Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2013, Article ID 264910, 9 pages
http://dx.doi.org/10.1155/2013/264910
Research Article
Implicit Iterative Scheme for a Countable Family of
Nonexpansive Mappings in 2-Uniformly Smooth Banach Spaces
Ming Tian and Xin Jin
College of Science, Civil Aviation University of China, Tianjin 300300, China
Correspondence should be addressed to Ming Tian; tianming1963@126.com
Received 1 January 2013; Accepted 22 March 2013
Academic Editor: Qamrul Hasan Ansari
Copyright © 2013 M. Tian and X. Jin. 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.
Implicit Mann process and Halpern-type iteration have been extensively studied by many others. In this paper, in order to find
a common fixed point of a countable family of nonexpansive mappings in the framework of Banach spaces, we propose a new
implicit iterative algorithm related to a strongly accretive and Lipschitzian continuous operator 𝐹 : 𝑥𝑛 = 𝛼𝑛 𝛾𝑉(𝑥𝑛 ) + 𝛽𝑛 𝑥𝑛−1 + ((1 −
𝛽𝑛 )𝐼 − 𝛼𝑛 𝜇𝐹)𝑇𝑛 𝑥𝑛 and get strong convergence under some mild assumptions. Our results improve and extend the corresponding
conclusions announced by many others.
1. Introduction
Let 𝑋 be a real 𝑞-uniformly smooth Banach space with
induced norm ‖ ⋅ ‖, 𝑞 > 1. Let 𝑋∗ be the dual space of 𝑋.
Let 𝐽𝑞 denote the generalized duality mapping from 𝑋 into
∗
2𝑋 given by 𝐽𝑞 (𝑥) = {𝑓 ∈ 𝑋∗ : ⟨𝑥, 𝑓⟩ = ‖𝑥‖𝑞 , ‖𝑓‖ = ‖𝑥‖𝑞−1 ,
𝑥 ∈ 𝑋}. In our paper, we consider real 2-uniformly smooth
Banach spaces, that is, 𝑞 = 2, so the normalized duality
mapping is 𝐽(𝑥) = {𝑓 ∈ 𝑋∗ : ⟨𝑥, 𝑓⟩ = ‖𝑥‖2 , ‖𝑓‖ = ‖𝑥‖, 𝑥 ∈
𝑋}. If 𝑋 is smooth, then 𝐽 is single valued. Throughout this
paper, we use Fix(𝑇) to denote the fixed points set of the
mapping 𝑇.
In what follows, we write 𝑥𝑛 ⇀ 𝑥 to indicate that the
sequence converges weakly to 𝑥. 𝑥𝑛 → 𝑥 implies that the
sequence converges strongly to 𝑥.
Given a nonlinear operator Γ : 𝑋 → 𝑋, it is well-known
that the generalized variational inequality problem VIP(Γ, 𝑋)
over 𝑋 is to find a 𝑥∗ ∈ 𝑋, such that
⟨Γ𝑥∗ , 𝑗 (𝑥 − 𝑥∗ )⟩ ≥ 0,
∀𝑥 ∈ 𝑋.
(1)
Scholars mainly proposed iterative algorithms to solve the
generalized variational inequalities, and some of them focused on the existence of the solutions of generalized variational inequalities; see [1, 2] and references therein.
Variational inequalities are developed from operator
equations and have been playing an essential role in management science, mechanics, and finance. As for mathematics,
variational inequality problems mainly originate from partial
differential equations, optimization problems; see [3–7] and
references therein.
Definition 1. A mapping 𝑇 is said to be
(1) 𝜂-strongly accretive if for each 𝑥, 𝑦 ∈ 𝑋, there exists a
𝑗(𝑥 − 𝑦) ∈ 𝐽(𝑥 − 𝑦) and 𝜂 > 0, such that
󵄩𝑞
󵄩
⟨𝑇𝑥 − 𝑇𝑦, 𝑗 (𝑥 − 𝑦)⟩ ≥ 𝜂󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩 ;
(2)
(2) 𝐿-Lipschitzian continuous if for each 𝑥, 𝑦 ∈ 𝑋, there
exists a constant 𝐿 > 0, such that
󵄩
󵄩
󵄩
󵄩󵄩
󵄩󵄩𝑇𝑥 − 𝑇𝑦󵄩󵄩󵄩 ≤ 𝐿 󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩 .
(3)
In particular, 𝑇 is called nonexpansive if 𝐿 = 1; it is said
to be contractive if 𝐿 < 1.
Yamada [3] introduced the hybrid steepest descent method:
𝑥𝑛+1 = (𝐼 − 𝜇𝜆 𝑛 𝐹) 𝑇𝑥𝑛 ,
∀𝑛 ≥ 0,
(4)
2
Abstract and Applied Analysis
where 𝑇 is a nonexpansive mapping in Hilbert spaces. Under
some appropriate conditions, Yamada [3] proved that the
sequence {𝑥𝑛 } generated by (4) converges strongly to the
unique solution 𝑥∗ ∈ Fix(𝑇) of the variational inequality:
VIP(𝐹, 𝑋) : ⟨𝐹𝑥∗ , 𝑥 − 𝑥∗ ⟩ ≥ 0, for all 𝑥 ∈ Fix(𝑇).
Moudafi [4] introduced the classical viscosity approximation method for nonexpansive mappings and defined a
sequence {𝑥𝑛 } by
𝑥𝑛+1 = 𝛼𝑛 𝑓 (𝑥𝑛 ) + (1 − 𝛼𝑛 ) 𝑇𝑥𝑛 ,
∀𝑛 ≥ 0,
(5)
where {𝛼𝑛 } is a sequence in (0, 1). Xu [6] proved that under
certain appropriate conditions on {𝛼𝑛 }, the sequence {𝑥𝑛 }
generated by (5) converges strongly to the unique solution
𝑥∗ ∈ 𝐶 of the variational inequality: ⟨(𝐼 − 𝑓)𝑥∗ , 𝑥 − 𝑥∗ ⟩ ≥ 0,
for all 𝑥 ∈ Fix(𝑇), (where 𝐶 = Fix(𝑇)) in Hilbert spaces as
well as in some Banach spaces.
Marino and Xu [7] considered the following general iterative method in Hilbert spaces:
𝑥𝑛+1 = 𝛼𝑛 𝛾𝑓 (𝑥𝑛 ) + (𝐼 − 𝛼𝑛 𝐴) 𝑇𝑥𝑛 ,
∀𝑛 ≥ 0,
(6)
where 𝐴 is a strongly positive bounded linear operator. It is
proved that if the sequence {𝛼𝑛 } satisfies appropriate conditions, the sequence {𝑥𝑛 } generated by (6) converges strongly
̃ ∈ 𝐶 of the variational inequality:
to the unique solution 𝑥
̃ ⟩ ≤ 0, for all 𝑥 ∈ 𝐶, where 𝐶 is the fixed
⟨(𝛾𝑓 − 𝐴)̃
𝑥, 𝑥 − 𝑥
points set of a nonexpansive mapping 𝑇.
Tian [8] considered the following general iterative algorithm (GIA) in Hilbert spaces:
𝑥𝑛+1 = 𝛼𝑛 𝛾𝑓 (𝑥𝑛 ) + (𝐼 − 𝜇𝛼𝑛 𝐹) 𝑇𝑥𝑛 ,
∀𝑛 ≥ 0.
(7)
It is proved that if the sequence {𝛼𝑛 } satisfies appropriate conditions, the sequence {𝑥𝑛 } generated by (7) converges strongly
̃ ∈ Fix(𝑇) of the variational inequalito the unique solution 𝑥
̃ ⟩ ≤ 0, for all 𝑥 ∈ Fix(𝑇).
ty: ⟨(𝛾𝑓 − 𝜇𝐹)̃
𝑥, 𝑥 − 𝑥
In 2001, Soltuz [9] introduced the following backward
Mann scheme iteration:
𝑥𝑛 = 𝛼𝑛 𝑥𝑛−1 + (1 − 𝛼𝑛 ) 𝑇𝑥𝑛 ,
∀𝑛 ≥ 1,
(8)
where 𝑇 is a nonexpansive mapping and got strong convergence in Hilbert spaces.
In order to find a common fixed point of a finite family
of nonexpansive mappings {𝑇𝑖 : 𝑖 ∈ 𝐽}, where 𝐽 stands for
{1, 2, 3, . . . , 𝑁}, in 2001, Xu and Ori [10] introduced the following implicit process:
𝑥𝑛 = 𝛼𝑛 𝑥𝑛−1 + (1 − 𝛼𝑛 ) 𝑇𝑛 𝑥𝑛 ,
∀𝑛 ≥ 1,
(9)
where 𝑇𝑛 = 𝑇𝑛(mod 𝑁) , and a weak convergence is obtained in
real Hilbert spaces.
Ceng et al. [11] introduced an iterative algorithm to find a
common fixed point of a finite family of nonexpansive semigroups in reflexive Banach spaces with a weak sequentially
continuous duality mapping, which satisfy the uniformly
asymptotical regularity condition:
𝑥𝑛+1 = (1 − 𝛼𝑛 − 𝛽𝑛 ) 𝑥𝑛 + 𝛼𝑛 𝑓 (𝑦𝑛 ) + 𝛽𝑛 𝑇𝑟𝑛 𝑦𝑛 ,
𝑦𝑛 = (1 − 𝛾𝑛 ) 𝑥𝑛 + 𝛾𝑛 𝑇𝑟𝑛 𝑥𝑛 .
(10)
Under some appropriate conditions one the parameter sequences {𝛼𝑛 }, {𝛽𝑛 }, {𝛾𝑛 }, and the sequence {𝑥𝑛 } generated
by (10) converges strongly to the approximate solution of a
variational inequality problem.
In order to find a common element of the solution set of
a general system of variational inequalities and the fixedpoint set of the mapping 𝑆, Ceng et al. [12] constructed a new
relaxed extragradient iterative method:
𝑢𝑛 = 𝑃𝐶 [𝑃𝐶 (𝑥𝑛 − 𝜇2 𝐵2 𝑥𝑛 ) − 𝜇1 𝐵1 𝑃𝐶 (𝑥𝑛 − 𝜇2 𝐵2 𝑥𝑛 )] ,
𝑧𝑛 = 𝑃𝐶 (𝑢𝑛 − 𝜆 𝑛 𝐴𝑢𝑛 ) ,
𝑦𝑛 = 𝛼𝑛 𝑓 (𝑥𝑛 ) + (1 − 𝛼𝑛 ) 𝑃𝐶 (𝑢𝑛 − 𝜆 𝑛 𝐴𝑧𝑛 ) ,
𝑥𝑛+1 = 𝛽𝑛 𝑥𝑛 + 𝛾𝑛 𝑦𝑛 + 𝛿𝑛 𝑆𝑦𝑛 ,
∀𝑛 ≥ 0.
(11)
Under mild assumptions, they obtained a strong convergence
theorem.
Yao et al. [13] introduced the following Halpern-type
implicit iterative method where 𝑇 is a continuous pseudocontraction:
𝑥𝑛 = 𝛼𝑛 𝑢 + 𝛽𝑛 𝑥𝑛−1 + 𝛾𝑛 𝑇𝑥𝑛 ,
∀𝑛 ≥ 1
(12)
and obtained a strong convergence theorem in Banach spaces.
Hu [14] introduced an iteration for a nonexpansive mapping in Banach spaces, which guarantee a uniformly Gêteaux
differentiable norm as follows:
𝑥𝑛+1 = 𝛼𝑛 𝑢 + 𝛽𝑛 𝑥𝑛 + 𝛾𝑛 𝑇𝑥𝑛 ,
∀𝑛 ≥ 0,
(13)
and several strong convergent theorems are obtained.
Very recently, Jung [15] proposed an iterative process in
the frame of Hilbert spaces as follows:
𝑥𝑛+1 = 𝛼𝑛 𝛾𝑓 (𝑥𝑛 ) + 𝛽𝑛 𝑥𝑛
+ ((1 − 𝛽𝑛 ) 𝐼 − 𝛼𝑛 𝜇𝐹) 𝑃𝐶𝑆𝑥𝑛 ,
∀𝑛 ≥ 0,
(14)
where 𝑆 is a mapping defined by 𝑆𝑥 = 𝑘𝑥 + (1 − 𝑘)𝑇𝑥 and 𝑇 is
a 𝑘-strictly pseudocontraction. Strong convergence theorems
are established.
Motivated and inspired by Soltuz [9], Xu and Ori [10],
Ceng et al. [11], Ceng et al. [12], Yao et al. [13], Hu [14], and
Jung [15], we consider the following new implicit iteration in
real 2-uniformly smooth Banach spaces:
𝑥𝑛 = 𝛼𝑛 𝛾𝑉 (𝑥𝑛 ) + 𝛽𝑛 𝑥𝑛−1
+ ((1 − 𝛽𝑛 ) 𝐼 − 𝛼𝑛 𝜇𝐹) 𝑇𝑛 𝑥𝑛 ,
∀𝑛 ≥ 1,
(15)
where {𝛼𝑛 } and {𝛽𝑛 } are real sequences in (0, 1), 𝑉 is an 𝐿Lipschitzian continuous with Lipschitzian constant 𝐿 > 0,
𝐹 is an 𝜂-strongly accretive and 𝜅-Lipschitzian continuous
mapping with 𝜅 > 0 and 𝜂 > 0, and {𝑇𝑖 }∞
𝑖=1 is a countable
family of nonexpansive mappings.
In this paper, we prove that the implicit iterative process
̃ of
(15) has strong convergence and find the unique solution 𝑥
variational inequality:
̃ , 𝑗 (𝑥 − 𝑥
̃ )⟩ ≥ 0,
⟨(𝜇𝐹 − 𝛾𝑉) 𝑥
∞
∀𝑥 ∈ 𝑆 = ⋂ Fix (𝑇𝑛 ) ≠ 0.
𝑛=1
(16)
Abstract and Applied Analysis
3
Our results improve and extend the corresponding conclusions announced by many others.
2. Preliminaries
Lemma 2 (see [17]). Let 𝑋 be a real 𝑞-uniformly smooth
Banach space for some 𝑞 > 1, then there exists some positive
constant 𝑑𝑞 , such that
󵄩𝑞
󵄩 󵄩𝑞
󵄩󵄩
𝑞
󵄩󵄩𝑥 + 𝑦󵄩󵄩󵄩 ≤ ‖𝑥‖ + 𝑞 ⟨𝑦, 𝑗𝑞 (𝑥)⟩ + 𝑑𝑞 󵄩󵄩󵄩𝑦󵄩󵄩󵄩 ,
Let 𝑆𝑋 = {𝑥 ∈ 𝑋 :‖ 𝑥 ‖= 1}. Then the norm of 𝑋 is said to be
Gâteaux differentiable if
󵄩
󵄩󵄩
󵄩𝑥 + 𝑡𝑦󵄩󵄩󵄩 − ‖𝑥‖
Δ = lim 󵄩
𝑡→0
𝑡
(17)
exists for each 𝑥, 𝑦 ∈ 𝑆𝑋 . In this case, 𝑋 is said to be smooth.
The norm of 𝑋 is called uniformly Gâteaux differentiable,
if for each 𝑦 ∈ 𝑆𝑋 , Δ is attained uniformly for 𝑥 ∈ 𝑆𝑋 .
The norm of 𝑋 is called Fréchet differentiable, if for each
𝑥 ∈ 𝑆𝑋 , Δ is attained uniformly for 𝑦 ∈ 𝑆𝑋 . The norm of
𝑋 is called uniformly Fréchet differentiable, if Δ is attained
uniformly for 𝑥, 𝑦 ∈ 𝑆𝑋 . It is well known that (uniformly)
Fréchet differentiability of the norm of 𝑋 implies (uniformly)
Gâteaux differentiability of the norm of 𝑋. If the norm on 𝑋
is uniformly Gâteaux differentiable, the generalized duality
mapping 𝐽𝑞 is single-valued and strong-weak∗ uniformly
continuous on any bounded subsets of 𝑋.
Let 𝜌𝑋 : [0, ∞) → ∞ be the modulus of smoothness of
𝑋 defined by
∀𝑥, 𝑦 ∈ 𝑋, 𝑗𝑞 (𝑥) ∈ 𝐽𝑞 (𝑥) ,
in particular, if 𝑋 is a real 2-uniformly smooth Banach space,
then there exists a best smooth constant 𝐾 > 0, such that
󵄩2
󵄩󵄩
󵄩 󵄩2
2
󵄩󵄩𝑥 + 𝑦󵄩󵄩󵄩 ≤ ‖𝑥‖ + 2 ⟨𝑦, 𝑗 (𝑥)⟩ + 2󵄩󵄩󵄩𝐾𝑦󵄩󵄩󵄩 ,
∀𝑥, 𝑦 ∈ 𝑋, 𝑗𝑞 (𝑥) ∈ 𝐽𝑞 (𝑥) .
(i) for all 𝑥, 𝑦 ∈ 𝑋, the following inequality holds:
󵄩
󵄩
Φ (󵄩󵄩󵄩𝑥 + 𝑦󵄩󵄩󵄩) ≤ Φ (‖𝑥‖) + ⟨𝑦, 𝑗𝑞 (𝑥 + 𝑦)⟩ ,
󵄩 󵄩
= {𝑓 ∈ 𝑋∗ : ⟨𝑥, 𝑓⟩ = ‖𝑥‖ 𝜑 (‖𝑥‖) , 󵄩󵄩󵄩𝑓󵄩󵄩󵄩 = 𝜑 (‖𝑥‖) , 𝑥 ∈ 𝑋} .
(19)
It is known that real 2-uniformly smooth Banach spaces have
a weakly continuous duality mapping with a guage function
𝜑(𝑡) = 𝑡, which is the same as the normalized duality mapping
𝑡
𝐽. Set Φ(𝑡) = ∫0 𝜑(𝜏)𝑑𝜏, for all 𝑡 ≥ 0, then 𝐽𝜑 (𝑥) = 𝜕Φ(‖𝑥‖),
where 𝜕 denotes the subdifferential in the sense of convex
analysis. In fact, for 0 ≤ 𝑘 ≤ 1, we have 𝜑(𝑘𝑡) ≤ 𝜑(𝑡) and
𝑘𝑡
𝑡
Φ (𝑘𝑡) = ∫ 𝜑 (𝜏) 𝑑𝜏 = 𝑘 ∫ 𝜑 (𝑘𝜏) 𝑑𝜏
0
0
𝑡
≤ 𝑘 ∫ 𝜑 (𝜏) 𝑑𝜏 = 𝑘Φ (𝑡) .
0
(20)
(23)
in particular, for all 𝑥, 𝑦 ∈ 𝑋, there holds:
󵄩2
󵄩󵄩
2
󵄩󵄩𝑥 + 𝑦󵄩󵄩󵄩 ≤ ‖𝑥‖ + 2 ⟨𝑦, 𝑗 (𝑥 + 𝑦)⟩ ;
(24)
(ii) assume that a sequence {𝑥𝑛 } ⊂ 𝑋 converges weakly to
a point 𝑥 ∈ 𝑋, then the following equation holds:
󵄩
󵄩
󵄩
󵄩
lim supΦ (󵄩󵄩󵄩𝑥𝑛 − 𝑦󵄩󵄩󵄩) = lim supΦ (󵄩󵄩󵄩𝑥𝑛 − 𝑥󵄩󵄩󵄩)
𝑛→∞
𝑛→∞
󵄩
󵄩
+ Φ (󵄩󵄩󵄩𝑦 − 𝑥󵄩󵄩󵄩) ,
(25)
∀𝑥, 𝑦 ∈ 𝑋.
Lemma 4 (see [19]). Assume that {𝑎𝑛 } is a sequence of nonnegative real numbers, such that
𝑎𝑛+1 ≤ (1 − 𝛾𝑛 ) 𝑎𝑛 + 𝛾𝑛 𝛿𝑛 ,
𝐽𝜑 (𝑥)
(22)
Lemma 3 (see [18]). Assume that a Banach space 𝑋 has a
weakly continuous duality mapping 𝐽𝑞 :
1 󵄩
󵄩
󵄩 󵄩
󵄩 󵄩
𝜌𝑋 (𝑡) = sup { (󵄩󵄩󵄩𝑥 + 𝑦󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩) − 1 : 𝑥 ∈ 𝑆𝑋 , 󵄩󵄩󵄩𝑦󵄩󵄩󵄩 ≤ 𝑡} .
2
(18)
A Banach space 𝑋 is said to be uniformly smooth if
𝜌𝑋 (𝑡)/𝑡 → 0 as 𝑡 → 0. A Banach space is said to be 𝑞-uniformly smooth, if there exists a fixed constant 𝑐 > 0, such that
𝜌𝑋 (𝑡) ≤ 𝑐𝑡𝑞 . It is well known that the 𝑋 is uniformly smooth if
and only if the norm of 𝑋 is uniformly Fréchet differentiable
[16].
The so-called gauge function 𝜑 is defined as follows: let
𝜑 : [0, ∞) := R+ → R+ be a continuous strictly increasing
function, such that 𝜑(0) = 0 and 𝜑(𝑡) → ∞ as 𝑡 → ∞. The
∗
duality mapping 𝐽𝜑 : 𝑋 → 2𝑋 associated with a gauge function 𝜑 is defined by
(21)
∀𝑛 ≥ 0,
(26)
where {𝛾𝑛 } is a sequence in (0, 1) and {𝛿𝑛 } is a sequence in R,
such that
(a) ∑∞
𝑛=0 𝛾𝑛 = ∞;
(b) lim sup𝑛 → ∞ 𝛿𝑛 ≤ 0 or ∑∞
𝑛=0 |𝛾𝑛 𝛿𝑛 | < ∞.
Then lim𝑛 → ∞ 𝑎𝑛 = 0.
Lemma 5. Let 𝑋 be a real 2-uniformly smooth Banach space.
Let 𝑇 be a nonexpansive mapping over 𝑋, and let 𝐹 : 𝑋 →
𝑋 be an 𝜂-strongly accretive and 𝜅-Lipschitzian continuous
mapping with 𝜅 > 0 and 𝜂 > 0. For 0 < 𝑡 < 𝜎 ≤ 1 and
𝜇 ∈ (0, min{1, 𝜂/𝐾2 𝜅2 }), set 𝜏 = 𝜇(𝜂 − 𝜇𝐾2 𝜅2 ), and define
a mapping 𝑇𝑡 : 𝑋 → 𝑋 by 𝑇𝑡 := 𝜎𝐼 − 𝑡𝜇𝐹. Then 𝑇𝑡 is a
contraction on 𝑋; that is, ‖ 𝑇𝑡 𝑥 − 𝑇𝑡 𝑦 ‖≤ (𝜎 − 𝑡𝜏) ‖ 𝑥 − 𝑦 ‖.
4
Abstract and Applied Analysis
Proof. From 0 < 𝜇 < 𝜂/𝐾2 𝜅2 , we have 𝜂 − 𝜇𝐾2 𝜅2 > 0. Setting
𝜂 < 1/2, we have 0 < 2(𝜂 − 𝜇𝐾2 𝜅2 ) < 1. For each 𝑥, 𝑦 ∈ 𝑋, by
Lemma 2, we have
󵄩󵄩 𝑡
󵄩2
󵄩󵄩𝑇 𝑥 − 𝑇𝑡 𝑦󵄩󵄩󵄩
󵄩
󵄩
󵄩2
󵄩󵄩
= 󵄩󵄩𝜎(𝑥 − 𝑦) − 𝑡𝜇(𝐹𝑥 − 𝐹𝑦)󵄩󵄩󵄩
󵄩2
󵄩
≤ 𝜎2 󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩 − 𝜎2𝑡𝜇 ⟨𝐹𝑥 − 𝐹𝑦, 𝑗 (𝑥 − 𝑦)⟩
󵄩2
󵄩
+ 2𝐾2 𝑡2 𝜇2 󵄩󵄩󵄩𝐹𝑥 − 𝐹𝑦󵄩󵄩󵄩
󵄩2
󵄩2
󵄩
󵄩
≤ 𝜎2 󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩 − 2𝜎𝑡𝜇𝜂󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩
󵄩2
󵄩
+ 2𝐾2 𝑡2 𝜇2 𝜅2 󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩
󵄩2
󵄩
≤ [𝜎2 − 2𝜎𝑡𝜇 (𝜂 − 𝜇𝐾2 𝜅2 )] 󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩
(27)
󵄩2
󵄩
= (𝜎2 − 2𝜎𝑡𝜏) 󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩
󵄩2
󵄩
≤ (𝜎 − 𝑡𝜏)2 󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩 .
󵄩
󵄩󵄩 𝑡
󵄩󵄩𝑇 𝑥 − 𝑇𝑡 𝑦󵄩󵄩󵄩 ≤ (𝜎 − 𝑡𝜏) 󵄩󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩󵄩 .
󵄩
󵄩
(28)
This completes the proof.
Proposition 9 (the path convergence). Let 𝑇 : 𝑋 → 𝑋 be
a nonexpansive mapping with Fix(𝑇) ≠ 0, and let 𝑉 : 𝑋 →
𝑋 be an 𝐿-Lipschitzian continuous mapping with Lipschitzian
constant 𝐿 > 0. 𝐹 : 𝑋 → 𝑋 is an 𝜂-strongly accretive and 𝜅Lipschitzian continuous mapping with 𝜅 > 0 and 𝜂 > 0. For 𝑡 ∈
(0, 1), let 𝜇 ∈ (0, min{1, 𝜂/𝐾2 𝜅2 }), and set 𝜏 = 𝜇(𝜂 − 𝜇𝐾2 𝜅2 )
and 0 < 𝛾 < 𝜏/𝐿. Then assume that {𝑥𝑡 } is defined by
Definition 6 (see [20]). Let 𝑋 be a real Banach space, let 𝐶 be
a nonempty subset of 𝑋, and let {𝑇𝑛 }∞
𝑛=1 be a countable family
Fix(𝑇
)
of mappings of 𝐶 with ⋂∞
𝑛 ≠ 0. Then {𝑇𝑛 } is said to
𝑛=1
satisfy the AKTT condition, if for any bounded subset 𝐷 of
𝐶, the following inequality holds:
∞
󵄩
󵄩
∑ sup {󵄩󵄩󵄩𝑇𝑛+1 𝑥 − 𝑇𝑛 𝑥󵄩󵄩󵄩 : 𝑥 ∈ 𝐷} < ∞.
(29)
𝑛=1
Lemma 7 (see [20]). Let 𝑋 be a Banach space, let 𝐶 be a
nonempty closed subset of 𝑋, and let {𝑇𝑛 } be a family of selfmappings of 𝐶 satisfying the AKTT condition. Then for each
𝑥 ∈ 𝐶, {𝑇𝑛 𝑥} converges strongly to a point in 𝐶. Moreover, let
the mapping 𝑇 be defined by
∀𝑥 ∈ 𝐶.
(30)
Then for any bounded subset 𝐷 of 𝐶, the following equality
holds:
󵄩
󵄩
lim sup sup {󵄩󵄩󵄩𝑇𝑥 − 𝑇𝑛 𝑥󵄩󵄩󵄩 : 𝑥 ∈ 𝐷} = 0.
(31)
𝑛→∞
Lemma 8 (see [1]). Suppose that 𝑞 > 1. Then the following
inequality holds:
1 𝑞 𝑞 − 1 𝑞/(𝑞−1)
,
𝑎 +
𝑏
𝑞
𝑞
for arbitrary positive real numbers 𝑎 and 𝑏.
(32)
(33)
̃ of 𝑇,
Then {𝑥𝑡 } converges strongly as 𝑡 → 0+ to a fixed point 𝑥
which is the unique solution of the variational inequality VIP:
̃ , 𝑗 (𝑥 − 𝑥
̃ )⟩ ≥ 0,
⟨(𝜇𝐹 − 𝛾𝑉) 𝑥
To deal with a family of mappings, we will introduce the
following concept called the AKTT condition.
𝑎𝑏 ≤
In order to obtain the main results, we divide this section
into 3 parts. In Proposition 9, we give the path convergence.
In Proposition 10, under the demiclosed assumption and
combined with Proposition 9, we find the unique solution
of a variational inequality. In Theorem 11, we prove that the
sequence {𝑥𝑛 } defined by the implicit scheme (15) converges
strongly to the unique solution of (16).
Throughout this paper, we assume that 𝑋 is a real 2uniformly smooth Banach space, which guarantees a weakly
continuous duality 𝐽 as proposed in Section 1.
𝑥𝑡 = 𝑡𝛾𝑉 (𝑥𝑡 ) + (𝐼 − 𝑡𝜇𝐹) 𝑇𝑥𝑡 .
Hence, it implies that
𝑇𝑥 = lim 𝑇𝑛 𝑥,
3. Main Results
∀𝑥 ∈ Fix(𝑇) .
(34)
Proof. Consider a mapping 𝑆𝑡 on 𝑋 defined by
𝑆𝑡 𝑥 = 𝑡𝛾𝑉 (𝑥𝑡 ) + (𝐼 − 𝑡𝜇𝐹) 𝑇𝑥𝑡 ,
∀𝑥 ∈ 𝑋.
(35)
It is easy to see that 𝑆𝑡 is a contraction. Indeed, for any 𝑥, 𝑦 ∈
𝑋, by Lemma 5, we have
󵄩
󵄩
󵄩
󵄩󵄩
󵄩󵄩𝑆𝑡 𝑥 − 𝑆𝑡 𝑦󵄩󵄩󵄩 ≤ 𝑡𝛾 󵄩󵄩󵄩𝑉 (𝑥) − 𝑉 (𝑦)󵄩󵄩󵄩
󵄩
󵄩
+ 󵄩󵄩󵄩(𝐼 − 𝑡𝜇𝐹) 𝑇𝑥 − (𝐼 − 𝑡𝜇𝐹) 𝑇𝑦󵄩󵄩󵄩
󵄩
󵄩
≤ [1 − 𝑡 (𝜏 − 𝛾𝐿)] 󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩 .
(36)
Hence, by the Banach contraction mapping principle, 𝑆𝑡 has a
unique fixed point, denoted by 𝑥𝑡 , which uniquely solves the
fixed point equation (33).
We divided the proof into several steps.
Step 1. We show the uniqueness of the solution of the variational inequality (34). Assume that both 𝑥1 ∈ Fix(𝑇) and
𝑥2 ∈ Fix(𝑇) are solutions of the variational inequality (34),
then we have
⟨(𝜇𝐹 − 𝛾𝑉) 𝑥1 , 𝑗𝑞 (𝑥2 − 𝑥1 )⟩ ≥ 0,
⟨(𝜇𝐹 − 𝛾𝑉) 𝑥2 , 𝑗 (𝑥1 − 𝑥2 )⟩ ≥ 0.
(37)
Adding up (37) yields
⟨(𝜇𝐹 − 𝛾𝑉) 𝑥1 − (𝜇𝐹 − 𝛾𝑉) 𝑥2 , 𝑗 (𝑥2 − 𝑥1 )⟩ ≥ 0.
(38)
Abstract and Applied Analysis
5
Indeed, from the given conditions 𝜇 ∈ (0, min{1, 𝜂/𝐾2 𝜅2 }),
and 𝜏 = 𝜇(𝜂 − 𝜇𝐾2 𝜅2 ), 0 < 𝛾 < 𝜏/𝐿, we have
From (42), we have
󵄩
󵄩
𝐵 (𝑇𝑥∗ ) = lim sup Φ (󵄩󵄩󵄩𝑥𝑛 − 𝑇𝑥∗ 󵄩󵄩󵄩)
𝑛→∞
⟨(𝜇𝐹 − 𝛾𝑉) 𝑥1 − (𝜇𝐹 − 𝛾𝑉) 𝑥2 , 𝑗 (𝑥1 − 𝑥2 )⟩
󵄩
󵄩
= lim sup Φ (󵄩󵄩󵄩𝑇𝑥𝑛 − 𝑇𝑥∗ + 𝑥𝑛 − 𝑇𝑥𝑛 󵄩󵄩󵄩)
= 𝜇 ⟨𝐹𝑥1 − 𝐹𝑥2 , 𝑗 (𝑥1 − 𝑥2 )⟩
𝑛→∞
− 𝛾 ⟨𝑉𝑥1 − 𝑉𝑥2 , 𝑗 (𝑥1 − 𝑥2 )⟩
(39)
󵄩2
󵄩2
󵄩
󵄩
≥ 𝜇𝜂󵄩󵄩󵄩𝑥1 − 𝑥2 󵄩󵄩󵄩 − 𝛾𝐿󵄩󵄩󵄩𝑥1 − 𝑥2 󵄩󵄩󵄩
󵄩2
󵄩
= (𝜇𝜂 − 𝛾𝐿) 󵄩󵄩󵄩𝑥1 − 𝑥2 󵄩󵄩󵄩 ≥ 0.
(45)
𝑛→∞
󵄩
󵄩
≤ lim sup Φ (󵄩󵄩󵄩𝑥𝑛 − 𝑥∗ 󵄩󵄩󵄩) = 𝐵 (𝑥∗ ) ,
𝑛→∞
and we also note that
Thus, we conclude that 𝑥1 = 𝑥2 . So the uniqueness of the
variational inequality (35) is guaranteed.
Step 2. We show that {𝑥𝑡 } is bounded. Taking 𝑝 ∈ Fix(𝑇), it
follows from Lemma 5 that
󵄩󵄩󵄩𝑥𝑡 − 𝑝󵄩󵄩󵄩2
󵄩
󵄩
󵄩
󵄩
󵄩
󵄩
𝐵 (𝑇𝑥∗ ) = lim sup Φ (󵄩󵄩󵄩𝑥𝑛 − 𝑥∗ 󵄩󵄩󵄩) + Φ (󵄩󵄩󵄩𝑇𝑥∗ − 𝑥∗ 󵄩󵄩󵄩)
𝑛→∞
󵄩
󵄩
= 𝐵 (𝑥∗ ) + Φ (󵄩󵄩󵄩𝑇𝑥∗ − 𝑥∗ 󵄩󵄩󵄩) ,
(46)
so, we obtain
󵄩
󵄩
Φ (󵄩󵄩󵄩𝑇𝑥∗ − 𝑥∗ 󵄩󵄩󵄩) ≤ 0.
(47)
This implies that 𝑇𝑥∗ = 𝑥∗ ; that is, 𝑥∗ ∈ Fix(𝑇).
By Lemma 5, we have
= ⟨𝑡𝛾𝑉(𝑥𝑡 ) + (𝐼 − 𝑡𝜇𝐹) 𝑇𝑥𝑡 − 𝑝, 𝑗 (𝑥𝑡 − 𝑝)⟩
= 𝑡 ⟨𝛾𝑉(𝑥𝑡 ) − 𝜇𝐹𝑝, 𝑗 (𝑥𝑡 − 𝑝)⟩
󵄩󵄩
∗ 󵄩2
󵄩󵄩𝑥𝑛 − 𝑥 󵄩󵄩󵄩
󵄩2
󵄩
= 󵄩󵄩󵄩𝑡𝑛 (𝛾𝑉(𝑥𝑛 ) − 𝜇𝐹𝑥∗ ) + (𝐼 − 𝑡𝑛 𝜇𝐹)𝑇𝑥𝑛 − (𝐼 − 𝑡𝑛 𝜇𝐹)𝑥∗ 󵄩󵄩󵄩
+ ⟨(𝐼 − 𝑡𝜇𝐹) 𝑇𝑥𝑡 − (𝐼 − 𝑡𝜇𝐹) 𝑝, 𝑗 (𝑥𝑡 − 𝑝)⟩
≤ 𝑡 ⟨𝛾𝑉(𝑥𝑡 ) − 𝛾𝑉(𝑝) + 𝛾𝑉(𝑝) − 𝜇𝐹𝑝, 𝑗 (𝑥𝑡 − 𝑝)⟩
= ⟨(𝐼 − 𝑡𝑛 𝜇𝐹) 𝑇𝑥𝑛 − (𝐼 − 𝑡𝑛 𝜇𝐹) 𝑥∗ , 𝑗 (𝑥𝑛 − 𝑥∗ )⟩
󵄩󵄩
󵄩
󵄩
+ 󵄩󵄩󵄩(𝐼 − 𝑡𝜇𝐹) 𝑇𝑥𝑡 − (𝐼 − 𝑡𝜇𝐹) 𝑝󵄩󵄩󵄩 󵄩󵄩󵄩𝑥𝑡 − 𝑝󵄩󵄩󵄩
+ 𝑡𝑛 ⟨𝛾𝑉 (𝑥𝑛 ) − 𝜇𝐹𝑥∗ , 𝑗 (𝑥𝑛 − 𝑥∗ )⟩
󵄩󵄩
󵄩
󵄩2
󵄩
󵄩
≤ 𝑡𝛾𝐿󵄩󵄩󵄩𝑥𝑡 − 𝑝󵄩󵄩󵄩 + 𝑡 󵄩󵄩󵄩𝛾𝑉(𝑝) − 𝜇𝐹𝑝󵄩󵄩󵄩 󵄩󵄩󵄩𝑥𝑡 − 𝑝󵄩󵄩󵄩
󵄩
󵄩2
+ (1 − 𝑡𝜏) 󵄩󵄩󵄩𝑥𝑡 − 𝑝󵄩󵄩󵄩
󵄩
󵄩2
≤ (1 − 𝑡 (𝜏 − 𝛾𝐿)) 󵄩󵄩󵄩𝑥𝑡 − 𝑝󵄩󵄩󵄩
󵄩󵄩
󵄩
󵄩
+ 𝑡 󵄩󵄩󵄩𝛾𝑉(𝑝) − 𝜇𝐹𝑝󵄩󵄩󵄩 󵄩󵄩󵄩𝑥𝑡 − 𝑝󵄩󵄩󵄩 .
󵄩
󵄩
= lim sup Φ (󵄩󵄩󵄩𝑇𝑥𝑛 − 𝑇𝑥∗ 󵄩󵄩󵄩)
󵄩2
󵄩
≤ (1 − 𝑡𝑛 𝜏) 󵄩󵄩󵄩𝑥𝑛 − 𝑥∗ 󵄩󵄩󵄩
+ 𝑡𝑛 ⟨𝛾𝑉 (𝑥𝑛 ) − 𝛾𝑉 (𝑥∗ ) , 𝑗 (𝑥𝑛 − 𝑥∗ )⟩
+ 𝑡𝑛 ⟨𝛾𝑉 (𝑥∗ ) − 𝜇𝐹𝑥∗ , 𝑗 (𝑥𝑛 − 𝑥∗ )⟩
(40)
󵄩2
󵄩2
󵄩
󵄩
≤ (1 − 𝑡𝑛 𝜏) 󵄩󵄩󵄩𝑥𝑛 − 𝑥∗ 󵄩󵄩󵄩 + 𝑡𝑛 𝛾𝐿󵄩󵄩󵄩𝑥𝑛 − 𝑥∗ 󵄩󵄩󵄩
+ 𝑡𝑛 ⟨𝛾𝑉 (𝑥∗ ) − 𝜇𝐹𝑥∗ , 𝑗 (𝑥𝑛 − 𝑥∗ )⟩
It follows that
1 󵄩󵄩
󵄩󵄩
󵄩
󵄩
󵄩󵄩𝑥𝑡 − 𝑝󵄩󵄩󵄩 ≤
󵄩𝛾𝑉 (𝑝) − 𝜇𝐹𝑝󵄩󵄩󵄩 .
𝜏 − 𝛾𝐿 󵄩
(41)
󵄩
󵄩2
= (1 − 𝑡𝑛 (𝜏 − 𝛾𝐿)) 󵄩󵄩󵄩𝑥𝑛 − 𝑥∗ 󵄩󵄩󵄩
+ 𝑡𝑛 ⟨𝛾𝑉 (𝑥∗ ) − 𝜇𝐹𝑥∗ , 𝑗 (𝑥𝑛 − 𝑥∗ )⟩ .
Hence {𝑥𝑡 } is bounded, so are {𝑉(𝑥𝑡 )} and {𝐹𝑇𝑥𝑡 )}.
(48)
Step 3. Next, we will show that {𝑥𝑡 } has a subsequence converging strongly to 𝑥∗ ∈ Fix(𝑇).
Assume 𝑡𝑛 → 0, and set 𝑥𝑛 := 𝑥𝑡𝑛 . By the definition of
{𝑥𝑛 }, we have
󵄩
󵄩
󵄩
󵄩󵄩
(42)
󵄩󵄩𝑥𝑛 − 𝑇𝑥𝑛 󵄩󵄩󵄩 = 𝑡𝑛 󵄩󵄩󵄩𝛾𝑉 (𝑥𝑛 ) − 𝜇𝐹𝑇𝑥𝑛 󵄩󵄩󵄩 󳨀→ 0.
Since {𝑥𝑛 } is bounded, there exists a subsequence {𝑥𝑛𝑘 } of
{𝑥𝑛 } converging weakly to 𝑥∗ ∈ 𝑋 as 𝑘 → ∞.
Set 𝑥𝑛 := 𝑥𝑛𝑘 . Define a mapping 𝐵 : 𝑋 → R by
This implies that
1
󵄩󵄩
∗ 󵄩2
⟨𝛾𝑉 (𝑥∗ ) − 𝜇𝐹𝑥∗ , 𝑗 (𝑥𝑛 − 𝑥∗ )⟩ .
󵄩󵄩𝑥𝑛 − 𝑥 󵄩󵄩󵄩 ≤
𝜏 − 𝛾𝐿
(49)
Since {𝑥𝑛 } is bounded, there exists a subsequence {𝑥𝑛𝑘 } of {𝑥𝑛 }
satisfying
(43)
1
󵄩2
󵄩󵄩
󵄩󵄩𝑥𝑛𝑘 − 𝑥∗ 󵄩󵄩󵄩 ≤
⟨𝛾𝑉 (𝑥∗ ) − 𝜇𝐹𝑥∗ , 𝑗 (𝑥𝑛𝑘 − 𝑥∗ )⟩ .
󵄩
󵄩
𝜏 − 𝛾𝐿
(50)
Again, 𝐽 is weakly continuous, by Lemma 3, and it follows that
󵄩
󵄩
𝐵 (𝑦) = 𝐵 (𝑥) + Φ (󵄩󵄩󵄩𝑦 − 𝑥∗ 󵄩󵄩󵄩) , ∀𝑦 ∈ 𝑋.
(44)
Since the mapping 𝐽 is single-valued and weakly continuous,
it follows from (50) that ‖𝑥𝑛𝑘 − 𝑥∗ ‖2 → 0 as 𝑘 → ∞. Thus,
there exists a subsequence, such that 𝑥𝑛𝑘 → 𝑥∗ .
󵄩
󵄩
𝐵 (𝑥) = lim sup Φ (󵄩󵄩󵄩𝑥𝑛 − 𝑥󵄩󵄩󵄩) ,
𝑛→∞
∀𝑥 ∈ 𝑋.
6
Abstract and Applied Analysis
󵄩2
󵄩
≤ 󵄩󵄩󵄩𝑥𝑡 − 𝑥𝑛 󵄩󵄩󵄩 + ⟨𝑇𝑥𝑛 − 𝑥𝑛 , 𝑗 (𝑥𝑡 − 𝑥𝑛 )⟩
Step 4. Finally, we show that 𝑥∗ is the unique solution of variational inequality (34).
Since 𝑥𝑡 = 𝑡𝛾𝑉(𝑥𝑡 ) + (𝐼 − 𝑡𝜇𝐹)𝑇𝑥𝑡 , we can derive that
1
(𝜇𝐹 − 𝛾𝑉) 𝑥𝑡 = − (𝐼 − 𝑇) 𝑥𝑡 + 𝜇 (𝐹𝑥𝑡 − 𝐹𝑇𝑥𝑡 ) .
𝑡
+ 𝑡 ⟨𝛾𝑉𝑥𝑡 − 𝜇𝐹𝑥𝑡 , 𝑗 (𝑥𝑡 − 𝑥𝑛 )⟩
+ 𝑡 ⟨𝜇𝐹𝑥𝑡 − 𝜇𝐹𝑇𝑥𝑡 , 𝑗 (𝑥𝑡 − 𝑥𝑛 )⟩
(51)
󵄩2 󵄩
󵄩󵄩
󵄩
󵄩
≤ 󵄩󵄩󵄩𝑥𝑡 − 𝑥𝑛 󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑇𝑥𝑛 − 𝑥𝑛 󵄩󵄩󵄩 󵄩󵄩󵄩𝑥𝑡 − 𝑥𝑛 󵄩󵄩󵄩
It follows that, for any 𝑥 ∈ Fix(𝑇),
+ 𝑡 ⟨𝛾𝑉𝑥𝑡 − 𝜇𝐹𝑥𝑡 , 𝑗 (𝑥𝑡 − 𝑥𝑛 )⟩
󵄩󵄩
󵄩
󵄩
+ 𝑡𝜇 󵄩󵄩󵄩𝐹𝑥𝑡 − 𝐹𝑇𝑥𝑡 󵄩󵄩󵄩 󵄩󵄩󵄩𝑥𝑡 − 𝑥𝑛 󵄩󵄩󵄩 ,
⟨(𝜇𝐹 − 𝛾𝑉) 𝑥𝑡 , 𝑗 (𝑥𝑡 − 𝑥)⟩
1
= − ⟨(𝐼 − 𝑇) 𝑥𝑡 − (𝐼 − 𝑇) 𝑥, 𝑗 (𝑥𝑡 − 𝑥)⟩
𝑡
(52)
+ 𝜇 ⟨𝐹𝑥𝑡 − 𝐹𝑇𝑥𝑡 , 𝑗 (𝑥𝑡 − 𝑥)⟩ .
(56)
which implies that
⟨(𝜇𝐹 − 𝛾𝑉) 𝑥𝑡 , 𝑗 (𝑥𝑡 − 𝑥𝑛 )⟩ ≤
Since 𝑇 is a nonexpansive mapping, for all 𝑥, 𝑦 ∈ 𝑋, we
conclude that
⟨(𝐼 − 𝑇) 𝑥 − (𝐼 − 𝑇) 𝑦, 𝑗 (𝑥 − 𝑦)⟩
= ⟨𝑥 − 𝑦, 𝑗 (𝑥 − 𝑦)⟩ − ⟨𝑇𝑥 − 𝑇𝑦, 𝑗 (𝑥 − 𝑦)⟩
(53)
It follows that
lim sup ⟨(𝜇𝐹 − 𝛾𝑉) 𝑥𝑡 , 𝑗 (𝑥𝑡 − 𝑥𝑛 )⟩
𝑛→∞
󵄩2 󵄩
󵄩2
󵄩
≥ 󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩 − 󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩 = 0.
󵄩
󵄩
󵄩
󵄩
≤ 𝜇 󵄩󵄩󵄩𝐹𝑥𝑡 − 𝐹𝑇𝑥𝑡 󵄩󵄩󵄩 lim sup 󵄩󵄩󵄩𝑥𝑡 − 𝑥𝑛 󵄩󵄩󵄩 .
∗
⟨(𝜇𝐹 − 𝛾𝑉) 𝑥 , 𝑗 (𝑥 − 𝑥)⟩ ≤ 0,
∗
(58)
𝑛→∞
Now replacing 𝑡 in (52) with 𝑡𝑛 and letting 𝑛 → ∞, from
(42), we have that 𝐹𝑥𝑡𝑛 − 𝐹𝑇𝑥𝑡𝑛 → 0, thus we can conclude
that
∗
1 󵄩󵄩
󵄩󵄩
󵄩
󵄩𝑇𝑥 − 𝑥𝑛 󵄩󵄩󵄩 󵄩󵄩󵄩𝑥𝑡 − 𝑥𝑛 󵄩󵄩󵄩
𝑡󵄩 𝑛
󵄩
󵄩󵄩
󵄩
+ 𝜇 󵄩󵄩󵄩𝐹𝑥𝑡 − 𝐹𝑇𝑥𝑡 󵄩󵄩󵄩 󵄩󵄩󵄩𝑥𝑡 − 𝑥𝑛 󵄩󵄩󵄩 .
(57)
∀𝑥 ∈ Fix (𝑇) .
(54)
̃ by uniqueness.
So, 𝑥 is a solution of (34). Hence, 𝑥 = 𝑥
̃ as 𝑡 → 0+ . This completes the proof.
Therefore, 𝑥𝑡 → 𝑥
Proposition 10 (the demiclosed result). Let 𝑇 : 𝑋 → 𝑋 be
a nonexpansive mapping with Fix(𝑇) ≠ 0, and let 𝑉 be an 𝐿Lipschitzian continuous self-mapping on 𝑋 with Lipschitzian
constant 𝐿 > 0. 𝐹 : 𝑋 → 𝑋 is an 𝜂-strongly accretive and
𝜅-Lipschitzian continuous mapping with 𝜅 > 0 and 𝜂 > 0.
Assume that the net {𝑥𝑡 } is defined as Proposition 9 which
̃ ∈ Fix(𝑇). Suppose that
converges strongly as 𝑡 → 0+ to 𝑥
the sequence {𝑥𝑛 } ⊂ 𝑋 is bounded and satisfies the condition
lim𝑛 → ∞ ‖ 𝑥𝑛 − 𝑇𝑥𝑛 ‖= 0 (the so-called demiclosed property).
Then the following inequality VIP holds:
𝑛→∞
= ⟨(𝑡𝛾𝑉𝑥𝑡 + (𝐼 − 𝑡𝜇𝐹) 𝑇𝑥𝑡 ) − 𝑥𝑛 , 𝑗 (𝑥𝑡 − 𝑥𝑛 )⟩
= ⟨𝑇𝑥𝑡 − 𝑇𝑥𝑛 + 𝑇𝑥𝑛 − 𝑥𝑛 + 𝑡 (𝛾𝑉𝑥𝑡 − 𝜇𝐹𝑥𝑡 )
+ 𝑡 (𝜇𝐹𝑥𝑡 − 𝜇𝐹𝑇𝑥𝑡 ) , 𝑗 (𝑥𝑡 − 𝑥𝑛 )⟩
𝑛→∞
(59)
On the other hand, since 𝑋 is a real 2-uniformly smooth
Banach space, and 𝐽 is single-valued and strong-weak∗ uniformly continuous on 𝑋, as 𝑡 → 0+ , we have
̃ , 𝑗𝑞 (̃
⟨(𝜇𝐹 − 𝛾𝑉) 𝑥
𝑥 − 𝑥𝑛 )⟩ − ⟨𝜇𝐹𝑥𝑡 − 𝛾𝑉𝑥𝑡 , 𝑗𝑞 (𝑥𝑡 − 𝑥𝑛 )⟩
̃ , 𝑗𝑞 (̃
𝑥 − 𝑥𝑛 ) − 𝑗𝑞 (𝑥𝑡 − 𝑥𝑛 )⟩
= ⟨(𝜇𝐹 − 𝛾𝑉) 𝑥
𝑥) , 𝑗𝑞 (𝑥𝑡 − 𝑥𝑛 )⟩
+ ⟨𝜇𝐹 (̃
𝑥) − 𝜇𝐹𝑥𝑡 + 𝛾𝑉 (𝑥𝑡 ) − 𝛾𝑉 (̃
̃ , 𝑗𝑞 (̃
𝑥 − 𝑥𝑛 ) − 𝑗𝑞 (𝑥𝑡 − 𝑥𝑛 )⟩
= ⟨(𝜇𝐹 − 𝛾𝑉) 𝑥
+ ⟨𝜇𝐹 (̃
𝑥) − 𝜇𝐹𝑥𝑡 , 𝑗𝑞 (𝑥𝑡 − 𝑥𝑛 )⟩
+ 𝛾 ⟨𝑉 (𝑥𝑡 ) − 𝑉 (̃
𝑥) , 𝑗𝑞 (𝑥𝑡 − 𝑥𝑛 )⟩ 󳨀→ 0.
(60)
(55)
Proof. Set 𝑎𝑛 (𝑡) = ‖𝑇𝑥𝑛 − 𝑥𝑛 ‖‖𝑥𝑡 − 𝑥𝑛 ‖. From the given condition and the boundness of {𝑥𝑡 } and {𝑥𝑛 }, it is obvious that
𝑎𝑛 (𝑡) → 0 when 𝑛 → ∞.
From (33) and the fact that 𝑇 is a nonexpansive mapping,
we obtain that
󵄩2
󵄩󵄩
󵄩󵄩𝑥𝑡 − 𝑥𝑛 󵄩󵄩󵄩
lim sup lim sup ⟨(𝜇𝐹 − 𝛾𝑉) 𝑥𝑡 , 𝑗 (𝑥𝑡 − 𝑥𝑛 )⟩ ≤ 0.
𝑡→0
∗
̃ , 𝑗 (𝑥𝑛 − 𝑥
̃ )⟩ ≤ 0.
lim sup ⟨(𝛾𝑉 − 𝜇𝐹) 𝑥
Taking the lim sup as 𝑡 → 0 and recalling (42) and the continuity of 𝐹, we conclude that
Thus, from (59) and (60), we obtain
̃ , 𝑗𝑞 (̃
lim sup ⟨(𝜇𝐹 − 𝛾𝑉) 𝑥
𝑥 − 𝑥𝑛 )⟩
𝑛→∞
̃ , 𝑗𝑞 (̃
= lim sup lim sup ⟨(𝜇𝐹 − 𝛾𝑉) 𝑥
𝑥 − 𝑥𝑛 )⟩ ≤ 0.
𝑡→0
(61)
𝑛→∞
So (55) is valid. This completes the proof.
Finally, we study the following implicit iterative method
process: the initial 𝑥0 ∈ 𝑋 is arbitrarily selected, and the iterative algorithm is recursively defined by
𝑥𝑛 = 𝛼𝑛 𝛾𝑉 (𝑥𝑛 ) + 𝛽𝑛 𝑥𝑛−1
+ ((1 − 𝛽𝑛 ) 𝐼 − 𝜇𝛼𝑛 𝐹) 𝑇𝑛 𝑥𝑛 ,
∀𝑛 ≥ 1,
(62)
Abstract and Applied Analysis
7
where the sequences {𝛼𝑛 } and {𝛽𝑛 } are sequences in (0, 1) and
satisfy the following conditions:
(C1) lim𝑛 → ∞ 𝛼𝑛 = lim𝑛 → ∞ 𝛽𝑛 = 0,
󵄩
󵄩󵄩
󵄩
+ 𝛽𝑛 󵄩󵄩󵄩𝑥𝑛−1 − 𝑝󵄩󵄩󵄩 󵄩󵄩󵄩𝑥𝑛 − 𝑝󵄩󵄩󵄩
󵄩
󵄩2
+ (1 − 𝛽𝑛 − 𝛼𝑛 𝜏) 󵄩󵄩󵄩𝑥𝑛 − 𝑝󵄩󵄩󵄩
(C2) ∑∞
𝑛=0 (𝛼𝑛 /(𝛼𝑛 + 𝛽𝑛 )) = ∞.
Theorem 11. Let {𝑇𝑖 }∞
𝑖=1 be a countable family of selfnonexpansive mappings on 𝑋, such that 𝑆 := ⋂∞
𝑛=1 Fix(𝑇𝑛 ) ≠ 0.
Let 𝑉 be an 𝐿-Lipschitzian continuous self-mapping on 𝑋 with
Lipschitzian constant 𝐿 > 0. 𝐹 : 𝑋 → 𝑋 is an 𝜂-strongly
accretive and 𝜅-Lipschitzian continuous mapping with 𝜅 > 0
and 𝜂 > 0. Suppose that the sequences {𝛼𝑛 } and {𝛽𝑛 } satisfy the
controlling conditions (C1)-(C2). Let 𝜇 ∈ (0, min{1, 𝜂/𝐾2 𝜅2 }),
and set 𝜏 = 𝜇(𝜂 − 𝜇𝐾2 𝜅2 ) and (𝜏 − 1)/𝐿 < 𝛾 < 𝜏/𝐿. Assume
that ({𝑇𝑛 }, 𝑇) satisfies the AKTT condition. Then {𝑥𝑛 } defined
̃ of {𝑇𝑛 }∞
by (62) converges strongly to a common fixed point 𝑥
𝑛=1
which equivalently solves the following variational inequality:
̃ , 𝑗 (̃
⟨(𝛾𝑉 − 𝜇𝐹) 𝑥
𝑥 − 𝑝)⟩ ≥ 0,
󵄩
󵄩2
≤ 𝛼𝑛 𝛾𝐿󵄩󵄩󵄩𝑥𝑛 − 𝑝󵄩󵄩󵄩 + 𝛼𝑛 ⟨𝛾𝑉 (𝑝) − 𝜇𝐹𝑝, 𝑗 (𝑥𝑛 − 𝑝)⟩
∀𝑝 ∈ 𝑆.
(63)
Proof. First we show that {𝑥𝑛 } is well defined. Consider a
mapping 𝑆𝑛 on 𝑋 defined by
𝑆𝑛 𝑥 = 𝛼𝑛 𝛾𝑉𝑥 + 𝛽𝑛 𝑥𝑛−1 + ((1 − 𝛽𝑛 ) 𝐼 − 𝜇𝛼𝑛 𝐹) 𝑇𝑛 𝑥.
(64)
It is easy to see that 𝑆𝑛 is a contraction. Indeed, for any 𝑥, 𝑦 ∈
𝑋, by Lemma 5, we have
󵄩󵄩
󵄩
󵄩󵄩𝑆𝑡 𝑥 − 𝑆𝑡 𝑦󵄩󵄩󵄩
󵄩
󵄩
≤ 𝛼𝑛 𝛾 󵄩󵄩󵄩𝑉 (𝑥) − 𝑉 (𝑦)󵄩󵄩󵄩
󵄩
󵄩
+ 󵄩󵄩󵄩((1 − 𝛽𝑛 ) 𝐼 − 𝛼𝑛 𝜇𝐹) 𝑇𝑥 − ((1 − 𝛽𝑛 ) 𝐼 − 𝛼𝑛 𝜇𝐹) 𝑇𝑦󵄩󵄩󵄩
󵄩
󵄩
≤ [1 − 𝛽𝑛 − 𝛼𝑛 (𝜏 − 𝛾𝐿)] 󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩 .
(65)
Hence, 𝑆𝑛 is a contraction. By the Banach contraction
mapping principle, we conclude that 𝑆𝑛 has a unique fixed
point, denoted by 𝑥𝑛 . So (62) is well defined.
Then we show that {𝑥𝑛 } is bounded. Taking any 𝑝 ∈ 𝑆, we
have
󵄩󵄩
󵄩2
󵄩󵄩𝑥𝑛 − 𝑝󵄩󵄩󵄩
󵄩
󵄩2
= (1 − 𝛽𝑛 − 𝛼𝑛 (𝜏 − 𝛾𝐿)) 󵄩󵄩󵄩𝑥𝑛 − 𝑝󵄩󵄩󵄩
󵄩󵄩
󵄩
󵄩
+ 𝛼𝑛 󵄩󵄩󵄩𝛾𝑉 (𝑝) − 𝜇𝐹𝑝󵄩󵄩󵄩 󵄩󵄩󵄩𝑥𝑛 − 𝑝󵄩󵄩󵄩
󵄩
󵄩󵄩
󵄩
+ 𝛽𝑛 󵄩󵄩󵄩𝑥𝑛−1 − 𝑝󵄩󵄩󵄩 󵄩󵄩󵄩𝑥𝑛 − 𝑝󵄩󵄩󵄩 ,
(66)
which implies that
𝛽𝑛
󵄩󵄩
󵄩
󵄩
󵄩󵄩
󵄩󵄩𝑥𝑛 − 𝑝󵄩󵄩󵄩 ≤
󵄩𝑥 − 𝑝󵄩󵄩󵄩
𝛽𝑛 + 𝛼𝑛 (𝜏 − 𝛾𝐿) 󵄩 𝑛−1
󵄩
󵄩
𝛼𝑛 (𝜏 − 𝛾𝐿) 󵄩󵄩󵄩𝛾𝑉 (𝑝) − 𝜇𝐹𝑝󵄩󵄩󵄩
+
.
𝜏 − 𝛾𝐿
𝛽𝑛 + 𝛼𝑛 (𝜏 − 𝛾𝐿)
By induction, it follows that
󵄩
󵄩
󵄩
󵄩
󵄩 󵄩󵄩𝛾𝑉 (𝑝) − 𝜇𝐹𝑝󵄩󵄩󵄩
󵄩󵄩
},
󵄩󵄩𝑥𝑛 − 𝑝󵄩󵄩󵄩 ≤ max {󵄩󵄩󵄩𝑥0 − 𝑝󵄩󵄩󵄩 , 󵄩
𝜏 − 𝛾𝐿
+ ⟨((1 − 𝛽𝑛 ) 𝐼 − 𝛼𝑛 𝜇𝐹) 𝑇𝑛 𝑥𝑛
− ((1 − 𝛽𝑛 ) 𝐼 − 𝛼𝑛 𝜇𝐹) 𝑝, 𝑗 (𝑥𝑛 − 𝑝)⟩
≤ 𝛼𝑛 𝛾 ⟨𝑉 (𝑥𝑛 ) − 𝑉 (𝑝) , 𝑗 (𝑥𝑛 − 𝑝)⟩
+ 𝛼𝑛 ⟨𝛾𝑉 (𝑝) − 𝜇𝐹𝑝, 𝑗 (𝑥𝑛 − 𝑝)⟩
+ 𝛽𝑛 ⟨𝑥𝑛−1 − 𝑝, 𝑗 (𝑥𝑛 − 𝑝)⟩
+ ⟨((1 − 𝛽𝑛 ) 𝐼 − 𝛼𝑛 𝜇𝐹) 𝑇𝑛 𝑥𝑛
− ((1 − 𝛽𝑛 ) 𝐼 − 𝛼𝑛 𝜇𝐹) 𝑝, 𝑗 (𝑥𝑛 − 𝑝)⟩
𝑛 ≥ 0.
(68)
Hence {𝑥𝑛 } is bounded, so are the {𝑉(𝑥𝑛 )} and {𝐹(𝑥𝑛 )}.
Next, we show that
󵄩
󵄩󵄩
󵄩󵄩𝑥𝑛 − 𝑇𝑥𝑛 󵄩󵄩󵄩 󳨀→ 0.
(69)
From (C1) and the definition of {𝑥𝑛 }, we observe that
󵄩󵄩
󵄩
󵄩󵄩𝑥𝑛 − 𝑇𝑛 𝑥𝑛 󵄩󵄩󵄩
󵄩
󵄩
= 󵄩󵄩󵄩𝛼𝑛 (𝛾𝑉 (𝑥𝑛 ) − 𝜇𝐹𝑇𝑛 𝑥𝑛 ) + 𝛽𝑛 (𝑥𝑛−1 − 𝑇𝑛 𝑥𝑛 )󵄩󵄩󵄩
󵄩
󵄩
≤ 𝛼𝑛 󵄩󵄩󵄩𝛾𝑉 (𝑥𝑛 ) − 𝜇𝐹𝑇𝑛 𝑥𝑛 󵄩󵄩󵄩
󵄩
󵄩
+ 𝛽𝑛 󵄩󵄩󵄩𝑥𝑛−1 − 𝑇𝑛 𝑥𝑛 󵄩󵄩󵄩 󳨀→ 0.
= 𝛼𝑛 ⟨𝛾𝑉 (𝑥𝑛 ) − 𝜇𝐹𝑝, 𝑗 (𝑥𝑛 − 𝑝)⟩
+ 𝛽𝑛 ⟨𝑥𝑛−1 − 𝑝, 𝑗 (𝑥𝑛 − 𝑝)⟩
(67)
(70)
By Lemma 7 and (70), we have
󵄩󵄩
󵄩 󵄩
󵄩
󵄩󵄩𝑥𝑛 − 𝑇𝑥𝑛 󵄩󵄩󵄩 = 󵄩󵄩󵄩𝑥𝑛 − 𝑇𝑛 𝑥𝑛 + 𝑇𝑛 𝑥𝑛 − 𝑇𝑥𝑛 󵄩󵄩󵄩
󵄩
󵄩
≤ 󵄩󵄩󵄩𝑥𝑛 − 𝑇𝑛 𝑥𝑛 󵄩󵄩󵄩
󵄩
󵄩
+ 󵄩󵄩󵄩𝑇𝑛 𝑥𝑛 − 𝑇𝑥𝑛 󵄩󵄩󵄩 󳨀→ 0.
(71)
Let 𝑥𝑡 be defined by (33), from Propositions 9 and 10, and
̃ ∈ 𝑆 := Fix(𝑇) =
we have that {𝑥𝑡 } converges strongly to 𝑥
Fix(𝑇
)
and
⋂∞
𝑖
𝑖=1
̃ , 𝑗 (𝑥𝑛 − 𝑥
̃ )⟩ ≤ 0.
lim sup ⟨(𝛾𝑉 − 𝜇𝐹) 𝑥
𝑛→∞
(72)
8
Abstract and Applied Analysis
̃ . As a matter of
As required, finally we show that 𝑥𝑛 → 𝑥
fact, by Lemmas 5 and 8, we have
2
󵄩󵄩
̃ 󵄩󵄩󵄩󵄩
󵄩󵄩𝑥𝑛 − 𝑥
̃
= ⟨((1 − 𝛽𝑛 ) 𝐼 − 𝛼𝑛 𝜇𝐹) 𝑇𝑛 𝑥𝑛 − ((1 − 𝛽𝑛 ) 𝐼 − 𝛼𝑛 𝜇𝐹) 𝑥
̃ ) + 𝛼𝑛 (𝛾𝑉 (𝑥𝑛 ) − 𝜇𝐹̃
̃ )⟩
+𝛽𝑛 (𝑥𝑛−1 − 𝑥
𝑥) , 𝑗 (𝑥𝑛 − 𝑥
≤ ⟨((1 − 𝛽𝑛 ) 𝐼 − 𝛼𝑛 𝜇𝐹) 𝑇𝑛 𝑥𝑛
̃ , 𝑗 (𝑥𝑛 − 𝑥
̃ )⟩
− ((1 − 𝛽𝑛 ) 𝐼 − 𝛼𝑛 𝜇𝐹) 𝑥
󵄩
󵄩
󵄩
󵄩
󵄩
󵄩
󵄩
󵄩
̃ 󵄩󵄩 󵄩󵄩𝑥𝑛 − 𝑥
̃ 󵄩󵄩
+ 𝛽𝑛 󵄩󵄩𝑥𝑛−1 − 𝑥
̃ )⟩
+ 𝛼𝑛 ⟨𝛾𝑉 (𝑥𝑛 ) − 𝜇𝐹̃
𝑥, 𝑗 (𝑥𝑛 − 𝑥
Remark 13. In 2008, Hu [14] introduced a modified Halperntype iteration for a single nonexpansive mapping in Banach
spaces which have a uniformly Gêteaux differentiable norm
as follows:
𝑥𝑛+1 = 𝛼𝑛 𝑢 + 𝛽𝑛 𝑥𝑛 + 𝛾𝑛 𝑇𝑥𝑛 .
(76)
̃ )⟩
+ 𝛼𝑛 ⟨𝛾𝑉 (̃
𝑥) − 𝜇𝐹̃
𝑥, 𝑗 (𝑥𝑛 − 𝑥
2
󵄩
̃ 󵄩󵄩󵄩󵄩
≤ (1 − 𝛽𝑛 − 𝛼𝑛 𝜏) 󵄩󵄩󵄩𝑥𝑛 − 𝑥
󵄩
̃ 󵄩󵄩󵄩󵄩 󵄩󵄩󵄩󵄩𝑥𝑛 − 𝑥
̃ 󵄩󵄩󵄩󵄩
+ 𝛽𝑛 󵄩󵄩󵄩𝑥𝑛−1 − 𝑥
2
󵄩
̃ 󵄩󵄩󵄩󵄩 + 𝛼𝑛 ⟨𝛾𝑉 (̃
̃ )⟩
+ 𝛼𝑛 𝛾𝐿󵄩󵄩󵄩𝑥𝑛 − 𝑥
𝑥) − 𝜇𝐹̃
𝑥, 𝑗 (𝑥𝑛 − 𝑥
2
󵄩
̃ 󵄩󵄩󵄩󵄩
≤ (1 − 𝛽𝑛 − 𝛼𝑛 (𝜏 − 𝛾𝐿)) 󵄩󵄩󵄩𝑥𝑛 − 𝑥
̃ )⟩
+ 𝛼𝑛 ⟨𝛾𝑉 (̃
𝑥) − 𝜇𝐹̃
𝑥, 𝑗 (𝑥𝑛 − 𝑥
2
󵄩
̃ 󵄩󵄩󵄩󵄩
≤ (1 − 𝛽𝑛 − 𝛼𝑛 (𝜏 − 𝛾𝐿)) 󵄩󵄩󵄩𝑥𝑛 − 𝑥
𝛽 󵄩
2
2
̃ 󵄩󵄩󵄩󵄩 + 󵄩󵄩󵄩󵄩𝑥𝑛 − 𝑥
̃ 󵄩󵄩󵄩󵄩 )
+ 𝑛 (󵄩󵄩󵄩𝑥𝑛−1 − 𝑥
2
̃ )⟩
+ 𝛼𝑛 ⟨𝛾𝑉 (̃
𝑥) − 𝜇𝐹̃
𝑥, 𝑗 (𝑥𝑛 − 𝑥
Remark 15. Ceng et al. [11] introduced the following iterative
algorithm to find a common fixed point of a finite family of
nonexpansive semigroups in reflexive Banach spaces:
𝑥𝑛+1 = (1 − 𝛼𝑛 − 𝛽𝑛 ) 𝑥𝑛 + 𝛼𝑛 𝑓 (𝑦𝑛 ) + 𝛽𝑛 𝑇𝑟𝑛 𝑦𝑛 ,
(77)
𝑦𝑛 = (1 − 𝛾𝑛 ) 𝑥𝑛 + 𝛾𝑛 𝑇𝑟𝑛 𝑥𝑛 .
𝛽𝑛
2
󵄩
̃ 󵄩󵄩󵄩󵄩
− 𝛼𝑛 (𝜏 − 𝛾𝐿)) 󵄩󵄩󵄩𝑥𝑛 − 𝑥
2
𝛽𝑛 󵄩󵄩
2
̃ 󵄩󵄩󵄩󵄩
󵄩𝑥 − 𝑥
2 󵄩 𝑛−1
̃ )⟩ ,
+ 𝛼𝑛 ⟨𝛾𝑉 (̃
𝑥) − 𝜇𝐹̃
𝑥, 𝑗 (𝑥𝑛 − 𝑥
+
(73)
which implies that
2
󵄩󵄩
̃ 󵄩󵄩󵄩󵄩
󵄩󵄩𝑥𝑛 − 𝑥
𝛽𝑛
2
󵄩󵄩
̃ 󵄩󵄩󵄩󵄩
󵄩󵄩𝑥𝑛−1 − 𝑥
𝛽𝑛 + 2𝛼𝑛 (𝜏 − 𝛾𝐿)
+
2𝛼𝑛
̃ )⟩
̃ , 𝑗 (𝑥𝑛 − 𝑥
⟨(𝛾𝑉 − 𝜇𝐹) 𝑥
𝛽𝑛 + 2𝛼𝑛 (𝜏 − 𝛾𝐿)
2𝛼𝑛 (𝜏 − 𝛾𝐿)
2
󵄩
̃ 󵄩󵄩󵄩󵄩
= [1 −
] 󵄩󵄩𝑥 − 𝑥
𝛽𝑛 + 2𝛼𝑛 (𝜏 − 𝛾𝐿) 󵄩 𝑛−1
+
̃ )⟩
̃ , 𝑗 (𝑥𝑛 − 𝑥
2𝛼𝑛 (𝜏 − 𝛾𝐿) ⟨(𝛾𝑉 − 𝜇𝐹) 𝑥
.
𝜏 − 𝛾𝐿
𝛽𝑛 + 2𝛼𝑛 (𝜏 − 𝛾𝐿)
Under some appropriate assumptions, he proved that the
sequence {𝑥𝑛 } defined by the iteration process (76) converges
strongly to the fixed point of 𝑇.
Corollary 14. If we take 𝛾 = 1, 𝐹 = 𝐼, 𝜇 = 1, and 𝛽𝑛 = 0 in
(62), we extend the classical viscosity approximation [4] under
a mild assumption: the contraction mapping 𝑓 is replaced by
an 𝐿-Lipschitzian continuous mapping 𝑉. Our proving process
needs no Banach limit and is different from the proving process
given by Xu [6] in some aspects.
󵄩
̃ 󵄩󵄩󵄩󵄩 󵄩󵄩󵄩󵄩𝑥𝑛 − 𝑥
̃ 󵄩󵄩󵄩󵄩
+ 𝛽𝑛 󵄩󵄩󵄩𝑥𝑛−1 − 𝑥
≤
Thus, (C2) yields that ∑∞
𝑛=0 (2𝛼𝑛 (𝜏 − 𝛾𝐿)/(𝛽𝑛 + 2𝛼𝑛 (𝜏 − 𝛾𝐿))) =
∞. Applying Lemma 4 and (72) to (74), we conclude that
̃.
𝑥𝑛 → 𝑥
This completes the proof.
Remark 12. Our result in Proposition 9 extends Theorem 3.1
of Tian [8] from real Hilbert spaces to real 2-uniformly
smooth Banach spaces. If we set 𝛽𝑛 = 0, our result in
Theorem 11 extends Theorem 3.2 of Tian [8] from real Hilbert
spaces to real 2-uniformly smooth Banach spaces as well as
from a single nonexpansive mapping to a countable family of
nonexpansive mappings.
2
󵄩
̃ 󵄩󵄩󵄩󵄩
≤ (1 − 𝛽𝑛 − 𝛼𝑛 𝜏) 󵄩󵄩󵄩𝑥𝑛 − 𝑥
󵄩
̃ 󵄩󵄩󵄩󵄩 󵄩󵄩󵄩󵄩𝑥𝑛 − 𝑥
̃ 󵄩󵄩󵄩󵄩
+ 𝛽𝑛 󵄩󵄩󵄩𝑥𝑛−1 − 𝑥
̃ )⟩
+ 𝛼𝑛 𝛾 ⟨𝑉 (𝑥𝑛 ) − 𝑉 (̃
𝑥) , 𝑗 (𝑥𝑛 − 𝑥
≤ (1 −
It is easily to see that
2𝛼 (𝜏 − 𝛾𝐿)
2𝛼𝑛 (𝜏 − 𝛾𝐿)
𝛼𝑛
= (𝜏 − 𝛾𝐿)
.
> 𝑛
2𝛽𝑛 + 2𝛼𝑛
𝛼𝑛 + 𝛽𝑛
𝛽𝑛 + 2𝛼𝑛 (𝜏 − 𝛾𝐿)
(75)
(74)
Under some appropriate conditions one the parameter
sequences {𝛼𝑛 }, {𝛽𝑛 }, {𝛾𝑛 }, and the sequence {𝑥𝑛 } converges
strongly to the approximate solution of a variational inequality problem.
If we set 𝛼𝑛 + 𝛽𝑛 = 1 and 𝛾𝑛 = 0, the algorithm is
simplified into viscosity-form iterative schemes for a finite
family of nonexpansive semigroups. Our algorithms are considered in full space and avoid the generalized projections or
sunny nonexpansive retractions in Banach space. For further
improving our works, in order to obtain more general results,
we should take the results given by Ceng et al. in [11] into
account.
Acknowledgments
Ming Tian was supported in part by the Fundamental
Research Funds for the Central Universities (no.
ZXH2012K001). Xin Jin was supported in part by Technology
Innovation Funds of Civil Aviation University of China for
Graduate (YJSCX12-18).
Abstract and Applied Analysis
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