Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2013, Article ID 602582, 6 pages
http://dx.doi.org/10.1155/2013/602582
Research Article
An Implicit Iteration Process for Common Fixed Points of
Two Infinite Families of Asymptotically Nonexpansive Mappings
in Banach Spaces
Wei-Qi Deng and Peng Bai
College of Statistics and Mathematics, Yunnan University of Finance and Economics, Kunming, Yunnan 650221, China
Correspondence should be addressed to Wei-Qi Deng; dwq1273@126.com
Received 30 October 2012; Accepted 18 January 2013
Academic Editor: Jinde Cao
Copyright © 2013 W.-Q. Deng and P. Bai. 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.
Let 𝐾 be a nonempty, closed, and convex subset of a real uniformly convex Banach space 𝐸. Let {𝑇𝜆 }𝜆∈Λ and {𝑆𝜆 }𝜆∈Λ be two infinite
families of asymptotically nonexpansive mappings from 𝐾 to itself with 𝐹 := {𝑥 ∈ 𝐾 : 𝑇𝜆 𝑥 = 𝑥 = 𝑆𝜆 𝑥, 𝜆 ∈ Λ} ≠ 0. For an arbitrary
∗ 𝑚𝑛−1
∗
)
𝑥𝑛−1 +𝛾𝑛 (𝑇𝑛∗ )𝑚𝑛 𝑦𝑛 , 𝑦𝑛 = 𝛼𝑛󸀠 𝑥𝑛 +𝛽𝑛󸀠 (𝑆𝑛−1
)𝑚𝑛−1 𝑥𝑛−1 +𝛾𝑛󸀠 (𝑆𝑛∗ )𝑚𝑛 𝑥𝑛 ,
initial point 𝑥0 ∈ 𝐾, {𝑥𝑛 } is defined as follows: 𝑥𝑛 = 𝛼𝑛 𝑥𝑛−1 +𝛽𝑛 (𝑇𝑛−1
∗
∗
𝑛 = 1, 2, 3, . . ., where 𝑇𝑛 = 𝑇𝜆𝑖 and 𝑆𝑛 = 𝑆𝜆𝑖 with 𝑖𝑛 and 𝑚𝑛 satisfying the positive integer equation: 𝑛 = 𝑖 + (𝑚 − 1)𝑚/2, 𝑚 ≥ 𝑖;
𝑛
𝑛
∞
󸀠
󸀠
󸀠
{𝑇𝜆𝑖 }∞
𝑖=1 and {𝑆𝜆 𝑖 }𝑖=1 are two countable subsets of {𝑇𝜆 }𝜆∈Λ and {𝑆𝜆 }𝜆∈Λ , respectively; {𝛼𝑛 }, {𝛽𝑛 }, {𝛾𝑛 }, {𝛼𝑛 }, {𝛽𝑛 }, and {𝛾𝑛 } are sequences
󸀠
󸀠
󸀠
in [𝛿, 1−𝛿] for some 𝛿 ∈ (0, 1), satisfying 𝛼𝑛 +𝛽𝑛 +𝛾𝑛 = 1 and 𝛼𝑛 +𝛽𝑛 +𝛾𝑛 = 1. Under some suitable conditions, a strong convergence
theorem for common fixed points of the mappings {𝑇𝜆 }𝜆∈Λ and {𝑆𝜆 }𝜆∈Λ is obtained. The results extend those of the authors whose
related researches are restricted to the situation of finite families of asymptotically nonexpansive mappings.
1. Introduction
Let 𝐾 be a nonempty, closed, and convex subset of a real
uniformly convex Banach space 𝐸. A mapping 𝑇 : 𝐾 → 𝐾 is
said to be nonexpansive if ‖𝑇𝑥−𝑇𝑦‖ ≤ ‖𝑥−𝑦‖ for all 𝑥, 𝑦 ∈ 𝐾.
𝑇 is said to be asymptotically nonexpansive if there exists a
sequence {𝑘𝑛 } ⊂ [1, ∞) with 𝑘𝑛 → 1 (𝑛 → ∞) such that
󵄩
󵄩󵄩 𝑛
󵄩
𝑛 󵄩
󵄩󵄩𝑇 𝑥 − 𝑇 𝑦󵄩󵄩󵄩 ≤ 𝑘𝑛 󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩 ,
∀𝑥, 𝑦 ∈ 𝐾, 𝑛 = 1, 2, 3, . . . .
(1)
It is obvious that a nonexpansive mapping is an asymptotically nonexpansive one, but the converse is not true. Denote
by 𝐹(𝑇) the set of fixed points of 𝑇, that is, 𝐹(𝑇) = {𝑥 ∈
𝐾 : 𝑇𝑥 = 𝑥}. Throughout this paper, we always assume that
𝐹(𝑇) ≠ 0. As an important generalization of nonexpansive
mappings, the class of asymptotically nonexpansive mappings was introduced by Goebel and Kirk [1] in 1972, who
proved that if 𝐾 is a nonempty, closed, and convex subset of
a real uniformly convex Banach space and 𝑇 : 𝐾 → 𝐾 is
an asymptotically nonexpansive mapping, then 𝑇 has a fixed
point.
Since then, iterative techniques for approximating fixed
points of asymptotically nonexpansive mappings have been
studied by various authors (see, e.g., [2–9]). However, these
researches are all restricted to the situation of at most finite
families of asymptotically nonexpansive mappings. For the
extension of finite families to infinite ones, we develop an
original method, namely, a specific way of choosing the
indexes, for the iterative approximation of common fixed
points of the involved mappings.
We now cite an announced result as the object of our
extension. In 2010, Wang et al. [10] constructed the following
iteration process for two asymptotically nonexpansive mappings and obtained some strong convergence theorems for
common fixed points of the given mappings in Banach spaces.
For an arbitrary initial point 𝑥0 ∈ 𝐾, {𝑥𝑛 }, {𝑦𝑛 } are defined as
follows:
𝑥𝑛 = 𝛼𝑛 𝑥𝑛−1 + 𝛽𝑛 𝑇𝑛−1 𝑥𝑛−1 + 𝛾𝑛 𝑇𝑛 𝑦𝑛 ,
𝑦𝑛 = 𝛼𝑛󸀠 𝑥𝑛 + 𝛽𝑛󸀠 𝑆𝑛−1 𝑥𝑛−1 + 𝛾𝑛󸀠 𝑆𝑛 𝑥𝑛 ,
𝑛 = 1, 2, 3, . . . ,
(2)
2
Journal of Applied Mathematics
where 𝑇, 𝑆 : 𝐾 → 𝐾 are two asymptotically nonexpansive
mappings; {𝛼𝑛 }, {𝛽𝑛 }, {𝛾𝑛 }, {𝛼𝑛󸀠 }, {𝛽𝑛󸀠 }, and {𝛾𝑛󸀠 } are real
sequences in [0, 1) satisfying 𝛼𝑛 +𝛽𝑛 +𝛾𝑛 = 1 and 𝛼𝑛󸀠 +𝛽𝑛󸀠 +𝛾𝑛󸀠 =
1.
In this paper, a modified iteration scheme of (2) is used for
approximating common fixed points of two infinite families
of asymptotically nonexpansive mappings; a strong convergence theorem is established in the framework of uniformly
convex Banach spaces. The results show the feasibility of the
newly developed technique and extend those of the authors
whose related researches are restricted to the situation of
finite families of such mappings.
2. Preliminaries
Throughout this paper, we use 𝐹 to denote the set of common
fixed points of two infinite families of asymptotically nonexpansive mappings {𝑇𝜆 }𝜆∈Λ and {𝑆𝜆 }𝜆∈Λ , that is, 𝐹 := {𝑥 ∈ 𝐾 :
𝑇𝜆 𝑥 = 𝑥 = 𝑆𝜆 , 𝜆 ∈ Λ}.
Let 𝐾 be a nonempty, closed, and convex subset of a real
and {𝑆𝜆 𝑖 }∞
be two countable
Banach space 𝐸. Let {𝑇𝜆 𝑖 }∞
𝑖=1
𝑖=1
subsets of {𝑇𝜆 }𝜆∈Λ and {𝑆𝜆 }𝜆∈Λ , respectively. In order to
approximate some member of 𝐹, we define, from an arbitrary
𝑥0 ∈ 𝐾, the following implicit iteration scheme:
𝑚𝑛−1
∗
𝑥𝑛 = 𝛼𝑛 𝑥𝑛−1 + 𝛽𝑛 (𝑇𝑛−1
)
𝑚𝑛−1
∗
)
𝑦𝑛 = 𝛼𝑛󸀠 𝑥𝑛 + 𝛽𝑛󸀠 (𝑆𝑛−1
𝑚
𝑥𝑛−1 + 𝛾𝑛 (𝑇𝑛∗ ) 𝑛 𝑦𝑛 ,
𝑚
𝑥𝑛−1 + 𝛾𝑛󸀠 (𝑆𝑛∗ ) 𝑛 𝑥𝑛 ,
(3)
𝑛 = 1, 2, 3, . . . ,
{𝛼𝑛󸀠 },
{𝛽𝑛󸀠 },
{𝛾𝑛󸀠 }
where {𝛼𝑛 }, {𝛽𝑛 }, {𝛾𝑛 },
and
are sequences in
[𝛿, 1 − 𝛿] for some 𝛿 ∈ (0, 1), satisfying 𝛼𝑛 + 𝛽𝑛 + 𝛾𝑛 = 1
and 𝛼𝑛󸀠 + 𝛽𝑛󸀠 + 𝛾𝑛󸀠 = 1; 𝑇𝑛∗ = 𝑇𝜆 𝑖 and 𝑆𝑛∗ = 𝑆𝜆 𝑖 with 𝑖𝑛 and
𝑛
𝑛
𝑚𝑛 being the solutions to the positive integer equation: 𝑛 =
𝑖 + (𝑚 − 1)𝑚/2 (𝑚 ≥ 𝑖, 𝑛 = 1, 2, 3, . . .), that is, for each 𝑛 ≥ 1,
there exist unique 𝑖𝑛 and 𝑚𝑛 such that
𝑖1 = 1,
𝑖2 = 1,
𝑖3 = 2,
𝑖4 = 1,
𝑖5 = 2,
𝑖6 = 3,
𝑖7 = 1,
𝑖8 = 2, . . . ,
𝑚1 = 1,
𝑚2 = 2,
𝑚3 = 2,
𝑚4 = 3,
𝑚5 = 3,
𝑚6 = 3,
𝑚7 = 4,
𝑚8 = 4, . . . .
𝐷(𝑇) such that {𝑥𝑛 } converges weakly to 𝑥∗ ∈ 𝐷(𝑇) and {𝑇𝑥𝑛 }
converges strongly to 𝑝, then 𝑇𝑥∗ = 𝑝.
We now need the following lemmas for our main results.
Lemma 1 (see [11]). Let {𝑎𝑛 }, {𝛿𝑛 }, and {𝑏𝑛 } be sequences of
nonnegative real numbers satisfying
𝑎𝑛+1 ≤ (1 + 𝛿𝑛 ) 𝑎𝑛 + 𝑏𝑛 ,
𝑛 = 1, 2, 3, . . . .
(6)
∞
If ∑∞
𝑛=1 𝛿𝑛 < ∞ and ∑𝑛=1 𝑏𝑛 < ∞, then lim𝑛 → ∞ 𝑎𝑛 exists.
Lemma 2 (see [6]). Let 𝐸 be a real uniformly convex Banach
space, and let 𝑎, and 𝑏 be two constants with 0 < 𝑎 < 𝑏 < 1.
Suppose that {𝑡𝑛 } ⊂ [𝑎, 𝑏] is a real sequence and {𝑥𝑛 } and {𝑦𝑛 }
are two sequences in 𝐸. Then, the conditions
󵄩
󵄩
lim 󵄩󵄩𝑡 𝑥 + (1 − 𝑡𝑛 ) 𝑦𝑛 󵄩󵄩󵄩 = 𝑑,
𝑛→∞ 󵄩 𝑛 𝑛
(7)
󵄩 󵄩
󵄩 󵄩
lim sup 󵄩󵄩󵄩𝑥𝑛 󵄩󵄩󵄩 ≤ 𝑑,
lim sup 󵄩󵄩󵄩𝑦𝑛 󵄩󵄩󵄩 ≤ 𝑑
𝑛→∞
𝑛→∞
imply that lim𝑛 → ∞ ‖𝑥𝑛 − 𝑦𝑛 ‖ = 0, where 𝑑 ≥ 0 is a constant.
Lemma 3 (see [2]). Let 𝐸 be a real uniformly convex Banach
space, 𝐾 a nonempty, closed, convex subset of 𝐸, and let 𝑇 :
𝐾 → 𝐸 be an asymptotically nonexpansive mapping with a
sequence {𝑘𝑛 } ⊂ [1, ∞) and 𝑘𝑛 → 1 as 𝑛 → ∞. Then, 𝐼 − 𝑇
is demiclosed at zero.
Lemma 4. The unique solutions to the positive integer equation
𝑛=𝑖+
(𝑚 − 1) 𝑚
,
2
𝑚 ≥ 𝑖, 𝑛 = 1, 2, 3, . . .
(8)
1
1
𝑚 = − [ − √ 2𝑛 + ] ,
2
4
(9)
are
𝑖=𝑛−
(𝑚 − 1) 𝑚
,
2
𝑛 = 1, 2, 3, . . . ,
where [𝑥] denotes the maximal integer that is not larger than
x.
Proof. It follows from (8) that
(4)
𝑖=𝑛−
𝑖 ≤ 𝑚, 𝑛 = 1, 2, 3, . . . ,
(10)
and hence
For convenience, we restate the following concepts and
results.
A Banach space 𝐸 is said to satisfy Opial’s condition if, for
any sequence {𝑥𝑛 } in 𝐸, 𝑥𝑛 ⇀ 𝑥 implies that
󵄩
󵄩
󵄩
󵄩
lim sup 󵄩󵄩󵄩𝑥𝑛 − 𝑥󵄩󵄩󵄩 < lim sup 󵄩󵄩󵄩𝑥𝑛 − 𝑦󵄩󵄩󵄩 ,
𝑛→∞
𝑛→∞
(𝑚 − 1) 𝑚
,
2
(5)
for all 𝑦 ∈ 𝐸 with 𝑦 ≠ 𝑥, where 𝑥𝑛 ⇀ 𝑥 denotes that {𝑥𝑛 }
converges weakly to 𝑥.
A mapping 𝑇 with domain 𝐷(𝑇) and range 𝑅(𝑇) in 𝐸 is
said to be demiclosed at 𝑝 if whenever {𝑥𝑛 } is a sequence in
(𝑚 − 1) 𝑚
≤ 𝑚,
2
𝑛 = 1, 2, 3, . . . ,
(11)
(𝑚 + 1) 𝑚
(𝑚 − 1) 𝑚
+1≤𝑛≤
,
2
2
𝑛 = 1, 2, 3, . . . ,
(12)
1≤𝑖=𝑛−
that is,
which implies that
7
1 2
(𝑚 − ) ≤ 2𝑛 − ,
2
4
1
1 2
(𝑚 + ) ≥ 2𝑛 + ,
2
4
𝑛 = 1, 2, 3, . . . .
(13)
Journal of Applied Mathematics
3
(𝑖𝑛 ) 󸀠
(𝑖𝑛 )
(𝛼𝑛 + 𝛾𝑛󸀠 𝑟𝑚
) > 0. We have
Note that 𝑎𝑛 := 1 − 𝛾𝑛 𝑘𝑚
𝑛
𝑛
Thus,
√ 2𝑛 +
1 1
7
1
− ≤ 𝑚 ≤ + √ 2𝑛 − ,
4 2
2
4
𝑛 = 1, 2, 3 . . . , (14)
𝑛
that is,
7 1
1
1
−√ 2𝑛 − − ≤ −𝑚 ≤ − √ 2𝑛 + ,
4 2
2
4
which implies that ‖𝑥𝑛 − 𝑞‖ ≤ (1 + 𝑏𝑛 )‖𝑥𝑛−1 − 𝑞‖, where
𝑛 = 1, 2, 3, . . . ,
(15)
while the difference of the two sides of the inequality above is
√2𝑛 + 1/4 + √2𝑛 − 7/4
∈ [0, 1) ,
(16)
𝑛 = 1, 2, 3, . . . .
Then, it follows from (15) that (9) holds obviously.
3. Main Results
Lemma 5. Let 𝐾 be a nonempty, closed, and convex subset
of a real uniformly convex Banach space 𝐸, and let {𝑇𝜆 𝑖 }∞
𝑖=1
and {𝑆𝜆 𝑖 }∞
be
two
countable
subsets
of
the
asymptotically
𝑖=1
nonexpansive mappings {𝑇𝜆 }𝜆∈Λ and {𝑆𝜆 }𝜆∈Λ from 𝐾 to itself,
respectively, with corresponding sequences {𝑘𝑛(𝑖) } ⊂ [1, ∞) and
∞
(𝑖)
{𝑟𝑛(𝑖) } ⊂ [1, ∞) such that ∑∞
𝑖=1 ∑𝑛=1 (𝑘𝑛 − 1) < ∞ and
∞
(𝑖)
∑∞
𝑖=1 ∑𝑛=1 (𝑟𝑛 − 1) < ∞. Suppose that {𝑥𝑛 } is generated by
(3), where 𝛾𝑛 𝑘𝑛(𝑖) (𝛼𝑛󸀠 + 𝛾𝑛󸀠 𝑟𝑛(𝑖) ) < 1 for all 𝑖 ≥ 1. If 𝐹 ≠ 0, then
lim𝑛 → ∞ ‖𝑥𝑛 − 𝑞‖ exists for each 𝑞 ∈ 𝐹.
(𝑖𝑛 )
(𝑖𝑛 )
(𝑖𝑛 )
(𝑖𝑛 )
= 𝑘𝑚
− 1 and 𝑣𝑚
= 𝑟𝑚
− 1 for each positive
Proof. Set 𝑢𝑚
𝑛
𝑛
𝑛
𝑛
integer 𝑛 ≥ 1, where 𝑖𝑛 and 𝑚𝑛 satisfy the positive integer
equation: 𝑛 = 𝑖 + (𝑚 − 1)𝑚/2(𝑚 ≥ 𝑖, 𝑚 ∈ N). For any 𝑞 ∈ 𝐹,
it follows from (3) that
󵄩
󵄩
󵄩󵄩
󸀠 󵄩
󸀠 (𝑖 ) 󵄩
󵄩󵄩𝑥 − 𝑞󵄩󵄩󵄩
󵄩󵄩𝑦𝑛 − 𝑞󵄩󵄩󵄩 ≤ 𝛼𝑛 󵄩󵄩󵄩𝑥𝑛 − 𝑞󵄩󵄩󵄩 + 𝛽𝑛 𝑟𝑚𝑛−1
󵄩
𝑛−1 󵄩 𝑛−1
(𝑖𝑛 ) 󵄩
󵄩󵄩𝑥 − 𝑞󵄩󵄩󵄩 = (𝛼󸀠 + 𝛾󸀠 𝑟(𝑖𝑛 ) ) 󵄩󵄩󵄩𝑥𝑛 − 𝑞󵄩󵄩󵄩
+ 𝛾𝑛󸀠 𝑟𝑚
𝑛
𝑛 𝑚𝑛 󵄩
󵄩
󵄩
𝑛 󵄩 𝑛−1
(𝑖𝑛−1 ) 󵄩
󵄩󵄩𝑥 − 𝑞󵄩󵄩󵄩 ,
+ 𝛽𝑛󸀠 𝑟𝑚
󵄩
𝑛−1 󵄩 𝑛−1
(17)
󵄩󵄩
󵄩
󵄩
󵄩
(𝑖 ) 󵄩
󵄩󵄩𝑥 − 𝑞󵄩󵄩󵄩
󵄩󵄩𝑥𝑛 − 𝑞󵄩󵄩󵄩 ≤ 𝛼𝑛 󵄩󵄩󵄩𝑥𝑛−1 − 𝑞󵄩󵄩󵄩 + 𝛽𝑛 𝑘𝑚𝑛−1
󵄩
𝑛−1 󵄩 𝑛−1
+
󵄩󵄩
󵄩
󵄩
(𝑖 ) 󵄩
) 󵄩󵄩󵄩𝑥𝑛−1 − 𝑞󵄩󵄩󵄩
󵄩󵄩𝑦𝑛 − 𝑞󵄩󵄩󵄩 = (𝛼𝑛 + 𝛽𝑛 𝑘𝑚𝑛−1
𝑛−1
(𝑖𝑛 ) 󵄩
󵄩󵄩𝑦 − 𝑞󵄩󵄩󵄩 .
+ 𝛾𝑛 𝑘𝑚
󵄩
𝑛 󵄩 𝑛
(18)
Substituting (17) into (18) yields that
[1 −
(𝑖𝑛 )
𝛾𝑛 𝑘𝑚
𝑛
(𝛼𝑛󸀠
+
󵄩
(𝑖𝑛 )
𝛾𝑛󸀠 𝑟𝑚
)] 󵄩󵄩󵄩𝑥𝑛
𝑛
(𝑖𝑛−1 )
(𝑖𝑛 ) (𝑖𝑛−1 )
+𝛾𝑛 𝛽𝑛󸀠 𝑣𝑚
+ 𝛾𝑛 𝛽𝑛󸀠 𝑢𝑚
𝑣
)
𝑛−1
𝑛 𝑚𝑛−1
(21)
−1
2
(𝑖𝑛 )
𝛾𝑛 𝑘𝑚
𝑛
(𝑖𝑛 )
(𝑖𝑛 )
(𝑖𝑛 ) (𝑖𝑛 )
(𝑖𝑛−1 )
𝑏𝑛 = (𝛾𝑛 𝛾𝑛󸀠 𝑣𝑚
+ 𝛾𝑛 𝑢𝑚
+ 𝛾𝑛 𝛾𝑛󸀠 𝑢𝑚
𝑣 + 𝛽𝑛 𝑢𝑚
𝑛
𝑛
𝑛 𝑚𝑛
𝑛−1
(𝑖𝑛 )
× (1 − 𝛾𝑛 (1 − 𝛽𝑛󸀠 ) − 𝛾𝑛 𝛾𝑛󸀠 𝑣𝑚
𝑛
1
7
1 − (√ 2𝑛 + − √ 2𝑛 − )
4
4
=1−
(𝑖𝑛−1 )
(𝑖𝑛 ) 󸀠 (𝑖𝑛−1 )
󵄩󵄩
󵄩 𝛼𝑛 + 𝛽𝑛 𝑘𝑚𝑛−1 + 𝛾𝑛 𝑘𝑚𝑛 𝛽𝑛 𝑟𝑚𝑛−1 󵄩󵄩
󵄩
󵄩󵄩𝑥𝑛 − 𝑞󵄩󵄩󵄩 ≤
󵄩󵄩𝑥𝑛−1 − 𝑞󵄩󵄩󵄩 , (20)
𝑎
󵄩
− 𝑞󵄩󵄩󵄩
󵄩
(𝑖𝑛−1 )
(𝑖𝑛 ) 󸀠 (𝑖𝑛−1 ) 󵄩
≤ (𝛼𝑛 + 𝛽𝑛 𝑘𝑚
+ 𝛾𝑛 𝑘𝑚
𝛽𝑛 𝑟𝑚𝑛−1 ) 󵄩󵄩󵄩𝑥𝑛−1 − 𝑞󵄩󵄩󵄩 .
𝑛−1
𝑛
(19)
(𝑖𝑛 )
(𝑖𝑛 ) (𝑖𝑛 )
−𝛾𝑛 𝑢𝑚
(1 − 𝛽𝑛󸀠 ) − 𝛾𝑛 𝛾𝑛󸀠 𝑢𝑚
𝑣 ) .
𝑛
𝑛 𝑚𝑛
∞
∞
∞
∞
(𝑖𝑛 )
(𝑖)
(𝑖)
Note that ∑∞
𝑛=1 𝑢𝑚𝑛 = ∑𝑖=1 ∑𝑛=𝑖 (𝑘𝑛 − 1) ≤ ∑𝑖=1 ∑𝑛=1 (𝑘𝑛 −
∞
∞
∞
∞
∞
(𝑖𝑛 )
(𝑖)
1) < ∞ and ∑𝑛=1 𝑣𝑚𝑛 = ∑𝑖=1 ∑𝑛=𝑖 (𝑟𝑛 − 1) ≤ ∑𝑖=1 ∑𝑛=1 (𝑟𝑛(𝑖) −
(𝑖𝑛 )
(𝑖𝑛 )
= lim𝑛 → ∞ 𝑣𝑚
= 0.
1) < ∞, which implies that lim𝑛 → ∞ 𝑢𝑚
𝑛
𝑛
Then, for a given 𝜖0 ∈ (0, 𝛿), there exists a positive 𝑛0 such
that
(𝑖𝑛 )
(𝑖𝑛 )
+ 𝛾𝑛 𝑢𝑚
(1 − 𝛽𝑛󸀠 )
𝛾𝑛 (1 − 𝛽𝑛󸀠 ) + 𝛾𝑛 𝛾𝑛󸀠 𝑣𝑚
𝑛
𝑛
(𝑖𝑛 ) (𝑖𝑛 )
+ 𝛾𝑛 𝛾𝑛󸀠 𝑢𝑚
𝑣 < 1 − 𝜖0 ,
𝑛 𝑚𝑛
(22)
as 𝑛 ≥ 𝑛0 . Then, it follows from (20) and (22) that
󵄩󵄩
󵄩
󵄩
󵄩
󵄩󵄩𝑥𝑛 − 𝑞󵄩󵄩󵄩 ≤ (1 + 𝑐𝑛 ) 󵄩󵄩󵄩𝑥𝑛−1 − 𝑞󵄩󵄩󵄩 ,
(23)
(𝑖𝑛 )
(𝑖𝑛 )
(𝑖𝑛 ) (𝑖𝑛 )
(𝑖𝑛−1 )
(𝑖𝑛−1 )
where 𝑐𝑛 = (1/𝜖0 )[𝑣𝑚
+ 𝑢𝑚
+ 𝑢𝑚
𝑣 + 𝑢𝑚
+ 𝑣𝑚
+
𝑛
𝑛
𝑛 𝑚𝑛
𝑛−1
𝑛−1
∞
(𝑖𝑛 ) (𝑖𝑛−1 )
𝑢𝑚𝑛 𝑣𝑚𝑛−1 ], and so ∑𝑛=1 𝑐𝑛 < ∞. Hence, it follows from (23)
and Lemma 1 that lim𝑛 → ∞ ‖𝑥𝑛 − 𝑞‖ exists for each 𝑞 ∈ 𝐹. The
proof is completed.
Remark 6. Because of the importance of the condition that 𝐹
is nonempty, we now give an example satisfying the lemma
with the set of common fixed points of {𝑇𝜆 }𝜆∈Λ and {𝑆𝜆 }𝜆∈Λ
being a non single point set. Let 𝐸 := R1 , and let 𝐾 := [−1, 1].
Define an infinite family of mappings {𝑇𝜆 }𝜆≥1 : 𝐾 → 𝐾 by
{ 1 𝑥𝜆 , 𝑥 ∈ [0, 1] ,
𝑇𝜆 (𝑥) = { 𝜆
𝑥 ∈ [−1, 0) ,
{𝑥,
(24)
and an infinite family of mappings {𝑆𝜆 }𝜆≥1 : 𝐾 → 𝐾 by
{ 1 sin 𝑥𝜆 , 𝑥 ∈ [0, 1] ,
𝑆𝜆 (𝑥) = { 𝜆
𝑥 ∈ [−1, 0) .
{𝑥,
(25)
Then, clearly, {𝑇𝜆 }𝜆≥1 and {𝑆𝜆 }𝜆≥1 are two infinite families of
asymptotically nonexpansive mappings with 𝐹 = [−1, 0].
, {𝑆𝜆 𝑖 }∞
, and {𝑥𝑛 } be the same as
Lemma 7. Let 𝐾, 𝐸, {𝑇𝜆 𝑖 }∞
𝑖=1
𝑖=1
those in Lemma 5. If 𝐹 ≠ 0, then for each 𝑖 ≥ 1, there exists a
subsequence {𝑥𝑛(𝑖) } of {𝑥𝑛 } such that lim𝑛 → ∞ ‖𝑥𝑛(𝑖) − 𝑇𝜆 𝑖 𝑥𝑛(𝑖) ‖ =
lim𝑛 → ∞ ‖𝑥𝑛(𝑖) − 𝑆𝜆 𝑖 𝑥𝑛(𝑖) ‖ = 0.
4
Journal of Applied Mathematics
Proof. By Lemma 5, we may assume that lim𝑛 → ∞ ‖𝑥𝑛 −𝑞‖ = 𝑑
for a given 𝑞 ∈ 𝐹, that is,
lim 𝑑
𝑛→∞ 𝑛
= 𝑑,
(26)
∗ 𝑚𝑛−1
where 𝑑𝑛 = ‖(1−𝛾𝑛 )[𝛼𝑛 (𝑥𝑛−1 −𝑞)/(1−𝛾𝑛 )+𝛽𝑛 ((𝑇𝑛−1
) 𝑥𝑛−1 −
∗ 𝑚𝑛
𝑞)/(1 − 𝛾𝑛 )] + 𝛾𝑛 ((𝑇𝑛 ) 𝑦𝑛 − 𝑞)‖. It follows from (17) that
󵄩
󵄩󵄩 ∗ 𝑚𝑛
󵄩󵄩(𝑇𝑛 ) 𝑦𝑛 − 𝑞󵄩󵄩󵄩
󵄩
󵄩
󵄩
(𝑖𝑛 ) 󵄩
≤ 𝑘𝑚𝑛 󵄩󵄩󵄩𝑦𝑛 − 𝑞󵄩󵄩󵄩
Combining (28) with (33), we have lim𝑛 → ∞ ‖𝑦𝑛 − 𝑞‖ = 𝑑.
Then, by ways similar to the preceding ones, it is easily shown
that
󵄩󵄩 𝛽󸀠
󵄩󵄩
𝛾𝑛󸀠
󵄩
󵄩󵄩
𝑚
∗
∗ 𝑚𝑛
) 𝑛−1 𝑥𝑛−1 +
(𝑆
)
𝑥
−
𝑥
lim 󵄩󵄩󵄩󵄩 𝑛 󸀠 (𝑆𝑛−1
󵄩󵄩 = 0,
𝑛
𝑛󵄩
𝑛
󸀠
𝑛→∞ 󵄩1 − 𝛼
1
−
𝛼
󵄩󵄩
𝑛
𝑛
󵄩
(34)
which means that
󵄩󵄩
lim 󵄩𝑦
𝑛→∞ 󵄩 𝑛
󵄩
(𝑖𝑛 )
(𝑖𝑛 ) 󵄩
(𝑖𝑛−1 ) 󵄩
󵄩󵄩𝑥 − 𝑞󵄩󵄩󵄩]
≤ 𝑘𝑚
[(𝛼𝑛󸀠 + 𝛾𝑛󸀠 𝑟𝑚
) 󵄩󵄩󵄩𝑥𝑛 − 𝑞󵄩󵄩󵄩 + 𝛽𝑛󸀠 𝑟𝑚
󵄩
𝑛
𝑛
𝑛−1 󵄩 𝑛−1
󵄩
󵄩
󵄩 󵄩
󵄩
(𝑖𝑛 ) 󵄩
= 𝑘𝑚
[󵄩󵄩󵄩𝑥𝑛 − 𝑞󵄩󵄩󵄩 + 𝛽𝑛󸀠 (󵄩󵄩󵄩𝑥𝑛 − 𝑞󵄩󵄩󵄩 − 󵄩󵄩󵄩𝑥𝑛−1 − 𝑞󵄩󵄩󵄩)
𝑛
(𝑖𝑛 )
+ 𝛾𝑛󸀠 𝑣𝑚
𝑛
󵄩󵄩
󵄩
󸀠 (𝑖 ) 󵄩
󵄩󵄩𝑥 − 𝑞󵄩󵄩󵄩] .
󵄩󵄩𝑥𝑛 − 𝑞󵄩󵄩󵄩 + 𝛽𝑛 𝑣𝑚𝑛−1
󵄩
𝑛−1 󵄩 𝑛−1
𝑚
󵄩󵄩
𝑛→∞
𝑛→∞
𝑛→∞
(𝑖𝑛−1 ) 󵄩
󵄩󵄩𝑥 − 𝑞󵄩󵄩󵄩 ≤ 𝑑,
≤ lim sup 𝑘𝑚
󵄩
𝑛−1 󵄩 𝑛−1
𝑛→∞
𝑛→∞
𝛽𝑛 (𝑖𝑛−1 ) 󵄩󵄩
󵄩
𝑢
) 󵄩𝑥 − 𝑞󵄩󵄩󵄩 = 𝑑,
1 − 𝛾𝑛 𝑚𝑛−1 󵄩 𝑛−1
(29)
󵄩󵄩 𝛼
󵄩󵄩
𝛽𝑛
𝑚
󵄩
󵄩
∗ 𝑚𝑛−1
(𝑇𝑛−1
) 𝑥𝑛−1 − (𝑇𝑛∗ ) 𝑛 𝑦𝑛 󵄩󵄩󵄩 = 0,
lim 󵄩󵄩󵄩 𝑛 𝑥𝑛−1 +
𝑛→∞ 󵄩1 − 𝛾
󵄩󵄩
1 − 𝛾𝑛
󵄩
𝑛
(30)
where 𝑒𝑛󸀠 = ‖𝛼𝑛 (𝑥𝑛−1 − 𝑞) + 𝛾𝑛 ((𝑇𝑛∗ ) 𝑛 𝑦𝑛 − 𝑞)‖/(1 − 𝛽𝑛 ). It then
follows from (36)–(38) and Lemma 2 that
󵄩󵄩 𝛼
󵄩󵄩
𝛾𝑛
𝑚
󵄩
󵄩
∗ 𝑚𝑛−1
(𝑇𝑛∗ ) 𝑛 𝑦𝑛 − (𝑇𝑛−1
) 𝑥𝑛−1 󵄩󵄩󵄩 = 0,
lim 󵄩󵄩󵄩 𝑛 𝑥𝑛−1 +
𝑛→∞ 󵄩1 − 𝛽
󵄩󵄩
1 − 𝛽𝑛
󵄩
𝑛
(39)
which implies that
󵄩󵄩
lim 󵄩󵄩𝑥𝑛
𝑛→∞ 󵄩
󵄩󵄩
lim 󵄩𝑥
𝑛→∞ 󵄩 𝑛
(31)
Now, we show that ‖𝑦𝑛 − 𝑞‖ → 𝑑 as 𝑛 → ∞. It follows from
(18) that
𝛼𝑛 + 𝛽𝑛 󵄩󵄩
󵄩 󵄩
󵄩 󵄩
󵄩
(󵄩󵄩𝑥𝑛 − 𝑞󵄩󵄩󵄩 − 󵄩󵄩󵄩𝑥𝑛−1 − 𝑞󵄩󵄩󵄩) + 󵄩󵄩󵄩𝑥𝑛 − 𝑞󵄩󵄩󵄩
𝛾𝑛
−
𝛽𝑛 (𝑖𝑛−1 ) 󵄩󵄩
󵄩
󵄩
(𝑖 ) 󵄩
𝑢
󵄩𝑥 − 𝑞󵄩󵄩󵄩 ≤ 𝑘𝑚𝑛𝑛 󵄩󵄩󵄩𝑦𝑛 − 𝑞󵄩󵄩󵄩 .
𝛾𝑛 𝑚𝑛−1 󵄩 𝑛−1
(32)
Taking lim inf on both sides in the inequality above yields that
󵄩
󵄩
󵄩
󵄩
𝑑 = lim inf 󵄩󵄩󵄩𝑥𝑛 − 𝑞󵄩󵄩󵄩 ≤ lim inf 󵄩󵄩󵄩𝑦𝑛 − 𝑞󵄩󵄩󵄩 .
𝑛→∞
𝑛→∞
∗
− (𝑇𝑛−1
)
(33)
𝑚𝑛−1
󵄩
𝑥𝑛−1 󵄩󵄩󵄩󵄩 = 0.
(40)
Then, it follows from (31), (35), and (40) that
which, in addition to (3), implies that
󵄩
𝑚
− (𝑇𝑛∗ ) 𝑛 𝑦𝑛 󵄩󵄩󵄩󵄩 = 0.
(38)
𝑚
𝑚
󵄩󵄩
𝛼𝑛 󵄩󵄩
󵄩
󵄩𝑥 − 𝑞󵄩󵄩󵄩
1 − 𝛽𝑛 󵄩 𝑛−1
𝛾
(𝑖𝑛 ) 󵄩
󵄩󵄩𝑦 − 𝑞󵄩󵄩󵄩) = 𝑑,
+ 𝑛 𝑘𝑚
󵄩
1 − 𝛽𝑛 𝑛 󵄩 𝑛
∗
where 𝑒𝑛 = ‖𝛼𝑛 (𝑥𝑛−1 − 𝑞) + 𝛽𝑛 ((𝑇𝑛−1
) 𝑛−1 𝑥𝑛−1 − 𝑞)‖/(1 − 𝛾𝑛 ).
It then follows from (26), (28), (29), and Lemma 2 that
lim 󵄩󵄩𝑥𝑛
𝑛→∞ 󵄩
(37)
𝑛→∞
lim sup 𝑒𝑛󸀠 ≤ lim sup (
𝛽𝑛 (𝑖𝑛−1 ) 󵄩󵄩
󵄩
𝑘
󵄩𝑥 − 𝑞󵄩󵄩󵄩)
1 − 𝛾𝑛 𝑚𝑛−1 󵄩 𝑛−1
(36)
󵄩 ∗ 𝑚𝑛−1
󵄩
lim sup 󵄩󵄩󵄩󵄩(𝑇𝑛−1
) 𝑥𝑛−1 − 𝑞󵄩󵄩󵄩󵄩
(28)
𝛼 󵄩
󵄩
lim sup 𝑒𝑛 ≤ lim sup ( 𝑛 󵄩󵄩󵄩𝑥𝑛−1 − 𝑞󵄩󵄩󵄩
1
−
𝛾
𝑛→∞
𝑛→∞
𝑛
= lim sup (1 +
𝑛→∞
𝑛→∞
Next, it follows from (26) that
+
󵄩
− 𝑞󵄩󵄩󵄩 = lim 𝑑𝑛󸀠 = 𝑑,
then
Taking lim sup on both sides in (27) yields that
󵄩
󵄩
𝑚
󵄩
󵄩
lim sup 󵄩󵄩󵄩󵄩(𝑇𝑛∗ ) 𝑛 𝑦𝑛 − 𝑞󵄩󵄩󵄩󵄩 ≤ lim sup 󵄩󵄩󵄩𝑦𝑛 − 𝑞󵄩󵄩󵄩 ≤ 𝑑.
(35)
Set 𝑑𝑛󸀠 = ‖(1−𝛽𝑛 )[𝛼𝑛 (𝑥𝑛−1 −𝑞)/(1−𝛽𝑛 )+𝛾𝑛 ((𝑇𝑛∗ ) 𝑛 𝑦𝑛 −𝑞)/(1−
∗ 𝑚𝑛−1
) 𝑥𝑛−1 − 𝑞)‖. Since
𝛽𝑛 )] + 𝛽𝑛 ((𝑇𝑛−1
lim 󵄩𝑥
𝑛→∞ 󵄩 𝑛
(27)
󵄩
− 𝑥𝑛 󵄩󵄩󵄩 = 0.
󵄩
− 𝑥𝑛−1 󵄩󵄩󵄩 = 0.
(41)
Similarly, we have
󵄩󵄩
󵄩
𝑚
− (𝑆𝑛∗ ) 𝑛 𝑥𝑛 󵄩󵄩󵄩󵄩 = 0,
󵄩
󵄩
𝑚
∗
lim 󵄩󵄩󵄩𝑦𝑛 − (𝑆𝑛−1
) 𝑛−1 𝑥𝑛−1 󵄩󵄩󵄩󵄩 = 0.
𝑛→∞ 󵄩
lim 󵄩󵄩𝑦𝑛
𝑛→∞ 󵄩
(42)
Next, for any 𝑖 ≥ 1, we consider the corresponding
subsequence {𝑥𝑙(𝑖) }𝑙∈Γ of {𝑥𝑛 }, where 𝑙 ∈ Γ𝑖 := {𝑙 : 𝑙 =
𝑖
𝑖 + (𝑗 − 1)𝑗/2, 𝑗 ≥ 𝑖, 𝑗 ∈ N}. For example, by the definition
of Γ1 , we have Γ1 = {1, 2, 4, 7, 11, 16, . . .} and 𝑖1 = 𝑖2 = 𝑖4 =
𝑖7 = 𝑖11 = 𝑖16 = ⋅ ⋅ ⋅ = 1. For simplicity, {𝑥𝑙(𝑖) }𝑙∈Γ𝑖 , {𝑦𝑙(𝑖) }𝑙∈Γ𝑖 ,
{𝑇𝑙∗(𝑖) }𝑙∈Γ𝑖 , and {𝑗𝑙(𝑖) }𝑙∈Γ𝑖 are written as {𝑥𝑛󸀠 }, {𝑦𝑛󸀠 }, {𝑇𝑛󸀠 } and {𝑚𝑛 },
respectively. Note that {𝑚𝑛 }𝑛∈Γ𝑖 = {𝑖, 𝑖 + 1, 𝑖 + 2, . . .}, that is,
Journal of Applied Mathematics
5
(𝑖 )
󸀠
𝑘1 𝑙 = 𝑘1(𝑖) , 𝑚𝑛 − 1 = 𝑚𝑛−1 , and 𝑇𝑛󸀠 = 𝑇𝜆 𝑖 = 𝑇𝑛−1
whenever
𝑙 ∈ Γ𝑖 . Then, we have
󵄩󵄩 󸀠
󵄩
󵄩󵄩𝑥𝑛 − 𝑇𝑛󸀠 𝑥𝑛 󵄩󵄩󵄩
󵄩
󵄩
󵄩󵄩 󸀠
𝑚𝑛 󵄩
󵄩
≤ 󵄩󵄩󵄩𝑥𝑛 − (𝑇𝑛󸀠 ) 𝑦𝑛󸀠 󵄩󵄩󵄩
󵄩
󵄩
󵄩󵄩 󸀠 𝑚𝑛 󸀠
󵄩󵄩 󵄩󵄩
󵄩󵄩
𝑚𝑛
𝑚𝑛
+ 󵄩󵄩󵄩(𝑇𝑛 ) 𝑦𝑛 − (𝑇𝑛󸀠 ) 𝑥𝑛󸀠 󵄩󵄩󵄩 + 󵄩󵄩󵄩(𝑇𝑛󸀠 ) 𝑥𝑛󸀠 − 𝑇𝑛󸀠 𝑥𝑛󸀠 󵄩󵄩󵄩
󵄩
󵄩 󵄩
󵄩
󵄩󵄩 󸀠
󵄩
𝑚
󵄩
󵄩
𝑛
󵄩
(𝑖) 󵄩 󸀠
󵄩󵄩𝑦 − 𝑥𝑛󸀠 󵄩󵄩󵄩
≤ 󵄩󵄩󵄩𝑥𝑛 − (𝑇𝑛󸀠 ) 𝑦𝑛󸀠 󵄩󵄩󵄩 + 𝑘𝑚
𝑛 󵄩 𝑛
󵄩
󵄩
󵄩
󵄩
𝑚𝑛 −1 󸀠
𝑚𝑛 󵄩
󵄩󵄩
󵄩󵄩 󵄩󵄩
+ 𝑘1(𝑖) 󵄩󵄩󵄩(𝑇𝑛󸀠 )
𝑥𝑛 − 𝑥𝑛󸀠 󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩𝑥𝑛󸀠 − (𝑇𝑛󸀠 ) 𝑦𝑛󸀠 󵄩󵄩󵄩
󵄩
󵄩 󵄩
󵄩
󵄩
󵄩
(𝑖) 󵄩 󸀠
󵄩󵄩𝑦 − 𝑥𝑛󸀠 󵄩󵄩󵄩 + 𝑘1(𝑖)
+ 𝑘𝑚
𝑛 󵄩 𝑛
󵄩
󵄩󵄩 󸀠 𝑚𝑛 −1 󸀠
𝑚𝑛−1 󸀠 󵄩
󸀠
󵄩󵄩󵄩
× [󵄩󵄩󵄩(𝑇𝑛 )
𝑥𝑛 − (𝑇𝑛−1
) 𝑥𝑛−1
󵄩󵄩
󵄩
󵄩󵄩 󸀠 𝑚𝑛−1 󸀠
󵄩
󵄩
+ 󵄩󵄩󵄩(𝑇𝑛−1 ) 𝑥𝑛−1 − 𝑥𝑛󸀠 󵄩󵄩󵄩]
󵄩
󵄩
󵄩󵄩 󸀠
󵄩
𝑚
󵄩
𝑛
󵄩
(𝑖) 󵄩 󸀠
󸀠󵄩
󵄩
≤ 󵄩󵄩󵄩𝑥𝑛 − (𝑇𝑛󸀠 ) 𝑦𝑛󸀠 󵄩󵄩󵄩 + 𝑘𝑚
󵄩󵄩𝑦 − 𝑥𝑛 󵄩󵄩󵄩
𝑛 󵄩 𝑛
󵄩
󵄩
󵄩
󵄩 󸀠
󵄩 󵄩󵄩󵄩 󸀠 𝑚𝑛−1 󸀠
(𝑖) 󵄩
󸀠 󵄩
󵄩󵄩𝑥 − 𝑥𝑛−1
󵄩󵄩 + 󵄩󵄩(𝑇𝑛−1 ) 𝑥𝑛−1 − 𝑥𝑛󸀠 󵄩󵄩󵄩󵄩] .
+ 𝑘1(𝑖) [𝑘𝑚
𝑛−1 󵄩 𝑛
󵄩 󵄩
󵄩
(43)
It hence follows from (31), (35), (40), and (41) that
lim𝑛 → ∞ ‖𝑥𝑛󸀠 − 𝑇𝑛󸀠 𝑥𝑛󸀠 ‖ = 0. That is, for each 𝑖 ≥ 1, there exists a
subsequence {𝑥𝑛(𝑖) } of {𝑥𝑛 } such that lim𝑛 → ∞ ‖𝑥𝑛(𝑖) −𝑇𝑛∗(𝑖) 𝑥𝑛(𝑖) ‖ =
0. Since 𝑇𝑛∗(𝑖) = 𝑇𝜆 𝑖 , we have, for each 𝑖 ≥ 1,
󵄩
lim 󵄩󵄩󵄩𝑥𝑛(𝑖)
𝑛→∞ 󵄩
󵄩
− 𝑇𝜆 𝑖 𝑥𝑛(𝑖) 󵄩󵄩󵄩󵄩 = 0.
(44)
Similarly, it can be shown that, for each 𝑖 ≥ 1,
󵄩
lim 󵄩󵄩󵄩𝑦𝑛(𝑖)
𝑛→∞ 󵄩
󵄩
− 𝑆𝜆 𝑖 𝑦𝑛(𝑖) 󵄩󵄩󵄩󵄩 = 0.
(45)
This completes the proof.
Remark 8. The key point of the proof of Lemma 7 lies in the
use of a specific way of choosing the indexes of the involved
mappings, which makes the generalization of finite families
of nonlinear mappings to infinite ones possible.
, {𝑆𝜆 𝑖 }∞
, and {𝑥𝑛 } be the same
Theorem 9. Let 𝐾, 𝐸, {𝑇𝜆 𝑖 }∞
𝑖=1
𝑖=1
or
as those in Lemma 5. If 𝐹 ≠ 0 and there exist a 𝑇𝜆 𝑖 ∈ {𝑇𝜆 𝑖 }∞
𝑖=1
0
∞
an 𝑆𝜆 𝑖 ∈ {𝑆𝜆 𝑖 }𝑖=1 and a nondecreasing function 𝑓 : [0, ∞) →
0
[0, ∞) with 𝑓(0) = 0 and 𝑓(𝑟) > 0 for all 𝑟 ∈ (0, ∞) such that
𝑓(𝑑(𝑥𝑛 , 𝐹)) ≤ ‖𝑥𝑛 − 𝑇𝜆 𝑖 𝑥𝑛 ‖ or 𝑓(𝑑(𝑥𝑛 , 𝐹)) ≤ ‖𝑥𝑛 − 𝑆𝜆 𝑖 𝑥𝑛 ‖ for
0
0
all 𝑛 ≥ 1, then {𝑥𝑛 } converges strongly to some point of 𝐹.
Proof. By Lemma 7, there exists a subsequence {𝑥𝑛(𝑖0 ) } of
{𝑥𝑛 } such that lim𝑛 → ∞ ‖𝑥𝑛(𝑖0 ) − 𝑇𝜆 𝑖 𝑥𝑛(𝑖0 ) ‖ = lim𝑛 → ∞ ‖𝑥𝑛(𝑖0 ) −
𝑆𝜆 𝑖 𝑥𝑛(𝑖0 ) ‖ = 0. Since
0
0
󵄩󵄩
󵄩󵄩
𝑓 (𝑑 (𝑥𝑛(𝑖0 ) , 𝐹)) ≤ 󵄩󵄩󵄩𝑥𝑛(𝑖0 ) − 𝑇𝜆 𝑖 𝑥𝑛(𝑖0 ) 󵄩󵄩󵄩 ,
0
󵄩
󵄩
(46)
or
󵄩󵄩
󵄩󵄩
𝑓 (𝑑 (𝑥𝑛(𝑖0 ) , 𝐹)) ≤ 󵄩󵄩󵄩𝑥𝑛(𝑖0 ) − 𝑆𝜆 𝑖 𝑥𝑛(𝑖0 ) 󵄩󵄩󵄩 ,
0
󵄩
󵄩
(47)
by taking lim sup as 𝑛 → ∞ on both sides in the inequality
above, we have
lim 𝑓 (𝑑 (𝑥𝑛(𝑖0 ) , 𝐹)) = 0,
𝑛→∞
(48)
which implies lim𝑛 → ∞ 𝑑(𝑥𝑛(𝑖0 ) , 𝐹) = 0 by the definition of the
function 𝑓.
Now, we will show that {𝑥𝑛(𝑖0 ) } is a Cauchy sequence. By
Lemma 5, there exists a constant 𝑀 > 0 such that ‖𝑥𝑛 − 𝑞‖ ≤
𝑀‖𝑥𝑚 − 𝑞‖ as 𝑛 > 𝑚. And for any 𝜖 > 0, there exists a positive
integer 𝑁 such that 𝑑(𝑥𝑛(𝑖0 ) , 𝐹) < 𝜖/2𝑀 for all 𝑛 ≥ 𝑁. Then,
for any 𝑞 ∈ 𝐹 and 𝑛, 𝑚 ≥ 𝑁, we have
󵄩󵄩 (𝑖0 )
(𝑖0 ) 󵄩
󵄩󵄩 ≤ 󵄩󵄩󵄩𝑥(𝑖0 ) − 𝑞󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑥(𝑖0 ) − 𝑞󵄩󵄩󵄩 ≤ 2𝑀 󵄩󵄩󵄩𝑥(𝑖0 ) − 𝑞󵄩󵄩󵄩 .
󵄩󵄩𝑥𝑛 − 𝑥𝑚
󵄩󵄩 𝑁
󵄩󵄩 󵄩󵄩 𝑛
󵄩󵄩 󵄩󵄩 𝑚
󵄩󵄩
󵄩󵄩
󵄩
(49)
Taking the infimum in the above inequalities for all 𝑞 ∈ 𝐹
yields that
󵄩󵄩 (𝑖0 )
(𝑖0 ) 󵄩
󵄩󵄩 ≤ 2𝑀𝑑 (𝑥(𝑖0 ) , 𝐹) < 𝜖,
󵄩󵄩𝑥𝑛 − 𝑥𝑚
󵄩󵄩
𝑁
󵄩
(50)
which implies that {𝑥𝑛(𝑖0 ) } is a Cauchy sequence. Therefore,
there exists a 𝑝 ∈ 𝐾 such that 𝑥𝑛(𝑖0 ) → 𝑝 as 𝑛 → ∞ since 𝐸
is complete. Furthermore, lim𝑛 → ∞ 𝑑(𝑥𝑛(𝑖0 ) , 𝐹) = 0 shows that
𝑑(𝑝, 𝐹) = 0, which implies that 𝑝 ∈ 𝐹 since 𝐹 is closed. It
follows from the existence of lim𝑛 → ∞ ‖𝑥𝑛 − 𝑝‖ that 𝑥𝑛 → 𝑝
as 𝑛 → ∞. This completes the proof.
Remark 10. The result of Theorem 9 extends that of Wang
et al. [10] whose related research is restricted to the situation
of two asymptotically nonexpansive mappings.
Acknowledgments
The authors are greatly grateful to the referees for their useful
suggestions by which the contents of this paper are improved.
This study is supported by the National Natural Science
Foundation of China (Grant no. 11061037).
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