Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2013, Article ID 854297, 14 pages
http://dx.doi.org/10.1155/2013/854297
Research Article
Implicit Relaxed and Hybrid Methods with Regularization for
Minimization Problems and Asymptotically Strict
Pseudocontractive Mappings in the Intermediate Sense
Lu-Chuan Ceng,1 Qamrul Hasan Ansari,2,3 and Ching-Feng Wen4
1
Department of Mathematics, Shanghai Normal University and Scientific Computing Key Laboratory of Shanghai Universities,
Shanghai 200234, China
2
Department of Mathematics, Aligarh Muslim University, Aligarh 202 002, India
3
Department of Mathematics and Statistics, King Fahd University of Petroleum & Minerals, Dhahran, Saudi Arabia
4
Center for Fundamental Science, Kaohsiung Medical University, Kaohsiung 80708, Taiwan
Correspondence should be addressed to Ching-Feng Wen; cfwen@kmu.edu.tw
Received 1 February 2013; Accepted 11 March 2013
Academic Editor: Jen-Chih Yao
Copyright © 2013 Lu-Chuan Ceng et al. 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 first introduce an implicit relaxed method with regularization for finding a common element of the set of fixed points of
an asymptotically strict pseudocontractive mapping 𝑆 in the intermediate sense and the set of solutions of the minimization
problem (MP) for a convex and continuously Frechet differentiable functional in the setting of Hilbert spaces. The implicit relaxed
method with regularization is based on three well-known methods: the extragradient method, viscosity approximation method,
and gradient projection algorithm with regularization. We derive a weak convergence theorem for two sequences generated by this
method. On the other hand, we also prove a new strong convergence theorem by an implicit hybrid method with regularization for
the MP and the mapping 𝑆. The implicit hybrid method with regularization is based on four well-known methods: the CQ method,
extragradient method, viscosity approximation method, and gradient projection algorithm with regularization.
1. Introduction
In 1972, Goebel and Kirk [1] established that every asymptotically nonexpansive mapping 𝑆 : 𝐶 → 𝐶 defined on a nonempty closed convex bounded subset of a uniformly convex
Banach space, that is, there exists a sequence {𝑘𝑛 } such that
lim𝑛 → ∞ 𝑘𝑛 = 1 and
󵄩
󵄩
󵄩󵄩 𝑛
𝑛 󵄩
󵄩󵄩𝑆 𝑥 − 𝑆 𝑦󵄩󵄩󵄩 ≤ 𝑘𝑛 󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩 ,
∀𝑛 ≥ 1, ∀𝑥, 𝑦 ∈ 𝐶
(1)
has a fixed point in 𝐶. It can be easily seen that every nonexpansive mapping is asymptotically nonexpansive, and every
asymptotically nonexpansive mapping is uniformly Lipschitzian; that is, there exists a constant L > 0 such that
󵄩
󵄩󵄩 𝑛
󵄩
𝑛 󵄩
󵄩󵄩𝑆 𝑥 − 𝑆 𝑦󵄩󵄩󵄩 ≤ L 󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩 ,
∀𝑛 ≥ 1, ∀𝑥, 𝑦 ∈ 𝐶.
(2)
Several researchers have weaken the assumption on the mapping 𝑆. Bruck et al. [2] introduced the following concept of
an asymptotically nonexpansive mapping in the intermediate
sense.
Definition 1. Let 𝐶 be a nonempty subset of a normed space
𝑋. A mapping 𝑆 : 𝐶 → 𝐶 is said to be asymptotically nonexpansive in the intermediate sense provided 𝑆 is uniformly
continuous and
󵄩
󵄩
󵄩 󵄩
lim sup sup (󵄩󵄩󵄩𝑆𝑛 𝑥 − 𝑆𝑛 𝑦󵄩󵄩󵄩 − 󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩) ≤ 0.
(3)
𝑛 → ∞ 𝑥,𝑦∈𝐶
They also studied iterative methods for the approximation
of fixed points of such mappings.
Recently, Kim and Xu [3] introduced the following concept of asymptotically 𝜅-strict pseudocontractive mappings
in setting of Hilbert spaces.
Definition 2. Let 𝐶 be a nonempty subset of a Hilbert space
𝐻. A mapping 𝑆 : 𝐶 → 𝐶 is said to be an asymptotically 𝜅strict pseudocontractive mapping with sequence {𝛾𝑛 } if there
2
Abstract and Applied Analysis
exists a constant 𝜅 ∈ [0, 1) and a sequence {𝛾𝑛 } in [0, ∞) with
lim𝑛 → ∞ 𝛾𝑛 = 0 such that
󵄩󵄩 𝑛
𝑛 󵄩2
󵄩󵄩𝑆 𝑥 − 𝑆 𝑦󵄩󵄩󵄩
󵄩2
󵄩
󵄩
󵄩2
≤ (1 + 𝛾𝑛 ) 󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩 + 𝜅󵄩󵄩󵄩𝑥 − 𝑆𝑛 𝑥 − (𝑦 − 𝑆𝑛 𝑦)󵄩󵄩󵄩 ,
∀𝑛 ≥ 1, ∀𝑥, 𝑦 ∈ 𝐶.
(4)
They studied weak and strong convergence theorems for
this class of mappings. It is important to note that every
asymptotically 𝜅-strict pseudocontractive mapping with sequence {𝛾𝑛 } is a uniformly L-Lipschitzian mapping with L =
sup{(𝜅 + √1 + (1 − 𝜅)𝛾𝑛 )/(1 + 𝜅) : 𝑛 ≥ 1}.
Very recently, Sahu et al. [4] considered the following
concept of asymptotically 𝜅-strict pseudocontractive mappings in the intermediate sense, which are not necessarily
Lipschitzian.
Definition 3. Let 𝐶 be a nonempty subset of a Hilbert space
𝐻. A mapping 𝑆 : 𝐶 → 𝐶 is said to be an asymptotically 𝜅strict pseudocontractive mapping in the intermediate sense
with sequence {𝛾𝑛 } if there exist a constant 𝜅 ∈ [0, 1) and a
sequence {𝛾𝑛 } in [0, ∞) with lim𝑛 → ∞ 𝛾𝑛 = 0 such that
󵄩2
󵄩
󵄩2
󵄩
lim sup sup (󵄩󵄩󵄩𝑆𝑛 𝑥 − 𝑆𝑛 𝑦󵄩󵄩󵄩 − (1 + 𝛾𝑛 ) 󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩
𝑛→∞
𝑥,𝑦∈𝐶
󵄩
󵄩2
− 𝜅󵄩󵄩󵄩𝑥 − 𝑆𝑛 𝑥 − (𝑦 − 𝑆𝑛 𝑦)󵄩󵄩󵄩 ) ≤ 0.
2. Preliminaries
(5)
Let
󵄩2
󵄩
󵄩2
󵄩
𝑐𝑛 := max {0, sup (󵄩󵄩󵄩𝑆𝑛 𝑥 − 𝑆𝑛 𝑦󵄩󵄩󵄩 − (1 + 𝛾𝑛 ) 󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩
𝑥,𝑦∈𝐶
󵄩
󵄩2
−𝜅󵄩󵄩󵄩𝑥 − 𝑆𝑛 𝑥 − (𝑦 − 𝑆𝑛 𝑦)󵄩󵄩󵄩 ) } .
(6)
Then, 𝑐𝑛 ≥ 0 (for all 𝑛 ≥ 1), 𝑐𝑛 → 0 (𝑛 → ∞), and (5)
reduces to the relation
󵄩2
󵄩󵄩 𝑛
󵄩
𝑛 󵄩2
󵄩󵄩𝑆 𝑥 − 𝑆 𝑦󵄩󵄩󵄩 ≤ (1 + 𝛾𝑛 ) 󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩
󵄩
󵄩2
+ 𝜅󵄩󵄩󵄩𝑥 − 𝑆𝑛 𝑥 − (𝑦 − 𝑆𝑛 𝑦)󵄩󵄩󵄩 + 𝑐𝑛 ,
∀𝑛 ≥ 1, ∀𝑥, 𝑦 ∈ 𝐶.
(7)
Whenever 𝑐𝑛 = 0 for all 𝑛 ≥ 1 in (7), then 𝑆 is an asymptotically 𝜅-strict pseudocontractive mapping with sequence {𝛾𝑛 }.
Let 𝑓 : 𝐶 → R be a convex and continuously Fréchet
differentiable functional. We consider the following minimization problem:
min𝑓 (𝑥) .
𝑥∈𝐶
compute the fixed point of a mapping which is also a solution
of some minimization problem; for further detail, we refer to
[5] and the references therein.
The main aim of this paper is to propose some iterative
schemes for finding a common solution of fixed point set of
an asymptotically 𝜅-strict pseudocontractive mapping and
the solution set of the minimization problem. In particular,
we introduce an implicit relaxed algorithm with regularization for finding a common element of the fixed point
set Fix(𝑆) of an asymptotically 𝜅-strict pseudocontractive
mapping 𝑆 and the solution set Γ of minimization problem
(8). This implicit relaxed method with regularization is based
on three well-known methods, namely, the extragradient
method [6], viscosity approximation method, and gradient
projection algorithm with regularization. We also propose
an implicit hybrid algorithm with regularization for finding
an element of Fix(𝑆) ∩ Γ. The implicit hybrid method with
regularization is based on four well-known methods, namely,
the CQ method, extragradient method, viscosity approximation method, and gradient projection algorithm with
regularization. The weak and strong convergence results of
these two algorithms are established, respectively.
(8)
We assume that the minimization problem (8) has a solution,
and the solution set of this problem is denoted by Γ.
To develop some new iterative methods for computing
the approximate solutions of the minimization problem is one
of the main areas of research in optimization and approximation theory. In the recent past, some study has also been
done in the direction to suggest some iterative algorithms to
Throughout the paper, unless otherwise specified, we use the
following assumptions, notations, and terminologies.
We assume that 𝐻 is a real Hilbert space whose inner
product and norm are denoted by ⟨⋅, ⋅⟩ and ‖ ⋅ ‖, respectively,
and 𝐶 is a nonempty closed convex subset of 𝐻. We write
𝑥𝑛 ⇀ 𝑥 to indicate that the sequence {𝑥𝑛 } converges weakly
to 𝑥 and 𝑥𝑛 → 𝑥 to indicate that the sequence {𝑥𝑛 } converges
strongly to 𝑥. Moreover, we use 𝜔𝑤 (𝑥𝑛 ) to denote the weak 𝜔limit set of the sequence {𝑥𝑛 }, that is,
𝜔𝑤 (𝑥𝑛 ) := {𝑥 ∈ 𝐻 : 𝑥𝑛𝑖 ⇀ 𝑥
for some subsequence {𝑥𝑛𝑖 } of {𝑥𝑛 }} .
(9)
The metric (or nearest point) projection from 𝐻 onto 𝐶 is
the mapping 𝑃𝐶 : 𝐻 → 𝐶 which assigns to each point 𝑥 ∈ 𝐻
the unique point 𝑃𝐶𝑥 ∈ 𝐶 satisfying the following property:
󵄩
󵄩
󵄩
󵄩󵄩
󵄩󵄩𝑥 − 𝑃𝐶𝑥󵄩󵄩󵄩 = inf 󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩 =: 𝑑 (𝑥, 𝐶) .
(10)
𝑦∈𝐶
We mention some important properties of projections in
the following proposition.
Proposition 4. For given 𝑥 ∈ 𝐻 and 𝑧 ∈ 𝐶,
(i) 𝑧 = 𝑃𝐶𝑥 ⇔ ⟨𝑥 − 𝑧, 𝑦 − 𝑧⟩ ≤ 0, for all 𝑦 ∈ 𝐶;
(ii) 𝑧 = 𝑃𝐶𝑥 ⇔ ‖𝑥 − 𝑧‖2 ≤ ‖𝑥 − 𝑦‖2 − ‖𝑦 − 𝑧‖2 , for all
𝑦 ∈ 𝐶;
(iii) ⟨𝑃𝐶𝑥 − 𝑃𝐶𝑦, 𝑥 − 𝑦⟩ ≥ ‖𝑃𝐶𝑥 − 𝑃𝐶𝑦‖2 , for all 𝑦 ∈ 𝐻.
Consequently, 𝑃𝐶 is nonexpansive.
Definition 5. A mapping 𝑆 : 𝐶 → 𝐻 is said to be
(a) monotone if
⟨𝑆𝑥 − 𝑆𝑦, 𝑥 − 𝑦⟩ ≥ 0,
∀𝑥, 𝑦 ∈ 𝐶;
(11)
Abstract and Applied Analysis
3
(b) 𝜂-strongly monotone if there exists a constant 𝜂 > 0
such that
󵄩2
󵄩
⟨𝑆𝑥 − 𝑆𝑦, 𝑥 − 𝑦⟩ ≥ 𝜂󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩 ,
∀𝑥, 𝑦 ∈ 𝐶;
(12)
(c) 𝛼-inverse-strongly monotone (𝛼-ism) if there exists a
constant 𝛼 > 0 such that
󵄩2
󵄩
⟨𝑆𝑥 − 𝑆𝑦, 𝑥 − 𝑦⟩ ≥ 𝛼󵄩󵄩󵄩𝑆𝑥 − 𝑆𝑦󵄩󵄩󵄩 ,
∀𝑥, 𝑦 ∈ 𝐶.
(13)
Obviously, if 𝑆 is 𝛼-inverse-strongly monotone, then it is
monotone and (1/𝛼)-Lipschitz continuous. It can be easily
seen that if 𝑆 : 𝐶 → 𝐶 is nonexpansive, then 𝐼 − 𝑆 is
monotone. It is also easy to see that a projection mapping 𝑃𝐶 is
1-ism. The inverse strongly monotone (also known as cocoercive) operators have been applied widely in solving practical
problems in various fields.
We need some facts and tools which are listed in the form
of the following lemmas.
Lemma 6. Let 𝑋 be a real inner product space. Then,
󵄩2
󵄩󵄩
2
󵄩󵄩𝑥 + 𝑦󵄩󵄩󵄩 ≤ ‖𝑥‖ + 2 ⟨𝑦, 𝑥 + 𝑦⟩ ,
∀𝑥, 𝑦 ∈ 𝑋.
Let 𝐴 : 𝐶 → 𝐻 be a nonlinear mapping. The classical
variational inequality problem (VIP) is to find 𝑥 ∈ 𝐶 such
that
∀𝑦 ∈ 𝐶.
(15)
The solution set of VIP is denoted by VI(𝐶, 𝐴).
The theory of variational inequalities is a well-established
subject in applied mathematics, nonlinear analysis, and
optimization. For further details on variational inequalities,
we refer to [8–13] and the references therein.
It is well known that the solution of a variational inequality can be characterized be a fixed point of a projection mapping. Therefore, by using Proposition 4(i), we have the following result.
Lemma 8. Let 𝐴 : 𝐶 → 𝐻 be a monotone mapping. Then,
𝑢 ∈ VI (𝐶, 𝐴) ⇐⇒ 𝑢 = 𝑃𝐶 (𝑢 − 𝜆𝐴𝑢) ,
𝑓𝑜𝑟 𝑠𝑜𝑚𝑒 𝜆 > 0.
(16)
Lemma 9. The following assertions hold:
(a) ‖𝑥 − 𝑦‖2 = ‖𝑥‖2 − ‖𝑦‖2 − 2⟨𝑥 − 𝑦, 𝑦⟩ for all 𝑥, 𝑦 ∈ 𝐻;
(b) ‖𝜆𝑥 + 𝜇𝑦 + ]𝑧‖2 = 𝜆‖𝑥‖2 + 𝜇‖𝑦‖2 + ]‖𝑧‖2 −
𝜆𝜇‖𝑥 − 𝑦‖2 −𝜇]‖𝑦 − 𝑧‖2 −𝜆‖𝑥 − 𝑧‖2 for all 𝑥, 𝑦, 𝑧 ∈ 𝐻
and 𝜆, 𝜇, ] ∈ [0, 1] with 𝜆 + 𝜇 + ] = 1 [14];
(c) if {𝑥𝑛 } is a sequence in 𝐻 such that 𝑥𝑛 ⇀ 𝑥, then
󵄩2
󵄩
󵄩
󵄩2
󵄩2 󵄩
lim sup󵄩󵄩󵄩𝑥𝑛 − 𝑦󵄩󵄩󵄩 = lim sup󵄩󵄩󵄩𝑥𝑛 − 𝑥󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩 ,
𝑛→∞
𝑛→∞
󵄩2
󵄩
{V ∈ 𝐶 : 󵄩󵄩󵄩𝑦 − V󵄩󵄩󵄩 ≤ ‖𝑥 − V‖2 + ⟨𝑧, V⟩ + 𝑎}
(18)
is convex (and closed).
Lemma 11 (see [4, Lemma 2.6]). Let 𝐶 be a nonempty subset
of a Hilbert space 𝐻 and 𝑆 : 𝐶 → 𝐶 an asymptotically
𝜅-strict pseudocontractive mapping in the intermediate sense
with sequence {𝛾𝑛 }. Then,
󵄩󵄩 𝑛
𝑛 󵄩
󵄩󵄩𝑆 𝑥 − 𝑆 𝑦󵄩󵄩󵄩
≤
1
󵄩
󵄩
(𝜅 󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩
1−𝜅
(19)
󵄩2
󵄩
+√(1 + (1 − 𝜅) 𝛾𝑛 ) 󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩 + (1 − 𝜅) 𝑐𝑛 )
for all 𝑥, 𝑦 ∈ 𝐶 and 𝑛 ≥ 1.
(14)
Lemma 7 (see [7, Proposition 2.4]). Let {𝑥𝑛 } be a bounded
sequence in a reflexive Banach space 𝑋. If 𝜔𝑤 ({𝑥𝑛 }) = {𝑥}, then
𝑥𝑛 ⇀ 𝑥.
⟨𝐴𝑥, 𝑦 − 𝑥⟩ ≥ 0,
Lemma 10 (see [4, Lemma 2.5]). For given points 𝑥, 𝑦, 𝑧 ∈ 𝐻
and given also a real number 𝑎 ∈ R, the set
∀𝑦 ∈ 𝐻.
(17)
Lemma 12 (see [4, Lemma 2.7]). Let 𝐶 be a nonempty subset
of a Hilbert space 𝐻 and 𝑆 : 𝐶 → 𝐶 a uniformly continuous
asymptotically 𝜅-strict pseudocontractive mapping in the intermediate sense with sequence {𝛾𝑛 }. Let {𝑥𝑛 } be a sequence in 𝐶
such that ‖𝑥𝑛 − 𝑥𝑛+1 ‖ → 0 and ‖𝑥𝑛 − 𝑆𝑛 𝑥𝑛 ‖ → 0 as 𝑛 → ∞.
Then, ‖𝑥𝑛 − 𝑆𝑥𝑛 ‖ → 0 as 𝑛 → ∞.
Lemma 13 (Demiclosedness Principle, [see [4, Proposition
3.1]]). Let 𝐶 be a nonempty closed convex subset of a Hilbert
space 𝐻 and 𝑆 : 𝐶 → 𝐶 be a continuous asymptotically
𝜅-strict pseudocontractive mapping in the intermediate sense
with sequence {𝛾𝑛 }. Then 𝐼 − 𝑆 is demiclosed at zero in the sense
that if {𝑥𝑛 } is a sequence in 𝐶 such that 𝑥𝑛 ⇀ 𝑥 ∈ 𝐶 and
lim sup𝑚 → ∞ lim sup𝑛 → ∞ ‖𝑥𝑛 − 𝑆𝑚 𝑥𝑛 ‖ = 0, then (𝐼 − 𝑆)𝑥 = 0.
Lemma 14 (see [4, Proposition 3.2]). Let 𝐶 be a nonempty
closed convex subset of a Hilbert space 𝐻 and 𝑆 : 𝐶 → 𝐶 a continuous asymptotically 𝜅-strict pseudocontractive mapping in
the intermediate sense with sequence {𝛾𝑛 } such that Fix(𝑆) ≠ 0.
Then, Fix(𝑆) is closed and convex.
To prove a weak convergence theorem by an implicit
relaxed method with regularization for the minimization
problem (8) and the fixed point problem of an asymptotically 𝜅-strict pseudocontractive mapping in the intermediate
sense, we need the following lemma due to Osilike et al. [15].
∞
∞
Lemma 15 (see [15, page 80]). Let {𝑎𝑛 }∞
𝑛=1 , {𝑏𝑛 }𝑛=1 , and {𝛿𝑛 }𝑛=1
be sequences of nonnegative real numbers satisfying the inequality
𝑎𝑛+1 ≤ (1 + 𝛿𝑛 ) 𝑎𝑛 + 𝑏𝑛 ,
∀𝑛 ≥ 1.
(20)
∞
If ∑∞
𝑛=1 𝛿𝑛 < ∞ and ∑𝑛=1 𝑏𝑛 < ∞, then lim𝑛 → ∞ 𝑎𝑛 exists. If,
∞
in addition, {𝑎𝑛 }𝑛=1 has a subsequence which converges to zero,
then lim𝑛 → ∞ 𝑎𝑛 = 0.
4
Abstract and Applied Analysis
∞
Corollary 16 (see [16, page 303]). Let {𝑎𝑛 }∞
𝑛=0 and {𝑏𝑛 }𝑛=0
be two sequences of nonnegative real numbers satisfying the
inequality
𝑎𝑛+1 ≤ 𝑎𝑛 + 𝑏𝑛 ,
∀𝑛 ≥ 0.
(21)
If ∑∞
𝑛=0 𝑏𝑛 converges, then lim𝑛 → ∞ 𝑎𝑛 exists.
𝑥𝑛 ⇀ 𝑥 󳨐⇒ 𝑥𝑛 󳨀→ 𝑥.
(22)
It is well known that every Hilbert space 𝐻 satisfies Opial’s
condition [18]; that is, for any sequence {𝑥𝑛 } with 𝑥𝑛 ⇀ 𝑥, we
have
󵄩
󵄩
󵄩
󵄩
lim inf 󵄩󵄩󵄩𝑥𝑛 − 𝑥󵄩󵄩󵄩 < lim inf 󵄩󵄩󵄩𝑥𝑛 − 𝑦󵄩󵄩󵄩 ,
𝑛→∞
𝑛→∞
∀𝑦 ∈ 𝐻 with 𝑦 ≠ 𝑥.
(23)
A set-valued mapping 𝑇 : 𝐻 → 2𝐻 is called monotone if
for all 𝑥, 𝑦 ∈ 𝐻, 𝑓 ∈ 𝑇𝑥 and 𝑔 ∈ 𝑇𝑦 imply ⟨𝑥−𝑦, 𝑓−𝑔⟩ ≥ 0. A
monotone mapping 𝑇 : 𝐻 → 2𝐻 is maximal if its graph 𝐺(𝑇)
is not properly contained in the graph of any other monotone
mapping. It is known that a monotone mapping 𝑇 is maximal
if and only if for (𝑥, 𝑓) ∈ 𝐻 × 𝐻, ⟨𝑥 − 𝑦, 𝑓 − 𝑔⟩ ≥ 0 for
all (𝑦, 𝑔) ∈ 𝐺(𝑇) implies 𝑓 ∈ 𝑇𝑥. Let 𝐴 : 𝐶 → 𝐻 be a
monotone and 𝐿-Lipschitz continuous mapping, and let 𝑁𝐶V
be the normal cone to 𝐶 at V ∈ 𝐶, that is, 𝑁𝐶V = {𝑤 ∈ 𝐻 :
⟨V − 𝑢, 𝑤⟩ ≥ 0, ∀ 𝑢 ∈ 𝐶}. Define
𝑇V = {
𝐴V + 𝑁𝐶V, if V ∈ 𝐶,
0,
if V ∉ 𝐶.
𝑥1 = 𝑥 ∈ 𝐶 𝑐ℎ𝑜𝑠𝑒𝑛 𝑎𝑟𝑏𝑖𝑡𝑟𝑎𝑟𝑦,
𝑦𝑛 = 𝑃𝐶 (𝑥𝑛 − 𝜆 𝑛 𝜇𝑛 ∇𝑓𝛼𝑛 (𝑥𝑛 ) − 𝜆 𝑛 (1 − 𝜇𝑛 ) ∇𝑓𝛼𝑛 (𝑦𝑛 )) ,
𝑡𝑛 = 𝑃𝐶 (𝑥𝑛 − 𝜆 𝑛 ∇𝑓𝛼𝑛 (𝑦𝑛 ) − 𝜆 𝑛 (1 − 𝜇𝑛 ) ∇𝑓𝛼𝑛 (𝑡𝑛 )) ,
Lemma 17 (see [17]). Every Hilbert space 𝐻 has the KadecKlee property; that is, given a sequence {𝑥𝑛 } ⊂ 𝐻 and a point
𝑥 ∈ 𝐻, we have
󵄩󵄩 󵄩󵄩
󵄩󵄩𝑥𝑛 󵄩󵄩 󳨀→ ‖𝑥‖ ,
and {𝑐𝑛 } be defined as in Definition 3. Let {𝑥𝑛 } and {𝑦𝑛 } be the
sequences generated by
(24)
It is known that in this case 𝑇 is maximal monotone, and 0 ∈
𝑇V if and only if V ∈ Ω; see [19].
3. Weak Convergence Theorem
In this section, we will prove a weak convergence theorem for an implicit relaxed method with regularization for
finding a common element of the set of fixed points of
an asymptotically 𝜅-strict pseudocontractive mapping 𝑆 :
𝐶 → 𝐶 in the intermediate sense and the set of solutions
of the minimization problem (8) for a convex functional
𝑓 : 𝐶 → R with 𝐿-Lipschitz continuous gradient ∇𝑓. This
implicit relaxed method with regularization is based on the
extragradient method, viscosity approximation method, and
gradient projection algorithm (GPA) with regularization.
Theorem 18. Let 𝐶 be a nonempty bounded closed convex
subset of a real Hilbert space 𝐻. Let 𝑓 : 𝐶 → R be a
convex functional with 𝐿-Lipschitz continuous gradient ∇𝑓,
𝑄 : 𝐶 → 𝐶 a 𝜌-contraction with coefficient 𝜌 ∈ [0, 1),
and 𝑆 : 𝐶 → 𝐶 a uniformly continuous asymptotically 𝜅strict pseudocontractive mapping in the intermediate sense
with sequence {𝛾𝑛 } such that Fix(𝑆) ∩ Γ is nonempty. Let {𝛾𝑛 }
𝑥𝑛+1 = (1 − 𝜎𝑛 − 𝛽𝑛 ) 𝑡𝑛 + 𝜎𝑛 𝑄𝑡𝑛 + 𝛽𝑛 𝑆𝑛 𝑡𝑛 ,
∀𝑛 ≥ 1,
(25)
where {𝛼𝑛 } is a sequence (0, ∞), {𝜆 𝑛 }, {𝜇𝑛 } are sequences in
(0, 1], and {𝜎𝑛 }, {𝛽𝑛 } are sequences in [0, 1] satisfying the
following conditions:
(i) 𝛿 ≤ 𝛽𝑛 and 𝜎𝑛 + 𝛽𝑛 ≤ 1 − 𝜅 − 𝜏, for all 𝑛 ≥ 1 for some
𝛿, 𝜏 ∈ (0, 1);
∞
∞
(ii) ∑∞
𝑛=1 𝛼𝑛 < ∞, ∑𝑛=1 𝛾𝑛 < ∞, ∑𝑛=1 𝜎𝑛 < ∞ and
∞
∑𝑛=1 𝑐𝑛 < ∞;
(iii) lim𝑛 → ∞ 𝜇𝑛 = 1;
(iv) 𝜆 𝑛 (𝛼𝑛 + 𝐿) < 1, for all 𝑛 ≥ 1 and {𝜆 𝑛 } ⊂ [𝑎, 𝑏] for some
𝑎, 𝑏 ∈ (0, (1/𝐿)).
Then, the sequences {𝑥𝑛 }, {𝑦𝑛 } converge weakly to some point
𝑢 ∈ Fix(𝑆) ∩ Γ.
Remark 19. Observe that for all 𝑥, 𝑦 ∈ 𝐶 and all 𝑛 ≥ 1
󵄩󵄩󵄩𝑃 (𝑥 − 𝜆 𝜇 ∇𝑓 (𝑥 ) − 𝜆 (1 − 𝜇 ) ∇𝑓 (𝑥))
󵄩󵄩 𝐶 𝑛
𝑛 𝑛 𝛼𝑛
𝑛
𝑛
𝑛
𝛼𝑛
󵄩
−𝑃𝐶 (𝑥𝑛 − 𝜆 𝑛 𝜇𝑛 ∇𝑓𝛼𝑛 (𝑥𝑛 ) − 𝜆 𝑛 (1 − 𝜇𝑛 ) ∇𝑓𝛼𝑛 (𝑦))󵄩󵄩󵄩󵄩
󵄩
≤ 󵄩󵄩󵄩󵄩(𝑥𝑛 − 𝜆 𝑛 𝜇𝑛 ∇𝑓𝛼𝑛 (𝑥𝑛 ) − 𝜆 𝑛 (1 − 𝜇𝑛 ) ∇𝑓𝛼𝑛 (𝑥))
󵄩
− (𝑥𝑛 − 𝜆 𝑛 𝜇𝑛 ∇𝑓𝛼𝑛 (𝑥𝑛 ) − 𝜆 𝑛 (1 − 𝜇𝑛 ) ∇𝑓𝛼𝑛 (𝑦))󵄩󵄩󵄩󵄩
󵄩
󵄩
= 𝜆 𝑛 (1 − 𝜇𝑛 ) 󵄩󵄩󵄩󵄩∇𝑓𝛼𝑛 (𝑥) − ∇𝑓𝛼𝑛 (𝑦)󵄩󵄩󵄩󵄩
󵄩
󵄩
≤ 𝜆 𝑛 (𝛼𝑛 + 𝐿) 󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩 .
(26)
By Banach contraction principle, we know that for each 𝑛 ≥ 1,
there exists a unique 𝑦𝑛 ∈ 𝐶 such that
𝑦𝑛 = 𝑃𝐶 (𝑥𝑛 − 𝜆 𝑛 𝜇𝑛 ∇𝑓𝛼𝑛 (𝑥𝑛 ) − 𝜆 𝑛 (1 − 𝜇𝑛 ) ∇𝑓𝛼𝑛 (𝑦𝑛 )) .
(27)
Also, observe that for all 𝑥, 𝑦 ∈ 𝐶 and all 𝑛 ≥ 1
󵄩󵄩
󵄩󵄩𝑃𝐶 (𝑥𝑛 − 𝜆 𝑛 ∇𝑓𝛼𝑛 (𝑦𝑛 ) − 𝜆 𝑛 (1 − 𝜇𝑛 ) ∇𝑓𝛼𝑛 (𝑥))
󵄩
󵄩
−𝑃𝐶 (𝑥𝑛 − 𝜆 𝑛 ∇𝑓𝛼𝑛 (𝑦𝑛 ) − 𝜆 𝑛 (1 − 𝜇𝑛 ) ∇𝑓𝛼𝑛 (𝑦))󵄩󵄩󵄩󵄩
󵄩
≤ 󵄩󵄩󵄩󵄩(𝑥𝑛 − 𝜆 𝑛 ∇𝑓𝛼𝑛 (𝑦𝑛 ) − 𝜆 𝑛 (1 − 𝜇𝑛 ) ∇𝑓𝛼𝑛 (𝑥))
󵄩
− (𝑥𝑛 − 𝜆 𝑛 ∇𝑓𝛼𝑛 (𝑦𝑛 ) − 𝜆 𝑛 (1 − 𝜇𝑛 ) ∇𝑓𝛼𝑛 (𝑦))󵄩󵄩󵄩󵄩
󵄩
󵄩
= 𝜆 𝑛 (1 − 𝜇𝑛 ) 󵄩󵄩󵄩󵄩∇𝑓𝛼𝑛 (𝑥) − ∇𝑓𝛼𝑛 (𝑦)󵄩󵄩󵄩󵄩
󵄩
󵄩
≤ 𝜆 𝑛 (𝛼𝑛 + 𝐿) 󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩 .
(28)
Abstract and Applied Analysis
5
Utilizing Banach contraction principle, for each 𝑛 ≥ 1, there
exists a unique 𝑡𝑛 ∈ 𝐶 such that
𝑡𝑛 = 𝑃𝐶 (𝑥𝑛 − 𝜆 𝑛 ∇𝑓𝛼𝑛 (𝑦𝑛 ) − 𝜆 𝑛 (1 − 𝜇𝑛 ) ∇𝑓𝛼𝑛 (𝑡𝑛 )) .
(29)
Proof of Theorem 18. Note that the 𝐿-Lipschitz continuity of
the gradient ∇𝑓 implies that ∇𝑓 is (1/𝐿)-ism [20], that is,
1󵄩
󵄩2
⟨∇𝑓 (𝑥) − ∇𝑓 (𝑦) , 𝑥 − 𝑦⟩ ≥ 󵄩󵄩󵄩∇𝑓 (𝑥) − ∇𝑓 (𝑦)󵄩󵄩󵄩 ,
𝐿
(30)
∀𝑥, 𝑦 ∈ 𝐶.
Thus, ∇𝑓 is monotone and 𝐿-Lipschitz continuous.
We divide the rest of the proof into several steps.
Step 1. lim𝑛 → ∞ ‖𝑥𝑛 − 𝑢‖ exists for each 𝑢 ∈ Fix(𝑆) ∩ Γ.
Indeed, note that 𝑡𝑛 = 𝑃𝐶(𝑥𝑛 − 𝜆 𝑛 ∇𝑓𝛼𝑛 (𝑦𝑛 ) − 𝜆 𝑛 (1 −
𝜇𝑛 )∇𝑓𝛼𝑛 (𝑡𝑛 )) for all 𝑛 ≥ 1. Take 𝑢 ∈ Fix(𝑆) ∩ Γ arbitrarily.
From Proposition 4(ii), monotonicity of ∇𝑓, and 𝑢 ∈ Γ, we
have
󵄩󵄩2
󵄩󵄩
󵄩󵄩𝑡𝑛 − 𝑢󵄩󵄩
󵄩
󵄩2
≤ 󵄩󵄩󵄩󵄩(𝑥𝑛 − 𝜆 𝑛 ∇𝑓𝛼𝑛 (𝑦𝑛 ) − 𝜆 𝑛 (1 − 𝜇𝑛 ) ∇𝑓𝛼𝑛 (𝑡𝑛 )) − 𝑢󵄩󵄩󵄩󵄩
󵄩2
󵄩
− 󵄩󵄩󵄩󵄩(𝑥𝑛 − 𝜆 𝑛 ∇𝑓𝛼𝑛 (𝑦𝑛 ) − 𝜆 𝑛 (1 − 𝜇𝑛 ) ∇𝑓𝛼𝑛 (𝑡𝑛 )) − 𝑡𝑛 󵄩󵄩󵄩󵄩
󵄩
󵄩2
= 󵄩󵄩󵄩󵄩𝑥𝑛 − 𝜆 𝑛 (1 − 𝜇𝑛 ) ∇𝑓𝛼𝑛 (𝑡𝑛 ) − 𝑢󵄩󵄩󵄩󵄩
󵄩2
󵄩
− 󵄩󵄩󵄩󵄩𝑥𝑛 − 𝜆 𝑛 (1 − 𝜇𝑛 ) ∇𝑓𝛼𝑛 (𝑡𝑛 ) − 𝑡𝑛 󵄩󵄩󵄩󵄩
+ 2𝜆 𝑛 ⟨∇𝑓𝛼𝑛 (𝑦𝑛 ) , 𝑢 − 𝑡𝑛 ⟩
󵄩2
󵄩
= 󵄩󵄩󵄩󵄩𝑥𝑛 − 𝜆 𝑛 (1 − 𝜇𝑛 ) ∇𝑓𝛼𝑛 (𝑡𝑛 ) − 𝑢󵄩󵄩󵄩󵄩
󵄩2
󵄩
− 󵄩󵄩󵄩󵄩𝑥𝑛 − 𝜆 𝑛 (1 − 𝜇𝑛 ) ∇𝑓𝛼𝑛 (𝑡𝑛 ) − 𝑡𝑛 󵄩󵄩󵄩󵄩
+ 2𝜆 𝑛 (⟨∇𝑓𝛼𝑛 (𝑦𝑛 ) , 𝑢 − 𝑦𝑛 ⟩ + ⟨∇𝑓𝛼𝑛 (𝑦𝑛 ) , 𝑦𝑛 − 𝑡𝑛 ⟩)
󵄩2
󵄩
= 󵄩󵄩󵄩󵄩𝑥𝑛 − 𝜆 𝑛 (1 − 𝜇𝑛 ) ∇𝑓𝛼𝑛 (𝑡𝑛 ) − 𝑢󵄩󵄩󵄩󵄩
󵄩2
󵄩
− 󵄩󵄩󵄩󵄩𝑥𝑛 − 𝜆 𝑛 (1 − 𝜇𝑛 ) ∇𝑓𝛼𝑛 (𝑡𝑛 ) − 𝑡𝑛 󵄩󵄩󵄩󵄩
+ 2𝜆 𝑛 (⟨∇𝑓𝛼𝑛 (𝑦𝑛 ) − ∇𝑓𝛼𝑛 (𝑢) , 𝑢 − 𝑦𝑛 ⟩
+ ⟨∇𝑓𝛼𝑛 (𝑢) , 𝑢 − 𝑦𝑛 ⟩ + ⟨∇𝑓𝛼𝑛 (𝑦𝑛 ) , 𝑦𝑛 − 𝑡𝑛 ⟩)
󵄩2
󵄩
≤ 󵄩󵄩󵄩󵄩𝑥𝑛 − 𝜆 𝑛 (1 − 𝜇𝑛 ) ∇𝑓𝛼𝑛 (𝑡𝑛 ) − 𝑢󵄩󵄩󵄩󵄩
󵄩2
󵄩
− 󵄩󵄩󵄩󵄩𝑥𝑛 − 𝜆 𝑛 (1 − 𝜇𝑛 ) ∇𝑓𝛼𝑛 (𝑡𝑛 ) − 𝑡𝑛 󵄩󵄩󵄩󵄩
+ 2𝜆 𝑛 (𝛼𝑛 ⟨𝑢, 𝑢 − 𝑦𝑛 ⟩ + ⟨∇𝑓𝛼𝑛 (𝑦𝑛 ) , 𝑦𝑛 − 𝑡𝑛 ⟩)
󵄩2
󵄩2 󵄩
󵄩
= 󵄩󵄩󵄩𝑥𝑛 − 𝑢󵄩󵄩󵄩 − 󵄩󵄩󵄩𝑥𝑛 − 𝑡𝑛 󵄩󵄩󵄩
− 2𝜆 𝑛 (1 − 𝜇𝑛 ) ⟨∇𝑓𝛼𝑛 (𝑡𝑛 ) , 𝑡𝑛 − 𝑢⟩
+ 2𝜆 𝑛 (𝛼𝑛 ⟨𝑢, 𝑢 − 𝑦𝑛 ⟩ + ⟨∇𝑓𝛼𝑛 (𝑦𝑛 ) , 𝑦𝑛 − 𝑡𝑛 ⟩)
󵄩2
󵄩2 󵄩
󵄩
= 󵄩󵄩󵄩𝑥𝑛 − 𝑢󵄩󵄩󵄩 − 󵄩󵄩󵄩𝑥𝑛 − 𝑦𝑛 󵄩󵄩󵄩
󵄩
󵄩2
− 2 ⟨𝑥𝑛 − 𝑦𝑛 , 𝑦𝑛 − 𝑡𝑛 ⟩ − 󵄩󵄩󵄩𝑦𝑛 − 𝑡𝑛 󵄩󵄩󵄩
+ 2𝜆 𝑛 (𝛼𝑛 ⟨𝑢, 𝑢 − 𝑦𝑛 ⟩ + ⟨∇𝑓𝛼𝑛 (𝑦𝑛 ) , 𝑦𝑛 − 𝑡𝑛 ⟩)
− 2𝜆 𝑛 (1 − 𝜇𝑛 )
× (⟨∇𝑓𝛼𝑛 (𝑡𝑛 ) − ∇𝑓𝛼𝑛 (𝑢) , 𝑡𝑛 − 𝑢⟩ + ⟨∇𝑓𝛼𝑛 (𝑢) , 𝑡𝑛 − 𝑢⟩)
󵄩2 󵄩
󵄩2
󵄩2 󵄩
󵄩
≤ 󵄩󵄩󵄩𝑥𝑛 − 𝑢󵄩󵄩󵄩 − 󵄩󵄩󵄩𝑥𝑛 − 𝑦𝑛 󵄩󵄩󵄩 − 󵄩󵄩󵄩𝑦𝑛 − 𝑡𝑛 󵄩󵄩󵄩
+ 2 ⟨𝑥𝑛 − 𝜆 𝑛 ∇𝑓𝛼𝑛 (𝑦𝑛 ) − 𝑦𝑛 , 𝑡𝑛 − 𝑦𝑛 ⟩
+ 2𝜆 𝑛 𝛼𝑛 (⟨𝑢, 𝑢 − 𝑦𝑛 ⟩ + (1 − 𝜇𝑛 ) ⟨𝑢, 𝑢 − 𝑡𝑛 ⟩)
󵄩2 󵄩
󵄩2
󵄩2 󵄩
󵄩
≤ 󵄩󵄩󵄩𝑥𝑛 − 𝑢󵄩󵄩󵄩 − 󵄩󵄩󵄩𝑥𝑛 − 𝑦𝑛 󵄩󵄩󵄩 − 󵄩󵄩󵄩𝑦𝑛 − 𝑡𝑛 󵄩󵄩󵄩
+ 2 ⟨𝑥𝑛 − 𝜆 𝑛 ∇𝑓𝛼𝑛 (𝑦𝑛 ) − 𝑦𝑛 , 𝑡𝑛 − 𝑦𝑛 ⟩
󵄩
󵄩 󵄩
󵄩
+ 2𝜆 𝑛 𝛼𝑛 ‖𝑢‖ (󵄩󵄩󵄩𝑦𝑛 − 𝑢󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑡𝑛 − 𝑢󵄩󵄩󵄩) .
(31)
Since 𝑦𝑛 = 𝑃𝐶(𝑥𝑛 − 𝜆 𝑛 𝜇𝑛 ∇𝑓𝛼𝑛 (𝑥𝑛 ) − 𝜆 𝑛 (1 − 𝜇𝑛 )∇𝑓𝛼𝑛 (𝑦𝑛 )) and
∇𝑓𝛼𝑛 is (𝛼𝑛 + 𝐿)-Lipschitz continuous, by Proposition 4(i), we
have
⟨𝑥𝑛 − 𝜆 𝑛 ∇𝑓𝛼𝑛 (𝑦𝑛 ) − 𝑦𝑛 , 𝑡𝑛 − 𝑦𝑛 ⟩
= ⟨𝑥𝑛 − 𝜆 𝑛 𝜇𝑛 ∇𝑓𝛼𝑛 (𝑥𝑛 ) − 𝜆 𝑛 (1 − 𝜇𝑛 ) ∇𝑓𝛼𝑛 (𝑦𝑛 )
−𝑦𝑛 , 𝑡𝑛 − 𝑦𝑛 ⟩
+ 𝜆 𝑛 𝜇𝑛 ⟨∇𝑓𝛼𝑛 (𝑥𝑛 ) − ∇𝑓𝛼𝑛 (𝑦𝑛 ) , 𝑡𝑛 − 𝑦𝑛 ⟩
≤ 𝜆 𝑛 𝜇𝑛 ⟨∇𝑓𝛼𝑛 (𝑥𝑛 ) − ∇𝑓𝛼𝑛 (𝑦𝑛 ) , 𝑡𝑛 − 𝑦𝑛 ⟩
󵄩
󵄩󵄩
󵄩
≤ 𝜆 𝑛 𝜇𝑛 󵄩󵄩󵄩󵄩∇𝑓𝛼𝑛 (𝑥𝑛 ) − ∇𝑓𝛼𝑛 (𝑦𝑛 )󵄩󵄩󵄩󵄩 󵄩󵄩󵄩𝑡𝑛 − 𝑦𝑛 󵄩󵄩󵄩
󵄩󵄩
󵄩
󵄩
≤ 𝜆 𝑛 (𝛼𝑛 + 𝐿) 󵄩󵄩󵄩𝑥𝑛 − 𝑦𝑛 󵄩󵄩󵄩 󵄩󵄩󵄩𝑡𝑛 − 𝑦𝑛 󵄩󵄩󵄩 .
(32)
So, we have
󵄩󵄩
󵄩2 󵄩
󵄩2
󵄩2 󵄩
󵄩2 󵄩
󵄩󵄩𝑡𝑛 − 𝑢󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩𝑥𝑛 − 𝑢󵄩󵄩󵄩 − 󵄩󵄩󵄩𝑥𝑛 − 𝑦𝑛 󵄩󵄩󵄩 − 󵄩󵄩󵄩𝑦𝑛 − 𝑡𝑛 󵄩󵄩󵄩
󵄩󵄩
󵄩
󵄩
+ 2𝜆 𝑛 (𝛼𝑛 + 𝐿) 󵄩󵄩󵄩𝑥𝑛 − 𝑦𝑛 󵄩󵄩󵄩 󵄩󵄩󵄩𝑡𝑛 − 𝑦𝑛 󵄩󵄩󵄩
󵄩
󵄩 󵄩
󵄩
+ 2𝜆 𝑛 𝛼𝑛 ‖𝑢‖ (󵄩󵄩󵄩𝑦𝑛 − 𝑢󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑡𝑛 − 𝑢󵄩󵄩󵄩)
󵄩
󵄩2 󵄩
󵄩2
󵄩2 󵄩
≤ 󵄩󵄩󵄩𝑥𝑛 − 𝑢󵄩󵄩󵄩 − 󵄩󵄩󵄩𝑥𝑛 − 𝑦𝑛 󵄩󵄩󵄩 − 󵄩󵄩󵄩𝑦𝑛 − 𝑡𝑛 󵄩󵄩󵄩
2󵄩
󵄩2 󵄩
󵄩2
+ 𝜆2𝑛 (𝛼𝑛 + 𝐿) 󵄩󵄩󵄩𝑥𝑛 − 𝑦𝑛 󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑡𝑛 − 𝑦𝑛 󵄩󵄩󵄩
󵄩
󵄩 󵄩
󵄩
+ 2𝜆 𝑛 𝛼𝑛 ‖𝑢‖ (󵄩󵄩󵄩𝑦𝑛 − 𝑢󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑡𝑛 − 𝑢󵄩󵄩󵄩)
2
󵄩
󵄩2
󵄩2
󵄩
= 󵄩󵄩󵄩𝑥𝑛 − 𝑢󵄩󵄩󵄩 + (𝜆2𝑛 (𝛼𝑛 + 𝐿) − 1) 󵄩󵄩󵄩𝑥𝑛 − 𝑦𝑛 󵄩󵄩󵄩
󵄩
󵄩 󵄩
󵄩
+ 2𝜆 𝑛 𝛼𝑛 ‖𝑢‖ (󵄩󵄩󵄩𝑦𝑛 − 𝑢󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑡𝑛 − 𝑢󵄩󵄩󵄩)
󵄩
󵄩
󵄩2
󵄩 󵄩
󵄩
≤ 󵄩󵄩󵄩𝑥𝑛 − 𝑢󵄩󵄩󵄩 + 2𝜆 𝑛 𝛼𝑛 ‖𝑢‖ (󵄩󵄩󵄩𝑦𝑛 − 𝑢󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑡𝑛 − 𝑢󵄩󵄩󵄩) .
(33)
6
Abstract and Applied Analysis
Therefore, from (33), 𝑥𝑛+1 = (1 − 𝜎𝑛 − 𝛽𝑛 )𝑡𝑛 + 𝜎𝑛 𝑓(𝑡𝑛 ) + 𝛽𝑛 𝑆𝑛 𝑡𝑛
and 𝑢 = 𝑆𝑢. By Lemma 9(b), we have
Step 2. lim𝑛 → ∞ ‖𝑥𝑛 − 𝑦𝑛 ‖ = 0 and lim𝑛 → ∞ ‖𝑥𝑛 − 𝑡𝑛 ‖ = 0.
Indeed, substituting (33) in (34), we get
󵄩2
󵄩󵄩
󵄩󵄩𝑥𝑛+1 − 𝑢󵄩󵄩󵄩
󵄩󵄩
󵄩2
󵄩󵄩𝑥𝑛+1 − 𝑢󵄩󵄩󵄩
󵄩
󵄩2
= 󵄩󵄩󵄩(1 − 𝜎𝑛 − 𝛽𝑛 ) (𝑡𝑛 − 𝑢) + 𝜎𝑛 (𝑄𝑡𝑛 − 𝑢) + 𝛽𝑛 (𝑆𝑛 𝑡𝑛 − 𝑢)󵄩󵄩󵄩
󵄩
󵄩
󵄩
󵄩2
󵄩2
󵄩2
≤ (1 − 𝜎𝑛 − 𝛽𝑛 ) 󵄩󵄩󵄩𝑡𝑛 − 𝑢󵄩󵄩󵄩 + 𝜎𝑛 󵄩󵄩󵄩𝑄𝑡𝑛 − 𝑢󵄩󵄩󵄩 + 𝛽𝑛 󵄩󵄩󵄩𝑆𝑛 𝑡𝑛 − 𝑢󵄩󵄩󵄩
󵄩
󵄩2
− (1 − 𝜎𝑛 − 𝛽𝑛 ) 𝛽𝑛 󵄩󵄩󵄩𝑡𝑛 − 𝑆𝑛 𝑡𝑛 󵄩󵄩󵄩
2
󵄩2
󵄩
+ (1 + 𝛾𝑛 ) (𝜆2𝑛 (𝛼𝑛 + 𝐿) − 1) 󵄩󵄩󵄩𝑥𝑛 − 𝑦𝑛 󵄩󵄩󵄩
󵄩
󵄩 󵄩
󵄩
+ 2𝛼𝑛 𝜆 𝑛 (1 + 𝛾𝑛 ) ‖𝑢‖ (󵄩󵄩󵄩𝑦𝑛 − 𝑢󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑡𝑛 − 𝑢󵄩󵄩󵄩)
󵄩
󵄩2
− (1 − 𝜎𝑛 − 𝛽𝑛 ) 𝛽𝑛 󵄩󵄩󵄩𝑡𝑛 − 𝑆𝑛 𝑡𝑛 󵄩󵄩󵄩
󵄩
󵄩2
+ 𝜎𝑛 󵄩󵄩󵄩𝑄𝑡𝑛 − 𝑢󵄩󵄩󵄩 + 𝑐𝑛 ,
󵄩
󵄩
󵄩2
󵄩2
= [1 − 𝜎𝑛 − 𝛽𝑛 + 𝛽𝑛 (1 + 𝛾𝑛 )] 󵄩󵄩󵄩𝑡𝑛 − 𝑢󵄩󵄩󵄩 + 𝜎𝑛 󵄩󵄩󵄩𝑄𝑡𝑛 − 𝑢󵄩󵄩󵄩
󵄩2
󵄩
+ 𝛽𝑛 (𝜅 − (1 − 𝜎𝑛 − 𝛽𝑛 )) 󵄩󵄩󵄩𝑡𝑛 − 𝑆𝑛 𝑡𝑛 󵄩󵄩󵄩 + 𝛽𝑛 𝑐𝑛
which hence implies that
2
󵄩2
󵄩
≤ (1 + 𝛾𝑛 ) (𝜆2𝑛 (𝛼𝑛 + 𝐿) − 1) 󵄩󵄩󵄩𝑥𝑛 − 𝑦𝑛 󵄩󵄩󵄩
󵄩2
󵄩
+ 𝛽𝑛 (𝜅 − (1 − 𝜎𝑛 − 𝛽𝑛 )) 󵄩󵄩󵄩𝑡𝑛 − 𝑆𝑛 𝑡𝑛 󵄩󵄩󵄩 + 𝑐𝑛
󵄩
󵄩
󵄩2 󵄩
󵄩2
󵄩2
≤ 󵄩󵄩󵄩𝑥𝑛 − 𝑢󵄩󵄩󵄩 − 󵄩󵄩󵄩𝑥𝑛+1 − 𝑢󵄩󵄩󵄩 + 𝛾𝑛 󵄩󵄩󵄩𝑥𝑛 − 𝑢󵄩󵄩󵄩
(38)
󵄩
󵄩 󵄩
󵄩
+ 2𝛼𝑛 𝜆 𝑛 (1 + 𝛾𝑛 ) ‖𝑢‖ (󵄩󵄩󵄩𝑦𝑛 − 𝑢󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑡𝑛 − 𝑢󵄩󵄩󵄩)
󵄩
󵄩
󵄩2
󵄩2
≤ (1 + 𝛾𝑛 ) 󵄩󵄩󵄩𝑡𝑛 − 𝑢󵄩󵄩󵄩 + 𝜎𝑛 󵄩󵄩󵄩𝑄𝑡𝑛 − 𝑢󵄩󵄩󵄩 + 𝑐𝑛
󵄩
󵄩
󵄩2
󵄩 󵄩
󵄩
≤ (1 + 𝛾𝑛 ) [󵄩󵄩󵄩𝑥𝑛 − 𝑢󵄩󵄩󵄩 + 2𝜆 𝑛 𝛼𝑛 ‖𝑢‖ (󵄩󵄩󵄩𝑦𝑛 − 𝑢󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑡𝑛 − 𝑢󵄩󵄩󵄩)]
󵄩
󵄩2
+ 𝜎𝑛 󵄩󵄩󵄩𝑄𝑡𝑛 − 𝑢󵄩󵄩󵄩 + 𝑐𝑛
󵄩
󵄩2
+ 𝜎𝑛 󵄩󵄩󵄩𝑄𝑡𝑛 − 𝑢󵄩󵄩󵄩 + 𝑐𝑛 .
Since lim𝑛 → ∞ ‖𝑥𝑛 − 𝑢‖ exists, 𝛼𝑛 → 0, 𝛾𝑛 → 0, 𝜎𝑛 → 0, and
𝑐𝑛 → 0, from the boundedness of 𝐶, it follows that
󵄩󵄩
lim 󵄩𝑥
𝑛→∞ 󵄩 𝑛
󵄩
󵄩2
= (1 + 𝛾𝑛 ) 󵄩󵄩󵄩𝑥𝑛 − 𝑢󵄩󵄩󵄩 + 2𝛼𝑛 𝜆 𝑛 (1 + 𝛾𝑛 ) ‖𝑢‖
󵄩
− 𝑦𝑛 󵄩󵄩󵄩 = 0.
(39)
Meantime, utilizing the arguments similar to those in (33), we
have
󵄩
󵄩 󵄩
󵄩
󵄩2
󵄩
× (󵄩󵄩󵄩𝑦𝑛 − 𝑢󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑡𝑛 − 𝑢󵄩󵄩󵄩) + 𝜎𝑛 󵄩󵄩󵄩𝑄𝑡𝑛 − 𝑢󵄩󵄩󵄩 + 𝑐𝑛 .
(34)
∞
∞
Since ∑∞
𝑛=1 𝛼𝑛 < ∞, ∑𝑛=1 𝛾𝑛 < ∞, ∑𝑛=1 𝜎𝑛 < ∞, and
∞
∑𝑛=1 𝑐𝑛 < ∞, from the boundedness of 𝐶, it follows that
∞
󵄩
󵄩 󵄩
󵄩
∑ {2𝛼𝑛 𝜆 𝑛 (1 + 𝛾𝑛 ) ‖𝑢‖ (󵄩󵄩󵄩𝑦𝑛 − 𝑢󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑡𝑛 − 𝑢󵄩󵄩󵄩)
(35)
󵄩
󵄩2
+𝜎𝑛 󵄩󵄩󵄩𝑄𝑡𝑛 − 𝑢󵄩󵄩󵄩 + 𝑐𝑛 } < ∞.
By Lemma 15, we have
󵄩
− 𝑢󵄩󵄩󵄩 exists for all 𝑢 ∈ Fix (𝑆) ∩ Γ.
(37)
2
󵄩2
󵄩
(1 + 𝛾𝑛 ) (𝑏2 (𝛼𝑛 + 𝐿) − 1) 󵄩󵄩󵄩𝑥𝑛 − 𝑦𝑛 󵄩󵄩󵄩
󵄩
󵄩
󵄩2
󵄩2
≤ (1 + 𝛾𝑛 ) 󵄩󵄩󵄩𝑡𝑛 − 𝑢󵄩󵄩󵄩 + 𝜎𝑛 󵄩󵄩󵄩𝑄𝑡𝑛 − 𝑢󵄩󵄩󵄩
󵄩󵄩
󵄩
󵄩 󵄩
󵄩
+2𝜆 𝑛 𝛼𝑛 ‖𝑢‖ (󵄩󵄩󵄩𝑦𝑛 − 𝑢󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑡𝑛 − 𝑢󵄩󵄩󵄩)]
󵄩
󵄩
󵄩2
󵄩2
= 󵄩󵄩󵄩𝑥𝑛 − 𝑢󵄩󵄩󵄩 + 𝛾𝑛 󵄩󵄩󵄩𝑥𝑛 − 𝑢󵄩󵄩󵄩
󵄩2
󵄩
󵄩2
󵄩
+ 𝛽𝑛 [(1 + 𝛾𝑛 ) 󵄩󵄩󵄩𝑡𝑛 − 𝑢󵄩󵄩󵄩 + 𝜅󵄩󵄩󵄩𝑡𝑛 − 𝑆𝑛 𝑡𝑛 󵄩󵄩󵄩 + 𝑐𝑛 ]
lim 󵄩𝑥
𝑛→∞ 󵄩 𝑛
2
󵄩2
󵄩
󵄩2
󵄩
≤ (1 + 𝛾𝑛 ) [󵄩󵄩󵄩𝑥𝑛 − 𝑢󵄩󵄩󵄩 + (𝜆2𝑛 (𝛼𝑛 + 𝐿) − 1) 󵄩󵄩󵄩𝑥𝑛 − 𝑦𝑛 󵄩󵄩󵄩
󵄩
󵄩2
+ 𝜎𝑛 󵄩󵄩󵄩𝑄𝑡𝑛 − 𝑢󵄩󵄩󵄩 + 𝑐𝑛
󵄩
󵄩
󵄩2
󵄩2
≤ (1 − 𝜎𝑛 − 𝛽𝑛 ) 󵄩󵄩󵄩𝑡𝑛 − 𝑢󵄩󵄩󵄩 + 𝜎𝑛 󵄩󵄩󵄩𝑄𝑡𝑛 − 𝑢󵄩󵄩󵄩
𝑛=1
󵄩
󵄩
󵄩2
󵄩2
≤ (1 + 𝛾𝑛 ) 󵄩󵄩󵄩𝑡𝑛 − 𝑢󵄩󵄩󵄩 + 𝜎𝑛 󵄩󵄩󵄩𝑄𝑡𝑛 − 𝑢󵄩󵄩󵄩 + 𝑐𝑛
(36)
󵄩󵄩
󵄩2 󵄩
󵄩2
󵄩2 󵄩
󵄩2 󵄩
󵄩󵄩𝑡𝑛 − 𝑢󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩𝑥𝑛 − 𝑢󵄩󵄩󵄩 − 󵄩󵄩󵄩𝑥𝑛 − 𝑦𝑛 󵄩󵄩󵄩 − 󵄩󵄩󵄩𝑦𝑛 − 𝑡𝑛 󵄩󵄩󵄩
󵄩󵄩
󵄩
󵄩
+ 2𝜆 𝑛 (𝛼𝑛 + 𝐿) 󵄩󵄩󵄩𝑥𝑛 − 𝑦𝑛 󵄩󵄩󵄩 󵄩󵄩󵄩𝑡𝑛 − 𝑦𝑛 󵄩󵄩󵄩
󵄩
󵄩 󵄩
󵄩
+ 2𝜆 𝑛 𝛼𝑛 ‖𝑢‖ (󵄩󵄩󵄩𝑦𝑛 − 𝑢󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑡𝑛 − 𝑢󵄩󵄩󵄩)
󵄩
󵄩2 󵄩
󵄩2
󵄩2 󵄩
≤ 󵄩󵄩󵄩𝑥𝑛 − 𝑢󵄩󵄩󵄩 − 󵄩󵄩󵄩𝑥𝑛 − 𝑦𝑛 󵄩󵄩󵄩 − 󵄩󵄩󵄩𝑦𝑛 − 𝑡𝑛 󵄩󵄩󵄩
2󵄩
󵄩2 󵄩
󵄩2
+ 𝜆2𝑛 (𝛼𝑛 + 𝐿) 󵄩󵄩󵄩𝑡𝑛 − 𝑦𝑛 󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑥𝑛 − 𝑦𝑛 󵄩󵄩󵄩
󵄩
󵄩 󵄩
󵄩
+ 2𝜆 𝑛 𝛼𝑛 ‖𝑢‖ (󵄩󵄩󵄩𝑦𝑛 − 𝑢󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑡𝑛 − 𝑢󵄩󵄩󵄩)
2
󵄩
󵄩2
󵄩2
󵄩
= 󵄩󵄩󵄩𝑥𝑛 − 𝑢󵄩󵄩󵄩 + (𝜆2𝑛 (𝛼𝑛 + 𝐿) − 1) 󵄩󵄩󵄩𝑡𝑛 − 𝑦𝑛 󵄩󵄩󵄩
󵄩
󵄩 󵄩
󵄩
+ 2𝜆 𝑛 𝛼𝑛 ‖𝑢‖ (󵄩󵄩󵄩𝑦𝑛 − 𝑢󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑡𝑛 − 𝑢󵄩󵄩󵄩) .
(40)
Abstract and Applied Analysis
7
Substituting (40) in (34), we get
which together with condition (i) yields
󵄩󵄩
󵄩2
󵄩󵄩𝑥𝑛+1 − 𝑢󵄩󵄩󵄩
󵄩
󵄩
󵄩2
󵄩2
≤ (1 + 𝛾𝑛 ) 󵄩󵄩󵄩𝑡𝑛 − 𝑢󵄩󵄩󵄩 + 𝜎𝑛 󵄩󵄩󵄩𝑄𝑡𝑛 − 𝑢󵄩󵄩󵄩 + 𝑐𝑛
2
󵄩2
󵄩
󵄩2
󵄩
≤ (1 + 𝛾𝑛 ) [󵄩󵄩󵄩𝑥𝑛 − 𝑢󵄩󵄩󵄩 + (𝜆2𝑛 (𝛼𝑛 + 𝐿) − 1) 󵄩󵄩󵄩𝑡𝑛 − 𝑦𝑛 󵄩󵄩󵄩
󵄩
󵄩
󵄩 󵄩
󵄩
󵄩2
+2𝜆 𝑛 𝛼𝑛 ‖𝑢‖ (󵄩󵄩󵄩𝑦𝑛 − 𝑢󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑡𝑛 − 𝑢󵄩󵄩󵄩)] + 𝜎𝑛 󵄩󵄩󵄩𝑄𝑡𝑛 − 𝑢󵄩󵄩󵄩 + 𝑐𝑛
󵄩
󵄩
󵄩2
󵄩2
= 󵄩󵄩󵄩𝑥𝑛 − 𝑢󵄩󵄩󵄩 + 𝛾𝑛 󵄩󵄩󵄩𝑥𝑛 − 𝑢󵄩󵄩󵄩
2
󵄩2
󵄩
+ (1 + 𝛾𝑛 ) (𝜆2𝑛 (𝛼𝑛 + 𝐿) − 1) 󵄩󵄩󵄩𝑡𝑛 − 𝑦𝑛 󵄩󵄩󵄩
󵄩
󵄩 󵄩
󵄩
+ 2𝛼𝑛 𝜆 𝑛 (1 + 𝛾𝑛 ) ‖𝑢‖ (󵄩󵄩󵄩𝑦𝑛 − 𝑢󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑡𝑛 − 𝑢󵄩󵄩󵄩)
󵄩
󵄩2
+ 𝜎𝑛 󵄩󵄩󵄩𝑄𝑡𝑛 − 𝑢󵄩󵄩󵄩 + 𝑐𝑛 ,
(41)
which hence implies that
󵄩2
󵄩
(1 + 𝛾𝑛 ) (𝑏 (𝛼𝑛 + 𝐿) − 1) 󵄩󵄩󵄩𝑡𝑛 − 𝑦𝑛 󵄩󵄩󵄩
2
2
2
󵄩2
󵄩
≤ (1 + 𝛾𝑛 ) (𝜆2𝑛 (𝛼𝑛 + 𝐿) − 1) 󵄩󵄩󵄩𝑡𝑛 − 𝑦𝑛 󵄩󵄩󵄩
󵄩
󵄩
󵄩2 󵄩
󵄩2
󵄩2
(42)
≤ 󵄩󵄩󵄩𝑥𝑛 − 𝑢󵄩󵄩󵄩 − 󵄩󵄩󵄩𝑥𝑛+1 − 𝑢󵄩󵄩󵄩 + 𝛾𝑛 󵄩󵄩󵄩𝑥𝑛 − 𝑢󵄩󵄩󵄩
󵄩
󵄩 󵄩
󵄩
+ 2𝛼𝑛 𝜆 𝑛 (1 + 𝛾𝑛 ) ‖𝑢‖ (󵄩󵄩󵄩𝑦𝑛 − 𝑢󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑡𝑛 − 𝑢󵄩󵄩󵄩)
󵄩
󵄩2
+ 𝜎𝑛 󵄩󵄩󵄩𝑄𝑡𝑛 − 𝑢󵄩󵄩󵄩 + 𝑐𝑛 .
Since lim𝑛 → ∞ ‖𝑥𝑛 − 𝑢‖ exists, 𝛼𝑛 → 0, 𝛾𝑛 → 0, 𝜎𝑛 → 0, and
𝑐𝑛 → 0, from the boundedness of 𝐶 it follows that
󵄩
󵄩
lim 󵄩󵄩𝑡 − 𝑦𝑛 󵄩󵄩󵄩 = 0.
(43)
𝑛→∞ 󵄩 𝑛
This together with ‖𝑥𝑛 − 𝑦𝑛 ‖ → 0 implies that
󵄩
󵄩
lim 󵄩󵄩𝑥 − 𝑡𝑛 󵄩󵄩󵄩 = 0.
𝑛→∞ 󵄩 𝑛
(44)
󵄩2
󵄩
𝛿𝜏󵄩󵄩󵄩𝑡𝑛 − 𝑆𝑛 𝑡𝑛 󵄩󵄩󵄩
󵄩2
󵄩
≤ 𝛽𝑛 ((1 − 𝜎𝑛 − 𝛽𝑛 ) − 𝜅) 󵄩󵄩󵄩𝑡𝑛 − 𝑆𝑛 𝑡𝑛 󵄩󵄩󵄩
󵄩
󵄩
󵄩2 󵄩
󵄩2
󵄩2
≤ 󵄩󵄩󵄩𝑥𝑛 − 𝑢󵄩󵄩󵄩 − 󵄩󵄩󵄩𝑥𝑛+1 − 𝑢󵄩󵄩󵄩 + 𝛾𝑛 󵄩󵄩󵄩𝑥𝑛 − 𝑢󵄩󵄩󵄩
(46)
󵄩
󵄩 󵄩
󵄩
+ 2𝛼𝑛 𝜆 𝑛 (1 + 𝛾𝑛 ) ‖𝑢‖ (󵄩󵄩󵄩𝑦𝑛 − 𝑢󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑡𝑛 − 𝑢󵄩󵄩󵄩)
󵄩
󵄩2
+ 𝜎𝑛 󵄩󵄩󵄩𝑄𝑡𝑛 − 𝑢󵄩󵄩󵄩 + 𝑐𝑛 .
Since lim𝑛 → ∞ ‖𝑥𝑛 − 𝑢‖ exists, 𝛼𝑛 → 0, 𝛾𝑛 → 0, 𝜎𝑛 → 0, and
𝑐𝑛 → 0, from the boundedness of 𝐶 it follows that
󵄩󵄩
󵄩
− 𝑆𝑛 𝑡𝑛 󵄩󵄩󵄩 = 0.
lim 󵄩𝑡
𝑛→∞ 󵄩 𝑛
Note that
󵄩󵄩󵄩𝑥𝑛+1 − 𝑥𝑛 󵄩󵄩󵄩
󵄩
󵄩
󵄩󵄩
󵄩 󵄩
󵄩
≤ 󵄩󵄩𝑥𝑛+1 − 𝑡𝑛 󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑡𝑛 − 𝑥𝑛 󵄩󵄩󵄩
󵄩
󵄩
󵄩 󵄩
= 󵄩󵄩󵄩𝜎𝑛 (𝑄𝑡𝑛 − 𝑡𝑛 ) + 𝛽𝑛 (𝑆𝑛 𝑡𝑛 − 𝑡𝑛 )󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑡𝑛 − 𝑥𝑛 󵄩󵄩󵄩
󵄩
󵄩 󵄩
󵄩 󵄩
󵄩
≤ 𝜎𝑛 󵄩󵄩󵄩𝑄𝑡𝑛 − 𝑡𝑛 󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑆𝑛 𝑡𝑛 − 𝑡𝑛 󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑡𝑛 − 𝑥𝑛 󵄩󵄩󵄩 .
(47)
(48)
From the boundedness of 𝐶, 𝜎𝑛 → 0, ‖𝑡𝑛 − 𝑥𝑛 ‖ → 0, and
‖𝑆𝑛 𝑡𝑛 − 𝑡𝑛 ‖ → 0, we deduce that
󵄩󵄩
lim 󵄩𝑥
𝑛 → ∞ 󵄩 𝑛+1
󵄩
− 𝑥𝑛 󵄩󵄩󵄩 = 0.
(49)
Furthermore, observe that
󵄩󵄩
󵄩
󵄩 󵄩
𝑛 󵄩
𝑛 󵄩 󵄩 𝑛
𝑛 󵄩
󵄩󵄩𝑥𝑛 − 𝑆 𝑥𝑛 󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩𝑥𝑛 − 𝑡𝑛 󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑡𝑛 − 𝑆 𝑡𝑛 󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑆 𝑡𝑛 − 𝑆 𝑥𝑛 󵄩󵄩󵄩 .
(50)
Utilizing Lemma 11, we have
Step 3. lim𝑛 → ∞ ‖𝑥𝑛+1 − 𝑥𝑛 ‖ = 0 and lim𝑛 → ∞ ‖𝑥𝑛 − 𝑆𝑥𝑛 ‖ = 0.
Indeed, from (33) and (34), we conclude that
󵄩2
󵄩󵄩
󵄩󵄩𝑥𝑛+1 − 𝑢󵄩󵄩󵄩
󵄩
󵄩
󵄩2
󵄩2
≤ (1 + 𝛾𝑛 ) 󵄩󵄩󵄩𝑡𝑛 − 𝑢󵄩󵄩󵄩 + 𝜎𝑛 󵄩󵄩󵄩𝑄𝑡𝑛 − 𝑢󵄩󵄩󵄩
󵄩2
󵄩
+ 𝛽𝑛 (𝜅 − (1 − 𝜎𝑛 − 𝛽𝑛 )) 󵄩󵄩󵄩𝑡𝑛 − 𝑆𝑛 𝑡𝑛 󵄩󵄩󵄩 + 𝑐𝑛
󵄩
󵄩
󵄩2
󵄩 󵄩
󵄩
≤ (1 + 𝛾𝑛 ) [󵄩󵄩󵄩𝑥𝑛 − 𝑢󵄩󵄩󵄩 + 2𝜆 𝑛 𝛼𝑛 ‖𝑢‖ (󵄩󵄩󵄩𝑦𝑛 − 𝑢󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑡𝑛 − 𝑢󵄩󵄩󵄩)]
󵄩
󵄩2
+ 𝜎𝑛 󵄩󵄩󵄩𝑄𝑡𝑛 − 𝑢󵄩󵄩󵄩
󵄩2
󵄩
+ 𝛽𝑛 (𝜅 − (1 − 𝜎𝑛 − 𝛽𝑛 )) 󵄩󵄩󵄩𝑡𝑛 − 𝑆𝑛 𝑡𝑛 󵄩󵄩󵄩 + 𝑐𝑛
󵄩
󵄩
󵄩2
󵄩2
= 󵄩󵄩󵄩𝑥𝑛 − 𝑢󵄩󵄩󵄩 + 𝛾𝑛 󵄩󵄩󵄩𝑥𝑛 − 𝑢󵄩󵄩󵄩
󵄩
󵄩 󵄩
󵄩
+ 2𝛼𝑛 𝜆 𝑛 (1 + 𝛾𝑛 ) ‖𝑢‖ (󵄩󵄩󵄩𝑦𝑛 − 𝑢󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑡𝑛 − 𝑢󵄩󵄩󵄩)
󵄩
󵄩2
󵄩2
󵄩
+ 𝜎𝑛 󵄩󵄩󵄩𝑄𝑡𝑛 − 𝑢󵄩󵄩󵄩 − 𝛽𝑛 ((1 − 𝜎𝑛 − 𝛽𝑛 ) − 𝜅) 󵄩󵄩󵄩𝑡𝑛 − 𝑆𝑛 𝑡𝑛 󵄩󵄩󵄩
+ 𝑐𝑛 ,
(45)
󵄩󵄩 𝑛
𝑛 󵄩
󵄩󵄩𝑆 𝑡𝑛 − 𝑆 𝑥𝑛 󵄩󵄩󵄩
≤
1
1−𝜅
󵄩
󵄩
× (𝜅 󵄩󵄩󵄩𝑡𝑛 − 𝑥𝑛 󵄩󵄩󵄩
󵄩2
󵄩
+√(1 + (1 − 𝜅) 𝛾𝑛 ) 󵄩󵄩󵄩𝑡𝑛 − 𝑥𝑛 󵄩󵄩󵄩 + (1 − 𝜅) 𝑐𝑛 )
(51)
for every 𝑛 = 1, 2, . . . . Hence, it follows from ‖𝑥𝑛 − 𝑡𝑛 ‖ → 0
that ‖𝑆𝑛 𝑡𝑛 −𝑆𝑛 𝑥𝑛 ‖ → 0. Thus, from (50) and ‖𝑡𝑛 −𝑆𝑛 𝑡𝑛 ‖ → 0,
we get ‖𝑥𝑛 − 𝑆𝑛 𝑥𝑛 ‖ → 0. Since ‖𝑥𝑛+1 − 𝑥𝑛 ‖ → 0, ‖𝑥𝑛 −
𝑆𝑛 𝑥𝑛 ‖ → 0 as 𝑛 → ∞, and 𝑆 is uniformly continuous, we
obtain from Lemma 12 that ‖𝑥𝑛 − 𝑆𝑥𝑛 ‖ → 0 as 𝑛 → ∞.
Step 4. 𝜔𝑤 (𝑥𝑛 ) ⊂ Fix(𝑆) ∩ Γ.
Indeed, from the boundedness of {𝑥𝑛 }, we know that
𝜔𝑤 ({𝑥𝑛 }) ≠ 0. Take 𝑢̂ ∈ 𝜔𝑤 ({𝑥𝑛 }) arbitrarily. Then, there exists
a subsequence {𝑥𝑛𝑖 } of {𝑥𝑛 } such that {𝑥𝑛𝑖 } converges weakly
to 𝑢̂. Note that 𝑆 is uniformly continuous and ‖𝑥𝑛 −𝑆𝑥𝑛 ‖ → 0.
8
Abstract and Applied Analysis
Hence, it is easy to see that ‖𝑥𝑛 − 𝑆𝑚 𝑥𝑛 ‖ → 0 for all 𝑚 ≥ 1.
By Lemma 13, we obtain 𝑢̂ ∈ Fix(𝑆). Furthermore, we show
𝑢̂ ∈ Γ. Since 𝑥𝑛 − 𝑡𝑛 → 0 and 𝑥𝑛 − 𝑦𝑛 → 0, we have 𝑡𝑛𝑖 ⇀ 𝑢̂
and 𝑦𝑛𝑖 ⇀ 𝑢̂. Let
𝑇V = {
∇𝑓 (V) + 𝑁𝐶V,
0,
if V ∈ 𝐶,
if V ∉ 𝐶,
(52)
where 𝑁𝐶V is the normal cone to 𝐶 at V ∈ 𝐶. We have
already mentioned that in this case the mapping 𝑇 is maximal
monotone, and 0 ∈ 𝑇V if and only if V ∈ VI(𝐶, ∇𝑓); see [19] for
more details. Let 𝐺(𝑇) be the graph of 𝑇 and let (V, 𝑤) ∈ 𝐺(𝑇).
Then, we have 𝑤 ∈ 𝑇V = ∇𝑓(V) + 𝑁𝐶V and hence 𝑤 − ∇𝑓(V) ∈
𝑁𝐶V. So, we have ⟨V − 𝑡, 𝑤 − ∇𝑓(V)⟩ ≥ 0 for all 𝑡 ∈ 𝐶. On the
other hand, from 𝑡𝑛 = 𝑃𝐶(𝑥𝑛 −𝜆 𝑛 ∇𝑓𝛼𝑛 (𝑦𝑛 )−𝜆 𝑛 (1−𝜇𝑛 )∇𝑓𝛼𝑛 (𝑡𝑛 ))
and V ∈ 𝐶, we have
⟨𝑥𝑛 − 𝜆 𝑛 ∇𝑓𝛼𝑛 (𝑦𝑛 ) − 𝜆 𝑛 (1 − 𝜇𝑛 ) ∇𝑓𝛼𝑛 (𝑡𝑛 ) − 𝑡𝑛 , 𝑡𝑛 − V⟩ ≥ 0,
(53)
and hence
𝑡 − 𝑥𝑛
+ ∇𝑓𝛼𝑛 (𝑦𝑛 ) + (1 − 𝜇𝑛 ) ∇𝑓𝛼𝑛 (𝑡𝑛 )⟩ ≥ 0.
⟨V − 𝑡𝑛 , 𝑛
𝜆𝑛
(54)
Therefore, from ⟨V−𝑡, 𝑤−∇𝑓(V)⟩ ≥ 0 for all 𝑡 ∈ 𝐶 and 𝑡𝑛𝑖 ∈ 𝐶,
we have
Since ‖∇𝑓(𝑡𝑛 )−∇𝑓(𝑦𝑛 )‖ → 0 (due to the Lipschitz continuity
of ∇𝑓), (𝑡𝑛 − 𝑥𝑛 )/(𝜆 𝑛 ) → 0 (due to {𝜆 𝑛 } ⊂ [𝑎, 𝑏]), 𝛼𝑛 → 0,
and 𝜇𝑛 → 1, we obtain ⟨V − 𝑢̂, 𝑤⟩ ≥ 0 as 𝑖 → ∞. Since
𝑇 is maximal monotone, we have 𝑢̂ ∈ 𝑇−1 0 and hence 𝑢̂ ∈
VI(𝐶, ∇𝑓). Clearly, 𝑢̂ ∈ Γ. Consequently, 𝑢̂ ∈ Fix(𝑆) ∩ Γ. This
implies that 𝜔𝑤 ({𝑥𝑛 }) ⊂ Fix(𝑆) ∩ Γ.
Step 5. {𝑥𝑛 } and {𝑦𝑛 } converge weakly to the same point 𝑢 ∈
Fix(𝑆) ∩ Γ.
Indeed, it is sufficient to show that 𝜔𝑤 (𝑥𝑛 ) is a single-point
set because 𝑥𝑛 − 𝑦𝑛 → 0 as 𝑛 → ∞. Since 𝜔𝑤 (𝑥𝑛 ) ≠ 0, let
us take two points 𝑢, 𝑢̂ ∈ 𝜔𝑤 (𝑥𝑛 ) arbitrarily. Then, there exist
two subsequences {𝑥𝑛𝑗 } and {𝑥𝑚𝑘 } of {𝑥𝑛 } such that 𝑥𝑛𝑗 ⇀ 𝑢
and 𝑥𝑚𝑘 ⇀ 𝑢̂, respectively. In terms of Step 4, we know that
𝑢, 𝑢̂ ∈ Fix(𝑆)∩Γ. Meantime, according to Step 1, we also know
that there exist both lim𝑛 → ∞ ‖𝑥𝑛 − 𝑢‖ and lim𝑛 → ∞ ‖𝑥𝑛 − 𝑢̂‖.
Let us show that 𝑢 = 𝑢̂. Assume that 𝑢 ≠ 𝑢̂. From the Opial
condition [18], it follows that
󵄩󵄩
lim 󵄩𝑥
𝑛→∞ 󵄩 𝑛
󵄩󵄩
󵄩󵄩
󵄩󵄩
󵄩󵄩
󵄩
− 𝑢󵄩󵄩󵄩 = lim inf 󵄩󵄩󵄩𝑥𝑛𝑗 − 𝑢󵄩󵄩󵄩 < lim inf 󵄩󵄩󵄩𝑥𝑛𝑗 − 𝑢̂󵄩󵄩󵄩
󵄩
󵄩
𝑗→∞ 󵄩
𝑗→∞ 󵄩
󵄩
󵄩
󵄩
󵄩
= lim 󵄩󵄩󵄩𝑥𝑛 − 𝑢̂󵄩󵄩󵄩 = lim inf 󵄩󵄩󵄩󵄩𝑥𝑚𝑘 − 𝑢̂󵄩󵄩󵄩󵄩
𝑛→∞
𝑘→∞
󵄩
󵄩
󵄩
󵄩
< lim inf 󵄩󵄩󵄩󵄩𝑥𝑚𝑘 − 𝑢󵄩󵄩󵄩󵄩 = lim 󵄩󵄩󵄩𝑥𝑛 − 𝑢󵄩󵄩󵄩 .
𝑛→∞
𝑘→∞
(56)
⟨V − 𝑡𝑛𝑖 , 𝑤⟩
This leads to a contradiction. Thus, we must have 𝑢 = 𝑢̂.
This implies that 𝜔𝑤 (𝑥𝑛 ) is a single-point set. Without loss
of generality, we may write 𝜔𝑤 (𝑥𝑛 ) = {𝑢}. Consequently,
by Lemma 7, we obtain that 𝑥𝑛 ⇀ 𝑢 ∈ Fix(𝑆) ∩ Γ. Since
𝑥𝑛 − 𝑦𝑛 → 0 as 𝑛 → ∞, we also have
≥ ⟨V − 𝑡𝑛𝑖 , ∇𝑓 (V)⟩
≥ ⟨V − 𝑡𝑛𝑖 , ∇𝑓 (V)⟩
− ⟨V − 𝑡𝑛𝑖 ,
𝑡𝑛𝑖 − 𝑥𝑛𝑖
𝜆 𝑛𝑖
+ ∇𝑓𝛼𝑛 (𝑦𝑛𝑖 )
𝑖
𝑦𝑛 ⇀ 𝑢 ∈ Fix (𝑆) ∩ Γ.
+ (1 − 𝜇𝑛𝑖 ) ∇𝑓𝛼𝑛 (𝑡𝑛𝑖 ) ⟩
𝑖
= ⟨V − 𝑡𝑛𝑖 , ∇𝑓 (V)⟩ − ⟨V − 𝑡𝑛𝑖 ,
This completes the proof.
𝑡𝑛𝑖 − 𝑥𝑛𝑖
𝜆 𝑛𝑖
+ ∇𝑓 (𝑦𝑛𝑖 )⟩
− 𝛼𝑛𝑖 ⟨V − 𝑡𝑛𝑖 , 𝑦𝑛𝑖 ⟩ − (1 − 𝜇𝑛𝑖 ) ⟨V − 𝑡𝑛𝑖 , ∇𝑓𝛼𝑛 (𝑡𝑛𝑖 )⟩
𝑖
= ⟨V − 𝑡𝑛𝑖 , ∇𝑓 (V) − ∇𝑓 (𝑡𝑛𝑖 )⟩
+ ⟨V − 𝑡𝑛𝑖 , ∇𝑓 (𝑡𝑛𝑖 ) − ∇𝑓 (𝑦𝑛𝑖 )⟩
− ⟨V − 𝑡𝑛𝑖 ,
𝑡𝑛𝑖 − 𝑥𝑛𝑖
𝜆 𝑛𝑖
(57)
⟩ − 𝛼𝑛𝑖 ⟨V − 𝑡𝑛𝑖 , 𝑦𝑛𝑖 ⟩
− (1 − 𝜇𝑛𝑖 ) ⟨V − 𝑡𝑛𝑖 , ∇𝑓𝛼𝑛 (𝑡𝑛𝑖 )⟩
𝑖
≥ ⟨V − 𝑡𝑛𝑖 , ∇𝑓 (𝑡𝑛𝑖 ) − ∇𝑓 (𝑦𝑛𝑖 )⟩ − ⟨V − 𝑡𝑛𝑖 ,
𝑡𝑛𝑖 − 𝑥𝑛𝑖
𝜆 𝑛𝑖
⟩
− 𝛼𝑛𝑖 ⟨V − 𝑡𝑛𝑖 , 𝑦𝑛𝑖 ⟩ − (1 − 𝜇𝑛𝑖 ) ⟨V − 𝑡𝑛𝑖 , ∇𝑓𝛼𝑛 (𝑡𝑛𝑖 )⟩ .
𝑖
(55)
4. Strong Convergence Theorem
In this section, we establish a strong convergence theorem for
an implicit hybrid method with regularization for finding a
common element of the set of fixed points of an asymptotically 𝜅-strict pseudocontractive mapping 𝑆 : 𝐶 → 𝐶 in the
intermediate sense and the set of solutions of the MP (8) for
a convex functional 𝑓 : 𝐶 → R with 𝐿-Lipschitz continuous
gradient ∇𝑓. This implicit hybrid method with regularization
is based on the CQ method, extragradient method, viscosity
approximation method, and gradient projection algorithm
(GPA) with regularization.
Theorem 20. Let 𝐶 be a nonempty closed convex subset of
a real Hilbert space 𝐻. Let 𝑓 : 𝐶 → R be a convex
functional with 𝐿-Lipschitz continuous gradient ∇𝑓, 𝑄 :
𝐶 → 𝐶 a 𝜌-contraction with coefficient 𝜌 ∈ [0, 1), and
𝑆 : 𝐶 → 𝐶 a uniformly continuous asymptotically 𝜅-strict
pseudocontractive mapping in the intermediate sense with
sequence {𝛾𝑛 } such that Fix(𝑆) ∩ Γ is nonempty and bounded.
Abstract and Applied Analysis
9
Let {𝛾𝑛 } and {𝑐𝑛 } be defined as in Definition 3. Let {𝑥𝑛 }, {𝑦𝑛 },
and {𝑧𝑛 } be the sequences generated by
𝑥1 = 𝑥 ∈ 𝐶 𝑐ℎ𝑜𝑠𝑒𝑛 𝑎𝑟𝑏𝑖𝑡𝑟𝑎𝑟𝑦,
𝑦𝑛 = 𝑃𝐶 (𝑥𝑛 − 𝜆 𝑛 𝜇𝑛 ∇𝑓𝛼𝑛 (𝑥𝑛 ) − 𝜆 𝑛 (1 − 𝜇𝑛 ) ∇𝑓𝛼𝑛 (𝑦𝑛 )) ,
𝑡𝑛 = 𝑃𝐶 (𝑥𝑛 − 𝜆 𝑛 ∇𝑓𝛼𝑛 (𝑦𝑛 ) − 𝜆 𝑛 (1 − 𝜇𝑛 ) ∇𝑓𝛼𝑛 (𝑡𝑛 )) ,
󵄩 󵄩2 󵄩 󵄩2
⟨2 (𝑥𝑛 − 𝑧𝑛 ) , 𝑧⟩ ≤ 󵄩󵄩󵄩𝑥𝑛 󵄩󵄩󵄩 − 󵄩󵄩󵄩𝑧𝑛 󵄩󵄩󵄩 + 𝜃𝑛 .
𝑧𝑛 = (1 − 𝜎𝑛 − 𝛽𝑛 ) 𝑡𝑛 + 𝜎𝑛 𝑄𝑡𝑛 + 𝛽𝑛 𝑆𝑛 𝑡𝑛 ,
󵄩
󵄩2 󵄩
󵄩2
𝐶𝑛 = {𝑧 ∈ 𝐶 : 󵄩󵄩󵄩𝑧𝑛 − 𝑧󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩𝑥𝑛 − 𝑧󵄩󵄩󵄩 + 𝜃𝑛 } ,
𝑄𝑛 = {𝑧 ∈ 𝐶 : ⟨𝑥𝑛 − 𝑧, 𝑥 − 𝑥𝑛 ⟩ ≥ 0} ,
𝑥𝑛+1 = 𝑃𝐶𝑛 ∩𝑄𝑛 𝑥,
∀𝑛 ≥ 1,
(58)
where {𝛼𝑛 } ⊂ (0, ∞), {𝜆 𝑛 }, {𝜇𝑛 } ⊂ (0, 1], {𝜎𝑛 }, {𝛽𝑛 } ⊂ [0, 1],
𝜃𝑛 = 𝑐𝑛 + (𝛼𝑛 + 𝛾𝑛 + 𝜎𝑛 )Δ 𝑛 and
Δ𝑛 =
sup
𝑧∈Fix(𝑆)∩Γ
{
Hence, it follows that ∇𝑓𝛼 = 𝛼𝐼 + ∇𝑓 is (1/(𝛼 + 𝐿))-ism. It is
clear that 𝐶𝑛 is closed and 𝑄𝑛 is closed and convex for every
𝑛 = 1, 2, . . . . As the defining inequality in 𝐶𝑛 is equivalent to
the inequality
2
33 󵄩󵄩
32 2
−1
󵄩2
󵄩󵄩𝑥𝑛 − 𝑧󵄩󵄩󵄩 + [((1 − 𝜌) + 𝛾𝑛 ) + 4 𝛼𝑛 ]
4
𝜇𝑛
𝜇𝑛
(62)
By Lemma 10, we also have that 𝐶𝑛 is convex for every 𝑛 =
1, 2, . . . . As 𝑄𝑛 = {𝑧 ∈ 𝐶 : ⟨𝑥𝑛 − 𝑧, 𝑥 − 𝑥𝑛 ⟩ ≥ 0}, we have
⟨𝑥𝑛 − 𝑧, 𝑥 − 𝑥𝑛 ⟩ ≥ 0 for all 𝑧 ∈ 𝑄𝑛 , and by Proposition 4(i),
we get 𝑥𝑛 = 𝑃𝑄𝑛 𝑥.
We divide the rest of the proof into several steps.
Step 1. {𝑥𝑛 }, {𝑦𝑛 }, and {𝑧𝑛 } are bounded.
Indeed, take 𝑢 ∈ Fix(𝑆)∩Γ arbitrarily. Taking into account
𝜆 𝑛 (𝛼𝑛 + 𝐿) < 1, for all 𝑛 ≥ 1, we deduce that
󵄩󵄩
󵄩2
󵄩󵄩𝑥𝑛 − 𝑢 − 𝜆 𝑛 𝜇𝑛 (∇𝑓𝛼𝑛 (𝑥𝑛 ) − ∇𝑓𝛼𝑛 (𝑢))󵄩󵄩󵄩
󵄩
󵄩
× (‖𝑧‖2 + ‖𝑄𝑧 − 𝑧‖2 ) } < ∞.
(59)
Assume that the following conditions hold:
(i) 𝛿 ≤ 𝛽𝑛 and 𝜎𝑛 + 𝛽𝑛 ≤ 1 − 𝜅, for all 𝑛 ≥ 1 for some
𝛿 ∈ (0, 1);
(ii) lim𝑛 → ∞ 𝛼𝑛 = 0 and lim𝑛 → ∞ 𝜎𝑛 = 0;
(iii) lim𝑛 → ∞ 𝜇𝑛 = 1;
(iv) 𝜆 𝑛 (𝛼𝑛 + 𝐿) < 1, for all 𝑛 ≥ 1 and {𝜆 𝑛 } ⊂ [𝑎, 𝑏] for some
𝑎, 𝑏 ∈ (0, (1/𝐿)).
Then, the sequences {𝑥𝑛 }, {𝑦𝑛 }, and {𝑧𝑛 } converge strongly to the
same point 𝑃Fix(𝑆)∩Γ 𝑥.
Proof. Utilizing the condition 𝜆 𝑛 (𝛼𝑛 + 𝐿) < 1 for all 𝑛 ≥ 1 and
repeating the same arguments as in Remark 19, we can see
that {𝑦𝑛 } and {𝑧𝑛 } are defined well. Note that the 𝐿-Lipschitz
continuity of the gradient ∇𝑓 implies that ∇𝑓 is (1/𝐿)-ism
[20], that is,
1󵄩
󵄩2
⟨∇𝑓 (𝑥) − ∇𝑓 (𝑦) , 𝑥 − 𝑦⟩ ≥ 󵄩󵄩󵄩∇𝑓 (𝑥) − ∇𝑓 (𝑦)󵄩󵄩󵄩 ,
𝐿
(60)
∀𝑥, 𝑦 ∈ 𝐶.
Observe that
(𝛼 + 𝐿) ⟨∇𝑓𝛼 (𝑥) − ∇𝑓𝛼 (𝑦) , 𝑥 − 𝑦⟩
󵄩2
󵄩
= (𝛼 + 𝐿) [𝛼󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩 + ⟨∇𝑓 (𝑥) − ∇𝑓 (𝑦) , 𝑥 − 𝑦⟩]
󵄩2
󵄩2
󵄩
󵄩
= 𝛼2 󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩 + 𝛼 ⟨∇𝑓 (𝑥) − ∇𝑓 (𝑦) , 𝑥 − 𝑦⟩ + 𝛼𝐿󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩
+ 𝐿 ⟨∇𝑓 (𝑥) − ∇𝑓 (𝑦) , 𝑥 − 𝑦⟩
󵄩2
󵄩
≥ 𝛼2 󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩 + 2𝛼 ⟨∇𝑓 (𝑥) − ∇𝑓 (𝑦) , 𝑥 − 𝑦⟩
󵄩2
󵄩
+ 󵄩󵄩󵄩∇𝑓 (𝑥) − ∇𝑓 (𝑦)󵄩󵄩󵄩
󵄩2
󵄩
= 󵄩󵄩󵄩𝛼 (𝑥 − 𝑦) + ∇𝑓 (𝑥) − ∇𝑓 (𝑦)󵄩󵄩󵄩
󵄩
󵄩2
= 󵄩󵄩󵄩∇𝑓𝛼 (𝑥) − ∇𝑓𝛼 (𝑦)󵄩󵄩󵄩 .
(61)
󵄩
󵄩2
≤ 󵄩󵄩󵄩𝑥𝑛 −𝑢󵄩󵄩󵄩 +𝜆 𝑛 𝜇𝑛 (𝜆 𝑛 𝜇𝑛 −
󵄩
󵄩2
≤ 󵄩󵄩󵄩𝑥𝑛 − 𝑢󵄩󵄩󵄩 + 𝜆 𝑛 (𝜆 𝑛 −
2
󵄩2
󵄩
) 󵄩󵄩󵄩∇𝑓 (𝑥 )−∇𝑓𝛼𝑛 (𝑢)󵄩󵄩󵄩󵄩
𝛼𝑛 + 𝐿 󵄩 𝛼𝑛 𝑛
2
󵄩2
󵄩
) 󵄩󵄩󵄩󵄩∇𝑓𝛼𝑛 (𝑥𝑛 ) − ∇𝑓𝛼𝑛 (𝑢)󵄩󵄩󵄩󵄩
𝛼𝑛 + 𝐿
󵄩
󵄩2
≤ 󵄩󵄩󵄩𝑥𝑛 − 𝑢󵄩󵄩󵄩 ,
󵄩󵄩
󵄩
󵄩󵄩𝑦𝑛 − 𝑢󵄩󵄩󵄩
󵄩
= 󵄩󵄩󵄩󵄩𝑃𝐶 (𝑥𝑛 − 𝜆 𝑛 𝜇𝑛 ∇𝑓𝛼𝑛 (𝑥𝑛 ) − 𝜆 𝑛 (1 − 𝜇𝑛 ) ∇𝑓𝛼𝑛 (𝑦𝑛 ))
󵄩
−𝑃𝐶 (𝑢 − 𝜆 𝑛 𝜇𝑛 ∇𝑓 (𝑢) − 𝜆 𝑛 (1 − 𝜇𝑛 ) ∇𝑓 (𝑢)) 󵄩󵄩󵄩󵄩
󵄩
≤ 󵄩󵄩󵄩󵄩(𝑥𝑛 − 𝜆 𝑛 𝜇𝑛 ∇𝑓𝛼𝑛 (𝑥𝑛 ) − 𝜆 𝑛 (1 − 𝜇𝑛 ) ∇𝑓𝛼𝑛 (𝑦𝑛 ))
󵄩
− (𝑢 − 𝜆 𝑛 𝜇𝑛 ∇𝑓 (𝑢) − 𝜆 𝑛 (1 − 𝜇𝑛 ) ∇𝑓 (𝑢)) 󵄩󵄩󵄩󵄩
󵄩
󵄩
≤ 󵄩󵄩󵄩󵄩𝑥𝑛 − 𝑢 − 𝜆 𝑛 𝜇𝑛 (∇𝑓𝛼𝑛 (𝑥𝑛 ) − ∇𝑓 (𝑢))󵄩󵄩󵄩󵄩
󵄩
󵄩
+ 𝜆 𝑛 (1 − 𝜇𝑛 ) 󵄩󵄩󵄩󵄩∇𝑓𝛼𝑛 (𝑦𝑛 ) − ∇𝑓 (𝑢)󵄩󵄩󵄩󵄩
󵄩
󵄩
≤ 󵄩󵄩󵄩󵄩𝑥𝑛 − 𝑢 − 𝜆 𝑛 𝜇𝑛 (∇𝑓𝛼𝑛 (𝑥𝑛 ) − ∇𝑓𝛼𝑛 (𝑢))󵄩󵄩󵄩󵄩
󵄩
󵄩
+ 𝜆 𝑛 𝜇𝑛 󵄩󵄩󵄩󵄩∇𝑓𝛼𝑛 (𝑢) − ∇𝑓 (𝑢)󵄩󵄩󵄩󵄩 + 𝜆 𝑛 (1 − 𝜇𝑛 )
󵄩 󵄩
󵄩
󵄩
× [󵄩󵄩󵄩󵄩∇𝑓𝛼𝑛 (𝑦𝑛 ) − ∇𝑓𝛼𝑛 (𝑢)󵄩󵄩󵄩󵄩 + 󵄩󵄩󵄩󵄩∇𝑓𝛼𝑛 (𝑢) − ∇𝑓 (𝑢)󵄩󵄩󵄩󵄩]
󵄩
󵄩
≤ 󵄩󵄩󵄩𝑥𝑛 − 𝑢󵄩󵄩󵄩 + 𝜆 𝑛 𝜇𝑛 𝛼𝑛 ‖𝑢‖
󵄩
󵄩
+ 𝜆 𝑛 (1 − 𝜇𝑛 ) [(𝛼𝑛 + 𝐿) 󵄩󵄩󵄩𝑦𝑛 − 𝑢󵄩󵄩󵄩 + 𝛼𝑛 ‖𝑢‖]
󵄩
󵄩
󵄩
󵄩
= 󵄩󵄩󵄩𝑥𝑛 − 𝑢󵄩󵄩󵄩 + 𝜆 𝑛 (1 − 𝜇𝑛 ) (𝛼𝑛 + 𝐿) 󵄩󵄩󵄩𝑦𝑛 − 𝑢󵄩󵄩󵄩
+ 𝜆 𝑛 𝛼𝑛 ‖𝑢‖
󵄩
󵄩
󵄩
󵄩
≤ 󵄩󵄩󵄩𝑥𝑛 − 𝑢󵄩󵄩󵄩 + (1 − 𝜇𝑛 ) 󵄩󵄩󵄩𝑦𝑛 − 𝑢󵄩󵄩󵄩 + 𝜆 𝑛 𝛼𝑛 ‖𝑢‖ ,
(63)
10
Abstract and Applied Analysis
which implies that
󵄩󵄩
󵄩 1 󵄩󵄩
󵄩
(󵄩𝑥 − 𝑢󵄩󵄩󵄩 + 𝜆 𝑛 𝛼𝑛 ‖𝑢‖) .
󵄩󵄩𝑦𝑛 − 𝑢󵄩󵄩󵄩 ≤
𝜇𝑛 󵄩 𝑛
Thus, from (64) and (66), it follows that
(64)
󵄩 󵄩
󵄩 1 󵄩󵄩
󵄩
󵄩󵄩
(󵄩𝑥 − 𝑢󵄩󵄩󵄩 + 𝜆 𝑛 𝛼𝑛 ‖𝑢‖)
󵄩󵄩𝑦𝑛 − 𝑢󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑡𝑛 − 𝑢󵄩󵄩󵄩 ≤
𝜇𝑛 󵄩 𝑛
1 + 2𝜇𝑛
𝜇𝑛2
+
Meantime, we also have
󵄩󵄩
󵄩
󵄩󵄩𝑡𝑛 − 𝑢󵄩󵄩󵄩
󵄩
= 󵄩󵄩󵄩󵄩𝑃𝐶 (𝑥𝑛 − 𝜆 𝑛 ∇𝑓𝛼𝑛 (𝑦𝑛 ) − 𝜆 𝑛 (1 − 𝜇𝑛 ) ∇𝑓𝛼𝑛 (𝑡𝑛 ))
󵄩
−𝑃𝐶 (𝑢 − 𝜆 𝑛 ∇𝑓 (𝑢) − 𝜆 𝑛 (1 − 𝜇𝑛 ) ∇𝑓 (𝑢)) 󵄩󵄩󵄩󵄩
󵄩
≤ 󵄩󵄩󵄩󵄩(𝑥𝑛 − 𝜆 𝑛 ∇𝑓𝛼𝑛 (𝑦𝑛 ) − 𝜆 𝑛 (1 − 𝜇𝑛 ) ∇𝑓𝛼𝑛 (𝑡𝑛 ))
󵄩
− (𝑢 − 𝜆 𝑛 ∇𝑓 (𝑢) − 𝜆 𝑛 (1 − 𝜇𝑛 ) ∇𝑓 (𝑢)) 󵄩󵄩󵄩󵄩
󵄩
󵄩
󵄩
󵄩
≤ 󵄩󵄩󵄩𝑥𝑛 − 𝑢󵄩󵄩󵄩 + 𝜆 𝑛 󵄩󵄩󵄩󵄩∇𝑓𝛼𝑛 (𝑦𝑛 ) − ∇𝑓 (𝑢)󵄩󵄩󵄩󵄩
󵄩
󵄩
+ 𝜆 𝑛 (1 − 𝜇𝑛 ) 󵄩󵄩󵄩󵄩∇𝑓𝛼𝑛 (𝑡𝑛 ) − ∇𝑓 (𝑢)󵄩󵄩󵄩󵄩
󵄩
󵄩
≤ 󵄩󵄩󵄩𝑥𝑛 − 𝑢󵄩󵄩󵄩
󵄩 󵄩
󵄩
󵄩
+ 𝜆 𝑛 [󵄩󵄩󵄩󵄩∇𝑓𝛼𝑛 (𝑦𝑛 ) − ∇𝑓𝛼𝑛 (𝑢)󵄩󵄩󵄩󵄩 + 󵄩󵄩󵄩󵄩∇𝑓𝛼𝑛 (𝑢) − ∇𝑓 (𝑢)󵄩󵄩󵄩󵄩]
=
1 + 3𝜇𝑛 󵄩󵄩
󵄩
(󵄩󵄩𝑥𝑛 − 𝑢󵄩󵄩󵄩 + 𝜆 𝑛 𝛼𝑛 ‖𝑢‖)
𝜇𝑛2
≤
4 󵄩󵄩
󵄩
(󵄩𝑥 − 𝑢󵄩󵄩󵄩 + 𝜆 𝑛 𝛼𝑛 ‖𝑢‖) ,
𝜇𝑛2 󵄩 𝑛
2
󵄩
󵄩 󵄩
󵄩 2 16 󵄩
󵄩
(󵄩󵄩󵄩𝑦𝑛 − 𝑢󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑡𝑛 − 𝑢󵄩󵄩󵄩) ≤ 4 (󵄩󵄩󵄩𝑥𝑛 − 𝑢󵄩󵄩󵄩 + 𝜆 𝑛 𝛼𝑛 ‖𝑢‖)
𝜇𝑛
≤
16 󵄩󵄩
󵄩2
(2󵄩𝑥 − 𝑢󵄩󵄩󵄩 + 2𝜆2𝑛 𝛼𝑛2 ‖𝑢‖2 )
𝜇𝑛4 󵄩 𝑛
≤
16 󵄩󵄩
󵄩2
(2󵄩𝑥 − 𝑢󵄩󵄩󵄩 + 2𝛼𝑛2 ‖𝑢‖2 )
𝜇𝑛4 󵄩 𝑛
≤
32 󵄩󵄩
󵄩2 32 2 2
󵄩󵄩𝑥𝑛 − 𝑢󵄩󵄩󵄩 + 4 𝛼𝑛 ‖𝑢‖ .
4
𝜇𝑛
𝜇𝑛
Repeating the same arguments as in (33) and (34), we can
obtain that
󵄩󵄩
󵄩2
󵄩󵄩𝑡𝑛 − 𝑢󵄩󵄩󵄩
2
󵄩2
󵄩
󵄩2
󵄩
≤ 󵄩󵄩󵄩𝑥𝑛 − 𝑢󵄩󵄩󵄩 + (𝜆2𝑛 (𝛼𝑛 + 𝐿) − 1) 󵄩󵄩󵄩𝑥𝑛 − 𝑦𝑛 󵄩󵄩󵄩
󵄩
󵄩 󵄩
󵄩
+ 2𝜆 𝑛 𝛼𝑛 ‖𝑢‖ (󵄩󵄩󵄩𝑦𝑛 − 𝑢󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑡𝑛 − 𝑢󵄩󵄩󵄩)
(69)
󵄩󵄩
󵄩2
󵄩󵄩𝑧𝑛 − 𝑢󵄩󵄩󵄩
󵄩󵄩
󵄩
󵄩󵄩𝑡𝑛 − 𝑢󵄩󵄩󵄩
󵄩
󵄩
󵄩2
󵄩2
≤ (1 + 𝛾𝑛 ) 󵄩󵄩󵄩𝑡𝑛 − 𝑢󵄩󵄩󵄩 + 𝜎𝑛 󵄩󵄩󵄩𝑄𝑡𝑛 − 𝑢󵄩󵄩󵄩
≤
1 󵄩󵄩
󵄩 󵄩
󵄩
[󵄩𝑥 − 𝑢󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑦𝑛 − 𝑢󵄩󵄩󵄩 + 2𝜆 𝑛 𝛼𝑛 ‖𝑢‖]
𝜇𝑛 󵄩 𝑛
≤
1 󵄩󵄩
󵄩 1 󵄩󵄩
󵄩
{󵄩𝑥 − 𝑢󵄩󵄩󵄩 +
(󵄩𝑥 − 𝑢󵄩󵄩󵄩 + 𝜆 𝑛 𝛼𝑛 ‖𝑢‖) + 2𝜆 𝑛 𝛼𝑛 ‖𝑢‖}
𝜇𝑛 󵄩 𝑛
𝜇𝑛 󵄩 𝑛
󵄩2
󵄩
+ 𝛽𝑛 (𝜅 − (1 − 𝜎𝑛 − 𝛽𝑛 )) 󵄩󵄩󵄩𝑡𝑛 − 𝑆𝑛 𝑡𝑛 󵄩󵄩󵄩 + 𝑐𝑛
1
1
1 󵄩
2
󵄩
= ( + 2 ) 󵄩󵄩󵄩𝑥𝑛 − 𝑢󵄩󵄩󵄩 + ( 2 + ) 𝜆 𝑛 𝛼𝑛 ‖𝑢‖
𝜇𝑛 𝜇𝑛
𝜇𝑛 𝜇𝑛
1 + 2𝜇𝑛 󵄩󵄩
󵄩
(󵄩󵄩𝑥𝑛 − 𝑢󵄩󵄩󵄩 + 𝜆 𝑛 𝛼𝑛 ‖𝑢‖) .
𝜇𝑛2
(68)
󵄩
󵄩
󵄩2
󵄩 󵄩
󵄩
≤ 󵄩󵄩󵄩𝑥𝑛 − 𝑢󵄩󵄩󵄩 + 2𝜆 𝑛 𝛼𝑛 ‖𝑢‖ (󵄩󵄩󵄩𝑦𝑛 − 𝑢󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑡𝑛 − 𝑢󵄩󵄩󵄩) ,
which hence implies that
≤
(67)
which hence implies that
+ 𝜆 𝑛 (1 − 𝜇𝑛 )
󵄩 󵄩
󵄩
󵄩
× [󵄩󵄩󵄩󵄩∇𝑓𝛼𝑛 (𝑡𝑛 ) − ∇𝑓𝛼𝑛 (𝑢)󵄩󵄩󵄩󵄩 + 󵄩󵄩󵄩󵄩∇𝑓𝛼𝑛 (𝑢) − ∇𝑓 (𝑢)󵄩󵄩󵄩󵄩]
󵄩
󵄩
󵄩
󵄩
≤ 󵄩󵄩󵄩𝑥𝑛 − 𝑢󵄩󵄩󵄩 + 𝜆 𝑛 [(𝛼𝑛 + 𝐿) 󵄩󵄩󵄩𝑦𝑛 − 𝑢󵄩󵄩󵄩 + 𝛼𝑛 ‖𝑢‖]
󵄩
󵄩
+ 𝜆 𝑛 (1 − 𝜇𝑛 ) [(𝛼𝑛 + 𝐿) 󵄩󵄩󵄩𝑡𝑛 − 𝑢󵄩󵄩󵄩 + 𝛼𝑛 ‖𝑢‖]
󵄩 󵄩
󵄩
󵄩
≤ 󵄩󵄩󵄩𝑥𝑛 − 𝑢󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑦𝑛 − 𝑢󵄩󵄩󵄩 + 𝜆 𝑛 𝛼𝑛 ‖𝑢‖
󵄩
󵄩
+ (1 − 𝜇𝑛 ) 󵄩󵄩󵄩𝑡𝑛 − 𝑢󵄩󵄩󵄩 + 𝜆 𝑛 (1 − 𝜇𝑛 ) 𝛼𝑛 ‖𝑢‖
󵄩
󵄩 󵄩
󵄩
󵄩
󵄩
≤ 󵄩󵄩󵄩𝑥𝑛 − 𝑢󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑦𝑛 − 𝑢󵄩󵄩󵄩 + 2𝜆 𝑛 𝛼𝑛 ‖𝑢‖ + (1 − 𝜇𝑛 ) 󵄩󵄩󵄩𝑡𝑛 − 𝑢󵄩󵄩󵄩 ,
(65)
󵄩
󵄩
(󵄩󵄩󵄩𝑥𝑛 − 𝑢󵄩󵄩󵄩 + 𝜆 𝑛 𝛼𝑛 ‖𝑢‖)
󵄩
󵄩
󵄩2
󵄩2
≤ (1 + 𝛾𝑛 ) 󵄩󵄩󵄩𝑡𝑛 − 𝑢󵄩󵄩󵄩 + 𝜎𝑛 󵄩󵄩󵄩𝑄𝑡𝑛 − 𝑢󵄩󵄩󵄩 + 𝑐𝑛
󵄩
󵄩
󵄩2
󵄩 󵄩
󵄩
≤ (1 + 𝛾𝑛 ) [󵄩󵄩󵄩𝑥𝑛 − 𝑢󵄩󵄩󵄩 + 2𝜆 𝑛 𝛼𝑛 ‖𝑢‖ (󵄩󵄩󵄩𝑦𝑛 − 𝑢󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑡𝑛 − 𝑢󵄩󵄩󵄩)]
󵄩
󵄩2
+ 𝜎𝑛 󵄩󵄩󵄩𝑄𝑡𝑛 − 𝑢󵄩󵄩󵄩 + 𝑐𝑛
󵄩
󵄩2
= (1 + 𝛾𝑛 ) 󵄩󵄩󵄩𝑥𝑛 − 𝑢󵄩󵄩󵄩 + 2𝛼𝑛 𝜆 𝑛 (1 + 𝛾𝑛 ) ‖𝑢‖
(66)
󵄩
󵄩 󵄩
󵄩
󵄩2
󵄩
× (󵄩󵄩󵄩𝑦𝑛 − 𝑢󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑡𝑛 − 𝑢󵄩󵄩󵄩) + 𝜎𝑛 󵄩󵄩󵄩𝑄𝑡𝑛 − 𝑢󵄩󵄩󵄩 + 𝑐𝑛
Abstract and Applied Analysis
11
󵄩2
󵄩
≤ (1 + 𝛾𝑛 ) 󵄩󵄩󵄩𝑥𝑛 − 𝑢󵄩󵄩󵄩
󵄩
󵄩2
= 󵄩󵄩󵄩𝑥𝑛 − 𝑢󵄩󵄩󵄩 + (𝛼𝑛 + 𝛾𝑛 + 𝜎𝑛 )
2
󵄩
󵄩 󵄩
󵄩2
+ 𝛼𝑛 [𝜆2𝑛 (1 + 𝛾𝑛 ) ‖𝑢‖2 + (󵄩󵄩󵄩𝑦𝑛 − 𝑢󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑡𝑛 − 𝑢󵄩󵄩󵄩) ]
2
󵄩
󵄩
+ 𝜎𝑛 (󵄩󵄩󵄩𝑄𝑡𝑛 − 𝑄𝑢󵄩󵄩󵄩 + ‖𝑄𝑢 − 𝑢‖) + 𝑐𝑛
󵄩
󵄩2
≤ (1 + 𝛾𝑛 ) 󵄩󵄩󵄩𝑥𝑛 − 𝑢󵄩󵄩󵄩
󵄩
󵄩 󵄩
󵄩2
+ 𝛼𝑛 [(1 + 𝛾𝑛 ) ‖𝑢‖ + (󵄩󵄩󵄩𝑦𝑛 − 𝑢󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑡𝑛 − 𝑢󵄩󵄩󵄩) ]
2
2
2
1
󵄩
󵄩
+ 𝜎𝑛 [𝜌 󵄩󵄩󵄩𝑡𝑛 − 𝑢󵄩󵄩󵄩 + (1 − 𝜌)
‖𝑄𝑢 − 𝑢‖] + 𝑐𝑛
1−𝜌
2
󵄩
󵄩 󵄩
󵄩2
+ 𝛼𝑛 [(1 + 𝛾𝑛 ) ‖𝑢‖2 + (󵄩󵄩󵄩𝑦𝑛 − 𝑢󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑡𝑛 − 𝑢󵄩󵄩󵄩) ]
󵄩
󵄩2
+ 𝜎𝑛 [𝜌󵄩󵄩󵄩𝑡𝑛 − 𝑢󵄩󵄩󵄩 + (1 − 𝜌)
1
− 𝑢‖2 ] + 𝑐𝑛
2 ‖𝑄𝑢
(1 − 𝜌)
2
󵄩
󵄩
󵄩2
󵄩2
≤ 󵄩󵄩󵄩𝑥𝑛 − 𝑢󵄩󵄩󵄩 + 𝛾𝑛 󵄩󵄩󵄩𝑥𝑛 − 𝑢󵄩󵄩󵄩 + 𝛼𝑛 (1 + 𝛾𝑛 ) ‖𝑢‖2
󵄩
󵄩 󵄩
󵄩2
+ (𝛼𝑛 + 𝛾𝑛 ) (󵄩󵄩󵄩𝑦𝑛 − 𝑢󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑡𝑛 − 𝑢󵄩󵄩󵄩)
−1
󵄩
󵄩2
+ 𝜎𝑛 󵄩󵄩󵄩𝑡𝑛 − 𝑢󵄩󵄩󵄩 + 𝜎𝑛 (1 − 𝜌) ‖𝑄𝑢 − 𝑢‖2 + 𝑐𝑛
2
󵄩
󵄩2
󵄩
󵄩2
≤ 󵄩󵄩󵄩𝑥𝑛 − 𝑢󵄩󵄩󵄩 + (𝛼𝑛 + 𝛾𝑛 ) 󵄩󵄩󵄩𝑥𝑛 − 𝑢󵄩󵄩󵄩 + 𝛼𝑛 (1 + 𝛾𝑛 ) ‖𝑢‖2
󵄩
󵄩 󵄩
󵄩2
+ (𝛼𝑛 + 𝛾𝑛 ) (󵄩󵄩󵄩𝑦𝑛 − 𝑢󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑡𝑛 − 𝑢󵄩󵄩󵄩)
󵄩
󵄩 󵄩
󵄩2
+ 𝜎𝑛 (󵄩󵄩󵄩𝑦𝑛 − 𝑢󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑡𝑛 − 𝑢󵄩󵄩󵄩)
−1
+ 𝛾𝑛 ) ‖𝑄𝑢 − 𝑢‖2 + 𝑐𝑛
󵄩
󵄩2
󵄩
󵄩2
≤ 󵄩󵄩󵄩𝑥𝑛 − 𝑢󵄩󵄩󵄩 + (𝛼𝑛 + 𝛾𝑛 ) 󵄩󵄩󵄩𝑥𝑛 − 𝑢󵄩󵄩󵄩
−1
+ 𝛼𝑛 ((1 − 𝜌)
2
+ 𝛾𝑛 ) ‖𝑢‖2
󵄩
󵄩 󵄩
󵄩2
+ (𝛼𝑛 + 𝛾𝑛 + 𝜎𝑛 ) (󵄩󵄩󵄩𝑦𝑛 − 𝑢󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑡𝑛 − 𝑢󵄩󵄩󵄩)
−1
+ 𝜎𝑛 ((1 − 𝜌)
+ 𝛾𝑛 ) ‖𝑄𝑢 − 𝑢‖2 + 𝑐𝑛
󵄩
󵄩2
≤ 󵄩󵄩󵄩𝑥𝑛 − 𝑢󵄩󵄩󵄩
󵄩
󵄩2 󵄩
󵄩 󵄩
󵄩2
+ (𝛼𝑛 + 𝛾𝑛 + 𝜎𝑛 ) [󵄩󵄩󵄩𝑥𝑛 − 𝑢󵄩󵄩󵄩 + (󵄩󵄩󵄩𝑦𝑛 − 𝑢󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑡𝑛 − 𝑢󵄩󵄩󵄩) ]
−1
+ 𝛼𝑛 ((1 − 𝜌)
−1
+ 𝜎𝑛 ((1 − 𝜌)
2
+ 𝛾𝑛 ) ‖𝑢‖2
2
+ 𝛾𝑛 ) ‖𝑄𝑢 − 𝑢‖ + 𝑐𝑛
󵄩
󵄩2
≤ 󵄩󵄩󵄩𝑥𝑛 − 𝑢󵄩󵄩󵄩 + (𝛼𝑛 + 𝛾𝑛 + 𝜎𝑛 )
󵄩
󵄩2 󵄩
󵄩 󵄩
󵄩2
× [󵄩󵄩󵄩𝑥𝑛 − 𝑢󵄩󵄩󵄩 + (󵄩󵄩󵄩𝑦𝑛 − 𝑢󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑡𝑛 − 𝑢󵄩󵄩󵄩) ]
−1
+ (𝛼𝑛 + 𝛾𝑛 + 𝜎𝑛 ) ((1 − 𝜌)
+ 𝑐𝑛
−1
+ ((1 − 𝜌)
2
+ 𝛾𝑛 ) (‖𝑢‖2 + ‖𝑄𝑢 − 𝑢‖2 )
2
+ 𝛾𝑛 ) (‖𝑢‖2 + ‖𝑄𝑢 − 𝑢‖2 )} + 𝑐𝑛
󵄩
󵄩2
≤ 󵄩󵄩󵄩𝑥𝑛 − 𝑢󵄩󵄩󵄩 + (𝛼𝑛 + 𝛾𝑛 + 𝜎𝑛 )
󵄩2 32 󵄩
󵄩2 32
󵄩
× {󵄩󵄩󵄩𝑥𝑛 − 𝑢󵄩󵄩󵄩 + 4 󵄩󵄩󵄩𝑥𝑛 − 𝑢󵄩󵄩󵄩 + 4 𝛼𝑛2 ‖𝑢‖2
𝜇𝑛
𝜇𝑛
−1
+ ((1 − 𝜌)
󵄩
󵄩2
≤ (1 + 𝛾𝑛 ) 󵄩󵄩󵄩𝑥𝑛 − 𝑢󵄩󵄩󵄩
+ 𝜎𝑛 ((1 − 𝜌)
󵄩2 󵄩
󵄩 󵄩
󵄩2
󵄩
× {󵄩󵄩󵄩𝑥𝑛 − 𝑢󵄩󵄩󵄩 + (󵄩󵄩󵄩𝑦𝑛 − 𝑢󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑡𝑛 − 𝑢󵄩󵄩󵄩)
2
+ 𝛾𝑛 ) (‖𝑢‖2 + ‖𝑄𝑢 − 𝑢‖2 ) } + 𝑐𝑛
󵄩
󵄩2
= 󵄩󵄩󵄩𝑥𝑛 − 𝑢󵄩󵄩󵄩 + (𝛼𝑛 + 𝛾𝑛 + 𝜎𝑛 )
×{
2
32 + 𝜇𝑛4 󵄩󵄩
32 2
−1
󵄩2
󵄩𝑥 − 𝑢󵄩󵄩󵄩 + [((1 − 𝜌) + 𝛾𝑛 ) + 4 𝛼𝑛 ]
𝜇𝑛4 󵄩 𝑛
𝜇𝑛
−1
× ‖𝑢‖2 + ((1 − 𝜌)
2
+ 𝛾𝑛 ) ‖𝑄𝑢 − 𝑢‖2 } + 𝑐𝑛
󵄩
󵄩2
≤ 󵄩󵄩󵄩𝑥𝑛 − 𝑢󵄩󵄩󵄩 + (𝛼𝑛 + 𝛾𝑛 + 𝜎𝑛 )
×{
2
32 2
33 󵄩󵄩
−1
󵄩2
󵄩𝑥 − 𝑢󵄩󵄩󵄩 + [((1 − 𝜌) + 𝛾𝑛 ) + 4 𝛼𝑛 ]
𝜇𝑛4 󵄩 𝑛
𝜇𝑛
× (‖𝑢‖2 + ‖𝑄𝑢 − 𝑢‖2 ) } + 𝑐𝑛
󵄩
󵄩2
≤ 󵄩󵄩󵄩𝑥𝑛 − 𝑢󵄩󵄩󵄩 + 𝑐𝑛 + (𝛼𝑛 + 𝛾𝑛 + 𝜎𝑛 ) Δ 𝑛 ,
(70)
for every 𝑛 = 1, 2, . . . and hence 𝑢 ∈ 𝐶𝑛 . So, Fix(𝑆) ∩ Γ ⊂
𝐶𝑛 for every 𝑛 = 1, 2, . . . . Next, let us show by mathematical
induction that {𝑥𝑛 } is well-defined and Fix(𝑆) ∩ Γ ⊂ 𝐶𝑛 ∩ 𝑄𝑛
for every 𝑛 = 1, 2, . . .. For 𝑛 = 1, we have 𝑄1 = 𝐶. Hence,
we obtain Fix(𝑆) ∩ Γ ⊂ 𝐶1 ∩ 𝑄1 . Suppose that 𝑥𝑘 is given and
Fix(𝑆) ∩ Γ ⊂ 𝐶𝑘 ∩ 𝑄𝑘 for some integer 𝑘 ≥ 1. Since Fix(𝑆) ∩ Γ
is nonempty, 𝐶𝑘 ∩ 𝑄𝑘 is a nonempty closed convex subset of
𝐶. So, there exists a unique element 𝑥𝑘+1 ∈ 𝐶𝑘 ∩ 𝑄𝑘 such that
𝑥𝑘+1 = 𝑃𝐶𝑘 ∩𝑄𝑘 𝑥. It is also obvious that there holds ⟨𝑥𝑘+1 −
𝑧, 𝑥 − 𝑥𝑘+1 ⟩ ≥ 0 for every 𝑧 ∈ 𝐶𝑘 ∩ 𝑄𝑘 . Since Fix(𝑆) ∩ Γ ⊂
𝐶𝑘 ∩𝑄𝑘 , we have ⟨𝑥𝑘+1 −𝑧, 𝑥−𝑥𝑘+1 ⟩ ≥ 0 for every 𝑧 ∈ Fix(𝑆)∩Γ
and hence Fix(𝑆)∩Γ ⊂ 𝑄𝑘+1 . Therefore, we obtain Fix(𝑆)∩Γ ⊂
𝐶𝑘+1 ∩ 𝑄𝑘+1 .
Step 2. lim𝑛 → ∞ ‖𝑥𝑛 − 𝑥𝑛+1 ‖ = 0 and lim𝑛 → ∞ ‖𝑥𝑛 − 𝑧𝑛 ‖ = 0.
Indeed, let 𝑞 = 𝑃Fix(𝑆)∩Γ 𝑥. From 𝑥𝑛+1 = 𝑃𝐶𝑛 ∩𝑄𝑛 𝑥 and 𝑞 ∈
Fix(𝑆) ∩ Γ ⊂ 𝐶𝑛 ∩ 𝑄𝑛 , we have
󵄩
󵄩󵄩
󵄩 󵄩
(71)
󵄩󵄩𝑥𝑛+1 − 𝑥󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩𝑞 − 𝑥󵄩󵄩󵄩
for every 𝑛 = 1, 2, . . . . Therefore, {𝑥𝑛 } is bounded. From (64),
(66), and (70), we also obtain that {𝑦𝑛 }, {𝑡𝑛 }, and {𝑧𝑛 } are
bounded. Since 𝑥𝑛+1 ∈ 𝐶𝑛 ∩ 𝑄𝑛 ⊂ 𝑄𝑛 and 𝑥𝑛 = 𝑃𝑄𝑛 𝑥, we
have
󵄩 󵄩
󵄩
󵄩󵄩
(72)
󵄩󵄩𝑥𝑛 − 𝑥󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩𝑥𝑛+1 − 𝑥󵄩󵄩󵄩
12
Abstract and Applied Analysis
for every 𝑛 = 1, 2, . . . . Therefore, there exists lim𝑛 → ∞ ‖𝑥𝑛 −𝑥‖.
Since 𝑥𝑛 = 𝑃𝑄𝑛 𝑥 and 𝑥𝑛+1 ∈ 𝑄𝑛 , using Proposition 4(ii), we
have
󵄩2 󵄩
󵄩󵄩
󵄩2 󵄩
󵄩2
󵄩󵄩𝑥𝑛+1 − 𝑥𝑛 󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩𝑥𝑛+1 − 𝑥󵄩󵄩󵄩 − 󵄩󵄩󵄩𝑥𝑛 − 𝑥󵄩󵄩󵄩
(73)
for every 𝑛 = 1, 2, . . . . This implies that
󵄩󵄩
lim 󵄩𝑥
𝑛 → ∞ 󵄩 𝑛+1
󵄩
− 𝑥𝑛 󵄩󵄩󵄩 = 0.
(74)
Since 𝑥𝑛+1 ∈ 𝐶𝑛 , we have
󵄩󵄩
󵄩2 󵄩
󵄩2
󵄩󵄩𝑧𝑛 − 𝑥𝑛+1 󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩𝑥𝑛 − 𝑥𝑛+1 󵄩󵄩󵄩 + 𝜃𝑛 ,
(75)
which implies that
󵄩󵄩
󵄩 󵄩
󵄩
󵄩󵄩𝑧𝑛 − 𝑥𝑛+1 󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩𝑥𝑛 − 𝑥𝑛+1 󵄩󵄩󵄩 + √𝜃𝑛 .
(76)
Hence, we get
󵄩
󵄩 󵄩
󵄩 󵄩
󵄩
󵄩
󵄩󵄩
󵄩󵄩𝑥𝑛 − 𝑧𝑛 󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩𝑥𝑛 − 𝑥𝑛+1 󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑥𝑛+1 − 𝑧𝑛 󵄩󵄩󵄩 ≤ 2 󵄩󵄩󵄩𝑥𝑛+1 − 𝑥𝑛 󵄩󵄩󵄩 + √𝜃𝑛
(77)
for every 𝑛 = 1, 2, . . . . From ‖𝑥𝑛+1 − 𝑥𝑛 ‖ → 0 and 𝜃𝑛 → 0,
we conclude that ‖𝑥𝑛 − 𝑧𝑛 ‖ → 0 as 𝑛 → ∞.
Step 3. lim𝑛 → ∞ ‖𝑥𝑛 − 𝑦𝑛 ‖ = 0, lim𝑛 → ∞ ‖𝑥𝑛 − 𝑡𝑛 ‖ = 0 and
lim𝑛 → ∞ ‖𝑥𝑛 − 𝑆𝑥𝑛 ‖ = 0.
Indeed, substituting (69) in (70), we get
󵄩󵄩
󵄩2
󵄩󵄩𝑧𝑛 − 𝑢󵄩󵄩󵄩
󵄩
󵄩2
󵄩2
󵄩
≤ (1 + 𝛾𝑛 ) 󵄩󵄩󵄩𝑡𝑛 − 𝑢󵄩󵄩󵄩 + 𝜎𝑛 󵄩󵄩󵄩𝑄𝑡𝑛 − 𝑢󵄩󵄩󵄩 + 𝑐𝑛
2
󵄩
󵄩2
≤ (1 + 𝛾𝑛 ) [󵄩󵄩󵄩𝑥𝑛 − 𝑢󵄩󵄩󵄩 + (𝜆2𝑛 (𝛼𝑛 + 𝐿) − 1)
󵄩2
󵄩
× 󵄩󵄩󵄩𝑥𝑛 − 𝑦𝑛 󵄩󵄩󵄩
󵄩
󵄩 󵄩
󵄩
+2𝜆 𝑛 𝛼𝑛 ‖𝑢‖ (󵄩󵄩󵄩𝑦𝑛 − 𝑢󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑡𝑛 − 𝑢󵄩󵄩󵄩) ]
which hence implies that
2
󵄩2
󵄩
(1 + 𝛾𝑛 ) (𝑏2 (𝛼𝑛 + 𝐿) − 1) 󵄩󵄩󵄩𝑥𝑛 − 𝑦𝑛 󵄩󵄩󵄩
2
󵄩
󵄩2
≤ (1 + 𝛾𝑛 ) (𝜆2𝑛 (𝛼𝑛 + 𝐿) − 1) 󵄩󵄩󵄩𝑥𝑛 − 𝑦𝑛 󵄩󵄩󵄩
󵄩
󵄩2 󵄩
󵄩2
󵄩2
󵄩
≤ 󵄩󵄩󵄩𝑥𝑛 − 𝑢󵄩󵄩󵄩 − 󵄩󵄩󵄩𝑧𝑛 − 𝑢󵄩󵄩󵄩 + 𝛾𝑛 󵄩󵄩󵄩𝑥𝑛 − 𝑢󵄩󵄩󵄩
󵄩
󵄩 󵄩
󵄩
+ 2𝛼𝑛 𝜆 𝑛 (1 + 𝛾𝑛 ) ‖𝑢‖ (󵄩󵄩󵄩𝑦𝑛 − 𝑢󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑡𝑛 − 𝑢󵄩󵄩󵄩)
󵄩
󵄩2
+ 𝜎𝑛 󵄩󵄩󵄩𝑄𝑡𝑛 − 𝑢󵄩󵄩󵄩 + 𝑐𝑛
󵄩
󵄩
󵄩
󵄩 󵄩
󵄩 󵄩
󵄩2
≤ (󵄩󵄩󵄩𝑥𝑛 − 𝑢󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑧𝑛 − 𝑢󵄩󵄩󵄩) 󵄩󵄩󵄩𝑥𝑛 − 𝑧𝑛 󵄩󵄩󵄩 + 𝛾𝑛 󵄩󵄩󵄩𝑥𝑛 − 𝑢󵄩󵄩󵄩
󵄩
󵄩 󵄩
󵄩
+ 2𝛼𝑛 𝜆 𝑛 (1 + 𝛾𝑛 ) ‖𝑢‖ (󵄩󵄩󵄩𝑦𝑛 − 𝑢󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑡𝑛 − 𝑢󵄩󵄩󵄩)
󵄩
󵄩2
+ 𝜎𝑛 󵄩󵄩󵄩𝑄𝑡𝑛 − 𝑢󵄩󵄩󵄩 + 𝑐𝑛 .
(79)
Since ‖𝑥𝑛 −𝑧𝑛 ‖ → 0, 𝛼𝑛 → 0, 𝛾𝑛 → 0, 𝜎𝑛 → 0, and 𝑐𝑛 → 0,
from the boundedness of {𝑥𝑛 }, {𝑦𝑛 }, {𝑡𝑛 }, and {𝑧𝑛 }, it follows
that
󵄩
󵄩
lim 󵄩󵄩𝑥 − 𝑦𝑛 󵄩󵄩󵄩 = 0.
(80)
𝑛→∞ 󵄩 𝑛
Meantime, utilizing the arguments similar to those in (69),
we have
󵄩󵄩
󵄩2
󵄩󵄩𝑡𝑛 − 𝑢󵄩󵄩󵄩
󵄩2 󵄩
󵄩2
󵄩
󵄩2 󵄩
≤ 󵄩󵄩󵄩𝑥𝑛 − 𝑢󵄩󵄩󵄩 − 󵄩󵄩󵄩𝑥𝑛 − 𝑦𝑛 󵄩󵄩󵄩 − 󵄩󵄩󵄩𝑦𝑛 − 𝑡𝑛 󵄩󵄩󵄩
󵄩󵄩
󵄩
󵄩
+ 2𝜆 𝑛 (𝛼𝑛 + 𝐿) 󵄩󵄩󵄩𝑥𝑛 − 𝑦𝑛 󵄩󵄩󵄩 󵄩󵄩󵄩𝑡𝑛 − 𝑦𝑛 󵄩󵄩󵄩
󵄩
󵄩 󵄩
󵄩
+ 2𝜆 𝑛 𝛼𝑛 ‖𝑢‖ (󵄩󵄩󵄩𝑦𝑛 − 𝑢󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑡𝑛 − 𝑢󵄩󵄩󵄩)
󵄩
󵄩2 󵄩
󵄩2
󵄩2 󵄩
(81)
≤ 󵄩󵄩󵄩𝑥𝑛 − 𝑢󵄩󵄩󵄩 − 󵄩󵄩󵄩𝑥𝑛 − 𝑦𝑛 󵄩󵄩󵄩 − 󵄩󵄩󵄩𝑦𝑛 − 𝑡𝑛 󵄩󵄩󵄩
2󵄩
󵄩2 󵄩
󵄩2
+ 𝜆2𝑛 (𝛼𝑛 + 𝐿) 󵄩󵄩󵄩𝑡𝑛 − 𝑦𝑛 󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑥𝑛 − 𝑦𝑛 󵄩󵄩󵄩
󵄩
󵄩 󵄩
󵄩
+ 2𝜆 𝑛 𝛼𝑛 ‖𝑢‖ (󵄩󵄩󵄩𝑦𝑛 − 𝑢󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑡𝑛 − 𝑢󵄩󵄩󵄩)
2
󵄩
󵄩2
󵄩2
󵄩
= 󵄩󵄩󵄩𝑥𝑛 − 𝑢󵄩󵄩󵄩 + (𝜆2𝑛 (𝛼𝑛 + 𝐿) − 1) 󵄩󵄩󵄩𝑡𝑛 − 𝑦𝑛 󵄩󵄩󵄩
󵄩
󵄩 󵄩
󵄩
+ 2𝜆 𝑛 𝛼𝑛 ‖𝑢‖ (󵄩󵄩󵄩𝑦𝑛 − 𝑢󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑡𝑛 − 𝑢󵄩󵄩󵄩) .
Substituting (81) in (70), we get
󵄩󵄩󵄩𝑧𝑛 − 𝑢󵄩󵄩󵄩2
󵄩
󵄩
󵄩
󵄩
󵄩2
󵄩2
≤ (1 + 𝛾𝑛 ) 󵄩󵄩󵄩𝑡𝑛 − 𝑢󵄩󵄩󵄩 + 𝜎𝑛 󵄩󵄩󵄩𝑄𝑡𝑛 − 𝑢󵄩󵄩󵄩 + 𝑐𝑛
2
󵄩2
󵄩
󵄩2
󵄩
≤ (1 + 𝛾𝑛 ) [󵄩󵄩󵄩𝑥𝑛 − 𝑢󵄩󵄩󵄩 + (𝜆2𝑛 (𝛼𝑛 + 𝐿) − 1) 󵄩󵄩󵄩𝑡𝑛 − 𝑦𝑛 󵄩󵄩󵄩
󵄩
󵄩 󵄩
󵄩
+2𝜆 𝑛 𝛼𝑛 ‖𝑢‖ (󵄩󵄩󵄩𝑦𝑛 − 𝑢󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑡𝑛 − 𝑢󵄩󵄩󵄩)]
󵄩
󵄩2
+ 𝜎𝑛 󵄩󵄩󵄩𝑄𝑡𝑛 − 𝑢󵄩󵄩󵄩 + 𝑐𝑛
󵄩
󵄩2
+ 𝜎𝑛 󵄩󵄩󵄩𝑄𝑡𝑛 − 𝑢󵄩󵄩󵄩 + 𝑐𝑛
2
󵄩
󵄩
󵄩2
󵄩2
= 󵄩󵄩󵄩𝑥𝑛 − 𝑢󵄩󵄩󵄩 + 𝛾𝑛 󵄩󵄩󵄩𝑥𝑛 − 𝑢󵄩󵄩󵄩 + (1 + 𝛾𝑛 ) (𝜆2𝑛 (𝛼𝑛 + 𝐿) − 1)
2
󵄩
󵄩
󵄩2
󵄩2
= 󵄩󵄩󵄩𝑥𝑛 − 𝑢󵄩󵄩󵄩 + 𝛾𝑛 󵄩󵄩󵄩𝑥𝑛 − 𝑢󵄩󵄩󵄩 + (1 + 𝛾𝑛 ) (𝜆2𝑛 (𝛼𝑛 + 𝐿) − 1)
󵄩2
󵄩
󵄩 󵄩
󵄩
󵄩
× 󵄩󵄩󵄩𝑥𝑛 − 𝑦𝑛 󵄩󵄩󵄩 + 2𝛼𝑛 𝜆 𝑛 (1 + 𝛾𝑛 ) ‖𝑢‖ (󵄩󵄩󵄩𝑦𝑛 − 𝑢󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑡𝑛 − 𝑢󵄩󵄩󵄩)
󵄩2
󵄩
󵄩 󵄩
󵄩
󵄩
× 󵄩󵄩󵄩𝑡𝑛 − 𝑦𝑛 󵄩󵄩󵄩 + 2𝛼𝑛 𝜆 𝑛 (1 + 𝛾𝑛 ) ‖𝑢‖ (󵄩󵄩󵄩𝑦𝑛 − 𝑢󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑡𝑛 − 𝑢󵄩󵄩󵄩)
󵄩
󵄩2
+ 𝜎𝑛 󵄩󵄩󵄩𝑄𝑡𝑛 − 𝑢󵄩󵄩󵄩 + 𝑐𝑛 ,
󵄩
󵄩2
+ 𝜎𝑛 󵄩󵄩󵄩𝑄𝑡𝑛 − 𝑢󵄩󵄩󵄩 + 𝑐𝑛 ,
(78)
(82)
Abstract and Applied Analysis
13
‖𝑥𝑛 − 𝑆𝑛 𝑥𝑛 ‖ → 0 as 𝑛 → ∞, and 𝑆 is uniformly continuous,
we obtain from Lemma 12 that ‖𝑥𝑛 − 𝑆𝑥𝑛 ‖ → 0 as 𝑛 → ∞.
which hence implies that
2
󵄩2
󵄩
(1 + 𝛾𝑛 ) (𝑏2 (𝛼𝑛 + 𝐿) − 1) 󵄩󵄩󵄩𝑡𝑛 − 𝑦𝑛 󵄩󵄩󵄩
≤ (1 +
𝛾𝑛 ) (𝜆2𝑛 (𝛼𝑛
Step 4. 𝜔𝑤 (𝑥𝑛 ) ⊂ Fix(𝑆) ∩ Γ.
Indeed, repeating the same arguments as in Step 4 of the
proof of Theorem 18, we can derive the desired conclusion.
󵄩2
󵄩
+ 𝐿) − 1) 󵄩󵄩󵄩𝑡𝑛 − 𝑦𝑛 󵄩󵄩󵄩
2
󵄩
󵄩
󵄩2 󵄩
󵄩2
󵄩2
≤ 󵄩󵄩󵄩𝑥𝑛 − 𝑢󵄩󵄩󵄩 − 󵄩󵄩󵄩𝑧𝑛 − 𝑢󵄩󵄩󵄩 + 𝛾𝑛 󵄩󵄩󵄩𝑥𝑛 − 𝑢󵄩󵄩󵄩
Step 5. {𝑥𝑛 }, {𝑦𝑛 }, and {𝑧𝑛 } converge strongly to 𝑞 = 𝑃Fix(𝑆)∩Γ 𝑥.
Indeed, take 𝑢̂ ∈ 𝜔𝑤 (𝑥𝑛 ) arbitrarily. Then, 𝑢̂ ∈ Fix(𝑆) ∩
Γ according to Step 4. Moreover, there exists a subsequence
{𝑥𝑛𝑖 } of {𝑥𝑛 } such that 𝑥𝑛𝑖 ⇀ 𝑢̂. Hence, from 𝑞 = 𝑃Fix(𝑆)∩Γ 𝑥,
𝑢̂ ∈ Fix(𝑆) ∩ Γ, and (71), we have
󵄩
󵄩 󵄩
󵄩
+ 2𝛼𝑛 𝜆 𝑛 (1 + 𝛾𝑛 ) ‖𝑢‖ (󵄩󵄩󵄩𝑦𝑛 − 𝑢󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑡𝑛 − 𝑢󵄩󵄩󵄩)
󵄩
󵄩2
+ 𝜎𝑛 󵄩󵄩󵄩𝑄𝑡𝑛 − 𝑢󵄩󵄩󵄩 + 𝑐𝑛
󵄩
󵄩
󵄩
󵄩󵄩
𝑢 − 𝑥‖ ≤ lim inf 󵄩󵄩󵄩󵄩𝑥𝑛𝑖 − 𝑥󵄩󵄩󵄩󵄩
󵄩󵄩𝑞 − 𝑥󵄩󵄩󵄩 ≤ ‖̂
𝑖→∞
󵄩
󵄩 󵄩
󵄩
≤ lim sup 󵄩󵄩󵄩󵄩𝑥𝑛𝑖 − 𝑥󵄩󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩𝑞 − 𝑥󵄩󵄩󵄩 .
󵄩
󵄩
󵄩
󵄩 󵄩
󵄩 󵄩
󵄩2
≤ (󵄩󵄩󵄩𝑥𝑛 − 𝑢󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑧𝑛 − 𝑢󵄩󵄩󵄩) 󵄩󵄩󵄩𝑥𝑛 − 𝑧𝑛 󵄩󵄩󵄩 + 𝛾𝑛 󵄩󵄩󵄩𝑥𝑛 − 𝑢󵄩󵄩󵄩
󵄩
󵄩 󵄩
󵄩
+ 2𝛼𝑛 𝜆 𝑛 (1 + 𝛾𝑛 ) ‖𝑢‖ (󵄩󵄩󵄩𝑦𝑛 − 𝑢󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑡𝑛 − 𝑢󵄩󵄩󵄩)
󵄩
󵄩2
+ 𝜎𝑛 󵄩󵄩󵄩𝑄𝑡𝑛 − 𝑢󵄩󵄩󵄩 + 𝑐𝑛 .
𝑖→∞
So, we obtain
󵄩󵄩
󵄩
− 𝑦𝑛 󵄩󵄩󵄩 = 0.
(84)
𝑖→∞ 󵄩
󵄩
󵄩
󵄩
󵄩
𝛿 󵄩󵄩󵄩𝑆𝑛 𝑡𝑛 − 𝑡𝑛 󵄩󵄩󵄩 ≤ 𝛽𝑛 󵄩󵄩󵄩𝑆𝑛 𝑡𝑛 − 𝑡𝑛 󵄩󵄩󵄩
(91)
≥ ⟨𝑞 − 𝑥𝑛𝑖 , 𝑥 − 𝑞⟩ .
(85)
Since 𝑧𝑛 = (1 − 𝜎𝑛 − 𝛽𝑛 )𝑡𝑛 + 𝜎𝑛 𝑄𝑡𝑛 + 𝛽𝑛 𝑆𝑛 𝑡𝑛 , we have
𝛽𝑛 (𝑆𝑛 𝑡𝑛 − 𝑡𝑛 ) = (𝑧𝑛 − 𝑡𝑛 ) − 𝜎𝑛 (𝑄𝑡𝑛 − 𝑡𝑛 ). Then,
(90)
From 𝑥𝑛𝑖 − 𝑥 ⇀ 𝑢̂ − 𝑥, we have 𝑥𝑛𝑖 − 𝑥 → 𝑢̂ − 𝑥 due to the
Kadec-Klee property of Hilbert spaces [17]. So, it is clear that
𝑥𝑛𝑖 → 𝑢̂. Since 𝑥𝑛 = 𝑃𝑄𝑛 𝑥 and 𝑞 ∈ Fix(𝑆)∩Γ ⊂ 𝐶𝑛 ∩𝑄𝑛 ⊂ 𝑄𝑛 ,
we have
󵄩2
󵄩
− 󵄩󵄩󵄩󵄩𝑞 − 𝑥𝑛𝑖 󵄩󵄩󵄩󵄩 = ⟨𝑞 − 𝑥𝑛𝑖 , 𝑥𝑛𝑖 − 𝑥⟩ + ⟨𝑞 − 𝑥𝑛𝑖 , 𝑥 − 𝑞⟩
This together with ‖𝑥𝑛 − 𝑦𝑛 ‖ → 0 implies that
󵄩
󵄩
lim 󵄩󵄩𝑥 − 𝑡𝑛 󵄩󵄩󵄩 = 0.
𝑛→∞ 󵄩 𝑛
󵄩
󵄩
lim 󵄩󵄩󵄩𝑥𝑛𝑖 − 𝑥󵄩󵄩󵄩󵄩 = ‖̂
𝑢 − 𝑥‖ .
(83)
Since ‖𝑥𝑛 −𝑧𝑛 ‖ → 0, 𝛼𝑛 → 0, 𝛾𝑛 → 0, 𝜎𝑛 → 0, and 𝑐𝑛 → 0,
from the boundedness of {𝑥𝑛 }, {𝑦𝑛 }, {𝑡𝑛 }, and {𝑧𝑛 }, it follows
that
lim 󵄩𝑡
𝑛→∞ 󵄩 𝑛
(89)
As 𝑖 → ∞, we obtain −‖𝑞 − 𝑢̂‖2 ≥ ⟨𝑞 − 𝑢̂, 𝑥 − 𝑞⟩ ≥ 0 by
𝑞 = 𝑃Fix(𝑆)∩Γ 𝑥 and 𝑢̂ ∈ Fix(𝑆) ∩ Γ. Hence, we have 𝑢̂ = 𝑞.
This implies that 𝑥𝑛 → 𝑞. It is easy to see that 𝑦𝑛 → 𝑞 and
𝑧𝑛 → 𝑞. This completes the proof.
󵄩
󵄩
󵄩
󵄩
≤ 󵄩󵄩󵄩𝑧𝑛 − 𝑡𝑛 󵄩󵄩󵄩 + 𝜎𝑛 󵄩󵄩󵄩𝑄𝑡𝑛 − 𝑡𝑛 󵄩󵄩󵄩
Acknowledgments
󵄩
󵄩 󵄩
󵄩
󵄩
󵄩
≤ 󵄩󵄩󵄩𝑧𝑛 − 𝑥𝑛 󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑥𝑛 − 𝑡𝑛 󵄩󵄩󵄩 + 𝜎𝑛 󵄩󵄩󵄩𝑄𝑡𝑛 − 𝑡𝑛 󵄩󵄩󵄩
In this research, first author was partially supported by the
National Science Foundation of China (11071169), Innovation
Program of Shanghai Municipal Education Commission
(09ZZ133), and Ph.D. Profram Foundation of Ministry of
Education of China (20123127110002). The research part of
the second author was done during his visit to KFUPM,
Dhahran, Saudi Arabia. The third author was partially supported by a Grant from the NSC 101-2115-M-037-001.
(86)
and hence ‖𝑡𝑛 − 𝑆𝑛 𝑡𝑛 ‖ → 0. Furthermore, observe that
󵄩
󵄩 󵄩
󵄩󵄩
𝑛 󵄩
𝑛 󵄩 󵄩 𝑛
𝑛 󵄩
󵄩󵄩𝑥𝑛 − 𝑆 𝑥𝑛 󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩𝑥𝑛 − 𝑡𝑛 󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑡𝑛 − 𝑆 𝑡𝑛 󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑆 𝑡𝑛 − 𝑆 𝑥𝑛 󵄩󵄩󵄩 .
(87)
Utilizing Lemma 11, we have
References
󵄩󵄩 𝑛
𝑛 󵄩
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1
󵄩
󵄩
≤
(𝜅 󵄩󵄩󵄩𝑡𝑛 − 𝑥𝑛 󵄩󵄩󵄩
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󵄩2
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(88)
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