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Convergence of the modified Mann's iterative method for asymptotically
kappa-strictly pseudocontractive mappings
Fixed Point Theory and Applications 2011, 2011:100 doi:10.1186/1687-1812-2011-100
Ying Zhang ()
Zhiwei Xie ()
ISSN 1687-1812
Article type Research
Submission date 4 May 2011
Acceptance date 9 December 2011
Publication date 9 December 2011
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Convergence of the modified Mann’s iterative
method for asymptotically κ-strictly
pseudocontractive mappings
Ying Zhang
∗,1,2
and Zhiwei Xie
3
1
School of Mathematics and Physics,
North China Electric Power University, Baoding, Hebei 071003, P.R. China

: x, f = x
2
= f
2
}, for all x ∈ E, where ·, · denotes the duality
1
pairing between E and E
∗
. In the sequel, we will denote the set of fixed points of a mapping
T : K → K by F (T ) = {x ∈ K : Tx = x}.
A mapping T : K → K is called asymptotically κ-strictly pseudocontractive with sequence
{κ
n
}
∞
n=1
⊆ [1, ∞) such that lim
n→∞
κ
n
= 1 (see, e.g., [1–3]) if for all x, y ∈ K, there exist a
constant κ ∈ [0, 1) and j(x − y) ∈ J(x − y) such that
T
n
x − T
n
y, j(x − y) ≤ κ
n
x − y
2

y
2
≤ λ
n
x − y
2
+ λx − y − (T
n
x − T
n
y)
2
,
where lim
n→∞
λ
n
= lim
n→∞
[1 + 2(κ
n
− 1)] = 1, λ = (1 − 2κ) ∈ [0, 1).
A mapping T with domain D(T) and range R(T ) in E is called strictly pseudocontractive of
Browder–Petryshyn type [4], if for all x, y ∈ D(T ), there exists κ ∈ [0, 1) and j(x−y) ∈ J(x − y)
such that
T x − Ty, j(x − y) ≤ x − y
2
− κx − y − (Tx − T y)
2
. (3)

κ-strictly pseudocontractive mappings in Hilbert space. They obtained a weak convergence
theorem of modified Mann iterative processes for this class of mappings. Moreover, a strong
convergence theorem was also established in a real Hilbert space by hybrid projection method.
They proved the following.
Theorem KX [6] Let K be a closed and convex subset of a Hilbert space H. Let T : K → K
be an asymptotically κ-strictly pseudocontractive mapping for some 0 ≤ κ < 1 with sequence
{κ
n
} ⊂ [1, ∞) such that

∞
n=1
(κ
n
−1) < ∞ and F (T ) = ∅. Let {x
n
}
∞
n=1
be a sequence generated
by the modified Mann’s iteration method:
x
n+1
= α
n
x
n
+ (1 − α
n
)T

1
n
, n ≥ 1. These are
motivations for us to improve the results. We prove the demiclosedness principle and weak
convergence theorem of the modified Mann’s algorithm for T in the framework of uniformly
convex Banach spaces which have the Fr´echet differentiable norm. Moreover, we also elicit a
necessary and sufficient condition that guarantees strong convergence of the modified Mann’s
iterative sequence to a fixed point of T in a real Banach spaces with the Fr ´echet differentiable
norm.
We will use the notation:
1.  for weak convergence.
2. ω
W
(x
n
) = {x : ∃x
n
j
 x} denotes the weak ω-limit set of {x
n
}.
2 Preliminaries
Let E be a real Banach space. The space E is called uniformly convex if for each  > 0,
there exists a δ > 0 such that for x, y ∈ E with x ≤ 1, y ≤ 1, x − y ≥ , we have

1
2
(x + y) ≤ 1 − δ. The modulus of convexity of E is defined by
δ
E

exists and is attained uniformly in y ∈ U. In this case, there exists an increasing function
b : [0, ∞) → [0, ∞) with lim
t→0
[b(t)/t] = 0 such that for all x, h ∈ E
1
2
x
2
+ h, j(x) ≤
1
2
x + h
2
≤
1
2
x
2
+ h, j(x) + b(h). (5)
It is well known (see, for example, [12, p. 107]) that uniformly smooth Banach space has a
Fr´echet differentiable norm.
Lemma 2.1 [2, p. 80] Let {a
n
}
∞
n=1
, {b
n
}
∞

n→∞
a
n
exists. If in addition {a
n
}
∞
n=1
has a
subsequence which converges strongly to zero, then lim
n→∞
a
n
= 0.
Lemma 2.2 [2, p. 78] Let E be a real Banach space, K a nonempty subset of E, and
T : K → K an asymptotically κ-strictly pseudocontractive mapping. Then, T is uniformly
L-Lipschitzian.
Lemma 2.3 [13, p. 29] Let K be a nonempty, closed, convex, and bounded subset of a
uniformly convex Banach space E, and let T : K → E be a nonexpansive mappings. Let {x
n
}
be a sequence in K such that {x
n
} converges weakly to some point x ∈ K. Then, there exists
an increasing continuous function h : [0, ∞) → [0, ∞) with h(0) = 0 depending on the diameter
of K such that
h(x − T x) ≤ lim inf
n→∞
x
n

∗
(t) = 0 and
x + h
2
≤ x
2
+ 2h, j(x) + hβ
∗
(h), ∀h ∈ E \ {0}. (6)
Remark 2.5 In a real Hilbert space, we can see that β
∗
(t) = t for t > 0. In our more general
setting, throughout this article we will still assume that
β
∗
(t) ≤ 2t,
where β
∗
is a function appearing in (6).
Then, we prove the demiclosedness principle of T in the uniformly convex Banach space
which has the Fr´echet differentiable norm.
Lemma 2 .6 Let E be a real uniformly convex Banach space which has the Fr´echet differ-
entiable norm. Let K be a nonempty, closed, and convex subset of E and T : K → K an
asymptotically κ-strictly pseudo contractive mapping with F (T) = ∅. Then, (I − T ) is demi-
closed at 0.
Proof. Let {x
n
} be a sequence in K which converges weakly to p ∈ K and {x
n
− T x

x, n ≥ 1,
Then for all x, y ∈ K,
T
α,n
x − T
α,n
y
2
= (x − y) − α[(I − T
n
)x − (I − T
n
)y]
2
≤ x − y
2
− 2α(I − T
n
)x − (I − T
n
)y, j(x − y)
+αx − y − (T
n
x − T
n
y)β
∗
[αx − y − (T
n
x − T

n
y)
2
≤ τ
2
n
x − y
2
,
where τ
n
= [1+2α(κ
n
−1)]
1
2
. (In fact, in (7) the domain of β
∗
(·) requires x−y−(T
n
x−T
n
y) =
0. But when x−y −(T
n
x−T
n
y) = 0, we have T
α,n
x−T

T
α,m
x, m ≥ 1.
Then, G
α,m
is nonexpansive and it follows from Lemma 2.3 that there exists an increasing
continuous function h : [0, ∞) → [0, ∞) with h(0) = 0 depending on the diameter of K such
that
h(p − G
α,m
p) ≤ lim inf
n→∞
x
n
− G
α,m
x
n
. (8)
Observe that
x
n
− G
α,m
x
n
 = x
n
−
1

x
n
 + (1 −
1
τ
m
)(τ
m
r + x
∗
), (9)
and as n → ∞
x
n
−T
α,m
x
n
 = αx
n
−T
m
x
n
 ≤
m

j=1
T
j−1

h(p − G
α,m
p) ≤ (1 −
1
τ
m
)(τ
m
r + x
∗
).
Observe that
p − G
α,m
p ≥ p − T
α,m
p − (1 −
1
τ
m
)T
α,m
p
≥ p − T
α,m
p − (1 −
1
τ
m
)(τ

τ
m
)(τ
m
r + x
∗
) → 0, as m → ∞.
Since T is continuous, we have (I − T )(p) = 0, completing the proof of Lemma 2.6. 
Lemma 2 .7 Let E be a real uniformly convex Banach space which has the Fr´echet differ-
entiable norm, and let K be a nonempty, closed, and convex subset of E. Let T : K → K be
an asymptotically κ-strictly pseudocontractive mapping with F (T ) = ∅. Let {x
n
}
∞
n=1
be the
sequence satisfying the following conditions:
(a) lim
n→∞
x
n
− p exists for every p ∈ F(T );
(b) lim
n→∞
x
n
− T x
n
 = 0;
(c) lim

∈ ω
W
(x
n
) and that {x
n
i
} and {x
m
j
} are subsequences of
{x
n
} such that x
n
i
 p
1
and x
m
j
 p
2
, respectively. Since E has the Fr´echet differentiable
norm, by setting x = p
1
− p
2
, h = t(x
n

≤
1
2
p
1
− p
2

2
+ tx
n
− p
1
, j(p
1
− p
2
) + b(tx
n
− p
1
),
where b is an increasing function. Since x
n
− p
1
 ≤ M, ∀n ≥ 1, for some M > 0, then
1
2
p

2

2
+ tx
n
− p
1
, j(p
1
− p
2
) + b(tM).
Therefore,
1
2
p
1
− p
2

2
+ t lim sup
n→∞
x
n
− p
1
, j(p
1
− p

1
− p
2
) + b(tM).
Hence, lim sup
n→∞
x
n
− p
1
, j(p
1
− p
2
) ≤ lim inf
n→∞
x
n
− p
1
, j(p
1
− p
2
)+
b(tM)
t
. Since lim
t→0
+

for all p ∈ ω
W
(x
n
). Set p = p
2
. We have p
2
− p
1
, j(p
1
− p
2
) = 0, that is, p
2
= p
1
. Hence,
ω
W
(x
n
) is singleton, so that {x
n
} converges weakly to a fixed point of T. 
3 Main results
Theorem 3.1 Let E be a real uniformly convex Banach space which has the Fr´echet differ-
entiable norm, and let K be a nonempty, closed, and convex subset of E. Let T : K → K
be an asymptotically κ-strictly pseudocontractive mapping for some 0 ≤ κ < 1 with sequence

n
(κ − α
n
) = ∞.
(11)
Given x
1
∈ K, then the sequence {x
n
}
∞
n=1
is generated by the modified Mann’s algorithm:
x
n+1
= (1 − α
n
)x
n
+ α
n
T
n
x
n
, (12)
converges weakly to a fixed point of T.
Proof. Pick a p ∈ F (T ). We firstly show that lim
n→∞
x

n
, j(x
n
− p) + α
n
x
n
− T
n
x
n
β
∗
(α
n
x
n
− T
n
x
n
)
≤ x
n
− p
2
− 2α
n
[κx
n

2
− 2α
n
(κ − α
n
)x
n
− T
n
x
n

2
. (13)
Obviously,
x
n+1
− p
2
≤ [1 + 2α
n
(κ
n
− 1)]x
n
− p
2
. (14)
Let δ
n

bounded, that is, there exists a constant M > 0 such that x
n
− p < M).
Then, we prove lim
n→∞
x
n
− T x
n
 = 0. In fact, it follows from (13) that
j

n=1
2α
n
(κ − α
n
)x
n
− T
n
x
n

2
≤
j

n=1
(x

2
) +
j

n=1
(δ
n
− 1)M
2
.
Then,
∞

n=1
2α
n
(κ − α
n
)x
n
− T
n
x
n

2
< x
1
− p
2

n
x
n
 = 0.
By Lemma 2.2 we know that T is uniformly L-Lipschitzian, then there exists a constant
L > 0, such that
x
n
− T x
n
 ≤ x
n
− T
n
x
n
 + T
n
x
n
− T x
n
 ≤ x
n
− T
n
x
n
 + LT
n−1

x
n
 + L
2
x
n
− x
n−1
 + LT
n−1
x
n−1
− x
n−1
 + Lx
n
− x
n−1

< x
n
− T
n
x
n
 + L(2 + L)T
n−1
x
n−1
− x

. It is obvious that lim
n→∞
σ
n
(0) = p
1
− p
2
 and
lim
n→∞
σ
n
(1) = lim
n→∞
x
n
− p
2
 exist. So, we only need to consider the case of t ∈ (0, 1).
Define T
n
: K → K by
T
n
x = (1 − α
n
)x + α
n
T

x − T
n
y)β
∗
[α
n
x − y − (T
n
x − T
n
y)]
≤ x − y
2
− 2α
n
[κx − y − (T
n
x − T
n
y)
2
− (κ
n
− 1)x − y
2
]
+2α
2
n
x − y − (T

≤ [1 + 2α
n
(κ
n
− 1)]x − y
2
. For the convenience
of the following discussing, set λ
n
= [1 + 2α
n
(κ
n
− 1)]
1
2
, then T
n
x − T
n
y ≤ λ
n
x − y.
Set S
n,m
= T
n+m−1
T
n+m−2
· · · T

1
) − tS
n,m
x
n
− (1 − t)S
n,m
p
1
. If x
n
− p
1
 = 0 for some n
0
,
then x
n
= p
1
for any n ≥ n
0
so that lim
n→∞
x
n
− p
1
 = 0, in fact {x
n

λ
j
)x
n
− p
1

z
n,m
=
S
n,m
(tx
n
+ (1 − t)p
1
) − S
n,m
x
n
(1 − t)(

n+m−1
j=n
λ
j
)x
n
− p
1

n
− p
1

and
tw
n,m
+ (1 − t)z
n,m
 =
S
n,m
x
n
− S
n,m
p
1

(

n+m−1
j=n
λ
j
)x
n
− p
1


− p
1






≤

n+m−1

j=n
λ
j

x
n
− p
1
 − S
n,m
x
n
− S
n,m
p
1
 =



n+m−1
j=n
λ
j
)λ
n−1
x
n−1
− p
1
 ≤ · · · ≤ (

n+m−1
j=n
λ
j
)(

n−1
j=1
λ
j
)x
1
− p
1
 = (

n+m−1


n+m−1

j=1
λ
j

x
1
− p
1

b
n,m






≤

n+m−1

j=n
λ
j

x
n

b
n,m
= 0 uniformly for all m ≥ 1. Observe that
σ
n+m
(t) ≤ tx
n+m
+ (1 − t)p
1
− p
2
+ (S
n,m
(tx
n
+ (1 − t)p
1
) − tS
n,m
x
n
− (1 − t)S
n,m
p
1
)
+S
n,m
(tx
n


tx
n
+ (1 − t)p
1
− p
2
 + b
n,m
=

n+m−1

j=n
λ
j

σ
n
(t) + b
n,m
.
Hence, lim sup
n→∞
σ
n
(t) ≤ lim inf
n→∞
σ
n

be the sequence generated by the modified Mann’s algorithm
(12). Then, the sequence {x
n
} converges strongly to a fixed point of T if and only if
lim inf
n→∞
d(x
n
, F (T )) = 0,
where d(x
n
, F (T )) = inf
p∈F (T )
x
n
− p.
Proof. In the real Banach space E with the Fr´echet differentiable norm, we still have
x
n+1
− p
2
≤ δ
n
x
n
− p
2
. (19)
as we have already proved in Theorem 3.1. Thus, [d(x
n+1

, F (T )) = 0.
Conversely, suppose lim inf
n→∞
d(x
n
, F (T )) = 0 , then the existence of lim
n→∞
d(x
n
, F (T ))
implies that lim
n→∞
d(x
n
, F (T )) = 0 . Thus, for arbitrary  > 0 there exists a positive integer
n
0
such that d(x
n
, F (T )) <

2
for any n ≥ n
0
.
From (19), we have
x
n+1
− p
2

2
(δ
n−2
− 1) + M
2
(δ
n−1
− 1)
≤ . . . ≤ x
l
− p
2
+ M
2
n−1

j=l
(δ
j
− 1), n − 1 ≥ l ≥ 1,
Since

∞
n=1
(δ
n
− 1) < ∞, then there exists a positive integer n
1
such that


2
+ M
2
n−1

j=N
(δ
j
− 1)]
1
2
+ [x
N
− p
2
+ M
2
m−1

j=N
(δ
j
− 1)]
1
2
≤ [x
N
− p
2
+ M

N
, F (T ))]
2
+ M
2
∞

j=N
(δ
j
− 1)}
1
2
+ {[d(x
N
, F (T ))]
2
+ M
2
∞

j=N
(δ
j
− 1)}
1
2
< 2[(

2

0 ≤ u − Tu ≤ u − x
n
 + x
n
− T x
n
 + Lx
n
− u → 0, as n → ∞.
Thus, u ∈ F(T ). 
Competing interests
The authors declare that they have no competing interests.
Author’s contributions
All authors read and approved the final manuscript.
Acknowledgment
This study was supported by the Youth Teacher Foundation of North China Electric Power
University.
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