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A NEW COMPOSITE IMPLICIT ITERATIVE PROCESS
FOR A FINITE FAMILY OF NONEXPANSIVE
MAPPINGS IN BANACH SPACES
FENG GU AND JING LU
Received 18 January 2006; Revised 22 August 2006; Accepted 23 August 2006
The purpose of this paper is to study the weak and strong convergence of implicit iter-
ation process with errors to a common fixed point for a finite family of nonexpansive
mappings in Banach spaces. The results presented in this paper extend and improve the
corresponding results of Chang and Cho (2003), Xu and Ori (2001), and Zhou and Chang
(2002).
Copyright © 2006 F. Gu and J. Lu. This is an open access article distributed under the
Creative Commons Attribution License, which permits unrest ricted use, distr ibution,
and reproduction in any medium, provided the original work is properly cited.
1. Introduction and preliminaries
Throughout this paper we assume that E is a real Banach space and T : E
→ E is a map-
ping. We denote by F(T)andD(T) the set of fixed points and the domain of T,respec-
tively.
Recall that E is said to satisfy Opial condition [11], if for each sequence
{x
n
} in E,the
condition that the sequence x
n
→ x weakly implies that
liminf
n→∞
x
n
− x
< liminf
n→∞
x
n
− y
(1.1)
for all y
∈ E with y = x. It is well known that (see, e.g., Dozo [9]) inequality (1.1)is
equivalent to
limsup
n→∞
x
n
− x
< limsup
n→∞
x
n
− y
. (1.2)
Definit ion 1.1. Let D be a closed subset of E and let T : D
→ D be a mapping.
(1) T is said to be demiclosed at the origin, if for each sequence
{x
n
} in D, the condi-
tions x
n
→ x
0
weakly and Tx
n
→ 0stronglyimplyTx
0
= 0.
Hindawi Publishing Corporation
Fixed Point Theory and Applications
Volume 2006, Article ID 82738, Pages 1–11
DOI 10.1155/FPTA/2006/82738
2 A new composite implicit iterative process
(2) T is said to be semicompact, if for any bounded sequence
{x
n
} in D such that
x
n
− Tx
n
→0(n →∞), then there exists a subsequence {x
n
i
}⊂{x
n
} such that
x
n
i
→ x
∗
∈ D.
(3) T is said to be nonexpansive,if
Tx− Ty≤x − y,foralln ≥ 1forallx, y ∈ D.
Let E be a Hilbet space, let K be a nonempty closed convex subset of E,andlet
{T
1
,T
2
, ,T
N
} : K → K be N nonexpansive mappings. In 2001, Xu and Ori [19]intro-
duced the following implicit iteration process
{x
n
} defined by
x
n
= α
n
x
n−1
+(1− α
n
)T
n(modN)
x
n
, ∀n ≥ 1, (1.3)
where x
0
∈ K is an initial point, {α
n
}
n≥1
is a real sequence in (0,1) and proved the weakly
convergence of the sequence
{x
n
} defined by (1.3) to a common fixed point p ∈ F =
N
i
=1
F(T
i
).
Recently concerning the convergence problems of an implicit (or nonimplicit) itera-
tive process to a common fixed point for a finite family of asymptotically nonexpansive
mappings (or nonexpansive mappings) in Hilbert spaces or uniformly convex Banach
spaces have been considered by several authors (see, e.g., Bauschke [1], Chang and Cho
[3], Chang et al. [4], Chidume et al. [5], Goebel and Kirk [6], G
´
ornicki [7], Halpern [8],
Lions [10], Reich [12], Rhoades [13], Schu [14], Shioji and Takahashi [15], Tan and Xu
[16, 17], Wittmann [18], Xu and Or i [19], and Zhou and Chang [20]).
In this paper, we introduce the fol low ing new implicit iterative sequence
{x
n
} with
errors:
x
1
= α
1
x
0
+ β
1
T
1
α
1
x
0
+
β
1
T
1
x
1
+ γ
1
v
1
+ γ
1
u
1
,
x
2
= α
2
x
1
+ β
2
T
2
α
2
x
1
+
β
2
T
2
x
2
+ γ
2
v
2
+ γ
2
u
2
,
.
.
.
x
N
= α
N
x
N−1
+ β
N
T
N
α
N
x
N−1
+
β
N
T
N
x
N
+ γ
N
v
N
+ γ
N
u
N
,
x
N+1
= α
N+1
x
N
+ β
N+1
T
1
α
N+1
x
N
+
β
N+1
T
1
x
N+1
+ γ
N+1
v
N+1
+ γ
N+1
u
N+1
,
.
.
.
x
2N
= α
2N
x
2N−1
+ β
2N
T
N
α
2N
x
2N−1
+
β
2N
T
N
x
2N
+ γ
2N
v
2N
+ γ
2N
u
2N
,
x
2N+1
= α
2N+1
x
2N
+ β
2N+1
T
1
α
2N+1
x
2N
+
β
2N+1
T
1
x
2N+1
+ γ
2N+1
v
2N+1
+ γ
2N+1
u
2N+1
,
.
.
.
(1.4)
for a finite family of nonexpansive mappings
{T
i
}
N
i
=1
: K → K,where{α
n
}, {β
n
}, {γ
n
},
{α
n
}, {
β
n
},and{γ
n
} are six sequences in [0, 1] satisfying α
n
+ β
n
+ γ
n
= α
n
+
β
n
+ γ
n
= 1
for all n
≥ 1, x
0
is a given point in K,aswellas{u
n
} and {v
n
} are two bounded sequences
F. Gu and J. Lu 3
in K, which can be written in the following compact form:
x
n
= α
n
x
n−1
+ β
n
T
n(modN)
y
n
+ γ
n
u
n
,
y
n
=
α
n
x
n−1
+
β
n
T
n(modN)
x
n
+ γ
n
v
n
, ∀n ≥ 1.
(1.5)
Especially, if
{T
i
}
N
i
=1
: K → K are N nonexpansive mapping s, {α
n
}, {β
n
}, {γ
n
} are three
sequences in [0, 1], and x
0
is a given point in K, then the sequence {x
n
} defined by
x
n
= α
n
x
n−1
+ β
n
T
n(modN)
x
n−1
+ γ
n
u
n
, ∀n ≥ 1 (1.6)
is called the explicit iterative sequence for a finite family of nonexpansive mappings
{T
i
}
N
i
=1
.
The purpose of this paper is to study the weak and strong convergence of iterative
sequence
{x
n
} defined by (1.5)and(1.6) to a common fixed point for a finite family
of nonexpansive mappings in Banach spaces. The results presented in this paper not only
generalized and extend the corresponding results of Chang and Cho [3], Xu and Ori [19],
and Zhou and Chang [20], but also in the case of γ
n
=
γ
n
= 0or
β
n
=
γ
n
= 0arealsonew.
In order to prove the main results of this paper, we need the following lemmas.
Lemma 1.2 [2]. Let E be a uniformly convex Banach space, let K be a nonempty closed
convex subset of E,andletT : K
→ K be a nonexpansive mapping with F(T) =∅. The n
I
− T is semiclosed at zero, that is, for each sequence {x
n
} in K, if {x
n
} converges weakly to
q
∈ K and {(I − T)x
n
} converges strongly to 0, then (I − T)q = 0.
Lemma 1.3 [17]. Let
{a
n
} and {b
n
} be two nonnegative real sequences satisfying the fol-
low ing condition: a
n+1
≤ a
n
+ b
n
for all n ≥ n
0
,wheren
0
is some nonnegative integer. If
∞
n=0
b
n
< ∞, then lim
n→∞
a
n
exists. If in addition {a
n
} has a subsequence which converges
strongly to zero, then lim
n→∞
a
n
= 0.
Lemma 1.4 [14]. Let E be a uniformly convex Banach space, let b and c be two constants with
0 <b<c<1.Supposethat
{t
n
} is a sequence in [b,c] and {x
n
} and {y
n
} are two sequence in
E such that lim
n→∞
t
n
x
n
+(1− t
n
)y
n
=d, limsup
n→∞
x
n
≤d,andlimsup
n→∞
y
n
≤
d hold for some d ≥ 0, then lim
n→∞
x
n
− y
n
=0.
Lemma 1.5. Let E be a real Banach space, let K beanonemptyclosedconvexsubsetofE,
and le t
{T
1
,T
2
, ,T
N
} : K → K be N nonexpansive mappings with F =
N
i
=1
F(T
i
) =∅.
Let
{u
n
} and {v
n
} betwoboundedsequencesinK,andlet{α
n
}, {β
n
}, {γ
n
}, {α
n
}, {
β
n
},
and
{γ
n
} be six sequences in [0,1] satisfying the following conditions:
(i) α
n
+ β
n
+ γ
n
=
α
n
+
β
n
+ γ
n
= 1,foralln ≥ 1;
(ii) τ
= sup{β
n
: n ≥ 1} < 1;
(iii)
∞
n=1
γ
n
< ∞,
∞
n=1
γ
n
< ∞.
If
{x
n
} is the implicit iterative sequence defined by (1.5), then for each p ∈ F =
N
i
=1
F(T
i
)
the limit lim
n→∞
x
n
− p exists.
4 A new composite implicit iterative process
Proof. Since F
=
N
n
=1
F(T
i
) =∅,foranygivenp ∈ F,itfollowsfrom(1.5)that
x
n
− p
=
1 − β
n
− γ
n
x
n−1
+ β
n
T
n(modN)
y
n
+ γ
n
u
n
− p
≤
1 − β
n
− γ
n
x
n−1
− p
+ β
n
T
n(modN)
y
n
− p
+ γ
n
u
n
− p
=
1 − β
n
− γ
n
x
n−1
− p
+ β
n
T
n(modN)
y
n
− T
n(modN)
p
+ γ
n
u
n
− p
≤
1 − β
n
x
n−1
− p
+ β
n
y
n
− p
+ γ
n
u
n
− p
.
(1.7)
Again it follows from (1.5)that
y
n
− p
=
1 −
β
n
− γ
n
x
n−1
+
β
n
T
n(modN)
x
n
+ γ
n
v
n
− p
≤
1 −
β
n
− γ
n
x
n−1
− p
+
β
n
T
n(modN)
x
n
− p
+ γ
n
v
n
− p
=
1 −
β
n
− γ
n
x
n−1
− p
+
β
n
T
n(modN)
x
n
− T
n(modN)
p
+ γ
n
v
n
− p
≤
1 −
β
n
x
n−1
− p
+
β
n
x
n
− p
+ γ
n
v
n
− p
.
(1.8)
Substituting (1.8)into(1.7), we obtain that
x
n
− p
≤
1 − β
n
β
n
x
n−1
− p
+ β
n
β
n
x
n
− p
+ β
n
γ
n
v
n
− p
+ γ
n
u
n
− p
.
(1.9)
Simplifying we have
1 − β
n
β
n
x
n
− p
≤
1 − β
n
β
n
x
n−1
− p
+ σ
n
, (1.10)
where σ
n
= β
n
γ
n
v
n
− p + γ
n
u
n
− p. By condition (iii) and the boundedness of the
sequences
{β
n
}, {u
n
− p},and{v
n
− p},wehave
∞
n=1
σ
n
< ∞. From condition (ii)
we know that
β
n
β
n
≤ β
n
≤ τ<1andso1− β
n
β
n
≥ 1 − τ>0; (1.11)
hence, from (1.10)wehave
x
n
− p
≤
x
n−1
− p
+
σ
n
1 − τ
=
x
n−1
− p
+ b
n
, (1.12)
where b
n
= σ
n
/(1 − τ)with
∞
i=1
b
n
< ∞.
Taki ng a
n
=x
n−1
− p in inequality (1.12), we have a
n+1
≤ a
n
+ b
n
,foralln ≥ 1, and
satisfied all conditions in Lemma 1.3. Therefore the limit lim
n→∞
x
n
− p exists. This
completes the proof of Lemma 1.5.
2. Main results
We are now in a position to prove our main results in this paper.
F. Gu and J. Lu 5
Theorem 2.1. Let E be a real Banach space, let K be a nonempty closed convex subset of
E,andlet
{T
1
,T
2
, ,T
N
} : K → K be N nonexpansive mappings with F =
N
i
=1
F(T
i
) =∅
(the set of common fixed points of {T
1
,T
2
, ,T
N
}). Let {u
n
} and {v
n
} be two bounded
sequences in K,andlet
{α
n
}, {β
n
}, {γ
n
}, {α
n
}, {
β
n
},and{γ
n
} be six s equences in [0,1]
satisfying the following conditions:
(i) α
n
+ β
n
+ γ
n
=
α
n
+
β
n
+ γ
n
= 1,foralln ≥ 1;
(ii) τ
= sup{β
n
: n ≥ 1} < 1;
(iii)
∞
n=1
γ
n
< ∞,
∞
n=1
γ
n
< ∞.
Then the implicit iterative sequence
{x
n
} defined by (1.5)convergesstronglytoacommon
fixed point p
∈ F =
N
i
=1
F(T
i
) if and only if
liminf
n→∞
d(x
n
,F) = 0. (2.1)
Proof. The necessity of condition (2.1)isobvious.
Next we prove the sufficiency of Theorem 2.1.Foranygivenp
∈ F,itfollowsfrom
(1.12)inLemma 1.5 that
x
n
− p
≤
x
n−1
− p
+ b
n
∀n ≥ 1, (2.2)
where b
n
= σ
n
/(1 − τ)with
∞
n=1
b
n
< ∞.Hence,wehave
d
x
n
,F
≤
d
x
n−1
,F
+ b
n
∀n ≥ 1. (2.3)
It follows from (2.3)andLemma 1.3 that the limit lim
n→∞
d(x
n
,F) exists. By condition
(2.1), we have lim
n→∞
d(x
n
,F) = 0.
Next we prove that the sequence
{x
n
} is a Cauchy sequence in K. In fact, for any posi-
tive int egers m and n,from(2.2), it follows that
x
n+m
− p
≤
x
n+m−1
− p
+ b
n+m
≤
x
n+m−2
− p
+ b
n+m−1
+ b
n+m
≤···≤
x
n
− p
+
n+m
i=n+1
b
i
≤
x
n
− p
+
∞
i=n+1
b
i
.
(2.4)
Since lim
n→∞
d(x
n
,F) = 0and
∞
n=1
b
n
< ∞,foranygiven > 0, there exists a positive
integer n
0
such that d(x
n
,F) < /8,
∞
i=n+1
b
i
< /2, for all n ≥ n
0
. Therefore there exists
p
1
∈ F such that x
n
− p
1
< /4, for all n ≥ n
0
. Consequently, for any n ≥ n
0
and for all
m
≥ 1, from (2.4), we have
x
n+m
− x
n
≤
x
n+m
− p
1
+
x
n
− p
1
≤
2
x
n
− p
1
+
∞
i=n+1
b
i
< 2 ·
4
+
2
= .
(2.5)
This implies that
{x
n
} is a Cauchy sequence in K. By the completeness of K,wecan
assume that lim
n→∞
x
n
= x
∗
∈ K. Moreover, since the set of fixed points of a nonexpansive
mapping is closed, so is F;thusx
∗
∈ F from lim
n→∞
d(x
n
,F) = 0, that is, x
∗
is a common
fixed point of T
1
,T
2
, ,T
N
. This completes the proof of Theorem 2.1.
6 A new composite implicit iterative process
Theorem 2.2. Let E be a real Banach space, let K be a nonempty closed convex subset of
E,andlet
{T
1
,T
2
, ,T
N
} : K → K be N nonexpansive mappings with F =
N
i
=1
F(T
i
) =∅
(the set of common fixed points of {T
1
,T
2
, ,T
N
}). Let {u
n
} be a bounded sequence in K,
and let
{α
n
}, {β
n
},and{γ
n
} be three sequences in [0,1] satisfying the follow ing conditions:
(i) α
n
+ β
n
+ γ
n
= 1,foralln ≥ 1;
(ii) τ
= sup{β
n
: n ≥ 1} < 1;
(iii)
∞
n=1
γ
n
< ∞.
Then the explicit iterative sequence
{x
n
} defined by (1.6)convergesstronglytoacommon
fixed point p
∈ F =
N
i
=1
F(T
i
) if and only if liminf
n→∞
d(x
n
,F) = 0.
Proof. Ta king
β
n
=
γ
n
= 0, for all n ≥ 1inTheorem 2.1, then the conclusion of
Theorem 2.2 can be obtained from Theorem 2.1 immediately. This completes the proof
of Theorem 2.2.
Theorem 2.3. Let E be a real uniformly convex Banach space satisfy ing Opial condition, let
K beanonemptyclosedconvexsubsetofE,andlet
{T
1
,T
2
, ,T
N
} : K → K be N nonex-
pansive mappings with F
=
N
i
=1
F(T
i
) =∅.Let{u
n
} and {v
n
} be two bounded sequences
in K,andlet
{α
n
}, {β
n
}, {γ
n
}, {α
n
}, {
β
n
},and{γ
n
} be six seque nces in [0,1] satisfying the
following conditions:
(i) α
n
+ β
n
+ γ
n
=
α
n
+
β
n
+ γ
n
= 1,foralln ≥ 1;
(ii) 0 <τ
1
= inf{β
n
: n ≥ 1}≤sup{β
n
: n ≥ 1}=τ
2
< 1;
(iii)
β
n
→ 0(n →∞);
(iv)
∞
n=1
γ
n
< ∞,
∞
n=1
γ
n
< ∞.
Then the implicit iterative sequence
{x
n
} defined by (1.5) converges weakly to a common
fixed point of
{T
1
,T
2
, ,T
N
}.
Proof. First, we prove that
lim
n→∞
x
n
− T
n(modN)+ j
x
n
=
0, ∀ j = 1,2, ,N. (2.6)
Let p
∈ F.Putd = lim
n→∞
x
n
− p.Itfollowsfrom(1.5)that
x
n
− p
=
1 − β
n
x
n−1
− p + γ
n
u
n
− x
n−1
+ β
n
T
n(modN)
y
n
− p + γ
n
u
n
− x
n−1
−→
d, n −→ ∞ .
(2.7)
Again since lim
n→∞
x
n
− p exists, so {x
n
} is a bounded sequence in K.Byvirtueof
condition (iv) and the boundedness of sequences
{x
n
} and {u
n
} we have
limsup
n→∞
x
n−1
− p + γ
n
u
n
− x
n−1
≤
limsup
n→∞
x
n−1
− p
+limsup
n→∞
γ
n
u
n
− x
n−1
=
d, p ∈ F.
(2.8)
F. Gu and J. Lu 7
It follows from (1.8) and condition (iii) that
limsup
n→∞
T
n(modN)
y
n
− p + γ
n
u
n
− x
n−1
≤
limsup
n→∞
y
n
− p
+limsup
n→∞
γ
n
u
n
− x
n−1
=
limsup
n→∞
y
n
− p
≤
limsup
n→∞
1 −
β
n
x
n−1
− p
+
β
n
x
n
− p
+ γ
n
v
n
− p
≤
limsup
n→∞
1 −
β
n
x
n−1
− p
+limsup
n→∞
β
n
x
n
− p
+limsup
n→∞
γ
n
v
n
− p
=
d, p ∈ F.
(2.9)
Therefore, from condition (ii), (2.7)–(2.9), and Lemma 1.4 we know that
lim
n→∞
T
n(modN)
y
n
− x
n−1
=
0. (2.10)
From (1.5)and(2.10)wehave
x
n
− x
n−1
=
β
n
T
n(modN)
y
n
− x
n−1
+ γ
n
u
n
− x
n−1
≤
β
n
T
n(modN)
y
n
− x
n−1
+ γ
n
u
n
− x
n−1
−→
0, n −→ ∞ ,
(2.11)
which implies that
lim
n−→ ∞
x
n
− x
n−1
=
0 (2.12)
and so
lim
n→∞
x
n
− x
n+ j
=
0 ∀ j = 1,2, ,N. (2.13)
On the other hand, we have
x
n
− T
n(modN)
x
n
≤
x
n
− x
n−1
+
x
n−1
− T
n(modN)
y
n
+
T
n(modN)
y
n
− T
n(modN)
x
n
.
(2.14)
Now, we consider the third term on the right-hand side of (2.14). From (1.5)wehave
T
n(modN)
y
n
− T
n(modN)
x
n
≤
y
n
− x
n
=
α
n
x
n−1
− x
n
+
β
n
T
n(modN)
x
n
− x
n
+ γ
n
v
n
− x
n
≤
α
n
x
n−1
− x
n
+
β
n
T
n(modN)
x
n
− x
n
+ γ
n
v
n
− x
n
.
(2.15)
Substituting (2.15)into(2.14), we obtain that
x
n
− T
n(modN)
x
n
≤
1+α
n
x
n
− x
n−1
+
x
n−1
− T
n(modN)
y
n
+
β
n
T
n(modN)
x
n
− x
n
+ γ
n
v
n
− x
n
.
(2.16)
8 A new composite implicit iterative process
Hence, by vir tue of conditions (iii), (iv), (2.10), (2.12) and the boundedness of sequences
{T
n(modN)
x
n
− x
n
} and {v
n
− x
n
} we have
lim
n→∞
x
n
− T
n(modN)
x
n
=
0. (2.17)
Therefore, from (2.13)and(2.17), for any j
= 1,2, ,N,wehave
x
n
− T
n(modN)+ j
x
n
≤
x
n
− x
n+ j
+
x
n+ j
− T
n(modN)+ j
x
n+ j
+
T
n(modN)+ j
x
n+ j
− T
n(modN)+ j
x
n
≤ 2
x
n
− x
n+ j
+
x
n+ j
− T
n(modN)+ j
x
n+ j
−→
0, n −→ ∞ .
(2.18)
That is, (2.6)holds.
Since E is uniformly convex, every bounded subset of E is weakly compact. Again since
{x
n
} is a bounded sequence in K, there exists a subsequence {x
n
k
}⊂{x
n
} such that {x
n
k
}
converges weakly to q ∈ K.
Without loss of generality, we can assume that n
k
= i(modN), where i is some positive
integer in
{1,2, ,N}. Otherwise, we can take a subsequence {x
n
k
j
}⊂{x
n
k
} such that
n
k
j
= i(modN). For any l ∈{1,2, ,N}, there exists an integer j ∈{1, 2, ,N} such that
n
k
+ j = l(modN). Hence, from (2.18)wehave
lim
k→∞
x
n
k
− T
l
x
n
k
=
0, l = 1,2, , N. (2.19)
By Lemma 1.2,weknowthatq
∈ F(T
l
). By the arbitrariness of l ∈{1,2, ,N},weknow
that q
∈ F =
N
j
=1
F(T
j
).
Finally, we prove that
{x
n
} converges weakly to q. In fact, suppose the contrary, then
there exists some subsequence
{x
n
j
}⊂{x
n
} such that {x
n
j
} converges weakly to q
1
∈ K
and q
1
= q. Then by the same method as given above, we can also prove that q
1
∈ F =
N
j
=1
F(T
j
).
Taki ng p
= q and p = q
1
and by using the same method given in the proof of
Lemma 1.5, we can prove that the following two limits exist and lim
n→∞
x
n
− q=d
1
and lim
n→∞
x
n
− q
1
=d
2
,whered
1
and d
2
are two nonnegative numbers. By virtue of
the Opial condition of E,wehave
d
1
= limsup
n
k
−→ ∞
x
n
k
− q
< limsup
n
k
→∞
x
n
k
− q
1
=
d
2
= limsup
n
j
→∞
x
n
j
− q
1
< limsup
n
j
→∞
x
n
j
− q
=
d
1
.
(2.20)
This is a contradiction. Hence q
1
= q. This implies that {x
n
} converges weakly to q. This
completes the proof of Theorem 2.3.
Theorem 2.4. Let E be a real uniformly convex Banach space satisfy ing Opial condition, let
K beanonemptyclosedconvexsubsetofE,andlet
{T
1
,T
2
, ,T
N
} : K → K be N nonex-
pansive mappings with F
=
N
i
=1
F(T
i
) =∅.Let{u
n
} be a bounded sequence in K,andlet
F. Gu and J. Lu 9
{α
n
}, {β
n
},and{γ
n
} be three sequences in [0,1] satisfying the following conditions:
(i) α
n
+ β
n
+ γ
n
= 1, ∀n ≥ 1;
(ii) 0 <τ
1
= inf{β
n
: n ≥ 1}≤sup{β
n
: n ≥ 1}=τ
2
< 1;
(iii)
∞
n=1
γ
n
< ∞.
Then the explicit iterative sequence
{x
n
} defined by (1.6) converges weakly to a c ommon
fixed point of
{T
1
,T
2
, ,T
N
}.
Proof. Ta king
β
n
=
γ
n
= 0, for all n ≥ 1inTheorem 2.3, then the conclusion of
Theorem 2.4 can be obtained from Theorem 2.3 immediately. This completes the proof
of Theorem 2.4.
Theorem 2.5. Let E be a real uniformly convex Banach space, let K be a nonempty closed
convex subset of E,andlet
{T
1
,T
2
, ,T
N
} : K → K be N nonexpansive mappings with F =
N
i
=1
F(T
i
) =∅and there exists an T
j
,1≤ j ≤ N, which is semicompact (without loss of
generality, assume that T
1
is semicompact). Let{u
n
} and {v
n
} be two bounded sequences in
K,andlet
{α
n
}, {β
n
}, {γ
n
}, {α
n
}, {
β
n
},and{γ
n
} be six sequences in [0,1] satisfying the
following conditions:
(i) α
n
+ β
n
+ γ
n
=
α
n
+
β
n
+ γ
n
= 1,foralln ≥ 1;
(ii) 0 <τ
1
= inf{β
n
: n ≥ 1}≤sup{β
n
: n ≥ 1}=τ
2
< 1;
(iii)
β
n
→ 0(n →∞);
(iv)
∞
n=1
γ
n
< ∞,
∞
n=1
γ
n
< ∞.
Then the implicit iterative sequence
{x
n
} defined by (1.5)convergesstronglytoacommon
fixed point of
{T
1
,T
2
, ,T
N
} in K.
Proof. For any given p
∈ F =
N
i
=1
F(T
i
), by the same m ethod as given in proving Lemma
1.5 and (2.19), we can prove that
lim
n→∞
x
n
− p
=
d, (2.21)
where d
≥ 0 is some nonnegative number, and
lim
k→∞
x
n
k
− T
l
x
n
k
=
0, l = 1,2, , N. (2.22)
Especially, we have
lim
k→∞
x
n
k
− T
1
x
n
k
=
0. (2.23)
By the assumption, T
1
is semicompact; therefore it follows from (2.23) that there exists a
subsequence
{x
n
k
i
}⊂{x
n
k
} such that x
n
k
i
→ x
∗
∈ K.Hencefrom(2.22)wehavethat
x
∗
− T
l
x
∗
=
lim
k
i
→∞
x
n
k
i
− T
l
x
n
k
i
=
0 ∀l = 1,2, , N, (2.24)
which implies that x
∗
∈ F =
N
i
=1
F(T
i
). Take p = x
∗
in (2.21), similarly we can prove
that lim
n→∞
x
n
− x
∗
=d
1
,whered
1
≥ 0 is some nonnegative number. From x
n
k
i
→ x
∗
we know that d
1
= 0, that is, x
n
→ x
∗
. This completes the proof of Theorem 2.5.
10 A new composite implicit iterative process
Theorem 2.6. Let E be a real uniformly convex Banach space, let K be a nonempty closed
convex subset of E,andlet
{T
1
,T
2
, ,T
N
} : K → K be N nonexpansive mappings with
F
=
N
i
=1
F(T
i
) =∅and the re exists an T
j
,1≤ j ≤ N, which is semicompact (without loss
of generality, assume that T
1
is semicompact). Let {u
n
} be a bounded sequence in K,andlet
{α
n
}, {β
n
},and{γ
n
} be three sequences in [0,1] satisfying the following conditions:
(i) α
n
+ β
n
+ γ
n
= 1,foralln ≥ 1;
(ii) 0 <τ
1
= inf{β
n
: n ≥ 1}≤sup{β
n
: n ≥ 1}=τ
2
< 1;
(iii)
∞
n=1
γ
n
< ∞.
Then the explicit iterative sequence
{x
n
} defined by (1.6)convergesstronglytoacommon
fixed point of
{T
1
,T
2
, ,T
N
} in K.
Proof. Ta king
β
n
=
γ
n
= 0, for all n ≥ 1inTheorem 2.5, then the conclusion of
Theorem 2.6 can be obtained from Theorem 2.5 immediately. This completes the proof
of Theorem 2.6.
Remark 2.7. Theorems 2.3–2.6 improve and extend the corresponding results in Chang
and Cho [3, Theorem 3.1] and Zhou and Chang [20, Theorem 3], and the implicit it-
erative process
{x
n
} defined by (1.3) is replaced by the more general implicit or explicit
iterative process
{x
n
} defined by (1.5)or(1.6).
Remark 2.8. Theorems 2.3–2.6 generalize and improve the main results of Xu and Ori
[19] in the following aspects.
(1) The class of Hilbert spaces is extended to that of Banach spaces satisfying Opial’s
or semicompactness condition.
(2) The implicit iterative process
{x
n
} defined by (1.3)isreplacedbythemoregeneral
implicit or explicit iterative process
{x
n
} defined by (1.5)or(1.6).
Remark 2.9. The iterative algorithm used in this paper is different from those in [1, 8, 10,
14, 18].
Acknowledgments
The present studies were supported by the Natural Science Foundation of Zhejiang
Province (Y605191), the Natural Science Foundation of Heilongjiang Province (A0211),
and the Scientific Research Foundation from Zhejiang Province Education Committee
(20051897).
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Feng Gu: Department of Mathematics, Institute of Applied Mathematics,
Hangzhou Teacher’s College, Hangzhou, Zhejiang 310036, China
E-mail address:
Jing Lu: Department of Mathematics, Institute of Applied Mathematics,
Hangzhou Teacher’s College, Hangzhou, Zhejiang 310036, China
E-mail address:
