Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2013, Article ID 378750, 19 pages
http://dx.doi.org/10.1155/2013/378750
Research Article
The Mann-Type Extragradient Iterative Algorithms with
Regularization for Solving Variational Inequality Problems,
Split Feasibility, and Fixed Point Problems
Lu-Chuan Ceng,1 Himanshu Gupta,2 and Ching-Feng Wen3
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
Center for Fundamental Science, Kaohsiung Medical University, Kaohsiung 807, Taiwan
Correspondence should be addressed to Ching-Feng Wen; cfwen@kmu.edu.tw
Received 3 December 2012; Accepted 31 December 2012
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.
The purpose of this paper is to introduce and analyze the Mann-type extragradient iterative algorithms with regularization for
finding a common element of the solution set Ξ of a general system of variational inequalities, the solution set Γ of a split feasibility
problem, and the fixed point set Fix(𝑆) of a strictly pseudocontractive mapping 𝑆 in the setting of the Hilbert spaces. These
iterative algorithms are based on the regularization method, the Mann-type iteration method, and the extragradient method due
to Nadezhkina and Takahashi (2006). Furthermore, we prove that the sequences generated by the proposed algorithms converge
weakly to an element of Fix(𝑆) ∩ Ξ ∩ Γ under mild conditions.
1. Introduction
Let H be a real Hilbert space with inner product ⟨⋅, ⋅⟩ and
norm ‖ ⋅ ‖. Let 𝐶 be a nonempty closed convex subset of H.
The projection (nearest point or metric projection) of H onto
𝐶 is denoted by 𝑃𝐶. Let 𝑆 : 𝐶 → 𝐶 be a mapping and Fix(𝑆)
be the set of fixed points of 𝑆. For a given nonlinear operator
𝐴 : 𝐶 → H, we consider the following variational inequality
problem (VIP) of finding 𝑥∗ ∈ 𝐶 such that
⟨𝐴𝑥∗ , 𝑥 − 𝑥∗ ⟩ ≥ 0,
∀𝑥 ∈ 𝐶.
(1)
The solution set of VIP (1) is denoted by VI(𝐶, 𝐴). The
theory of variational inequalities has been studied quite
extensively and has emerged as an important tool in the study
of a wide class of problems from mechanics, optimization,
engineering, science, and social sciences. It is well known that
the VIP is equivalent to a fixed point problem. This alternative
formulation has been used to suggest and analyze projection
iterative method for solving variational inequalities under
the conditions that the involved operator must be strongly
monotone and Lipschitz continuous. In the recent past,
several people have studied and proposed several iterative
methods to find a solution of variational inequalities which is
also a fixed point of a nonexpansive mapping or strict pseudocontractive mapping; see, for example, [1–9] and the references therein.
For finding an element of Fix(𝑆) ∩ VI(𝐶, 𝐴) when 𝐶 is
closed and convex, 𝑆 is nonexpansive, and 𝐴 is 𝛼-inverse
strongly monotone, Takahashi and Toyoda [10] introduced
the following Mann-type iterative algorithm:
𝑥𝑛+1 = 𝛼𝑛 𝑥𝑛 + (1 − 𝛼𝑛 ) 𝑆𝑃𝐶 (1 − 𝜆 𝑛 𝐴𝑥𝑛 ) ,
∀𝑛 ≥ 0,
(2)
where 𝑃𝐶 is the metric projection of H onto 𝐶, 𝑥0 = 𝑥 ∈
𝐶, {𝛼𝑛 } is a sequence in (0, 1), and {𝜆 𝑛 } is a sequence in
(0, 2𝛼). They showed that if Fix(𝑆)∩VI(𝐶, 𝐴) ≠ 0, then the sequence {𝑥𝑛 } converges weakly to some 𝑧 ∈ Fix(𝑆) ∩ VI(𝐶, 𝐴).
Nadezhkina and Takahashi [9] and Zeng and Yao [8] proposed extragradient methods motivated by Korpelevič [11]
for finding a common element of the fixed point set of a
nonexpansive mapping and the solution set of a variational
inequality problem. Further, these iterative methods are
2
Abstract and Applied Analysis
extended in [12] to develop a new iterative method for finding
elements in Fix(𝑆) ∩ VI(𝐶, 𝐴).
Let 𝐵1 , 𝐵2 : 𝐶 → H be two mappings. Recently, Ceng
et al. [4] introduced and considered the following problem of
finding (𝑥∗ , 𝑦∗ ) ∈ 𝐶 × 𝐶 such that
⟨𝜇1 𝐵1 𝑦∗ + 𝑥∗ − 𝑦∗ , 𝑥 − 𝑥∗ ⟩ ≥ 0,
∀𝑥 ∈ 𝐶,
⟨𝜇2 𝐵2 𝑥∗ + 𝑦∗ − 𝑥∗ , 𝑥 − 𝑦∗ ⟩ ≥ 0,
∀𝑥 ∈ 𝐶,
(3)
𝑦𝑛 = 𝛼𝑛 𝑄𝑥𝑛
(4)
where 𝑦 = 𝑃𝐶(𝑥 − 𝜇2 𝐵2 𝑥).
(5)
− 𝜇1 𝐵1 𝑃𝐶 (𝑧𝑛 − 𝜇2 𝐵2 𝑧𝑛 )] ,
𝑥𝑛+1 = 𝛽𝑛 𝑥𝑛 + 𝛾𝑛 𝑦𝑛 + 𝛿𝑛 𝑆𝑦𝑛 ,
Lemma 1 (see [4]). For given 𝑥, 𝑦 ∈ 𝐶, (𝑥, 𝑦) is a solution
of problem (3) if and only if 𝑥 is a fixed point of the mapping
𝐺 : 𝐶 → 𝐶 defined by
∀𝑥 ∈ 𝐶,
𝑧𝑛 = 𝑃𝐶 (𝑥𝑛 − 𝜆 𝑛 𝐴𝑥𝑛 ) ,
+ (1 − 𝛼𝑛 ) 𝑃𝐶 [𝑃𝐶 (𝑧𝑛 − 𝜇2 𝐵2 𝑧𝑛 )
which is called a general system of variational inequalities
(GSVI), where 𝜇1 > 0 and 𝜇2 > 0 are two constants. The
set of solutions of problem (3) is denoted by GSVI(𝐶, 𝐵1 , 𝐵2 ).
In particular, if 𝐵1 = 𝐵2 , then problem (3) reduces to the
new system of variational inequalities (NSVI), introduced
and studied by Verma [13]. Further, if 𝑥∗ = 𝑦∗ , then the NSVI
reduces to VIP (1).
Recently, Ceng et al. [4] transformed problem (3) into a
fixed point problem in the following way.
𝐺 (𝑥) = 𝑃𝐶 [𝑃𝐶 (𝑥 − 𝜇2 𝐵2 𝑥) − 𝜇1 𝐵1 𝑃𝐶 (𝑥 − 𝜇2 𝐵2 𝑥)] ,
𝜌 ∈ [0, 1/2). For given 𝑥0 ∈ 𝐶 arbitrarily, let the sequences
{𝑥𝑛 }, {𝑦𝑛 }, and {𝑧𝑛 } be generated iteratively by
∀𝑛 ≥ 0,
where 𝜇𝑖 ∈ (0, 2𝛽𝑖 ) for 𝑖 = 1, 2, {𝜆 𝑛 } ⊂ (0, 2𝛼] and {𝛼𝑛 }, {𝛽𝑛 },
{𝛾𝑛 }, {𝛿𝑛 } ⊂ [0, 1] such that
(i) 𝛽𝑛 + 𝛾𝑛 + 𝛿𝑛 = 1 and (𝛾𝑛 + 𝛿𝑛 )𝑘 ≤ 𝛾𝑛 for all 𝑛 ≥ 0;
(ii) lim𝑛 → ∞ 𝛼𝑛 = 0 and ∑∞
𝑛=0 𝛼𝑛 = ∞;
(iii) 0 < lim inf 𝑛 → ∞ 𝛽𝑛 ≤ lim sup𝑛 → ∞ 𝛽𝑛 < 1 and
lim inf 𝑛 → ∞ 𝛿𝑛 > 0;
(iv) lim𝑛 → ∞ (𝛾𝑛+1 /(1 − 𝛽𝑛+1 ) − 𝛾𝑛 /(1 − 𝛽𝑛 )) = 0;
(v) 0 < lim inf 𝑛 → ∞ 𝜆 𝑛 ≤ lim sup𝑛 → ∞ 𝜆 𝑛 < 2𝛼 and
lim𝑛 → ∞ |𝜆 𝑛+1 − 𝜆 𝑛 | = 0.
Then the sequence {𝑥𝑛 } generated by (5) converges strongly to
𝑥 = 𝑃Fix(𝑆)∩Ξ∩VI(𝐶,𝐴) 𝑄𝑥 and (𝑥, 𝑦) is a solution of GSVI (3),
where 𝑦 = 𝑃𝐶(𝑥 − 𝜇2 𝐵2 𝑥).
On the other hand, let 𝐶 and 𝑄 be nonempty closed convex subsets of real Hilbert spaces H1 and H2 , respectively.
The split feasibility problem (SFP) is to find a point 𝑥∗ with
the following property:
𝑥∗ ∈ 𝐶,
𝐴𝑥∗ ∈ 𝑄,
(6)
In particular, if the mapping 𝐵𝑖 : 𝐶 → H is 𝛽𝑖 -inverse
strongly monotone for 𝑖 = 1, 2, then the mapping 𝐺 is
nonexpansive provided 𝜇𝑖 ∈ (0, 2𝛽𝑖 ) for 𝑖 = 1, 2.
Utilizing Lemma 1, they introduced and studied a relaxed
extragradient method for solving GSVI (3).
Throughout this paper, unless otherwise specified, the set
of fixed points of the mapping 𝐺 is denoted by Ξ. Based on the
relaxed extragradient method and viscosity approximation
method, Yao et al. [7] proposed and analyzed an iterative
algorithm for finding a common solution of GSVI (3) and
fixed point problem of a strictly pseudocontractive mapping
𝑆 : 𝐶 → 𝐶, where 𝐶 is a nonempty bounded closed convex
subset of a real Hilbert space H.
Subsequently, Ceng at al. [14] further presented and
analyzed an iterative scheme for finding a common element
of the solution set of VIP (1), the solution set of GSVI (3),
and fixed point set of a strictly pseudocontractive mapping
𝑆 : 𝐶 → 𝐶.
where 𝐴 ∈ 𝐵(H1 , H2 ) and 𝐵(H1 , H2 ) denotes the family of
all bounded linear operators from H1 to H2 .
In 1994, the SFP was first introduced by Censor and
Elfving [15], in finite-dimensional Hilbert spaces, for modeling inverse problems which arise from phase retrievals
and in medical image reconstruction. A number of image
reconstruction problems can be formulated as the SFP; see,
for example, [16] and the references therein. Recently, it is
found that the SFP can also be applied to study intensitymodulated radiation therapy; see, for example, [17–19] and
the references therein. In the recent past, a wide variety
of iterative methods have been used in signal processing
and image reconstruction and for solving the SFP; see, for
example, [16–26] and the references therein. A special case
of the SFP is the following convex constrained linear inverse
problem [27] of finding an element 𝑥 such that
Theorem 2 (see [14, Theorem 3.1]). Let 𝐶 be a nonempty
closed convex subset of a real Hilbert space H. Let 𝐴 : 𝐶 → H
be 𝛼-inverse strongly monotone, and let 𝐵𝑖 : 𝐶 → H be
𝛽𝑖 -inverse strongly monotone for 𝑖 = 1, 2. Let 𝑆 : 𝐶 → 𝐶
be a 𝑘-strictly pseudocontractive mapping such that Fix(𝑆) ∩
Ξ ∩ VI(𝐶, 𝐴) ≠ 0. Let 𝑄 : 𝐶 → 𝐶 be a 𝜌-contraction with
It has been extensively investigated in the literature using the
projected Landweber iterative method [28]. Comparatively,
the SFP has received much less attention so far, due to
the complexity resulting from the set 𝑄. Therefore, whether
various versions of the projected Landweber iterative method
[28] can be extended to solve the SFP remains an interesting
𝑥 ∈ 𝐶,
𝐴𝑥 = 𝑏.
(7)
Abstract and Applied Analysis
3
open topic. For example, it is yet not clear whether the dual
approach to (7) of [29] can be extended to the SFP. The
original algorithm given in [15] involves the computation of
the inverse 𝐴−1 (assuming the existence of the inverse of 𝐴),
and thus has not become popular. A seemingly more popular
algorithm that solves the SFP is the 𝐶𝑄 algorithm of Byrne
[16, 21] which is found to be a gradient-projection method
(GPM) in convex minimization. It is also a special case of the
proximal forward-backward splitting method [30]. The 𝐶𝑄
algorithm only involves the computation of the projections
𝑃𝐶 and 𝑃𝑄 onto the sets 𝐶 and 𝑄, respectively, and is therefore
implementable in the case where 𝑃𝐶 and 𝑃𝑄 have closed-form
expressions; for example, 𝐶 and 𝑄 are closed balls or halfspaces. However, it remains a challenge how to implement the
𝐶𝑄 algorithm in the case where the projections 𝑃𝐶 and/or 𝑃𝑄
fail to have closed-form expressions, though theoretically we
can prove the (weak) convergence of the algorithm.
Very recently, Xu [20] gave a continuation of the study
on the 𝐶𝑄 algorithm and its convergence. He applied Mann’s
algorithm to the SFP and purposed an averaged 𝐶𝑄 algorithm
which was proved to be weakly convergent to a solution of the
SFP. He also established the strong convergence result, which
shows that the minimum-norm solution can be obtained.
Furthermore, Korpelevič [11] introduced the so-called
extragradient method for finding a solution of a saddle point
problem. He proved that the sequences generated by the
proposed iterative algorithm converge to a solution of the
saddle point problem.
Throughout this paper, assume that the SFP is consistent;
that is, the solution set Γ of the SFP is nonempty. Let 𝑓 :
H1 → R be a continuous differentiable function. The minimization problem
1󵄩
󵄩2
min𝑓 (𝑥) := 󵄩󵄩󵄩𝐴𝑥 − 𝑃𝑄𝐴𝑥󵄩󵄩󵄩
𝑥∈𝐶
2
(8)
is ill posed. Therefore, Xu [20] considered the following
Tikhonov regularization problem:
1󵄩
󵄩2 1
min𝑓𝛼 (𝑥) := 󵄩󵄩󵄩𝐴𝑥 − 𝑃𝑄𝐴𝑥󵄩󵄩󵄩 + 𝛼‖𝑥‖2 ,
𝑥∈𝐶
2
2
(9)
where 𝛼 > 0 is the regularization parameter. The regularized
minimization (9) has a unique solution which is denoted by
𝑥𝛼 . The following results are easy to prove.
Proposition 3 (see [31, Proposition 3.1]). Given 𝑥∗ ∈ H1 , the
following statements are equivalent:
(i) 𝑥∗ solves the SFP;
(ii) 𝑥∗ solves the fixed point equation
𝑃𝐶 (𝐼 − 𝜆∇𝑓) 𝑥∗ = 𝑥∗ ,
(10)
where 𝜆 > 0, ∇𝑓 = 𝐴∗ (𝐼 − 𝑃𝑄)𝐴 and 𝐴∗ is the adjoint
of 𝐴;
(iii) 𝑥∗ solves the variational inequality problem (VIP) of
finding 𝑥∗ ∈ 𝐶 such that
⟨∇𝑓 (𝑥∗ ) , 𝑥 − 𝑥∗ ⟩ ≥ 0,
∀𝑥 ∈ 𝐶.
(11)
It is clear from Proposition 3 that
Γ = Fix (𝑃𝐶 (𝐼 − 𝜆∇𝑓)) = VI (𝐶, ∇𝑓)
(12)
for all 𝜆 > 0, where Fix(𝑃𝐶(𝐼 − 𝜆∇𝑓)) and VI(𝐶, ∇𝑓) denote
the set of fixed points of 𝑃𝐶(𝐼 − 𝜆∇𝑓) and the solution set of
VIP (11), respectively.
Proposition 4 (see [31]). The following statements hold:
(i) the gradient
∇𝑓𝛼 = ∇𝑓 + 𝛼𝐼 = 𝐴∗ (𝐼 − 𝑃𝑄) 𝐴 + 𝛼𝐼
(13)
is (𝛼+‖𝐴‖2 )-Lipschitz continuous and 𝛼-strongly monotone;
(ii) the mapping 𝑃𝐶(𝐼 − 𝜆∇𝑓𝛼 ) is a contraction with coefficient
2
1
√ 1 − 𝜆 (2𝛼 − 𝜆(‖𝐴‖2 + 𝛼) ) (≤ √1 − 𝛼𝜆 ≤ 1 − 𝛼𝜆) ,
2
(14)
where 0 < 𝜆 ≤ 𝛼/(‖𝐴‖2 + 𝛼)2 ;
(iii) if the SFP is consistent, then the strong lim𝛼 → 0 𝑥𝛼 exists
and is the minimum-norm solution of the SFP.
Very recently, by combining the regularization method
and extragradient method due to Nadezhkina and Takahashi
[32], Ceng et al. [31] proposed an extragradient algorithm
with regularization and proved that the sequences generated
by the proposed algorithm converge weakly to an element of
Fix(𝑆) ∩ Γ, where 𝑆 : 𝐶 → 𝐶 is a nonexpansive mapping.
Theorem 5 (see [31, Theorem 3.1]). Let 𝑆 : 𝐶 → 𝐶 be a
nonexpansive mapping such that Fix(𝑆) ∩ Γ ≠ 0. Let {𝑥𝑛 } and
{𝑦𝑛 } be the sequences in 𝐶 generated by the following extragradient algorithm:
𝑥0 = 𝑥 ∈ 𝐶
chosen arbitrarily,
𝑦𝑛 = 𝑃𝐶 (𝑥𝑛 − 𝜆 𝑛 ∇𝑓𝛼𝑛 (𝑥𝑛 )) ,
𝑥𝑛+1 = 𝛽𝑛 𝑥𝑛 + (1 − 𝛽𝑛 ) 𝑆𝑃𝐶 (𝑥𝑛 − 𝜆 𝑛 ∇𝑓𝛼𝑛 (𝑦𝑛 )) ,
∀𝑛 ≥ 0,
(15)
2
where ∑∞
𝑛=0 𝛼𝑛 < ∞, {𝜆 𝑛 } ⊂ [𝑎, 𝑏] for some 𝑎, 𝑏 ∈ (0, 1/‖𝐴‖ )
and {𝛽𝑛 } ⊂ [𝑐, 𝑑] for some 𝑐, 𝑑 ∈ (0, 1). Then, both sequences
{𝑥𝑛 } and {𝑦𝑛 } converge weakly to an element 𝑥̂ ∈ Fix(𝑆) ∩ Γ.
Motivated and inspired by the research going on this area,
we propose and analyze the following Mann-type extragradient iterative algorithms with regularization for finding a
common element of the solution set of the GSVI (3), the
solution set of the SFP (6), and the fixed point set of a strictly
pseudocontractive mapping 𝑆 : 𝐶 → 𝐶.
Algorithm 6. Let 𝜇𝑖 ∈ (0, 2𝛽𝑖 ) for 𝑖 = 1, 2, {𝛼𝑛 } ⊂ (0, ∞),
{𝜆 𝑛 } ⊂ (0, 1/‖𝐴‖2 ) and {𝜎𝑛 }, {𝜏𝑛 }, {𝛽𝑛 }, {𝛾𝑛 }, {𝛿𝑛 } ⊂ [0, 1] such
that 𝜎𝑛 + 𝜏𝑛 ≤ 1 and 𝛽𝑛 + 𝛾𝑛 + 𝛿𝑛 = 1 for all 𝑛 ≥ 0. For
4
Abstract and Applied Analysis
given 𝑥0 ∈ 𝐶 arbitrarily, let {𝑥𝑛 }, {𝑦𝑛 }, {𝑧𝑛 } be the sequences
generated by the Mann-type extragradient iterative scheme
with regularization
𝑧𝑛 = 𝑃𝐶 (𝑥𝑛 − 𝜆 𝑛 ∇𝑓𝛼𝑛 (𝑥𝑛 )) ,
𝑦𝑛 = 𝜎𝑛 𝑥𝑛 + 𝜏𝑛 𝑃𝐶 (𝑥𝑛 − 𝜆 𝑛 ∇𝑓𝛼𝑛 (𝑧𝑛 ))
+ (1 − 𝜎𝑛 − 𝜏𝑛 ) 𝑃𝐶 [𝑃𝐶 (𝑧𝑛 − 𝜇2 𝐵2 𝑧𝑛 )
(16)
∀𝑛 ≥ 0.
Under appropriate assumptions, it is proven that all the
sequences {𝑥𝑛 }, {𝑦𝑛 }, {𝑧𝑛 } converge weakly to an element 𝑥 ∈
Fix(𝑆) ∩ Ξ ∩ Γ. Furthermore, (𝑥, 𝑦) is a solution of the GSVI
(3), where 𝑦 = 𝑃𝐶(𝑥 − 𝜇2 𝐵2 𝑥).
Algorithm 7. Let 𝜇𝑖 ∈ (0, 2𝛽𝑖 ) for 𝑖 = 1, 2, {𝛼𝑛 } ⊂ (0, ∞),
{𝜆 𝑛 } ⊂ (0, 1/‖𝐴‖2 ) and {𝜎𝑛 }, {𝛽𝑛 }, {𝛾𝑛 }, {𝛿𝑛 } ⊂ [0, 1] such that
𝛽𝑛 + 𝛾𝑛 + 𝛿𝑛 = 1 for all 𝑛 ≥ 0. For given 𝑥0 ∈ 𝐶 arbitrarily, let
𝑢𝑛 } be the sequences generated by the Mann-type
{𝑥𝑛 }, {𝑢𝑛 }, {̃
extragradient iterative scheme with regularization
𝑢𝑛 = 𝑃𝐶 [𝑃𝐶 (𝑥𝑛 − 𝜇2 𝐵2 𝑥𝑛 ) − 𝜇1 𝐵1 𝑃𝐶 (𝑥𝑛 − 𝜇2 𝐵2 𝑥𝑛 )] ,
𝑢̃𝑛 = 𝑃𝐶 (𝑢𝑛 − 𝜆 𝑛 ∇𝑓𝛼𝑛 (𝑢𝑛 )) ,
𝑢𝑛 )) ,
𝑦𝑛 = 𝜎𝑛 𝑥𝑛 + (1 − 𝜎𝑛 ) 𝑃𝐶 (𝑢𝑛 − 𝜆 𝑛 ∇𝑓𝛼𝑛 (̃
𝑥𝑛+1 = 𝛽𝑛 𝑥𝑛 + 𝛾𝑛 𝑦𝑛 + 𝛿𝑛 𝑆𝑦𝑛 ,
Let H be a real Hilbert space, whose inner product and norm
are denoted by ⟨⋅, ⋅⟩ and ‖⋅‖, respectively. Let 𝐾 be a nonempty,
closed, and convex subset of H. Now we present some known
definitions and results which will be used in the sequel.
The metric (or nearest point) projection from H onto 𝐾
is the mapping 𝑃𝐾 : H → 𝐾 which assigns to each point
𝑥 ∈ H the unique point 𝑃𝐾 𝑥 ∈ 𝐾 satisfying the property
󵄩
󵄩󵄩
󵄩
󵄩
󵄩󵄩𝑥 − 𝑃𝐾 𝑥󵄩󵄩󵄩 = inf 󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩 =: 𝑑 (𝑥, 𝐾) .
𝑦∈𝐾
− 𝜇1 𝐵1 𝑃𝐶 (𝑧𝑛 − 𝜇2 𝐵2 𝑧𝑛 )] ,
𝑥𝑛+1 = 𝛽𝑛 𝑥𝑛 + 𝛾𝑛 𝑦𝑛 + 𝛿𝑛 𝑆𝑦𝑛 ,
2. Preliminaries
(18)
Some important properties of projections are gathered in
the following proposition.
Proposition 8. For given 𝑥 ∈ H and 𝑧 ∈ 𝐾:
(i) 𝑧 = 𝑃𝐾 𝑥 ⇔ ⟨𝑥 − 𝑧, 𝑦 − 𝑧⟩ ≤ 0, for all 𝑦 ∈ 𝐾;
(ii) 𝑧 = 𝑃𝐾 𝑥 ⇔ ‖𝑥 − 𝑧‖2 ≤ ‖𝑥 − 𝑦‖2 − ‖𝑦 − 𝑧‖2 , for all
𝑦 ∈ 𝐾;
(iii) ⟨𝑃𝐾 𝑥 − 𝑃𝐾 𝑦, 𝑥 − 𝑦⟩ ≥ ‖𝑃𝐾 𝑥 − 𝑃𝐾 𝑦‖2 , for all 𝑦 ∈
H, which hence implies that 𝑃𝐾 is nonexpansive and
monotone.
Definition 9. A mapping 𝑇 : H → H is said to be
(a) nonexpansive if
(17)
∀𝑛 ≥ 0.
Also, under mild conditions, it is shown that all the
𝑢𝑛 } converge weakly to an element 𝑥 ∈
sequences {𝑥𝑛 }, {𝑢𝑛 }, {̃
Fix(𝑆) ∩ Ξ ∩ Γ. Furthermore, (𝑥, 𝑦) is a solution of the GSVI
(3), where 𝑦 = 𝑃𝐶(𝑥 − 𝜇2 𝐵2 𝑥).
Observe that both [20, Theorem 5.7] and [31, Theorem
3.1] are weak convergence results for solving the SFP and so
are our results as well. But our problem of finding an element of Fix(𝑆) ∩ Ξ ∩ Γ is more general than the corresponding
ones in [20, Theorem 5.7] and [31, Theorem 3.1], respectively.
Hence, there is no doubt that our weak convergence results
are very interesting and quite valuable. Because the Manntype extragradient iterative schemes (16) and (17) with regularization involve two inverse strongly monotone mappings
𝐵1 and 𝐵2 , a 𝑘-strictly pseudocontractive self-mapping 𝑆 and
several parameter sequences, they are more flexible and more
subtle than the corresponding ones in [20, Theorem 5.7]
and [31, Theorem 3.1], respectively. Furthermore, the hybrid
extragradient iterative scheme (5) is extended to develop the
Mann-type extragradient iterative schemes (16) and (17) with
regularization. In our results, the Mann-type extragradient
iterative schemes (16) and (17) with regularization lack the
requirement of boundedness for the domain in which various
mappings are defined; see, for example, Yao et al. [7, Theorem
3.2]. Therefore, our results represent the modification, supplementation, extension, and improvement of [20, Theorem
5.7], [31, Theorem 3.1], [14, Theorem 3.1], and [7, Theorem
3.2].
󵄩
󵄩 󵄩
󵄩󵄩
󵄩󵄩𝑇𝑥 − 𝑇𝑦󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩 ,
∀𝑥, 𝑦 ∈ H;
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(b) firmly nonexpansive if 2𝑇 − 𝐼 is nonexpansive, or
equivalently,
󵄩2
󵄩
⟨𝑥 − 𝑦, 𝑇𝑥 − 𝑇𝑦⟩ ≥ 󵄩󵄩󵄩𝑇𝑥 − 𝑇𝑦󵄩󵄩󵄩 ,
∀𝑥, 𝑦 ∈ H;
(20)
alternatively, 𝑇 is firmly nonexpansive if and only if 𝑇
can be expressed as
𝑇=
1
(𝐼 + 𝑆) ,
2
(21)
where 𝑆 : H → H is nonexpansive; projections are
firmly nonexpansive.
Definition 10. Let 𝑇 be a nonlinear operator with domain
𝐷(𝑇) ⊆ H and range 𝑅(𝑇) ⊆ H.
(a) 𝑇 is said to be monotone if
⟨𝑥 − 𝑦, 𝑇𝑥 − 𝑇𝑦⟩ ≥ 0,
∀𝑥, 𝑦 ∈ 𝐷 (𝑇) .
(22)
(b) Given a number 𝛽 > 0, 𝑇 is said to be 𝛽-strongly
monotone if
󵄩2
󵄩
⟨𝑥 − 𝑦, 𝑇𝑥 − 𝑇𝑦⟩ ≥ 𝛽󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩 ,
∀𝑥, 𝑦 ∈ 𝐷 (𝑇) .
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(c) Given a number 𝜈 > 0, 𝑇 is said to be 𝜈-inverse
strongly monotone (𝜈-ism) if
󵄩2
󵄩
⟨𝑥 − 𝑦, 𝑇𝑥 − 𝑇𝑦⟩ ≥ 𝜈󵄩󵄩󵄩𝑇𝑥 − 𝑇𝑦󵄩󵄩󵄩 ,
∀𝑥, 𝑦 ∈ 𝐷 (𝑇) .
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Abstract and Applied Analysis
5
It can be easily seen that if 𝑆 is nonexpansive, then 𝐼 − 𝑆 is
monotone. It is also easy to see that a projection 𝑃𝐾 is 1-ism.
Inverse strongly monotone (also referred to as cocoercive) operators have been applied widely in solving practical
problems in various fields, for instance, in traffic assignment
problems; see, for example, [33, 34].
Definition 11. A mapping 𝑇 : H → H is said to be an averaged mapping if it can be written as the average of the identity
𝐼 and a nonexpansive mapping, that is,
𝑇 ≡ (1 − 𝛼) 𝐼 + 𝛼𝑆,
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where 𝛼 ∈ (0, 1) and 𝑆 : H → H is nonexpansive. More
precisely, when the last equality holds, we say that 𝑇 is 𝛼averaged. Thus, firmly nonexpansive mappings (in particular,
projections) are 1/2-averaged maps.
Proposition 12 (see [21]). Let 𝑇 : H → H be a given
mapping.
(i) 𝑇 is nonexpansive if and only if the complement 𝐼 − 𝑇
is 1/2-ism.
(ii) If 𝑇 is 𝜈-ism, then for 𝛾 > 0, 𝛾𝑇 is 𝜈/𝛾-ism.
(iii) 𝑇 is averaged if and only if the complement 𝐼 − 𝑇 is
𝜈-ism for some 𝜈 > 1/2. Indeed, for 𝛼 ∈ (0, 1), 𝑇 is
𝛼-averaged if and only if 𝐼 − 𝑇 is 1/2𝛼-ism.
Proposition 13 (see [21, 35]). Let 𝑆, 𝑇, 𝑉 : H → H be given
operators.
(i) If 𝑇 = (1 − 𝛼)𝑆 + 𝛼𝑉 for some 𝛼 ∈ (0, 1) and if 𝑆 is
averaged and 𝑉 is nonexpansive, then 𝑇 is averaged.
(ii) 𝑇 is firmly nonexpansive if and only if the complement
𝐼 − 𝑇 is firmly nonexpansive.
(iii) If 𝑇 = (1 − 𝛼)𝑆 + 𝛼𝑉 for some 𝛼 ∈ (0, 1) and if 𝑆 is
firmly nonexpansive and 𝑉 is nonexpansive, then 𝑇 is
averaged.
(iv) The composite of finitely many averaged mappings is
averaged. That is, if each of the mappings {𝑇𝑖 }𝑁
𝑖=1 is
averaged, then so is the composite 𝑇1 ∘ 𝑇2 ∘ ⋅ ⋅ ⋅ ∘ 𝑇𝑁. In
particular, if 𝑇1 is 𝛼1 -averaged and 𝑇2 is 𝛼2 -averaged,
where 𝛼1 , 𝛼2 ∈ (0, 1), then the composite 𝑇1 ∘ 𝑇2 is 𝛼averaged, where 𝛼 = 𝛼1 + 𝛼2 − 𝛼1 𝛼2 .
(v) If the mappings {𝑇𝑖 }𝑁
𝑖=1 are averaged and have a
common fixed point, then
𝑁
⋂ Fix (𝑇𝑖 ) = Fix (𝑇1 ⋅ ⋅ ⋅ 𝑇𝑁) .
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𝑖=1
The notation Fix(𝑇) denotes the set of all fixed points of
the mapping 𝑇, that is, Fix(𝑇) = {𝑥 ∈ H : 𝑇𝑥 = 𝑥}.
It is clear that in a real Hilbert space H, 𝑆 : 𝐶 → 𝐶
is 𝑘-strictly pseudocontractive if and only if there holds the
following inequality:
⟨𝑆𝑥 − 𝑆𝑦, 𝑥 − 𝑦⟩
󵄩2
󵄩2 1 − 𝑘 󵄩󵄩
󵄩
≤ 󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩 −
󵄩(𝐼 − 𝑆) 𝑥 − (𝐼 − 𝑆) 𝑦󵄩󵄩󵄩 ,
2 󵄩
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∀𝑥, 𝑦 ∈ 𝐶.
This immediately implies that if 𝑆 is a 𝑘-strictly pseudocontractive mapping, then 𝐼 − 𝑆 is (1 − 𝑘)/2-inverse strongly
monotone; for further detail, we refer to [9] and the references
therein. It is well known that the class of strict pseudocontractions strictly includes the class of nonexpansive mappings.
The following elementary result in the real Hilbert spaces
is quite well known.
Lemma 14 (see [36]). Let H be a real Hilbert space. Then, for
all 𝑥, 𝑦 ∈ H and 𝜆 ∈ [0, 1],
󵄩 󵄩2
󵄩2
󵄩󵄩
2
󵄩󵄩𝜆𝑥 + (1 − 𝜆)𝑦󵄩󵄩󵄩 = 𝜆‖𝑥‖ + (1 − 𝜆) 󵄩󵄩󵄩𝑦󵄩󵄩󵄩
󵄩2
󵄩
− 𝜆 (1 − 𝜆) 󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩 .
(28)
Lemma 15 (see [37, Proposition 2.1]). Let 𝐶 be a nonempty
closed convex subset of a real Hilbert space H and 𝑆 : 𝐶 → 𝐶
be a mapping.
(i) If 𝑆 is a 𝑘-strict pseudocontractive mapping, then 𝑆
satisfies the Lipschitz condition
󵄩
󵄩 1 + 𝑘 󵄩󵄩
󵄩󵄩
󵄩𝑥 − 𝑦󵄩󵄩󵄩 ,
󵄩󵄩𝑆𝑥 − 𝑆𝑦󵄩󵄩󵄩 ≤
1−𝑘󵄩
∀𝑥, 𝑦 ∈ 𝐶.
(29)
(ii) If 𝑆 is a 𝑘-strict pseudocontractive mapping, then the
mapping 𝐼 − 𝑆 is semiclosed at 0, that is, if {𝑥𝑛 } is a
sequence in 𝐶 such that 𝑥𝑛 → 𝑥̃ weakly and (𝐼 −
𝑆)𝑥𝑛 → 0 strongly, then (𝐼 − 𝑆)𝑥̃ = 0.
(iii) If 𝑆 is 𝑘-(quasi-)strict pseudocontraction, then the fixed
point set Fix(𝑆) of 𝑆 is closed and convex so that the
projection 𝑃Fix(𝑆) is well defined.
The following lemma plays a key role in proving weak
convergence of the sequences generated by our algorithms.
∞
∞
Lemma 16 (see [38, p. 80]). Let {𝑎𝑛 }∞
𝑛=0 , {𝑏𝑛 }𝑛=0 , and {𝛿𝑛 }𝑛=0 be
sequences of nonnegative real numbers satisfying the inequality
𝑎𝑛+1 ≤ (1 + 𝛿𝑛 ) 𝑎𝑛 + 𝑏𝑛 ,
∀𝑛 ≥ 0.
(30)
∞
If ∑∞
𝑛=0 𝛿𝑛 < ∞ and ∑𝑛=0 𝑏𝑛 < ∞, then lim𝑛 → ∞ 𝑎𝑛 exists. If,
∞
in addition, {𝑎𝑛 }𝑛=0 has a subsequence which converges to zero,
then lim𝑛 → ∞ 𝑎𝑛 = 0.
∞
Corollary 17 (see [39, p. 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.
If ∑∞
𝑛=0 𝑏𝑛 converges, then lim𝑛 → ∞ 𝑎𝑛 exists.
(31)
6
Abstract and Applied Analysis
Lemma 18 (see [7]). Let 𝐶 be a nonempty closed convex subset
of a real Hilbert space H. Let 𝑆 : 𝐶 → 𝐶 be a 𝑘-strictly
pseudocontractive mapping. Let 𝛾 and 𝛿 be two nonnegative
real numbers such that (𝛾 + 𝛿)𝑘 ≤ 𝛾. Then
󵄩
󵄩
󵄩
󵄩󵄩
󵄩󵄩𝛾 (𝑥 − 𝑦) + 𝛿 (𝑆𝑥 − 𝑆𝑦)󵄩󵄩󵄩 ≤ (𝛾 + 𝛿) 󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩 ,
∀𝑥, 𝑦 ∈ 𝐶.
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The following lemma is an immediate consequence of an
inner product.
Lemma 19. In a real Hilbert space H, there holds the inequality
󵄩2
󵄩󵄩
2
󵄩󵄩𝑥 + 𝑦󵄩󵄩󵄩 ≤ ‖𝑥‖ + 2 ⟨𝑦, 𝑥 + 𝑦⟩ ,
∀𝑥, 𝑦 ∈ H.
(33)
Let 𝐾 be a nonempty closed convex subset of a real Hilbert
space H and let 𝐹 : 𝐾 → H be a monotone mapping. The
variational inequality problem (VIP) is to find 𝑥 ∈ 𝐾 such
that
⟨𝐹𝑥, 𝑦 − 𝑥⟩ ≥ 0,
∀𝑦 ∈ 𝐾.
(34)
The solution set of the VIP is denoted by VI(𝐾, 𝐹). It is
well known that
𝑥 ∈ VI (𝐾, 𝐹) ⇐⇒ 𝑥 = 𝑃𝐾 (𝑥 − 𝜆𝐹𝑥) ,
∀𝜆 > 0.
Ξ ∩ Γ ≠ 0. For given 𝑥0 ∈ 𝐶 arbitrarily, let {𝑥𝑛 }, {𝑦𝑛 }, {𝑧𝑛 } be the
sequences generated by the Mann-type extragradient iterative
algorithm (16) with regularization, where 𝜇𝑖 ∈ (0, 2𝛽𝑖 ) for 𝑖 =
1, 2, {𝛼𝑛 } ⊂ (0, ∞), {𝜆 𝑛 } ⊂ (0, 1/‖𝐴‖2 ) and {𝜎𝑛 }, {𝜏𝑛 }, {𝛽𝑛 },
{𝛾𝑛 }, {𝛿𝑛 } ⊂ [0, 1] such that
(i) ∑∞
𝑛=0 𝛼𝑛 < ∞;
(ii) 𝛽𝑛 + 𝛾𝑛 + 𝛿𝑛 = 1 and (𝛾𝑛 + 𝛿𝑛 )𝑘 ≤ 𝛾𝑛 for all 𝑛 ≥ 0;
(iii) 𝜎𝑛 + 𝜏𝑛 ≤ 1 for all 𝑛 ≥ 0;
(iv) 0 < lim inf 𝑛 → ∞ 𝜏𝑛 ≤ lim sup𝑛 → ∞ (𝜎𝑛 + 𝜏𝑛 ) < 1;
(v) 0 < lim inf 𝑛 → ∞ 𝛽𝑛 ≤ lim sup𝑛 → ∞ 𝛽𝑛 < 1 and
lim inf 𝑛 → ∞ 𝛿𝑛 > 0;
(vi) 0 < lim inf 𝑛 → ∞ 𝜆 𝑛 ≤ lim sup𝑛 → ∞ 𝜆 𝑛 < 1/‖𝐴‖2 .
Then all the sequences {𝑥𝑛 }, {𝑦𝑛 }, {𝑧𝑛 } converge weakly to an
element 𝑥 ∈ Fix(𝑆) ∩ Ξ ∩ Γ. Furthermore, (𝑥, 𝑦) is a solution of
GSVI (3), where 𝑦 = 𝑃𝐶(𝑥 − 𝜇2 𝐵2 𝑥).
Proof. First, taking into account 0 < lim inf 𝑛 → ∞ 𝜆 𝑛 ≤
lim sup𝑛 → ∞ 𝜆 𝑛 < 1/‖𝐴‖2 , without loss of generality we may
assume that {𝜆 𝑛 } ⊂ [𝑎, 𝑏] for some 𝑎, 𝑏 ∈ (0, 1/‖𝐴‖2 ).
Now, let us show that 𝑃𝐶(𝐼 − 𝜆∇𝑓𝛼 ) is 𝜁-averaged for each
𝜆 ∈ (0, 2/(𝛼 + ‖𝐴‖2 )), where
(35)
𝜁=
H
A set-valued mapping 𝑇 : H → 2 is called monotone
if for all 𝑥, 𝑦 ∈ H, 𝑓 ∈ 𝑇𝑥 and 𝑔 ∈ 𝑇𝑦 imply that ⟨𝑥 − 𝑦, 𝑓 −
𝑔⟩ ≥ 0. A monotone set-valued mapping 𝑇 : H → 2H is
called maximal if its graph Gph(𝑇) is not properly contained
in the graph of any other monotone set-valued mapping. It is
known that a monotone set-valued mapping 𝑇 : H → 2H is
maximal if and only if for (𝑥, 𝑓) ∈ H × H, ⟨𝑥 − 𝑦, 𝑓 − 𝑔⟩ ≥ 0
for every (𝑦, 𝑔) ∈ Gph(𝑇) implies that 𝑓 ∈ 𝑇𝑥. Let 𝐹 : 𝐾 →
H be a monotone and Lipschitz continuous mapping and let
𝑁𝐾 𝑣 be the normal cone to 𝐾 at 𝑣 ∈ 𝐾, that is,
𝑁𝐾 𝑣 = {𝑤 ∈ H : ⟨𝑣 − 𝑢, 𝑤⟩ ≥ 0, ∀𝑢 ∈ 𝐾} .
(36)
2 + 𝜆 (𝛼 + ‖𝐴‖2 )
4
∈ (0, 1) .
(38)
Indeed, it is easy to see that ∇𝑓 = 𝐴∗ (𝐼 − 𝑃𝑄)𝐴 is 1/‖𝐴‖2 ism, that is,
⟨∇𝑓 (𝑥) − ∇𝑓 (𝑦) , 𝑥 − 𝑦⟩ ≥
1 󵄩󵄩
󵄩2
󵄩󵄩∇𝑓 (𝑥) − ∇𝑓 (𝑦)󵄩󵄩󵄩 . (39)
‖𝐴‖2
Observe that
(𝛼 + ‖𝐴‖2 ) ⟨∇𝑓𝛼 (𝑥) − ∇𝑓𝛼 (𝑦) , 𝑥 − 𝑦⟩
󵄩2
󵄩
= (𝛼 + ‖𝐴‖2 ) [𝛼󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩
+ ⟨∇𝑓 (𝑥) − ∇𝑓 (𝑦) , 𝑥 − 𝑦⟩ ]
Define
𝑇𝑣 = {
𝐹𝑣 + 𝑁𝐾 𝑣,
0,
if 𝑣 ∈ 𝐾,
if 𝑣 ∉ 𝐾.
(37)
It is known that in this case the mapping 𝑇 is maximal
monotone, and 0 ∈ 𝑇𝑣 if and only if 𝑣 ∈ VI(𝐾, 𝐹); for further
details, we refer to [40] and the references therein.
3. Main Results
In this section, we first prove the weak convergence of the
sequences generated by the Mann-type extragradient iterative
algorithm (16) with regularization.
Theorem 20. Let 𝐶 be a nonempty closed convex subset of a
real Hilbert space H1 . Let 𝐴 ∈ 𝐵(H1 , H2 ) and 𝐵𝑖 : 𝐶 → H1
be 𝛽𝑖 -inverse strongly monotone for 𝑖 = 1, 2. Let 𝑆 : 𝐶 → 𝐶
be a 𝑘-strictly pseudocontractive mapping such that Fix(𝑆) ∩
󵄩2
󵄩
= 𝛼2 󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩 + 𝛼 ⟨∇𝑓 (𝑥) − ∇𝑓 (𝑦) , 𝑥 − 𝑦⟩
󵄩2
󵄩
+ 𝛼‖𝐴‖2 󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩
2
(40)
+ ‖𝐴‖ ⟨∇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)󵄩󵄩󵄩 .
Hence, it follows that ∇𝑓𝛼 = 𝛼𝐼 + 𝐴∗ (𝐼 − 𝑃𝑄)𝐴 is
1/(𝛼 + ‖𝐴‖2 )-ism. Thus, 𝜆∇𝑓𝛼 is 1/𝜆(𝛼 + ‖𝐴‖2 )-ism according
to Proposition 12(ii). By Proposition 12(iii), the complement
Abstract and Applied Analysis
7
𝐼 − 𝜆∇𝑓𝛼 is 𝜆(𝛼 + ‖𝐴‖2 )/2-averaged. Therefore, noting that
𝑃𝐶 is 1/2-averaged and utilizing Proposition 13(iv), we know
that for each 𝜆 ∈ (0, 2/(𝛼 + ‖𝐴‖2 )), 𝑃𝐶(𝐼 − 𝜆∇𝑓𝛼 ) is 𝜁-averaged
with
𝜁=
=
2
2
1 𝜆 (𝛼 + ‖𝐴‖ ) 1 𝜆 (𝛼 + ‖𝐴‖ )
+
− ⋅
2
2
2
2
2 + 𝜆 (𝛼 + ‖𝐴‖2 )
4
(41)
∈ (0, 1) .
This shows that 𝑃𝐶(𝐼 − 𝜆∇𝑓𝛼 ) is nonexpansive. Furthermore,
for {𝜆 𝑛 } ⊂ [𝑎, 𝑏] with 𝑎, 𝑏 ∈ (0, 1/‖𝐴‖2 ), we have
𝑎 ≤ inf 𝜆 𝑛 ≤ sup𝜆 𝑛 ≤ 𝑏 <
𝑛≥0
𝑛≥0
1
1
= lim
.
2
𝑛
→
∞
𝛼𝑛 + ‖𝐴‖2
‖𝐴‖
(42)
𝑛≥0
𝑛≥0
1
,
𝛼𝑛 + ‖𝐴‖2
∀𝑛 ≥ 0.
𝜁𝑛 =
=
1
+
2
2
2 + 𝜆 𝑛 (𝛼𝑛 + ‖𝐴‖2 )
4
2
1 𝜆 𝑛 (𝛼𝑛 + ‖𝐴‖ )
− ⋅
2
2
(43)
− 𝑃𝐶 (𝐼 − 𝜆 𝑛 ∇𝑓) 𝑝, 𝑧𝑛 − 𝑝⟩
(47)
󵄩
󵄩
× 󵄩󵄩󵄩𝑧𝑛 − 𝑝󵄩󵄩󵄩
󵄩
󵄩2
≤ 󵄩󵄩󵄩𝑥𝑛 − 𝑝󵄩󵄩󵄩
󵄩
󵄩󵄩
󵄩
+ 2 󵄩󵄩󵄩󵄩(𝐼 − 𝜆 𝑛 ∇𝑓𝛼𝑛 ) 𝑝 − (𝐼 − 𝜆 𝑛 ∇𝑓) 𝑝󵄩󵄩󵄩󵄩 󵄩󵄩󵄩𝑧𝑛 − 𝑝󵄩󵄩󵄩
󵄩
󵄩2
󵄩
󵄩 󵄩󵄩
= 󵄩󵄩󵄩𝑥𝑛 − 𝑝󵄩󵄩󵄩 + 2𝜆 𝑛 𝛼𝑛 󵄩󵄩󵄩𝑝󵄩󵄩󵄩 󵄩󵄩󵄩𝑧𝑛 − 𝑝󵄩󵄩󵄩 .
(44)
∈ (0, 1) .
This immediately implies that 𝑃𝐶(𝐼 − 𝜆 𝑛 ∇𝑓𝛼𝑛 ) is nonexpansive for all 𝑛 ≥ 0.
Next we divide the remainder of the proof into several
steps.
Step 1. {𝑥𝑛 } is bounded.
Indeed, take 𝑝 ∈ Fix(𝑆) ∩ Ξ ∩ Γ arbitrarily. Then 𝑆𝑝 = 𝑝,
𝑃𝐶(𝐼 − 𝜆∇𝑓)𝑝 = 𝑝 for 𝜆 ∈ (0, 2/‖𝐴‖2 ), and
𝑝 = 𝑃𝐶 [𝑃𝐶 (𝑝 − 𝜇2 𝐵2 𝑝) − 𝜇1 𝐵1 𝑃𝐶 (𝑝 − 𝜇2 𝐵2 𝑝)] .
+ 2 ⟨𝑃𝐶 (𝐼 − 𝜆 𝑛 ∇𝑓𝛼𝑛 ) 𝑝
󵄩
󵄩
+ 2 󵄩󵄩󵄩󵄩𝑃𝐶 (𝐼 − 𝜆 𝑛 ∇𝑓𝛼𝑛 ) 𝑝 − 𝑃𝐶 (𝐼 − 𝜆 𝑛 ∇𝑓) 𝑝󵄩󵄩󵄩󵄩
Consequently, it follows that for each integer 𝑛 ≥ 0, 𝑃𝐶(𝐼 −
𝜆 𝑛 ∇𝑓𝛼𝑛 ) is 𝜁𝑛 -averaged with
𝜆 𝑛 (𝛼𝑛 + ‖𝐴‖2 )
󵄩2
󵄩󵄩
󵄩󵄩𝑧𝑛 − 𝑝󵄩󵄩󵄩
󵄩
󵄩2
= 󵄩󵄩󵄩󵄩𝑃𝐶 (𝐼 − 𝜆 𝑛 ∇𝑓𝛼𝑛 ) 𝑥𝑛 − 𝑃𝐶 (𝐼 − 𝜆 𝑛 ∇𝑓) 𝑝󵄩󵄩󵄩󵄩
󵄩
= 󵄩󵄩󵄩󵄩𝑃𝐶 (𝐼 − 𝜆 𝑛 ∇𝑓𝛼𝑛 ) 𝑥𝑛 − 𝑃𝐶 (𝐼 − 𝜆 𝑛 ∇𝑓𝛼𝑛 ) 𝑝
󵄩2
+ 𝑃𝐶 (𝐼 − 𝜆 𝑛 ∇𝑓𝛼𝑛 ) 𝑝 − 𝑃𝐶 (𝐼 − 𝜆 𝑛 ∇𝑓) 𝑝󵄩󵄩󵄩󵄩
󵄩
󵄩2
≤ 󵄩󵄩󵄩󵄩𝑃𝐶 (𝐼 − 𝜆 𝑛 ∇𝑓𝛼𝑛 ) 𝑥𝑛 − 𝑃𝐶 (𝐼 − 𝜆 𝑛 ∇𝑓𝛼𝑛 ) 𝑝󵄩󵄩󵄩󵄩
󵄩2
󵄩
≤ 󵄩󵄩󵄩𝑥𝑛 − 𝑝󵄩󵄩󵄩
Without loss of generality, we may assume that
𝑎 ≤ inf 𝜆 𝑛 ≤ sup𝜆 𝑛 ≤ 𝑏 <
Utilizing Lemma 19, we also have
(45)
From (16), it follows that
󵄩󵄩
󵄩
󵄩󵄩𝑧𝑛 − 𝑝󵄩󵄩󵄩
󵄩
󵄩
= 󵄩󵄩󵄩󵄩𝑃𝐶 (𝐼 − 𝜆 𝑛 ∇𝑓𝛼𝑛 ) 𝑥𝑛 − 𝑃𝐶 (𝐼 − 𝜆 𝑛 ∇𝑓) 𝑝󵄩󵄩󵄩󵄩
󵄩
󵄩
≤ 󵄩󵄩󵄩󵄩𝑃𝐶 (𝐼 − 𝜆 𝑛 ∇𝑓𝛼𝑛 ) 𝑥𝑛 − 𝑃𝐶 (𝐼 − 𝜆 𝑛 ∇𝑓𝛼𝑛 ) 𝑝󵄩󵄩󵄩󵄩
(46)
󵄩
󵄩
+ 󵄩󵄩󵄩󵄩𝑃𝐶 (𝐼 − 𝜆 𝑛 ∇𝑓𝛼𝑛 ) 𝑝 − 𝑃𝐶 (𝐼 − 𝜆 𝑛 ∇𝑓) 𝑝󵄩󵄩󵄩󵄩
󵄩
󵄩
󵄩 󵄩
≤ 󵄩󵄩󵄩𝑥𝑛 − 𝑝󵄩󵄩󵄩 + 󵄩󵄩󵄩󵄩(𝐼 − 𝜆 𝑛 ∇𝑓𝛼𝑛 ) 𝑝 − (𝐼 − 𝜆 𝑛 ∇𝑓) 𝑝󵄩󵄩󵄩󵄩
󵄩
󵄩 󵄩
󵄩
≤ 󵄩󵄩󵄩𝑥𝑛 − 𝑝󵄩󵄩󵄩 + 𝜆 𝑛 𝛼𝑛 󵄩󵄩󵄩𝑝󵄩󵄩󵄩 .
For simplicity, we write 𝑞 = 𝑃𝐶(𝑝 − 𝜇2 𝐵2 𝑝), 𝑧̃𝑛 = 𝑃𝐶(𝑧𝑛 −
𝜇2 𝐵2 𝑧𝑛 ),
𝑢𝑛 = 𝑃𝐶 [𝑃𝐶 (𝑧𝑛 − 𝜇2 𝐵2 𝑧𝑛 ) − 𝜇1 𝐵1 𝑃𝐶 (𝑧𝑛 − 𝜇2 𝐵2 𝑧𝑛 )] ,
𝑢𝑛 = 𝑃𝐶 (𝑥𝑛 − 𝜆 𝑛 ∇𝑓𝛼𝑛 (𝑧𝑛 ))
(48)
for each 𝑛 ≥ 0. Then 𝑦𝑛 = 𝜎𝑛 𝑥𝑛 +𝜏𝑛 𝑢𝑛 +(1−𝜎𝑛 −𝜏𝑛 )𝑢𝑛 for each
𝑛 ≥ 0. Since 𝐵𝑖 : 𝐶 → H1 is 𝛽𝑖 -inverse strongly monotone
and 0 < 𝜇𝑖 < 2𝛽𝑖 for 𝑖 = 1, 2, we know that for all 𝑛 ≥ 0,
󵄩2
󵄩󵄩
󵄩󵄩𝑢𝑛 − 𝑝󵄩󵄩󵄩
󵄩
= 󵄩󵄩󵄩𝑃𝐶 [𝑃𝐶 (𝑧𝑛 − 𝜇2 𝐵2 𝑧𝑛 )
󵄩2
− 𝜇1 𝐵1 𝑃𝐶 (𝑧𝑛 − 𝜇2 𝐵2 𝑧𝑛 )] − 𝑝󵄩󵄩󵄩
󵄩
= 󵄩󵄩󵄩𝑃𝐶 [𝑃𝐶 (𝑧𝑛 − 𝜇2 𝐵2 𝑧𝑛 ) − 𝜇1 𝐵1 𝑃𝐶 (𝑧𝑛 − 𝜇2 𝐵2 𝑧𝑛 )]
󵄩2
− 𝑃𝐶 [𝑃𝐶 (𝑝 − 𝜇2 𝐵2 𝑝) − 𝜇1 𝐵1 𝑃𝐶 (𝑝 − 𝜇2 𝐵2 𝑝)]󵄩󵄩󵄩
󵄩
≤ 󵄩󵄩󵄩[𝑃𝐶 (𝑧𝑛 − 𝜇2 𝐵2 𝑧𝑛 ) − 𝜇1 𝐵1 𝑃𝐶 (𝑧𝑛 − 𝜇2 𝐵2 𝑧𝑛 )]
󵄩2
− [𝑃𝐶 (𝑝 − 𝜇2 𝐵2 𝑝) − 𝜇1 𝐵1 𝑃𝐶 (𝑝 − 𝜇2 𝐵2 𝑝)]󵄩󵄩󵄩
󵄩
= 󵄩󵄩󵄩[𝑃𝐶 (𝑧𝑛 − 𝜇2 𝐵2 𝑧𝑛 ) − 𝑃𝐶 (𝑝 − 𝜇2 𝐵2 𝑝)]
󵄩2
− 𝜇1 [𝐵1 𝑃𝐶 (𝑧𝑛 − 𝜇2 𝐵2 𝑧𝑛 ) − 𝐵1 𝑃𝐶 (𝑝 − 𝜇2 𝐵2 𝑝)]󵄩󵄩󵄩
8
Abstract and Applied Analysis
󵄩
󵄩2
≤ 󵄩󵄩󵄩𝑃𝐶 (𝑧𝑛 − 𝜇2 𝐵2 𝑧𝑛 ) − 𝑃𝐶 (𝑝 − 𝜇2 𝐵2 𝑝)󵄩󵄩󵄩
Further, by Proposition 8(i), we have
󵄩
− 𝜇1 (2𝛽1 − 𝜇1 ) 󵄩󵄩󵄩𝐵1 𝑃𝐶 (𝑧𝑛 − 𝜇2 𝐵2 𝑧𝑛 )
⟨𝑥𝑛 − 𝜆 𝑛 ∇𝑓𝛼𝑛 (𝑧𝑛 ) − 𝑧𝑛 , 𝑢𝑛 − 𝑧𝑛 ⟩
󵄩2
− 𝐵1 𝑃𝐶 (𝑝 − 𝜇2 𝐵2 𝑝)󵄩󵄩󵄩
= ⟨𝑥𝑛 − 𝜆 𝑛 ∇𝑓𝛼𝑛 (𝑥𝑛 ) − 𝑧𝑛 , 𝑢𝑛 − 𝑧𝑛 ⟩
󵄩
󵄩2
≤ 󵄩󵄩󵄩(𝑧𝑛 − 𝜇2 𝐵2 𝑧𝑛 ) − (𝑝 − 𝜇2 𝐵2 𝑝)󵄩󵄩󵄩
+ ⟨𝜆 𝑛 ∇𝑓𝛼𝑛 (𝑥𝑛 ) − 𝜆 𝑛 ∇𝑓𝛼𝑛 (𝑧𝑛 ) , 𝑢𝑛 − 𝑧𝑛 ⟩
󵄩
󵄩2
− 𝜇1 (2𝛽1 − 𝜇1 ) 󵄩󵄩󵄩𝐵1 𝑧̃𝑛 − 𝐵1 𝑞󵄩󵄩󵄩
≤ ⟨𝜆 𝑛 ∇𝑓𝛼𝑛 (𝑥𝑛 ) − 𝜆 𝑛 ∇𝑓𝛼𝑛 (𝑧𝑛 ) , 𝑢𝑛 − 𝑧𝑛 ⟩
󵄩
󵄩2
= 󵄩󵄩󵄩(𝑧𝑛 − 𝑝) − 𝜇2 (𝐵2 𝑧𝑛 − 𝐵2 𝑝)󵄩󵄩󵄩
(51)
󵄩
󵄩󵄩
󵄩
≤ 𝜆 𝑛 󵄩󵄩󵄩󵄩∇𝑓𝛼𝑛 (𝑥𝑛 ) − ∇𝑓𝛼𝑛 (𝑧𝑛 )󵄩󵄩󵄩󵄩 󵄩󵄩󵄩𝑢𝑛 − 𝑧𝑛 󵄩󵄩󵄩
󵄩
󵄩2
− 𝜇1 (2𝛽1 − 𝜇1 ) 󵄩󵄩󵄩𝐵1 𝑧̃𝑛 − 𝐵1 𝑞󵄩󵄩󵄩
󵄩
󵄩2
󵄩
󵄩2
≤ 󵄩󵄩󵄩𝑧𝑛 − 𝑝󵄩󵄩󵄩 − 𝜇2 (2𝛽2 − 𝜇2 ) 󵄩󵄩󵄩𝐵2 𝑧𝑛 − 𝐵2 𝑝󵄩󵄩󵄩
󵄩
󵄩2 󵄩
󵄩2
− 𝜇1 (2𝛽1 − 𝜇1 ) 󵄩󵄩󵄩𝐵1 𝑧̃𝑛 − 𝐵1 𝑞󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩𝑧𝑛 − 𝑝󵄩󵄩󵄩 .
󵄩󵄩
󵄩
󵄩
≤ 𝜆 𝑛 (𝛼𝑛 + ‖𝐴‖2 ) 󵄩󵄩󵄩𝑥𝑛 − 𝑧𝑛 󵄩󵄩󵄩 󵄩󵄩󵄩𝑢𝑛 − 𝑧𝑛 󵄩󵄩󵄩 .
So, from (46), we obtain
(49)
Furthermore, by Proposition 8(ii), we have
󵄩󵄩
󵄩2
󵄩󵄩𝑢𝑛 − 𝑝󵄩󵄩󵄩
󵄩2
󵄩
󵄩2 󵄩
≤ 󵄩󵄩󵄩󵄩𝑥𝑛 − 𝜆 𝑛 ∇𝑓𝛼𝑛 (𝑧𝑛 ) − 𝑝󵄩󵄩󵄩󵄩 − 󵄩󵄩󵄩󵄩𝑥𝑛 − 𝜆 𝑛 ∇𝑓𝛼𝑛 (𝑧𝑛 ) − 𝑢𝑛 󵄩󵄩󵄩󵄩
󵄩2
󵄩
󵄩2 󵄩
= 󵄩󵄩󵄩𝑥𝑛 − 𝑝󵄩󵄩󵄩 − 󵄩󵄩󵄩𝑥𝑛 − 𝑢𝑛 󵄩󵄩󵄩 + 2𝜆 𝑛 ⟨∇𝑓𝛼𝑛 (𝑧𝑛 ) , 𝑝 − 𝑢𝑛 ⟩
󵄩2
󵄩2 󵄩
󵄩
= 󵄩󵄩󵄩𝑥𝑛 − 𝑝󵄩󵄩󵄩 − 󵄩󵄩󵄩𝑥𝑛 − 𝑢𝑛 󵄩󵄩󵄩
󵄩󵄩
󵄩2 󵄩
󵄩2
󵄩2 󵄩
󵄩2 󵄩
󵄩󵄩𝑢𝑛 − 𝑝󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩𝑥𝑛 − 𝑝󵄩󵄩󵄩 − 󵄩󵄩󵄩𝑥𝑛 − 𝑧𝑛 󵄩󵄩󵄩 − 󵄩󵄩󵄩𝑧𝑛 − 𝑢𝑛 󵄩󵄩󵄩
+ 2 ⟨𝑥𝑛 − 𝜆 𝑛 ∇𝑓𝛼𝑛 (𝑧𝑛 ) − 𝑧𝑛 , 𝑢𝑛 − 𝑧𝑛 ⟩
+ 2𝜆 𝑛 𝛼𝑛 ⟨𝑝, 𝑝 − 𝑧𝑛 ⟩
󵄩2 󵄩
󵄩2
󵄩2 󵄩
󵄩
≤ 󵄩󵄩󵄩𝑥𝑛 − 𝑝󵄩󵄩󵄩 − 󵄩󵄩󵄩𝑥𝑛 − 𝑧𝑛 󵄩󵄩󵄩 − 󵄩󵄩󵄩𝑧𝑛 − 𝑢𝑛 󵄩󵄩󵄩
󵄩󵄩
󵄩
󵄩
+ 2𝜆 𝑛 (𝛼𝑛 + ‖𝐴‖2 ) 󵄩󵄩󵄩𝑥𝑛 − 𝑧𝑛 󵄩󵄩󵄩 󵄩󵄩󵄩𝑢𝑛 − 𝑧𝑛 󵄩󵄩󵄩
󵄩 󵄩󵄩
󵄩
+ 2𝜆 𝑛 𝛼𝑛 󵄩󵄩󵄩𝑝󵄩󵄩󵄩 󵄩󵄩󵄩𝑧𝑛 − 𝑝󵄩󵄩󵄩
+ 2𝜆 𝑛 (⟨∇𝑓𝛼𝑛 (𝑧𝑛 ) − ∇𝑓𝛼𝑛 (𝑝) , 𝑝 − 𝑧𝑛 ⟩
󵄩
󵄩2 󵄩
󵄩2
󵄩2 󵄩
≤ 󵄩󵄩󵄩𝑥𝑛 − 𝑝󵄩󵄩󵄩 − 󵄩󵄩󵄩𝑥𝑛 − 𝑧𝑛 󵄩󵄩󵄩 − 󵄩󵄩󵄩𝑧𝑛 − 𝑢𝑛 󵄩󵄩󵄩
+ ⟨∇𝑓𝛼𝑛 (𝑝) , 𝑝 − 𝑧𝑛 ⟩
2󵄩
󵄩2
+ 𝜆2𝑛 (𝛼𝑛 + ‖𝐴‖2 ) 󵄩󵄩󵄩𝑥𝑛 − 𝑧𝑛 󵄩󵄩󵄩
+ ⟨∇𝑓𝛼𝑛 (𝑧𝑛 ) , 𝑧𝑛 − 𝑢𝑛 ⟩)
󵄩2
󵄩 󵄩󵄩
󵄩
󵄩
+ 󵄩󵄩󵄩𝑧𝑛 − 𝑢𝑛 󵄩󵄩󵄩 + 2𝜆 𝑛 𝛼𝑛 󵄩󵄩󵄩𝑝󵄩󵄩󵄩 󵄩󵄩󵄩𝑧𝑛 − 𝑝󵄩󵄩󵄩
󵄩2
󵄩2 󵄩
󵄩
≤ 󵄩󵄩󵄩𝑥𝑛 − 𝑝󵄩󵄩󵄩 − 󵄩󵄩󵄩𝑥𝑛 − 𝑢𝑛 󵄩󵄩󵄩
+ 2𝜆 𝑛 (⟨∇𝑓𝛼𝑛 (𝑝) , 𝑝 − 𝑧𝑛 ⟩ + ⟨∇𝑓𝛼𝑛 (𝑧𝑛 ) , 𝑧𝑛 − 𝑢𝑛 ⟩)
󵄩2
󵄩2 󵄩
󵄩
= 󵄩󵄩󵄩𝑥𝑛 − 𝑝󵄩󵄩󵄩 − 󵄩󵄩󵄩𝑥𝑛 − 𝑢𝑛 󵄩󵄩󵄩
+ 2𝜆 𝑛 [ ⟨(𝛼𝑛 𝐼 + ∇𝑓) 𝑝, 𝑝−𝑧𝑛 ⟩ + ⟨∇𝑓𝛼𝑛 (𝑧𝑛 ) , 𝑧𝑛 −𝑢𝑛 ⟩ ]
󵄩2
󵄩2 󵄩
󵄩
≤ 󵄩󵄩󵄩𝑥𝑛 − 𝑝󵄩󵄩󵄩 − 󵄩󵄩󵄩𝑥𝑛 − 𝑢𝑛 󵄩󵄩󵄩
󵄩
󵄩 󵄩󵄩
󵄩2
󵄩
= 󵄩󵄩󵄩𝑥𝑛 − 𝑝󵄩󵄩󵄩 + 2𝜆 𝑛 𝛼𝑛 󵄩󵄩󵄩𝑝󵄩󵄩󵄩 󵄩󵄩󵄩𝑧𝑛 − 𝑝󵄩󵄩󵄩
2
󵄩2
󵄩
+ (𝜆2𝑛 (𝛼𝑛 + ‖𝐴‖2 ) − 1) 󵄩󵄩󵄩𝑥𝑛 − 𝑧𝑛 󵄩󵄩󵄩
󵄩 󵄩󵄩
󵄩
󵄩2
󵄩
≤ 󵄩󵄩󵄩𝑥𝑛 − 𝑝󵄩󵄩󵄩 + 2𝜆 𝑛 𝛼𝑛 󵄩󵄩󵄩𝑝󵄩󵄩󵄩 󵄩󵄩󵄩𝑧𝑛 − 𝑝󵄩󵄩󵄩
󵄩
󵄩 󵄩 󵄩
󵄩 󵄩
󵄩2
󵄩
≤ 󵄩󵄩󵄩𝑥𝑛 − 𝑝󵄩󵄩󵄩 + 2𝜆 𝑛 𝛼𝑛 󵄩󵄩󵄩𝑝󵄩󵄩󵄩 [󵄩󵄩󵄩𝑥𝑛 − 𝑝󵄩󵄩󵄩 + 𝜆 𝑛 𝛼𝑛 󵄩󵄩󵄩𝑝󵄩󵄩󵄩]
󵄩
󵄩 󵄩󵄩
󵄩 󵄩2
󵄩2
󵄩
≤ 󵄩󵄩󵄩𝑥𝑛 − 𝑝󵄩󵄩󵄩 + 4𝜆 𝑛 𝛼𝑛 󵄩󵄩󵄩𝑝󵄩󵄩󵄩 󵄩󵄩󵄩𝑥𝑛 − 𝑝󵄩󵄩󵄩 + 4𝜆2𝑛 𝛼𝑛2 󵄩󵄩󵄩𝑝󵄩󵄩󵄩
+ 2𝜆 𝑛 [𝛼𝑛 ⟨𝑝, 𝑝 − 𝑧𝑛 ⟩ + ⟨∇𝑓𝛼𝑛 (𝑧𝑛 ) , 𝑧𝑛 − 𝑢𝑛 ⟩]
󵄩 󵄩2
󵄩
󵄩
= (󵄩󵄩󵄩𝑥𝑛 − 𝑝󵄩󵄩󵄩 + 2𝜆 𝑛 𝛼𝑛 󵄩󵄩󵄩𝑝󵄩󵄩󵄩) .
󵄩2
󵄩2 󵄩
󵄩
= 󵄩󵄩󵄩𝑥𝑛 − 𝑝󵄩󵄩󵄩 − 󵄩󵄩󵄩𝑥𝑛 − 𝑧𝑛 󵄩󵄩󵄩
󵄩2
󵄩
− 2 ⟨𝑥𝑛 − 𝑧𝑛 , 𝑧𝑛 − 𝑢𝑛 ⟩ − 󵄩󵄩󵄩𝑧𝑛 − 𝑢𝑛 󵄩󵄩󵄩
(52)
Hence, it follows from (46), (49), and (52) that
+ 2𝜆 𝑛 [𝛼𝑛 ⟨𝑝, 𝑝 − 𝑧𝑛 ⟩ + ⟨∇𝑓𝛼𝑛 (𝑧𝑛 ) , 𝑧𝑛 − 𝑢𝑛 ⟩]
󵄩2 󵄩
󵄩2
󵄩2 󵄩
󵄩
= 󵄩󵄩󵄩𝑥𝑛 − 𝑝󵄩󵄩󵄩 − 󵄩󵄩󵄩𝑥𝑛 − 𝑧𝑛 󵄩󵄩󵄩 − 󵄩󵄩󵄩𝑧𝑛 − 𝑢𝑛 󵄩󵄩󵄩
+ 2 ⟨𝑥𝑛 − 𝜆 𝑛 ∇𝑓𝛼𝑛 (𝑧𝑛 ) − 𝑧𝑛 , 𝑢𝑛 − 𝑧𝑛 ⟩
+ 2𝜆 𝑛 𝛼𝑛 ⟨𝑝, 𝑝 − 𝑧𝑛 ⟩ .
(50)
󵄩󵄩
󵄩
󵄩󵄩𝑦𝑛 − 𝑝󵄩󵄩󵄩
󵄩
󵄩
= 󵄩󵄩󵄩𝜎𝑛 (𝑥𝑛 − 𝑝) + 𝜏𝑛 (𝑢𝑛 − 𝑝) + (1 − 𝜎𝑛 − 𝜏𝑛 ) (𝑢𝑛 − 𝑝)󵄩󵄩󵄩
󵄩
󵄩
󵄩
󵄩
󵄩
󵄩
≤ 𝜎𝑛 󵄩󵄩󵄩𝑥𝑛 − 𝑝󵄩󵄩󵄩 + 𝜏𝑛 󵄩󵄩󵄩𝑢𝑛 − 𝑝󵄩󵄩󵄩 + (1 − 𝜎𝑛 − 𝜏𝑛 ) 󵄩󵄩󵄩𝑢𝑛 − 𝑝󵄩󵄩󵄩
󵄩
󵄩
󵄩
󵄩
󵄩
󵄩
≤ 𝜎𝑛 󵄩󵄩󵄩𝑥𝑛 − 𝑝󵄩󵄩󵄩 + 𝜏𝑛 󵄩󵄩󵄩𝑢𝑛 − 𝑝󵄩󵄩󵄩 + (1 − 𝜎𝑛 − 𝜏𝑛 ) 󵄩󵄩󵄩𝑧𝑛 − 𝑝󵄩󵄩󵄩
Abstract and Applied Analysis
9
󵄩
󵄩 󵄩
󵄩
󵄩
󵄩
≤ 𝜎𝑛 󵄩󵄩󵄩𝑥𝑛 − 𝑝󵄩󵄩󵄩 + 𝜏𝑛 (󵄩󵄩󵄩𝑥𝑛 − 𝑝󵄩󵄩󵄩 + 2𝜆 𝑛 𝛼𝑛 󵄩󵄩󵄩𝑝󵄩󵄩󵄩)
󵄩
󵄩2
≤ 𝛽𝑛 󵄩󵄩󵄩𝑥𝑛 − 𝑝󵄩󵄩󵄩 + (𝛾𝑛 + 𝛿𝑛 )
󵄩 󵄩
󵄩
󵄩
+ (1 − 𝜎𝑛 − 𝜏𝑛 ) (󵄩󵄩󵄩𝑥𝑛 − 𝑝󵄩󵄩󵄩 + 𝜆 𝑛 𝛼𝑛 󵄩󵄩󵄩𝑝󵄩󵄩󵄩)
󵄩
󵄩
󵄩2
󵄩2
× [𝜎𝑛 󵄩󵄩󵄩𝑥𝑛 − 𝑝󵄩󵄩󵄩 + 𝜏𝑛 󵄩󵄩󵄩𝑢𝑛 − 𝑝󵄩󵄩󵄩
󵄩
󵄩 󵄩
󵄩
= 󵄩󵄩󵄩𝑥𝑛 − 𝑝󵄩󵄩󵄩 + [2𝜏𝑛 + (1 − 𝜎𝑛 − 𝜏𝑛 )] 𝜆 𝑛 𝛼𝑛 󵄩󵄩󵄩𝑝󵄩󵄩󵄩
󵄩
󵄩 󵄩
󵄩
≤ 󵄩󵄩󵄩𝑥𝑛 − 𝑝󵄩󵄩󵄩 + 2𝜆 𝑛 𝛼𝑛 󵄩󵄩󵄩𝑝󵄩󵄩󵄩 .
󵄩
󵄩2
+ (1 − 𝜎𝑛 − 𝜏𝑛 ) 󵄩󵄩󵄩𝑢𝑛 − 𝑝󵄩󵄩󵄩 ]
(53)
Since (𝛾𝑛 + 𝛿𝑛 )𝑘 ≤ 𝛾𝑛 for all 𝑛 ≥ 0, utilizing Lemma 18, we
obtain from (53)
󵄩󵄩
󵄩
󵄩󵄩𝑥𝑛+1 − 𝑝󵄩󵄩󵄩
󵄩
󵄩
= 󵄩󵄩󵄩𝛽𝑛 (𝑥𝑛 − 𝑝) + 𝛾𝑛 (𝑦𝑛 − 𝑝) + 𝛿𝑛 (𝑆𝑦𝑛 − 𝑝)󵄩󵄩󵄩
󵄩
󵄩 󵄩
󵄩
≤ 𝛽𝑛 󵄩󵄩󵄩𝑥𝑛 − 𝑝󵄩󵄩󵄩 + 󵄩󵄩󵄩𝛾𝑛 (𝑦𝑛 − 𝑝) + 𝛿𝑛 (𝑆𝑦𝑛 − 𝑝)󵄩󵄩󵄩
󵄩
󵄩
󵄩
󵄩
≤ 𝛽𝑛 󵄩󵄩󵄩𝑥𝑛 − 𝑝󵄩󵄩󵄩 + (𝛾𝑛 + 𝛿𝑛 ) 󵄩󵄩󵄩𝑦𝑛 − 𝑝󵄩󵄩󵄩
󵄩
󵄩
≤ 𝛽𝑛 󵄩󵄩󵄩𝑥𝑛 − 𝑝󵄩󵄩󵄩 + (𝛾𝑛 + 𝛿𝑛 )
󵄩
󵄩 󵄩
󵄩
× [󵄩󵄩󵄩𝑥𝑛 − 𝑝󵄩󵄩󵄩 + 2𝜆 𝑛 𝛼𝑛 󵄩󵄩󵄩𝑝󵄩󵄩󵄩]
󵄩
󵄩 󵄩
󵄩
≤ 󵄩󵄩󵄩𝑥𝑛 − 𝑝󵄩󵄩󵄩 + 2𝑏 󵄩󵄩󵄩𝑝󵄩󵄩󵄩 𝛼𝑛 .
󵄩
󵄩2
≤ 𝛽𝑛 󵄩󵄩󵄩𝑥𝑛 − 𝑝󵄩󵄩󵄩 + (𝛾𝑛 + 𝛿𝑛 )
󵄩󵄩 1
󵄩󵄩2
󵄩
󵄩
× 󵄩󵄩󵄩
[𝛾𝑛 (𝑦𝑛 − 𝑝) + 𝛿𝑛 (𝑆𝑦𝑛 − 𝑝)]󵄩󵄩󵄩
󵄩󵄩 𝛾𝑛 + 𝛿𝑛
󵄩󵄩
󵄩
󵄩2
󵄩
󵄩2
≤ 𝛽𝑛 󵄩󵄩󵄩𝑥𝑛 − 𝑝󵄩󵄩󵄩 + (𝛾𝑛 + 𝛿𝑛 ) 󵄩󵄩󵄩𝑦𝑛 − 𝑝󵄩󵄩󵄩
2
󵄩2
󵄩
+ (𝜆2𝑛 (𝛼𝑛 + ‖𝐴‖2 ) − 1) 󵄩󵄩󵄩𝑥𝑛 − 𝑧𝑛 󵄩󵄩󵄩 ]
󵄩
󵄩2
− 𝜇1 (2𝛽1 − 𝜇1 ) 󵄩󵄩󵄩𝐵1 𝑧̃𝑛 − 𝐵1 𝑞󵄩󵄩󵄩 ]}
(54)
󵄩
󵄩2
≤ 𝛽𝑛 󵄩󵄩󵄩𝑥𝑛 − 𝑝󵄩󵄩󵄩 + (𝛾𝑛 + 𝛿𝑛 )
󵄩
󵄩2
× {𝜎𝑛 󵄩󵄩󵄩𝑥𝑛 − 𝑝󵄩󵄩󵄩
󵄩 󵄩󵄩
󵄩
󵄩2
󵄩
+ 𝜏𝑛 [󵄩󵄩󵄩𝑥𝑛 − 𝑝󵄩󵄩󵄩 + 2𝜆 𝑛 𝛼𝑛 󵄩󵄩󵄩𝑝󵄩󵄩󵄩 󵄩󵄩󵄩𝑧𝑛 − 𝑝󵄩󵄩󵄩
2
󵄩2
󵄩
+ (𝜆2𝑛 (𝛼𝑛 + ‖𝐴‖2 ) − 1) 󵄩󵄩󵄩𝑥𝑛 − 𝑧𝑛 󵄩󵄩󵄩 ]
(55)
Step 2. Consider lim𝑛 → ∞ ‖𝐵2 𝑧𝑛 − 𝐵2 𝑝‖ = 0, lim𝑛 → ∞ ‖𝐵1 𝑧̃𝑛 −
𝐵1 𝑞‖ = 0 and lim𝑛 → ∞ ‖𝑥𝑛 −𝑧𝑛 ‖ = 0, where 𝑞 = 𝑃𝐶(𝑝−𝜇2 𝐵2 𝑝).
Indeed, utilizing Lemma 18 and the convexity of ‖ ⋅ ‖2 , we
obtain from (16) and (47)–(52) that
󵄩
󵄩2
= 󵄩󵄩󵄩𝛽𝑛 (𝑥𝑛 − 𝑝) + 𝛾𝑛 (𝑦𝑛 − 𝑝) + 𝛿𝑛 (𝑆𝑦𝑛 − 𝑝)󵄩󵄩󵄩
󵄩 󵄩󵄩
󵄩
󵄩2
󵄩
+ 𝜏𝑛 [󵄩󵄩󵄩𝑥𝑛 − 𝑝󵄩󵄩󵄩 + 2𝜆 𝑛 𝛼𝑛 󵄩󵄩󵄩𝑝󵄩󵄩󵄩 󵄩󵄩󵄩𝑧𝑛 − 𝑝󵄩󵄩󵄩
󵄩2
󵄩
󵄩2
󵄩
× [󵄩󵄩󵄩𝑧𝑛 − 𝑝󵄩󵄩󵄩 − 𝜇2 (2𝛽2 − 𝜇2 ) 󵄩󵄩󵄩𝐵2 𝑧𝑛 − 𝐵2 𝑝󵄩󵄩󵄩
and the sequence {𝑥𝑛 } is bounded. Taking into account that
𝑃𝐶, ∇𝑓𝛼𝑛 , 𝐵1 and 𝐵2 are Lipschitz continuous, we can easily
see that {𝑧𝑛 }, {𝑢𝑛 }, {𝑢𝑛 }, {𝑦𝑛 }, and {̃𝑧𝑛 } are bounded, where
𝑧̃𝑛 = 𝑃𝐶(𝑧𝑛 − 𝜇2 𝐵2 𝑧𝑛 ) for all 𝑛 ≥ 0.
󵄩󵄩
󵄩2
󵄩󵄩𝑥𝑛+1 − 𝑝󵄩󵄩󵄩
󵄩
󵄩2
× {𝜎𝑛 󵄩󵄩󵄩𝑥𝑛 − 𝑝󵄩󵄩󵄩
+ (1 − 𝜎𝑛 − 𝜏𝑛 )
∞
Since ∑∞
𝑛=0 𝛼𝑛 < ∞, it is clear that ∑𝑛=0 2𝑏‖𝑝‖𝛼𝑛 < ∞. Thus,
by Corollary 17, we conclude that
󵄩
󵄩
lim 󵄩󵄩𝑥 − 𝑝󵄩󵄩󵄩 exists for each 𝑝 ∈ Fix (𝑆) ∩ Ξ ∩ Γ,
𝑛→∞ 󵄩 𝑛
󵄩
󵄩2
≤ 𝛽𝑛 󵄩󵄩󵄩𝑥𝑛 − 𝑝󵄩󵄩󵄩 + (𝛾𝑛 + 𝛿𝑛 )
+ (1 − 𝜎𝑛 − 𝜏𝑛 )
󵄩 󵄩󵄩
󵄩2
󵄩
󵄩
× [󵄩󵄩󵄩𝑥𝑛 − 𝑝󵄩󵄩󵄩 + 2𝜆 𝑛 𝛼𝑛 󵄩󵄩󵄩𝑝󵄩󵄩󵄩 󵄩󵄩󵄩𝑧𝑛 − 𝑝󵄩󵄩󵄩
󵄩
󵄩2
− 𝜇2 (2𝛽2 − 𝜇2 ) 󵄩󵄩󵄩𝐵2 𝑧𝑛 − 𝐵2 𝑝󵄩󵄩󵄩
󵄩
󵄩2
− 𝜇1 (2𝛽1 − 𝜇1 ) 󵄩󵄩󵄩𝐵1 𝑧̃𝑛 − 𝐵1 𝑞󵄩󵄩󵄩 ]}
󵄩2
󵄩
= 󵄩󵄩󵄩𝑥𝑛 − 𝑝󵄩󵄩󵄩 + (𝛾𝑛 + 𝛿𝑛 )
󵄩 󵄩󵄩
󵄩
× { (1 − 𝜎𝑛 ) 2𝜆 𝑛 𝛼𝑛 󵄩󵄩󵄩𝑝󵄩󵄩󵄩 󵄩󵄩󵄩𝑧𝑛 − 𝑝󵄩󵄩󵄩
2
󵄩2
󵄩
+ 𝜏𝑛 (𝜆2𝑛 (𝛼𝑛 + ‖𝐴‖2 ) − 1) 󵄩󵄩󵄩𝑥𝑛 − 𝑧𝑛 󵄩󵄩󵄩
− (1 − 𝜎𝑛 − 𝜏𝑛 )
󵄩
󵄩2
× [𝜇2 (2𝛽2 − 𝜇2 ) 󵄩󵄩󵄩𝐵2 𝑧𝑛 − 𝐵2 𝑝󵄩󵄩󵄩
󵄩
󵄩2
+ 𝜇1 (2𝛽1 − 𝜇1 ) 󵄩󵄩󵄩𝐵1 𝑧̃𝑛 − 𝐵1 𝑞󵄩󵄩󵄩 ]}
󵄩 󵄩󵄩
󵄩2
󵄩
󵄩
≤ 󵄩󵄩󵄩𝑥𝑛 − 𝑝󵄩󵄩󵄩 + 2𝜆 𝑛 𝛼𝑛 󵄩󵄩󵄩𝑝󵄩󵄩󵄩 󵄩󵄩󵄩𝑧𝑛 − 𝑝󵄩󵄩󵄩
2 󵄩
󵄩2
− (𝛾𝑛 + 𝛿𝑛 ) {𝜏𝑛 (1 − 𝜆2𝑛 (𝛼𝑛 + ‖𝐴‖2 ) ) 󵄩󵄩󵄩𝑥𝑛 − 𝑧𝑛 󵄩󵄩󵄩
+ (1 − 𝜎𝑛 − 𝜏𝑛 )
󵄩
󵄩2
× [𝜇2 (2𝛽2 − 𝜇2 ) 󵄩󵄩󵄩𝐵2 𝑧𝑛 − 𝐵2 𝑝󵄩󵄩󵄩
󵄩
󵄩2
+ 𝜇1 (2𝛽1 − 𝜇1 ) 󵄩󵄩󵄩𝐵1 𝑧̃𝑛 − 𝐵1 𝑞󵄩󵄩󵄩 ] } .
(56)
10
Abstract and Applied Analysis
Therefore,
≤
2 󵄩
󵄩2
(𝛾𝑛 + 𝛿𝑛 ) {𝜏𝑛 (1 − 𝜆2𝑛 (𝛼𝑛 + ‖𝐴‖2 ) ) 󵄩󵄩󵄩𝑥𝑛 − 𝑧𝑛 󵄩󵄩󵄩
=
+ (1 − 𝜎𝑛 − 𝜏𝑛 )
󵄩
󵄩2
× [𝜇2 (2𝛽2 − 𝜇2 ) 󵄩󵄩󵄩𝐵2 𝑧𝑛 − 𝐵2 𝑝󵄩󵄩󵄩
󵄩
󵄩2
−𝜇22 󵄩󵄩󵄩𝐵2 𝑧𝑛 − 𝐵2 𝑝󵄩󵄩󵄩 ]
󵄩
󵄩 󵄩󵄩
󵄩2 󵄩
󵄩2
󵄩
≤ 󵄩󵄩󵄩𝑥𝑛 − 𝑝󵄩󵄩󵄩 − 󵄩󵄩󵄩𝑥𝑛+1 − 𝑝󵄩󵄩󵄩 + 2𝜆 𝑛 𝛼𝑛 󵄩󵄩󵄩𝑝󵄩󵄩󵄩 󵄩󵄩󵄩𝑧𝑛 − 𝑝󵄩󵄩󵄩 .
≤
Since 𝛼𝑛 → 0, lim𝑛 → ∞ ‖𝑥𝑛 −𝑝‖ exists, lim inf 𝑛 → ∞ (𝛾𝑛 +𝛿𝑛 ) >
0, {𝜆 𝑛 } ⊂ [𝑎, 𝑏] and 0 < lim inf 𝑛 → ∞ 𝜏𝑛 ≤ lim sup𝑛 → ∞ (𝜎𝑛 +
𝜏𝑛 ) < 1, it follows that
󵄩󵄩
󵄩
− 𝑧𝑛 󵄩󵄩󵄩 = 0,
lim 󵄩𝐵 𝑧̃
𝑛→∞ 󵄩 1 𝑛
󵄩󵄩
󵄩
− 𝐵1 𝑞󵄩󵄩󵄩 = 0,
󵄩󵄩
󵄩
− 𝐵2 𝑝󵄩󵄩󵄩 = 0.
lim 󵄩𝐵 𝑧
𝑛→∞ 󵄩 2 𝑛
1 󵄩󵄩
󵄩2 󵄩
󵄩2
[󵄩𝑧 − 𝑝󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑧̃𝑛 − 𝑞󵄩󵄩󵄩
2 󵄩 𝑛
󵄩
󵄩2
− 󵄩󵄩󵄩𝑧𝑛 − 𝑧̃𝑛 − (𝑝 − 𝑞)󵄩󵄩󵄩
󵄩
󵄩󵄩
󵄩
+2𝜇2 󵄩󵄩󵄩𝑧𝑛 − 𝑧̃𝑛 − (𝑝 − 𝑞)󵄩󵄩󵄩 󵄩󵄩󵄩𝐵2 𝑧𝑛 − 𝐵2 𝑝󵄩󵄩󵄩 ] ,
(60)
that is,
(58)
Step 3. Consider lim𝑛 → ∞ ‖𝑆𝑦𝑛 − 𝑦𝑛 ‖ = 0.
Indeed, observe that
󵄩󵄩
󵄩
󵄩󵄩𝑢𝑛 − 𝑧𝑛 󵄩󵄩󵄩
󵄩
󵄩
= 󵄩󵄩󵄩󵄩𝑃𝐶 (𝑥𝑛 − 𝜆 𝑛 ∇𝑓𝛼𝑛 (𝑧𝑛 )) − 𝑃𝐶 (𝑥𝑛 − 𝜆 𝑛 ∇𝑓𝛼𝑛 (𝑥𝑛 ))󵄩󵄩󵄩󵄩
󵄩
󵄩
≤ 󵄩󵄩󵄩󵄩(𝑥𝑛 − 𝜆 𝑛 ∇𝑓𝛼𝑛 (𝑧𝑛 )) − (𝑥𝑛 − 𝜆 𝑛 ∇𝑓𝛼𝑛 (𝑥𝑛 ))󵄩󵄩󵄩󵄩
󵄩
󵄩
= 𝜆 𝑛 󵄩󵄩󵄩󵄩∇𝑓𝛼𝑛 (𝑧𝑛 ) − ∇𝑓𝛼𝑛 (𝑥𝑛 )󵄩󵄩󵄩󵄩
󵄩
󵄩
≤ 𝜆 𝑛 (𝛼𝑛 + ‖𝐴‖2 ) 󵄩󵄩󵄩𝑧𝑛 − 𝑥𝑛 󵄩󵄩󵄩 .
(59)
󵄩󵄩̃
󵄩2
󵄩󵄩𝑧𝑛 − 𝑞󵄩󵄩󵄩
󵄩
󵄩2 󵄩
󵄩2
≤ 󵄩󵄩󵄩𝑧𝑛 − 𝑝󵄩󵄩󵄩 − 󵄩󵄩󵄩𝑧𝑛 − 𝑧̃𝑛 − (𝑝 − 𝑞)󵄩󵄩󵄩
󵄩
󵄩󵄩
󵄩
+ 2𝜇2 󵄩󵄩󵄩𝑧𝑛 − 𝑧̃𝑛 − (𝑝 − 𝑞)󵄩󵄩󵄩 󵄩󵄩󵄩𝐵2 𝑧𝑛 − 𝐵2 𝑝󵄩󵄩󵄩 .
This together with ‖𝑧𝑛 − 𝑥𝑛 ‖ → 0 implies that
lim𝑛 → ∞ ‖𝑢𝑛 − 𝑧𝑛 ‖ = 0 and hence lim𝑛 → ∞ ‖𝑢𝑛 − 𝑥𝑛 ‖ = 0. By
firm nonexpansiveness of 𝑃𝐶, we have
󵄩󵄩
󵄩2
󵄩󵄩𝑢𝑛 − 𝑝󵄩󵄩󵄩
󵄩
󵄩2
= 󵄩󵄩󵄩𝑃𝐶 (̃𝑧𝑛 − 𝜇1 𝐵1 𝑧̃𝑛 ) − 𝑃𝐶 (𝑞 − 𝜇1 𝐵1 𝑞)󵄩󵄩󵄩
≤ ⟨(̃𝑧𝑛 − 𝜇1 𝐵1 𝑧̃𝑛 ) − (𝑞 − 𝜇1 𝐵1 𝑞) , 𝑢𝑛 − 𝑝⟩
=
=
󵄩󵄩̃
󵄩2
󵄩󵄩𝑧𝑛 − 𝑞󵄩󵄩󵄩
󵄩
󵄩2
= 󵄩󵄩󵄩𝑃𝐶 (𝑧𝑛 − 𝜇2 𝐵2 𝑧𝑛 ) − 𝑃𝐶 (𝑝 − 𝜇2 𝐵2 𝑝)󵄩󵄩󵄩
1 󵄩󵄩
󵄩2 󵄩
󵄩2
[󵄩󵄩𝑧̃𝑛 − 𝑞 − 𝜇1 (𝐵1 𝑧̃𝑛 − 𝐵1 𝑞)󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑢𝑛 − 𝑝󵄩󵄩󵄩
2
󵄩
󵄩2
− 󵄩󵄩󵄩(̃𝑧𝑛 − 𝑞) − 𝜇1 (𝐵1 𝑧̃𝑛 − 𝐵1 𝑞) − (𝑢𝑛 − 𝑝)󵄩󵄩󵄩 ]
1 󵄩󵄩
󵄩2 󵄩
󵄩2
[󵄩𝑧̃ − 𝑞󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑢𝑛 − 𝑝󵄩󵄩󵄩
2 󵄩 𝑛
󵄩
󵄩2
− 󵄩󵄩󵄩(̃𝑧𝑛 − 𝑢𝑛 ) − 𝜇1 (𝐵1 𝑧̃𝑛 − 𝐵1 𝑞) + (𝑝 − 𝑞)󵄩󵄩󵄩 ] (62)
1 󵄩󵄩
󵄩2 󵄩
󵄩2
[󵄩𝑧̃ − 𝑞󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑢𝑛 − 𝑝󵄩󵄩󵄩
2 󵄩 𝑛
󵄩
󵄩2
− 󵄩󵄩󵄩𝑧̃𝑛 − 𝑢𝑛 + (𝑝 − 𝑞)󵄩󵄩󵄩
+ 2𝜇1 ⟨̃𝑧𝑛 − 𝑢𝑛 + (𝑝 − 𝑞) , 𝐵1 𝑧̃𝑛 − 𝐵1 𝑞⟩
󵄩
󵄩2
− 𝜇12 󵄩󵄩󵄩𝐵1 𝑧̃𝑛 − 𝐵1 𝑞󵄩󵄩󵄩 ]
≤ ⟨(𝑧𝑛 − 𝜇2 𝐵2 𝑧𝑛 ) − (𝑝 − 𝜇2 𝐵2 𝑝) , 𝑧̃𝑛 − 𝑞⟩
1 󵄩󵄩
󵄩2 󵄩
󵄩2
[󵄩𝑧 − 𝑝 − 𝜇2 (𝐵2 𝑧𝑛 − 𝐵2 𝑝)󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑧̃𝑛 − 𝑞󵄩󵄩󵄩
2 󵄩 𝑛
󵄩
󵄩2
− 󵄩󵄩󵄩(𝑧𝑛 − 𝑝) − 𝜇2 (𝐵2 𝑧𝑛 − 𝐵2 𝑝) − (̃𝑧𝑛 − 𝑞)󵄩󵄩󵄩 ]
(61)
Moreover, using the argument technique similar to the
previous one, we derive
≤
=
1 󵄩󵄩
󵄩2 󵄩
󵄩2 󵄩
󵄩2
[󵄩𝑧 − 𝑝󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑧̃𝑛 − 𝑞󵄩󵄩󵄩 − 󵄩󵄩󵄩𝑧𝑛 − 𝑧̃𝑛 − (𝑝 − 𝑞)󵄩󵄩󵄩
2 󵄩 𝑛
+ 2𝜇2 ⟨𝑧𝑛 − 𝑧̃𝑛 − (𝑝 − 𝑞) , 𝐵2 𝑧𝑛 − 𝐵2 𝑝⟩
(57)
󵄩
󵄩2
+ 𝜇1 (2𝛽1 − 𝜇1 ) 󵄩󵄩󵄩𝐵1 𝑧̃𝑛 − 𝐵1 𝑞󵄩󵄩󵄩 ] }
lim 󵄩𝑥
𝑛→∞ 󵄩 𝑛
1 󵄩󵄩
󵄩2 󵄩
󵄩2
[󵄩𝑧 − 𝑝󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑧̃𝑛 − 𝑞󵄩󵄩󵄩
2 󵄩 𝑛
󵄩
󵄩2
− 󵄩󵄩󵄩(𝑧𝑛 − 𝑧̃𝑛 ) − 𝜇2 (𝐵2 𝑧𝑛 − 𝐵2 𝑝) − (𝑝 − 𝑞)󵄩󵄩󵄩 ]
≤
1 󵄩󵄩
󵄩2 󵄩
󵄩2 󵄩
󵄩2
[󵄩󵄩𝑧̃𝑛 − 𝑞󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑢𝑛 − 𝑝󵄩󵄩󵄩 − 󵄩󵄩󵄩𝑧̃𝑛 − 𝑢𝑛 + (𝑝 − 𝑞)󵄩󵄩󵄩
2
󵄩
󵄩󵄩
󵄩
+ 2𝜇1 󵄩󵄩󵄩𝑧̃𝑛 − 𝑢𝑛 + (𝑝 − 𝑞)󵄩󵄩󵄩 󵄩󵄩󵄩𝐵1 𝑧̃𝑛 − 𝐵1 𝑞󵄩󵄩󵄩] ,
Abstract and Applied Analysis
11
󵄩
󵄩󵄩
󵄩
+ 2𝜇1 󵄩󵄩󵄩𝑧̃𝑛 − 𝑢𝑛 + (𝑝 − 𝑞)󵄩󵄩󵄩 󵄩󵄩󵄩𝐵1 𝑧̃𝑛 − 𝐵1 𝑞󵄩󵄩󵄩
that is,
󵄩󵄩
󵄩2
󵄩󵄩𝑢𝑛 − 𝑝󵄩󵄩󵄩
󵄩
󵄩2 󵄩
󵄩2
≤ 󵄩󵄩󵄩𝑧̃𝑛 − 𝑞󵄩󵄩󵄩 − 󵄩󵄩󵄩𝑧̃𝑛 − 𝑢𝑛 + (𝑝 − 𝑞)󵄩󵄩󵄩
󵄩
󵄩2
− (1 − 𝜎𝑛 − 𝜏𝑛 ) (󵄩󵄩󵄩𝑧𝑛 − 𝑧̃𝑛 − (𝑝 − 𝑞)󵄩󵄩󵄩
󵄩
󵄩󵄩
󵄩
+ 2𝜇1 󵄩󵄩󵄩𝑧̃𝑛 − 𝑢𝑛 + (𝑝 − 𝑞)󵄩󵄩󵄩 󵄩󵄩󵄩𝐵1 𝑧̃𝑛 − 𝐵1 𝑞󵄩󵄩󵄩 .
Utilizing (47), (52), (61), and (63), we have
󵄩󵄩
󵄩2
󵄩󵄩𝑦𝑛 − 𝑝󵄩󵄩󵄩
󵄩
= 󵄩󵄩󵄩𝜎𝑛 (𝑥𝑛 − 𝑝) + 𝜏𝑛 (𝑢𝑛 − 𝑝)
󵄩2
+ (1 − 𝜎𝑛 − 𝜏𝑛 ) (𝑢𝑛 − 𝑝)󵄩󵄩󵄩
󵄩
󵄩
󵄩2
󵄩2
≤ 𝜎𝑛 󵄩󵄩󵄩𝑥𝑛 − 𝑝󵄩󵄩󵄩 + 𝜏𝑛 󵄩󵄩󵄩𝑢𝑛 − 𝑝󵄩󵄩󵄩
󵄩
󵄩2
+ (1 − 𝜎𝑛 − 𝜏𝑛 ) 󵄩󵄩󵄩𝑢𝑛 − 𝑝󵄩󵄩󵄩
󵄩
󵄩2
≤ 𝜎𝑛 󵄩󵄩󵄩𝑥𝑛 − 𝑝󵄩󵄩󵄩
󵄩 󵄩󵄩
󵄩
󵄩2
󵄩
+ 𝜏𝑛 (󵄩󵄩󵄩𝑥𝑛 − 𝑝󵄩󵄩󵄩 + 2𝜆 𝑛 𝛼𝑛 󵄩󵄩󵄩𝑝󵄩󵄩󵄩 󵄩󵄩󵄩𝑧𝑛 − 𝑝󵄩󵄩󵄩)
+ (1 − 𝜎𝑛 − 𝜏𝑛 )
󵄩2 󵄩
󵄩2
󵄩
× [ 󵄩󵄩󵄩𝑧̃𝑛 − 𝑞󵄩󵄩󵄩 − 󵄩󵄩󵄩𝑧̃𝑛 − 𝑢𝑛 + (𝑝 − 𝑞)󵄩󵄩󵄩
󵄩
󵄩󵄩
󵄩
+ 2𝜇1 󵄩󵄩󵄩𝑧̃𝑛 − 𝑢𝑛 + (𝑝 − 𝑞)󵄩󵄩󵄩 󵄩󵄩󵄩𝐵1 𝑧̃𝑛 − 𝐵1 𝑞󵄩󵄩󵄩 ]
󵄩2
󵄩
󵄩2
󵄩
󵄩
󵄩 󵄩󵄩
≤ 𝜎𝑛 󵄩󵄩󵄩𝑥𝑛 − 𝑝󵄩󵄩󵄩 + 𝜏𝑛 (󵄩󵄩󵄩𝑥𝑛 − 𝑝󵄩󵄩󵄩 + 2𝜆 𝑛 𝛼𝑛 󵄩󵄩󵄩𝑝󵄩󵄩󵄩 󵄩󵄩󵄩𝑧𝑛 − 𝑝󵄩󵄩󵄩)
+ (1 − 𝜎𝑛 − 𝜏𝑛 )
󵄩
󵄩2 󵄩
󵄩2
× {󵄩󵄩󵄩𝑧𝑛 − 𝑝󵄩󵄩󵄩 − 󵄩󵄩󵄩𝑧𝑛 − 𝑧̃𝑛 − (𝑝 − 𝑞)󵄩󵄩󵄩
󵄩
󵄩󵄩
󵄩
+ 2𝜇2 󵄩󵄩󵄩𝑧𝑛 − 𝑧̃𝑛 − (𝑝 − 𝑞)󵄩󵄩󵄩 󵄩󵄩󵄩𝐵2 𝑧𝑛 − 𝐵2 𝑝󵄩󵄩󵄩
󵄩
󵄩2
− 󵄩󵄩󵄩𝑧̃𝑛 − 𝑢𝑛 + (𝑝 − 𝑞)󵄩󵄩󵄩
󵄩
󵄩󵄩
󵄩
+ 2𝜇1 󵄩󵄩󵄩𝑧̃𝑛 − 𝑢𝑛 + (𝑝 − 𝑞)󵄩󵄩󵄩 󵄩󵄩󵄩𝐵1 𝑧̃𝑛 − 𝐵1 𝑞󵄩󵄩󵄩 }
󵄩
󵄩 󵄩󵄩
󵄩2
󵄩
󵄩2
󵄩
≤ 𝜎𝑛 󵄩󵄩󵄩𝑥𝑛 − 𝑝󵄩󵄩󵄩 + 𝜏𝑛 (󵄩󵄩󵄩𝑥𝑛 − 𝑝󵄩󵄩󵄩 + 2𝜆 𝑛 𝛼𝑛 󵄩󵄩󵄩𝑝󵄩󵄩󵄩 󵄩󵄩󵄩𝑧𝑛 − 𝑝󵄩󵄩󵄩)
+ (1 − 𝜎𝑛 − 𝜏𝑛 )
󵄩 󵄩󵄩
󵄩2
󵄩
󵄩
× {󵄩󵄩󵄩𝑥𝑛 − 𝑝󵄩󵄩󵄩 + 2𝜆 𝑛 𝛼𝑛 󵄩󵄩󵄩𝑝󵄩󵄩󵄩 󵄩󵄩󵄩𝑧𝑛 − 𝑝󵄩󵄩󵄩
󵄩
󵄩
󵄩2
󵄩
− 󵄩󵄩󵄩𝑧𝑛 − 𝑧̃𝑛 − (𝑝 − 𝑞)󵄩󵄩󵄩 + 2𝜇2 󵄩󵄩󵄩𝑧𝑛 − 𝑧̃𝑛 − (𝑝 − 𝑞)󵄩󵄩󵄩
󵄩
󵄩 󵄩
󵄩2
× 󵄩󵄩󵄩𝐵2 𝑧𝑛 − 𝐵2 𝑝󵄩󵄩󵄩 − 󵄩󵄩󵄩𝑧̃𝑛 − 𝑢𝑛 + (𝑝 − 𝑞)󵄩󵄩󵄩
󵄩
󵄩󵄩
󵄩
+ 2𝜇1 󵄩󵄩󵄩𝑧̃𝑛 − 𝑢𝑛 + (𝑝 − 𝑞)󵄩󵄩󵄩 󵄩󵄩󵄩𝐵1 𝑧̃𝑛 − 𝐵1 𝑞󵄩󵄩󵄩 }
󵄩
󵄩 󵄩󵄩
󵄩2
󵄩
≤ 󵄩󵄩󵄩𝑥𝑛 − 𝑝󵄩󵄩󵄩 + 2𝜆 𝑛 𝛼𝑛 󵄩󵄩󵄩𝑝󵄩󵄩󵄩 󵄩󵄩󵄩𝑧𝑛 − 𝑝󵄩󵄩󵄩
󵄩
󵄩󵄩
󵄩
+ 2𝜇2 󵄩󵄩󵄩𝑧𝑛 − 𝑧̃𝑛 − (𝑝 − 𝑞)󵄩󵄩󵄩 󵄩󵄩󵄩𝐵2 𝑧𝑛 − 𝐵2 𝑝󵄩󵄩󵄩
󵄩
󵄩2
+ 󵄩󵄩󵄩𝑧̃𝑛 − 𝑢𝑛 + (𝑝 − 𝑞)󵄩󵄩󵄩 ) .
(63)
(64)
Thus, utilizing Lemma 14, from (16) and (64) it follows that
󵄩󵄩
󵄩2
󵄩󵄩𝑥𝑛+1 − 𝑝󵄩󵄩󵄩
󵄩
󵄩2
= 󵄩󵄩󵄩𝛽𝑛 (𝑥𝑛 − 𝑝) + 𝛾𝑛 (𝑦𝑛 − 𝑝) + 𝛿𝑛 (𝑆𝑦𝑛 − 𝑝)󵄩󵄩󵄩
󵄩󵄩
1
󵄩
= 󵄩󵄩󵄩𝛽𝑛 (𝑥𝑛 − 𝑝) + (1 − 𝛽𝑛 ) ⋅
󵄩󵄩
1 − 𝛽𝑛
󵄩󵄩2
󵄩
× [𝛾𝑛 (𝑦𝑛 − 𝑝)+𝛿𝑛 (𝑆𝑦𝑛 −𝑝)] 󵄩󵄩󵄩
󵄩󵄩
󵄩
󵄩2
= 𝛽𝑛 󵄩󵄩󵄩𝑥𝑛 − 𝑝󵄩󵄩󵄩 + (1 − 𝛽𝑛 )
󵄩󵄩2
󵄩󵄩 1
󵄩
󵄩
[𝛾𝑛 (𝑦𝑛 − 𝑝) + 𝛿𝑛 (𝑆𝑦𝑛 − 𝑝)]󵄩󵄩󵄩
× 󵄩󵄩󵄩
󵄩󵄩
󵄩󵄩 1 − 𝛽𝑛
− 𝛽𝑛 (1 − 𝛽𝑛 )
󵄩󵄩2
󵄩󵄩 1
󵄩
󵄩
[𝛾𝑛 (𝑥𝑛 − 𝑦𝑛 ) + 𝛿𝑛 (𝑥𝑛 − 𝑆𝑦𝑛 )]󵄩󵄩󵄩
× 󵄩󵄩󵄩
󵄩󵄩
󵄩󵄩 1 − 𝛽𝑛
󵄩
󵄩2
󵄩
󵄩2
≤ 𝛽𝑛 󵄩󵄩󵄩𝑥𝑛 − 𝑝󵄩󵄩󵄩 + (1 − 𝛽𝑛 ) 󵄩󵄩󵄩𝑦𝑛 − 𝑝󵄩󵄩󵄩 − 𝛽𝑛 (1 − 𝛽𝑛 )
󵄩󵄩2
󵄩󵄩 1
󵄩
󵄩
[𝛾𝑛 (𝑥𝑛 − 𝑦𝑛 ) + 𝛿𝑛 (𝑥𝑛 − 𝑆𝑦𝑛 )]󵄩󵄩󵄩
× 󵄩󵄩󵄩
󵄩󵄩
󵄩󵄩 1 − 𝛽𝑛
󵄩
󵄩2
≤ 𝛽𝑛 󵄩󵄩󵄩𝑥𝑛 − 𝑝󵄩󵄩󵄩 + (1 − 𝛽𝑛 )
󵄩 󵄩󵄩
󵄩2
󵄩
󵄩
× {󵄩󵄩󵄩𝑥𝑛 − 𝑝󵄩󵄩󵄩 + 2𝜆 𝑛 𝛼𝑛 󵄩󵄩󵄩𝑝󵄩󵄩󵄩 󵄩󵄩󵄩𝑧𝑛 − 𝑝󵄩󵄩󵄩
󵄩
󵄩󵄩
󵄩
+ 2𝜇2 󵄩󵄩󵄩𝑧𝑛 − 𝑧̃𝑛 − (𝑝 − 𝑞)󵄩󵄩󵄩 󵄩󵄩󵄩𝐵2 𝑧𝑛 − 𝐵2 𝑝󵄩󵄩󵄩
󵄩
󵄩󵄩
󵄩
+ 2𝜇1 󵄩󵄩󵄩𝑧̃𝑛 − 𝑢𝑛 + (𝑝 − 𝑞)󵄩󵄩󵄩 󵄩󵄩󵄩𝐵1 𝑧̃𝑛 − 𝐵1 𝑞󵄩󵄩󵄩
󵄩
󵄩2
− (1 − 𝜎𝑛 − 𝜏𝑛 ) (󵄩󵄩󵄩𝑧𝑛 − 𝑧̃𝑛 − (𝑝 − 𝑞)󵄩󵄩󵄩
󵄩
󵄩2
+ 󵄩󵄩󵄩𝑧̃𝑛 − 𝑢𝑛 + (𝑝 − 𝑞)󵄩󵄩󵄩 )}
− 𝛽𝑛 (1 − 𝛽𝑛 )
󵄩󵄩2
󵄩󵄩 1
󵄩
󵄩
[𝛾𝑛 (𝑥𝑛 − 𝑦𝑛 ) + 𝛿𝑛 (𝑥𝑛 − 𝑆𝑦𝑛 )]󵄩󵄩󵄩
× 󵄩󵄩󵄩
󵄩󵄩
󵄩󵄩 1 − 𝛽𝑛
󵄩 󵄩󵄩
󵄩
󵄩2
󵄩
≤ 󵄩󵄩󵄩𝑥𝑛 − 𝑝󵄩󵄩󵄩 + 2𝜆 𝑛 𝛼𝑛 󵄩󵄩󵄩𝑝󵄩󵄩󵄩 󵄩󵄩󵄩𝑧𝑛 − 𝑝󵄩󵄩󵄩
󵄩
󵄩󵄩
󵄩
+ 2𝜇2 󵄩󵄩󵄩𝑧𝑛 − 𝑧̃𝑛 − (𝑝 − 𝑞)󵄩󵄩󵄩 󵄩󵄩󵄩𝐵2 𝑧𝑛 − 𝐵2 𝑝󵄩󵄩󵄩
󵄩
󵄩󵄩
󵄩
+ 2𝜇1 󵄩󵄩󵄩𝑧̃𝑛 − 𝑢𝑛 + (𝑝 − 𝑞)󵄩󵄩󵄩 󵄩󵄩󵄩𝐵1 𝑧̃𝑛 − 𝐵1 𝑞󵄩󵄩󵄩
󵄩
󵄩2
− (1 − 𝛽𝑛 ) (1 − 𝜎𝑛 − 𝜏𝑛 ) (󵄩󵄩󵄩𝑧𝑛 − 𝑧̃𝑛 − (𝑝 − 𝑞)󵄩󵄩󵄩
󵄩
󵄩2
+ 󵄩󵄩󵄩𝑧̃𝑛 − 𝑢𝑛 + (𝑝 − 𝑞)󵄩󵄩󵄩 )
12
Abstract and Applied Analysis
− 𝛽𝑛 (1 − 𝛽𝑛 )
󵄩󵄩2
󵄩󵄩 1
󵄩
󵄩
[𝛾𝑛 (𝑥𝑛 − 𝑦𝑛 ) + 𝛿𝑛 (𝑥𝑛 − 𝑆𝑦𝑛 )]󵄩󵄩󵄩 ,
× 󵄩󵄩󵄩
󵄩󵄩
󵄩󵄩 1 − 𝛽𝑛
(65)
which hence implies that
(1 − 𝛽𝑛 ) (1 − 𝜎𝑛 − 𝜏𝑛 )
󵄩2 󵄩
󵄩2
󵄩
× (󵄩󵄩󵄩𝑧𝑛 − 𝑧̃𝑛 − (𝑝 − 𝑞)󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑧̃𝑛 − 𝑢𝑛 + (𝑝 − 𝑞)󵄩󵄩󵄩 )
󵄩󵄩 1
󵄩󵄩󵄩2
󵄩
+ 𝛽𝑛 (1 − 𝛽𝑛 ) 󵄩󵄩󵄩
[𝛾𝑛 (𝑥𝑛 − 𝑦𝑛 ) + 𝛿𝑛 (𝑥𝑛 − 𝑆𝑦𝑛 )]󵄩󵄩󵄩
󵄩󵄩 1 − 𝛽𝑛
󵄩󵄩
󵄩 󵄩󵄩
󵄩
󵄩2 󵄩
󵄩2
󵄩
≤ 󵄩󵄩󵄩𝑥𝑛 − 𝑝󵄩󵄩󵄩 − 󵄩󵄩󵄩𝑥𝑛+1 − 𝑝󵄩󵄩󵄩 + 2𝜆 𝑛 𝛼𝑛 󵄩󵄩󵄩𝑝󵄩󵄩󵄩 󵄩󵄩󵄩𝑧𝑛 − 𝑝󵄩󵄩󵄩
󵄩
󵄩󵄩
󵄩
+ 2𝜇2 󵄩󵄩󵄩𝑧𝑛 − 𝑧̃𝑛 − (𝑝 − 𝑞)󵄩󵄩󵄩 󵄩󵄩󵄩𝐵2 𝑧𝑛 − 𝐵2 𝑝󵄩󵄩󵄩
󵄩
󵄩󵄩
󵄩
+ 2𝜇1 󵄩󵄩󵄩𝑧̃𝑛 − 𝑢𝑛 + (𝑝 − 𝑞)󵄩󵄩󵄩 󵄩󵄩󵄩𝐵1 𝑧̃𝑛 − 𝐵1 𝑞󵄩󵄩󵄩 .
(66)
Since 0 < lim inf 𝑛 → ∞ 𝛽𝑛 ≤ lim sup𝑛 → ∞ 𝛽𝑛 < 1,
lim sup𝑛 → ∞ (𝜎𝑛 + 𝜏𝑛 ) < 1, {𝜆 𝑛 } ⊂ [𝑎, 𝑏], 𝛼𝑛 → 0, ‖𝐵2 𝑧𝑛 −
𝐵2 𝑝‖ → 0, ‖𝐵1 𝑧̃𝑛 − 𝐵1 𝑞‖ → 0 and lim𝑛 → ∞ ‖𝑥𝑛 − 𝑝‖ exists,
it follows from the boundedness of {𝑢𝑛 }, {𝑧𝑛 } and {̃𝑧𝑛 } that
lim𝑛 → ∞ ‖𝑧𝑛 − 𝑧̃𝑛 − (𝑝 − 𝑞)‖ = 0,
󵄩󵄩
lim 󵄩𝑧̃
𝑛→∞ 󵄩 𝑛
󵄩
− 𝑢𝑛 + (𝑝 − 𝑞)󵄩󵄩󵄩 = 0,
󵄩
󵄩
lim 󵄩󵄩𝛾 (𝑥𝑛 − 𝑦𝑛 ) + 𝛿𝑛 (𝑥𝑛 − 𝑆𝑦𝑛 )󵄩󵄩󵄩 = 0.
𝑛→∞ 󵄩 𝑛
Indeed, since {𝑥𝑛 } is bounded, there exists a subsequence
{𝑥𝑛𝑖 } of {𝑥𝑛 } that converges weakly to some 𝑥 ∈ 𝐶. We obtain
that 𝑥 ∈ Fix(𝑆)∩Ξ∩Γ. Taking into account that ‖𝑥𝑛 −𝑦𝑛 ‖ → 0
and ‖𝑥𝑛 − 𝑧𝑛 ‖ → 0 as 𝑛 → ∞, we deduce that 𝑦𝑛𝑖 → 𝑥
weakly and 𝑧𝑛𝑖 → 𝑥 weakly. First, it is clear from Lemma 15
and ‖𝑆𝑦𝑛 − 𝑦𝑛 ‖ → 0 that 𝑥 ∈ Fix(𝑆). Now let us show that
𝑥 ∈ Ξ. Note that
󵄩󵄩
󵄩
󵄩󵄩𝑧𝑛 − 𝐺 (𝑧𝑛 )󵄩󵄩󵄩
󵄩
󵄩
= 󵄩󵄩󵄩𝑧𝑛 − 𝑃𝐶 [𝑃𝐶 (𝑧𝑛 − 𝜇2 𝐵2 𝑧𝑛 ) − 𝜇1 𝐵1 𝑃𝐶 (𝑧𝑛 − 𝜇2 𝐵2 𝑧𝑛 )]󵄩󵄩󵄩
󵄩
󵄩
= 󵄩󵄩󵄩𝑧𝑛 − 𝑢𝑛 󵄩󵄩󵄩 󳨀→ 0,
(73)
as 𝑛 → ∞ where 𝐺 : 𝐶 → 𝐶 is defined as that in Lemma 1.
According to Lemma 15, we get 𝑥 ∈ Ξ. Further, let us show
that 𝑥 ∈ Γ. As a matter of fact, define
∇𝑓 (𝑣) + 𝑁𝐶𝑣, if 𝑣 ∈ 𝐶,
𝑇𝑣 = {
0,
if 𝑣 ∉ 𝐶,
(74)
where 𝑁𝐶𝑣 = {𝑤 ∈ H1 : ⟨𝑣 − 𝑢, 𝑤⟩ ≥ 0, ∀𝑢 ∈ 𝐶}. Then, 𝑇 is
maximal monotone and 0 ∈ 𝑇𝑣 if and only if 𝑣 ∈ VI(𝐶, ∇𝑓);
see [40] for more details. Let (𝑣, 𝑤) ∈ Gph(𝑇). Then, we have
(67)
𝑤 ∈ 𝑇𝑣 = ∇𝑓 (𝑣) + 𝑁𝐶𝑣,
(75)
𝑤 − ∇𝑓 (𝑣) ∈ 𝑁𝐶𝑣.
(76)
and hence
Consequently, it immediately follows that
󵄩󵄩
lim 󵄩𝑧
𝑛→∞ 󵄩 𝑛
󵄩
− 𝑢𝑛 󵄩󵄩󵄩 = 0,
󵄩󵄩
lim 󵄩𝑢
𝑛→∞ 󵄩 𝑛
󵄩
− 𝑢𝑛 󵄩󵄩󵄩 = 0.
(68)
Also, note that
󵄩󵄩
󵄩
󵄩󵄩𝑦𝑛 − 𝑢𝑛 󵄩󵄩󵄩
(69)
󵄩
󵄩
󵄩
󵄩
≤ 𝜎𝑛 󵄩󵄩󵄩𝑥𝑛 − 𝑢𝑛 󵄩󵄩󵄩 + (1 − 𝜎𝑛 − 𝜏𝑛 ) 󵄩󵄩󵄩𝑢𝑛 − 𝑢𝑛 󵄩󵄩󵄩 󳨀→ 0.
This together with ‖𝑥𝑛 − 𝑢𝑛 ‖ → 0 implies that
󵄩󵄩
lim 󵄩𝑥
𝑛→∞ 󵄩 𝑛
󵄩
− 𝑦𝑛 󵄩󵄩󵄩 = 0.
we have
󵄩
󵄩
lim 󵄩󵄩𝑆𝑦 − 𝑦𝑛 󵄩󵄩󵄩 = 0.
𝑛→∞ 󵄩 𝑛
⟨𝑣 − 𝑢, 𝑤 − ∇𝑓 (𝑣)⟩ ≥ 0,
(72)
∀𝑢 ∈ 𝐶.
(77)
On the other hand, from
𝑧𝑛 = 𝑃𝐶 (𝑥𝑛 − 𝜆 𝑛 ∇𝑓𝛼𝑛 (𝑥𝑛 )) ,
(70)
Since
󵄩󵄩
󵄩
󵄩󵄩𝛿𝑛 (𝑆𝑦𝑛 − 𝑥𝑛 )󵄩󵄩󵄩
󵄩
󵄩
= 󵄩󵄩󵄩𝛾𝑛 (𝑥𝑛 − 𝑦𝑛 ) + 𝛿𝑛 (𝑥𝑛 − 𝑆𝑦𝑛 ) + 𝛾𝑛 (𝑦𝑛 − 𝑥𝑛 )󵄩󵄩󵄩 (71)
󵄩
󵄩
󵄩
󵄩
≤ 󵄩󵄩󵄩𝛾𝑛 (𝑥𝑛 − 𝑦𝑛 ) + 𝛿𝑛 (𝑥𝑛 − 𝑆𝑦𝑛 )󵄩󵄩󵄩 + 𝛾𝑛 󵄩󵄩󵄩𝑥𝑛 − 𝑦𝑛 󵄩󵄩󵄩 ,
󵄩
󵄩
lim 󵄩󵄩𝑆𝑦 − 𝑥𝑛 󵄩󵄩󵄩 = 0,
𝑛→∞ 󵄩 𝑛
So, we have
𝑣 ∈ 𝐶,
(78)
⟨𝑥𝑛 − 𝜆 𝑛 ∇𝑓𝛼𝑛 (𝑥𝑛 ) − 𝑧𝑛 , 𝑧𝑛 − 𝑣⟩ ≥ 0,
(79)
we have
and hence,
⟨𝑣 − 𝑧𝑛 ,
𝑧𝑛 − 𝑥𝑛
+ ∇𝑓𝛼𝑛 (𝑥𝑛 )⟩ ≥ 0.
𝜆𝑛
(80)
Therefore, from
Step 4. {𝑥𝑛 }, {𝑦𝑛 }, and {𝑧𝑛 } converge weakly to an element 𝑥 ∈
Fix(𝑆) ∩ Ξ ∩ Γ.
𝑤 − ∇𝑓 (𝑣) ∈ 𝑁𝐶𝑣,
𝑧𝑛𝑖 ∈ 𝐶,
(81)
Abstract and Applied Analysis
13
H1 be 𝛽𝑖 -inverse strongly monotone for 𝑖 = 1, 2. Let 𝑆 :
𝐶 → 𝐶 be a 𝑘-strictly pseudocontractive mapping such that
Fix(𝑆)∩Ξ∩Γ ≠ 0. For given 𝑥0 ∈ 𝐶 arbitrarily, let the sequences
{𝑥𝑛 }, {𝑦𝑛 }, {𝑧𝑛 } be generated iteratively by
we have
⟨𝑣 − 𝑧𝑛𝑖 , 𝑤⟩
≥ ⟨𝑣 − 𝑧𝑛𝑖 , ∇𝑓 (𝑣)⟩
𝑧𝑛 = 𝑃𝐶 (𝑥𝑛 − 𝜆 𝑛 ∇𝑓𝛼𝑛 (𝑥𝑛 )) ,
≥ ⟨𝑣 − 𝑧𝑛𝑖 , ∇𝑓 (𝑣)⟩
− ⟨𝑣 − 𝑧𝑛𝑖 ,
𝑧𝑛𝑖 − 𝑥𝑛𝑖
𝜆 𝑛𝑖
𝑦𝑛 = 𝜏𝑛 𝑃𝐶 (𝑥𝑛 − 𝜆 𝑛 ∇𝑓𝛼𝑛 (𝑧𝑛 ))
+ ∇𝑓𝛼𝑛 (𝑥𝑛𝑖 )⟩
+ (1 − 𝜏𝑛 ) 𝑃𝐶 [𝑃𝐶 (𝑧𝑛 − 𝜇2 𝐵2 𝑧𝑛 )
𝑖
−𝜇1 𝐵1 𝑃𝐶 (𝑧𝑛 − 𝜇2 𝐵2 𝑧𝑛 )] ,
= ⟨𝑣 − 𝑧𝑛𝑖 , ∇𝑓 (𝑣)⟩
− ⟨𝑣 − 𝑧𝑛𝑖 ,
𝑧𝑛𝑖 − 𝑥𝑛𝑖
𝜆 𝑛𝑖
𝑥𝑛+1 = 𝛽𝑛 𝑥𝑛 + 𝛾𝑛 𝑦𝑛 + 𝛿𝑛 𝑆𝑦𝑛 ,
+ ∇𝑓 (𝑥𝑛𝑖 )⟩
(82)
− 𝛼𝑛𝑖 ⟨𝑣 − 𝑧𝑛𝑖 , 𝑥𝑛𝑖 ⟩
𝜆 𝑛𝑖
(iii) 0 < lim inf 𝑛 → ∞ 𝜏𝑛 ≤ lim sup𝑛 → ∞ 𝜏𝑛 < 1;
(iv) 0 < lim inf 𝑛 → ∞ 𝛽𝑛 ≤ lim sup𝑛 → ∞ 𝛽𝑛 < 1 and
lim inf 𝑛 → ∞ 𝛿𝑛 > 0;
⟩ − 𝛼𝑛𝑖 ⟨𝑣 − 𝑧𝑛𝑖 , 𝑥𝑛𝑖 ⟩
(v) 0 < lim inf 𝑛 → ∞ 𝜆 𝑛 ≤ lim sup𝑛 → ∞ 𝜆 𝑛 < 1/‖𝐴‖2 .
≥ ⟨𝑣 − 𝑧𝑛𝑖 , ∇𝑓 (𝑧𝑛𝑖 ) − ∇𝑓 (𝑥𝑛𝑖 )⟩
− ⟨𝑣 − 𝑧𝑛𝑖 ,
𝑧𝑛𝑖 − 𝑥𝑛𝑖
𝜆 𝑛𝑖
Then all the sequences {𝑥𝑛 }, {𝑦𝑛 }, {𝑧𝑛 } converge weakly to an
element 𝑥 ∈ Fix(𝑆) ∩ Ξ ∩ Γ. Furthermore, (𝑥, 𝑦) is a solution of
the GSVI (3), where 𝑦 = 𝑃𝐶(𝑥 − 𝜇2 𝐵2 𝑥).
⟩ − 𝛼𝑛𝑖 ⟨𝑣 − 𝑧𝑛𝑖 , 𝑥𝑛𝑖 ⟩ .
Hence, we get
⟨𝑣 − 𝑥, 𝑤⟩ ≥ 0,
as 𝑖 󳨀→ ∞.
(83)
Since 𝑇 is maximal monotone, we have 𝑥 ∈ 𝑇−1 0, and hence,
𝑥 ∈ VI(𝐶, ∇𝑓). Thus, it is clear that 𝑥 ∈ Γ. Therefore, 𝑥 ∈
Fix(𝑆) ∩ Ξ ∩ Γ.
Let {𝑥𝑛𝑗 } be another subsequence of {𝑥𝑛 } such that {𝑥𝑛𝑗 }
converges weakly to 𝑥̂ ∈ 𝐶. Then, 𝑥̂ ∈ Fix(𝑆) ∩ Ξ ∩ Γ. Let us
̂ Assume that 𝑥 ≠ 𝑥.
̂ From the Opial condition
show that 𝑥 = 𝑥.
[41], we have
󵄩
󵄩
lim 󵄩󵄩𝑥 − 𝑥󵄩󵄩󵄩
𝑛→∞ 󵄩 𝑛
󵄩
󵄩
󵄩
󵄩
= lim inf 󵄩󵄩󵄩󵄩𝑥𝑛𝑖 − 𝑥󵄩󵄩󵄩󵄩 < lim inf 󵄩󵄩󵄩󵄩𝑥𝑛𝑖 − 𝑥̂󵄩󵄩󵄩󵄩
𝑖→∞
𝑖→∞
󵄩󵄩
󵄩󵄩
󵄩
󵄩
= lim inf 󵄩󵄩󵄩𝑥𝑛 − 𝑥̂󵄩󵄩󵄩 = lim inf 󵄩󵄩󵄩𝑥𝑛𝑗 − 𝑥̂󵄩󵄩󵄩
𝑛→∞
󵄩
𝑗→∞ 󵄩
where 𝜇𝑖 ∈ (0, 2𝛽𝑖 ) for 𝑖 = 1, 2, {𝛼𝑛 } ⊂ (0, ∞), {𝜆 𝑛 } ⊂ (0, 1/
‖𝐴‖2 ) and {𝜏𝑛 }, {𝛽𝑛 }, {𝛾𝑛 }, {𝛿𝑛 } ⊂ [0, 1] such that
(ii) 𝛽𝑛 + 𝛾𝑛 + 𝛿𝑛 = 1 and (𝛾𝑛 + 𝛿𝑛 )𝑘 ≤ 𝛾𝑛 for all 𝑛 ≥ 0;
+ ⟨𝑣 − 𝑧𝑛𝑖 , ∇𝑓 (𝑧𝑛𝑖 ) − ∇𝑓 (𝑥𝑛𝑖 )⟩
𝑧𝑛𝑖 − 𝑥𝑛𝑖
∀𝑛 ≥ 0,
(i) ∑∞
𝑛=0 𝛼𝑛 < ∞;
= ⟨𝑣 − 𝑧𝑛𝑖 , ∇𝑓 (𝑣) − ∇𝑓 (𝑧𝑛𝑖 )⟩
− ⟨𝑣 − 𝑧𝑛𝑖 ,
(85)
Proof. In Theorem 20, put 𝜎𝑛 = 0 for all 𝑛 ≥ 0. Then, in this
case, Theorem 20 reduces to Corollary 21.
Next, utilizing Corollary 21, we give the following result.
Corollary 22. Let 𝐶 be a nonempty closed convex subset of
a real Hilbert space H1 . Let 𝐴 ∈ 𝐵(H1 , H2 ) and 𝑆 : 𝐶 →
𝐶 be a nonexpansive mapping such that Fix(𝑆) ∩ Γ ≠ 0. For
given 𝑥0 ∈ 𝐶 arbitrarily, let the sequences {𝑥𝑛 }, {𝑦𝑛 }, {𝑧𝑛 } be
generated iteratively by
𝑧𝑛 = 𝑃𝐶 (𝑥𝑛 − 𝜆 𝑛 ∇𝑓𝛼𝑛 (𝑥𝑛 )) ,
𝑦𝑛 = (1 − 𝜏𝑛 ) 𝑧𝑛 + 𝜏𝑛 𝑃𝐶 (𝑥𝑛 − 𝜆 𝑛 ∇𝑓𝛼𝑛 (𝑧𝑛 )) ,
(84)
󵄩󵄩
󵄩󵄩
󵄩
󵄩
< lim inf 󵄩󵄩󵄩𝑥𝑛𝑗 − 𝑥󵄩󵄩󵄩 = lim 󵄩󵄩󵄩𝑥𝑛 − 𝑥󵄩󵄩󵄩 .
󵄩 𝑛→∞
𝑗→∞ 󵄩
̂
This leads to a contradiction. Consequently, we have 𝑥 = 𝑥.
This implies that {𝑥𝑛 } converges weakly to 𝑥 ∈ Fix(𝑆) ∩ Ξ ∩ Γ.
Further, from ‖𝑥𝑛 − 𝑦𝑛 ‖ → 0 and ‖𝑥𝑛 − 𝑧𝑛 ‖ → 0, it follows
that both {𝑦𝑛 } and {𝑧𝑛 } converge weakly to 𝑥. This completes
the proof.
Corollary 21. Let 𝐶 be a nonempty closed convex subset of a
real Hilbert space H1 . Let 𝐴 ∈ 𝐵(H1 , H2 ) and 𝐵𝑖 : 𝐶 →
𝑥𝑛+1 = 𝛽𝑛 𝑥𝑛 + (1 − 𝛽𝑛 ) 𝑆𝑦𝑛 ,
(86)
∀𝑛 ≥ 0,
where {𝛼𝑛 } ⊂ (0, ∞), {𝜆 𝑛 } ⊂ (0, 1/‖𝐴‖2 ) and {𝜏𝑛 }, {𝛽𝑛 } ⊂ [0, 1]
such that
(i) ∑∞
𝑛=0 𝛼𝑛 < ∞;
(ii) 0 < lim inf 𝑛 → ∞ 𝜏𝑛 ≤ lim sup𝑛 → ∞ 𝜏𝑛 < 1;
(iii) 0 < lim inf 𝑛 → ∞ 𝛽𝑛 ≤ lim sup𝑛 → ∞ 𝛽𝑛 < 1;
(iv) 0 < lim inf 𝑛 → ∞ 𝜆 𝑛 ≤ lim sup𝑛 → ∞ 𝜆 𝑛 < 1/‖𝐴‖2 .
Then all the sequences {𝑥𝑛 }, {𝑦𝑛 }, {𝑧𝑛 } converge weakly to an
element 𝑥 ∈ Fix(𝑆) ∩ Γ.
14
Abstract and Applied Analysis
Proof. In Corollary 21, put 𝐵1 = 𝐵2 = 0 and 𝛾𝑛 = 0. Then,
Ξ = 𝐶, 𝛽𝑛 + 𝛿𝑛 = 1 for all 𝑛 ≥ 0, and the iterative scheme (85)
is equivalent to
𝑥𝑛+1 = 𝛽𝑛 𝑥𝑛 + 𝛿𝑛 𝑆𝑦𝑛 ,
(87)
∀𝑛 ≥ 0.
This is equivalent to (86). Since 𝑆 is a nonexpansive mapping,
𝑆 must be a 𝑘-strictly pseudocontractive mapping with 𝑘 = 0.
In this case, it is easy to see that all the conditions (i)–(v) in
Corollary 21 are satisfied. Therefore, in terms of Corollary 21,
we obtain the desired result.
Now, we are in a position to prove the weak convergence
of the sequences generated by the Mann-type extragradient
iterative algorithm (17) with regularization.
Theorem 23. Let 𝐶 be a nonempty closed convex subset of
a real Hilbert space H1 . Let 𝐴 ∈ 𝐵(H1 , H2 ) and 𝐵𝑖 :
𝐶 → H1 be 𝛽𝑖 -inverse strongly monotone for 𝑖 = 1, 2.
Let 𝑆 : 𝐶 → 𝐶 be a 𝑘-strictly pseudocontractive mapping
such that Fix(𝑆) ∩ Ξ ∩ Γ ≠ 0. For given 𝑥0 ∈ 𝐶 arbitrarily,
𝑢𝑛 } be the sequences generated by the Mannlet {𝑥𝑛 }, {𝑢𝑛 }, {̃
type extragradient iterative algorithm (17) with regularization,
where 𝜇𝑖 ∈ (0, 2𝛽𝑖 ) for 𝑖 = 1, 2, {𝛼𝑛 } ⊂ (0, ∞), {𝜆 𝑛 } ⊂
(0, 1/‖𝐴‖2 ) and {𝜎𝑛 }, {𝛽𝑛 }, {𝛾𝑛 }, {𝛿𝑛 } ⊂ [0, 1] such that
(i) ∑∞
𝑛=0 𝛼𝑛 < ∞;
(ii) 𝛽𝑛 + 𝛾𝑛 + 𝛿𝑛 = 1 and (𝛾𝑛 + 𝛿𝑛 )𝑘 ≤ 𝛾𝑛 for all 𝑛 ≥ 0;
(iii) lim sup𝑛 → ∞ 𝜎𝑛 < 1;
(iv) 0 < lim inf 𝑛 → ∞ 𝛽𝑛 ≤ lim sup𝑛 → ∞ 𝛽𝑛 < 1 and
lim inf 𝑛 → ∞ 𝛿𝑛 > 0;
(v) 0 < lim inf 𝑛 → ∞ 𝜆 𝑛 ≤ lim sup𝑛 → ∞ 𝜆 𝑛 < 1/‖𝐴‖2 .
𝑢𝑛 } converge weakly to an
Then the sequences {𝑥𝑛 }, {𝑢𝑛 }, {̃
element 𝑥 ∈ Fix(𝑆) ∩ Ξ ∩ Γ. Furthermore, (𝑥, 𝑦) is a solution of
the GSVI (3), where 𝑦 = 𝑃𝐶(𝑥 − 𝜇2 𝐵2 𝑥).
Proof. First, taking into account 0 < lim inf 𝑛 → ∞ 𝜆 𝑛 ≤
lim sup𝑛 → ∞ 𝜆 𝑛 < 1/‖𝐴‖2 , without loss of generality, we may
assume that {𝜆 𝑛 } ⊂ [𝑎, 𝑏] for some 𝑎, 𝑏 ∈ (0, 1/‖𝐴‖2 ). Repeating the same argument as that in the proof of Theorem 20,
we can show that 𝑃𝐶(𝐼 − 𝜆∇𝑓𝛼 ) is 𝜁-averaged for each
𝜆 ∈ (0, 1/(𝛼 + ‖𝐴‖2 )), where 𝜁 = (2 + 𝜆(𝛼 + ‖𝐴‖2 ))/4.
Further, repeating the same argument as that in the proof of
Theorem 20, we can also show that for each integer 𝑛 ≥ 0,
𝑃𝐶(𝐼−𝜆 𝑛 ∇𝑓𝛼𝑛 ) is 𝜁𝑛 -averaged with 𝜁𝑛 = (2+𝜆 𝑛 (𝛼𝑛 +‖𝐴‖2 ))/4 ∈
(0, 1).
Next we divide the remainder of the proof into several
steps.
Step 1. {𝑥𝑛 } is bounded.
Indeed, take 𝑝 ∈ Fix(𝑆) ∩ Ξ ∩ Γ arbitrarily. Then 𝑆𝑝 = 𝑝,
𝑃𝐶(𝐼 − 𝜆∇𝑓)𝑝 = 𝑝 for 𝜆 ∈ (0, 2/‖𝐴‖2 ), and
𝑝 = 𝑃𝐶 [𝑃𝐶 (𝑝 − 𝜇2 𝐵2 𝑝) − 𝜇1 𝐵1 𝑃𝐶 (𝑝 − 𝜇2 𝐵2 𝑝)] .
𝑞 = 𝑃𝐶 (𝑝 − 𝜇2 𝐵2 𝑝) ,
𝑥̃𝑛 = 𝑃𝐶 (𝑥𝑛 − 𝜇2 𝐵2 𝑥𝑛 ) ,
𝑧𝑛 = 𝑃𝐶 (𝑥𝑛 − 𝜆 𝑛 ∇𝑓𝛼𝑛 (𝑥𝑛 )) ,
𝑦𝑛 = 𝜏𝑛 𝑃𝐶 (𝑥𝑛 − 𝜆 𝑛 ∇𝑓𝛼𝑛 (𝑧𝑛 )) + (1 − 𝜏𝑛 ) 𝑧𝑛 ,
For simplicity, we write
(88)
(89)
𝑢𝑛 = 𝑃𝐶 (𝑢𝑛 − 𝜆 𝑛 ∇𝑓𝛼𝑛 (̃
𝑢𝑛 )) ,
for each 𝑛 ≥ 0. Then 𝑦𝑛 = 𝜎𝑛 𝑥𝑛 + (1 − 𝜎𝑛 )𝑢𝑛 for each 𝑛 ≥ 0.
Utilizing the arguments similar to those of (46) and (47) in
the proof of Theorem 20, from (17) we can obtain
󵄩󵄩󵄩𝑢̃𝑛 − 𝑝󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩𝑢𝑛 − 𝑝󵄩󵄩󵄩 + 𝜆 𝑛 𝛼𝑛 󵄩󵄩󵄩𝑝󵄩󵄩󵄩 ,
(90)
󵄩 󵄩
󵄩 󵄩
󵄩
󵄩
󵄩󵄩 ̃
󵄩 󵄩󵄩
󵄩2 󵄩
󵄩2
󵄩
(91)
󵄩󵄩𝑢𝑛 − 𝑝󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩𝑢𝑛 − 𝑝󵄩󵄩󵄩 + 2𝜆 𝑛 𝛼𝑛 󵄩󵄩󵄩𝑝󵄩󵄩󵄩 󵄩󵄩󵄩𝑢̃𝑛 − 𝑝󵄩󵄩󵄩 .
Since 𝐵𝑖 : 𝐶 → H1 is 𝛽𝑖 -inverse strongly monotone and
0 < 𝜇𝑖 < 2𝛽𝑖 for 𝑖 = 1, 2, utilizing the argument similar to
that of (49) in the proof of Theorem 20, we can obtain that
for all 𝑛 ≥ 0,
󵄩2
󵄩󵄩
󵄩󵄩𝑢𝑛 − 𝑝󵄩󵄩󵄩
󵄩
󵄩2
󵄩
󵄩2
(92)
≤ 󵄩󵄩󵄩𝑥𝑛 − 𝑝󵄩󵄩󵄩 − 𝜇2 (2𝛽2 − 𝜇2 ) 󵄩󵄩󵄩𝐵2 𝑥𝑛 − 𝐵2 𝑝󵄩󵄩󵄩
󵄩
󵄩2 󵄩
󵄩2
− 𝜇1 (2𝛽1 − 𝜇1 ) 󵄩󵄩󵄩𝐵1 𝑥̃𝑛 − 𝐵1 𝑞󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩𝑥𝑛 − 𝑝󵄩󵄩󵄩 .
Utilizing the argument similar to that of (52) in the proof of
Theorem 20, from (90) we can obtain
󵄩󵄩
󵄩2
󵄩󵄩𝑢𝑛 − 𝑝󵄩󵄩󵄩
󵄩 󵄩󵄩
󵄩
󵄩2
󵄩
≤ 󵄩󵄩󵄩𝑢𝑛 − 𝑝󵄩󵄩󵄩 + 2𝜆 𝑛 𝛼𝑛 󵄩󵄩󵄩𝑝󵄩󵄩󵄩 󵄩󵄩󵄩𝑢̃𝑛 − 𝑝󵄩󵄩󵄩
(93)
2
󵄩2
󵄩
+ (𝜆2𝑛 (𝛼𝑛 + ‖𝐴‖2 ) − 1) 󵄩󵄩󵄩𝑢𝑛 − 𝑢̃𝑛 󵄩󵄩󵄩
󵄩 󵄩2
󵄩
󵄩
≤ (󵄩󵄩󵄩𝑢𝑛 − 𝑝󵄩󵄩󵄩 + 2𝜆 𝑛 𝛼𝑛 󵄩󵄩󵄩𝑝󵄩󵄩󵄩) .
Hence, it follows from (92) and (93) that
󵄩󵄩
󵄩
󵄩󵄩𝑦𝑛 − 𝑝󵄩󵄩󵄩
󵄩
󵄩
= 󵄩󵄩󵄩𝜎𝑛 (𝑥𝑛 − 𝑝) + (1 − 𝜎𝑛 ) (𝑢𝑛 − 𝑝)󵄩󵄩󵄩
󵄩
󵄩
󵄩
󵄩
≤ 𝜎𝑛 󵄩󵄩󵄩𝑥𝑛 − 𝑝󵄩󵄩󵄩 + (1 − 𝜎𝑛 ) 󵄩󵄩󵄩𝑢𝑛 − 𝑝󵄩󵄩󵄩
󵄩
󵄩
≤ 𝜎𝑛 󵄩󵄩󵄩𝑥𝑛 − 𝑝󵄩󵄩󵄩 + (1 − 𝜎𝑛 )
󵄩 󵄩
󵄩
󵄩
× (󵄩󵄩󵄩𝑥𝑛 − 𝑝󵄩󵄩󵄩 + 2𝜆 𝑛 𝛼𝑛 󵄩󵄩󵄩𝑝󵄩󵄩󵄩)
󵄩
󵄩 󵄩
󵄩
≤ 󵄩󵄩󵄩𝑥𝑛 − 𝑝󵄩󵄩󵄩 + 2𝜆 𝑛 𝛼𝑛 󵄩󵄩󵄩𝑝󵄩󵄩󵄩 .
(94)
Since (𝛾𝑛 +𝛿𝑛 )𝑘 ≤ 𝛾𝑛 for all 𝑛 ≥ 0, by Lemma 18 we can readily
see from (94) that
󵄩
󵄩󵄩
󵄩󵄩𝑥𝑛+1 − 𝑝󵄩󵄩󵄩
󵄩
󵄩
󵄩
󵄩
≤ 𝛽𝑛 󵄩󵄩󵄩𝑥𝑛 − 𝑝󵄩󵄩󵄩 + (𝛾𝑛 + 𝛿𝑛 ) 󵄩󵄩󵄩𝑦𝑛 − 𝑝󵄩󵄩󵄩
(95)
󵄩 󵄩
󵄩
󵄩
≤ 󵄩󵄩󵄩𝑥𝑛 − 𝑝󵄩󵄩󵄩 + 2𝑏 󵄩󵄩󵄩𝑝󵄩󵄩󵄩 𝛼𝑛 .
∞
Since ∑∞
𝑛=0 𝛼𝑛 < ∞, it is clear that ∑𝑛=0 2𝑏‖𝑝‖𝛼𝑛 < ∞. Thus,
by Corollary 17 we conclude that
󵄩
󵄩
lim 󵄩󵄩𝑥 − 𝑝󵄩󵄩󵄩 exists for each 𝑝 ∈ Fix (𝑆) ∩ Ξ ∩ Γ, (96)
𝑛→∞ 󵄩 𝑛
Abstract and Applied Analysis
15
and the sequence {𝑥𝑛 } is bounded. Since 𝑃𝐶, ∇𝑓𝛼𝑛 , 𝐵1 and 𝐵2
are Lipschitz continuous, it is easy to see that {𝑢𝑛 }, {̃
𝑢𝑛 }, {𝑢𝑛 },
{𝑦𝑛 } and {𝑥̃𝑛 } are bounded, where 𝑥̃𝑛 = 𝑃𝐶(𝑥𝑛 − 𝜇2 𝐵2 𝑥𝑛 ) for
all 𝑛 ≥ 0.
Step 2. Consider lim𝑛 → ∞ ‖𝐵2 𝑥𝑛 − 𝐵2 𝑝‖ = 0, lim𝑛 → ∞ ‖𝐵1 𝑥̃𝑛 −
𝐵1 𝑞‖ = 0 and lim𝑛 → ∞ ‖𝑢𝑛 − 𝑢̃𝑛 ‖ = 0, where 𝑞 = 𝑃𝐶(𝑝−𝜇2 𝐵2 𝑝).
Indeed, utilizing Lemma 18 and the convexity of ‖ ⋅ ‖2 , we
obtain from (17), (92), and (93) that
󵄩2
󵄩󵄩
󵄩󵄩𝑥𝑛+1 − 𝑝󵄩󵄩󵄩
󵄩
󵄩2
󵄩
󵄩2
≤ 𝛽𝑛 󵄩󵄩󵄩𝑥𝑛 − 𝑝󵄩󵄩󵄩 + (𝛾𝑛 + 𝛿𝑛 ) 󵄩󵄩󵄩𝑦𝑛 − 𝑝󵄩󵄩󵄩
󵄩
󵄩2
≤ 𝛽𝑛 󵄩󵄩󵄩𝑥𝑛 − 𝑝󵄩󵄩󵄩 + (𝛾𝑛 + 𝛿𝑛 )
󵄩
󵄩2
󵄩
󵄩2
× [𝜎𝑛 󵄩󵄩󵄩𝑥𝑛 − 𝑝󵄩󵄩󵄩 + (1 − 𝜎𝑛 ) 󵄩󵄩󵄩𝑢𝑛 − 𝑝󵄩󵄩󵄩 ]
󵄩󵄩
lim 󵄩𝐵 𝑥̃
𝑛→∞ 󵄩 1 𝑛
󵄩󵄩
lim 󵄩𝐵 𝑥
𝑛→∞ 󵄩 2 𝑛
󵄩
− 𝐵1 𝑞󵄩󵄩󵄩 = 0,
(99)
󵄩
− 𝐵2 𝑝󵄩󵄩󵄩 = 0.
Step 3. Consider lim𝑛 → ∞ ‖𝑆𝑦𝑛 − 𝑦𝑛 ‖ = 0.
Indeed, utilizing the Lipschitz continuity of ∇𝑓𝛼𝑛 , we have
󵄩
󵄩󵄩
󵄩󵄩𝑢𝑛 − 𝑢̃𝑛 󵄩󵄩󵄩
󵄩
󵄩
= 󵄩󵄩󵄩󵄩𝑃𝐶 (𝑢𝑛 − 𝜆 𝑛 ∇𝑓𝛼𝑛 (̃
𝑢𝑛 )) − 𝑃𝐶 (𝑢𝑛 − 𝜆 𝑛 ∇𝑓𝛼𝑛 (𝑢𝑛 ))󵄩󵄩󵄩󵄩
󵄩
󵄩
≤ 𝜆 𝑛 (𝛼𝑛 + ‖𝐴‖2 ) 󵄩󵄩󵄩𝑢̃𝑛 − 𝑢𝑛 󵄩󵄩󵄩 .
(100)
󵄩
󵄩2
≤ 𝛽𝑛 󵄩󵄩󵄩𝑥𝑛 − 𝑝󵄩󵄩󵄩 + (𝛾𝑛 + 𝛿𝑛 )
󵄩
󵄩2
× {𝜎𝑛 󵄩󵄩󵄩𝑥𝑛 − 𝑝󵄩󵄩󵄩 + (1 − 𝜎𝑛 )
󵄩
󵄩 󵄩󵄩
󵄩2
󵄩
× [󵄩󵄩󵄩𝑢𝑛 − 𝑝󵄩󵄩󵄩 + 2𝜆 𝑛 𝛼𝑛 󵄩󵄩󵄩𝑝󵄩󵄩󵄩 󵄩󵄩󵄩𝑢̃𝑛 − 𝑝󵄩󵄩󵄩
2
󵄩2
󵄩
+ (𝜆2𝑛 (𝛼𝑛 + ‖𝐴‖2 ) − 1) 󵄩󵄩󵄩𝑢𝑛 − 𝑢̃𝑛 󵄩󵄩󵄩 ]}
󵄩
󵄩2
≤ 𝛽𝑛 󵄩󵄩󵄩𝑥𝑛 − 𝑝󵄩󵄩󵄩 + (𝛾𝑛 + 𝛿𝑛 )
󵄩
󵄩2
× {𝜎𝑛 󵄩󵄩󵄩𝑥𝑛 − 𝑝󵄩󵄩󵄩 + (1 − 𝜎𝑛 )
󵄩2
󵄩
󵄩2
󵄩
× [󵄩󵄩󵄩𝑥𝑛 − 𝑝󵄩󵄩󵄩 − 𝜇2 (2𝛽2 − 𝜇2 ) 󵄩󵄩󵄩𝐵2 𝑥𝑛 − 𝐵2 𝑝󵄩󵄩󵄩
󵄩 󵄩󵄩
󵄩
󵄩2
󵄩
−𝜇1 (2𝛽1 −𝜇1 ) 󵄩󵄩󵄩𝐵1 𝑥̃𝑛 −𝐵1 𝑞󵄩󵄩󵄩 +2𝜆 𝑛 𝛼𝑛 󵄩󵄩󵄩𝑝󵄩󵄩󵄩 󵄩󵄩󵄩𝑢̃𝑛 −𝑝󵄩󵄩󵄩
2
󵄩2
󵄩
+ (𝜆2𝑛 (𝛼𝑛 + ‖𝐴‖2 ) − 1) 󵄩󵄩󵄩𝑢𝑛 − 𝑢̃𝑛 󵄩󵄩󵄩 ]}
󵄩
󵄩 󵄩󵄩
󵄩2
󵄩
≤ 󵄩󵄩󵄩𝑥𝑛 − 𝑝󵄩󵄩󵄩 + 2𝜆 𝑛 𝛼𝑛 󵄩󵄩󵄩𝑝󵄩󵄩󵄩 󵄩󵄩󵄩𝑢̃𝑛 − 𝑝󵄩󵄩󵄩 − (𝛾𝑛 + 𝛿𝑛 ) (1 − 𝜎𝑛 )
󵄩
󵄩2
× {𝜇 (2𝛽2 − 𝜇2 ) 󵄩󵄩󵄩𝐵2 𝑥𝑛 − 𝐵2 𝑝󵄩󵄩󵄩
2
󵄩
󵄩2
+ 𝜇1 (2𝛽1 − 𝜇1 ) 󵄩󵄩󵄩𝐵1 𝑥̃𝑛 − 𝐵1 𝑞󵄩󵄩󵄩
2 󵄩
󵄩2
+ (1 − 𝜆2𝑛 (𝛼𝑛 + ‖𝐴‖2 ) ) 󵄩󵄩󵄩𝑢𝑛 − 𝑢̃𝑛 󵄩󵄩󵄩 } .
(97)
Therefore,
󵄩
󵄩2
(𝛾𝑛 + 𝛿𝑛 ) (1 − 𝜎𝑛 ) {𝜇2 (2𝛽2 − 𝜇2 ) 󵄩󵄩󵄩𝐵2 𝑥𝑛 − 𝐵2 𝑝󵄩󵄩󵄩
󵄩
󵄩2
+ 𝜇1 (2𝛽1 − 𝜇1 ) 󵄩󵄩󵄩𝐵1 𝑥̃𝑛 − 𝐵1 𝑞󵄩󵄩󵄩
2 󵄩
󵄩2
+ (1 − 𝜆2𝑛 (𝛼𝑛 + ‖𝐴‖2 ) ) 󵄩󵄩󵄩𝑢𝑛 − 𝑢̃𝑛 󵄩󵄩󵄩 }
󵄩
󵄩 󵄩󵄩
󵄩2 󵄩
󵄩2
󵄩
≤ 󵄩󵄩󵄩𝑥𝑛 − 𝑝󵄩󵄩󵄩 − 󵄩󵄩󵄩𝑥𝑛+1 − 𝑝󵄩󵄩󵄩 + 2𝜆 𝑛 𝛼𝑛 󵄩󵄩󵄩𝑝󵄩󵄩󵄩 󵄩󵄩󵄩𝑢̃𝑛 − 𝑝󵄩󵄩󵄩 .
Since 𝛼𝑛 → 0, lim sup𝑛 → ∞ 𝜎𝑛 < 1, lim𝑛 → ∞ ‖𝑥𝑛 − 𝑝‖ exists,
lim inf 𝑛 → ∞ (𝛾𝑛 + 𝛿𝑛 ) > 0 and {𝜆 𝑛 } ⊂ [𝑎, 𝑏] for some 𝑎, 𝑏 ∈
𝑢𝑛 } that
(0, 1/‖𝐴‖2 ), it follows from the boundedness of {̃
󵄩󵄩
󵄩
󵄩
lim 󵄩𝑢 − 𝑢̃𝑛 󵄩󵄩 = 0,
𝑛→∞ 󵄩 𝑛
(98)
This together with ‖̃
𝑢𝑛 − 𝑢𝑛 ‖ → 0 implies that lim𝑛 → ∞ ‖𝑢𝑛 −
𝑢̃𝑛 ‖ = 0 and hence lim𝑛 → ∞ ‖𝑢𝑛 − 𝑢𝑛 ‖ = 0. Utilizing the
arguments similar to those of (61) and (63) in the proof of
Theorem 20, we get
󵄩2 󵄩
󵄩2 󵄩
󵄩2
󵄩󵄩 ̃
󵄩󵄩𝑥𝑛 − 𝑞󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩𝑥𝑛 − 𝑝󵄩󵄩󵄩 − 󵄩󵄩󵄩𝑥𝑛 − 𝑥̃𝑛 − (𝑝 − 𝑞)󵄩󵄩󵄩
󵄩
󵄩󵄩
󵄩
+ 2𝜇2 󵄩󵄩󵄩𝑥𝑛 − 𝑥̃𝑛 − (𝑝 − 𝑞)󵄩󵄩󵄩 󵄩󵄩󵄩𝐵2 𝑥𝑛 − 𝐵2 𝑝󵄩󵄩󵄩 ,
(101)
󵄩󵄩
󵄩2 󵄩
󵄩2 󵄩
󵄩2
󵄩󵄩𝑢𝑛 − 𝑝󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩𝑥̃𝑛 − 𝑞󵄩󵄩󵄩 − 󵄩󵄩󵄩𝑥̃𝑛 − 𝑢𝑛 + (𝑝 − 𝑞)󵄩󵄩󵄩
󵄩
󵄩󵄩
󵄩
+ 2𝜇1 󵄩󵄩󵄩𝑥̃𝑛 − 𝑢𝑛 + (𝑝 − 𝑞)󵄩󵄩󵄩 󵄩󵄩󵄩𝐵1 𝑥̃𝑛 − 𝐵1 𝑞󵄩󵄩󵄩 .
Utilizing (91) and (101), we have
󵄩󵄩󵄩𝑢𝑛 − 𝑝󵄩󵄩󵄩2
󵄩
󵄩
󵄩2
󵄩
= 󵄩󵄩󵄩𝑢̃𝑛 − 𝑝 + 𝑢𝑛 − 𝑢̃𝑛 󵄩󵄩󵄩
󵄩
󵄩2
≤ 󵄩󵄩󵄩𝑢̃𝑛 − 𝑝󵄩󵄩󵄩 + 2 ⟨𝑢𝑛 − 𝑢̃𝑛 , 𝑢𝑛 − 𝑝⟩
󵄩󵄩
󵄩2
󵄩
󵄩
󵄩
≤ 󵄩󵄩󵄩𝑢̃𝑛 − 𝑝󵄩󵄩󵄩 + 2 󵄩󵄩󵄩𝑢𝑛 − 𝑢̃𝑛 󵄩󵄩󵄩 󵄩󵄩󵄩𝑢𝑛 − 𝑝󵄩󵄩󵄩
󵄩
󵄩 󵄩󵄩
󵄩2
󵄩
≤ 󵄩󵄩󵄩𝑢𝑛 − 𝑝󵄩󵄩󵄩 + 2𝜆 𝑛 𝛼𝑛 󵄩󵄩󵄩𝑝󵄩󵄩󵄩 󵄩󵄩󵄩𝑢̃𝑛 − 𝑝󵄩󵄩󵄩
󵄩
󵄩󵄩
󵄩
+ 2 󵄩󵄩󵄩𝑢𝑛 − 𝑢̃𝑛 󵄩󵄩󵄩 󵄩󵄩󵄩𝑢𝑛 − 𝑝󵄩󵄩󵄩
󵄩
󵄩2 󵄩
󵄩2
≤ 󵄩󵄩󵄩𝑥̃𝑛 − 𝑞󵄩󵄩󵄩 − 󵄩󵄩󵄩𝑥̃𝑛 − 𝑢𝑛 + (𝑝 − 𝑞)󵄩󵄩󵄩
(102)
󵄩
󵄩󵄩
󵄩
+ 2𝜇1 󵄩󵄩󵄩𝑥̃𝑛 − 𝑢𝑛 + (𝑝 − 𝑞)󵄩󵄩󵄩 󵄩󵄩󵄩𝐵1 𝑥̃𝑛 − 𝐵1 𝑞󵄩󵄩󵄩
󵄩
󵄩 󵄩󵄩
󵄩󵄩
󵄩
󵄩
+ 2𝜆 𝑛 𝛼𝑛 󵄩󵄩󵄩𝑝󵄩󵄩󵄩 󵄩󵄩󵄩𝑢̃𝑛 − 𝑝󵄩󵄩󵄩 + 2 󵄩󵄩󵄩𝑢𝑛 − 𝑢̃𝑛 󵄩󵄩󵄩 󵄩󵄩󵄩𝑢𝑛 − 𝑝󵄩󵄩󵄩
󵄩
󵄩2 󵄩
󵄩2
≤ 󵄩󵄩󵄩𝑥𝑛 − 𝑝󵄩󵄩󵄩 − 󵄩󵄩󵄩𝑥𝑛 − 𝑥̃𝑛 − (𝑝 − 𝑞)󵄩󵄩󵄩
󵄩󵄩
󵄩
󵄩
+ 2𝜇2 󵄩󵄩󵄩𝑥𝑛 − 𝑥̃𝑛 − (𝑝 − 𝑞)󵄩󵄩󵄩 󵄩󵄩󵄩𝐵2 𝑥𝑛 − 𝐵2 𝑝󵄩󵄩󵄩
󵄩
󵄩2
− 󵄩󵄩󵄩𝑥̃𝑛 − 𝑢𝑛 + (𝑝 − 𝑞)󵄩󵄩󵄩
󵄩
󵄩󵄩
󵄩
+ 2𝜇1 󵄩󵄩󵄩𝑥̃𝑛 − 𝑢𝑛 + (𝑝 − 𝑞)󵄩󵄩󵄩 󵄩󵄩󵄩𝐵1 𝑥̃𝑛 − 𝐵1 𝑞󵄩󵄩󵄩
󵄩
󵄩 󵄩󵄩
󵄩󵄩
󵄩
󵄩
+ 2𝜆 𝑛 𝛼𝑛 󵄩󵄩󵄩𝑝󵄩󵄩󵄩 󵄩󵄩󵄩𝑢̃𝑛 − 𝑝󵄩󵄩󵄩 + 2 󵄩󵄩󵄩𝑢𝑛 − 𝑢̃𝑛 󵄩󵄩󵄩 󵄩󵄩󵄩𝑢𝑛 − 𝑝󵄩󵄩󵄩 .
16
Abstract and Applied Analysis
Thus, utilizing Lemma 14, from (17) and (102), it follows that
󵄩󵄩
󵄩2
󵄩󵄩𝑥𝑛+1 − 𝑝󵄩󵄩󵄩
󵄩
󵄩2
󵄩
󵄩2
≤ 𝛽𝑛 󵄩󵄩󵄩𝑥𝑛 − 𝑝󵄩󵄩󵄩 + (1 − 𝛽𝑛 ) 󵄩󵄩󵄩𝑦𝑛 − 𝑝󵄩󵄩󵄩 − 𝛽𝑛 (1 − 𝛽𝑛 )
󵄩󵄩2
󵄩󵄩󵄩 1
󵄩
[𝛾𝑛 (𝑥𝑛 − 𝑦𝑛 ) + 𝛿𝑛 (𝑥𝑛 − 𝑆𝑦𝑛 )]󵄩󵄩󵄩
× 󵄩󵄩󵄩
󵄩󵄩
󵄩󵄩 1 − 𝛽𝑛
󵄩
󵄩2
≤ 𝛽𝑛 󵄩󵄩󵄩𝑥𝑛 − 𝑝󵄩󵄩󵄩 + (1 − 𝛽𝑛 )
󵄩
󵄩2
󵄩
󵄩2
× [𝜎𝑛 󵄩󵄩󵄩𝑥𝑛 − 𝑝󵄩󵄩󵄩 + (1 − 𝜎𝑛 ) 󵄩󵄩󵄩𝑢𝑛 − 𝑝󵄩󵄩󵄩 ]
󵄩󵄩 1
󵄩󵄩󵄩2
󵄩
− 𝛽𝑛 (1 − 𝛽𝑛 ) 󵄩󵄩󵄩
[𝛾𝑛 (𝑥𝑛 − 𝑦𝑛 ) + 𝛿𝑛 (𝑥𝑛 − 𝑆𝑦𝑛 )]󵄩󵄩󵄩
󵄩󵄩 1 − 𝛽𝑛
󵄩󵄩
󵄩
󵄩2
≤ 𝛽𝑛 󵄩󵄩󵄩𝑥𝑛 − 𝑝󵄩󵄩󵄩 + (1 − 𝛽𝑛 )
󵄩
󵄩2
× {𝜎𝑛 󵄩󵄩󵄩𝑥𝑛 − 𝑝󵄩󵄩󵄩
󵄩
󵄩2 󵄩
󵄩2
+ (1 − 𝜎𝑛 ) [󵄩󵄩󵄩𝑥𝑛 − 𝑝󵄩󵄩󵄩 − 󵄩󵄩󵄩𝑥𝑛 − 𝑥̃𝑛 − (𝑝 − 𝑞)󵄩󵄩󵄩
󵄩
󵄩󵄩
󵄩
+ 2𝜇2 󵄩󵄩󵄩𝑥𝑛 − 𝑥̃𝑛 − (𝑝−𝑞)󵄩󵄩󵄩 󵄩󵄩󵄩𝐵2 𝑥𝑛 −𝐵2 𝑝󵄩󵄩󵄩
󵄩
󵄩2
− 󵄩󵄩󵄩𝑥̃𝑛 − 𝑢𝑛 + (𝑝 − 𝑞)󵄩󵄩󵄩
󵄩
󵄩
+ 2𝜇1 󵄩󵄩󵄩𝑥̃𝑛 − 𝑢𝑛 + (𝑝 − 𝑞)󵄩󵄩󵄩
󵄩
󵄩 󵄩󵄩
󵄩
󵄩
× 󵄩󵄩󵄩𝐵1 𝑥̃𝑛 − 𝐵1 𝑞󵄩󵄩󵄩 + 2𝜆 𝑛 𝛼𝑛 󵄩󵄩󵄩𝑝󵄩󵄩󵄩 󵄩󵄩󵄩𝑢̃𝑛 − 𝑝󵄩󵄩󵄩
󵄩
󵄩 󵄩
󵄩
+ 2 󵄩󵄩󵄩𝑢𝑛 − 𝑢̃𝑛 󵄩󵄩󵄩 󵄩󵄩󵄩𝑢𝑛 − 𝑝󵄩󵄩󵄩 ] }
− 𝛽𝑛 (1 − 𝛽𝑛 )
󵄩󵄩 1
󵄩󵄩2
󵄩
󵄩
× 󵄩󵄩󵄩
[𝛾𝑛 (𝑥𝑛 − 𝑦𝑛 ) + 𝛿𝑛 (𝑥𝑛 − 𝑆𝑦𝑛 )]󵄩󵄩󵄩
󵄩󵄩 1 − 𝛽𝑛
󵄩󵄩
󵄩
󵄩2
≤ 󵄩󵄩󵄩𝑥𝑛 − 𝑝󵄩󵄩󵄩 − (1 − 𝛽𝑛 ) (1 − 𝜎𝑛 )
󵄩2 󵄩
󵄩2
󵄩
× (󵄩󵄩󵄩𝑥𝑛 − 𝑥̃𝑛 − (𝑝 − 𝑞)󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑥̃𝑛 − 𝑢𝑛 + (𝑝 − 𝑞)󵄩󵄩󵄩 )
󵄩
󵄩󵄩
󵄩
+ 2𝜇2 󵄩󵄩󵄩𝑥𝑛 − 𝑥̃𝑛 − (𝑝 − 𝑞)󵄩󵄩󵄩 󵄩󵄩󵄩𝐵2 𝑥𝑛 − 𝐵2 𝑝󵄩󵄩󵄩
󵄩
󵄩󵄩
󵄩
+ 2𝜇1 󵄩󵄩󵄩𝑥̃𝑛 − 𝑢𝑛 + (𝑝 − 𝑞)󵄩󵄩󵄩 󵄩󵄩󵄩𝐵1 𝑥̃𝑛 − 𝐵1 𝑞󵄩󵄩󵄩
󵄩
󵄩 󵄩󵄩
󵄩󵄩
󵄩
󵄩
+ 2𝜆 𝑛 𝛼𝑛 󵄩󵄩󵄩𝑝󵄩󵄩󵄩 󵄩󵄩󵄩𝑢̃𝑛 − 𝑝󵄩󵄩󵄩 + 2 󵄩󵄩󵄩𝑢𝑛 − 𝑢̃𝑛 󵄩󵄩󵄩 󵄩󵄩󵄩𝑢𝑛 − 𝑝󵄩󵄩󵄩
󵄩󵄩 1
󵄩󵄩2
󵄩
󵄩
− 𝛽𝑛 (1 − 𝛽𝑛 ) 󵄩󵄩󵄩
[𝛾𝑛 (𝑥𝑛 − 𝑦𝑛 ) + 𝛿𝑛 (𝑥𝑛 − 𝑆𝑦𝑛 )]󵄩󵄩󵄩 ,
󵄩󵄩
󵄩󵄩 1 − 𝛽𝑛
(103)
which hence implies that
(1 − 𝛽𝑛 ) (1 − 𝜎𝑛 )
󵄩
󵄩2 󵄩
󵄩2
× (󵄩󵄩󵄩𝑥𝑛 − 𝑥̃𝑛 − (𝑝 − 𝑞)󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑥̃𝑛 − 𝑢𝑛 + (𝑝 − 𝑞)󵄩󵄩󵄩 )
󵄩󵄩 1
󵄩󵄩󵄩2
󵄩
+ 𝛽𝑛 (1 − 𝛽𝑛 ) 󵄩󵄩󵄩
[𝛾𝑛 (𝑥𝑛 − 𝑦𝑛 ) + 𝛿𝑛 (𝑥𝑛 − 𝑆𝑦𝑛 )]󵄩󵄩󵄩
󵄩󵄩 1 − 𝛽𝑛
󵄩󵄩
󵄩 󵄩󵄩
󵄩
󵄩2 󵄩
󵄩2
󵄩
≤ 󵄩󵄩󵄩𝑥𝑛 − 𝑝󵄩󵄩󵄩 − 󵄩󵄩󵄩𝑥𝑛+1 − 𝑝󵄩󵄩󵄩 + 2𝜆 𝑛 𝛼𝑛 󵄩󵄩󵄩𝑝󵄩󵄩󵄩 󵄩󵄩󵄩𝑢̃𝑛 − 𝑝󵄩󵄩󵄩
󵄩
󵄩󵄩
󵄩
+ 2𝜇2 󵄩󵄩󵄩𝑥𝑛 − 𝑥̃𝑛 − (𝑝 − 𝑞)󵄩󵄩󵄩 󵄩󵄩󵄩𝐵2 𝑥𝑛 − 𝐵2 𝑝󵄩󵄩󵄩
󵄩
󵄩󵄩
󵄩
+ 2𝜇1 󵄩󵄩󵄩𝑥̃𝑛 − 𝑢𝑛 + (𝑝 − 𝑞)󵄩󵄩󵄩 󵄩󵄩󵄩𝐵1 𝑥̃𝑛 − 𝐵1 𝑞󵄩󵄩󵄩
󵄩
󵄩󵄩
󵄩
+ 2 󵄩󵄩󵄩𝑢𝑛 − 𝑢̃𝑛 󵄩󵄩󵄩 󵄩󵄩󵄩𝑢𝑛 − 𝑝󵄩󵄩󵄩 .
(104)
Since 0 < lim inf 𝑛 → ∞ 𝛽𝑛 ≤ lim sup𝑛 → ∞ 𝛽𝑛 < 1,
lim sup𝑛 → ∞ 𝜎𝑛 < 1, {𝜆 𝑛 } ⊂ [𝑎, 𝑏], 𝛼𝑛 → 0, ‖𝐵2 𝑥𝑛 − 𝐵2 𝑝‖ →
0, ‖𝐵1 𝑥̃𝑛 − 𝐵1 𝑞‖ → 0, ‖𝑢𝑛 − 𝑢̃𝑛 ‖ → 0 and lim𝑛 → ∞ ‖𝑥𝑛 − 𝑝‖
𝑢𝑛 }
exists, it follows from the boundedness of {𝑥𝑛 }, {𝑥̃𝑛 }, {𝑢𝑛 }, {̃
and {𝑢𝑛 } that lim𝑛 → ∞ ‖𝑥𝑛 − 𝑥̃𝑛 − (𝑝 − 𝑞)‖ = 0:
󵄩
󵄩
lim 󵄩󵄩𝑥̃ − 𝑢𝑛 + (𝑝 − 𝑞)󵄩󵄩󵄩 = 0,
𝑛→∞ 󵄩 𝑛
(105)
󵄩
󵄩
lim 󵄩󵄩󵄩𝛾𝑛 (𝑥𝑛 − 𝑦𝑛 ) + 𝛿𝑛 (𝑥𝑛 − 𝑆𝑦𝑛 )󵄩󵄩󵄩 = 0.
𝑛→∞
Consequently, it immediately follows that
󵄩
󵄩
󵄩
󵄩
lim 󵄩󵄩𝑥 − 𝑢𝑛 󵄩󵄩󵄩 = 0.
lim 󵄩󵄩𝑥 − 𝑢𝑛 󵄩󵄩󵄩 = 0,
𝑛→∞ 󵄩 𝑛
𝑛→∞ 󵄩 𝑛
(106)
This together with ‖𝑦𝑛 − 𝑢𝑛 ‖ ≤ 𝜎𝑛 ‖𝑥𝑛 − 𝑢𝑛 ‖ → 0 implies that
󵄩
󵄩
lim 󵄩󵄩𝑥 − 𝑦𝑛 󵄩󵄩󵄩 = 0.
(107)
𝑛→∞ 󵄩 𝑛
Since
󵄩󵄩
󵄩󵄩
󵄩󵄩𝛿𝑛 (𝑆𝑦𝑛 − 𝑥𝑛 )󵄩󵄩
󵄩
󵄩
= 󵄩󵄩󵄩𝛾𝑛 (𝑥𝑛 − 𝑦𝑛 ) + 𝛿𝑛 (𝑥𝑛 − 𝑆𝑦𝑛 ) + 𝛾𝑛 (𝑦𝑛 − 𝑥𝑛 )󵄩󵄩󵄩 (108)
󵄩
󵄩
󵄩
󵄩
≤ 󵄩󵄩󵄩𝛾𝑛 (𝑥𝑛 − 𝑦𝑛 ) + 𝛿𝑛 (𝑥𝑛 − 𝑆𝑦𝑛 )󵄩󵄩󵄩 + 𝛾𝑛 󵄩󵄩󵄩𝑥𝑛 − 𝑦𝑛 󵄩󵄩󵄩 ,
we have
󵄩
󵄩
lim 󵄩󵄩𝑆𝑦𝑛 − 𝑥𝑛 󵄩󵄩󵄩 = 0,
𝑛→∞ 󵄩
󵄩
󵄩
lim 󵄩󵄩𝑆𝑦𝑛 − 𝑦𝑛 󵄩󵄩󵄩 = 0.
𝑛→∞ 󵄩
(109)
Step 4. {𝑥𝑛 }, {𝑢𝑛 } and {̃
𝑢𝑛 } converge weakly to an element 𝑥 ∈
Fix(𝑆) ∩ Ξ ∩ Γ.
Indeed, since {𝑥𝑛 } is bounded, there exists a subsequence
{𝑥𝑛𝑖 } of {𝑥𝑛 } that converges weakly to some 𝑥 ∈ 𝐶. We obtain
that 𝑥 ∈ Fix(𝑆)∩Ξ∩Γ. Taking into account that ‖𝑥𝑛 −𝑢𝑛 ‖ → 0
and ‖̃
𝑢𝑛 − 𝑢𝑛 ‖ → 0 and ‖𝑥𝑛 − 𝑦𝑛 ‖ → 0, we deduce that
𝑦𝑛𝑖 → 𝑥 weakly and 𝑢̃𝑛𝑖 → 𝑥 weakly.
First, it is clear from Lemma 15 and ‖𝑆𝑦𝑛 − 𝑦𝑛 ‖ → 0 that
𝑥 ∈ Fix(𝑆). Now let us show that 𝑥 ∈ Ξ. Note that
󵄩󵄩󵄩𝑥𝑛 − 𝐺 (𝑥𝑛 )󵄩󵄩󵄩
󵄩
󵄩
󵄩󵄩
= 󵄩󵄩𝑥𝑛 − 𝑃𝐶 [𝑃𝐶 (𝑥𝑛 − 𝜇2 𝐵2 𝑥𝑛 )
(110)
󵄩
− 𝜇1 𝐵1 𝑃𝐶 (𝑥𝑛 − 𝜇2 𝐵2 𝑥𝑛 )]󵄩󵄩󵄩
󵄩
󵄩
= 󵄩󵄩󵄩𝑥𝑛 − 𝑢𝑛 󵄩󵄩󵄩 󳨀→ 0,
as 𝑛 → ∞, where 𝐺 : 𝐶 → 𝐶 is defined as that in Lemma 1.
According to Lemma 15, we get 𝑥 ∈ Ξ. Further, let us show
that 𝑥 ∈ Γ. As a matter of fact, define
∇𝑓 (𝑣) + 𝑁𝐶𝑣, if 𝑣 ∈ 𝐶,
𝑇𝑣 = {
0,
if 𝑣 ∉ 𝐶,
(111)
Abstract and Applied Analysis
17
where 𝑁𝐶𝑣 = {𝑤 ∈ H1 : ⟨𝑣−𝑢, 𝑤⟩ ≥ 0, ∀𝑢 ∈ 𝐶}. Utilizing the
argument similar to that of Step 4 in the proof of Theorem 20,
from the relation
𝑢̃𝑛 = 𝑃𝐶 (𝑢𝑛 − 𝜆 𝑛 ∇𝑓𝛼𝑛 (𝑢𝑛 )) ,
𝑣 ∈ 𝐶,
(iii) 0 < lim inf 𝑛 → ∞ 𝜆 𝑛 ≤ lim sup𝑛 → ∞ 𝜆 𝑛 < 1/‖𝐴‖2 .
𝑢𝑛 } converge weakly to an
Then, both the sequences {𝑥𝑛 } and {̃
element 𝑥 ∈ Fix(𝑆) ∩ Γ.
(113)
Proof. In Corollary 24, put 𝐵1 = 𝐵2 = 0 and 𝛾𝑛 = 0. Then,
Ξ = 𝐶, 𝛽𝑛 + 𝛿𝑛 = 1 for all 𝑛 ≥ 0, and the iterative scheme (114)
is equivalent to
It is easy to see that 𝑥 ∈ Γ. Therefore, 𝑥 ∈ Fix(𝑆)∩Ξ∩Γ. Finally,
utilizing the Opial condition [41], we infer that {𝑥𝑛 } converges
weakly to 𝑥 ∈ Fix(𝑆)∩Ξ∩Γ. Further, from ‖𝑥𝑛 −𝑢𝑛 ‖ → 0 and
𝑢𝑛 } converge
‖𝑥𝑛 − 𝑢̃𝑛 ‖ → 0, it follows that both {𝑢𝑛 } and {̃
weakly to 𝑥. This completes the proof.
Corollary 24. Let 𝐶 be a nonempty closed convex subset of a
real Hilbert space H1 . Let 𝐴 ∈ 𝐵(H1 , H2 ) and 𝐵𝑖 : 𝐶 → H1
be 𝛽𝑖 -inverse strongly monotone for 𝑖 = 1, 2. Let 𝑆 : 𝐶 → 𝐶 be
a 𝑘-strictly pseudocontractive mapping such that Fix(𝑆) ∩ Ξ ∩
Γ ≠ 0. For given 𝑥0 ∈ 𝐶 arbitrarily, let the sequences {𝑥𝑛 }, {𝑢𝑛 },
{̃
𝑢𝑛 } be generated iteratively by
𝑢𝑛 = 𝑃𝐶 [𝑃𝐶 (𝑥𝑛 − 𝜇2 𝐵2 𝑥𝑛 ) − 𝜇1 𝐵1 𝑃𝐶 (𝑥𝑛 − 𝜇2 𝐵2 𝑥𝑛 )] ,
𝑢̃𝑛 = 𝑃𝐶 (𝑢𝑛 − 𝜆 𝑛 ∇𝑓𝛼𝑛 (𝑢𝑛 )) ,
(ii) 0 < lim inf 𝑛 → ∞ 𝛽𝑛 ≤ lim sup𝑛 → ∞ 𝛽𝑛 < 1;
(112)
we can easily conclude that
⟨𝑣 − 𝑥, 𝑤⟩ ≥ 0.
(i) ∑∞
𝑛=0 𝛼𝑛 < ∞;
𝑢𝑛 = 𝑥𝑛 ,
𝑢̃𝑛 = 𝑃𝐶 (𝑢𝑛 − 𝜆 𝑛 ∇𝑓𝛼𝑛 (𝑢𝑛 )) ,
𝑢𝑛 )) ,
𝑦𝑛 = 𝑃𝐶 (𝑢𝑛 − 𝜆 𝑛 ∇𝑓𝛼𝑛 (̃
𝑥𝑛+1 = 𝛽𝑛 𝑥𝑛 + 𝛿𝑛 𝑆𝑦𝑛 ,
(116)
∀𝑛 ≥ 0.
This is equivalent to (115). Since 𝑆 is a nonexpansive mapping,
𝑆 must be a 𝑘-strictly pseudocontractive mapping with 𝑘 = 0.
In this case, it is easy to see that all the conditions (i)–(iv) in
Corollary 24 are satisfied. Therefore, in terms of Corollary 24,
we obtain the desired result.
(114)
Remark 26. Compared with the Ceng and Yao [31, Theorem
3.1], our Corollary 25 coincides essentially with [31, Theorem
3.1]. This shows that our Theorem 23 includes [31, Theorem
3.1] as a special case.
where 𝜇𝑖 ∈ (0, 2𝛽𝑖 ) for 𝑖 = 1, 2, {𝛼𝑛 } ⊂ (0, ∞), {𝜆 𝑛 } ⊂
(0, 1/‖𝐴‖2 ) and {𝛽𝑛 }, {𝛾𝑛 }, {𝛿𝑛 } ⊂ [0, 1] such that
Remark 27. Our Theorems 20 and 23 improve, extend, and
develop [20, Theorem 5.7], [31, Theorem 3.1], [7, Theorem
3.2], and [14, Theorem 3.1] in the following aspects.
𝑢𝑛 )) ,
𝑦𝑛 = 𝑃𝐶 (𝑢𝑛 − 𝜆 𝑛 ∇𝑓𝛼𝑛 (̃
𝑥𝑛+1 = 𝛽𝑛 𝑥𝑛 + 𝛾𝑛 𝑦𝑛 + 𝛿𝑛 𝑆𝑦𝑛 ,
∀𝑛 ≥ 0,
(i) ∑∞
𝑛=0 𝛼𝑛 < ∞;
(ii) 𝛽𝑛 + 𝛾𝑛 + 𝛿𝑛 = 1 and (𝛾𝑛 + 𝛿𝑛 )𝑘 ≤ 𝛾𝑛 for all 𝑛 ≥ 0;
(iii) 0 < lim inf 𝑛 → ∞ 𝛽𝑛 ≤ lim sup𝑛 → ∞ 𝛽𝑛 < 1 and
lim inf 𝑛 → ∞ 𝛿𝑛 > 0;
(iv) 0 < lim inf 𝑛 → ∞ 𝜆 𝑛 ≤ lim sup𝑛 → ∞ 𝜆 𝑛 < 1/‖𝐴‖2 .
Then the sequences {𝑥𝑛 }, {𝑢𝑛 }, {̃
𝑢𝑛 } converge weakly to an
element 𝑥 ∈ Fix(𝑆) ∩ Ξ ∩ Γ. Furthermore, (𝑥, 𝑦) is a solution of
GSVI (3), where 𝑦 = 𝑃𝐶(𝑥 − 𝜇2 𝐵2 𝑥).
Next, utilizing Corollary 24, we derive the following
result.
Corollary 25. Let 𝐶 be a nonempty closed convex subset of a
real Hilbert space H1 . Let 𝐴 ∈ 𝐵(H1 , H2 ) and 𝑆 : 𝐶 → 𝐶
be a nonexpansive mapping such that Fix(𝑆) ∩ Γ ≠ 0. For given
𝑢𝑛 } be generated
𝑥0 ∈ 𝐶 arbitrarily, let the sequences {𝑥𝑛 }, {̃
iteratively by
𝑢̃𝑛 = 𝑃𝐶 (𝑥𝑛 − 𝜆 𝑛 ∇𝑓𝛼𝑛 (𝑥𝑛 )) ,
𝑢𝑛 )) ,
𝑥𝑛+1 = 𝛽𝑛 𝑥𝑛 + (1 − 𝛽𝑛 ) 𝑆𝑃𝐶 (𝑥𝑛 − 𝜆 𝑛 ∇𝑓𝛼𝑛 (̃
(115)
∀𝑛 ≥ 0,
where {𝛼𝑛 } ⊂ (0, ∞), {𝜆 𝑛 } ⊂ (0, 1/‖𝐴‖2 ) and {𝛽𝑛 } ⊂ [0, 1] such
that
(i) Compared with the relaxed extragradient iterative
algorithm in [7, Theorem 3.2], our Mann-type
extragradient iterative algorithms with regularization
remove the requirement of boundedness for the
domain 𝐶 in which various mappings are defined.
(ii) Because [31, Theorem 3.1] is the supplementation,
improvement, and extension of [20, Theorem 5.7] and
our Theorem 23 includes [31, Theorem 3.1] as a special
case, beyond question our results are very interesting
and quite valuable.
(iii) The problem of finding an element of Fix(𝑆) ∩ Ξ ∩ Γ
in our Theorems 20 and 23 is more general than the
corresponding problems in [20, Theorem 5.7] and [31,
Theorem 3.1], respectively.
(iv) The hybrid extragradient method for finding an
element of Fix(𝑆) ∩ Ξ ∩ VI(𝐶, 𝐴) in [14, Theorem 3.1]
is extended to develop our Mann-type extragradient
iterative algorithms (16) and (17) with regularization
for finding an element of Fix(𝑆) ∩ Ξ ∩ Γ.
(v) The proof of our results are very different from that
of [14, Theorem 3.1] because our argument technique
depends on the Opial condition, the restriction on
the regularization parameter sequence {𝛼𝑛 }, and the
properties of the averaged mappings 𝑃𝐶(𝐼 − 𝜆 𝑛 ∇𝑓𝛼𝑛 )
to a great extent.
18
Abstract and Applied Analysis
(vi) Because our iterative algorithms (16) and (17) involve
two inverse strongly monotone mappings 𝐵1 and 𝐵2 ,
a 𝑘-strictly pseudocontractive self-mapping 𝑆 and
several parameter sequences, they are more flexible
and more subtle than the corresponding ones in [20,
Theorem 5.7] and [31, Theorem 3.1], respectively.
[13]
[14]
Acknowledgments
In this research, the first author was partially supported by
the National Science Foundation of China (11071169) and
Ph.D. Program Foundation of Ministry of Education of China
(20123127110002). The third author was partially supported by
Grant NSC 101-2115-M-037-001.
[16]
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