xnj − T xnj T xnj − p −→ 0
2.37
that is, limj → ∞ xnj − p 0. By the Lipschitz continuity of T , it follows that
p − T p ≤ p − xnj xnj − T p −→ 0
j −→ ∞ .
2.38
This means that p ∈ FT . By Lemmas 1.9ii and 2.1i, the sequence {xn } converges strongly
to p ∈ FT .
Theorem 2.6. Let K be a nonempty closed convex subset of a real Banach space E. Let T : K → K
be a demicompact and strictly pseudocontractive mapping with FT /
∅. Let {xn } be defined as in
Lemma 2.1. Then {xn } converges strongly to a fixed point of T .
Proof. From Lemma 2.11, it follows that limn → ∞ xn − q exists, for any q ∈ FT . By
Lemma 2.2, there exists a subsequence {xnj } of {xn } such that limj → ∞ xnj − T xnj 0. Since
{xnj } is bounded together with T being demicompact, there exists a subsequence of {xnj }
which converges strongly to some p ∈ K. Taking into account that limj → ∞ xnj − T xnj 0
and the Lipschitz continuity of T , we have p ∈ FT . By Lemma 1.9, {xn } converges strongly
to p ∈ FT .
Acknowledgments
This work was supported by National Natural Science Foundations of China 60970149 and
the Natural Science Foundations of Jiangxi Province 2009GZS0021, 2007GQS2063.
Fixed Point Theory and Applications
13
References
1 F. E. Browder and W. V. Petryshyn, “Construction of fixed points of nonlinear mappings in Hilbert
space,” Journal of Mathematical Analysis and Applications, vol. 20, pp. 197–228, 1967.
2 L. S. Liu, “Ishikawa and Mann iterative process with errors for nonlinear strongly accretive mappings
in Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 194, no. 1, pp. 114–125, 1995.
3 Y. Xu, “Ishikawa and Mann iterative processes with errors for nonlinear strongly accretive operator
equations,” Journal of Mathematical Analysis and Applications, vol. 224, no. 1, pp. 91–101, 1998.
4 S. Ishikawa, “Fixed points by a new iteration method,” Proceedings of the American Mathematical Society,
vol. 44, no. 1, pp. 147–150, 1974.
5 W. R. Mann, “Mean value methods in iteration,” Proceedings of the American Mathematical Society, vol.
4, pp. 506–510, 1953.
6 B. E. Rhoades, “Fixed point iterations using infinite matrices,” Transactions of the American
Mathematical Society, vol. 196, pp. 161–176, 1974.
7 M. O. Osilike and A. Udomene, “Demiclosedness principle and convergence theorems for strictly
pseudocontractive mappings of Browder-Petryshyn type,” Journal of Mathematical Analysis and
Applications, vol. 256, no. 2, pp. 431–445, 2001.
8 L.-C. Zeng, N.-C. Wong, and J.-C. Yao, “Strong convergence theorems for strictly pseudocontractive
mappings of Browder-Petryshyn type,” Taiwanese Journal of Mathematics, vol. 10, no. 4, pp. 837–849,
2006.
9 L. C. Zeng, G. M. Lee, and N. C. Wong, “Ishikawa iteration with errors for approximating fixed points
of strictly pseudocontractive mappings of Browder-Petryshyn type,” Taiwanese Journal of Mathematics,
vol. 10, no. 1, pp. 87–99, 2006.
10 Y. C. Tang and J. G. Peng, “On the strong convergence of the Ishikawa process with errors for a strictly
pseudocontractive mapping of Browder-Petryshyn type,” submitted.
11 M. O. Osilike and S. C. Aniagbosor, “Weak and strong convergence theorems for fixed points of
asymptotically nonexpansive mappings,” Mathematical and Computer Modelling, vol. 32, no. 10, pp.
1181–1191, 2000.
12 S.-S. Chang, “Some problems and results in the study of nonlinear analysis,” Nonlinear Analysis:
Theory, Methods & Applications, vol. 30, no. 7, pp. 4197–4208, 1997.
13 T. Kato, “Nonlinear semigroups and evolution equations,” Journal of the Mathematical Society of Japan,
vol. 19, pp. 508–520, 1964.
14 H. F. Senter and W. G. Dotson Jr., “Approximating fixed points of nonexpansive mappings,”
Proceedings of the American Mathematical Society, vol. 44, no. 2, pp. 375–380, 1974.