Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2013, Article ID 259470, 7 pages
http://dx.doi.org/10.1155/2013/259470
Research Article
Convergence Theorems for Common Fixed Points of a Finite
Family of Relatively Nonexpansive Mappings in Banach Spaces
Yuanheng Wang1 and Weifeng Xuan2
1
2
Department of Mathematics, Zhejiang Normal University, Zhejiang 321004, China
Department of Mathematics, Nanjing University, Nanjing 210093, China
Correspondence should be addressed to Yuanheng Wang; wangyuanheng@yahoo.com.cn
Received 15 January 2013; Accepted 18 February 2013
Academic Editor: Yisheng Song
Copyright © 2013 Y. Wang and W. Xuan. 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 establish some strong convergence theorems for a common fixed point of a finite family of relatively nonexpansive mappings
by using a new hybrid iterative method in mathematical programming and the generalized projection method in a Banach space.
Our results improve and extend the corresponding results by many others.
1. Introduction
Let 𝐸 be a smooth Banach space and 𝐸∗ the dual of 𝐸. The
function Φ : 𝐸 × 𝐸 → 𝑅 is defined by
󵄩 󵄩2
𝜙 (𝑦, 𝑥) = 󵄩󵄩󵄩𝑦󵄩󵄩󵄩 − 2 ⟨𝑦, 𝐽𝑥⟩ + ‖𝑥‖2 ,
(1)
for all 𝑥, 𝑦 ∈ 𝐸, where 𝐽 is the normalized duality mapping
from 𝐸 to 𝐸∗ . Let 𝐶 be a closed convex subset of 𝐸, and let 𝑇
be a mapping from 𝐶 into itself. We denote by 𝐹(𝑇) the set of
fixed points of 𝑇. A point 𝑝 in 𝐶 is said to be an asymptotic
fixed point of 𝑇 (see [1]), if 𝐶 contains a sequence {𝑥𝑛 } which
converges weakly to 𝑝 such that the strong lim𝑛 → ∞ (𝑥𝑛 −
𝑇𝑥𝑛 ) = 0. The set of asymptotic fixed points of 𝑇 will be
̂
denoted by 𝐹(𝑇).
A mapping 𝑇 from 𝐶 into itself is called
nonexpansive, if
󵄩󵄩󵄩𝑇𝑥 − 𝑇𝑦󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩 ,
(2)
󵄩
󵄩 󵄩
󵄩
for all 𝑥, 𝑦 ∈ 𝐶, and relatively nonexpansive (see [2]), if
̂
𝐹(𝑇)
= 𝐹(𝑇) and
𝜙 (𝑝, 𝑇𝑥) ≤ 𝜙 (𝑝, 𝑥) ,
(3)
for all 𝑥 ∈ 𝐶 and 𝑝 ∈ 𝐹(𝑇). The iterative methods for approximation of fixed points of nonexpansive mappings, relatively
nonexpansive mappings, and other generational nonexpansive mappings have been studied by many researchers; see [3–
13].
Actually, Mann [14] firstly introduced Mann iteration
process in 1953, which is defined as follows:
𝑥0 = 𝑥 ∈ 𝐶,
𝑥𝑛+1 = 𝛼𝑛 𝑥𝑛 + (1 − 𝛼𝑛 ) 𝑇𝑥𝑛 ,
𝑛 ≥ 0.
(4)
It is very useful to approximate a fixed point of a nonexpansive mapping. However, as we all know, it has only weak
convergence in a Hilbert space (see [15]). As a matter of
fact, the process (3) may fail to converge for a Lipschitz
pseudocontractive mapping in a Hilbert space (see [16]). For
example, Reich [17] proved that if 𝐸 is a uniformly convex
Banach space with Fréchet differentiable norm and if {𝛼𝑛 } is
chosen such that ∑∞
𝑛=0 𝛼𝑛 (1−𝛼𝑛 ) = ∞, then the sequence {𝑥𝑛 }
defined by (3) converges weakly to a fixed point of 𝑇.
Some have made attempts to modify the Mann iteration
methods, so that strong convergence is guaranteed. Nakajo
and Takahashi [18] proposed the following modification
of the Mann iteration method for a single nonexpansive
mapping 𝑇 in a Hilbert space 𝐻:
𝑥0 = 𝑥 ∈ 𝐶,
𝑦𝑛 = 𝛼𝑛 𝑥𝑛 + (1 − 𝛼𝑛 ) 𝑇𝑥𝑛 ,
2
Abstract and Applied Analysis
󵄩 󵄩
󵄩
󵄩
𝐶𝑛 = {𝑧 ∈ 𝐶 : 󵄩󵄩󵄩𝑧 − 𝑦𝑛 󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩𝑧 − 𝑥𝑛 󵄩󵄩󵄩} ,
𝑄𝑛 = {𝑧 ∈ 𝐶 : ⟨𝑥𝑛 − 𝑧, 𝑥 − 𝑥𝑛 ⟩ ≥ 0} ,
𝑥𝑛+1 = 𝑃𝐶𝑛 ⋂ 𝑄𝑛 𝑥,
𝑛 = 0, 1, 2, . . . ,
(5)
where 𝑃𝐾 denotes the metric projection from 𝐻 onto a closed
convex subset of 𝐻. They proved that if the sequence {𝛼𝑛 } is
bounded above from one, then {𝑥𝑛 } defined by (5) converges
strongly to 𝑃𝐹(𝑇) 𝑥.
The ideas to generate the process (5) from Hilbert spaces
to Banach spaces have been made. By using the properties
available on uniformly convex and uniformly smooth Banach
spaces, Matsushita and Takahashi [10] presented their idea
of the following method for a single relatively nonexpansive
mapping 𝑇 in a Banach space 𝐸:
(6)
𝑛 = 0, 1, 2, . . . ,
where 𝐽 is the duality mapping on 𝐸 and Π𝐹(𝑇) 𝑥 is the
generalized projection from 𝐶 onto 𝐹(𝑇).
In 2007 and 2008, Plubing and Ungchittrakool [19, 20]
improved and generalized the process (6) to the new general
process of two relatively nonexpansive mappings in a Banach
space:
𝑥0 = 𝑥 ∈ 𝐶,
𝑦𝑛 = 𝐽−1 (𝛼𝑛 𝐽𝑥𝑛 + (1 − 𝛼𝑛 ) 𝐽𝑧𝑛 ) ,
(7)
𝐻𝑛 = {𝑧 ∈ 𝐶 : 𝜙 (𝑧, 𝑦𝑛 ) ≤ 𝜙 (𝑧, 𝑥𝑛 )} ,
𝑥0 = 𝑥 ∈ 𝐶,
𝑦𝑛 = 𝐽−1 (𝛼𝑛 𝐽𝑥 + (1 − 𝛼𝑛 ) 𝐽𝑧𝑛 ) ,
𝑧𝑛 = 𝐽−1 (𝛽𝑛(1) 𝐽𝑥𝑛 + 𝛽𝑛(2) 𝐽𝑇𝑥𝑛 + 𝛽𝑛(3) 𝐽𝑆𝑥𝑛 ) ,
(8)
+ 𝛼𝑛 (‖𝑥‖2 + 2 ⟨𝐽𝑥𝑛 − 𝐽𝑥, 𝑧⟩)} ,
𝑊𝑛 = {𝑧 ∈ 𝐶 : ⟨𝑥𝑛 − 𝑧, 𝐽𝑥 − 𝐽𝑥𝑛 ⟩ ≥ 0} ,
𝑥𝑛+1 = Π𝐻𝑛 ⋂ 𝑊𝑛 𝑥,
(9)
𝑊𝑛 = {𝑧 ∈ 𝐶 : ⟨𝑥𝑛 − 𝑧, 𝐽𝑥 − 𝐽𝑥𝑛 ⟩ ≥ 0} ,
𝑛 = 0, 1, 2, . . . ,
where Π𝐻𝑛 ⋂ 𝑊𝑛 is the generalized projection from 𝐶
onto the intersection set 𝐻𝑛 ⋂ 𝑊𝑛 ; {𝛼𝑛 }, {𝛽𝑛 }, {𝛾𝑛 }, {𝜆(0)
𝑛 },
(𝑁)
},
.
.
.
,
{𝜆
}
are
the
sequences
in
[0,
1]
with
𝛼
+
𝛽
{𝜆(1)
𝑛
𝑛 +
𝑛
𝑛
𝑁
(𝑖)
𝛾𝑛 = 1 and ∑𝑖=0 𝜆 𝑛 = 1 for all 𝑛 ≥ 0. We prove, under certain
appropriate assumptions on the sequences, that {𝑥𝑛 } defined
by (9) converges strongly to 𝑃𝐹 𝑥, where 𝑃𝐹 is the generalized
projection from 𝐶 to 𝐹.
Obviously, the process (9) reduces to become (7) when
𝑁 = 2, 𝛼𝑛 = 0 and become (8) when 𝑁 = 2, 𝛽𝑛 = 0.
So, our results extend and improve the corresponding ones
announced by Nakajo and Takahashi [18], Plubtieng and
Ungchittrakool [19, 20], Matsushita and Takahashi [10], and
Martinez-Yanes and Xu [21].
This section collects some definitions and lemmas which will
be used in the proofs for the main results in the next section.
Throughout this paper, let 𝐸 be a real Banach space. Let 𝐽
∗
denote the normalized duality mapping from 𝐸 into 2𝐸 given
by
󵄩 󵄩
󵄩 󵄩
𝐽 (𝑥) = {𝑓 ∈ 𝐸∗ , ⟨𝑥, 𝑓⟩ = ‖𝑥‖ 󵄩󵄩󵄩𝑓󵄩󵄩󵄩 , ‖𝑥‖ = 󵄩󵄩󵄩𝑓󵄩󵄩󵄩} , ∀𝑥 ∈ 𝐸,
(10)
𝑛 = 0, 1, 2, . . . ,
𝐻𝑛 = {𝑧 ∈ 𝐶 : 𝜙 (𝑧, 𝑦𝑛 ) ≤ 𝜙 (𝑧, 𝑥𝑛 )
𝑖=1
2. Preliminaries
𝑊𝑛 = {𝑧 ∈ 𝐶 : ⟨𝑥𝑛 − 𝑧, 𝐽𝑥 − 𝐽𝑥𝑛 ⟩ ≥ 0} ,
𝑥𝑛+1 = Π𝐻𝑛 ⋂ 𝑊𝑛 𝑥,
𝑁
(𝑖)
𝑧𝑛 = 𝐽−1 (𝜆(0)
𝑛 𝐽𝑥𝑛 + ∑𝜆 𝑛 𝐽𝑇𝑖 𝑥𝑛 ) ,
𝑥𝑛+1 = Π𝐻𝑛 ⋂ 𝑊𝑛 𝑥,
𝑊𝑛 = {𝑧 ∈ 𝐶 : ⟨𝑥𝑛 − 𝑧, 𝐽𝑥 − 𝐽𝑥𝑛 ⟩ ≥ 0} ,
𝑧𝑛 = 𝐽−1 (𝛽𝑛(1) 𝐽𝑥𝑛 + 𝛽𝑛(2) 𝐽𝑇𝑥𝑛 + 𝛽𝑛(3) 𝐽𝑆𝑥𝑛 ) ,
𝑦𝑛 = 𝐽−1 (𝛼𝑛 𝐽𝑥 + 𝛽𝑛 𝐽𝑥𝑛 + 𝛾𝑛 𝐽𝑧𝑛 ) ,
+ 𝛼𝑛 (‖𝑥‖2 + 2 ⟨𝐽𝑥𝑛 − 𝐽𝑥, 𝑧⟩)} ,
𝑦𝑛 = 𝐽−1 (𝛼𝑛 𝐽𝑥𝑛 + (1 − 𝛼𝑛 ) 𝐽𝑇𝑥𝑛 ) ,
𝑥𝑛+1 = Π𝐻𝑛 ⋂ 𝑊𝑛 𝑥,
𝑥0 = 𝑥 ∈ 𝐶,
𝐻𝑛 = {𝑧 ∈ 𝐶 : 𝜙 (𝑧, 𝑦𝑛 ) ≤ 𝜙 (𝑧, 𝑥𝑛 )
𝑥0 = 𝑥 ∈ 𝐶,
𝐻𝑛 = {𝑧 ∈ 𝐶 : 𝜙 (𝑧, 𝑦𝑛 ) ≤ 𝜙 (𝑧, 𝑥𝑛 )} ,
mappings 𝑆 and 𝑇 provided that the sequences satisfy some
appropriate conditions.
Inspired and motivated by these facts, in this paper,
we aim to improve and generalize the process (7) and (8)
to the new general process of a finite family of relatively
nonexpansive mappings in a Banach space. Let 𝐶 be a closed
convex subset of a Banach space 𝐸 and let 𝑇1 , 𝑇2 , . . . , 𝑇𝑁 :
𝐶 → 𝐶 be relatively nonexpansive mappings such that 𝐹 :=
∩𝑁
𝑖=1 𝐹(𝑇𝑖 ) ≠ 𝜙. Define {𝑥𝑛 } in the following way:
𝑛 = 0, 1, 2, . . . .
They proved that both iterations (7) and (8) converge strongly
to a common fixed point of two relatively nonexpansive
where 𝐸∗ denotes the dual space of 𝐸 and ⟨⋅, ⋅⟩ denotes the
generalized duality pairing.
A Banach space 𝐸 is said to be strictly convex if ‖𝑥 +
𝑦‖/2 < 1 for ‖𝑥‖ = ‖𝑦‖ = 1 and 𝑥 ≠ 𝑦. It is also said to
be uniformly convex if lim𝑛 → ∞ ‖𝑥𝑛 − 𝑦𝑛 ‖ = 0 for any two
sequences {𝑥𝑛 }, {𝑦𝑛 } in 𝐸 such that ‖𝑥𝑛 ‖ = ‖𝑦𝑛 ‖ = 1 and
lim𝑛 → ∞ (‖𝑥𝑛 + 𝑦𝑛 ‖/2) = 1. Let 𝑈 = {𝑥 ∈ 𝐸 : ‖𝑥‖ = 1}
be the unit sphere of 𝐸, then the Banach space 𝐸 is said to
be smooth provided that lim𝑡 → 0 ((‖𝑥 + 𝑡𝑦‖ − ‖𝑥‖)/𝑡) exists
Abstract and Applied Analysis
3
for each 𝑥, 𝑦 ∈ 𝑈. It is also said to be uniformly smooth if
the limit is attainted uniformly for each 𝑥, 𝑦 ∈ 𝑈. It is well
known that if 𝐸 is smooth, then the duality mapping 𝐽 is single
valued. It is also known that if 𝐸 is uniformly smooth, then
𝐽 is uniformly norm-to-norm continuous on each bounded
subset of 𝐸. Some properties of the duality mapping have been
given in [22]. A Banach space 𝐸 is said to have Kadec-Klee
property if a sequence {𝑥𝑛 } of 𝐸 satisfying that 𝑥𝑛 ⇀ 𝑥 ∈ 𝐸
and ‖𝑥𝑛 ‖ → ‖𝑥‖, then 𝑥𝑛 → 𝑥. It is known that if 𝐸 is
uniformly convex, then 𝐸 has the Kadec-Klee property; see
[22] for more details.
Let 𝐸 be a smooth Banach space. The function Φ : 𝐸 ×
𝐸 → 𝑅 is defined by
󵄩 󵄩2
𝜙 (𝑦, 𝑥) = 󵄩󵄩󵄩𝑦󵄩󵄩󵄩 − 2 ⟨𝑦, 𝐽𝑥⟩ + ‖𝑥‖2 ,
(11)
for all 𝑥, 𝑦 ∈ 𝐸. It is obvious from the definition of the
function 𝜙 that
(1) (‖𝑥‖ − ‖𝑦‖)2 ≤ 𝜙(𝑦, 𝑥) ≤ (‖𝑦‖2 + ‖𝑥‖2 ),
(2) 𝜙(𝑥, 𝑦) = 𝜙(𝑥, 𝑧) + 𝜙(𝑧, 𝑦) + 2⟨𝑥 − 𝑧, 𝐽𝑧 − 𝐽𝑥⟩,
(3) 𝜙(𝑥, 𝑦) = ⟨𝑥, 𝐽𝑥 − 𝐽𝑦⟩ + ⟨𝑦 − 𝑥, 𝐽𝑦⟩ ≤ ‖𝑥‖‖𝐽𝑥 − 𝐽𝑦‖ +
‖𝑦 − 𝑥‖‖𝑦‖,
for all 𝑥, 𝑦 ∈ 𝐸; see [4, 7, 23] for more details.
Lemma 1 (see [4]). If 𝐸 is a strictly convex and smooth Banach
space, then for 𝑥, 𝑦 ∈ 𝐸, 𝜙(𝑥, 𝑦) = 0 if and only if 𝑥 = 𝑦.
Lemma 2 (see [23]). Let 𝐸 be a uniformly convex and smooth
Banach space and let {𝑦𝑛 }, {𝑧𝑛 } be two sequences of 𝐸. If
𝜙(𝑦𝑛 , 𝑧𝑛 ) → 0 and either {𝑦𝑛 } or {𝑧𝑛 } is bounded, then
𝑦𝑛 − 𝑧𝑛 → 0.
Let 𝐶 be a closed convex subset of 𝐸. Suppose that
𝐸 is reflexive, strictly convex, and smooth. Then, for any
𝑥 ∈ 𝐸, there exists a point 𝑥0 ∈ 𝐶 such that 𝜙(𝑥0 , 𝑥) =
min𝑦∈𝐶𝜙(𝑦, 𝑥). The mapping Π𝐶 : 𝐸 → 𝐶 defined by
Π𝐶𝑥 = 𝑥0 is called the generalized projection (see [4, 7, 23]).
Lemma 3 (see [7]). Let 𝐶 be a closed convex subset of a smooth
Banach space 𝐸 and 𝑥 ∈ 𝐸. Then, 𝑥0 = Π𝐶𝑥 if and only if
⟨𝑥0 − 𝑦, 𝐽𝑥 − 𝐽𝑥0 ⟩ ≥ 0,
∀𝑦 ∈ 𝐶.
(12)
Lemma 4 (see [7]). Let 𝐸 be a reflexive, strictly convex, and
smooth Banach space and let 𝐶 be a closed convex subset of 𝐸
and 𝑥 ∈ 𝐸. Then, 𝜙(𝑦, Π𝐶𝑥) + 𝜙(Π𝐶𝑥, 𝑥) ≤ 𝜙(𝑦, 𝑥) for all y ∈
C.
Lemma 5 (see [24]). Let 𝐸 be a uniformly convex Banach
space and 𝐵𝑟 (0) = {𝑥 ∈ 𝐸 : ‖𝑥‖ ≤ 𝑟} a closed ball of 𝐸. Then,
there exists a continuous strictly increasing convex function
𝑔 : [0, ∞) → [0, ∞) with 𝑔(0) = 0 such that
󵄩
󵄩2
󵄩 󵄩2
󵄩
󵄩󵄩
2
2
󵄩󵄩𝜆𝑥 + 𝜇𝑦 + ]𝑧󵄩󵄩󵄩 ≤ 𝜆‖𝑥‖ + 𝜇󵄩󵄩󵄩𝑦󵄩󵄩󵄩 + ]‖𝑧‖ − 𝜆𝜇𝑔 (󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩) ,
(13)
for all 𝑥, 𝑦, 𝑧 ∈ 𝐵𝑟 (0) and 𝜆, 𝜇, ] ∈ [0, 1] with 𝜆 + 𝜇 + ] = 1.
Lemma 6 (see [19]). Let 𝐸 be a uniformly convex and
uniformly smooth Banach space and let 𝐶 be a closed convex
subset of 𝐸. Then, for points 𝑤, 𝑥, 𝑦, 𝑧 ∈ 𝐸 and a real number
𝑎 ∈ 𝑅, the set 𝐾 := {V ∈ 𝐶 : 𝜙(V, 𝑦) ≤ 𝜙(V, 𝑥)+⟨V, 𝐽𝑧−𝐽𝑤⟩+𝑎}
is closed and convex.
3. Main Results
In this section, we will prove the strong convergence theorem
for a common fixed point of a finite family of relatively
nonexpansive mappings in a Banach space by using the
hybrid method in mathematical programming. Let us prove
a proposition first.
Proposition 7. Let 𝐸 be a uniformly convex Banach space and
𝐵𝑟 (0) = {𝑥 ∈ 𝐸 : ‖𝑥‖ ≤ 𝑟} a closed ball of 𝐸. Then, there exists
a continuous strictly increasing convex function 𝑔 : [0, ∞) →
[0, ∞) with 𝑔(0) = 0 such that
󵄩󵄩2
󵄩󵄩 𝑛
𝑛
𝑛 𝑛
󵄩
󵄩󵄩
󵄩󵄩∑𝜆 𝑖 𝑥𝑖 󵄩󵄩󵄩 ≤ ∑𝜆 𝑖 󵄩󵄩󵄩𝑥𝑖 󵄩󵄩󵄩2 − 1 𝑔 (∑∑ 󵄩󵄩󵄩󵄩𝑥𝑖 − 𝑥𝑗 󵄩󵄩󵄩󵄩) ,
󵄩󵄩
󵄩󵄩
󵄩 󵄩
󵄩
󵄩
𝑛2
󵄩󵄩
󵄩󵄩𝑖=1
𝑖=1
𝑖=1 𝑗=1
(14)
for all n ≥ 3, 𝑥𝑖 ∈ 𝐵𝑟 (0) and 𝜆 𝑖 ∈ [0, 1] with ∑𝑛𝑖=1 𝜆 𝑖 = 1, 𝑖 =
1, 2, . . . , 𝑛.
Proof. If 𝜆 3 +𝜆 4 ⋅ ⋅ ⋅ +𝜆 𝑛 ≠ 0, using Lemma 5 and the convexity
of ‖ ⋅ ‖2 , we have
󵄩󵄩 𝑛
󵄩󵄩2
󵄩󵄩
󵄩
󵄩󵄩∑𝜆 𝑖 𝑥𝑖 󵄩󵄩󵄩
󵄩󵄩
󵄩󵄩
󵄩󵄩𝑖=1
󵄩󵄩
󵄩󵄩
= 󵄩󵄩𝜆 1 𝑥1 + 𝜆 2 𝑥2 + (𝜆 3 + ⋅ ⋅ ⋅ + 𝜆 𝑛 )
×(
󵄩󵄩2
𝜆 𝑛 𝑥𝑛
𝜆 3 𝑥3
󵄩
+ ⋅⋅⋅ +
)󵄩󵄩󵄩
𝜆3 + ⋅ ⋅ ⋅ + 𝜆𝑛
𝜆 3 + ⋅ ⋅ ⋅ + 𝜆 𝑛 󵄩󵄩
󵄩 󵄩2
󵄩 󵄩2
≤ 𝜆 1 󵄩󵄩󵄩𝑥1 󵄩󵄩󵄩 + 𝜆 2 󵄩󵄩󵄩𝑥2 󵄩󵄩󵄩 + (𝜆 3 + ⋅ ⋅ ⋅ + 𝜆 𝑛 )
(15)
󵄩󵄩2
󵄩󵄩
𝜆 3 𝑥3
𝜆 𝑛 𝑥𝑛
󵄩󵄩
󵄩
+ ⋅⋅⋅ +
× 󵄩󵄩󵄩
󵄩
󵄩󵄩 𝜆 3 + ⋅ ⋅ ⋅ + 𝜆 𝑛
𝜆 3 + ⋅ ⋅ ⋅ + 𝜆 𝑛 󵄩󵄩󵄩
󵄩
󵄩
− 𝜆 1 𝜆 2 𝑔 (󵄩󵄩󵄩𝑥1 − 𝑥2 󵄩󵄩󵄩)
𝑛
󵄩
󵄩 󵄩2
󵄩
≤ ∑𝜆 𝑖 󵄩󵄩󵄩𝑥𝑖 󵄩󵄩󵄩 − 𝜆 1 𝜆 2 𝑔 (󵄩󵄩󵄩𝑥1 − 𝑥2 󵄩󵄩󵄩) .
𝑖=1
If 𝜆 3 + 𝜆 4 ⋅ ⋅ ⋅ + 𝜆 𝑛 = 0, the last inequality above also holds
obviously. By the same argument in the proof above, we
obtain
󵄩󵄩2
󵄩󵄩 𝑛
𝑛
󵄩
󵄩󵄩
󵄩󵄩∑𝜆 𝑖 𝑥𝑖 󵄩󵄩󵄩 ≤ ∑𝜆 𝑖 󵄩󵄩󵄩𝑥𝑖 󵄩󵄩󵄩2 − 𝜆 𝑖 𝜆 𝑗 𝑔 (󵄩󵄩󵄩󵄩𝑥𝑖 − 𝑥𝑗 󵄩󵄩󵄩󵄩) ,
󵄩 󵄩
󵄩
󵄩
󵄩󵄩󵄩
󵄩󵄩󵄩𝑖=1
𝑖=1
󵄩
󵄩
(16)
4
Abstract and Applied Analysis
Let 𝑢 ∈ 𝐹 and let 𝑛 ≥ 0. Then, by the convexity of ‖ ⋅ ‖2 ,
we have
for all 𝑖, 𝑗 ∈ {1, 2, . . . , 𝑛}. Then,
󵄩󵄩 𝑛
󵄩󵄩2
𝑛
𝑛 𝑛
󵄩󵄩
󵄩󵄩
󵄩
󵄩
󵄩 󵄩2
𝑛2 󵄩󵄩󵄩∑𝜆 𝑖 𝑥𝑖 󵄩󵄩󵄩 ≤ 𝑛2 ∑𝜆 𝑖 󵄩󵄩󵄩𝑥𝑖 󵄩󵄩󵄩 − ∑ ∑𝜆 𝑖 𝜆 𝑗 𝑔 (󵄩󵄩󵄩󵄩𝑥𝑖 − 𝑥𝑗 󵄩󵄩󵄩󵄩)
󵄩󵄩𝑖=1
󵄩󵄩
𝑖=1
𝑖=1 𝑗=1
󵄩
󵄩
𝜙 (𝑢, 𝑧𝑛 )
𝑁
𝑛
𝑛 𝑛
󵄩
󵄩
󵄩 󵄩2
≤ 𝑛2 ∑𝜆 𝑖 󵄩󵄩󵄩𝑥𝑖 󵄩󵄩󵄩 − 𝑔 (∑∑𝜆 𝑖 𝜆 𝑗 󵄩󵄩󵄩󵄩𝑥𝑖 − 𝑥𝑗 󵄩󵄩󵄩󵄩) .
𝑖=1
𝑖=1 𝑗=1
(17)
(𝑖)
= 𝜙 (𝑢, 𝐽−1 (𝜆(0)
𝑛 𝐽𝑥𝑛 + ∑𝜆 𝑛 𝐽𝑇𝑖 𝑥𝑛 ))
𝑖=1
𝑁
(𝑖)
= ‖𝑢‖2 − 2 ⟨𝑢, 𝜆(0)
𝑛 𝐽𝑥𝑛 + ∑𝜆 𝑛 𝐽𝑇𝑖 𝑥𝑛 ⟩
𝑖=1
󵄩󵄩
󵄩󵄩2
𝑁
󵄩󵄩
󵄩󵄩
(𝑖)
󵄩
+ 󵄩󵄩󵄩𝜆(0)
𝐽𝑥
+
𝜆
𝐽𝑇
𝑥
∑
𝑛
𝑖 𝑛󵄩
𝑛
𝑛
󵄩󵄩
󵄩󵄩
󵄩󵄩
𝑖=1
󵄩
So,
󵄩󵄩 𝑛
󵄩󵄩2
𝑛
𝑛 𝑛
󵄩󵄩
󵄩
󵄩󵄩∑𝜆 𝑖 𝑥𝑖 󵄩󵄩󵄩 ≤ ∑𝜆 𝑖 󵄩󵄩󵄩𝑥𝑖 󵄩󵄩󵄩2 − 1 𝑔 (∑ ∑𝜆 𝑖 𝜆 𝑗 󵄩󵄩󵄩󵄩𝑥𝑖 − 𝑥𝑗 󵄩󵄩󵄩󵄩) . (18)
󵄩 󵄩
󵄩
󵄩
󵄩󵄩󵄩𝑖=1
󵄩󵄩󵄩
𝑛2
𝑖=1
𝑖=1 𝑗=1
󵄩
󵄩
𝑁
(𝑖)
≤ ‖𝑢‖2 − 2𝜆(0)
𝑛 ⟨𝑢, 𝐽𝑥𝑛 ⟩ − 2∑𝜆 𝑛 ⟨𝑢, 𝐽𝑇𝑖 𝑥𝑛 ⟩
(19)
𝑖=1
𝑁
󵄩󵄩 󵄩󵄩2
󵄩󵄩2
(𝑖) 󵄩
󵄩
+ 𝜆(0)
𝑛 󵄩
󵄩𝐽𝑥𝑛 󵄩󵄩 + ∑𝜆 𝑛 󵄩󵄩𝐽𝑇𝑖 𝑥𝑛 󵄩󵄩
𝑖=1
Theorem 8. Let 𝐸 be a uniformly convex and uniformly
smooth Banach space, and let 𝐶 be a nonempty closed convex
subset of 𝐸 and 𝑇1 , 𝑇2 , . . . , 𝑇𝑁 : 𝐶 → 𝐶 relatively
nonexpansive mappings such that 𝐹 := ∩𝑁
𝑖=1 𝐹(𝑇𝑖 ) ≠ 𝜙. The
sequence {𝑥𝑛 } is given by (9) with the following restrictions:
𝑁
(𝑖)
= 𝜆(0)
𝑛 𝜙 (𝑢, 𝑥𝑛 ) + ∑𝜆 𝑛 𝜙 (𝑢, 𝑇𝑖 𝑥𝑛 )
𝑖=1
𝑁
(𝑖)
≤ 𝜆(0)
𝑛 𝜙 (𝑢, 𝑥𝑛 ) + ∑𝜆 𝑛 𝜙 (𝑢, 𝑥𝑛 )
𝑖=1
(a) 𝛼𝑛 + 𝛽𝑛 + 𝛾𝑛 = 1, 0 ≤ 𝛼𝑛 < 1, 0 ≤ 𝛽𝑛 < 1, 0 < 𝛾𝑛 ≤ 1
for all 𝑛 ≥ 0;
= 𝜙 (𝑢, 𝑥𝑛 ) ,
and then,
(b) lim 𝑛 → ∞ 𝛼𝑛 = 0 and lim sup 𝑛 → ∞ 𝛽𝑛 < 1;
(c) 𝜆𝑖𝑛 ∈ [0, 1] with
𝑛 ≥ 0;
0
𝑛 → ∞𝜆𝑛
∑𝑁
𝑖=0
𝜙 (𝑢, 𝑦𝑛 )
𝜆𝑖𝑛 = 1, 𝑖 = 0, 1, 2, . . . , 𝑁, for all
= 0 and lim inf
(d) lim
1, 2, . . . , 𝑁; or
𝑖 𝑗
𝑛 → ∞𝜆𝑛𝜆𝑛
> 0, 𝑖, 𝑗 =
(d󸀠 ) lim inf 𝑛 → ∞ 𝜆0𝑛 𝜆𝑖𝑛 > 0, 𝑖 = 1, 2, . . . , 𝑁.
= 𝜙 (𝑢, 𝐽−1 (𝛼𝑛 𝐽𝑥 + 𝛽𝑛 𝐽𝑥𝑛 + 𝛾𝑛 𝐽𝑧𝑛 ))
= ‖𝑢‖2 − 2 ⟨𝑢, 𝛼𝑛 𝐽𝑥 + 𝛽𝑛 𝐽𝑥𝑛 + 𝛾𝑛 𝐽𝑧𝑛 ⟩
󵄩2
󵄩
+ 󵄩󵄩󵄩𝛼𝑛 𝐽𝑥 + 𝛽𝑛 𝐽𝑥𝑛 + 𝛾𝑛 𝐽𝑧𝑛 󵄩󵄩󵄩
≤ ‖𝑢‖2 − 2𝛼𝑛 ⟨𝑢, 𝐽𝑥⟩ − 2𝛽𝑛 ⟨𝑢, 𝐽𝑥𝑛 ⟩ − 2𝛾𝑛 ⟨𝑢, 𝐽𝑧𝑛 ⟩
󵄩 󵄩2
󵄩 󵄩2
+ 𝛼𝑛 ‖𝑥‖2 + 𝛽𝑛 󵄩󵄩󵄩𝑥𝑛 󵄩󵄩󵄩 + 𝛾𝑛 󵄩󵄩󵄩𝑧𝑛 󵄩󵄩󵄩
Then, the sequence {𝑥𝑛 } converges strongly to Π𝐹 𝑥, where Π𝐹 is
the generalized projection from 𝐶 onto 𝐹.
= 𝛼𝑛 𝜙 (𝑢, 𝑥) + 𝛽𝑛 𝜙 (𝑢, 𝑥𝑛 ) + 𝛾𝑛 𝜙 (𝑢, 𝑧𝑛 )
Proof. We split the proof into seven steps.
≤ 𝛼𝑛 𝜙 (𝑢, 𝑥) + (1 − 𝛼𝑛 ) 𝜙 (𝑢, 𝑥𝑛 )
Step 1. Show that 𝑃𝐹 is well defined for every 𝑥 ∈ 𝐶.
It is easy to know that 𝐹(𝑇𝑖 ), 𝑖 = 1, 2, . . . , 𝑁 are closed
convex sets and so is 𝐹. What is more, 𝐹 is nonempty by our
assumption. Therefore, 𝑃𝐹 is well defined for every 𝑥 ∈ 𝐶.
Step 2. Show that 𝐻𝑛 and 𝑊𝑛 are closed and convex for all
𝑛 ≥ 0.
From the definition of 𝑊𝑛 , it is obvious 𝑊𝑛 is closed and
convex for each 𝑛 ≥ 0. By Lemma 6, we also know that 𝐻𝑛 is
closed and convex for each 𝑛 ≥ 0.
Step 3. Show that 𝐹 ⊂ 𝐻𝑛 ⋂ 𝑊𝑛 for all 𝑛 ≥ 0.
(20)
= 𝜙 (𝑢, 𝑥𝑛 ) + 𝛼𝑛 (𝜙 (𝑢, 𝑥) − 𝜙 (𝑢, 𝑥𝑛 ))
≤ 𝜙 (𝑢, 𝑥𝑛 ) + 𝛼𝑛 (‖𝑥‖2 + 2 ⟨𝐽𝑥𝑛 − 𝐽𝑥, 𝑧⟩) .
Thus, we have 𝑢 ∈ 𝐻𝑛 . Therefore, we obtain 𝐹 ⊂ 𝐻𝑛 for all
𝑛 ≥ 0.
Next, we prove 𝐹 ⊂ 𝑊𝑛 for all 𝑛 ≥ 0. We prove this by
induction. For 𝑛 = 0, we have 𝐹 ⊂ 𝐶 = 𝑊0 . Assume that
𝐹 ⊂ 𝑊𝑛 . Since 𝑥𝑛+1 is the projection of 𝑥 onto 𝐻𝑛 ⋂ 𝑊𝑛 , by
Lemma 3, we have
⟨𝑥𝑛+1 − 𝑧, 𝐽𝑥 − 𝐽𝑥𝑛+1 ⟩ ≥ 0,
(21)
Abstract and Applied Analysis
5
for any 𝑧 ∈ 𝐻𝑛 ⋂ 𝑊𝑛 . As 𝐹 ⊂ 𝐻𝑛 ⋂ 𝑊𝑛 by the induction
assumption, 𝐹 ⊂ 𝑊𝑛 holds, in particular, for all 𝑢 ∈ 𝐹. This
together with the definition of 𝑊𝑛+1 implies that 𝐹 ⊂ 𝑊𝑛+1 .
Hence, 𝐹 ⊂ 𝐻𝑛 ⋂ 𝑊𝑛 for all 𝑛 ≥ 0.
Step 4. Show that ‖𝑥𝑛+1 − 𝑥𝑛 ‖ → 0 as 𝑛 → ∞.
In view of (19) and Lemma 4, we have 𝑥𝑛 = 𝑃𝑊𝑛 𝑥, which
means that, for any 𝑧 ∈ 𝑊𝑛 ,
𝜙 (𝑥𝑛 , 𝑥) ≤ 𝜙 (𝑧, 𝑥) .
(22)
Since 𝑥𝑛+1 ∈ 𝑊𝑛 and 𝑢 ∈ 𝐹 ⊂ 𝑊𝑛 , we obtain
𝜙 (𝑥𝑛 , 𝑥) ≤ 𝜙 (𝑥𝑛+1 , 𝑥) ,
𝜙 (𝑥𝑛 , 𝑥) ≤ 𝜙 (𝑢, 𝑥) ,
𝑁
(23)
(24)
+ 𝛼𝑛 (‖𝑥‖ + 2 ⟨𝐽𝑥𝑛 − 𝐽𝑥, 𝑥𝑛+1 ⟩) 󳨀→ 0,
󵄩󵄩
󵄩󵄩2
𝑁
󵄩󵄩 (0)
󵄩󵄩
(𝑖)
󵄩
+ 󵄩󵄩𝜆 𝑛 𝐽𝑥𝑛 + ∑𝜆 𝑛 𝐽𝑇𝑖 𝑥𝑛 󵄩󵄩󵄩
󵄩󵄩
󵄩󵄩
𝑖=1
󵄩
󵄩
𝑁
󵄩 󵄩2
(𝑖)
≤ 󵄩󵄩󵄩𝑝󵄩󵄩󵄩 − 2𝜆(0)
𝑛 ⟨𝑝, 𝐽𝑥𝑛 ⟩ − 2∑𝜆 𝑛 ⟨𝑝, 𝐽𝑇𝑖 𝑥𝑛 ⟩
𝑖=1
𝑁
Step 5. Show that ‖𝑥𝑛 − 𝑧𝑛 ‖ → 0 as 𝑛 → ∞.
From 𝑥𝑛+1 = 𝑃𝐻𝑛 ⋂ 𝑊𝑛 𝑥 ∈ 𝐻𝑛 , we have
2
󵄩 󵄩2
(𝑖)
= 󵄩󵄩󵄩𝑝󵄩󵄩󵄩 − 2 ⟨𝑝, 𝜆(0)
𝑛 𝐽𝑥𝑛 + ∑𝜆 𝑛 𝐽𝑇𝑖 𝑥𝑛 ⟩
𝑖=1
as 𝑛 → ∞. By using Lemma 2, we obtain ‖𝑥𝑛+1 − 𝑥𝑛 ‖ → 0
as 𝑛 → ∞.
𝜙 (𝑥𝑛+1 , 𝑦𝑛 ) ≤ 𝜙 (𝑥𝑛+1 , 𝑥𝑛 )
Step 6. Show that ‖𝑥𝑛 − 𝑇𝑖 𝑥𝑛 ‖ → 0, 𝑖 = 1, 2, . . . , 𝑁.
Since {𝑥𝑛 } is bounded and 𝜙(𝑝, 𝑇𝑖 𝑥𝑛 ) ≤ 𝜙(𝑝, 𝑥𝑛 ),
where 𝑝 ∈ 𝐹, 𝑖 = 1, 2, . . . , 𝑁, we also obtain that
{𝐽𝑥𝑛 }, {𝐽𝑇1 𝑥𝑛 }, . . . , {𝐽𝑇𝑁𝑥𝑛 } are bounded, and hence, there
exists 𝑟 > 0 such that {𝐽𝑥𝑛 }, {𝐽𝑇1 𝑥𝑛 }, . . . , {𝐽𝑇𝑁𝑥𝑛 } ⊂ 𝐵𝑟 (0).
Therefore, Proposition 7 can be applied and we observe that
𝜙 (𝑝, 𝑧𝑛 )
for all 𝑛 ≥ 0. Consequently, lim𝑛 → ∞ 𝜙(𝑥𝑛 , 𝑥) exists and {𝑥𝑛 }
is bounded. By using Lemma 4, we have
𝜙 (𝑥𝑛+1 , 𝑥𝑛 ) ≤ 𝜙 (𝑥𝑛+1 , 𝑥) − 𝜙 (𝑥𝑛 , 𝑥) 󳨀→ 0,
as 𝑛 → ∞. Using Lemma 2, we obtain ‖𝑥𝑛 − 𝑧𝑛 ‖ → 0 as
𝑛 → ∞.
󵄩󵄩 󵄩󵄩2
󵄩󵄩2
(𝑖) 󵄩
󵄩
+ 𝜆(0)
𝑛 󵄩
󵄩𝑥𝑛 󵄩󵄩 + ∑𝜆 𝑛 󵄩󵄩𝑇𝑖 𝑥𝑛 󵄩󵄩
𝑖=1
(25)
−
as 𝑛 → ∞. By Lemma 2, we also have ‖𝑥𝑛+1 − 𝑦𝑛 ‖ → 0, and
then,
󵄩 󵄩
󵄩 󵄩
󵄩
󵄩󵄩
(26)
󵄩󵄩𝑥𝑛 − 𝑦𝑛 󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩𝑦𝑛 − 𝑥𝑛+1 󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑥𝑛 − 𝑥𝑛+1 󵄩󵄩󵄩 󳨀→ 0,
as 𝑛 → ∞. We observe that
𝑁 𝑁
1
(𝑗) 󵄩
󵄩󵄩𝐽𝑇 𝑥 − 𝐽𝑇 𝑥 󵄩󵄩󵄩
𝑔
(
∑
∑𝜆(𝑖)
𝑗 𝑛󵄩
𝑛 𝜆𝑛 󵄩
2
󵄩 𝑖 𝑛
󵄩
𝑁
𝑖=1 𝑗=1
𝑁
(𝑖) 󵄩
󵄩𝐽𝑥𝑛 − 𝐽𝑇𝑖 𝑥𝑛 󵄩󵄩󵄩)
+2∑𝜆(0)
𝑛 𝜆𝑛 󵄩
󵄩
󵄩
𝑖=1
𝑁
(𝑖)
= 𝜆(0)
𝑛 𝜙 (𝑝, 𝑥𝑛 ) + ∑𝜆 𝑛 𝜙 (𝑝, 𝑇𝑖 𝑥𝑛 )
𝜙 (𝑧𝑛 , 𝑥𝑛 )
𝑖=1
= 𝜙 (𝑧𝑛 , 𝑦𝑛 ) + 𝜙 (𝑦𝑛 , 𝑥𝑛 ) + 2 ⟨𝑧𝑛 − 𝑦𝑛 , 𝐽𝑦𝑛 − 𝐽𝑥𝑛 ⟩
−
󵄩󵄩
󵄩
󵄩
≤ 𝜙 (𝑧𝑛 , 𝑦𝑛 ) + 𝜙 (𝑦𝑛 , 𝑥𝑛 ) + 2 󵄩󵄩󵄩𝑧𝑛 − 𝑦𝑛 󵄩󵄩󵄩 󵄩󵄩󵄩𝐽𝑦𝑛 − 𝐽𝑥𝑛 󵄩󵄩󵄩 ,
𝜙 (𝑧𝑛 , 𝑦𝑛 )
(27)
󵄩 󵄩2
= 󵄩󵄩󵄩𝑧𝑛 󵄩󵄩󵄩 − 2 ⟨𝑧𝑛 , 𝛼𝑛 𝐽𝑥 + 𝛽𝑛 𝐽𝑥𝑛 + 𝛾𝑛 𝐽𝑧𝑛 ⟩
𝑁 𝑁
1
(𝑗) 󵄩
󵄩󵄩𝐽𝑇 𝑥 − 𝐽𝑇 𝑥 󵄩󵄩󵄩
𝑔
(
∑
∑𝜆(𝑖)
𝑗 𝑛󵄩
𝑛 𝜆𝑛 󵄩
2
󵄩 𝑖 𝑛
󵄩
𝑁
𝑖=1 𝑗=1
𝑁
(𝑖) 󵄩
󵄩𝐽𝑥𝑛 − 𝐽𝑇𝑖 𝑥𝑛 󵄩󵄩󵄩)
+2∑𝜆(0)
𝑛 𝜆𝑛 󵄩
󵄩
󵄩
𝑖=1
󵄩2
󵄩
+ 󵄩󵄩󵄩𝛼𝑛 𝐽𝑥 + 𝛽𝑛 𝐽𝑥𝑛 + 𝛾𝑛 𝐽𝑧𝑛 󵄩󵄩󵄩
≤ 𝜙 (𝑝, 𝑥𝑛 ) −
≤ 𝛼𝑛 𝜙 (𝑧𝑛 , 𝑥) + 𝛽𝑛 𝜙 (𝑧𝑛 , 𝑥𝑛 ) .
𝑁 𝑁
1
(𝑗) 󵄩
󵄩󵄩𝐽𝑇 𝑥 − 𝐽𝑇 𝑥 󵄩󵄩󵄩
𝑔 (∑∑ 𝜆(𝑖)
𝑗 𝑛󵄩
𝑛 𝜆𝑛 󵄩
2
󵄩 𝑖 𝑛
󵄩
𝑁
𝑖=1 𝑗=1
𝑁
(𝑖) 󵄩
󵄩𝐽𝑥𝑛 − 𝐽𝑇𝑖 𝑥𝑛 󵄩󵄩󵄩) ,
+2∑𝜆(0)
𝑛 𝜆𝑛 󵄩
󵄩
󵄩
So,
𝜙 (𝑧𝑛 , 𝑥𝑛 ) ≤ 𝛼𝑛 𝜙 (𝑧𝑛 , 𝑥) + 𝛽𝑛 𝜙 (𝑧𝑛 , 𝑥𝑛 )
󵄩󵄩
󵄩
󵄩
+ 𝜙 (𝑦𝑛 , 𝑥𝑛 ) + 2 󵄩󵄩󵄩𝑧𝑛 − 𝑦𝑛 󵄩󵄩󵄩 󵄩󵄩󵄩𝐽𝑦𝑛 − 𝐽𝑥𝑛 󵄩󵄩󵄩 .
𝑖=1
(28)
Since lim𝑛 → ∞ 𝛼𝑛 = 0, lim sup𝑛 → ∞ 𝛽𝑛 < 1, 𝜙(𝑦𝑛 , 𝑥𝑛 ) → 0,
and ‖𝑧𝑛 − 𝑦𝑛 ‖‖𝐽𝑦𝑛 − 𝐽𝑥𝑛 ‖ → 0 as 𝑛 → ∞, we have
𝜙 (𝑧𝑛 , 𝑥𝑛 ) ≤
(30)
𝛼𝑛
1
𝜙 (𝑧𝑛 , 𝑥) +
𝜙 (𝑦𝑛 , 𝑥𝑛 )
1 − 𝛽𝑛
1 − 𝛽𝑛
+
2 󵄩󵄩
󵄩󵄩
󵄩
󵄩𝑧 − 𝑦𝑛 󵄩󵄩󵄩 󵄩󵄩󵄩𝐽𝑦𝑛 − 𝐽𝑥𝑛 󵄩󵄩󵄩 󳨀→ 0,
1 − 𝛽𝑛 󵄩 𝑛
(29)
where 𝑔 : [0, ∞) → [0, ∞) is a continuous strictly
increasing convex function with 𝑔(0) = 0. And
𝜙 (𝑝, 𝑥𝑛 ) − 𝜙 (𝑝, 𝑧𝑛 )
󵄩 󵄩2
󵄩 󵄩2 󵄩 󵄩 2
󵄩 󵄩2
= 󵄩󵄩󵄩𝑝󵄩󵄩󵄩 − 2 ⟨𝑝, 𝐽𝑥𝑛 ⟩ + 󵄩󵄩󵄩𝑥𝑛 󵄩󵄩󵄩 − 󵄩󵄩󵄩𝑝󵄩󵄩󵄩 + 2 ⟨𝑝, 𝐽𝑧𝑛 ⟩ − 󵄩󵄩󵄩𝑧𝑛 󵄩󵄩󵄩
󵄩 󵄩 󵄩2 󵄩 󵄩2
󵄩 󵄩󵄩
≤ 2 󵄩󵄩󵄩𝑝󵄩󵄩󵄩 󵄩󵄩󵄩𝐽𝑧𝑛 − 𝐽𝑥𝑛 󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑥𝑛 󵄩󵄩󵄩 − 󵄩󵄩󵄩𝑧𝑛 󵄩󵄩󵄩 󳨀→ 0,
(31)
6
Abstract and Applied Analysis
as 𝑛 → ∞. From the properties of the mapping 𝑔, we have
lim 𝜆(0) 𝜆(𝑖)
𝑛
𝑛→∞ 𝑛
lim 𝜆(𝑖) 𝜆(𝑗)
𝑛→∞ 𝑛 𝑛
󵄩󵄩
󵄩
󵄩󵄩𝑥𝑛 − 𝑇𝑖 𝑥𝑛 󵄩󵄩󵄩 = 0,
󵄩󵄩
󵄩
󵄩󵄩𝑇𝑖 𝑥𝑛 − 𝑇𝑗 𝑥𝑛 󵄩󵄩󵄩 = 0,
󵄩
󵄩
(32)
󵄩󵄩
󵄩
− 𝑇𝑗 𝑥𝑛 󵄩󵄩󵄩󵄩 = 0,
𝜙 (̂
𝑥, 𝑥)
𝑥, 𝐽𝑥⟩ + ‖𝑥‖2
= ‖̂
𝑥‖2 − 2 ⟨̂
for all 𝑖, 𝑗 ∈ {1, 2, . . . , 𝑁}. From the condition (d󸀠 ), we have
‖𝑥𝑛 − 𝑇𝑖 𝑥𝑛 ‖ → 0 immediately, as 𝑛 → ∞, 𝑖 = 1, 2, . . . , 𝑁;
from the condition (d), we can also have ‖𝑥𝑛 − 𝑇𝑖 𝑥𝑛 ‖ → 0, as
(𝑗)
𝑛 → ∞, 𝑖 = 1, 2, . . . , 𝑁. In fact, since lim inf 𝑛 → ∞ 𝜆(𝑖)
𝑛 𝜆𝑛 >
0, it follows that
lim 󵄩󵄩𝑇𝑖 𝑥𝑛
𝑛→∞ 󵄩
On the other hand, from weakly lower semicontinuity of the
norm, we have
(33)
for all 𝑖, 𝑗 ∈ {1, 2, . . . , 𝑁}. Next, we note by the convexity of
‖ ⋅ ‖2 and (9) that
𝜙 (𝑇𝑗 𝑥𝑛 , 𝑧𝑛 )
𝑁
󵄩2
󵄩
(𝑖)
= 󵄩󵄩󵄩󵄩𝑇𝑗 𝑥𝑛 󵄩󵄩󵄩󵄩 − 2 ⟨𝑇𝑗 𝑥𝑛 , 𝜆(0)
𝑛 𝐽𝑥𝑛 + ∑𝜆 𝑛 𝐽𝑇𝑖 𝑥𝑛 ⟩
󵄩 󵄩2
󵄩 󵄩2
≤ lim inf (󵄩󵄩󵄩󵄩𝑥𝑛𝑘 󵄩󵄩󵄩󵄩 − 2 ⟨󵄩󵄩󵄩󵄩𝑥𝑛𝑘 󵄩󵄩󵄩󵄩 , 𝐽𝑥⟩ + ‖𝑥‖2 )
𝑛→∞
(37)
= lim inf 𝜙 (𝑥𝑛𝑘 , 𝑥) ≤ lim sup 𝜙 (𝑥𝑛𝑘 , 𝑥) ≤ 𝜙 (𝜔, 𝑥) .
𝑛→∞
𝑛→∞
̂ = 𝜔, and hence,
From the definition of Π𝐹 𝑥, we obtain 𝑥
lim𝑛 → ∞ 𝜙(𝑥𝑛𝑘 , 𝑥) = 𝜙(𝜔, 𝑥). So, we have lim𝑘 → ∞ ‖𝑥𝑛𝑘 ‖ =
‖𝜔‖. Using the Kadec-klee property of 𝐸, we obtain that
{𝑥𝑛𝑘 } converges strongly to Π𝐹 𝑥. Since {𝑥𝑛𝑘 } is an arbitrary
weakly convergent sequence of {𝑥𝑛 }, we can conclude that
{𝑥𝑛 } converges strongly to Π𝐹 𝑥.
Corollary 9. Let 𝐶 be a nonempty closed convex subset of
a Hilbert space 𝐻 and 𝑇1 , 𝑇2 , . . . , 𝑇𝑁 : 𝐶 → 𝐶 relatively
nonexpansive mappings such that 𝐹 := ∩𝑁
𝑖=1 𝐹(𝑇𝑖 ) ≠ 𝜙. The
sequence {𝑥𝑛 } is given by (9) with the following restrictions:
(a) 𝛼𝑛 + 𝛽𝑛 + 𝛾𝑛 = 1, 0 ≤ 𝛼𝑛 < 1, 0 ≤ 𝛽𝑛 < 1, 0 ≤ 𝛾𝑛 ≤ 1
for all n ≥ 0;
𝑖=1
󵄩󵄩
󵄩󵄩2
𝑁
󵄩󵄩 (0)
󵄩󵄩
(𝑖)
󵄩
+ 󵄩󵄩𝜆 𝑛 𝐽𝑥𝑛 + ∑𝜆 𝑛 𝐽𝑇𝑖 𝑥𝑛 󵄩󵄩󵄩
󵄩󵄩
󵄩󵄩
𝑖=1
󵄩
󵄩
(b) lim 𝑛 → ∞ 𝛼𝑛 = 0 and lim sup 𝑛 → ∞ 𝛽𝑛 < 1;
𝑁
󵄩2
󵄩
≤ 󵄩󵄩󵄩󵄩𝑇𝑗 𝑥𝑛 󵄩󵄩󵄩󵄩 − 2𝜆(0)
⟨𝑇
𝑥
,
𝐽𝑥
⟩
−
2
𝜆(𝑖)
∑
𝑗 𝑛
𝑛
𝑛
𝑛 ⟨𝑇𝑗 𝑥𝑛 , 𝐽𝑇𝑖 𝑥𝑛 ⟩
𝑖=1
𝑁
󵄩󵄩 󵄩󵄩2
󵄩󵄩2
(𝑖) 󵄩
󵄩
+ 𝜆(0)
𝑛 󵄩
󵄩𝑥𝑛 󵄩󵄩 + ∑𝜆 𝑛 󵄩󵄩𝑇𝑖 𝑥𝑛 󵄩󵄩
𝑁
Then, the sequence {𝑥𝑛 } converges strongly to 𝑃𝐹 𝑥, where 𝑃𝐹 is
the metric projection from 𝐶 onto 𝐹.
(𝑖)
= 𝜆(0)
𝑛 𝜙 (𝑇𝑗 𝑥𝑛 , 𝑥𝑛 ) + ∑𝜆 𝑛 𝜙 (𝑇𝑗 𝑥𝑛 , 𝑇𝑖 𝑥𝑛 ) 󳨀→ 0,
𝑖=1
(34)
as 𝑛 → ∞. By Lemma 2, we have lim𝑛 → ∞ ‖𝑇𝑖 𝑥𝑛 − 𝑧𝑛 ‖ = 0
and
(35)
as 𝑛 → ∞ for all 𝑖 ∈ {1, 2, . . . , 𝑁}.
Step 7. Show that 𝑥𝑛 → Π𝐹 𝑥, as 𝑛 → ∞.
From the result of Step 6, we know that if {𝑥𝑛𝑘 } is a
̂ ∈ 𝐶, then 𝑥
̂ ∈
subsequence of {𝑥𝑛 } such that {𝑥𝑛𝑘 } ⇀ 𝑥
̂ 𝑖 ) = ∩𝑁 𝐹(𝑇𝑖 ). Because 𝐸 is a uniformly convex and
𝐹(𝑇
∩𝑁
𝑖=1
𝑖=1
uniformly smooth Banach space and {𝑥𝑛 } is bounded, so we
can assume {𝑥𝑛𝑘 } is a subsequence of {𝑥𝑛 } such that {𝑥𝑛𝑘 } ⇀
̂ ∈ 𝐹 and 𝜔 = Π𝐹 𝑥. For any 𝑛 ≥ 1, from 𝑥𝑛+1 = Π𝐻𝑛 ⋂ 𝑊𝑛 𝑥
𝑥
and 𝜔 ∈ 𝐹 ⊂ 𝐻𝑛 ⋂ 𝑊𝑛 , we have
𝜙 (𝑥𝑛+1 , 𝑥) ≤ 𝜙 (𝜔, 𝑥) .
(d) lim 𝑛 → ∞ 𝜆0𝑛 = 0 and lim inf 𝑛 → ∞ 𝜆𝑖𝑛 𝜆𝑗𝑛 > 0, 𝑖, 𝑗 =
1, 2, . . . , 𝑁; or
(d󸀠 ) lim inf 𝑛 → ∞ 𝜆0𝑛 𝜆𝑖𝑛 > 0, 𝑖 = 1, 2, . . . , 𝑁.
𝑖=1
󵄩 󵄩
󵄩 󵄩
󵄩
󵄩󵄩
󵄩󵄩𝑇𝑖 𝑥𝑛 − 𝑥𝑛 󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩𝑇𝑖 𝑥𝑛 − 𝑧𝑛 󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑥𝑛 − 𝑧𝑛 󵄩󵄩󵄩 󳨀→ 0,
𝑖
(c) 𝜆𝑖𝑛 ∈ [0, 1] with ∑𝑁
𝑖=0 𝜆 𝑛 = 1, 𝑖 = 0, 1, 2, . . . , 𝑁, for all
𝑛 ≥ 0;
(36)
Proof. It is true because the generalized projection Π𝐹 is just
the metric projection 𝑃𝐹 in Hilbert spaces.
Remark 10. The results of Nakajo and Takahashi [18] and
Song et al. [11] are the special cases of our results in
Corollary 9. And in our results of Theorem 8, if 𝑇1 = 𝑇2 =
⋅ ⋅ ⋅ = 𝑇𝑁, 𝜆(0)
𝑛 and 𝛼𝑛 = 0 for all 𝑛 ≥ 0, then, we obtain
Theorem 4.1 of Matsushita and Takahashi [10]; if 𝑇1 = 𝑇2 =
⋅ ⋅ ⋅ = 𝑇𝑁−1 and 𝛼𝑛 = 0 for all 𝑛 ≥ 0, then, we obtain Theorem
3.1 of Plubtieng and Ungchittrakool [19]; if 𝑇1 = 𝑇2 = ⋅ ⋅ ⋅ =
𝑇𝑁−1 and 𝛽𝑛 = 0 for all 𝑛 ≥ 0, then, we obtain Theorem 3.2
of Plubtieng and Ungchittrakool [19]. So, our results improve
and extend the corresponding results by many others.
Acknowledgments
This work was partially supported by the Natural Science Foundation of Zhejiang Province (Y6100696) and the
National Natural Science Foundation of China (11071169,
11271330).
Abstract and Applied Analysis
7
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