Journal of Inequalities in Pure and
Applied Mathematics
STRONG CONVERGENCE THEOREMS FOR ITERATIVE SCHEMES
WITH ERRORS FOR ASYMPTOTICALLY DEMICONTRACTIVE
MAPPINGS IN ARBITRARY REAL NORMED LINEAR SPACES
volume 3, issue 5, article 83,
2002.
1
2
YEOL JE CHO , HAIYUN ZHOU AND SHIN MIN KANG
1 Department
of Mathematics,
Gyeongsang National University,
Chinju 660-701, Korea.
EMail: yjcho@nongae.gsnu.ac.kr
2 Department
of Mathematics,
Shijiazhuang Mechanical Engineering College,
Shijiazhuang 050003,
People’s Republic of China.
EMail: luyao_846@163.com
1
Received 30 April, 2002;
accepted 5 September, 2002.
Communicated by: S.S. Dragomir
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
040-02
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Abstract
In the present paper, by virtue of new analysis technique, we will establish several strong convergence theorems for the modified Ishikawa and Mann iteration
schemes with errors for a class of asymptotically demicontractive mappings in
arbitrary real normed linear spaces. Our results extend, generalize and improve the corresponding results obtained by Igbokwe [1], Liu [2], Osilike [3] and
others.
2000 Mathematics Subject Classification: Primary 47H17; Secondary 47H05, 47H10
Key words: Asymptotically demicontractive mapping; Modified Mann and Ishikawa
iteration schemes with errors; arbitrary linear space.
Strong Convergence Theorems
for Iterative Schemes with
Errors for Asymptotically
Demicontractive Mappings in
Arbitrary Real Normed Linear
Spaces
This work was supported by grant No. (2000-1-10100-003-3) from the Basic Research Program of the Korea Science & Engineering Foundation.
Yeol Je Cho, Haiyun Zhou and
Shin Min Kang
Contents
Title Page
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2
The Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
References
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1.
Introduction
Let X be a real normed linear space and let J denote the normalized duality
∗
mapping from X into 2X given by
J(x) = {f ∈ X ∗ : hx, f i = kxk2 = kf k2 },
x ∈ X,
where X ∗ denotes the dual space of X and h·, ·i denotes the generalized duality
pairing of elements between X and X ∗ .
Let F (T ) denote the set of all fixed points of a mapping T . Let C be a
nonempty subset of X.
A mapping T : C → C is said to be k-strictly asymptotically pseudocontractive with a sequence {kn } ⊂ [0, ∞), kn ≥ 1 and kn → 1 as n → ∞ if there
exists k ∈ [0, 1) such that
(1.1)
n
n
2
kT x − T yk ≤
kn2 kx
2
n
n
2
− yk + k k(x − T x) − (y − T y)k
for all n ≥ 1 and x, y ∈ C.
The mapping T is said to be asymptotically demicontractive with a sequence
{kn } ⊂ [1, ∞) and limn→∞ kn = 1 if F (T ) 6= ∅ and there exists k ∈ [0, 1) such
that
(1.2)
kT n x − pk2 ≤ kn2 kx − pk2 + kkx − T n xk2
for all n ≥ 1, x ∈ C and p ∈ F (T ).
The classes of k-strictly asymptotically pseudocontractive and asymptotically demicontractive mappings, as a natural extension to the class of asymptotically non-expansive mappings, were first introduced in Hilbert spaces by Liu
Strong Convergence Theorems
for Iterative Schemes with
Errors for Asymptotically
Demicontractive Mappings in
Arbitrary Real Normed Linear
Spaces
Yeol Je Cho, Haiyun Zhou and
Shin Min Kang
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[5] in 1996. By using the modified Mann iterates introduced by Schu [4, 5],
he established several strong convergence results concerning an iterative approximation to fixed points of k-strictly asymptotically pseudocontractive and
asymptotically demicontractive mappings in Hilbert spaces. In 1998, Osilike
[3], by virtue of normalized duality mapping, first extended the concepts of
k-strictly asymptotically pseudocontractive and asymptotically demicontractive
maps from Hilbert spaces to the much more general Banach spaces, and then
proved the corresponding convergence theorems which generalized the results
of Liu [2].
A mapping T : C → C is said to be k-strictly asymptotically pseudocontractive with a sequence {kn } ⊂ [0, ∞), kn ≥ 1 and kn → 1 as n → ∞ if there
exist k ∈ [0, 1) and j(x − y) ∈ J(x − y) such that
(1.3) h(I − T n )x − (I − T n )y, j(x − y)i
1
1
≥ (1 − k)k(I − T n )x − (I − T n )yk2 − (kn2 − 1)kx − yk2
2
2
for all n ≥ 1 and x, y ∈ C.
The mapping T is called an asymptotically demicontractive mapping with a
sequence {kn } ⊂ [0, ∞), limn→∞ kn = 1 if F (T ) 6= ∅ and there exist k ∈ [0, 1)
and j(x − y) ∈ J(x − y) such that
(1.4)
1
1
hx − T n x, j(x − p)i ≥ (1 − k)kx − T n xk2 − (kn2 − 1)kx − pk2
2
2
for all n ≥ 1, x ∈ C and p ∈ F (T ).
Strong Convergence Theorems
for Iterative Schemes with
Errors for Asymptotically
Demicontractive Mappings in
Arbitrary Real Normed Linear
Spaces
Yeol Je Cho, Haiyun Zhou and
Shin Min Kang
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Furthermore, T is said to be uniformly L-Lipschitzian if there is a constant
L ≥ 1 such that
(1.5)
kT n x − T n yk ≤ Lkx − yk
for all x, y ∈ C and n ≥ 1.
Remark 1.1. The definitions above may be stated in the setting of a real normed
linear space. In the case of X being a Hilbert space, (1.1) and (1.2) are equivalent to (1.3) and (1.4), respectively.
Recall that there are two iterative schemes with errors which have been used
extensively by various authors.
Let X be a normed linear space, C be a nonempty convex subset of X and
T : C → C be a given mapping. Then the modified Ishikawa iteration scheme
{xn } with errors is defined by

x1 ∈ C,





yn = (1 − βn )xn + βn T n (xn ) + vn ,
n ≥ 1,





xn+1 = (1 − αn )xn + αn T n (yn ) + un , n ≥ 1,
where {αn }, {βn } are some suitable sequences in [0, 1] and {un }, {vn } are two
summable sequences in X.
With X, C, {αn } and x1 as above, the modified Mann iteration scheme {xn }
with errors is defined by

 x1 ∈ C,
(1.6)

xn+1 = (1 − αn )xn + αn T n (xn ) + un , n ≥ 1.
Strong Convergence Theorems
for Iterative Schemes with
Errors for Asymptotically
Demicontractive Mappings in
Arbitrary Real Normed Linear
Spaces
Yeol Je Cho, Haiyun Zhou and
Shin Min Kang
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Let X, C and T be as in above. Let {an }, {bn }, {cn }, {a0n }, {b0n } and {c0n }
be real sequences in [0, 1] satisfying an +bn +cn = 1 = a0n +b0n +c0n and let {un }
and {vn } be bounded sequences in C. Define the modified Ishikawa iteration
schemes {xn } with errors generated from an arbitrary x1 ∈ C as follows:

 yn = an xn + bn T n (xn ) + cn un , n ≥ 1,
(1.7)

xn+1 = a0n xn + b0n T n yn + c0n vn , n ≥ 1.
In particular, if we set bn = cn = 0 in (1.7), we obtain the modified Mann
iteration scheme {xn } with errors given by

 x1 ∈ C,
(1.8)

xn+1 = a0n xn + b0n T n xn + c0n vn , n ≥ 1.
Osilike [3] proved the following convergence theorems for k-strictly asymptotically demicontractive mappings:
Theorem 1.1. [3] Let q > 1 and let E be a real q-uniformly smooth Banach space. Let K be a nonempty closed convex and bounded subset of E
and T : K → K be a completely continuous and uniformly L-Lipschitzian
asymptotically demicontractive mapping
with a sequence {kn } ⊂ [1, ∞) for all
P∞
n ≥ 1, kn → 1 as n → ∞ and n=1 (kn 2 − 1) < ∞. Let {αn } and {βn } be
real sequences in [0, 1] satisfying the conditions:
(i) 0 < ≤ cq αn
some > 0,
q−1
≤
1
{q(1
2
−(q−2)
− k)(1 + L)
} − for all n ≥ 1 and for
Strong Convergence Theorems
for Iterative Schemes with
Errors for Asymptotically
Demicontractive Mappings in
Arbitrary Real Normed Linear
Spaces
Yeol Je Cho, Haiyun Zhou and
Shin Min Kang
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(ii)
P∞
n=1
βn < ∞.
Then the sequence {xn } defined by (1.6) with un ≡ 0 and vn ≡ 0 for all
n ≥ 1 converges strongly to a fixed point of T .
Very recently, Igbokwe [2] extended the above Theorem 1.1 to Banach spaces.
More precisely, he proved the following results:
Theorem 1.2. [2] Let E be a real Banach space and K be a nonempty closed
convex subset of E. Let T : K → K be a completely continuous and uniformly L-Lipschitzian asymptotically demicontractive P
mapping with a sequence
2
{kn } ⊂ [1, ∞) for all n ≥ 1, kn → 1 as n → ∞ and ∞
n=1 (kn − 1) < ∞. Let
the sequence {xn } be defined by (1.7) with the restrictions that
(i) an + bn + cn = 1 = a0n + b0n + c0n ,
P
0
(ii) ∞
n=0 bn = ∞,
P
P∞ 0
P∞
P∞
0 2
(iii) ∞
n=0 (b ) < ∞,
n=0 cn < ∞,
n=0 bn < ∞ and
n=0 cn < ∞.
Then the modified Ishikawa iteration {xn } defined by (1.6) and (1.7) converges strongly to a fixed point p of T .
It is our purpose in this paper to extend and improve the above Theorem
1.2 from Banach spaces to real normed linear spaces. In the case of Banach
spaces, we use the Condition (A) to replace the assumption that T is completely
continuous.
In the sequel, we will need the following lemmas:
Strong Convergence Theorems
for Iterative Schemes with
Errors for Asymptotically
Demicontractive Mappings in
Arbitrary Real Normed Linear
Spaces
Yeol Je Cho, Haiyun Zhou and
Shin Min Kang
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Lemma 1.3. Let X be a normed linear space and C be a nonempty convex
subset of X. Let T : C → C be a uniformly L-Lipschitzian mapping and the
sequence {xn } be defined by (1.7). Then we have
(1.9)
kT xn − xn k ≤ kT n xn − xn k + L(1 + L2 )kT n−1 xn−1 − xn−1 k
+ L(1 + L)c0n−1 kvn−1 − xn−1 k
+ L2 (1 + L)cn−1 kun−1 − xn k
+ Lc0n−1 kxn−1 − T n−1 xn−1 k, n ≥ 1.
Proof. See Igbokwe [1, Lemma 1].
Lemma 1.4. Let X be a normed linear space and C be a nonempty convex
subset of X. Let T : C → C be a uniformly L-Lipschitzian and asymptotically
P∞ demicontractive mapping with a sequence {kn } such that kn ≥ 1 and
n=1 (kn − 1) < ∞. Let the sequence {xn } be defined by (1.7) with the restrictions
∞
X
n=1
b0n = ∞,
∞
X
2
(b0n )
< ∞,
n=1
∞
X
c0n < ∞,
n=1
Then we have the following conclusions:
(i) limn→∞ kxn − pk exists for any p ∈ F (T ).
∞
X
n=1
Strong Convergence Theorems
for Iterative Schemes with
Errors for Asymptotically
Demicontractive Mappings in
Arbitrary Real Normed Linear
Spaces
Yeol Je Cho, Haiyun Zhou and
Shin Min Kang
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(ii) limn→∞ d(xn , F (T )) exists.
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(iii) lim inf n→∞ kxn − T xn k = 0.
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Proof. It is very clear that (1.7) is equivalent to the following:

 yn = (1 − bn )xn + bn T n (xn ) + cn (un − xn ), n ≥ 1,
(1.10)

xn+1 = (1 − b0n )xn + b0n T n yn + c0n (vn − xn ), n ≥ 1.
For any p ∈ F (T ), let M > 0 be such that
M = max{sup{kun − pk}, sup{kvn − pk}}.
Observe first that
(1.11) kyn − pk ≤ (1 − bn )kxn − pk + bn Lkxn − pk + cn (M + kxn − pk)
≤ (1 + L)kxn − pk + M
and
(1.12)
Yeol Je Cho, Haiyun Zhou and
Shin Min Kang
kT n yn − xn k ≤ Lkyn − pk + kxn − pk
≤ L[(1 + L)kxn − pk + M ] + kxn − pk
≤ [1 + L(1 + L)]kxn − pk + M L.
Observe also that
(1.13)
Strong Convergence Theorems
for Iterative Schemes with
Errors for Asymptotically
Demicontractive Mappings in
Arbitrary Real Normed Linear
Spaces
b0n [kT n yn
kxn+1 − yn k ≤ kxn − yn k +
− pk + kyn − pk]
0
+ cn [kvn − pk + kyn − pk]
≤ [bn (1 + L) + cn ]kxn − pk + (cn + c0n )M
+ [b0n (1 + L) + c0n ][(1 + L)kxn − pk + M ]
≤ σn kxn − pk + ςn ,
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where
σn = [bn (1 + L) + cn ] + [b0n (1 + L) + c0n ](1 + L)
and
ςn = M [b0n (1 + L) + 2c0n + cn ].
Thus we have
(1.14)
kxn+1 − xn k ≤ b0n kT n yn − xn k + c0n (M + kxn − pk)
≤ b0n σn kxn − pk + ςn + c0n (M + kxn − pk).
Using iterates (1.10), we have
(1.15) kxn+1 − pk2 ≤ kxn − pkkxn+1 − pk − b0n hxn − T n yn , j(xn+1 − p)i
+ c0n hvn − xn , j(xn+1 − p)ik
1
1
≤ kxn − pk2 + kxn+1 − pk2
2
2
− b0n hxn+1 − T n xn+1 , j(xn+1 − p)i
+ b0n hxn+1 − xn + T n yn − T n xn+1 , j(xn+1 − p)i
+ c0n (M + kxn − pk)kxn+1 − pk,
which implies that
(1.16) kxn+1 − pk2 ≤ kxn − pk2 − (1 − k)b0n kxn+1 − T n xn+1 k2
+ b0n (kn2 − 1)kxn+1 − pk2 + 2b0n (kxn+1 − xn k
+ kT n xn+1 − T n yn k)kxn+1 − pk
+ 2c0n (M + kxn − pk)kxn+1 − pk.
Strong Convergence Theorems
for Iterative Schemes with
Errors for Asymptotically
Demicontractive Mappings in
Arbitrary Real Normed Linear
Spaces
Yeol Je Cho, Haiyun Zhou and
Shin Min Kang
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Substituting (1.12) – (1.14) in (1.16) and, after some calculations, we obtain
kxn+1 − pk2 ≤ (1 + γn )kxn − pk2 − (1 − k)b0n kxn+1 − T n xn+1 k2
P
for all very large n, where the sequence {γn } satisfies that ∞
n=1 γn < ∞. A
direct induction of (1.17) leads to
(1.17)
kxn+1 − pk2 ≤ kxn − pk2 + M γn − b0n kxn+1 − T n xn+1 k2 ,
(1.18)
which implies that limn→∞ kxn − pk exists by Tan and Xu [7, Lemma 1] and so
this proves the claim (i). The claim (ii) follows from (1.18).
Now, we prove the claim (iii). It follows from (1.18) that
∞
X
b0n kxn+1 − T n xn+1 k2 < ∞
n=1
and hence
Strong Convergence Theorems
for Iterative Schemes with
Errors for Asymptotically
Demicontractive Mappings in
Arbitrary Real Normed Linear
Spaces
Yeol Je Cho, Haiyun Zhou and
Shin Min Kang
lim inf kxn+1 − T n xn+1 k = 0
n→∞
0
n bn
since
= ∞. Therefore, we have lim inf n→∞ kxn − T n xn k = 0 by (1.7)
and so lim inf n→∞ kxn − T xn k = 0 by Lemma 1.3. This completes the proof.
P
A mapping T : C → C with a nonempty fixed point set F (T ) in C will
be said to satisfy the Condition (A) on C if there is a nondecreasing function
f : [0, ∞) → [0, ∞) with f (0) = 0 and f (r) > 0 for all r ∈ (0, ∞) such that
kx − T xk ≥ f (d(x, F (T )))
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for all x ∈ C.
Page 11 of 16
2.
The Main Results
Now we prove the main results of this paper.
Theorem 2.1. Let X be a real normed linear space, C be a nonempty closed
convex subset of X and T : C → C be a completely continuous and uniformly L-Lipschitzian asymptotically demicontractive
mapping with a sequence
P
(k
−
1)
< ∞. Let the sequence
{kn } ⊂ [1, ∞) such that kn ≥ 1 and ∞
n=1 n
{xn } be defined by (1.7) with the restrictions
∞
X
n=1
b0n = ∞,
∞
X
2
b0n < ∞,
n=1
∞
X
c0n < ∞,
n=1
∞
X
cn < ∞.
n=1
Then {xn } converges strongly to a fixed point p of T .
Proof. It follows from Lemma 1.4 that
lim inf kxn − T xn k = 0.
n→∞
Since T is completely continuous, we see that there exists an infinite subsequence {xnk } such that {xnk } converges strongly for some p ∈ C and T p = p.
This shows that p ∈ F (T ). However, limn→∞ kxn − pk exists for any p ∈ F (T )
and so we must have that the sequence {xn } converges strongly to p. This
completes the proof.
Theorem 2.2. Let X be a real Banach space, C be a nonempty closed convex
subset of X and T : C → C be a uniformly L-Lipschitzian and asymptotically
demicontractive mapping with a sequence {kn } ⊂ [1, ∞) such that kn ≥ 1
Strong Convergence Theorems
for Iterative Schemes with
Errors for Asymptotically
Demicontractive Mappings in
Arbitrary Real Normed Linear
Spaces
Yeol Je Cho, Haiyun Zhou and
Shin Min Kang
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P
and ∞
n=1 (kn − 1) < ∞. Let the sequence {xn } be defined by (1.7) with the
restrictions
∞
∞
∞
∞
X
X
X
X
2
b0n = ∞,
b0n < ∞,
c0n < ∞,
cn < ∞.
n=1
n=1
n=1
n=1
Suppose in addition that T satisfies the Condition (A), then the sequence {xn }
converges strongly to a fixed point p of T .
Proof. By Lemma 1.4, we see that
lim inf kxn − T xn k = 0.
n→∞
Since T satisfies the Condition (A), we have
lim inf f (d(xn , F (T ))) = 0
n→∞
and hence
Strong Convergence Theorems
for Iterative Schemes with
Errors for Asymptotically
Demicontractive Mappings in
Arbitrary Real Normed Linear
Spaces
Yeol Je Cho, Haiyun Zhou and
Shin Min Kang
lim inf d(xn , F (T )) = 0.
n→∞
By Lemma 1.4 (ii), we conclude that d(xn , F (T )) → 0 as n → ∞.
Now we can take an infinite subsequence {xnj } of {xP
n } and a sequence
−j
{pj } ⊂ F (T ) such that kxnj − pj k ≤ 2 . Set M = exp { ∞
n=1 γn } and write
nj+1 = nj + l for some l ≥ 1. Then we have
(2.1)
kxnj+1 − pj k = kxnj +l − pj k
≤ [1 + γnj +l−1 ]kxnj +l−1 − pj k
( l−1
)
X
M
≤ exp
γnj +m kxnj − pj k ≤ j .
2
m=0
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It follows from (2.1) that
2M + 1
.
2j+1
Hence {pj } is a Cauchy sequence. Assume that pj → p as j → ∞. Then
p ∈ F (T ) since F (T ) is closed and this in turn implies that xj → p as j → ∞.
This completes the proof.
kpj+1 − pj k ≤
Remark 2.1. We remark that, if T : C → C is completely continuous, then it
must be demicompact (cf. [6]) and, if T is continuous and demicompact, it must
satisfy the Condition (A) (cf. [6]). In view of this observation, our Theorem 2.1
improves Theorem 1.2 in the following aspects:
(i) X may be not a Banach space.
(ii) T may be not completely continuous.
Strong Convergence Theorems
for Iterative Schemes with
Errors for Asymptotically
Demicontractive Mappings in
Arbitrary Real Normed Linear
Spaces
Yeol Je Cho, Haiyun Zhou and
Shin Min Kang
(iii) Our proof methods are simpler than those of Igbokwe [1, Theorem 2].
As corollaries of Theorems 2.1 and 2.2, we have the following:
Corollary 2.3. Let X be a real normed linear space, C be a nonempty closed
convex subset of X and T : C → C be a completely continuous and uniformly L-Lipschitzian asymptotically demicontractive
mapping with a sequence
P
{kn } ⊂ [0, ∞) such that kn ≥ 1 and ∞
(k
−
1)
< ∞. Let the sequence
n=1 n
{xn } be defined by (1.8) with the restrictions
∞
X
n=1
b0n = ∞,
∞
X
n=1
2
b0n < ∞,
∞
X
n=1
Then {xn } converges strongly to a fixed point p of T .
c0n < ∞.
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Proof. By taking bn , cn ≡ 0 for n ≥ 1 in Theorem 2.1, we can obtain the
desired conclusion.
Corollary 2.4. Let X be a real Banch space, C be a nonempty closed convex
subset of X and T : C → C be a uniformly L-Lipschitzian asymptotically
demicontractive
mapping with a sequence {kn } ⊂ [0, ∞) such that kn ≥ 1
P∞
and n=1 (kn − 1) < ∞. Let the sequence {xn } be defined by (1.8) with the
restrictions
∞
∞
∞
X
X
X
0
2
bn = ∞,
bn < ∞,
c0n < ∞.
n=1
n=1
n=1
If T satisfies the Condition (A) on the sequence {xn }, then {xn } converges
strongly to a fixed point p of T .
Proof. It follows from Theorem 2.2 by taking bn , cn ≡ 0 for all n ≥ 1.
Remark 2.2. Using the same methods as in Lemma 1.4, Theorems 2.1 and
2.2, we can prove several convergence results similar to Theorems 2.1 and 2.2
concerning on the modified Ishikawa iteration schemes with errors defined by
(1.6).
Remark 2.3. Igbokwe [1, Corollary 1] has shown that, if T : C → C is asymptotically pseudocontractive, then it must be uniformly L-Lipschitzian and hence
our Theorems 2.1 and 2.2 hold for asymptotically pseudocontractive mappings
with a nonempty fixed point set F (T ).
Strong Convergence Theorems
for Iterative Schemes with
Errors for Asymptotically
Demicontractive Mappings in
Arbitrary Real Normed Linear
Spaces
Yeol Je Cho, Haiyun Zhou and
Shin Min Kang
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References
[1] D.I. IGBOKWE, Approximation of fixed points of asymptotically demicontractive mappings in arbitrary Banach spaces, J.
Ineq. Pure and Appl. Math., 3(1) (2002), Article 3. [ONLINE:
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Strong Convergence Theorems
for Iterative Schemes with
Errors for Asymptotically
Demicontractive Mappings in
Arbitrary Real Normed Linear
Spaces
Yeol Je Cho, Haiyun Zhou and
Shin Min Kang
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