Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2013, Article ID 230392, 17 pages
http://dx.doi.org/10.1155/2013/230392
Research Article
A New Computational Technique for Common Solutions
between Systems of Generalized Mixed Equilibrium and Fixed
Point Problems
Pongsakorn Sunthrayuth and Poom Kumam
Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT),
Bang Mod, Bangkok 10140, Thailand
Correspondence should be addressed to Poom Kumam; poom.kum@kmutt.ac.th
Received 18 February 2013; Accepted 1 April 2013
Academic Editor: Wei-Shih Du
Copyright © 2013 P. Sunthrayuth and P. Kumam. 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 introduce a new iterative algorithm for finding a common element of a fixed point problem of amenable semigroups of
nonexpansive mappings, the set solutions of a system of a general system of generalized equilibria in a real Hilbert space. Then, we
prove the strong convergence of the proposed iterative algorithm to a common element of the above three sets under some suitable
conditions. As applications, at the end of the paper, we apply our results to find the minimum-norm solutions which solve some
quadratic minimization problems. The results obtained in this paper extend and improve many recent ones announced by many
others.
1. Introduction
Throughout this paper, we denoted by R the set of all real
numbers. We always assume that 𝐻 is a real Hilbert space
with inner product ⟨⋅, ⋅⟩ and induced norm ‖ ⋅ ‖ and 𝐶 is a
nonempty, closed, and convex subset of 𝐻. 𝑃𝐶 denotes the
metric projection of 𝐻 onto 𝐶. A mapping 𝑇 : 𝐶 → 𝐶 is
said to be 𝐿-Lipschitzian if there exists a constant 𝐿 > 0 such
that
󵄩󵄩󵄩𝑇𝑥 − 𝑇𝑦󵄩󵄩󵄩 ≤ 𝐿 󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩 , ∀𝑥, 𝑦 ∈ 𝐶.
(1)
󵄩
󵄩
󵄩
󵄩
If 0 < 𝐿 < 1, then 𝑇 is a contraction, and if 𝐿 = 1, then 𝑇 is
a nonexpansive mapping. We denote by Fix(𝑇) the set of all
fixed points set of the mapping 𝑇; that is, Fix(𝑇) = {𝑥 ∈ 𝐶 :
𝑇𝑥 = 𝑥}.
A mapping 𝐹 : 𝐶 → 𝐻 is said to be monotone if
Let 𝜑 : 𝐶 → R be a real-valued function, Θ : 𝐶 × 𝐶 →
R an equilibrium bifunction, and Ψ : 𝐶 → 𝐻 a nonlinear
mapping. The generalized mixed equilibrium problem is to find
𝑥∗ ∈ 𝐶 such that
Θ (𝑥∗ , 𝑦) + 𝜑 (𝑦) − 𝜑 (𝑥∗ ) + ⟨Ψ𝑥∗ , 𝑦 − 𝑥∗ ⟩ ≥ 0,
∀𝑦 ∈ 𝐶,
(4)
which was introduced and studied by Peng and Yao [1]. The
set of solutions of problem (4) is denoted by GMEP(Θ, 𝜑, Ψ).
As special cases of problem (4), we have the following results.
(1) If Ψ = 0, then problem (4) reduces to mixed equilibrium problem. Find 𝑥∗ ∈ 𝐶 such that
Θ (𝑥∗ , 𝑦) + 𝜑 (𝑦) − 𝜑 (𝑥∗ ) ≥ 0,
∀𝑦 ∈ 𝐶,
(5)
(2)
which was considered by Ceng and Yao [2]. The set of
solutions of problem (5) is denoted by MEP(Θ).
A mapping 𝐹 : 𝐶 → 𝐻 is said to be strongly monotone if
there exists 𝜂 > 0 such that
(2) If 𝜑 = 0, then problem (4) reduces to generalized
equilibrium problem. Find 𝑥∗ ∈ 𝐶 such that
⟨𝐹𝑥 − 𝐹𝑦, 𝑥 − 𝑦⟩ ≥ 0,
∀𝑥, 𝑦 ∈ 𝐶.
󵄩2
󵄩
⟨𝐹𝑥 − 𝐹𝑦, 𝑥 − 𝑦⟩ ≥ 𝜂󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩 ,
∀𝑥, 𝑦 ∈ 𝐶.
(3)
Θ (𝑥∗ , 𝑦) + ⟨Ψ𝑥∗ , 𝑦 − 𝑥∗ ⟩ ≥ 0,
∀𝑦 ∈ 𝐶,
(6)
2
Journal of Applied Mathematics
which was considered by S. Takahashi and W. Takahashi [3]. The set of solutions of problem (6) is
denoted by GEP(Θ, Ψ).
(3) If Ψ = 𝜑 = 0, then problem (4) reduces to equilibrium
problem. Find 𝑥∗ ∈ 𝐶 such that
Θ (𝑥∗ , 𝑦) ≥ 0,
∀𝑦 ∈ 𝐶.
(7)
The set of solutions of problem (7) is denoted by
EP(Θ).
(4) If Θ = 𝜑 = 0, then problem (4) reduces to classical
variational inequality problem. Find 𝑥∗ ∈ 𝐶 such that
⟨Ψ𝑥∗ , 𝑦 − 𝑥∗ ⟩ ≥ 0,
∀𝑦 ∈ 𝐶.
(8)
The set of solutions of problem (8) is denoted by
VI(𝐶, Ψ). It is known that 𝑥∗ ∈ 𝐶 is a solution of the
problem (8) if and only if 𝑥∗ is a fixed point of the
mapping 𝑃𝐶(𝐼 − 𝜆Ψ), where 𝜆 > 0 is a constant and 𝐼
is the identity mapping.
The problem (4) is very general in the sense that it includes
several problems, namely, fixed point problems, optimization
problems, saddle point problems, complementarity problems, variational inequality problems, minimax problems,
Nash equilibrium problems in noncooperative games, and
others as special cases. Numerous problems in physics, optimization, and economics reduce to find a solution of problem
(4) (see, e.g., [4–9]). Several iterative methods to solve the
fixed point problems, variational inequality problems, and
equilibrium problems are proposed in the literature (see, e.g.,
[1–3, 10–18]) and the references therein.
Let 𝐴 1 , 𝐴 2 : 𝐶 → 𝐻 be two mappings. Ceng and Yao [12]
considered the following problem of finding (𝑥∗ , 𝑦∗ ) ∈ 𝐶 × 𝐶
such that
𝐺2 (𝑥∗ , 𝑥) + ⟨𝐴 2 𝑦∗ , 𝑥 − 𝑥∗ ⟩ +
𝐺1 (𝑦∗ , 𝑦) + ⟨𝐴 1 𝑥∗ , 𝑦 − 𝑦∗ ⟩ +
1
⟨𝑥∗ − 𝑦∗ , 𝑥 − 𝑥∗ ⟩ ≥ 0,
𝜆2
∀𝑥 ∈ 𝐶,
⟨𝜆 1 𝐴 1 𝑥∗ + 𝑦∗ − 𝑥∗ , 𝑦 − 𝑦∗ ⟩ ≥ 0,
∀𝑦 ∈ 𝐶,
(11)
where 𝜆 1 > 0 and 𝜆 2 > 0 are two constants, which is
introduced and considered by Ceng et al. [19].
In 2010, Ceng and Yao [12] proposed the following relaxed
extragradient-like method for finding a common solution
of generalized mixed equilibrium problems, a system of
generalized equilibria (9), and a fixed point problem of a 𝑘strictly pseudocontractive self-mapping 𝑆 on 𝐶 as follows:
(𝑥𝑛 − 𝑟𝑛 Ψ𝑥𝑛 ) ,
𝑧𝑛 = 𝑆𝑟(Θ,𝜑)
𝑛
𝐺
𝐺
𝐺
𝑦𝑛 = 𝑆𝜆 1 [𝑆𝜆 2 (𝑧𝑛 − 𝜆 2 𝐴 2 𝑧𝑛 ) − 𝜆 1 𝐴 1 𝑆𝜆 2 (𝑧𝑛 − 𝜆 2 𝐴 2 𝑧𝑛 )] ,
1
2
2
𝑥𝑛+1 = 𝛼𝑛 𝑢 + 𝛽𝑛 𝑥𝑛 + 𝛾𝑛 𝑦𝑛 + 𝛿𝑛 𝑆𝑦𝑛 ,
∀𝑛 ≥ 0,
(12)
where Ψ, 𝐴 1 , 𝐴 2 : 𝐶 → 𝐻 are 𝛼-inverse strongly monotone, 𝛼̃1 -inverse strongly monotone, and 𝛼̃2 -inverse strongly
monotone, respectively. They proved strong convergence of
the related extragradient-like algorithm (12) under some
appropriate conditions {𝛼𝑛 }, {𝛽𝑛 }, {𝛾𝑛 }, and {𝛿𝑛 } ⊂ [0, 1]
̂ where
satisfying 𝛼𝑛 +𝛽𝑛 +𝛾𝑛 +𝛿𝑛 = 1, for all 𝑛 ≥ 0, to 𝑥̂ = 𝑃Ω 𝑥,
Ω = Fix(𝑆) ∩ GMEP(Θ, 𝜑, Ψ) ∩ Fix(𝐾), with the mapping
𝐾 : 𝐶 → 𝐶 defined by
𝐺
𝐺
𝐺
𝐾𝑥 = 𝑆𝜆 1 [𝑆𝜆 2 (𝑥 − 𝜆 2 𝐴 2 𝑥) − 𝜆 1 𝐴 1 𝑆𝜆 2 (𝑥 − 𝜆 2 𝐴 2 𝑥)] ,
1
⟨𝑦∗ − 𝑥∗ , 𝑦 − 𝑦∗ ⟩ ≥ 0,
𝜆1
∀𝑥 ∈ 𝐶.
which is called a general system of generalized equilibria,
where 𝜆 1 > 0 and 𝜆 2 > 0 are two constants. In particular,
if 𝐺1 = 𝐺2 = 𝐺 and 𝐴 1 = 𝐴 2 = 𝐴, then problem (9) reduces
to the following problem of finding (𝑥∗ , 𝑦∗ ) ∈ 𝐶×𝐶 such that
1
⟨𝑥∗ − 𝑦∗ , 𝑥 − 𝑥∗ ⟩ ≥ 0,
𝐺 (𝑥 , 𝑥) + ⟨𝐴𝑦 , 𝑥 − 𝑥 ⟩ +
𝜆2
∗
⟨𝜆 2 𝐴 2 𝑦∗ + 𝑥∗ − 𝑦∗ , 𝑥 − 𝑥∗ ⟩ ≥ 0,
∀𝑥 ∈ 𝐶,
∀𝑦 ∈ 𝐶,
(9)
∗
which is called a new system of generalized equilibria, where
𝜆 1 > 0 and 𝜆 2 > 0 are two constants.
If 𝐺1 = 𝐺2 = Θ, 𝐴 1 = 𝐴 2 = 𝐴, and 𝑥∗ = 𝑦∗ , then
problem (9) reduces to problem (7).
If 𝐺1 = 𝐺2 = 0, then problem (9) reduces to a general
system of variational inequalities. Find (𝑥∗ , 𝑦∗ ) ∈ 𝐶 × 𝐶 such
that
1
2
2
(13)
Very recently, Ceng et al. [11] introduced an iterative method
for finding fixed points of a nonexpansive mapping 𝑇 on a
nonempty, closed, and convex subset 𝐶 in a real Hilbert space
𝐻 as follows:
𝑥𝑛+1 = 𝑃𝐶 [𝛼𝑛 𝛾𝑉𝑥𝑛 + (𝐼 − 𝛼𝑛 𝜇𝐹) 𝑇𝑥𝑛 ] ,
∀𝑛 ≥ 0,
(14)
∗
∀𝑥 ∈ 𝐶,
𝐺 (𝑦∗ , 𝑦) + ⟨𝐴𝑥∗ , 𝑦 − 𝑦∗ ⟩ +
1
⟨𝑦∗ − 𝑥∗ , 𝑦 − 𝑦∗ ⟩ ≥ 0,
𝜆1
∀𝑦 ∈ 𝐶,
(10)
where 𝑃𝐶 is a metric projection from 𝐻 onto 𝐶, 𝑉 is an
𝐿-Lipschitzian mapping with a constant 𝐿 ≥ 0, and 𝐹
is a 𝜅-Lipschitzian and 𝜂-strongly monotone operator with
constants 𝜅, 𝜂 > 0 and 0 < 𝜇 < 2𝜂/𝜅2 . Then, they proved that
the sequences generated by (14) converge strongly to a unique
solution of variational inequality as follows:
⟨(𝜇𝐹 − 𝛾𝑉) 𝑥∗ , 𝑥∗ − 𝑥⟩ ≥ 0,
∀𝑥 ∈ Fix (𝑇) .
(15)
Journal of Applied Mathematics
3
In this paper, motivated and inspired by the previous
facts, we first introduce the following problem of finding
∗
) ∈ 𝐶 × 𝐶 × ⋅ ⋅ ⋅ × 𝐶 such that
(𝑥1∗ , 𝑥2∗ , . . . , 𝑥𝑀
∗
𝐺𝑀 (𝑥1∗ , 𝑥1 ) + ⟨𝐴 𝑀𝑥𝑀
, 𝑥1 − 𝑥1∗ ⟩
+
1
∗
⟨𝑥∗ − 𝑥𝑀
, 𝑥1 − 𝑥1∗ ⟩ ≥ 0,
𝜆𝑀 1
∀𝑥1 ∈ 𝐶,
∗
∗
∗
𝐺𝑀−1 (𝑥𝑀
, 𝑥𝑀) + ⟨𝐴 𝑀−1 𝑥𝑀−1
, 𝑥𝑀 − 𝑥𝑀
⟩
+
1
∗
∗
⟨𝑥∗ − 𝑥𝑀−1
, 𝑥𝑀 − 𝑥𝑀
⟩ ≥ 0,
𝜆 𝑀−1 𝑀
∀𝑥𝑀 ∈ 𝐶,
..
.
𝐺2 (𝑥3∗ , 𝑥3 ) + ⟨𝐴 2 𝑥2∗ , 𝑥3 − 𝑥3∗ ⟩ +
1
⟨𝑥∗ − 𝑥2∗ , 𝑥3 − 𝑥3∗ ⟩ ≥ 0,
𝜆2 3
∀𝑥3 ∈ 𝐶,
𝐺1 (𝑥2∗ , 𝑥2 ) + ⟨𝐴 1 𝑥1∗ , 𝑥2 − 𝑥2∗ ⟩ +
1
⟨𝑥∗ − 𝑥1∗ , 𝑥2 − 𝑥2∗ ⟩ ≥ 0,
𝜆1 2
∀𝑥2 ∈ 𝐶,
(16)
which is called a more general system of generalized equilibria
in Hilbert spaces, where 𝜆 𝑖 > 0 for all 𝑖 ∈ {1, 2, . . . , 𝑀}. In
particular, if 𝑀 = 2, 𝑥1∗ = 𝑥∗ , 𝑥2∗ = 𝑦∗ , 𝑥1 = 𝑥, and
𝑥2 = 𝑦, then problem (16) reduces to problem (9). Finally,
by combining the relaxed extragradient-like algorithm (12)
with the general iterative algorithm (14), we introduce a new
iterative method for finding a common element of a fixed
point problem of a nonexpansive semigroup, the set solutions
of a general system of generalized equilibria in a real Hilbert
space. We prove the strong convergence of the proposed
iterative algorithm to a common element of the previous
three sets under some suitable conditions. Furthermore, we
apply our results to finding the minimum-norm solutions
which solve some quadratic minimization problem. The main
result extends various results existing in the current literature.
Let 𝑋 be a translation invariant subspace of ℓ∞ (𝑆) (i.e.,
𝑙(𝑠)𝑋 ⊂ 𝑋 and 𝑟(𝑠)𝑋 ⊂ 𝑋 for each 𝑠 ∈ 𝑆) containing 1. Then, a
mean 𝜇 on 𝑋 is said to be left invariant (resp., right invariant)
if 𝜇(𝑙(𝑠)𝑓) = 𝜇(𝑓) (resp., 𝜇(𝑟(𝑠)𝑓) = 𝜇(𝑓)) for each 𝑠 ∈ 𝑆
and 𝑓 ∈ 𝑋. A mean 𝜇 on 𝑋 is said to be invariant if 𝜇 is
both left and right invariant [20–22]. 𝑆 is said to be left (resp.,
right) amenable if 𝑋 has a left (resp., right) invariant mean.
𝑆 is a amenable if 𝑆 is left and right amenable. In this case,
ℓ∞ (𝑆) also has an invariant mean. As is well known, ℓ∞ (𝑆) is
amenable when 𝑆 is commutative semigroup; see [23]. A net
{𝜇𝛼 } of means on 𝑋 is said to be left regular if
󵄩󵄩 ∗
󵄩
lim
󵄩𝑙 𝜇 − 𝜇𝛼 󵄩󵄩󵄩 = 0,
𝛼 󵄩𝑠 𝛼
(19)
for each 𝑠 ∈ 𝑆, where 𝑙𝑠∗ is the adjoint operator of 𝑙𝑠 .
Let 𝐶 be a nonempty, closed, and convex subset of 𝐻. A
family S = {𝑇(𝑠) : 𝑠 ∈ 𝑆} is called a nonexpansive semigroup
on 𝐶 if for each 𝑠 ∈ 𝑆, the mapping 𝑇(𝑠) : 𝐶 → 𝐶 is
nonexpansive and 𝑇(𝑠𝑡) = 𝑇(𝑡𝑠) for each 𝑠, 𝑡 ∈ 𝑆. We denote
by Fix(S) the set of common fixed point of S; that is,
Fix (S) = ⋂ Fix (𝑇 (𝑠)) = ⋂ {𝑥 ∈ 𝐶 : 𝑇 (𝑠) 𝑥 = 𝑥} .
𝑠∈𝑆
𝑠∈𝑆
(20)
Throughout this paper, the open ball of radius 𝑟 centered at 0
is denoted by 𝐵𝑟 , and for a subset 𝐷 of 𝐻 by co 𝐷, we denote
the closed convex hull of 𝐷. For 𝜖 > 0 and a mapping 𝑇 :
𝐷 → 𝐻, the set of 𝜖-approximate fixed point of 𝑇 is denoted
by 𝐹𝜖 (𝑇, 𝐷); that is, 𝐹𝜖 (𝑇, 𝐷) = {𝑥 ∈ 𝐷 : ‖𝑥 − 𝑇𝑥‖ ≤ 𝜖}.
In order to prove our main results, we need the following
lemmas.
Lemma 1 (see [23–25]). Let 𝑓 be a function of a semigroup
𝑆 into a Banach space 𝐸 such that the weak closure of {𝑓(𝑡) :
𝑡 ∈ 𝑆} is weakly compact, and let 𝑋 be a subspace of ℓ∞ (𝑆)
containing all the functions 𝑡 󳨃→ ⟨𝑓(𝑡), 𝑥∗ ⟩ with 𝑥∗ ∈ 𝐸∗ .
Then, for any 𝜇 ∈ 𝑋∗ , there exists a unique element 𝑓𝜇 in 𝐸
such that
⟨𝑓𝜇 , 𝑥∗ ⟩ = 𝜇𝑡 ⟨𝑓 (𝑡) , 𝑥∗ ⟩ ,
(21)
for all 𝑥∗ ∈ 𝐸∗ . Moreover, if 𝜇 is a mean on 𝑋, then
2. Preliminaries
Let 𝑆 be a semigroup. We denote by ℓ∞ the Banach space
of all bounded real-valued functionals on 𝑆 with supremum
norm. For each 𝑠 ∈ 𝑆, we define the left and right translation
operators 𝑙(𝑠) and 𝑟(𝑠) on ℓ∞ (𝑆) by
(𝑙 (𝑠) 𝑓) (𝑡) = 𝑓 (𝑠𝑡) ,
(𝑟 (s) 𝑓) (𝑡) = 𝑓 (𝑡𝑠) ,
(17)
for each 𝑡 ∈ 𝑆 and 𝑓 ∈ ℓ∞ (𝑆), respectively. Let 𝑋 be a subspace
of ℓ∞ (𝑆) containing 1. An element 𝜇 in the dual space 𝑋∗ of
𝑋 is said to be a mean on 𝑋 if ‖𝜇‖ = 𝜇(1) = 1. It is well known
that 𝜇 is a mean on 𝑋 if and only if
inf 𝑓 (𝑠) ≤ 𝜇 (𝑓) ≤ sup𝑓 (𝑠) ,
𝑠∈𝑆
𝑠∈𝑆
(18)
for each 𝑓 ∈ 𝑋. We often write 𝜇𝑡 (𝑓(𝑡)) instead of 𝜇(𝑓) for
𝜇 ∈ 𝑋∗ and 𝑓 ∈ 𝑋.
∫ 𝑓 (𝑡) 𝑑𝜇 (𝑡) ∈ co {𝑓 (𝑡) : 𝑡 ∈ 𝑆} .
(22)
One can write 𝑓𝜇 by ∫ 𝑓(𝑡)𝑑𝜇(𝑡).
Lemma 2 (see [23–25]). Let 𝐶 be a closed and convex subset
of a Hilbert space 𝐻, S = {𝑇(𝑠) : 𝑠 ∈ 𝑆} a nonexpansive
semigroup from 𝐶 into 𝐶 such that Fix(S) ≠ 0, and 𝑋 a
subspace of ℓ∞ containing 1, the mapping 𝑡 󳨃→ ⟨𝑇(𝑡)𝑥, 𝑦⟩ an
element of 𝑋 for each 𝑥 ∈ 𝐶 and 𝑦 ∈ 𝐻, and 𝜇 a mean on 𝑋.
If one writes 𝑇(𝜇)𝑥 instead of ∫ 𝑇𝑡 𝑥𝑑𝜇(𝑡), then the following hold:
(i) 𝑇(𝜇) is nonexpansive mapping from C into C;
(ii) 𝑇(𝜇)𝑥 = 𝑥 for each 𝑥 ∈ Fix(S);
(iii) 𝑇(𝜇)𝑥 ∈ co{𝑇𝑡 𝑥 : 𝑡 ∈ 𝑆} for each 𝑥 ∈ 𝐶;
4
Journal of Applied Mathematics
(iv) if 𝜇 is left invariant, then 𝑇(𝜇) is a nonexpansive retraction from 𝐶 onto Fix(S).
Let 𝐻 be a real Hilbert space with inner product ⟨⋅, ⋅⟩
and norm ‖ ⋅ ‖, and let 𝐶 be a nonempty, closed, and convex
subset of 𝐻. We denote the strong convergence and the weak
convergence of {𝑥𝑛 } to 𝑥 ∈ 𝐻 by 𝑥𝑛 → 𝑥 and 𝑥𝑛 ⇀ 𝑥,
respectively. Also, a mapping 𝐼 : 𝐶 → 𝐶 denotes the identity
mapping. For every point 𝑥 ∈ 𝐻, there exists a unique nearest
point of 𝐶, denoted by 𝑃𝐶𝑥, such that
󵄩󵄩󵄩𝑥 − 𝑃𝐶𝑥󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩 , ∀𝑦 ∈ 𝐶.
(23)
󵄩
󵄩
󵄩 󵄩
Such a projection 𝑃𝐶 is called the metric projection of 𝐻 onto
𝐶. We know that 𝑃𝐶 is a firmly nonexpansive mapping of 𝐻
onto 𝐶; that is,
󵄩
󵄩2
⟨𝑥 − 𝑦, 𝑃𝐶𝑥 − 𝑃𝐶𝑦⟩ ≥ 󵄩󵄩󵄩𝑃𝐶𝑥 − 𝑃𝐶𝑦󵄩󵄩󵄩 ,
∀𝑥, 𝑦 ∈ 𝐻.
(24)
It is known that
𝑧 = 𝑃𝐶𝑥 ⇐⇒ ⟨𝑥 − 𝑧, 𝑦 − 𝑧⟩ ≤ 0,
∀𝑥 ∈ 𝐻, 𝑦 ∈ 𝐶. (25)
(26)
(27)
for all 𝑥, 𝑦 ∈ 𝐻 and 𝜆 ∈ [0, 1].
If 𝐴 : 𝐶 → 𝐻 is 𝛼-inverse strongly monotone, then it
is obvious that 𝐴 is 1/𝛼-Lipschitz continuous. We also have
that, for all 𝑥, 𝑦 ∈ 𝐶 and 𝜆 > 0,
󵄩󵄩
󵄩2
󵄩󵄩(𝐼 − 𝜆𝐴) 𝑥 − (𝐼 − 𝜆) 𝑦󵄩󵄩󵄩
󵄩2
󵄩
= 󵄩󵄩󵄩(𝑥 − 𝑦) − 𝜆(𝐴𝑥 − 𝐴𝑦)󵄩󵄩󵄩
(28)
󵄩2
󵄩2
󵄩
󵄩
= 󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩 − 2𝜆 ⟨𝐴𝑥 − 𝐴𝑦, 𝑥 − 𝑦⟩ + 𝜆2 󵄩󵄩󵄩𝐴𝑥 − 𝐴𝑦󵄩󵄩󵄩
󵄩2
󵄩2
󵄩
󵄩
≤ 󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩 + 𝜆 (𝜆 − 2𝛼) 󵄩󵄩󵄩𝐴𝑥 − 𝐴𝑦󵄩󵄩󵄩 .
In particular, if 𝜆 < 2𝛼, then 𝐼−𝜆𝐴 is a nonexpansive mapping
from 𝐶 to 𝐻.
For solving the equilibrium problem, let us assume that
the bifunction Θ : 𝐶 × 𝐶 → R satisfies the following
conditions:
(A1) Θ(𝑥, 𝑥) = 0 for all 𝑥 ∈ 𝐶;
(A2) Θ is monotone, that is, Θ(𝑥, 𝑦) + Θ(𝑦, 𝑥) ≤ 0 for each
𝑥, 𝑦 ∈ 𝐶;
(A3) Θ is upper semicontinuous, that is, for each 𝑥, 𝑦, 𝑧 ∈
𝐶,
lim sup Θ (𝑡𝑧 + (1 − 𝑡) 𝑥, 𝑦) ≤ Θ (𝑥, 𝑦) ;
𝑡 → 0+
Θ (𝑧, 𝑦𝑥 ) + 𝜑 (𝑦𝑥 ) − 𝜑 (𝑧) +
1
⟨𝑦 − 𝑧, 𝑧 − 𝑥⟩ < 0;
𝑟 𝑥
(30)
(B2) 𝐶 is a bounded set.
Lemma 3 (see [1]). Let 𝐶 be a nonempty, closed, and convex
subset of a real Hilbert space 𝐻. Let Θ : 𝐶 × 𝐶 → R be a
bifunction satisfying conditions (A1) − (A4), and let 𝜑 : 𝐶 →
R be a lower semicontinuous and convex function. For 𝑟 > 0
and 𝑥 ∈ 𝐻, define a mapping 𝑆𝑟(Θ,𝜑) : 𝐻 → 𝐶 as follows:
𝑆𝑟(Θ,𝜑) (𝑥)
= {𝑦 ∈ 𝐶 : Θ (𝑦, 𝑧) + 𝜑 (𝑧) − 𝜑 (𝑦)
(31)
1
+ ⟨𝑦 − 𝑧, 𝑧 − 𝑥⟩ ≥ 0, ∀𝑧 ∈ 𝐶} .
𝑟
Assume that either (B1) or (B2) holds. Then, the following hold:
In a real Hilbert space 𝐻, it is well known that
󵄩2
󵄩󵄩
󵄩 󵄩2
2
󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩 = ‖𝑥‖ − 󵄩󵄩󵄩𝑦󵄩󵄩󵄩 − 2 ⟨𝑥 − 𝑦, 𝑦⟩ ,
󵄩2
󵄩 󵄩2
󵄩󵄩
2
󵄩󵄩𝜆𝑥 + (1 − 𝜆) 𝑦󵄩󵄩󵄩 = 𝜆‖𝑥‖ + (1 − 𝜆) 󵄩󵄩󵄩𝑦󵄩󵄩󵄩
󵄩2
󵄩
− 𝜆 (1 − 𝜆) 󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩 ,
(B1) for each 𝑥 ∈ 𝐻 and 𝑟 > 0, there exists a bounded
subset 𝐷𝑥 ⊂ 𝐶 and 𝑦𝑥 ∈ 𝐶 such that for all 𝑧 ∈ 𝐶 \ 𝐷𝑥 ,
(29)
(A4) Θ(𝑥, ⋅) is convex and weakly lower semicontinuous
for each 𝑥 ∈ 𝐶;
(i) 𝑆𝑟(Θ,𝜑) ≠ 0 for all 𝑥 ∈ 𝐻 and 𝑆𝑟(Θ,𝜑) is single valued;
(ii) 𝑆𝑟(Θ,𝜑) is firmly nonexpansive, that is, for all 𝑥, 𝑦 ∈ 𝐻,
󵄩󵄩 (Θ,𝜑)
󵄩2
󵄩󵄩𝑆𝑟 𝑥 − 𝑆𝑟(Θ,𝜑) 𝑦󵄩󵄩󵄩 ≤ ⟨𝑆𝑟(Θ,𝜑) 𝑥 − 𝑆𝑟(Θ,𝜑) 𝑦, 𝑥 − 𝑦⟩ ;
󵄩
󵄩
(32)
(iii) Fix(𝑆𝑟(Θ,𝜑) ) = MEP(Θ, 𝜑);
(iv) MEP(Θ, 𝜑) is closed and convex.
Remark 4. If 𝜑 = 0, then 𝑆𝑟(Θ,𝜑) is rewritten as 𝑆𝑟Θ (see [12,
Lemma 2.1] for more details).
Lemma 5 (see [26]). Let {𝑥𝑛 } and {𝑙𝑛 } be bounded sequences
in a Banach space 𝑋, and let {𝛽𝑛 } be a sequence in [0, 1] with
0 < lim inf 𝑛 → ∞ 𝛽𝑛 ≤ lim sup𝑛 → ∞ 𝛽𝑛 < 1. Suppose that 𝑥𝑛+1 =
(1−𝛽𝑛 )𝑙𝑛 +𝛽𝑛 𝑥𝑛 for all integers 𝑛 ≥ 0 and lim supn → ∞ (‖𝑙𝑛+1 −
𝑙𝑛 ‖ − ‖𝑥𝑛+1 − 𝑥𝑛 ‖) ≤ 0. Then, lim𝑛 → ∞ ‖𝑙𝑛 − 𝑥𝑛 ‖ = 0.
Lemma 6 (Demiclosedness Principle [27]). Let 𝐶 be a
nonempty, closed, and convex subset of a real Hilbert space 𝐻.
Let 𝑇 : 𝐶 → 𝐶 be a nonexpansive mapping with Fix(𝑇) ≠ 0.
If {𝑥𝑛 } is a sequence in 𝐶 that converges weakly to 𝑥 and if
{(𝐼 − 𝑇)𝑥𝑛 } converges strongly to 𝑦, then (𝐼 − 𝑇)𝑥 = 𝑦; in
particular, if 𝑦 = 0, then 𝑥 ∈ Fix(𝑇).
Lemma 7 (see [28]). Assume that {𝑎𝑛 } is a sequence of nonnegative real numbers such that
𝑎𝑛+1 ≤ (1 − 𝜎𝑛 ) 𝑎𝑛 + 𝛿𝑛 ,
(33)
where {𝜎𝑛 } is a sequence in (0, 1) and {𝛿𝑛 } is a sequence in R
such that
(i) ∑∞
𝑛=0 𝜎𝑛 = ∞;
(ii) lim sup𝑛 → ∞ (𝛿𝑛 /𝜎𝑛 ) ≤ 0 or ∑∞
𝑛=0 |𝛿𝑛 | < ∞.
Then, lim𝑛 → ∞ 𝑎𝑛 = 0.
Journal of Applied Mathematics
5
The following lemma can be found in [29, 30]. For the
sake of the completeness, we include its proof in a real Hilbert
space version.
Lemma 8. Let 𝐶 be a nonempty, closed, and convex subset of
a real Hilbert space 𝐻. Let 𝐹 : 𝐶 → 𝑋 be a 𝜅-Lipschitzian
and 𝜂-strongly monotone operator. Let 0 < 𝜇 < 2𝜂/𝜅2 and
𝜏 = 𝜇(𝜂 − 𝜇𝜅2 /2). Then, for each 𝑡 ∈ (0, min{1, 1/2𝜏}), the
mapping 𝑆 : 𝐶 → 𝐻 defined by 𝑆 := 𝐼 − 𝑡𝜇𝐹 is a contraction
with constant 1 − 𝑡𝜏.
Proof. Since 0 < 𝜇 < 2𝜂/𝜅2 and 𝑡 ∈ (0, min{1, 1/2𝜏}), this
implies that 1 − 𝑡𝜏 ∈ (0, 1). For all 𝑥, 𝑦 ∈ 𝐶, we have
󵄩2
󵄩2 󵄩
󵄩󵄩
󵄩󵄩𝑆𝑥 − 𝑆𝑦󵄩󵄩󵄩 = 󵄩󵄩󵄩(𝐼 − 𝑡𝜇𝐹) 𝑥 − (𝐼 − 𝑡𝜇𝐹) 𝑦󵄩󵄩󵄩
󵄩2
󵄩
= 󵄩󵄩󵄩(𝑥 − 𝑦) 𝑥 − 𝑡𝜇 (𝐹𝑥 − 𝐹𝑦)󵄩󵄩󵄩
󵄩2
󵄩
= 󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩 − 2𝑡𝜇 ⟨𝐹𝑥 − 𝐹𝑦, 𝑥 − 𝑦⟩
󵄩2
󵄩
+ 𝑡2 𝜇2 󵄩󵄩󵄩𝐹𝑥 − 𝐹𝑦󵄩󵄩󵄩
󵄩2
󵄩2
󵄩
󵄩
≤ 󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩 − 2𝑡𝜇𝜂󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩
󵄩2
󵄩
+ 𝑡2 𝜇2 𝜅2 󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩
󵄩2
󵄩
≤ [1 − 𝑡𝜇 (2𝜂 − 𝜇𝜅2 )] 󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩
󵄩
󵄩󵄩
󵄩󵄩𝐾𝑥 − 𝐾𝑦󵄩󵄩󵄩
󵄩
󵄩
= 󵄩󵄩󵄩󵄩𝑄𝑀𝑥 − 𝑄𝑀𝑦󵄩󵄩󵄩󵄩
󵄩 𝐺
󵄩
𝐺
= 󵄩󵄩󵄩󵄩𝑆𝜆 𝑀 (𝐼 − 𝜆 𝑀𝐴 𝑀) 𝑄𝑀−1 𝑥 − 𝑆𝜆 𝑀 (𝐼 − 𝜆 𝑀𝐴 𝑀) 𝑄𝑀−1 𝑦󵄩󵄩󵄩󵄩
𝑀
𝑀
󵄩
󵄩
≤ 󵄩󵄩󵄩󵄩(𝐼 − 𝜆 𝑀𝐴 𝑀) 𝑄𝑀−1 𝑥 − (𝐼 − 𝜆 𝑀𝐴 𝑀) 𝑄𝑀−1 𝑦󵄩󵄩󵄩󵄩
󵄩
󵄩
≤ 󵄩󵄩󵄩󵄩𝑄𝑀−1 𝑥 − 𝑄𝑀−1 𝑦󵄩󵄩󵄩󵄩
(37)
which implies that 𝐾 is nonexpansive. This completes the
proof.
(34)
Lemma 10. Let 𝐶 be a nonempty, closed, and convex subset of
a real Hilbert space 𝐻. Let 𝐴 𝑖 : 𝐶 → 𝐻 (𝑖 = 1, 2, . . . , 𝑀)
∗
) ∈ 𝐶×
be a nonlinear mapping. For given (𝑥1∗ , 𝑥2∗ , . . . , 𝑥𝑀
𝐺
∗
,
𝐶 × ⋅ ⋅ ⋅ × 𝐶, where 𝑥∗ = 𝑥1∗ , 𝑥𝑖∗ = 𝑆𝜆 𝑖−1 (𝐼 − 𝜆 𝑖−1 𝐴 𝑖−1 )𝑥𝑖−1
𝐺
𝑖−1
∗
. Then,
for 𝑖 = 2, 3, . . . , 𝑀, and 𝑥1∗ = 𝑆𝜆 𝑀 (𝐼 − 𝜆 𝑀𝐴 𝑀)𝑥𝑀
𝑀
∗ ∗
∗
(𝑥1 , 𝑥2 , . . . , 𝑥𝑀) is a solution of the problem (16) if and only
if 𝑥∗ is a fixed point of the mapping 𝐾 defined as in Lemma 9.
𝜇𝜅2
󵄩2
󵄩
)] 󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩
2
∗
Proof. Let (𝑥1∗ , 𝑥2∗ , . . . , 𝑥𝑀
) ∈ 𝐶 × 𝐶 × ⋅ ⋅ ⋅ × 𝐶 be a solution of
the problem (16). Then, we have
∗
, 𝑥1 − 𝑥1∗ ⟩
𝐺𝑀 (𝑥1∗ , 𝑥1 ) + ⟨𝐴 𝑀𝑥𝑀
It follows that
󵄩
󵄩
󵄩
󵄩󵄩
󵄩󵄩𝑆𝑥 − 𝑆𝑦󵄩󵄩󵄩 ≤ (1 − 𝑡𝜏) 󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩 .
+
(35)
Hence, we have that 𝑆 := 𝐼−𝑡𝜇𝐹 is a contraction with constant
1 − 𝑡𝜏. This completes the proof.
Lemma 9. Let 𝐶 be a nonempty, closed, and convex subset of
a real Hilbert space 𝐻. Let 𝐴 𝑖 : 𝐶 → 𝐻 (𝑖 = 1, 2, . . . , 𝑀)
be a finite family of 𝛼𝑖 -inverse strongly monotone operator. Let
𝐾 : 𝐶 → 𝐶 be a mapping defined by
𝐺
𝐾𝑥 := 𝑆𝜆 𝑀 (𝐼 − 𝜆 𝑀𝐴 𝑀) 𝑆𝜆 𝑀−1
𝑀
1
𝜆 1 𝐴 1 ) for 𝑖 = 1, 2, . . . , 𝑀 and 𝑄0 = 𝐼. Then, 𝐾 = 𝑄𝑀. For all
𝑥, 𝑦 ∈ 𝐶, it follows from (28) that
󵄩
󵄩
= 󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩 ,
󵄩2
󵄩
= (1 − 𝑡𝜏)2 󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩 .
𝐺
𝐺
𝑖−1
󵄩
󵄩
≤ 󵄩󵄩󵄩󵄩𝑄0 𝑥 − 𝑄0 𝑦󵄩󵄩󵄩󵄩
2
≤ [1 − 𝑡𝜇 (𝜂 −
𝑖
..
.
𝜇𝜅2
󵄩2
󵄩
)] 󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩
2
= [1 − 2𝑡𝜇 (𝜂 −
𝐺
𝐺
Proof. Put 𝑄𝑖 = 𝑆𝜆 𝑖 (𝐼 − 𝜆 𝑖 𝐴 𝑖 )𝑆𝜆 𝑖−1 (𝐼 − 𝜆 𝑖−1 𝐴 𝑖−1 ) ⋅ ⋅ ⋅ 𝑆𝜆 1 (𝐼 −
𝐺
× (𝐼 − 𝜆 𝑀−1 𝐴 𝑀−1 ) ⋅ ⋅ ⋅ 𝑆𝜆 1 (𝐼 − 𝜆 1 𝐴 1 ) 𝑥,
1
∀𝑥 ∈ 𝐶.
(36)
If 0 < 𝜆 𝑖 < 2𝛼𝑖 for all 𝑖 = 1, 2, . . . , 𝑀, then 𝐾 : 𝐶 → 𝐶 is
nonexpansive.
∀𝑥1 ∈ 𝐶,
∗
∗
∗
𝐺𝑀−1 (𝑥𝑀
, 𝑥𝑀) + ⟨𝐴 𝑀−1 𝑥𝑀−1
, 𝑥𝑀 − 𝑥𝑀
⟩
+
1
∗
∗
⟨𝑥∗ − 𝑥𝑀−1
, 𝑥𝑀 − 𝑥𝑀
⟩ ≥ 0,
𝜆 𝑀−1 𝑀
∀𝑥𝑀 ∈ 𝐶,
..
.
𝐺2 (𝑥3∗ , 𝑥3 ) + ⟨𝐴 2 𝑥2∗ , 𝑥3 − 𝑥3∗ ⟩
+
𝑀−1
1
∗
⟨𝑥∗ − 𝑥𝑀
, 𝑥1 − 𝑥1∗ ⟩ ≥ 0,
𝜆𝑀 1
1
⟨𝑥∗ − 𝑥2∗ , 𝑥3 − 𝑥3∗ ⟩ ≥ 0,
𝜆2 3
∀𝑥3 ∈ 𝐶,
𝐺1 (𝑥2∗ , 𝑥2 ) + ⟨𝐴 1 𝑥1∗ , 𝑥2 − 𝑥2∗ ⟩
+
1
⟨𝑥∗ − 𝑥1∗ , 𝑥2 − 𝑥2∗ ⟩ ≥ 0,
𝜆1 2
⇕
∀𝑥2 ∈ 𝐶,
6
Journal of Applied Mathematics
𝐺
∗
𝑥1∗ = 𝑆𝜆 𝑀 (𝐼 − 𝜆 𝑀𝐴 𝑀) 𝑥𝑀
𝑀
(𝐶1) lim𝑛 → ∞ 𝛼𝑛 = 0 and ∑∞
𝑛=1 𝛼𝑛 = ∞;
𝐺
(𝐶2) 0 < lim inf 𝑛 → ∞ 𝛽𝑛 ≤ lim sup𝑛 → ∞ 𝛽𝑛 < 1;
𝑥2∗ = 𝑆𝜆 1 (𝐼 − 𝜆 1 𝐴 1 ) 𝑥1∗ ,
(𝐶3) lim inf 𝑛 → ∞ 𝑟𝑘,𝑛 > 0 and lim𝑛 → ∞ (𝑟𝑘,𝑛 /𝑟𝑘,𝑛+1 ) = 1 for
all 𝑘 ∈ {1, 2, . . . , 𝑁}.
1
..
.
Then, the sequence {𝑥𝑛 } defined by (39) converges strongly to
𝑥̂ ∈ F as 𝑛 → ∞, where 𝑥̂ solves uniquely the variational
inequality
𝐺
∗
∗
𝑥𝑀−1
= 𝑆𝜆 𝑀−2 (𝐼 − 𝜆 𝑀−2 𝐴 𝑀−2 ) 𝑥𝑀−2
,
𝑀−2
∗
𝑥𝑀
=
𝐺
𝑆𝜆 𝑀−1
𝑀−1
∗
(𝐼 − 𝜆 𝑀−1 𝐴 𝑀−1 ) 𝑥𝑀−1
,
̂ 𝑥̂ − V⟩ ≤ 0,
⟨(𝜇𝐹 − 𝛾𝑉) 𝑥,
𝐺
𝑥∗ = 𝑆𝜆 𝑀 (𝐼 − 𝜆 𝑀𝐴 𝑀) 𝑆𝜆 𝑀−1
𝑀
𝑀−1
𝐺
× (𝐼 − 𝜆 𝑀−1 𝐴 𝑀−1 ) ⋅ ⋅ ⋅ 𝑆𝜆 1 (𝐼 − 𝜆 1 𝐴 1 ) 𝑥∗ = 𝐾𝑥∗ .
1
(38)
Proof. Note that from condition (𝐶1), we may assume,
without loss of generality, that 𝛼𝑛 ≤ min{1, 1/2𝜏} for all 𝑛 ∈ N.
First, we show that {𝑥𝑛 } is bounded. Set
𝑘 ,𝜑𝑘 )
𝑘−1 ,𝜑𝑘−1 )
(𝐼 − 𝑟𝑘,𝑛 Ψ𝑘 ) 𝑆𝑟(Θ
𝐺𝑛𝑘 := 𝑆𝑟(Θ
𝑘,𝑛
𝑘−1,𝑛
1 ,𝜑1 )
× (𝐼 − 𝑟𝑘−1,𝑛 Ψ𝑘−1 ) ⋅ ⋅ ⋅ 𝑆𝑟(Θ
(𝐼 − 𝑟1,𝑛 Ψ1 ) ,
1,𝑛
This completes the proof.
∀𝑘 ∈ {1, 2, . . . , 𝑁} , 𝑛 ∈ N,
3. Main Results
Theorem 11. Let 𝐶 be a nonempty, closed, and convex subset of
a real Hilbert space 𝐻. Let Θ𝑘 : 𝐶×𝐶 → R (𝑘 = 1, 2, . . . , 𝑁) a
finite family of bifunctions which satisfy (A1)–(A4), 𝜑𝑘 : 𝐶 →
R (𝑘 = 1, 2, . . . , 𝑁) a finite family of lower semicontinuous
and convex functions, and Ψ𝑘 : 𝐶 → 𝐻 (𝑘 = 1, 2, . . . , 𝑁)
a finite family of a 𝜇𝑘 -inverse strongly monotone mapping and
𝐴 𝑘 : 𝐶 → 𝐻 (𝑘 = 1, 2, . . . , 𝑀) a finite family of an 𝛼𝑘 inverse strongly monotone mapping. Let 𝑆 be a semigroup, and
let S = {𝑇(𝑡) : 𝑡 ∈ 𝑆} be a nonexpansive semigroup on
𝐶 such that Fix(S) ≠ 0. Let 𝑋 be a left invariant subspace of
ℓ∞ (𝑆) such that 1 ∈ 𝑋 and the function 𝑡 → ⟨𝑇(𝑡)𝑥, 𝑦⟩ is an
element of 𝑋 for 𝑥 ∈ 𝐶 and 𝑦 ∈ 𝐻. Let {𝜇𝑛 } be a left regular
sequence of means on X such that lim𝑛 → ∞ ‖𝜇𝑛+1 − 𝜇𝑛 ‖ = 0.
Let 𝐹 : 𝐶 → 𝐻 be a 𝜅-Lipschitzian and 𝜂-strongly monotone
operator with constants 𝜅, 𝜂 > 0, and let 𝑉 : 𝐶 → 𝐻 be an 𝐿Lipschitzian mapping with a constant 𝐿 ≥ 0. Let 0 < 𝜇 < 2𝜂/𝜅2
and 0 ≤ 𝛾𝐿 < 𝜏, where 𝜏 = 𝜇(𝜂 − 𝜇𝜅2 /2). Assume that
F := ⋂𝑁
𝑘=1 GMEP(Θ𝑘 , 𝜑𝑘 , Ψ𝑘 ) ∩ (𝐾) ∩ Fix(S) ≠ 0, where 𝐾 is
defined as in Lemma 9. For given 𝑥1 ∈ 𝐶, let {𝑥𝑛 } be a sequence
defined by
𝑢𝑛 =
𝑁 ,𝜑𝑁 )
𝑆𝑟(Θ
𝑁,𝑛
(𝐼 −
𝑁−1 ,𝜑𝑁−1 )
𝑟𝑁,𝑛 Ψ𝑁) 𝑆𝑟(Θ
𝑁−1,𝑛
1 ,𝜑1 )
× (𝐼 − 𝑟𝑁−1,𝑛 Ψ𝑁−1 ) ⋅ ⋅ ⋅ 𝑆𝑟(Θ
(𝐼 − 𝑟1,𝑛 Ψ1 ) 𝑥𝑛 ,
1,𝑛
𝐺
𝐺
𝑦𝑛 = 𝑆𝜆 𝑀 (𝐼 − 𝜆 𝑀𝐴 𝑀) 𝑆𝜆 𝑀−1
𝑀
(40)
̂
Equivalently, one has 𝑥̂ = 𝑃F (𝐼 − 𝜇𝐹 + 𝛾𝑉)𝑥.
⇕
𝐺
∀V ∈ F.
𝑀−1
𝐺
× (𝐼 − 𝜆 𝑀−1 𝐴 𝑀−1 ) ⋅ ⋅ ⋅ 𝑆𝜆 1 (𝐼 − 𝜆 1 𝐴 1 ) 𝑢𝑛 ,
1
𝑥𝑛+1 = 𝛽𝑛 𝑥𝑛 + (1 − 𝛽𝑛 ) 𝑃𝐶 [𝛼𝑛 𝛾𝑉𝑥𝑛 + (𝐼 − 𝛼𝑛 𝜇𝐹) 𝑇 (𝜇𝑛 ) 𝑦𝑛 ] ,
∀𝑛 ≥ 1,
(39)
where {𝛼𝑛 }, {𝛽𝑛 } are sequences in (0, 1), and {𝑟𝑘,𝑛 }𝑁
𝑘=1 is a
𝑁
sequence such that {𝑟𝑘,𝑛 }𝑘=1 ⊂ [𝑎𝑘 , 𝑏𝑘 ] ⊂ (0, 2𝛾𝑘 ) satisfying the
following conditions:
𝐺
𝐺
𝐺
𝑄𝑖 := 𝑆𝜆 𝑖 (𝐼 − 𝜆 𝑖 𝐴 𝑖 ) 𝑆𝜆 𝑖−1 (𝐼 − 𝜆 𝑖−1 𝐴 𝑖−1 ) ⋅ ⋅ ⋅ 𝑆𝜆 1 (𝐼 − 𝜆 1 𝐴 1 ) ,
𝑖
𝑖−1
1
∀𝑖 ∈ {1, 2, . . . , 𝑀} ,
(41)
𝐺𝑛0 = 𝑄0 = 𝐼. Then, we have 𝑢𝑛 = 𝐺𝑛𝑁𝑥𝑛 and 𝑦𝑛 =
𝑄𝑀𝑢𝑛 . From Lemmas 3 and 9, we have that 𝐺𝑛𝑁 and 𝑄𝑀 are
nonexpansive. Take 𝑥∗ ∈ F; we have
󵄩 𝑁
󵄩
󵄩󵄩
∗󵄩
𝑁 ∗󵄩
∗󵄩
(42)
󵄩󵄩𝑢𝑛 − 𝑥 󵄩󵄩󵄩 = 󵄩󵄩󵄩󵄩𝐺𝑛 𝑥𝑛 − 𝐺𝑛 𝑥 󵄩󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩𝑥𝑛 − 𝑥 󵄩󵄩󵄩 .
By Lemma 10, we have 𝑥∗ = 𝑄𝑀𝑥∗ . It follows from (42) that
󵄩 𝑀
󵄩󵄩
∗󵄩
𝑀 ∗󵄩
󵄩󵄩𝑦𝑛 − 𝑥 󵄩󵄩󵄩 = 󵄩󵄩󵄩󵄩𝑄 𝑢𝑛 − 𝑄 𝑥 󵄩󵄩󵄩󵄩
󵄩
󵄩
(43)
≤ 󵄩󵄩󵄩𝑢𝑛 − 𝑥∗ 󵄩󵄩󵄩
󵄩
󵄩
≤ 󵄩󵄩󵄩𝑥𝑛 − 𝑥∗ 󵄩󵄩󵄩 .
Set
𝑧𝑛 := 𝑃𝐶 [𝛼𝑛 𝛾𝑉𝑥𝑛 + (𝐼 − 𝛼𝑛 𝜇𝐹) 𝑇 (𝜇𝑛 ) 𝑦𝑛 ] ,
∀𝑛 ∈ N.
(44)
Then, we can rewrite (39) as 𝑥𝑛+1 = 𝛽𝑛 𝑥𝑛 + (1 − 𝛽𝑛 )𝑧𝑛 . From
Lemma 8 and (43), we have
󵄩󵄩
∗󵄩
󵄩󵄩𝑧𝑛 − 𝑥 󵄩󵄩󵄩
󵄩
󵄩
= 󵄩󵄩󵄩𝑃𝐶 [𝛼𝑛 𝛾𝑉𝑥𝑛 + (𝐼 − 𝛼𝜇𝐹) 𝑇 (𝜇𝑛 ) 𝑦𝑛 ] − 𝑃𝐶𝑥∗ 󵄩󵄩󵄩
󵄩
󵄩
≤ 󵄩󵄩󵄩𝛼𝑛 (𝛾𝑉𝑥𝑛 − 𝜇𝐹𝑥∗ ) + (𝐼 − 𝛼𝑛 𝜇𝐹) (𝑇 (𝜇𝑛 ) 𝑦𝑛 − 𝑥∗ )󵄩󵄩󵄩
󵄩
󵄩
󵄩
󵄩
≤ 𝛼𝑛 󵄩󵄩󵄩𝛾𝑉𝑥𝑛 − 𝜇𝐹𝑥∗ 󵄩󵄩󵄩 + (1 − 𝛼𝑛 𝜏) 󵄩󵄩󵄩𝑇 (𝜇𝑛 ) 𝑦𝑛 − 𝑥∗ 󵄩󵄩󵄩
󵄩
󵄩
󵄩
󵄩
≤ 𝛼𝑛 𝛾 󵄩󵄩󵄩𝑉𝑥𝑛 − 𝑉𝑥∗ 󵄩󵄩󵄩 + 𝛼𝑛 󵄩󵄩󵄩𝛾𝑉𝑥∗ − 𝜇𝐹𝑥∗ 󵄩󵄩󵄩
󵄩
󵄩
+ (1 − 𝛼𝑛 𝜏) 󵄩󵄩󵄩𝑦𝑛 − 𝑥∗ 󵄩󵄩󵄩
󵄩
󵄩
󵄩
󵄩
≤ (1 − 𝛼𝑛 (𝜏 − 𝛾𝐿)) 󵄩󵄩󵄩𝑥𝑛 − 𝑥∗ 󵄩󵄩󵄩 + 𝛼𝑛 󵄩󵄩󵄩𝛾𝑉𝑥∗ − 𝜇𝐹𝑥∗ 󵄩󵄩󵄩 .
(45)
Journal of Applied Mathematics
7
It follows from (45) that
󵄩󵄩
∗󵄩
󵄩󵄩𝑥𝑛+1 − 𝑥 󵄩󵄩󵄩
󵄩
󵄩
= 󵄩󵄩󵄩𝛽𝑛 (𝑥𝑛 − 𝑥∗ ) + (1 − 𝛽𝑛 ) (𝑧𝑛 − 𝑥∗ )󵄩󵄩󵄩
󵄩
󵄩
󵄩
󵄩
≤ 𝛽𝑛 󵄩󵄩󵄩𝑥𝑛 − 𝑥∗ 󵄩󵄩󵄩 + (1 − 𝛽𝑛 ) 󵄩󵄩󵄩𝑧𝑛 − 𝑥∗ 󵄩󵄩󵄩
󵄩
󵄩
≤ 𝛽𝑛 󵄩󵄩󵄩𝑥𝑛 − 𝑥∗ 󵄩󵄩󵄩 + (1 − 𝛽𝑛 )
󵄩
󵄩
󵄩
󵄩
× [(1 − 𝛼𝑛 (𝜏 − 𝛾𝐿)) 󵄩󵄩󵄩𝑥𝑛 − 𝑥∗ 󵄩󵄩󵄩 + 𝛼𝑛 󵄩󵄩󵄩𝛾𝑉𝑥∗ − 𝜇𝐹𝑥∗ 󵄩󵄩󵄩]
󵄩
󵄩
= (1 − 𝛼𝑛 (1 − 𝛽𝑛 ) (𝜏 − 𝛾𝐿)) 󵄩󵄩󵄩𝑥𝑛 − 𝑥∗ 󵄩󵄩󵄩
󵄩
󵄩󵄩
󵄩𝛾𝑉𝑥∗ − 𝜇𝐹𝑥∗ 󵄩󵄩󵄩
+ 𝛼𝑛 (1 − 𝛽𝑛 ) (𝜏 − 𝛾𝐿) 󵄩
.
𝜏 − 𝛾𝐿
(46)
𝑘
𝜔𝑛 ∈
In fact, since 𝐷𝑛𝑘 𝜔𝑛 ∈ GMEP(Θ𝑘 , 𝜑𝑘 , Ψ𝑘 ) and 𝐷𝑛+1
GMEP(Θ𝑘 , 𝜑k , Ψ𝑘 ), we have
Θ𝑘 (𝐷𝑛𝑘 𝜔𝑛 , 𝑦) + 𝜑𝑘 (𝑦) − 𝜑𝑘 (𝐷𝑛𝑘 𝜔𝑛 )
+ ⟨Ψ𝑘 𝜔𝑛 , 𝑦 − 𝐷𝑛𝑘 𝜔𝑛 ⟩
+
𝑘
𝑘
Θ𝑘 (𝐷𝑛+1
𝜔𝑛 , 𝑦) + 𝜑𝑘 (𝑦) − 𝜑𝑘 (𝐷𝑛+1
𝜔𝑛 )
𝑘
𝜔𝑛 ⟩
+ ⟨Ψ𝑘 𝜔𝑛 , 𝑦 − 𝐷𝑛+1
By induction, we have
Hence, {𝑥𝑛 } is bounded, and so are {𝑉𝑥𝑛 } and {𝐹𝑇(𝜇𝑛 )𝑦𝑛 }.
Next, we show that
󵄩󵄩
lim 󵄩𝑥
𝑛 → ∞ 󵄩 𝑛+1
1
𝑟𝑘,𝑛+1
𝑘
𝑘
⟨𝑦 − 𝐷𝑛+1
𝜔𝑛 , 𝐷𝑛+1
𝜔𝑛 − 𝜔𝑛 ⟩ ≥ 0,
󵄩
− 𝑥𝑛 󵄩󵄩󵄩 = 0.
𝑘
Substituting 𝑦 = 𝐷𝑛+1
𝜔𝑛 in (54) and 𝑦 = 𝐷𝑛𝑘 𝜔𝑛 in (55), then
add these two inequalities, and using (A2), we obtain
𝑘
𝜔𝑛 − 𝐷𝑛𝑘 𝜔𝑛 ,
⟨𝐷𝑛+1
(48)
Observe that
󵄩
󵄩
lim 󵄩󵄩𝑇 (𝜇𝑛+1 ) 𝑦𝑛 − 𝑇 (𝜇𝑛 ) 𝑦𝑛 󵄩󵄩󵄩 = 0.
𝑛→∞ 󵄩
−
(49)
Indeed,
(55)
∀𝑦 ∈ 𝐶.
∀𝑛 ≥ 1.
(47)
(54)
∀𝑦 ∈ 𝐶,
+
󵄩󵄩
∗
∗󵄩
󵄩
󵄩
󵄩󵄩
∗󵄩
∗ 󵄩 󵄩𝛾𝑉𝑥 − 𝜇𝐹𝑥 󵄩
󵄩} ,
󵄩󵄩𝑥𝑛 − 𝑥 󵄩󵄩󵄩 ≤ max {󵄩󵄩󵄩𝑥1 − 𝑥 󵄩󵄩󵄩 , 󵄩
𝜏 − 𝛾𝐿
1
⟨𝑦 − 𝐷𝑛𝑘 𝜔𝑛 , 𝐷𝑛𝑘 𝜔𝑛 − 𝜔𝑛 ⟩ ≥ 0,
𝑟𝑘,𝑛
1
𝑟𝑘,𝑛+1
1
(𝐷𝑛𝑘 𝜔𝑛 − 𝜔𝑛 )
𝑟𝑘,𝑛
𝑘
(𝐷𝑛+1
𝜔𝑛
(56)
− 𝜔𝑛 )⟩ ≥ 0.
Hence,
󵄩󵄩
󵄩
󵄩󵄩𝑇 (𝜇𝑛+1 ) 𝑦𝑛 − 𝑇 (𝜇𝑛 ) 𝑦𝑛 󵄩󵄩󵄩
󵄨
󵄨
= sup 󵄨󵄨󵄨⟨𝑇 (𝜇𝑛+1 ) 𝑦𝑛 − 𝑇 (𝜇𝑛 ) 𝑦𝑛 , 𝑧⟩󵄨󵄨󵄨
𝑘
𝑘
𝑘
⟨𝐷𝑛+1
𝜔𝑛 − 𝐷𝑛𝑘 𝜔𝑛 , 𝐷𝑛𝑘 𝜔𝑛 − 𝐷𝑛+1
𝜔𝑛 + 𝐷𝑛+1
𝜔𝑛
‖𝑧‖=1
󵄨
󵄨
= sup 󵄨󵄨󵄨(𝜇𝑛+1 )𝑠 ⟨𝑇 (𝑠) 𝑦𝑛 , 𝑧⟩ − (𝜇𝑛 )𝑠 ⟨𝑇 (𝑠) 𝑦𝑛 , 𝑧⟩󵄨󵄨󵄨
(50)
‖𝑧‖=1
󵄩
󵄩
󵄩
󵄩
≤ 󵄩󵄩󵄩𝜇𝑛+1 − 𝜇𝑛 󵄩󵄩󵄩 sup 󵄩󵄩󵄩𝑇 (𝑠) 𝑦𝑛 󵄩󵄩󵄩 .
Since {𝑦𝑛 } is bounded and lim𝑛 → ∞ ‖𝜇𝑛+1 − 𝜇𝑛 ‖ = 0, then (49)
holds. We observe that
󵄩 󵄩 𝑀
󵄩󵄩
𝑀 󵄩
󵄩󵄩𝑦𝑛+1 − 𝑦𝑛 󵄩󵄩󵄩 = 󵄩󵄩󵄩󵄩𝑄 𝑢𝑛+1 − 𝑄 𝑢𝑛 󵄩󵄩󵄩󵄩
(51)
󵄩
󵄩
≤ 󵄩󵄩󵄩𝑢𝑛+1 − 𝑢𝑛 󵄩󵄩󵄩 .
Let {𝜔𝑛 } be a bounded sequence in 𝐶. Now, we show that
󵄩
𝑁
− 𝐺𝑛+1
𝜔𝑛 󵄩󵄩󵄩󵄩 = 0.
(57)
we derive from (57) that
𝑠∈𝑆
󵄩
lim 󵄩󵄩󵄩𝐺𝑛𝑁𝜔𝑛
𝑛→∞ 󵄩
𝑟𝑘,𝑛
𝑘
−𝜔𝑛 −
(𝐷𝑛+1
𝜔𝑛 − 𝜔𝑛 )⟩ ≥ 0;
𝑟𝑛,𝑘+1
󵄩󵄩 𝑘
󵄩2
󵄩󵄩𝐷𝑛+1 𝜔𝑛 − 𝐷𝑛𝑘 𝜔𝑛 󵄩󵄩󵄩
󵄩
󵄩
𝑘
≤ ⟨𝐷𝑛+1
𝜔𝑛 − 𝐷𝑛𝑘 𝜔𝑛 , (1 −
𝑟𝑘,𝑛
𝑘
) (𝐷𝑛+1
𝜔𝑛 − 𝜔𝑛 )⟩
𝑟𝑘,𝑛+1
󵄨
𝑟𝑘,𝑛 󵄨󵄨󵄨󵄨 󵄩󵄩 𝑘
󵄩 𝑘
󵄩 󵄨󵄨
󵄩
󵄨󵄨 󵄩󵄩𝐷𝑛+1 𝜔𝑛 − 𝜔𝑛 󵄩󵄩󵄩 ,
≤ 󵄩󵄩󵄩󵄩𝐷𝑛+1
𝜔𝑛 − 𝐷𝑛𝑘 𝜔𝑛 󵄩󵄩󵄩󵄩 󵄨󵄨󵄨󵄨1 −
󵄩
󵄨
󵄩
𝑟𝑘,𝑛+1 󵄨󵄨
󵄨󵄨
(52)
𝑘 ,𝜑𝑘 )
For the previous purpose, put 𝐷𝑛𝑘 = 𝑆𝑟(Θ
(𝐼 − 𝑟𝑘,𝑛 Ψ𝑘 ), and
𝑘,𝑛
we first show that
󵄩 𝑘
󵄩
𝜔𝑛 − 𝐷𝑛𝑘 𝜔𝑛 󵄩󵄩󵄩󵄩 = 0, ∀𝑘 ∈ {1, 2, . . . , 𝑁} . (53)
lim 󵄩󵄩󵄩𝐷𝑛+1
𝑛→∞ 󵄩
(58)
which implies that
󵄨
𝑟 󵄨󵄨󵄨 󵄩 𝑘
󵄩󵄩 𝑘
󵄩 󵄨󵄨
󵄩
󵄩󵄩𝐷𝑛+1 𝜔𝑛 − 𝐷𝑛𝑘 𝜔𝑛 󵄩󵄩󵄩 ≤ 󵄨󵄨󵄨1 − 𝑘,𝑛 󵄨󵄨󵄨 󵄩󵄩󵄩𝐷𝑛+1
𝜔𝑛 − 𝜔𝑛 󵄩󵄩󵄩󵄩 .
󵄩
󵄩
󵄨
󵄩 󵄨󵄨
𝑟𝑘,𝑛+1 󵄨󵄨
󵄨
(59)
8
Journal of Applied Mathematics
Noticing that condition (𝐶3) implies that (53) holds, from the
definition of 𝐺𝑛𝑁 and the nonexpansiveness of 𝐷𝑛𝑘 , we have
󵄩󵄩 𝑁
󵄩
𝑁
󵄩󵄩𝐺𝑛 𝜔𝑛 − 𝐺𝑛+1
𝜔𝑛 󵄩󵄩󵄩󵄩
󵄩
󵄩
󵄩
𝑁
𝑁−1
= 󵄩󵄩󵄩󵄩𝐷𝑛𝑁𝐺𝑛𝑁−1 𝜔𝑛 − 𝐷𝑛+1
𝐺𝑛+1
𝜔𝑛 󵄩󵄩󵄩󵄩
󵄩
󵄩
𝑁
≤ 󵄩󵄩󵄩󵄩𝐷𝑛𝑁𝐺𝑛𝑁−1 𝜔𝑛 − 𝐷𝑛+1
𝐺𝑛𝑁−1 𝜔𝑛 󵄩󵄩󵄩󵄩
󵄩 𝑁 𝑁−1
󵄩
𝑁
𝑁−1
+ 󵄩󵄩󵄩󵄩𝐷𝑛+1
𝐺𝑛 𝜔𝑛 − 𝐷𝑛+1
𝐺𝑛+1
𝜔𝑛 󵄩󵄩󵄩󵄩
󵄩
󵄩
𝑁
≤ 󵄩󵄩󵄩󵄩𝐷𝑛𝑁𝐺𝑛𝑁−1 𝜔𝑛 − 𝐷𝑛+1
𝐺𝑛𝑁−1 𝜔𝑛 󵄩󵄩󵄩󵄩
󵄩
󵄩
𝑁−1
+ 󵄩󵄩󵄩󵄩𝐺𝑛𝑁−1 𝜔𝑛 − 𝐺𝑛+1
𝜔𝑛 󵄩󵄩󵄩󵄩
󵄩
󵄩
𝑁
≤ 󵄩󵄩󵄩󵄩𝐷𝑛𝑁𝐺𝑛𝑁−1 𝜔𝑛 − 𝐷𝑛+1
𝐺𝑛𝑁−1 𝜔𝑛 󵄩󵄩󵄩󵄩
󵄩
󵄩
𝑁−1 𝑁−2
+ 󵄩󵄩󵄩󵄩𝐷𝑛𝑁−1 𝐺𝑛𝑁−2 𝜔𝑛 − 𝐷𝑛+1
𝐺𝑛+1 𝜔𝑛 󵄩󵄩󵄩󵄩
󵄩
󵄩
𝑁−2
+ 󵄩󵄩󵄩󵄩𝐺𝑛𝑁−2 𝜔𝑛 − 𝐺𝑛+1
𝜔𝑛 󵄩󵄩󵄩󵄩
󵄩
󵄩
𝑁
≤ 󵄩󵄩󵄩󵄩𝐷𝑛𝑁𝐺𝑛𝑁−1 𝜔𝑛 − 𝐷𝑛+1
𝐺𝑛𝑁−1 𝜔𝑛 󵄩󵄩󵄩󵄩
󵄩
󵄩
𝑁−1 𝑁−2
+ 󵄩󵄩󵄩󵄩𝐷𝑛𝑁−1 𝐺𝑛𝑁−2 𝜔𝑛 − 𝐷𝑛+1
𝐺𝑛+1 𝜔𝑛 󵄩󵄩󵄩󵄩 + ⋅ ⋅ ⋅
󵄩
󵄩 󵄩
󵄩
2
1
+ 󵄩󵄩󵄩󵄩𝐷𝑛2 𝜔𝑛 − 𝐷𝑛+1
𝜔𝑛 󵄩󵄩󵄩󵄩 + 󵄩󵄩󵄩󵄩𝐷𝑛1 𝜔𝑛 − 𝐷𝑛+1
𝜔𝑛 󵄩󵄩󵄩󵄩 ,
(63)
It follows from (51), (61), and (63) that
(60)
𝑛≥1
󵄩
󵄩
󵄩
󵄩
𝛾 󵄩󵄩󵄩𝑉𝑥𝑛 󵄩󵄩󵄩 + 𝜇 󵄩󵄩󵄩𝐹𝑇 (𝑡𝑛 ) 𝑦𝑛 󵄩󵄩󵄩} .
From definition of {𝑧𝑛 }, we note that
󵄩
󵄩󵄩
󵄩󵄩𝑧𝑛+1 − 𝑧𝑛 󵄩󵄩󵄩
󵄩
= 󵄩󵄩󵄩𝑃𝐶 [𝛼𝑛+1 𝛾𝑉𝑥𝑛+1 + (𝐼 − 𝛼𝑛+1 𝜇𝐹) 𝑇 (𝜇𝑛+1 ) 𝑦𝑛+1 ]
󵄩
−𝑃𝐶 [𝛼𝑛 𝛾𝑉𝑥𝑛 + (𝐼 − 𝛼𝑛 𝜇𝐹) 𝑇 (𝜇𝑛 ) 𝑦𝑛 ]󵄩󵄩󵄩
󵄩
󵄩
≤ 𝛼𝑛+1 󵄩󵄩󵄩𝛾𝑉𝑥𝑛+1 − 𝜇𝐹𝑇 (𝜇𝑛+1 ) 𝑦𝑛+1 󵄩󵄩󵄩
󵄩
󵄩
+ 𝛼𝑛 󵄩󵄩󵄩𝛾𝑉𝑥𝑛+1 − 𝜇𝐹𝑇 (𝜇𝑛 ) 𝑦𝑛 󵄩󵄩󵄩
󵄩
󵄩
+ 󵄩󵄩󵄩𝑇 (𝜇𝑛+1 ) 𝑦𝑛+1 − 𝑇 (𝜇𝑛 ) 𝑦𝑛 󵄩󵄩󵄩
󵄩
󵄩 𝑁
󵄩󵄩
󵄩
𝑁
󵄩󵄩𝑧𝑛+1 − 𝑧𝑛 󵄩󵄩󵄩 ≤ (𝛼𝑛+1 + 𝛼𝑛 ) 𝑀1 + 󵄩󵄩󵄩󵄩𝐺𝑛 𝑥𝑛 − 𝐺𝑛+1 𝑥𝑛 󵄩󵄩󵄩󵄩
󵄩
󵄩 󵄩
󵄩
+ 󵄩󵄩󵄩𝑥𝑛+1 − 𝑥𝑛 󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑇 (𝜇𝑛+1 ) 𝑦𝑛 − 𝑇 (𝜇𝑛 ) 𝑦𝑛 󵄩󵄩󵄩 .
(64)
From condition (𝐶1), (49), and (52), we have
󵄩 󵄩
󵄩
󵄩
lim sup (󵄩󵄩󵄩𝑧𝑛+1 − 𝑧𝑛 󵄩󵄩󵄩 − 󵄩󵄩󵄩𝑥𝑛+1 − 𝑥𝑛 󵄩󵄩󵄩) ≤ 0.
𝑛→∞
(65)
Hence, by Lemma 5, we obtain
󵄩󵄩
lim 󵄩𝑧
𝑛→∞ 󵄩 𝑛
󵄩
− 𝑥𝑛 󵄩󵄩󵄩 = 0.
(66)
Consequently,
󵄩󵄩
lim 󵄩𝑥
𝑛 → ∞ 󵄩 𝑛+1
󵄩
󵄩
󵄩
− 𝑥𝑛 󵄩󵄩󵄩 = lim (1 − 𝛽𝑛 ) 󵄩󵄩󵄩𝑧𝑛 − 𝑥𝑛 󵄩󵄩󵄩 = 0.
𝑛→∞
(67)
From condition (𝐶1), we have
(61)
Put a constant 𝑀1 > 0 such that
󵄩
󵄩
󵄩
󵄩
𝑀1 = sup {𝛾 󵄩󵄩󵄩𝑉𝑥𝑛+1 󵄩󵄩󵄩 + 𝜇 󵄩󵄩󵄩𝐹𝑇 (𝑡𝑛+1 ) 𝑦𝑛+1 󵄩󵄩󵄩 ,
󵄩
󵄩
+ 𝛼𝑛 󵄩󵄩󵄩𝛾𝑉𝑥𝑛+1 − 𝜇𝐹𝑇 (𝜇𝑛 ) 𝑦𝑛 󵄩󵄩󵄩
󵄩
󵄩
+ 󵄩󵄩󵄩𝑇 (𝜇𝑛+1 ) 𝑦𝑛+1 − 𝑇 (𝜇𝑛+1 ) 𝑦𝑛 󵄩󵄩󵄩
󵄩
󵄩
+ 󵄩󵄩󵄩𝑇 (𝜇𝑛+1 ) 𝑦𝑛 − 𝑇 (𝜇𝑛 ) 𝑦𝑛 󵄩󵄩󵄩
󵄩
󵄩
≤ (𝛼𝑛+1 + 𝛼𝑛 ) 𝑀1 + 󵄩󵄩󵄩𝑦𝑛+1 − 𝑦𝑛 󵄩󵄩󵄩
󵄩
󵄩
+ 󵄩󵄩󵄩𝑇 (𝜇𝑛+1 ) 𝑦𝑛 − 𝑇 (𝜇𝑛 ) 𝑦𝑛 󵄩󵄩󵄩 .
for which (52) follows by (53). Since 𝑢𝑛 = 𝐺𝑛𝑁𝑥𝑛 and 𝑢𝑛+1 =
𝑁
𝐺𝑛+1
𝑥𝑛+1 , we have
󵄩󵄩
󵄩
󵄩󵄩𝑢𝑛 − 𝑢𝑛+1 󵄩󵄩󵄩
󵄩
󵄩
𝑁
= 󵄩󵄩󵄩󵄩𝐺𝑛𝑁𝑥𝑛 − 𝐺𝑛+1
𝑥𝑛+1 󵄩󵄩󵄩󵄩
󵄩
󵄩 󵄩 𝑁
󵄩
𝑁
𝑁
≤ 󵄩󵄩󵄩󵄩𝐺𝑛𝑁𝑥𝑛 − 𝐺𝑛+1
𝑥𝑛 󵄩󵄩󵄩󵄩 + 󵄩󵄩󵄩󵄩𝐺𝑛+1
𝑥𝑛 − 𝐺𝑛+1
𝑥𝑛+1 󵄩󵄩󵄩󵄩
󵄩
󵄩 󵄩
󵄩
𝑁
≤ 󵄩󵄩󵄩󵄩𝐺𝑛𝑁𝑥𝑛 − 𝐺𝑛+1
𝑥𝑛 󵄩󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑥𝑛 − 𝑥𝑛+1 󵄩󵄩󵄩 .
󵄩
󵄩
≤ 𝛼𝑛+1 󵄩󵄩󵄩𝛾𝑉𝑥𝑛+1 − 𝜇𝐹𝑇 (𝜇𝑛+1 ) 𝑦𝑛+1 󵄩󵄩󵄩
󵄩󵄩
󵄩
󵄩󵄩𝑧𝑛 − 𝑇 (𝜇𝑛 ) 𝑦𝑛 󵄩󵄩󵄩
󵄩
󵄩
= 󵄩󵄩󵄩𝑃𝐶 [𝛼𝑛 𝛾𝑉𝑥𝑛 + (𝐼 − 𝛼𝑛 𝜇𝐹) 𝑇 (𝜇𝑛 ) 𝑦𝑛 ] − 𝑃𝐶𝑇 (𝜇𝑛 ) 𝑦𝑛 󵄩󵄩󵄩
󵄩
󵄩
≤ 𝛼𝑛 󵄩󵄩󵄩𝛾𝑉𝑥𝑛 − 𝜇𝐹𝑇 (𝜇𝑛 ) 𝑦𝑛 󵄩󵄩󵄩 󳨀→ 0 as 𝑛 󳨀→ ∞.
(68)
From (66) and (68), we have
(62)
󵄩󵄩
󵄩 󵄩
󵄩
󵄩󵄩𝑥𝑛 − 𝑇 (𝜇𝑛 ) 𝑦𝑛 󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩𝑥𝑛 − 𝑧𝑛 󵄩󵄩󵄩
󵄩
󵄩
+ 󵄩󵄩󵄩𝑧𝑛 − 𝑇 (𝜇𝑛 ) 𝑦𝑛 󵄩󵄩󵄩 󳨀→ 0 as 𝑛 󳨀→ ∞.
(69)
Set 𝑧𝑛 = 𝑃𝐶V𝑛 , where V𝑛 = 𝛼𝑛 𝛾𝑉𝑥𝑛 + (𝐼 − 𝛼𝑛 𝜇𝐹)𝑇(𝜇𝑛 )𝑦𝑛 . From
(25), we have
󵄩󵄩
∗󵄩
∗
∗
󵄩󵄩𝑧𝑛 − 𝑥 󵄩󵄩󵄩 = ⟨V𝑛 − 𝑥 , 𝑧𝑛 − 𝑥 ⟩
+ ⟨𝑃𝐶V𝑛 − V𝑛 , 𝑃𝐶V𝑛 − 𝑥∗ ⟩
≤ ⟨V𝑛 − 𝑥∗ , 𝑧𝑛 − 𝑥∗ ⟩
= 𝛼𝑛 ⟨𝛾𝑉𝑥𝑛 − 𝜇𝐹𝑥∗ , 𝑧𝑛 − 𝑥∗ ⟩
+ ⟨(𝐼 − 𝛼𝑛 𝜇𝐹) (𝑇 (𝜇𝑛 ) 𝑦𝑛 − 𝑥∗ ) , 𝑧𝑛 − 𝑥∗ ⟩
Journal of Applied Mathematics
9
󵄩󵄩
󵄩
󵄩
≤ (1 − 𝛼𝑛 𝜏) 󵄩󵄩󵄩𝑦𝑛 − 𝑥∗ 󵄩󵄩󵄩 󵄩󵄩󵄩𝑧𝑛 − 𝑥∗ 󵄩󵄩󵄩
From (72) and (75), we have
󵄩󵄩
∗ 󵄩2
󵄩󵄩𝑥𝑛+1 − 𝑥 󵄩󵄩󵄩
+ 𝛼𝑛 ⟨𝛾𝑉𝑥𝑛 − 𝜇𝐹𝑥∗ , 𝑧𝑛 − 𝑥∗ ⟩
≤
󵄩2
󵄩
󵄩
󵄩2
≤ 𝛽𝑛 󵄩󵄩󵄩𝑥𝑛 − 𝑥∗ 󵄩󵄩󵄩 + (1 − 𝛽𝑛 ) 󵄩󵄩󵄩󵄩𝐺𝑛𝑁𝑥𝑛 − 𝑥∗ 󵄩󵄩󵄩󵄩
(1 − 𝛼𝑛 𝜏) 󵄩󵄩
󵄩2 󵄩
󵄩2
(󵄩󵄩𝑦𝑛 − 𝑥∗ 󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑧𝑛 − 𝑥∗ 󵄩󵄩󵄩 )
2
+ 2𝛼𝑛 (1 − 𝛽𝑛 ) 𝑀2
+ 𝛼𝑛 ⟨𝛾𝑉𝑥𝑛 − 𝜇𝐹𝑥∗ , 𝑧𝑛 − 𝑥∗ ⟩
(1 − 𝛼𝑛 𝜏) 󵄩󵄩
∗ 󵄩2
≤
󵄩󵄩𝑦𝑛 − 𝑥 󵄩󵄩󵄩 +
2
󵄩2
󵄩
󵄩
󵄩2
≤ 𝛽𝑛 󵄩󵄩󵄩𝑥𝑛 − 𝑥∗ 󵄩󵄩󵄩 + (1 − 𝛽𝑛 ) 󵄩󵄩󵄩󵄩𝐺𝑛𝑘 𝑥𝑛 − 𝑥∗ 󵄩󵄩󵄩󵄩
1 󵄩󵄩
∗ 󵄩2
󵄩𝑧 − 𝑥 󵄩󵄩󵄩
2󵄩 𝑛
+ 2𝛼𝑛 (1 − 𝛽𝑛 ) 𝑀2 .
+ 𝛼𝑛 ⟨𝛾𝑉𝑥𝑛 − 𝜇𝐹𝑥∗ , 𝑧𝑛 − 𝑥∗ ⟩ .
(70)
It follows that
󵄩󵄩
󵄩
∗ 󵄩2
∗ 󵄩2
∗
∗
󵄩󵄩𝑧𝑛 − 𝑥 󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩𝑦𝑛 − 𝑥 󵄩󵄩󵄩 + 2𝛼𝑛 ⟨𝛾𝑉𝑥𝑛 − 𝜇𝐹𝑥 , 𝑧𝑛 − 𝑥 ⟩
󵄩2
󵄩
󵄩󵄩
󵄩
󵄩
≤ 󵄩󵄩󵄩𝑦𝑛 − 𝑥∗ 󵄩󵄩󵄩 + 2𝛼𝑛 󵄩󵄩󵄩𝛾𝑉𝑥𝑛 − 𝜇𝐹𝑥∗ 󵄩󵄩󵄩 󵄩󵄩󵄩𝑧𝑛 − 𝑥∗ 󵄩󵄩󵄩 .
(71)
󵄩󵄩
∗ 󵄩2
󵄩󵄩𝑥𝑛+1 − 𝑥 󵄩󵄩󵄩
󵄩
󵄩2
≤ 𝛽𝑛 󵄩󵄩󵄩𝑥𝑛 − 𝑥∗ 󵄩󵄩󵄩 + (1 − 𝛽𝑛 )
󵄩2
󵄩
󵄩
󵄩2
× {󵄩󵄩󵄩𝑥𝑛 − 𝑥∗ 󵄩󵄩󵄩 + 𝑟𝑘,𝑛 (𝑟𝑘,𝑛 − 2𝜇𝑘 ) 󵄩󵄩󵄩󵄩Ψ𝑘 𝐺𝑛𝑘−1 𝑥𝑛 − Ψ𝑘 𝑥∗ 󵄩󵄩󵄩󵄩 }
󵄩2
󵄩
= 󵄩󵄩󵄩𝑥𝑛 − 𝑥∗ 󵄩󵄩󵄩 + (1 − 𝛽𝑛 ) 𝑟𝑘,𝑛 (𝑟𝑘,𝑛 − 2𝜇𝑘 )
󵄩󵄩
∗ 󵄩2
󵄩󵄩𝑥𝑛+1 − 𝑥 󵄩󵄩󵄩
󵄩
󵄩2
= 󵄩󵄩󵄩𝛽𝑛 (𝑥𝑛 − 𝑥∗ ) + (1 − 𝛽𝑛 ) (𝑧𝑛 − 𝑥∗ )󵄩󵄩󵄩
󵄩
󵄩2
󵄩2
󵄩
≤ 𝛽𝑛 󵄩󵄩󵄩𝑥𝑛 − 𝑥∗ 󵄩󵄩󵄩 + (1 − 𝛽𝑛 ) 󵄩󵄩󵄩𝑧𝑛 − 𝑥∗ 󵄩󵄩󵄩
󵄩
󵄩2
≤ 𝛽𝑛 󵄩󵄩󵄩𝑥𝑛 − 𝑥∗ 󵄩󵄩󵄩 + (1 − 𝛽𝑛 )
󵄩2
󵄩
󵄩󵄩
󵄩
󵄩
× {󵄩󵄩󵄩𝑦𝑛 − 𝑥∗ 󵄩󵄩󵄩 + 2𝛼𝑛 󵄩󵄩󵄩𝛾𝑉𝑥𝑛 − 𝜇𝐹𝑥∗ 󵄩󵄩󵄩 󵄩󵄩󵄩𝑧𝑛 − 𝑥∗ 󵄩󵄩󵄩}
󵄩2
󵄩
× 󵄩󵄩󵄩󵄩Ψ𝑘 𝐺𝑛𝑘−1 𝑥𝑛 − Ψ𝑘 𝑥∗ 󵄩󵄩󵄩󵄩
+ 2𝛼𝑛 (1 − 𝛽𝑛 ) 𝑀2 ,
(77)
which in turn implies that
󵄩
󵄩2
󵄩2
󵄩
≤ 𝛽𝑛 󵄩󵄩󵄩𝑥𝑛 − 𝑥∗ 󵄩󵄩󵄩 + (1 − 𝛽𝑛 ) 󵄩󵄩󵄩𝑦𝑛 − 𝑥∗ 󵄩󵄩󵄩 + 2𝛼𝑛 (1 − 𝛽𝑛 ) 𝑀2 ,
(72)
where 𝑀2 > 0 is an appropriate constant such that 𝑀2 =
sup𝑛≥1 {‖𝛾𝑉𝑥𝑛 − 𝜇𝐹𝑥∗ ‖‖𝑧𝑛 − 𝑥∗ ‖}.
Next, we show that
󵄩
󵄩
lim 󵄩󵄩󵄩𝐺𝑛𝑘−1 𝑥𝑛 − 𝐺𝑛𝑘 𝑥𝑛 󵄩󵄩󵄩󵄩 = 0, ∀𝑘 ∈ {1, 2, . . . , 𝑁} . (73)
𝑛→∞ 󵄩
From (28), we have
󵄩󵄩 𝑘
󵄩2
󵄩󵄩𝐺𝑛 𝑥𝑛 − 𝑥∗ 󵄩󵄩󵄩
󵄩
󵄩
󵄩󵄩 𝑘 ,𝜑𝑘 )
󵄩󵄩2
𝑘−1
(Θ𝑘 ,𝜑𝑘 )
∗󵄩
= 󵄩󵄩󵄩𝑆𝑟(Θ
(𝐼
−
𝑟
Ψ
)
𝐺
𝑥
−
𝑆
(𝐼
−
𝑟
Ψ
)
𝑥
󵄩󵄩
𝑘,𝑛
𝑘
𝑛
𝑘,𝑛
𝑘
𝑛
𝑟
𝑘,𝑛
󵄩
󵄩 𝑘,𝑛
2
󵄩
󵄩󵄩
≤ 󵄩󵄩󵄩(𝐼 − 𝑟𝑘,𝑛 Ψ𝑘 ) 𝐺𝑛𝑘−1 − (𝐼 − 𝑟𝑘,𝑛 Ψ𝑘 ) 𝑥∗ 󵄩󵄩󵄩󵄩
󵄩2
󵄩2
󵄩
󵄩
≤ 󵄩󵄩󵄩󵄩𝐺𝑛𝑘−1 𝑥𝑛 − 𝑥∗ 󵄩󵄩󵄩󵄩 + 𝑟𝑘,𝑛 (𝑟𝑘,𝑛 − 2𝜇𝑘 ) 󵄩󵄩󵄩󵄩Ψ𝑘 𝐺𝑛𝑘−1 𝑥𝑛 − Ψ𝑘 𝑥∗ 󵄩󵄩󵄩󵄩
󵄩2
󵄩
󵄩2
󵄩
≤ 󵄩󵄩󵄩𝑥𝑛 − 𝑥∗ 󵄩󵄩󵄩 + 𝑟𝑘,𝑛 (𝑟𝑘,𝑛 − 2𝜇𝑘 ) 󵄩󵄩󵄩󵄩Ψ𝑘 𝐺𝑛𝑘−1 𝑥𝑛 − Ψ𝑘 𝑥∗ 󵄩󵄩󵄩󵄩 .
(74)
󵄩2 󵄩
󵄩2
󵄩
= 󵄩󵄩󵄩󵄩𝐺𝑛𝑁𝑥𝑛 − 𝑥∗ 󵄩󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩󵄩𝐺𝑛𝑘 𝑥𝑛 − 𝑥∗ 󵄩󵄩󵄩󵄩 .
Substituting (74) into (72), we have
+ 2𝛼𝑛 (1 − 𝛽𝑛 ) 𝑀2
By the convexity of ‖ ⋅ ‖2 and (71), we have
From (42), for all 𝑘 ∈ {1, 2, . . . , 𝑁}, we note that
󵄩 𝑀
󵄩
󵄩󵄩
∗ 󵄩2
∗󵄩
∗ 󵄩2
󵄩󵄩𝑦𝑛 − 𝑥 󵄩󵄩󵄩 = 󵄩󵄩󵄩󵄩𝑄 𝑢𝑛 − 𝑥 󵄩󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩𝑢𝑛 − 𝑥 󵄩󵄩󵄩
(76)
(75)
󵄩2
󵄩
(1 − 𝛽𝑛 ) 𝑟𝑘,𝑛 (2𝜇𝑘 − 𝑟𝑘,𝑛 ) 󵄩󵄩󵄩󵄩Ψ𝑘 𝐺𝑛𝑘−1 𝑥𝑛 − Ψ𝑘 𝑥∗ 󵄩󵄩󵄩󵄩
󵄩2 󵄩
󵄩2
󵄩
≤ 󵄩󵄩󵄩𝑥𝑛 − 𝑥∗ 󵄩󵄩󵄩 − 󵄩󵄩󵄩𝑥𝑛+1 − 𝑥∗ 󵄩󵄩󵄩 + 2𝛼𝑛 (1 − 𝛽𝑛 ) 𝑀2
󵄩 󵄩
󵄩 󵄩
󵄩
󵄩
≤ (󵄩󵄩󵄩𝑥𝑛 − 𝑥∗ 󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑥𝑛+1 − 𝑥∗ 󵄩󵄩󵄩) 󵄩󵄩󵄩𝑥𝑛+1 − 𝑥𝑛 󵄩󵄩󵄩
(78)
+ 2𝛼𝑛 (1 − 𝛽𝑛 ) 𝑀2 .
Since lim inf 𝑛 → ∞ (1 − 𝛽𝑛 ) > 0, 0 < 𝑟𝑘,𝑛 < 2𝜇𝑘 , for all 𝑘 ∈
{1, 2, . . . , 𝑁}, from (𝐶1) and (67), we obtain that
󵄩
󵄩
lim 󵄩󵄩󵄩Ψ𝑘 𝐺𝑛𝑘−1 𝑥𝑛 − Ψ𝑘 𝑥∗ 󵄩󵄩󵄩󵄩 = 0, ∀𝑘 ∈ {1, 2, . . . , 𝑁} . (79)
𝑛→∞ 󵄩
On the other hand, from Lemma 3 and (26), we have
󵄩󵄩 𝑘
󵄩2
󵄩󵄩𝐺𝑛 𝑥𝑛 − 𝑥∗ 󵄩󵄩󵄩
󵄩
󵄩
󵄩󵄩2
󵄩󵄩 𝑘 ,𝜑𝑘 )
𝑘 ,𝜑𝑘 )
= 󵄩󵄩󵄩𝑆𝑟(Θ
(𝐼 − 𝑟𝑘,𝑛 Ψ𝑘 ) 𝐺𝑛𝑘−1 𝑥𝑛 − 𝑆𝑟(Θ
(𝐼 − 𝑟𝑘,𝑛 Ψ𝑘 ) 𝑥∗ 󵄩󵄩󵄩
𝑘,𝑛
󵄩
󵄩 𝑘,𝑛
≤ ⟨(𝐼 − 𝑟𝑘,𝑛 Ψ𝑘 ) 𝐺𝑛𝑘−1 𝑥𝑛 − (𝐼 − 𝑟𝑘,𝑛 Ψ𝑘 ) 𝑥∗ , 𝐺𝑛𝑘 𝑥𝑛 − 𝑥∗ ⟩
=
1 󵄩󵄩
󵄩2
(󵄩󵄩󵄩(𝐼 − 𝑟𝑘,𝑛 Ψ𝑘 ) 𝐺𝑛𝑘−1 𝑥𝑛 − (𝐼 − 𝑟𝑘,𝑛 Ψ𝑘 ) 𝑥∗ 󵄩󵄩󵄩󵄩
2
󵄩2
󵄩
+ 󵄩󵄩󵄩󵄩𝐺𝑛𝑘 𝑥𝑛 − 𝑥∗ 󵄩󵄩󵄩󵄩
󵄩
− 󵄩󵄩󵄩󵄩(𝐼 − 𝑟𝑘,𝑛 Ψ𝑘 ) 𝐺𝑛𝑘−1 𝑥𝑛 − (𝐼 − 𝑟𝑘,𝑛 Ψ𝑘 ) 𝑥∗
󵄩2
− (𝐺𝑛𝑘 𝑥𝑛 − 𝑥∗ )󵄩󵄩󵄩󵄩 )
10
Journal of Applied Mathematics
≤
1 󵄩󵄩 𝑘−1
󵄩2 󵄩
󵄩2
(󵄩󵄩𝐺 𝑥 − 𝑥∗ 󵄩󵄩󵄩󵄩 + 󵄩󵄩󵄩󵄩𝐺𝑛𝑘 − 𝑥∗ 󵄩󵄩󵄩󵄩
2 󵄩 𝑛 𝑛
󵄩
󵄩2
−󵄩󵄩󵄩󵄩𝐺𝑛𝑘−1 𝑥𝑛 − 𝐺𝑛𝑘 𝑥𝑛 − 𝑟𝑘,𝑛 (Ψ𝑘 𝐺𝑛𝑘−1 𝑥𝑛 − Ψ𝑘 𝑥∗ )󵄩󵄩󵄩󵄩 ) ,
(80)
󵄩 󵄩
󵄩 󵄩
󵄩
󵄩
≤ (󵄩󵄩󵄩𝑥𝑛 − 𝑥∗ 󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑥𝑛+1 − 𝑥∗ 󵄩󵄩󵄩) 󵄩󵄩󵄩𝑥𝑛+1 − 𝑥𝑛 󵄩󵄩󵄩
󵄩󵄩
󵄩
󵄩
+ 2𝑟𝑘,𝑛 (1 − 𝛽𝑛 ) 󵄩󵄩󵄩󵄩𝐺𝑛𝑘−1 𝑥𝑛 − 𝐺𝑛𝑘 𝑥𝑛 󵄩󵄩󵄩󵄩 󵄩󵄩󵄩󵄩Ψ𝑘 𝐺𝑛𝑘−1 𝑥𝑛 − Ψ𝑘 𝑥∗ 󵄩󵄩󵄩󵄩
+ 2𝛼𝑛 (1 − 𝛽𝑛 ) 𝑀2 .
(83)
which in turn implies that
󵄩2
󵄩󵄩 𝑘
󵄩󵄩𝐺𝑛 𝑥𝑛 − 𝑥∗ 󵄩󵄩󵄩
󵄩
󵄩
󵄩2
󵄩󵄩 𝑘−1
≤ 󵄩󵄩󵄩𝐺𝑛 𝑥𝑛 − 𝑥∗ 󵄩󵄩󵄩󵄩
󵄩
󵄩2
− 󵄩󵄩󵄩󵄩𝐺𝑛𝑘−1 𝑥𝑛 − 𝐺𝑛𝑘 − 𝑟𝑘,𝑛 (Ψ𝑘 𝐺𝑛𝑘−1 − Ψ𝑘 𝑥∗ )󵄩󵄩󵄩󵄩
󵄩2 󵄩
󵄩
󵄩
= 󵄩󵄩󵄩󵄩𝐺𝑛𝑘−1 𝑥𝑛 − 𝑥∗ 󵄩󵄩󵄩󵄩 − 󵄩󵄩󵄩󵄩𝐺𝑛𝑘−1 𝑥𝑛 − 𝐺𝑛𝑘 𝑥𝑛 󵄩󵄩󵄩󵄩
󵄩2
2 󵄩
󵄩󵄩󵄩Ψ𝑘 𝐺𝑛𝑘−1 𝑥𝑛 − Ψ𝑘 𝑥∗ 󵄩󵄩󵄩
− 𝑟𝑘,𝑛
󵄩
󵄩
Since lim inf 𝑛 → ∞ (1 − 𝛽𝑛 ) > 0, from (𝐶1), (67), and (79), we
obtain that (73) holds. Consequently,
󵄩󵄩 0
󵄩
󵄩󵄩𝐺𝑛 − 𝐺𝑛𝑁𝑥𝑛 󵄩󵄩󵄩
󵄩
󵄩
󵄩󵄩 0
󵄩
󵄩
1 󵄩
≤ 󵄩󵄩󵄩𝐺𝑛 𝑥𝑛 − 𝐺𝑛 𝑥𝑛 󵄩󵄩󵄩󵄩 + 󵄩󵄩󵄩󵄩𝐺𝑛1 𝑥𝑛 − 𝐺𝑛2 𝑥𝑛 󵄩󵄩󵄩󵄩 + ⋅ ⋅ ⋅
󵄩
󵄩
+ 󵄩󵄩󵄩󵄩𝐺𝑛𝑁−1 𝑥𝑛 − 𝐺𝑛𝑁𝑥𝑛 󵄩󵄩󵄩󵄩 󳨀→ 0 as 𝑛 󳨀→ ∞.
(84)
󵄩
󵄩󵄩
󵄩󵄩𝑥𝑛 − 𝑢𝑛 󵄩󵄩󵄩 =
(81)
+ 2𝑟𝑘,𝑛 ⟨𝐺𝑛𝑘−1 𝑥𝑛 − 𝐺𝑛𝑘 𝑥𝑛 , Ψ𝑘 𝐺𝑛𝑘−1 𝑥𝑛 − Ψ𝑘 𝑥∗ ⟩
Next, we show that
󵄩
󵄩
lim 󵄩󵄩𝐴 𝑖 𝑄𝑖−1 𝑢𝑛 − 𝐴 𝑖 𝑄𝑖−1 𝑥∗ 󵄩󵄩󵄩󵄩 = 0,
󵄩
𝑛→∞ 󵄩
󵄩2 󵄩
󵄩2
󵄩
≤ 󵄩󵄩󵄩󵄩𝐺𝑛𝑘−1 𝑥𝑛 − 𝑥∗ 󵄩󵄩󵄩󵄩 − 󵄩󵄩󵄩󵄩𝐺𝑛𝑘−1 𝑥𝑛 − 𝐺𝑛𝑘 𝑥𝑛 󵄩󵄩󵄩󵄩
󵄩
󵄩󵄩
󵄩
+ 2𝑟𝑘,𝑛 󵄩󵄩󵄩󵄩𝐺𝑛𝑘−1 𝑥𝑛 − 𝐺𝑛𝑘 𝑥𝑛 󵄩󵄩󵄩󵄩 󵄩󵄩󵄩󵄩Ψ𝑘 𝐺𝑛𝑘−1 𝑥𝑛 − Ψ𝑘 𝑥∗ 󵄩󵄩󵄩󵄩
󵄩2
󵄩
󵄩2 󵄩
≤ 󵄩󵄩󵄩𝑥𝑛 − 𝑥∗ 󵄩󵄩󵄩 − 󵄩󵄩󵄩󵄩𝐺𝑛𝑘−1 𝑥𝑛 − 𝐺𝑛𝑘 𝑥𝑛 󵄩󵄩󵄩󵄩
󵄩
󵄩󵄩
󵄩
+ 2𝑟𝑘,𝑛 󵄩󵄩󵄩󵄩𝐺𝑛𝑘−1 𝑥𝑛 − 𝐺𝑛𝑘 𝑥𝑛 󵄩󵄩󵄩󵄩 󵄩󵄩󵄩󵄩Ψ𝑘 𝐺𝑛𝑘−1 𝑥𝑛 − Ψ𝑘 𝑥∗ 󵄩󵄩󵄩󵄩 .
∀𝑖 ∈ {1, 2, . . . , 𝑀} .
(85)
From (28), we have
󵄩󵄩 𝑀
󵄩2
󵄩󵄩𝑄 𝑢𝑛 − 𝑄𝑀𝑥∗ 󵄩󵄩󵄩
󵄩
󵄩
󵄩2
󵄩󵄩 𝐺𝑀
𝐺
= 󵄩󵄩󵄩𝑆𝜆 (𝐼 − 𝜆 𝑀𝐴 𝑀)𝑄𝑀−1 𝑢𝑛 − 𝑆𝜆 𝑀 (𝐼 − 𝜆 𝑀𝐴 𝑀)𝑄𝑀−1 𝑥∗ 󵄩󵄩󵄩󵄩
𝑀
𝑀
󵄩2
󵄩
≤ 󵄩󵄩󵄩󵄩(𝐼 − 𝜆 𝑀𝐴 𝑀)𝑄𝑀−1 𝑢𝑛 − (𝐼 − 𝜆 𝑀𝐴 𝑀) 𝑄𝑀−1 𝑥∗ 󵄩󵄩󵄩󵄩
󵄩2
󵄩
≤ 󵄩󵄩󵄩󵄩𝑄𝑀−1 𝑢𝑛 − 𝑄𝑀−1 𝑥∗ 󵄩󵄩󵄩󵄩 + 𝜆 𝑀 (𝜆 𝑀 − 2𝛼𝑀)
󵄩2
󵄩
× 󵄩󵄩󵄩󵄩𝐴 𝑀𝑄𝑀−1 𝑢𝑛 − 𝐴 𝑀𝑄𝑀−1 𝑥∗ 󵄩󵄩󵄩󵄩 .
(86)
Substituting (81) into (76), we have
󵄩󵄩
∗ 󵄩2
󵄩󵄩𝑥𝑛+1 − 𝑥 󵄩󵄩󵄩
󵄩
󵄩2
≤ 𝛽𝑛 󵄩󵄩󵄩𝑥𝑛 − 𝑥∗ 󵄩󵄩󵄩 + (1 − 𝛽𝑛 )
󵄩2
󵄩
󵄩2 󵄩
× {󵄩󵄩󵄩𝑥𝑛 − 𝑥∗ 󵄩󵄩󵄩 − 󵄩󵄩󵄩󵄩𝐺𝑛𝑘−1 𝑥𝑛 − 𝐺𝑛𝑘 𝑥𝑛 󵄩󵄩󵄩󵄩
󵄩
󵄩󵄩
󵄩
+2𝑟𝑘,𝑛 󵄩󵄩󵄩󵄩𝐺𝑛𝑘−1 𝑥𝑛 − 𝐺𝑛𝑘 𝑥𝑛 󵄩󵄩󵄩󵄩 󵄩󵄩󵄩󵄩Ψ𝑘 𝐺𝑛𝑘−1 𝑥𝑛 − Ψ𝑘 𝑥∗ 󵄩󵄩󵄩󵄩 }
By induction, we have
+ 2𝛼𝑛 (1 − 𝛽𝑛 ) 𝑀2
󵄩2
󵄩󵄩 𝑀
󵄩󵄩𝑄 𝑢𝑛 − 𝑄𝑀𝑥∗ 󵄩󵄩󵄩
󵄩
󵄩
󵄩
󵄩
󵄩2
󵄩
= 󵄩󵄩󵄩𝑥𝑛 − 𝑥∗ 󵄩󵄩󵄩 − (1 − 𝛽𝑛 ) 󵄩󵄩󵄩󵄩𝐺𝑛𝑘−1 𝑥𝑛 − 𝐺𝑛𝑘 𝑥𝑛 󵄩󵄩󵄩󵄩
󵄩󵄩
󵄩
󵄩
+ 2𝑟𝑘,𝑛 (1 − 𝛽𝑛 ) 󵄩󵄩󵄩󵄩𝐺𝑛𝑘−1 𝑥𝑛 − 𝐺𝑛𝑘 𝑥𝑛 󵄩󵄩󵄩󵄩 󵄩󵄩󵄩󵄩Ψ𝑘 𝐺𝑛𝑘−1 𝑥𝑛 − Ψ𝑘 𝑥∗ 󵄩󵄩󵄩󵄩
+ 2𝛼𝑛 (1 − 𝛽𝑛 ) 𝑀2 ,
(82)
𝑀
󵄩2
󵄩
󵄩
󵄩
≤ 󵄩󵄩󵄩𝑢𝑛 − 𝑥∗ 󵄩󵄩󵄩 + ∑𝜆 𝑖 (𝜆 𝑖 − 2𝛼𝑖 ) 󵄩󵄩󵄩󵄩𝐴 𝑖 𝑄𝑖−1 𝑢𝑛 − 𝐴 𝑖 𝑄𝑖−1 𝑥∗ 󵄩󵄩󵄩󵄩
𝑖=1
𝑀
󵄩2
󵄩
󵄩
󵄩
≤ 󵄩󵄩󵄩𝑥𝑛 − 𝑥∗ 󵄩󵄩󵄩 + ∑𝜆 𝑖 (𝜆 𝑖 − 2𝛼𝑖 ) 󵄩󵄩󵄩󵄩𝐴 𝑖 𝑄𝑖−1 𝑢𝑛 − 𝐴 𝑖 𝑄𝑖−1 𝑥∗ 󵄩󵄩󵄩󵄩 .
𝑖=1
(87)
which in turn implies that
󵄩
󵄩
(1 − 𝛽𝑛 ) 󵄩󵄩󵄩󵄩𝐺𝑛𝑘−1 𝑥𝑛 − 𝐺𝑛𝑘 𝑥𝑛 󵄩󵄩󵄩󵄩
󵄩
󵄩2 󵄩
󵄩2
≤ 󵄩󵄩󵄩𝑥𝑛 − 𝑥∗ 󵄩󵄩󵄩 − 󵄩󵄩󵄩𝑥𝑛+1 − 𝑥∗ 󵄩󵄩󵄩
󵄩󵄩
󵄩
󵄩
+ 2𝑟𝑘,𝑛 (1 − 𝛽𝑛 ) 󵄩󵄩󵄩󵄩𝐺𝑛𝑘−1 𝑥𝑛 − 𝐺𝑛𝑘 𝑥𝑛 󵄩󵄩󵄩󵄩 󵄩󵄩󵄩󵄩Ψ𝑘 𝐺𝑛𝑘−1 𝑥𝑛 − Ψ𝑘 𝑥∗ 󵄩󵄩󵄩󵄩
+ 2𝛼𝑛 (1 − 𝛽𝑛 ) 𝑀2
From (72) and (75), we have
󵄩 𝑀
󵄩󵄩
󵄩
∗ 󵄩2
∗ 󵄩2
𝑀 ∗ 󵄩2
󵄩󵄩𝑥𝑛+1 − 𝑥 󵄩󵄩󵄩 ≤ 𝛽𝑛 󵄩󵄩󵄩𝑥𝑛 − 𝑥 󵄩󵄩󵄩 + (1 − 𝛽𝑛 ) 󵄩󵄩󵄩󵄩𝑄 𝑢𝑛 − 𝑄 𝑥 󵄩󵄩󵄩󵄩
+ 2𝛼𝑛 (1 − 𝛽𝑛 ) 𝑀2 .
(88)
Journal of Applied Mathematics
11
Substituting (87) into (88), we have
≤
󵄩󵄩
∗ 󵄩2
󵄩󵄩𝑥𝑛+1 − 𝑥 󵄩󵄩󵄩
󵄩
󵄩2
≤ 𝛽𝑛 󵄩󵄩󵄩𝑥𝑛 − 𝑥∗ 󵄩󵄩󵄩 + (1 − 𝛽𝑛 )
(91)
󵄩2
󵄩
× {󵄩󵄩󵄩𝑥𝑛 − 𝑥∗ 󵄩󵄩󵄩
𝑀
󵄩2
󵄩
+∑𝜆 𝑖 (𝜆 𝑖 − 2𝛼𝑖 ) 󵄩󵄩󵄩󵄩𝐴 𝑖 𝑄𝑖−1 𝑢𝑛 − 𝐴 𝑖 𝑄𝑖−1 𝑥∗ 󵄩󵄩󵄩󵄩 }
𝑖=1
+ 2𝛼𝑛 (1 − 𝛽𝑛 ) 𝑀2
󵄩2
󵄩
= 󵄩󵄩󵄩𝑥𝑛 − 𝑥∗ 󵄩󵄩󵄩
𝑀
󵄩2
󵄩
+ (1 − 𝛽𝑛 ) ∑𝜆 𝑖 (𝜆 𝑖 − 2𝛼𝑖 ) 󵄩󵄩󵄩󵄩𝐴 𝑖 𝑄𝑖−1 𝑢𝑛 − 𝐴 𝑖 𝑄𝑖−1 𝑥∗ 󵄩󵄩󵄩󵄩
𝑖=1
+ 2𝛼𝑛 (1 − 𝛽𝑛 ) 𝑀2 ,
(89)
which in turn implies that
󵄩2
󵄩󵄩 𝑀
󵄩󵄩𝑄 𝑢𝑛 − 𝑄𝑀𝑥∗ 󵄩󵄩󵄩
󵄩
󵄩
󵄩2
󵄩
≤ 󵄩󵄩󵄩󵄩𝑄𝑀−1 𝑢𝑛 − 𝑄𝑀−1 𝑥∗ 󵄩󵄩󵄩󵄩
󵄩
− 󵄩󵄩󵄩󵄩𝑄𝑀−1 𝑢𝑛 − 𝑄𝑀𝑢𝑛 + 𝑄𝑀𝑥∗ − 𝑄𝑀−1 𝑥∗
󵄩2
−𝜆 𝑀(𝐴 𝑀𝑄𝑀−1 𝑢𝑛 − 𝐴 𝑀𝑄𝑀−1 𝑥∗ )󵄩󵄩󵄩󵄩
󵄩2
󵄩
= 󵄩󵄩󵄩󵄩𝑄𝑀−1 𝑢𝑛 − 𝑄𝑀−1 𝑥∗ 󵄩󵄩󵄩󵄩
󵄩2
󵄩
− 󵄩󵄩󵄩󵄩𝑄𝑀−1 𝑢𝑛 − 𝑄𝑀𝑢𝑛 − 𝑄𝑀𝑥∗ − 𝑄𝑀−1 𝑥∗ 󵄩󵄩󵄩󵄩
󵄩
󵄩2
− 𝜆2𝑀󵄩󵄩󵄩󵄩𝐴 𝑀𝑄𝑀−1 𝑢𝑛 − 𝐴 𝑀𝑄𝑀−1 𝑥∗ 󵄩󵄩󵄩󵄩
(92)
+ 2𝜆 𝑀 ⟨𝑄𝑀−1 𝑢𝑛 − 𝑄𝑀𝑢𝑛 + 𝑄𝑀𝑥∗ − 𝑄𝑀−1 𝑥∗ ,
which in turn implies that
𝐴 𝑀𝑄𝑀−1 𝑢𝑛 − 𝐴 𝑀𝑄𝑀−1 𝑥∗ ⟩
𝑀
󵄩2
󵄩
(1 − 𝛽𝑛 ) ∑𝜆 𝑖 (2𝛼𝑖 − 𝜆 𝑖 ) 󵄩󵄩󵄩󵄩𝐴 𝑖 𝑄𝑖−1 𝑢𝑛 − 𝐴 𝑖 𝑄𝑖−1 𝑥∗ 󵄩󵄩󵄩󵄩
󵄩2
󵄩
≤ 󵄩󵄩󵄩󵄩𝑄𝑀−1 𝑢𝑛 − 𝑄𝑀−1 𝑥∗ 󵄩󵄩󵄩󵄩
𝑖=1
󵄩2 󵄩
󵄩2
󵄩
≤ 󵄩󵄩󵄩𝑥𝑛 − 𝑥∗ 󵄩󵄩󵄩 − 󵄩󵄩󵄩𝑥𝑛+1 − 𝑥∗ 󵄩󵄩󵄩 + 2𝛼𝑛 (1 − 𝛽𝑛 ) 𝑀2
(90)
󵄩 󵄩
󵄩 󵄩
󵄩
󵄩
≤ (󵄩󵄩󵄩𝑥𝑛 − 𝑥∗ 󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑥𝑛+1 − 𝑥∗ 󵄩󵄩󵄩) 󵄩󵄩󵄩𝑥𝑛+1 − 𝑥𝑛 󵄩󵄩󵄩
+ 2𝛼𝑛 (1 − 𝛽𝑛 ) 𝑀2 .
Since lim inf 𝑛 → ∞ (1 − 𝛽𝑛 ) > 0, from (𝐶1) and (67), we obtain
that (85) holds.
On the other hand, from (24) and (26), we have
󵄩2
󵄩󵄩 𝑀
󵄩󵄩𝑄 𝑢𝑛 − 𝑄𝑀𝑥∗ 󵄩󵄩󵄩
󵄩
󵄩
󵄩2
󵄩 𝐺
𝐺
= 󵄩󵄩󵄩󵄩𝑆𝜆 𝑀 (𝐼 − 𝜆 𝑀𝐴 𝑀)𝑄𝑀−1 𝑢𝑛 − 𝑆𝜆 𝑀 (𝐼 − 𝜆 𝑀𝐴 𝑀)𝑄𝑀−1 𝑥∗ 󵄩󵄩󵄩󵄩
𝑀
𝑀
≤ ⟨(𝐼 − 𝜆 𝑀𝐴 𝑀) 𝑄𝑀−1 𝑢𝑛 − (𝐼 − 𝜆 𝑀𝐴 𝑀) 𝑄𝑀−1 𝑥∗ ,
𝑀
1 󵄩󵄩 𝑀−1
󵄩2 󵄩
󵄩2
𝑢𝑛 − 𝑄𝑀−1 𝑥∗ 󵄩󵄩󵄩󵄩 + 󵄩󵄩󵄩󵄩𝑄𝑀𝑢𝑛 − 𝑄𝑀𝑥∗ 󵄩󵄩󵄩󵄩
(󵄩󵄩𝑄
2 󵄩
󵄩
− 󵄩󵄩󵄩󵄩𝑄𝑀−1 𝑢𝑛 − 𝑄𝑀𝑢𝑛 + 𝑄𝑀𝑥∗ − 𝑄𝑀−1 𝑥∗
󵄩2
−𝜆 𝑀 (𝐴 𝑀𝑄𝑀−1 𝑢𝑛 − 𝐴 𝑀𝑄𝑀−1 𝑥∗ )󵄩󵄩󵄩󵄩 ) ,
󵄩2
󵄩
− 󵄩󵄩󵄩󵄩𝑄𝑀−1 𝑢𝑛 − 𝑄𝑀𝑢𝑛 + 𝑄𝑀𝑥∗ − 𝑄𝑀−1 𝑥∗ 󵄩󵄩󵄩󵄩
󵄩
󵄩
+ 2𝜆 𝑀 󵄩󵄩󵄩󵄩𝑄𝑀−1 𝑢𝑛 − 𝑄𝑀𝑢𝑛 + 𝑄𝑀𝑥∗ − 𝑄𝑀−1 𝑥∗ 󵄩󵄩󵄩󵄩
󵄩
󵄩
× 󵄩󵄩󵄩󵄩𝐴 𝑀𝑄𝑀−1 𝑢𝑛 − 𝐴 𝑀𝑄𝑀−1 𝑥∗ 󵄩󵄩󵄩󵄩 .
By induction, we have
󵄩󵄩 𝑀
󵄩2
󵄩󵄩𝑄 𝑢𝑛 − 𝑄𝑀𝑥∗ 󵄩󵄩󵄩
󵄩
󵄩
󵄩
󵄩
≤ 󵄩󵄩󵄩𝑢𝑛 − 𝑥∗ 󵄩󵄩󵄩
𝑀
󵄩
󵄩2
− ∑󵄩󵄩󵄩󵄩𝑄𝑖−1 𝑢𝑛 − 𝑄𝑖 𝑢𝑛 + 𝑄𝑖 𝑥∗ − 𝑄𝑖−1 𝑥∗ 󵄩󵄩󵄩󵄩
𝑖=1
𝑁
󵄩
󵄩
+ ∑2𝜆 𝑖 󵄩󵄩󵄩󵄩𝑄𝑖−1 𝑢𝑛 − 𝑄𝑖 𝑢𝑛 + 𝑄𝑖 𝑥∗ − 𝑄𝑖−1 𝑥∗ 󵄩󵄩󵄩󵄩
𝑖=1
𝑀 ∗
𝑄 𝑢𝑛 − 𝑄 𝑥 ⟩
1 󵄩
󵄩2
= (󵄩󵄩󵄩󵄩(𝐼 − 𝜆 𝑀𝐴 𝑀) 𝑄𝑀−1 𝑢𝑛 − (𝐼 − 𝜆 𝑀𝐴 𝑀) 𝑄𝑀−1 𝑥∗ 󵄩󵄩󵄩󵄩
2
󵄩2
󵄩
+ 󵄩󵄩󵄩󵄩𝑄𝑀𝑢𝑛 − 𝑄𝑀𝑥∗ 󵄩󵄩󵄩󵄩
󵄩
− 󵄩󵄩󵄩󵄩(𝐼 − 𝜆 𝑀𝐴 𝑀) 𝑄𝑀−1 𝑢𝑛
󵄩2
−(𝐼 − 𝜆 𝑀𝐴 𝑀)𝑥∗ − (𝑄𝑀𝑢𝑛 − 𝑄𝑀𝑥∗ )󵄩󵄩󵄩󵄩 )
󵄩
󵄩
× 󵄩󵄩󵄩󵄩𝐴 𝑖 𝑄𝑖−1 𝑢𝑛 − 𝐴 𝑖 𝑄𝑖−1 𝑥∗ 󵄩󵄩󵄩󵄩
𝑀
󵄩
󵄩2
󵄩
󵄩
≤ 󵄩󵄩󵄩𝑥𝑛 − 𝑥∗ 󵄩󵄩󵄩 − ∑󵄩󵄩󵄩󵄩𝑄𝑖−1 𝑢𝑛 − 𝑄𝑖 𝑢𝑛 + 𝑄𝑖 𝑥∗ − 𝑄𝑖−1 𝑥∗ 󵄩󵄩󵄩󵄩
𝑖=1
𝑁
󵄩
󵄩
+ ∑2𝜆 𝑖 󵄩󵄩󵄩󵄩𝑄𝑖−1 𝑢𝑛 − 𝑄𝑖 𝑢𝑛 + 𝑄𝑖 𝑥∗ − 𝑄𝑖−1 𝑥∗ 󵄩󵄩󵄩󵄩
𝑖=1
󵄩
󵄩
× 󵄩󵄩󵄩󵄩𝐴 𝑖 𝑄𝑖−1 𝑢𝑛 − 𝐴 𝑖 𝑄𝑖−1 𝑥∗ 󵄩󵄩󵄩󵄩 .
(93)
12
Journal of Applied Mathematics
Since lim inf 𝑛 → ∞ (1 − 𝛽𝑛 ) > 0, from (𝐶1), (67), and (85), we
obtain that
Substituting (93) into (88), we have
󵄩󵄩
∗ 󵄩2
󵄩󵄩𝑥𝑛+1 − 𝑥 󵄩󵄩󵄩
󵄩
󵄩2
≤ 𝛽𝑛 󵄩󵄩󵄩𝑥𝑛 − 𝑥∗ 󵄩󵄩󵄩 + (1 − 𝛽𝑛 )
󵄩
󵄩
lim 󵄩󵄩𝑄𝑖−1 𝑢𝑛 − 𝑄𝑖 𝑢𝑛 + 𝑄𝑖 𝑥∗ − 𝑄𝑖−1 𝑥∗ 󵄩󵄩󵄩󵄩 = 0,
󵄩
𝑛→∞ 󵄩
∀𝑖 ∈ {1, 2, . . . , 𝑀} .
𝑀
󵄩
󵄩2
󵄩
󵄩
× {󵄩󵄩󵄩𝑥𝑛 − 𝑥∗ 󵄩󵄩󵄩 − ∑󵄩󵄩󵄩󵄩𝑄𝑖−1 𝑢𝑛 − 𝑄𝑖 𝑢𝑛 + 𝑄𝑖 𝑥∗ − 𝑄𝑖−1 𝑥∗ 󵄩󵄩󵄩󵄩
𝑖=1
(96)
Consequently,
𝑀
󵄩
󵄩
+ ∑2𝜆 𝑖 󵄩󵄩󵄩󵄩𝑄𝑖−1 𝑢𝑛 − 𝑄𝑖 𝑢𝑛 + 𝑄𝑖 𝑥∗ − 𝑄𝑖−1 𝑥∗ 󵄩󵄩󵄩󵄩
󵄩󵄩
󵄩 󵄩 0
𝑀 󵄩
󵄩󵄩𝑢𝑛 − 𝑦𝑛 󵄩󵄩󵄩 = 󵄩󵄩󵄩󵄩𝑄 𝑢𝑛 − 𝑄 𝑢𝑛 󵄩󵄩󵄩󵄩
𝑖=1
𝑀
󵄩
󵄩
≤ ∑ 󵄩󵄩󵄩󵄩𝑄𝑖−1 𝑢𝑛 − 𝑄𝑖 𝑢𝑛 + 𝑄𝑖 𝑥∗ − 𝑄𝑖−1 𝑥∗ 󵄩󵄩󵄩󵄩
󵄩
󵄩
× 󵄩󵄩󵄩󵄩𝐴 𝑖 𝑄𝑖−1 𝑢𝑛 − 𝐴 𝑖 𝑄𝑖−1 𝑥∗ 󵄩󵄩󵄩󵄩 }
(97)
𝑖=1
󳨀→ 0 as 𝑛 󳨀→ ∞.
+ 2𝛼𝑛 (1 − 𝛽𝑛 ) 𝑀2
󵄩2
󵄩
≤ 󵄩󵄩󵄩𝑥𝑛 − 𝑥∗ 󵄩󵄩󵄩
It follows from (84) and (97) that
𝑀
󵄩
󵄩2
− (1 − 𝛽𝑛 ) ∑󵄩󵄩󵄩󵄩𝑄𝑖−1 𝑢𝑛 − 𝑄𝑖 𝑢𝑛 + 𝑄𝑖 𝑥∗ − 𝑄𝑖−1 𝑥∗ 󵄩󵄩󵄩󵄩
󵄩󵄩
󵄩 󵄩
󵄩 󵄩
󵄩
󵄩󵄩𝑥𝑛 − 𝑦𝑛 󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩𝑥𝑛 − 𝑢𝑛 󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑢𝑛 − 𝑦𝑛 󵄩󵄩󵄩
𝑖=1
󳨀→ 0
+ (1 − 𝛽𝑛 )
𝑀
󵄩
󵄩
× ∑2𝜆 𝑖 󵄩󵄩󵄩󵄩𝑄𝑖−1 𝑢𝑛 − 𝑄𝑖 𝑢𝑛 + 𝑄𝑖 𝑥∗ − 𝑄𝑖−1 𝑥∗ 󵄩󵄩󵄩󵄩
(98)
as 𝑛 󳨀→ ∞.
Next, we show that
󵄩󵄩
𝑖=1
lim 󵄩𝑥
𝑛→∞ 󵄩 𝑛
󵄩
󵄩
× 󵄩󵄩󵄩󵄩𝐴 𝑖 𝑄𝑖−1 𝑢𝑛 − 𝐴 𝑖 𝑄𝑖−1 𝑥∗ 󵄩󵄩󵄩󵄩
󵄩
− 𝑇 (𝑡) 𝑥𝑛 󵄩󵄩󵄩 = 0,
∀𝑡 ∈ 𝑆.
(99)
Put
+ 2𝛼𝑛 (1 − 𝛽𝑛 ) 𝑀2 ,
(94)
󵄩
󵄩
𝑀∗ = max {󵄩󵄩󵄩𝑥1 − 𝑥∗ 󵄩󵄩󵄩 ,
which in turn implies that
1 󵄩󵄩
∗
∗󵄩
󵄩𝛾𝑉𝑥 − 𝜇𝐹𝑥 󵄩󵄩󵄩} .
𝜏 − 𝛾𝐿 󵄩
(100)
Set 𝐷 = {𝑦 ∈ 𝐶 : ‖𝑦 − 𝑥∗ ‖ ≤ 𝑀∗ }. We remark that 𝐷 is
nonempty, bounded, closed, and convex set, and {𝑥𝑛 }, {𝑦𝑛 },
and {𝑧𝑛 } are in 𝐷. We will show that
𝑀
󵄩
󵄩2
(1 − 𝛽𝑛 ) ∑󵄩󵄩󵄩󵄩𝑄𝑖−1 𝑢𝑛 − 𝑄𝑖 𝑢𝑛 + 𝑄𝑖 𝑥∗ − 𝑄𝑖−1 𝑥∗ 󵄩󵄩󵄩󵄩
𝑖=1
󵄩2 󵄩
󵄩2
󵄩
≤ 󵄩󵄩󵄩𝑥𝑛 − 𝑥∗ 󵄩󵄩󵄩 − 󵄩󵄩󵄩𝑥𝑛+1 − 𝑥∗ 󵄩󵄩󵄩
󵄩
󵄩
lim sup sup 󵄩󵄩󵄩𝑇 (𝜇𝑛 ) 𝑦 − 𝑇 (𝑡) 𝑇 (𝜇𝑛 ) 𝑦󵄩󵄩󵄩 = 0,
𝑀
󵄩
󵄩
+ (1 − 𝛽𝑛 ) ∑2𝜆 𝑖 󵄩󵄩󵄩󵄩𝑄𝑖−1 𝑢𝑛 − 𝑄𝑖 𝑢𝑛 + 𝑄𝑖 𝑥∗ − 𝑄𝑖−1 𝑥∗ 󵄩󵄩󵄩󵄩
𝑖=1
󵄩
󵄩
× 󵄩󵄩󵄩󵄩𝐴 𝑖 𝑄𝑖−1 𝑢𝑛 − 𝐴 𝑖 𝑄𝑖−1 𝑥∗ 󵄩󵄩󵄩󵄩
+ 2𝛼𝑛 (1 − 𝛽𝑛 ) 𝑀2
𝑛 → ∞ 𝑦∈𝐷
To complete our proof, we follow the proof line as in [31] (see
also [23, 32, 33]). Let 𝜖 > 0. By [34, Theorem 1.2], there exists
𝛿 > 0 such that
co 𝐹𝛿 (𝑇 (𝑡) ; 𝐷) + 𝐵𝛿 ⊂ 𝐹𝜖 (𝑇 (𝑡) ; 𝐷) ,
󵄩 󵄩
󵄩 󵄩
󵄩
󵄩
≤ (󵄩󵄩󵄩𝑥𝑛 − 𝑥∗ 󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑥𝑛+1 − 𝑥∗ 󵄩󵄩󵄩) 󵄩󵄩󵄩𝑥𝑛+1 − 𝑥𝑛 󵄩󵄩󵄩
𝑀
󵄩
󵄩
+ (1 − 𝛽𝑛 ) ∑2𝜆 𝑖 󵄩󵄩󵄩󵄩𝑄𝑖−1 𝑢𝑛 − 𝑄𝑖 𝑢𝑛 + 𝑄𝑖 𝑥∗ − 𝑄𝑖−1 𝑥∗ 󵄩󵄩󵄩󵄩
𝑖=1
󵄩
󵄩
× 󵄩󵄩󵄩󵄩𝐴 𝑖 𝑄𝑖−1 𝑢𝑛 − 𝐴 𝑖 𝑄𝑖−1 𝑥∗ 󵄩󵄩󵄩󵄩
∀𝑡 ∈ 𝑆. (101)
∀𝑡 ∈ 𝑆.
Also by [34, Corollary 1.1], there exists a natural number 𝑁
such that
󵄩󵄩
󵄩󵄩
󵄩󵄩 1 𝑁
󵄩󵄩
1 𝑁
𝑖
𝑖
󵄩󵄩
󵄩
󵄩󵄩 𝑁 + 1 ∑𝑇 (𝑡 𝑠) 𝑦 − 𝑇 (𝑡) ( 𝑁 + 1 ∑𝑇 (𝑡 ) 𝑦)󵄩󵄩󵄩 ≤ 𝛿,
󵄩󵄩
󵄩󵄩
𝑖=0
𝑖=0
+ 2𝛼𝑛 (1 − 𝛽𝑛 ) 𝑀2 .
(95)
(102)
(103)
Journal of Applied Mathematics
13
for all 𝑡, 𝑠 ∈ 𝑆 and 𝑦 ∈ 𝐷. Let 𝑡 ∈ 𝑆. Since {𝜇𝑛 } is strongly left
regular, there exists 𝑛0 ∈ N such that ‖𝜇𝑛 − 𝑙𝑡∗𝑖 𝜇𝑛 ‖ ≤ 𝛿/(𝑀∗ +
‖𝑤‖) for all 𝑛 ≥ 𝑛0 and 𝑖 = 1, 2, . . . , 𝑁. Then, we have
󵄩󵄩
󵄩󵄩󵄩
󵄩󵄩
1 𝑁
󵄩
sup 󵄩󵄩󵄩𝑇 (𝜇𝑛 ) 𝑦 − ∫
∑𝑇 (𝑡𝑖 𝑠) 𝑦𝑑𝜇𝑛 (𝑠)󵄩󵄩󵄩
󵄩󵄩
𝑁 + 1 𝑖=0
𝑦∈𝐷 󵄩
󵄩󵄩
󵄩
󵄨󵄨
󵄨󵄨
= sup sup 󵄨󵄨󵄨 ⟨𝑇 (𝜇𝑛 ) 𝑦, 𝑧⟩
𝑦∈𝐷 ‖𝑧‖=1 󵄨󵄨󵄨
𝑥𝑛+1 = 𝛽𝑛 𝑥𝑛 + (1 − 𝛽𝑛 ) 𝑇 (𝜇𝑛 ) 𝑦𝑛
= 𝑇 (𝜇𝑛 ) 𝑦𝑛 + 𝛽𝑛 (𝑥𝑛 − 𝑇 (𝜇𝑛 ) 𝑦𝑛 )
for all 𝑛 > 𝑘0 . Hence, lim sup𝑛 → ∞ ‖𝑥𝑛 − 𝑇(𝑡)𝑥𝑛 ‖ ≤ 𝜖. Since
𝜖 > 0 is arbitrary, we obtain that (99) holds.
Next, we show that
󵄨󵄨
󵄨󵄨 1 𝑁
= sup sup 󵄨󵄨󵄨
∑(𝜇𝑛 )𝑠 ⟨𝑇 (𝑠) 𝑦, 𝑧⟩
𝑦∈𝐷 ‖𝑧‖=1 󵄨󵄨󵄨 𝑁 + 1 𝑖=0
≤
̂ 𝑧𝑛 − 𝑥⟩
̂ ≤ 0,
lim sup ⟨𝛾𝑉𝑥̂ − 𝜇𝐹𝑥,
𝑛→∞
󵄨󵄨
󵄨󵄨
1 𝑁
∑(𝜇𝑛 )𝑠 ⟨𝑇 (𝑡𝑖 ) 𝑦, 𝑧⟩󵄨󵄨󵄨
󵄨󵄨
𝑁 + 1 𝑖=0
󵄨
̂ 𝑥𝑛 − 𝑥⟩
̂ = lim ⟨𝛾𝑉𝑥̂ − 𝜇𝐹𝑥,
̂ 𝑥𝑛𝑖 − 𝑥⟩
̂ .
lim sup ⟨𝛾𝑉𝑥̂ − 𝜇𝐹𝑥,
1
𝑁+1
(110)
𝑖=0 𝑦∈𝐷 ‖𝑧‖=1
󵄩󵄩
∗ 󵄩
∗
󵄩𝜇𝑛 − 𝑙𝑡𝑖 𝜇𝑛 󵄩󵄩󵄩 (𝑀 + ‖𝑤‖) ≤ 𝛿,
𝑖=1,2,3,...,𝑁 󵄩
∀𝑛 ≥ 𝑛0 .
(104)
On the other hand, by Lemma 2, we have
∫
1 𝑁
∑𝑇 (𝑡𝑖 𝑠) 𝑦 𝑑𝜇𝑛 (𝑠)
𝑁 + 1 𝑖=0
∈ co {
𝑁
Since {𝑥𝑛 } is bounded, there exists a subsequence {𝑥𝑛𝑖 } of {𝑥𝑛 }
such that 𝑥𝑛𝑖 ⇀ V. Now, we show that V ∈ F.
(i) We first show that V ∈ Fix(S). From (99), we have
‖𝑥𝑛 −𝑇(𝑡)𝑥𝑛 ‖ → 0 as 𝑛 → ∞ for all 𝑡 ∈ 𝑆. Then, from
Demiclosedness Principle 2.6, we get V ∈ Fix(S).
(ii) We show that V ∈ Fix(𝐾), where 𝐾 is defined as in
Lemma 9. Then, from (97), we have
󵄩 󵄩
󵄩 󵄩
󵄩
󵄩󵄩
󵄩󵄩𝑦𝑛 − 𝐾𝑦𝑛 󵄩󵄩󵄩 = 󵄩󵄩󵄩𝐾𝑢𝑛 − 𝐾𝑦𝑛 󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩𝑢𝑛 − 𝑦𝑛 󵄩󵄩󵄩
(105)
󳨀→ 0 as 𝑛 󳨀→ ∞,
1
∑𝑇 (𝑡𝑖 ) 𝑇 (𝑠) 𝑦 : 𝑠 ∈ 𝑆} .
𝑁 + 1 𝑖=0
(iii) We show that V ∈ ⋂𝑁
𝑘=1 GMEP(Θ𝑘 , 𝜑𝑘 , Ψ𝑘 ). Note
𝑘 ,𝜑𝑘 )
(𝐼
−
𝑟𝑘,𝑛 Ψ𝑘 )𝐺𝑛𝑘−1 𝑥𝑛 , for all 𝑘 ∈
that 𝐺𝑛𝑘 𝑥𝑛 = 𝑆𝑟(Θ
𝑘,𝑛
{1, 2, . . . , 𝑁}. Then, we have
𝑁
1
∑𝑇 (𝑡𝑖 𝑠) 𝑦 𝑑𝜇𝑛 (𝑠)
𝑁 + 1 𝑖=1
+ (𝑇 (𝜇𝑛 ) 𝑦 − ∫
1 𝑁
∑𝑇 (𝑡𝑖 𝑠) 𝑦 𝑑𝜇𝑛 (𝑠))
𝑁 + 1 𝑖=1
Θ𝑘 (𝐺𝑛𝑘 𝑥𝑛 , 𝑦) + 𝜑𝑘 (𝑦) − 𝜑𝑘 (𝐺𝑛𝑘 )
+ ⟨Ψ𝑘 𝐺𝑛𝑘−1 𝑥𝑛 , 𝑦 − 𝐺𝑛𝑘 𝑥𝑛 ⟩
𝑁
1
∈ co {
∑𝑇 (𝑡𝑖 ) (𝑇 (𝑠) 𝑦) : 𝑠 ∈ 𝑆} + 𝐵𝛿
𝑁 + 1 𝑖=0
⊂ co 𝐹𝛿 (𝑇 (𝑡) ; 𝐷) + 𝐵𝛿 ,
(106)
for all 𝑦 ∈ 𝐷 and 𝑛 ≥ 𝑛0 . Therefore,
󵄩
󵄩
lim sup sup 󵄩󵄩󵄩𝑇 (𝜇𝑛 ) 𝑦 − 𝑇 (𝑡) 𝑇 (𝜇𝑛 ) 𝑦󵄩󵄩󵄩 ≤ 𝜖.
𝑛→∞
𝑦∈𝐷
(111)
and from (98), we also have ‖𝑥𝑛 − 𝐾𝑥𝑛 ‖ → 0. By
Demiclosedness Principle 2.6, we get V ∈ Fix(𝐾).
Combining (103)–(105), we have
𝑇 (𝜇𝑛 ) 𝑦 =
𝑖→∞
𝑛→∞
󵄨
󵄨
× ∑ sup sup 󵄨󵄨󵄨(𝜇𝑛 )𝑠 ⟨𝑇 (𝑠) 𝑦, 𝑧⟩ − (𝑙𝑡∗𝑖 𝜇𝑛 )𝑠 ⟨𝑇 (𝑠) 𝑦, 𝑧⟩󵄨󵄨󵄨
max
(109)
̂ To show this, we choose a
where 𝑥̂ = 𝑃F (𝐼 − 𝜇𝐹 + 𝛾𝑉)𝑥.
subsequence {𝑥𝑛𝑖 } of {𝑥𝑛 } such that
𝑁
≤
(108)
∈ 𝐹𝛿 (𝑇 (𝑡) ; 𝐷) + 𝐵𝛿 ⊂ 𝐹𝜖 (𝑇 (𝑡) ; 𝐷) ,
󵄨󵄨
1 𝑁
󵄨󵄨
𝑖
− ⟨∫
∑𝑇 (𝑡 𝑠) 𝑦 𝑑𝜇𝑛 (𝑠) , 𝑧⟩󵄨󵄨󵄨
󵄨󵄨
𝑁 + 1 𝑖=0
󵄨
−
(101) and condition (𝐶2), there exists 𝑎, 𝑏 ∈ (0, 1) such that
0 < 𝑎 ≤ 𝛽𝑛 ≤ 𝑏 < 1 and 𝑇(𝜇𝑛 )𝑦 ∈ 𝐹𝛿 (𝑇(𝑡); 𝐷) for all 𝑦 ∈ 𝐷.
From (69), there exists 𝑘0 ∈ N such that ‖𝑥𝑛 − 𝑇(𝜇𝑛 )𝑦𝑛 ‖ < 𝛿/𝑏
for all 𝑛 > 𝑘0 . Then, from (102) and (106), we have
(107)
Since 𝜖 > 0 is arbitrary, we obtain that (101) holds. Let 𝑡 ∈ 𝑆
and 𝜖 > 0. Then, there exists 𝛿 > 0 satisfying (102). From
+
(112)
1
⟨𝑦 − 𝐺𝑛𝑘 𝑥𝑛 , 𝐺𝑛𝑘 𝑥𝑛 − 𝐺𝑛𝑘−1 𝑥𝑛 ⟩ ≥ 0.
𝑟𝑘,𝑛
Replacing 𝑛 by 𝑛𝑖 in the last inequality and using (𝐴2), we
have
𝑥𝑛𝑖 , 𝑦 − 𝐺𝑛𝑘𝑖 𝑥𝑛𝑖 ⟩
𝜑𝑘 (𝑦) − 𝜑𝑘 (𝐺𝑛𝑘𝑖 𝑥𝑛𝑖 ) + ⟨Ψ𝑘 𝐺𝑛𝑘−1
𝑖
+
1
⟨𝑦 − 𝐺𝑛𝑘𝑖 𝑥𝑛𝑖 , 𝐺𝑛𝑘𝑖 𝑥𝑛𝑖 − 𝐺𝑛𝑘−1
𝑥𝑛𝑖 ⟩ ≥ Θ𝑘 (𝑦, 𝐺𝑛𝑘𝑖 𝑥𝑛𝑖 ) .
𝑖
𝑟𝑘,𝑛
(113)
14
Journal of Applied Mathematics
Let 𝑦𝑡 = 𝑡𝑦 + (1 − 𝑡)V for all 𝑡 ∈ (0, 1] and 𝑦 ∈ 𝐶. This implies
that 𝑦𝑡 ∈ 𝐶. Then, we have
⟨𝑦𝑡 − 𝐺𝑛𝑘𝑖 𝑥𝑛𝑖 , Ψ𝑘 𝑦𝑡 ⟩
Finally, we show that 𝑥𝑛 → 𝑥̂ as 𝑛 → ∞. Notice that 𝑧𝑛 =
𝑃𝐶V𝑛 , where V𝑛 = 𝛼𝑛 𝛾𝑉𝑥𝑛 + (𝐼 − 𝛼𝑛 𝜇𝐹)𝑇(𝜇𝑛 )𝑦𝑛 . Then, from
(25), we have
󵄩2
󵄩󵄩
̂ 𝑧𝑛 − 𝑥⟩
̂ + ⟨𝑃𝐶V𝑛 − V𝑛 , 𝑃𝐶V𝑛 − 𝑥⟩
̂
󵄩󵄩𝑧𝑛 − 𝑥̂󵄩󵄩󵄩 = ⟨V𝑛 − 𝑥,
≥ 𝜑𝑘 (𝐺𝑛𝑘𝑖 𝑥𝑛𝑖 ) − 𝜑𝑘 (𝑦𝑡 ) + ⟨𝑦𝑡 − 𝐺𝑛𝑘𝑖 𝑥𝑛𝑖 , Ψ𝑘 𝑦𝑡 ⟩
̂ 𝑧𝑛 − 𝑥⟩
̂
≤ ⟨V𝑛 − 𝑥,
− ⟨𝑦𝑡 − 𝐺𝑛𝑘𝑖 𝑥𝑛𝑖 , Ψ𝑘 𝐺𝑛𝑘−1 𝑥𝑛𝑖 ⟩
− ⟨𝑦𝑡 − 𝐺𝑛𝑘𝑖 𝑥𝑛𝑖 ,
𝑥𝑛𝑖
𝐺𝑛𝑘𝑖 𝑥𝑛𝑖 − 𝐺𝑛𝑘−1
𝑖
𝑟𝑘,𝑛𝑖
̂ 𝑧𝑛 − 𝑥⟩
̂
= 𝛼𝑛 𝛾⟨𝑉𝑥𝑛 − 𝑉𝑥,
⟩ + Θ𝑘 (𝑦𝑡 , 𝐺𝑛𝑘𝑖 𝑥𝑛𝑖 )
̂ 𝑧𝑛 − 𝑥⟩
̂
+ 𝛼𝑛 ⟨𝛾𝑉𝑥̂ − 𝜇𝐹𝑥,
̂ , 𝑧𝑛 − 𝑥⟩
̂
+ ⟨(𝐼 − 𝛼𝑛 𝜇𝐹) (𝑇 (𝜇𝑛 ) 𝑦𝑛 − 𝑥)
= 𝜑𝑘 (𝐺𝑛𝑘𝑖 𝑥𝑛𝑖 ) − 𝜑𝑘 (𝑦𝑡 ) + ⟨𝑦𝑡 − 𝐺𝑛𝑘𝑖 𝑥𝑛𝑖 , Ψ𝑘 𝑦𝑡 − Ψ𝑘 𝐺𝑛𝑘𝑖 𝑥𝑛𝑖 ⟩
+ ⟨𝑦𝑡 −
𝐺𝑛𝑘𝑖 𝑥𝑛𝑖 , Ψ𝑘 𝐺𝑛𝑘𝑖 𝑥𝑛𝑖
− ⟨𝑦𝑡 − 𝐺𝑛𝑘𝑖 𝑥𝑛𝑖 ,
𝐺𝑛𝑘𝑖 𝑥𝑛𝑖
−
−
󵄩
󵄩2
≤ (1 − 𝛼𝑛 (1 − 𝛽𝑛 ) (𝜏 − 𝛾𝐿)) 󵄩󵄩󵄩𝑥𝑛 − 𝑥̂󵄩󵄩󵄩
Ψ𝑘 𝐺𝑛𝑘−1
𝑥𝑛𝑖 ⟩
𝑖
𝐺𝑛𝑘−1
𝑥𝑛𝑖
𝑖
𝑟𝑘,𝑛𝑖
̂ 𝑧𝑛 − 𝑥⟩
̂ .
+ 𝛼𝑛 ⟨𝛾𝑉𝑥̂ − 𝜇𝐹𝑥,
⟩ + Θ𝑘 (𝑦𝑡 , 𝐺𝑛𝑘𝑖 𝑥𝑛𝑖 ) .
(114)
From (73), we have ‖Ψ𝑘 𝐺𝑛𝑘𝑖 𝑥𝑛𝑖 − Ψ𝑘 𝐺𝑛𝑘−1
𝑥𝑛𝑖 ‖ → 0 as 𝑖 →
𝑖
∞. Furthermore, by the monotonicity of Ψ𝑘 , we obtain ⟨𝑦𝑡 −
𝐺𝑛𝑘𝑖 𝑥𝑛𝑖 , Ψ𝑘 𝑦𝑡 − Ψ𝑘 𝐺𝑛𝑘𝑖 𝑥𝑛𝑖 ⟩ ≥ 0. Then, from (𝐴4), we obtain
⟨𝑦𝑡 − V, Ψ𝑘 𝑦𝑡 ⟩ ≥ 𝜑𝑘 (V) − 𝜑𝑘 (𝑦𝑡 ) + Θ𝑘 (𝑦𝑡 , V) .
(115)
It follows from (121) that
󵄩
󵄩2
󵄩2
󵄩
󵄩2
󵄩󵄩
󵄩󵄩𝑥𝑛+1 − 𝑥̂󵄩󵄩󵄩 ≤ 𝛽𝑛 󵄩󵄩󵄩𝑥𝑛 − 𝑥̂󵄩󵄩󵄩 + (1 − 𝛽𝑛 ) 󵄩󵄩󵄩𝑧𝑛 − 𝑥̂󵄩󵄩󵄩
󵄩
󵄩2
≤ 𝛽𝑛 󵄩󵄩󵄩𝑥𝑛 − 𝑥̂󵄩󵄩󵄩 + (1 − 𝛽𝑛 )
󵄩
󵄩2
× {(1 − 𝛼𝑛 (1 − 𝛽𝑛 ) (𝜏 − 𝛾𝐿)) 󵄩󵄩󵄩𝑥𝑛 − 𝑥̂󵄩󵄩󵄩
̂ 𝑧𝑛 − 𝑥⟩
̂ }
+𝛼𝑛 ⟨𝛾𝑉𝑥̂ − 𝜇𝐹𝑥,
Using (𝐴1), (𝐴4), and (115), we also obtain
󵄩
󵄩2
≤ (1 − 𝛼𝑛 (1 − 𝛽𝑛 ) (𝜏 − 𝛾𝐿)) 󵄩󵄩󵄩𝑥𝑛 − 𝑥̂󵄩󵄩󵄩
0 = Θ𝑘 (𝑦𝑡 , 𝑦𝑡 ) + 𝜑𝑘 (𝑦𝑡 ) − 𝜑𝑘 (𝑦𝑡 )
̂ 𝑧𝑛 − 𝑥⟩.
̂
+ 𝛼𝑛 (1 − 𝛽𝑛 ) ⟨𝛾𝑉𝑥̂ − 𝜇𝐹𝑥,
≤ 𝑡Θ𝑘 (𝑦𝑡 , 𝑦) + (1 − 𝑡) Θ𝑘 (𝑦𝑡 , V) + (1 − 𝑡) 𝜑𝑘 (V) − 𝜑𝑘 (𝑦𝑡 )
(122)
≤ 𝑡 [Θ𝑘 (𝑦𝑡 , 𝑦) + 𝜑𝑘 (𝑦) − 𝜑𝑘 (𝑦𝑡 )] + (1 − 𝑡) ⟨𝑦𝑡 − V, Ψ𝑘 𝑦𝑡 ⟩
= 𝑡 [Θ𝑘 (𝑦𝑡 , 𝑦) + 𝜑𝑘 (𝑦) − 𝜑𝑘 (𝑦𝑡 )] + (1 − 𝑡) 𝑡 ⟨𝑦 − V, Ψ𝑘 𝑦𝑡 ⟩ ,
(116)
Put 𝜎𝑛 := 𝛼𝑛 (1 − 𝛽𝑛 )(𝜏 − 𝛾𝐿) and 𝛿𝑛 := 𝛼𝑛 (1 − 𝛽𝑛 )⟨𝛾𝑉𝑥̂ −
̂ Then, (122) reduces to formula
̂ 𝑧𝑛 − 𝑥⟩.
𝜇𝐹𝑥,
and, hence,
0 ≤ Θ𝑘 (𝑦𝑡 , 𝑦) + 𝜑𝑘 (𝑦) − 𝜑𝑘 (𝑦𝑡 ) + (1 − 𝑡) ⟨𝑦 − V, Ψ𝑘 𝑦𝑡 ⟩ .
(117)
Letting 𝑡 → 0 and using (𝐴3), we have, for each 𝑦 ∈ 𝐶,
0 ≤ Θ𝑘 (V, 𝑦) + 𝜑𝑘 (𝑦) − 𝜑𝑘 (V) + ⟨𝑦 − V, Ψ𝑘 V⟩ .
(121)
(118)
This implies that V ∈ GMEP(Θ𝑘 , 𝜑𝑘 , Ψ𝑘 ). Hence, V
⋂𝑁
𝑘=1 GMEP(Θ𝑘 , 𝜑𝑘 , Ψ𝑘 ). Therefore,
∈
𝑁
V ∈ F := ⋂ GMEP (Θ𝑘 , 𝜑𝑘 , Ψ𝑘 ) ∩ Fix (𝐾) ∩ Fix (S) . (119)
󵄩󵄩
󵄩2
󵄩
󵄩2
󵄩󵄩𝑥𝑛+1 − 𝑥̂󵄩󵄩󵄩 ≤ (1 − 𝜎𝑛 ) 󵄩󵄩󵄩𝑥𝑛 − 𝑥̂󵄩󵄩󵄩 + 𝛿𝑛 .
(123)
It is easily seen that ∑∞
𝑛=1 𝜎𝑛 = ∞, and (using (120))
lim sup
𝑛→∞
𝛿𝑛
1
̂ 𝑧𝑛 − 𝑥⟩
̂ ≤ 0.
=
lim sup ⟨𝛾𝑉𝑥̂ − 𝜇𝐹𝑥,
𝜎𝑛 𝜏 − 𝛾𝐿 𝑛 → ∞
(124)
Hence, by Lemma 7, we conclude that 𝑥𝑛 → 𝑥̂ as 𝑛 → ∞.
This completes the proof.
𝑘=1
From (66) and (110), we obtain
̂ 𝑧𝑛 − 𝑥⟩
̂ = lim sup ⟨𝛾𝑉𝑥̂ − 𝜇𝐹𝑥,
̂ 𝑥𝑛 − 𝑥⟩
̂
lim sup ⟨𝛾𝑉𝑥̂ − 𝜇𝐹𝑥,
𝑛→∞
𝑛→∞
̂ 𝑥𝑛𝑖 − 𝑥⟩
̂
= lim ⟨𝛾𝑉𝑥̂ − 𝜇𝐹𝑥,
𝑖→∞
̂ V − 𝑥⟩
̂ ≤ 0.
= ⟨𝛾𝑉𝑥̂ − 𝜇𝐹𝑥,
(120)
Using the results proved in [35] (see also [32]), we obtain
the following results.
Corollary 12. Let 𝐶, 𝐻, Θ𝑘 , 𝜑𝑘 , Ψ𝑘 , 𝐴 𝑘 , 𝐹, and 𝑉 be the
same as in Theorem 11. Let 𝑆 and 𝑇 be nonexpansive mappings
on 𝐶 with 𝑆𝑇 = 𝑇𝑆. Assume that F := Fix(𝑆) ∩ Fix(𝑇) ∩
⋂∞
𝑛=1 GMEP(Θ𝑘 , 𝜑𝑘 , Ψ𝑘 ) ∩ Fix(𝐾) ≠ 0, where 𝐾 is defined as in
Journal of Applied Mathematics
15
Lemma 9. Let {𝛼𝑛 }, {𝛽𝑛 }, and {𝑟𝑘,𝑛 }𝑁
𝑘=1 be sequences satisfying
(𝐶1)–(𝐶3). Then, the sequence {𝑥𝑛 } defined by
𝑢𝑛 =
𝑁 ,𝜑𝑁 )
𝑆𝑟(Θ
𝑁,𝑛
× (𝐼 −
𝑦𝑛 =
𝐺
𝑆𝜆 𝑀
𝑀
(𝐼 −
(𝐼 −
× (𝐼 −
𝑁 ,𝜑𝑁 )
𝑁−1 ,𝜑𝑁−1 )
(𝐼 − 𝑟𝑁,𝑛 Ψ𝑁) 𝑆𝑟(Θ
𝑢𝑛 = 𝑆𝑟(Θ
𝑁,𝑛
N−1,𝑛
𝑁−1 ,𝜑𝑁−1 )
𝑟𝑁,𝑛 Ψ𝑁) 𝑆𝑟(Θ
𝑁−1,𝑛
1 ,𝜑1 )
𝑟𝑁−1,𝑛 Ψ𝑁−1 ) ⋅ ⋅ ⋅ 𝑆𝑟(Θ
1,𝑛
be sequences satisfying (𝐶1)–(𝐶3). Then, the sequence {𝑥𝑛 }
defined by
(𝐼 − 𝑟1,𝑛 Ψ1 ) 𝑥𝑛 ,
𝐺
𝜆 𝑀𝐴 𝑀) 𝑆𝜆 𝑀−1
𝑀−1
1 ,𝜑1 )
× (𝐼 − 𝑟𝑁−1,𝑛 Ψ𝑁−1 ) ⋅ ⋅ ⋅ 𝑆𝑟(Θ
(𝐼 − 𝑟1,𝑛 Ψ1 ) 𝑥𝑛 ,
1,𝑛
𝐺
𝐺
𝑦𝑛 = 𝑆𝜆 𝑀 (𝐼 − 𝜆 𝑀𝐴 𝑀) 𝑆𝜆 𝑀−1
𝑀
𝑀−1
𝐺
𝐺
𝜆 𝑀−1 𝐴 𝑀−1 ) ⋅ ⋅ ⋅ 𝑆𝜆 1
1
(𝐼 − 𝜆 1 𝐴 1 ) 𝑢𝑛 ,
𝑥𝑛+1 = 𝛽𝑛 𝑥𝑛 + (1 − 𝛽𝑛 )
1 𝑛−1 𝑛−1
× 𝑃𝐶 [𝛼𝑛 𝛾𝑉𝑥𝑛 + (𝐼 − 𝛼𝑛 𝜇𝐹) 2 ∑ ∑ 𝑆𝑖 𝑇𝑗 𝑦𝑛 ] ,
𝑛 𝑖=0 𝑗=0
[
]
∀𝑛 ≥ 1,
(125)
converges strongly to 𝑥̂ ∈ F, where 𝑥̂ solves uniquely the
variational inequality (40).
Corollary 13. Let 𝐶, 𝐻, Θ𝑘 , 𝜑𝑘 , Ψ𝑘 , 𝐴 𝑘 , 𝐹, and 𝑉 be the
same as in Theorem 11. Let S = {𝑇(𝑡) : 𝑡 > 0} be a
strongly continuous nonexpansive semigroup on 𝐶. Assume
that Ω := Fix(S) ∩ ⋂∞
𝑛=1 GMEP(Θ𝑘 , 𝜑𝑘 , Ψ𝑘 ) ∩ Fix(𝐾) ≠ 0,
where 𝐾 is defined as in Lemma 9. Let {𝛼𝑛 }, {𝛽𝑛 }, and {𝑟𝑘,𝑛 }𝑁
𝑘=1
be sequences satisfying (𝐶1)–(𝐶3). Then, the sequence {𝑥𝑛 }
defined by
× (𝐼 − 𝜆 𝑀−1 𝐴 𝑀−1 ) ⋅ ⋅ ⋅ 𝑆𝜆 1 (𝐼 − 𝜆 1 𝐴 1 ) 𝑢𝑛 ,
1
𝑥𝑛+1 = 𝛽𝑛 𝑥𝑛 + (1 − 𝛽𝑛 ) 𝑃𝐶
× [𝛼𝑛 𝛾𝑉𝑥𝑛 + (𝐼 − 𝛼𝑛 𝜇𝐹)
∞
× (𝑎𝑛 ∫ exp (−𝑎𝑛 𝑠) 𝑇 (𝑠) 𝑦𝑛 ) 𝑑𝑠] ,
0
(127)
where {𝑎𝑛 } is a decreasing sequence in (0, ∞) with lim𝑛 → ∞ 𝑎𝑛 =
0, converges strongly to 𝑥̂ ∈ Ω, where 𝑥̂ solves uniquely the
variational inequality (40).
4. Some Applications
In this section, as applications, we will apply Theorem 11 to
find minimum-norm solutions 𝑥̂ = 𝑃Ω (0) of some variational
inequalities. Namely, find a point 𝑥̂ which solves uniquely the
following quadratic minimization problem:
̂ 2 = min‖𝑥‖2 .
‖𝑥‖
(128)
𝑥∈Ω
𝑁 ,𝜑𝑁 )
𝑁−1 ,𝜑𝑁−1 )
(𝐼 − 𝑟𝑁,𝑛 Ψ𝑁) 𝑆𝑟(Θ
𝑢𝑛 = 𝑆𝑟(Θ
𝑁,𝑛
𝑁−1,𝑛
1 ,𝜑1 )
× (𝐼 − 𝑟𝑁−1,𝑛 Ψ𝑁−1 ) ⋅ ⋅ ⋅ 𝑆𝑟(Θ
(𝐼 − 𝑟1,𝑛 Ψ1 ) 𝑥𝑛 ,
1,𝑛
𝐺
𝐺
𝑦𝑛 = 𝑆𝜆 𝑀 (𝐼 − 𝜆 𝑀𝐴 𝑀) 𝑆𝜆 𝑀−1
𝑀
𝑀−1
𝐺
× (𝐼 − 𝜆 𝑀−1 𝐴 𝑀−1 ) ⋅ ⋅ ⋅ 𝑆𝜆 1 (𝐼 − 𝜆 1 𝐴 1 ) 𝑢𝑛 ,
1
𝑥𝑛+1 = 𝛽𝑛 𝑥𝑛 + (1 − 𝛽𝑛 )
× 𝑃𝐶 [𝛼𝑛 𝛾𝑉𝑥𝑛 + (𝐼 − 𝛼𝑛 𝜇𝐹)
1 𝑡𝑛
∫ 𝑇 (𝑠) 𝑦𝑛 𝑑𝑠] ,
𝑡𝑛 0
∀𝑛 ≥ 1,
(126)
where {𝑡𝑛 } is an increasing sequence in (0, ∞) with
lim𝑛 → ∞ (𝑡𝑛 /𝑡𝑛+1 ) = 1, converges strongly to 𝑥̂ ∈ F,
where 𝑥̂ solves uniquely the variational inequality (40).
Corollary 14. Let 𝐶, 𝐻, Θ𝑘 , 𝜑𝑘 , Ψ𝑘 , 𝐴 𝑘 , 𝐹, and 𝑉 be the
same as in Theorem 11. Let S = {𝑇(𝑡) : 𝑡 > 0} be a
strongly continuous nonexpansive semigroup on 𝐶. Assume
that Ω := Fix(S) ∩ ⋂∞
𝑛=1 GMEP(Θ𝑘 , 𝜑𝑘 , Ψ𝑘 ) ∩ Fix(𝐾) ≠ 0,
where 𝐾 is defined as in Lemma 9. Let {𝛼𝑛 }, {𝛽𝑛 }, and {𝑟𝑘,𝑛 }𝑁
𝑘=1
∀𝑛 ≥ 1,
Minimum-norm solutions have been applied widely in several branches of pure and applied sciences, for example,
defining the pseudoinverse of a bounded linear operator,
signal processing, and many other problems in a convex
polyhedron and a hyperplane (see [36, 37]).
Recently, some iterative methods have been studied to
find the minimum-norm fixed point of nonexpansive mappings and their generalizations (see, e.g. [38–49] and the
references therein).
Using Theorem 11 and Corollaries 12, 13, and 14, we
immediately have the following results, respectively.
Theorem 15. Let 𝐶 and 𝐻 be the same as in Theorem 11. Let
S = {𝑇(𝑡) : 𝑡 ∈ 𝑆} be a nonexpansive semigroup on 𝐶 such
that F := Fix(S) ≠ 0. Let {𝛼𝑛 } and {𝛽𝑛 } be sequences satisfying
(𝐶1)–(𝐶3). Then, the sequence {𝑥𝑛 } defined by
𝑥𝑛+1 = 𝛽𝑛 𝑥𝑛 + (1 − 𝛽𝑛 ) 𝑃𝐶 [(1 − 𝛼𝑛 ) 𝑇 (𝜇𝑛 ) 𝑥𝑛 ] ,
∀𝑛 ≥ 1,
(129)
converges strongly to 𝑥̂ ∈ F, where 𝑥̂ = 𝑃F (0) is the minimumnorm fixed point of F, where 𝑥̂ solves uniquely the quadratic
minimization problem (128).
Theorem 16. Let 𝐶 and 𝐻 be the same as in Corollary 12. Let
𝑆 and 𝑇 be nonexpansive mappings on 𝐶 with 𝑆𝑇 = 𝑇𝑆 such
16
Journal of Applied Mathematics
that F := Fix(𝑆) ∩ Fix(𝑇) ≠ 0. Let {𝛼𝑛 } and {𝛽𝑛 } be sequences
satisfying (𝐶1)–(𝐶3). Then, the sequence {𝑥𝑛 } defined by
𝑥𝑛+1 = 𝛽𝑛 𝑥𝑛 + (1 − 𝛽𝑛 ) 𝑃𝐶
1 𝑛−1 𝑛−1
× [(1 − 𝛼𝑛 ) 2 ∑ ∑ 𝑆𝑖 𝑇𝑗 𝑥𝑛 ] ,
𝑛 𝑖=0 𝑗=0
[
]
∀𝑛 ≥ 1,
(130)
converges strongly to 𝑥̂ ∈ F, where 𝑥̂ = 𝑃F (0) is the minimumnorm fixed point of F, where 𝑥̂ solves uniquely the quadratic
minimization problem (128).
Theorem 17. Let 𝐶 and 𝐻 be the same as in Corollary 13. Let
S = {𝑇(𝑡) : 𝑡 > 0} be a strongly continuous nonexpansive
semigroup on 𝐶 such that F := Fix(S) ≠ 0. Let {𝛼𝑛 } and
{𝛽𝑛 } be sequences satisfying (𝐶1)–(𝐶3). Then, the sequence {𝑥𝑛 }
defined by
𝑥𝑛+1 = 𝛽𝑛 𝑥𝑛 + (1 − 𝛽𝑛 ) 𝑃𝐶 [(1 − 𝛼𝑛 )
1 𝑡𝑛
∫ 𝑇 (𝑠) 𝑥𝑛 𝑑𝑠] ,
𝑡𝑛 0
∀𝑛 ≥ 1,
(131)
where {𝑡𝑛 } is an increasing sequence in (0, ∞) with
lim𝑛 → ∞ (𝑡𝑛 /𝑡𝑛+1 ) = 1, converges strongly to 𝑥̂ ∈ F,
where 𝑥̂ = 𝑃F (0) is the minimum-norm fixed point of F,
where 𝑥̂ solves uniquely the quadratic minimization problem
(128).
Theorem 18. Let 𝐶 and 𝐻 be the same as in Corollary 14. Let
S = {𝑇(𝑡) : 𝑡 > 0} be a nonexpansive semigroup on 𝐶 such
that F := Fix(S) ≠ 0. Let {𝛼𝑛 } and {𝛽𝑛 } be sequences satisfying
(𝐶1)–(𝐶3). Then, the sequence {𝑥𝑛 } defined by
𝑥𝑛+1 = 𝛽𝑛 𝑥𝑛 + (1 − 𝛽𝑛 ) 𝑃𝐶
∞
× [(1 − 𝛼𝑛 ) (𝑎𝑛 ∫ exp (−𝑎𝑛 𝑠) 𝑇 (𝑠) 𝑥𝑛 𝑑𝑠)] ,
0
∀𝑛 ≥ 1,
(132)
where {𝑎𝑛 } is a decreasing sequence in (0, ∞) with lim𝑛 → ∞ 𝑎𝑛 =
0, converges strongly to 𝑥̂ ∈ F, where 𝑥̂ = 𝑃F (0) is the
minimum-norm fixed point of F, where 𝑥̂ solves uniquely the
quadratic minimization problem (128).
Acknowledgment
The authors were supported by the Higher Education
Research Promotion and National Research University
Project of Thailand, Office of the Higher Education Commission (Grant no. NRU56000508).
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