Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2013, Article ID 573583, 6 pages
http://dx.doi.org/10.1155/2013/573583
Research Article
A New Iterative Method for Solving a System of
Generalized Mixed Equilibrium Problems for
a Countable Family of Generalized Quasi-𝜙-Asymptotically
Nonexpansive Mappings
Wei-Qi Deng
College of Statistics and Mathematics, Yunnan University of Finance and Economics, Kunming, Yunnan 650221, China
Correspondence should be addressed to Wei-Qi Deng; dwq1273@126.com
Received 3 October 2012; Accepted 4 January 2013
Academic Editor: Satit Saejung
Copyright © 2013 Wei-Qi Deng. 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.
By using a specific way of choosing the indexes, we introduce an up-to-date iterative algorithm for approximating common
fixed points of a countable family of generalized quasi-𝜙-asymptotically nonexpansive mappings and obtain a strong convergence
theorem under some suitable conditions. As application, an iterative solution to a system of generalized mixed equilibrium problems
is studied. The results extend those of other authors, in which the involved mappings consist of just finite families.
It is obvious from the definition of 𝜙 that
1. Introduction
Throughout this paper, we assume that 𝐸 is a real Banach
space with its dual 𝐸∗ , 𝐶 is a nonempty closed convex subset
∗
of 𝐸, and 𝐽 : 𝐸 → 2𝐸 is the normalized duality mapping
defined by
󵄩 󵄩2
𝐽𝑥 = {𝑓 ∈ 𝐸∗ : ⟨𝑥, 𝑓⟩ = ‖𝑥‖2 = 󵄩󵄩󵄩𝑓󵄩󵄩󵄩 } ,
∀𝑥 ∈ 𝐸.
(1)
󵄩 󵄩2
󵄩 󵄩2
(‖𝑥‖ − 󵄩󵄩󵄩𝑦󵄩󵄩󵄩) ≤ 𝜙 (𝑥, 𝑦) ≤ (‖𝑥‖ + 󵄩󵄩󵄩𝑦󵄩󵄩󵄩) .
(4)
(2) A mapping 𝑇 : 𝐶 → 𝐶 is said to be uniformly LLipschitz continuous, if there exists a constant 𝐿 > 0 such that
󵄩
󵄩
󵄩󵄩 𝑛
𝑛 󵄩
󵄩󵄩𝑇 𝑥 − 𝑇 𝑦󵄩󵄩󵄩 ≤ 𝐿 󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩 ,
∀𝑛 ≥ 1, 𝑥, 𝑦 ∈ 𝐶.
(5)
In the sequel, we use 𝐹(𝑇) to denote the set of fixed points of
a mapping 𝑇.
Example 2. Let 𝐶 be a unit ball in a real Hilbert space 𝑙2 , and
let 𝑇 : 𝐶 → 𝐶 be a mapping defined by
Definition 1. (1) [1] A mapping 𝑇 : 𝐶 → 𝐶 is said to be
generalized quasi-𝜙-asymptotically nonexpansive in the light
of [1], if 𝐹(𝑇) ≠ 0, and there exist nonnegative real sequences
{𝜈𝑛 } and {𝜇𝑛 } with 𝜈𝑛 , 𝜇𝑛 → 0 (as 𝑛 → ∞) such that
𝑇 (𝑥1 , 𝑥2 , . . .) = (0, 𝑥12 , 𝑎2 𝑥2 , 𝑎3 𝑥3 , . . .) ,
𝜙 (𝑝, 𝑇𝑛 𝑥) ≤ (1+𝜈𝑛 ) 𝜙 (𝑝, 𝑥)+𝜇𝑛 ,
∀𝑛 ≥ 1, 𝑥 ∈ 𝐶, 𝑝 ∈ 𝐹 (𝑇) ,
(2)
where 𝜙 : 𝐸 × 𝐸 → R denotes the Lyapunov functional
defined by
󵄩 󵄩2
𝜙 (𝑥, 𝑦) = ‖𝑥‖2 − 2 ⟨𝑥, 𝐽𝑦⟩ + 󵄩󵄩󵄩𝑦󵄩󵄩󵄩 ,
∀𝑥, 𝑦 ∈ 𝐸.
(3)
(6)
where {𝑎𝑖 } is a sequence in (0, 1) satisfying ∏∞
𝑖=2 𝑎𝑖 = 1/2. It
is shown by Goebel and Kirk [2] that
𝜙 (𝑝, 𝑇𝑛 𝑦) ≤ (1+𝜈𝑛 ) 𝜙 (𝑝, 𝑦)+𝜇𝑛 ,
∀𝑛 ≥ 1, 𝑦 ∈ 𝐶, 𝑝 ∈ 𝐹 (𝑇) ,
(7)
where 𝜙(𝑥, 𝑦) = ‖𝑥 − 𝑦‖2 , 𝜈𝑛 = (2∏𝑛𝑖=2 𝑎𝑖 )2 − 1, for all 𝑛 ≥
1, and {𝜇𝑛 } is a nonnegative real sequence with 𝜇𝑛 → 0 as
𝑛 → ∞. This shows that the mapping 𝑇 defined earlier is a
generalized quasi-𝜙-asymptotically nonexpansive mapping.
2
Abstract and Applied Analysis
Let 𝜃 : 𝐶 × 𝐶 → R be a bifunction, 𝜓 : 𝐶 → R a real
valued function, and 𝐵 : 𝐶 → 𝐸∗ a nonlinear mapping. The
so-called generalized mixed equilibrium problem (GMEP) is
to find a 𝑢 ∈ 𝐶 such that
𝜃 (𝑢, 𝑦) + ⟨𝑦 − 𝑢, 𝐵𝑢⟩ + 𝜓 (𝑦) − 𝜓 (𝑢) ≥ 0,
∀𝑦 ∈ 𝐶,
(8)
whose set of solutions is denoted by Ω(𝜃, 𝐵, 𝜓).
The equilibrium problem is a unifying model for several
problems arising in physics, engineering, science optimization, economics, transportation, network and structural analysis, Nash equilibrium problems in noncooperative games,
and others. It has been shown that variational inequalities
and mathematical programming problems can be viewed as
a special realization of the abstract equilibrium problems.
Many authors have proposed some useful methods to solve
the (equilibrium problem) EP, (generalized equilibrium problem) GEP, (mixed equilibrium problem) MEP, and GMEP.
Concerning the weak and strong convergence of iterative
sequences to approximate a common element of the set of
solutions for the GMEP, the set of solutions to variational
inequality problems and the set of common fixed points
for relatively nonexpansive mappings, quasi-𝜙-nonexpansive
mappings, and quasi-𝜙-asymptotically nonexpansive mappings have been studied by many authors in the setting of
Hilbert or Banach spaces (e.g., see [3–16] and the references
therein).
Inspired and motivated by the study mentioned earlier,
in this paper, by using a specific way of choosing the indexes,
we propose an up-to-date iteration scheme for approximating
common fixed points of a countable family of generalized
quasi-𝜙-asymptotically nonexpansive mappings and obtain a
strong convergence theorem for solving a system of generalized mixed equilibrium problems. The results extend those of
the authors, in which the involved mappings consist of just
finite families.
2. Preliminaries
𝐸 is said to have uniformly Gâteaux differentiable norm if for
each 𝑦 ∈ 𝑆(𝐸), the limit (11) is attained uniformly for 𝑥 ∈ 𝑆(𝐸).
The norm of 𝐸 is said to be Fréchet differentiable if for each
𝑥 ∈ 𝑆(𝐸), the limit (11) is attained uniformly for 𝑦 ∈ 𝑆(𝐸). The
norm of 𝐸 is said to be uniformly Fréchet differentiable (and
𝐸 is said to be uniformly smooth) if the limit (11) is attained
uniformly for 𝑥, 𝑦 ∈ 𝑆(𝐸). Note that 𝐸 (𝐸∗ , resp.) is uniformly
convex ⇔ 𝐸∗ (𝐸, resp.) is uniformly smooth.
Following Alber [17], the generalized projection Π𝐶 : 𝐸 →
𝐶 is defined by
Π𝐶 (𝑥) = arg inf 𝜙 (𝑦, 𝑥) ,
𝑦∈𝐶
∀𝑥 ∈ 𝐸.
(12)
Lemma 3 (see [17]). Let E be a smooth, strictly convex, and
reflexive Banach space, and let C be a nonempty closed convex
subset of E. Then, the following conclusions hold:
(1) 𝜙(𝑥, Π𝐶𝑦) + 𝜙(Π𝐶𝑦, 𝑦) ≤ 𝜙(𝑥, 𝑦) for all 𝑥 ∈ 𝐶 and
𝑦 ∈ 𝐸;
(2) if 𝑥 ∈ 𝐸 and 𝑧 ∈ 𝐶, then 𝑧 = Π𝐶𝑥 ⇔ ⟨𝑧 − 𝑦, 𝐽𝑥 − 𝐽𝑧⟩ ≥
0, for all 𝑦 ∈ 𝐶;
(3) for 𝑥, 𝑦 ∈ 𝐸, 𝜙(𝑥, 𝑦) = 0 if and only if 𝑥 = 𝑦.
Remark 4. The following basic properties for a Banach space
𝐸 can be found in Cioranescu [18].
(i) If 𝐸 is uniformly smooth, then 𝐽 is uniformly continuous on each bounded subset of 𝐸.
(ii) If 𝐸 is reflexive and strictly convex, then 𝐽−1 is normweak continuous.
(iii) If 𝐸 is a smooth, strictly convex, and reflexive Banach
space, then the normalized duality mapping 𝐽 : 𝐸 →
∗
2𝐸 is single valued, one-to-one, and onto.
(iv) A Banach space 𝐸 is uniformly smooth if and only if
𝐸∗ is uniformly convex.
(v) Each uniformly convex Banach space 𝐸 has the KadecKlee property; that is, for any sequence {𝑥𝑛 } ⊂ 𝐸, if
𝑥𝑛 ⇀ 𝑥 ∈ 𝐸 and ‖𝑥𝑛 ‖ → ‖𝑥‖, then 𝑥𝑛 → 𝑥, where
𝑥𝑛 ⇀ 𝑥 denotes that {𝑥𝑛 } converges weakly to 𝑥.
A Banach space 𝐸 is strictly convex if the following implication
holds for 𝑥, 𝑦 ∈ 𝐸:
󵄩󵄩 𝑥 + 𝑦 󵄩󵄩
󵄩 󵄩
󵄩󵄩 < 1.
(9)
‖𝑥‖ = 󵄩󵄩󵄩𝑦󵄩󵄩󵄩 = 1, 𝑥 ≠ 𝑦 󳨐⇒ 󵄩󵄩󵄩󵄩
󵄩
󵄩 2 󵄩󵄩
Lemma 5 (see [19]). Let E be a real uniformly smooth and
strictly convex Banach space with Kadec-Klee property, and let
C be a nonempty closed convex subset of E. Let {𝑥𝑛 } and {𝑦𝑛 }
be two sequences in C such that 𝑥𝑛 → 𝑝 and 𝜙(𝑥𝑛 , 𝑦𝑛 ) → 0,
where 𝜙 is the function defined by (3); then, 𝑦𝑛 → 𝑝.
It is also said to be uniformly convex if for any 𝜖 > 0, there
exists a 𝛿 > 0 such that
󵄩󵄩 𝑥 + 𝑦 󵄩󵄩
󵄩
󵄩󵄩
󵄩 󵄩
󵄩󵄩 ≤ 1 − 𝛿.
‖𝑥‖ = 󵄩󵄩󵄩𝑦󵄩󵄩󵄩 = 1,
󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩 ≥ 𝜖 󳨐⇒ 󵄩󵄩󵄩󵄩
󵄩
󵄩 2 󵄩󵄩
(10)
Lemma 6 (see [1]). Let E and C be the same as those in
Lemma 5. Let 𝑇 : 𝐶 → 𝐶 be a closed and generalized quasi𝜙-asymptotically nonexpansive mapping with nonnegative real
sequences {𝜈𝑛 } and {𝜇𝑛 }; then, the fixed point set F(T) of T is a
closed and convex subset of C.
It is known that if 𝐸 is uniformly convex Banach space, then
𝐸 is reflexive and strictly convex. A Banach space 𝐸 is said to
be smooth if
󵄩
󵄩󵄩
󵄩𝑥 + 𝑡𝑦󵄩󵄩󵄩 − ‖𝑥‖
(11)
lim 󵄩
𝑡→0
𝑡
Lemma 7 (see [20]). Let 𝐸 be a real uniformly convex Banach
space, and let 𝐵𝑟 (0) be the closed ball of 𝐸 with center at the
origin and radius 𝑟 > 0. Then, there exists a continuous strictly
increasing convex function 𝑔 : [0, ∞) → [0, ∞) with 𝑔(0) =
0 such that
󵄩
󵄩2
󵄩 󵄩2
󵄩
󵄩󵄩
2
(13)
󵄩󵄩𝛼𝑥 + 𝛽𝑦󵄩󵄩󵄩 ≤ 𝛼‖𝑥‖ + 𝛽󵄩󵄩󵄩𝑦󵄩󵄩󵄩 − 𝛼𝛽𝑔 (󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩)
for all 𝑥, 𝑦, ∈ 𝐵𝑟 (0), and 𝛼, 𝛽 ∈ [0, 1] with 𝛼 + 𝛽 = 1.
exists for each 𝑥, 𝑦 ∈ 𝑆(𝐸) := {𝑥 ∈ 𝐸 : ‖𝑥‖ = 1}. In this case,
the norm of 𝐸 is said to be Gâteaux differentiable. The space
Abstract and Applied Analysis
3
3. Main Results
Theorem 8. Let E be a real uniformly smooth and strictly
convex Banach space with Kadec-Klee property, C a nonempty
closed convex subset of E, and 𝑇𝑖 : 𝐶 → 𝐶, 𝑖 = 1, 2,
. . . a countable family of closed and generalized quasi-𝜙asymptotically nonexpansive mappings with nonnegative real
sequences {𝜈𝑛(𝑖) } and {𝜇𝑛(𝑖) } satisfying 𝜈𝑛(𝑖) → 0 and 𝜇𝑛(𝑖) → 0
(as 𝑛 → ∞ and for each 𝑖 ≥ 1), and each 𝑇𝑖 is uniformly
𝐿 𝑖 -Lipschitz continuous. Let {𝛼𝑛 } be a sequence in [0, 𝜖] for
some 𝜖 ∈ (0, 1), and let {𝛽𝑛 } be a sequence in [0, 1] satisfying
0 < lim inf 𝑛 → ∞ 𝛽𝑛 (1 − 𝛽𝑛 ). Let {𝑥𝑛 } be the sequence generated
by
𝑥1 ∈ 𝐶;
−1
𝑧𝑛 = 𝐽 [𝛽𝑛 𝐽𝑥𝑛 + (1 −
𝑚
𝛽𝑛 ) 𝐽𝑇𝑖𝑛 𝑛 𝑥𝑛 ] ,
(14)
𝑚
𝑚
󵄩 󵄩2
= 󵄩󵄩󵄩𝑝󵄩󵄩󵄩 − 2 ⟨𝑝, 𝛽𝑛 𝐽𝑥𝑛 + (1 − 𝛽𝑛 ) 𝐽𝑇𝑖𝑛 𝑛 𝑥𝑛 ⟩
∀𝑛 ≥ 1,
(𝑖𝑛 )
(𝑖𝑛 )
sup𝑝∈𝐹 𝜙(𝑝, 𝑥𝑛 ) + 𝜇𝑚
, Π𝐶𝑛+1 is the generalized
where 𝜉𝑛 := 𝜈𝑚
𝑛
𝑛
projection of 𝐸 onto 𝐶𝑛+1 , and 𝑖𝑛 and 𝑚𝑛 satisfy the positive
integer equation: 𝑛 = 𝑖 + (𝑚 − 1)𝑚/2, 𝑚 ≥ 𝑖 (𝑚 ≥ 𝑖, 𝑛 =
1, 2, . . .); that is, for each 𝑛 ≥ 1, there exist unique 𝑖𝑛 and 𝑚𝑛
such that
𝑖6 = 3,
𝑚1 = 1,
𝑚2 = 2,
𝑚6 = 3,
𝑖3 = 2,
𝑖7 = 1,
𝑚3 = 2,
𝑚7 = 4,
𝑖4 = 1,
𝑖5 = 2,
𝑚5 = 3,
(15)
If 𝐹 := ⋂∞
𝑖=1 𝐹(𝑇𝑖 ) is nonempty and bounded, then {𝑥𝑛 } converges strongly to Π𝐹 𝑥1 .
Proof. We divide the proof into several steps.
(I) 𝐹 and 𝐶𝑛 (for all 𝑛 ≥ 1) both are closed and convex
subsets in 𝐶.
In fact, it follows from Lemma 6 that each 𝐹(𝑇𝑖 ) is a closed
and convex subset of 𝐶, so is 𝐹. In addition, with 𝐶1 (=𝐶)
being closed and convex, we may assume that 𝐶𝑛 is closed
and convex for some 𝑛 ≥ 2. In view of the definition of 𝜙, we
have that
(16)
where 𝜑(𝑣) = 2⟨𝑣, 𝐽𝑥𝑛 − 𝐽𝑦𝑛 ⟩, and 𝑎 = ‖𝑥𝑛 ‖2 − ‖𝑦𝑛 ‖2 + 𝜉𝑛 . This
shows that 𝐶𝑛+1 is closed and convex.
(II) 𝐹 is a subset of ⋂∞
𝑛=1 𝐶𝑛 .
󵄩 󵄩2
≤ 󵄩󵄩󵄩𝑝󵄩󵄩󵄩 − 2𝛽𝑛 ⟨𝑝, 𝐽𝑥𝑛 ⟩ − 2 (1 − 𝛽𝑛 )
󵄩 𝑚 󵄩2
𝑚
󵄩 󵄩2
× ⟨𝑝, 𝐽𝑇𝑖𝑛 𝑛 𝑥𝑛 ⟩ + 𝛽𝑛 󵄩󵄩󵄩𝑥𝑛 󵄩󵄩󵄩 + (1 − 𝛽𝑛 ) 󵄩󵄩󵄩󵄩𝑇𝑖𝑛 𝑛 𝑥𝑛 󵄩󵄩󵄩󵄩
󵄩
󵄩
𝑚
− 𝛽𝑛 (1 − 𝛽𝑛 ) 𝑔 (󵄩󵄩󵄩󵄩𝐽𝑥𝑛 − 𝐽𝑇𝑖𝑛 𝑛 𝑥𝑛 󵄩󵄩󵄩󵄩)
𝑚
𝑚8 = 4, . . . .
𝐶𝑛+1 = {𝑣 ∈ 𝐶 : 𝜑 (𝑣) ≤ 𝑎} ∩ 𝐶𝑛 ,
󵄩2
󵄩
𝑚
+ 󵄩󵄩󵄩󵄩𝛽𝑛 𝐽𝑥𝑛 + (1 − 𝛽𝑛 ) 𝐽𝑇𝑖𝑛 𝑛 𝑥𝑛 󵄩󵄩󵄩󵄩
= 𝛽𝑛 𝜙 (𝑝, 𝑥𝑛 ) + (1 − 𝛽𝑛 ) 𝜙 (𝑝, 𝑇𝑖𝑛 𝑛 𝑥𝑛 )
𝑖8 = 2, . . . ;
𝑚4 = 3,
Furthermore, it follows from Lemma 7 that for any 𝑝 ∈ 𝐹, we
have that
𝜙 (𝑝, 𝑧𝑛 ) = 𝜙 (𝑝, 𝐽−1 [𝛽𝑛 𝐽𝑥𝑛 + (1 − 𝛽𝑛 ) 𝐽𝑇𝑖𝑛 𝑛 𝑥𝑛 ])
𝐶𝑛+1 = {𝑣 ∈ 𝐶𝑛 : 𝜙 (𝑣, 𝑦𝑛 ) ≤ 𝜙 (𝑣, 𝑥𝑛 ) + 𝜉𝑛 } ,
𝑖2 = 1,
󵄩 󵄩2
= 󵄩󵄩󵄩𝑝󵄩󵄩󵄩 − 2 ⟨𝑝, 𝛼𝑛 𝐽𝑥𝑛 + (1 − 𝛼𝑛 ) 𝐽𝑧𝑛 ⟩
󵄩2
󵄩
+ 󵄩󵄩󵄩𝛼𝑛 𝐽𝑥𝑛 + (1 − 𝛼𝑛 ) 𝐽𝑧𝑛 󵄩󵄩󵄩
󵄩 󵄩2
≤ 󵄩󵄩󵄩𝑝󵄩󵄩󵄩 − 2𝛼𝑛 ⟨𝑝, 𝐽𝑥𝑛 ⟩ − 2 (1 − 𝛼𝑛 ) ⟨𝑝, 𝐽𝑧𝑛 ⟩
󵄩 󵄩2
󵄩 󵄩2
+ 𝛼𝑛 󵄩󵄩󵄩𝑥𝑛 󵄩󵄩󵄩 + (1 − 𝛼𝑛 ) 󵄩󵄩󵄩𝑧𝑛 󵄩󵄩󵄩
(17)
𝑦𝑛 = 𝐽−1 [𝛼𝑛 𝐽𝑥𝑛 + (1 − 𝛼𝑛 ) 𝐽𝑧𝑛 ] ,
𝑖1 = 1,
𝜙 (𝑝, 𝑦𝑛 ) = 𝜙 (𝑝, 𝐽−1 [𝛼𝑛 𝐽𝑥𝑛 + (1 − 𝛼𝑛 ) 𝐽𝑥𝑛 ])
= 𝛼𝑛 𝜙 (𝑝, 𝑥𝑛 ) + (1 − 𝛼𝑛 ) 𝜙 (𝑝, 𝑧𝑛 ) .
𝐶1 = 𝐶,
𝑥𝑛+1 = Π𝐶𝑛+1 𝑥1 ,
It is obvious that 𝐹 ⊂ 𝐶1 . Suppose that 𝐹 ⊂ 𝐶𝑛 for some
𝑛 ≥ 2. Since 𝐸 is uniformly smooth, 𝐸∗ is uniformly convex.
Then, for any 𝑝 ∈ 𝐹 ⊂ 𝐶𝑛 , we have that
󵄩
󵄩
𝑚
− 𝛽𝑛 (1 − 𝛽𝑛 ) 𝑔 (󵄩󵄩󵄩󵄩𝐽𝑥𝑛 − 𝐽𝑇𝑖𝑛 𝑛 𝑥𝑛 󵄩󵄩󵄩󵄩)
≤ 𝛽𝑛 𝜙 (𝑝, 𝑥𝑛 ) + (1 − 𝛽𝑛 )
(𝑖𝑛 )
(𝑖𝑛 )
sup𝜙 (𝑝, 𝑥𝑛 ) + 𝜇𝑚
]
× [𝜙 (𝑝, 𝑥𝑛 )+𝜈𝑚
𝑛
𝑛
𝑝∈𝐹
󵄩
󵄩
𝑚
− 𝛽𝑛 (1 − 𝛽𝑛 ) 𝑔 (󵄩󵄩󵄩󵄩𝐽𝑥𝑛 − 𝐽𝑇𝑖𝑛 𝑛 𝑥𝑛 󵄩󵄩󵄩󵄩)
(𝑖𝑛 )
(𝑖𝑛 )
≤ 𝜙 (𝑝, 𝑥𝑛 ) + 𝜈𝑚
sup𝜙 (𝑝, 𝑥𝑛 ) + 𝜇𝑚
𝑛
𝑛
𝑝∈𝐹
󵄩
󵄩
𝑚
− 𝛽𝑛 (1 − 𝛽𝑛 ) 𝑔 (󵄩󵄩󵄩󵄩𝐽𝑥𝑛 − 𝐽𝑇𝑖𝑛 𝑛 𝑥𝑛 󵄩󵄩󵄩󵄩)
󵄩
󵄩
𝑚
= 𝜙 (𝑝, 𝑥𝑛 ) + 𝜉𝑛 − 𝛽𝑛 (1 − 𝛽𝑛 ) 𝑔 (󵄩󵄩󵄩󵄩𝐽𝑥𝑛 − 𝐽𝑇𝑖𝑛 𝑛 𝑥𝑛 󵄩󵄩󵄩󵄩) .
(18)
Substituting (18) into (17) and simplifying it, we have that
𝜙 (𝑝, 𝑦𝑛 ) ≤ 𝜙 (𝑝, 𝑥𝑛 ) + (1 − 𝛼𝑛 ) 𝜉𝑛 ≤ 𝜙 (𝑝, 𝑥𝑛 ) + 𝜉𝑛 .
(19)
This implies that 𝑝 ∈ 𝐶𝑛+1 , and so 𝐹 ⊂ 𝐶𝑛+1 .
(III) 𝑥𝑛 → 𝑥∗ ∈ 𝐶 as 𝑛 → ∞.
In fact, since 𝑥𝑛 = Π𝐶𝑛 𝑥1 , from Lemma 3 (2), we have
that ⟨𝑥𝑛 − 𝑦, 𝐽𝑥1 − 𝐽𝑥𝑛 ⟩ ≥ 0, for all 𝑦 ∈ 𝐶𝑛 . Again, since
4
Abstract and Applied Analysis
𝐹 ⊂ ⋂∞
𝑛=1 𝐶𝑛 , we have that ⟨𝑥𝑛 − 𝑝, 𝐽𝑥1 − 𝐽𝑥𝑛 ⟩ ≥ 0, for all
𝑝 ∈ 𝐹. It follows from Lemma 3 (1) that for each 𝑝 ∈ 𝐹 and
for each 𝑛 ≥ 1,
𝜙 (𝑥𝑛 , 𝑥1 ) = 𝜙 (Π𝐶𝑛 𝑥1 , 𝑥1 ) ≤ 𝜙 (𝑝, 𝑥1 )−𝜙 (𝑝, 𝑥𝑛 ) ≤ 𝜙 (𝑝, 𝑥1 ) ,
(20)
which implies that {𝜙(𝑥𝑛 , 𝑥1 )} is bounded, so is {𝑥𝑛 }. Since for
all 𝑛 ≥ 1, 𝑥𝑛 = Π𝐶𝑛 𝑥1 and 𝑥𝑛+1 = Π𝐶𝑛+1 𝑥1 ∈ 𝐶𝑛+1 ⊂ 𝐶𝑛 , we
have 𝜙(𝑥𝑛 , 𝑥1 ) ≤ 𝜙(𝑥𝑛+1 , 𝑥1 ). This implies that {𝜙(𝑥𝑛 , 𝑥1 )} is
nondecreasing; hence, the limit
lim 𝜙 (𝑥𝑛 , 𝑥1 ) exists.
Since 𝐸 is reflexive, there exists a subsequence {𝑥𝑛𝑖 } of {𝑥𝑛 }
such that 𝑥𝑛𝑖 ⇀ 𝑥∗ ∈ 𝐶 as 𝑖 → ∞. Since 𝐶𝑛 is closed and
convex and 𝐶𝑛+1 ⊂ 𝐶𝑛 , this implies that 𝐶𝑛 is weakly, closed
and 𝑥∗ ∈ 𝐶𝑛 for each 𝑛 ≥ 1. In view of 𝑥𝑛𝑖 = Π𝐶𝑛 𝑥1 , we have
𝑖
that
∀𝑖 ≥ 1.
(22)
Since the norm ‖ ⋅ ‖ is weakly lower semicontinuous, we have
that
󵄩 󵄩2
󵄩 󵄩2
lim inf 𝜙 (𝑥𝑛𝑖 , 𝑥1 ) = lim inf (󵄩󵄩󵄩󵄩𝑥𝑛𝑖 󵄩󵄩󵄩󵄩 − 2 ⟨𝑥𝑛𝑖 , 𝐽𝑥1 ⟩ + 󵄩󵄩󵄩𝑥1 󵄩󵄩󵄩 )
𝑖→∞
lim 𝑥
𝑛→∞ 𝑛
(𝑖)
(𝑖)
𝜉𝑘 = 𝜈𝑚
sup𝜙 (𝑝, 𝑥𝑘 ) + 𝜇𝑚
,
𝑘
𝑘
𝑝∈𝐹
Since 𝑥𝑛+1 ∈ 𝐶𝑛+1 , it follows from (14), (27), and (29) that
𝜙 (𝑥𝑘+1 , 𝑦𝑘 ) ≤ 𝜙 (𝑥𝑘+1 , 𝑥𝑘 ) + 𝜉𝑘 󳨀→ 0
𝜙 (𝑥∗ , 𝑥1 ) ≤ lim inf 𝜙 (𝑥𝑛𝑖 , 𝑥1) ≤ lim sup𝜙 (𝑥𝑛𝑖 , 𝑥1) ≤ 𝜙 (𝑥∗ , 𝑥1) .
𝑖→∞
(24)
This implies that lim𝑖 → ∞ 𝜙(𝑥𝑛𝑖 , 𝑥1 ) = 𝜙(𝑥∗ , 𝑥1 ), and so
‖𝑥𝑛𝑖 ‖ → ‖𝑥∗ ‖ as 𝑖 → ∞. Since 𝑥𝑛𝑖 ⇀ 𝑥∗ , by virtue of KadecKlee property of 𝐸, we obtain that
lim 𝑥𝑛𝑖 = 𝑥∗ .
(25)
𝑖→∞
Since {𝜙(𝑥𝑛 , 𝑥1 )} is convergent, this, together with
lim𝑖 → ∞ 𝜙(𝑥𝑛𝑖 , 𝑥1 ) = 𝜙(𝑥∗ , 𝑥1 ), shows that lim𝑛 → ∞ 𝜙(𝑥𝑛 ,
𝑥1 ) = 𝜙(𝑥∗ , 𝑥1 ). If there exists some subsequence {𝑥𝑛𝑗 } of
{𝑥𝑛 } such that 𝑥𝑛𝑗 → 𝑦 as 𝑗 → ∞, then, from Lemma 3 (1),
we have that
as K𝑖 ∋ 𝑘 → ∞. Since 𝑥𝑘 → 𝑥 as K𝑖 ∋ 𝑘 → ∞, it follows
from (30) and Lemma 5 that
lim 𝑦𝑘 = 𝑥∗ .
𝑚
𝜙 (𝑝, 𝑦𝑘 ) ≤ 𝜙 (𝑝, 𝑥𝑘 ) + 𝜉𝑘 − (1 − 𝛼𝑘 ) 𝛽𝑘 (1 − 𝛽𝑘 ) 𝑔
that is,
𝑚
󵄩
󵄩
(1 − 𝛼𝑘 ) 𝛽𝑘 (1 − 𝛽𝑘 ) 𝑔 (󵄩󵄩󵄩𝐽𝑥𝑘 − 𝐽𝑇𝑖 𝑘 𝑥𝑘 󵄩󵄩󵄩)
≤ 𝜙 (𝑝, 𝑥𝑘 ) + 𝜉𝑘 − 𝜙 (𝑝, 𝑦𝑘 ) 󳨀→ 0
(K𝑖 ∋ 𝑘 󳨀→ ∞) .
(33)
This, together with assumption conditions imposed on the
sequences {𝛼𝑛 } and {𝛽𝑛 }, shows that limK𝑖 ∋𝑘 → ∞ 𝑔(‖𝐽𝑥𝑘 −
𝑚
𝐽𝑇𝑖 𝑘 𝑥𝑘 ‖) = 0. In view of property of 𝑔, we have that
lim
𝑚
󵄩󵄩
󵄩
󵄩𝐽𝑥𝑘 − 𝐽𝑇𝑖 𝑘 𝑥𝑘 󵄩󵄩󵄩 = 0.
K𝑖 ∋𝑘 → ∞ 󵄩
(34)
𝑚
In addition, 𝐽𝑥𝑘 → 𝐽𝑥∗ implies that limK𝑖 ∋𝑘 → ∞ 𝐽𝑇𝑖 𝑘 𝑥𝑘 =
𝐽𝑥∗ . From Remark 4 (ii), it yields that, as K𝑖 ∋ 𝑘 → ∞,
𝑚
𝑗
(32)
𝑚
󵄩
󵄩
× (󵄩󵄩󵄩𝐽𝑥𝑘 − 𝐽𝑇𝑖 𝑘 𝑥𝑘 󵄩󵄩󵄩) ;
𝑇𝑖 𝑘 𝑥𝑘 ⇀ 𝑥∗ ,
∀𝑖 ≥ 1.
(35)
Again, since for each 𝑖 ≥ 1, as K𝑖 ∋ 𝑘 → ∞,
󵄨󵄨󵄩󵄩 𝑚𝑘 󵄩󵄩 󵄩󵄩 ∗ 󵄩󵄩󵄨󵄨 󵄨󵄨󵄩󵄩 𝑚𝑘 󵄩󵄩 󵄩󵄩 ∗ 󵄩󵄩󵄨󵄨 󵄩󵄩 𝑚𝑘
∗󵄩
󵄨󵄨󵄩󵄩𝑇𝑖 𝑥𝑘 󵄩󵄩 − 󵄩󵄩𝑥 󵄩󵄩󵄨󵄨 = 󵄨󵄨󵄩󵄩𝐽𝑇𝑖 𝑥𝑘 󵄩󵄩 − 󵄩󵄩𝐽𝑥 󵄩󵄩󵄨󵄨 ≤ 󵄩󵄩𝐽𝑇𝑖 𝑥𝑘 − 𝐽𝑥 󵄩󵄩󵄩 󳨀→ 0.
(36)
≤ lim (𝜙 (𝑥𝑛𝑖 , 𝑥1 ) − 𝜙 (Π𝐶𝑛 𝑥1 , 𝑥1 ))
𝑗
= lim (𝜙 (𝑥𝑛𝑖 , 𝑥1 ) − 𝜙 (𝑥𝑛𝑗 , 𝑥1 ))
𝑖,𝑗 → ∞
∗
(31)
K𝑖 ∋𝑘 → ∞
𝜙 (𝑥∗ , 𝑦) = lim 𝜙 (𝑥𝑛𝑖 , 𝑥𝑛𝑗 ) = lim 𝜙 (𝑥𝑛𝑖 , Π𝐶𝑛 𝑥1 )
𝑖,𝑗 → ∞
(30)
∗
and so
𝑖,𝑗 → ∞
(28)
Note that {𝑚𝑘 }𝑘∈K𝑖 = {𝑖, 𝑖 + 1, 𝑖 + 2, . . .}; that is, 𝑚𝑘 ↑ ∞ as
K𝑖 ∋ 𝑘 → ∞. It follows from (27) and (28) that
lim 𝜉𝑘 = 0.
(29)
K𝑖 ∋𝑘 → ∞
𝑚
(23)
𝑖,𝑗 → ∞
∀𝑘 ∈ K𝑖 .
Note that 𝑇𝑖𝑘 𝑘 = 𝑇𝑖 𝑘 whenever 𝑘 ∈ K𝑖 for each 𝑖 ≥ 1. From
(17) and (18), for any 𝑝 ∈ 𝐹, we have that
= 𝜙 (𝑥∗ , 𝑥1 ) ,
𝑖→∞
(27)
Set K𝑖 = {𝑘 ≥ 1 : 𝑘 = 𝑖 + (𝑚 − 1)𝑚/2, 𝑚 ≥ 𝑖, 𝑚 ∈ N}
(𝑖𝑘 )
(𝑖)
(𝑖𝑘 )
(𝑖)
for each 𝑖 ≥ 1. Note that 𝜈𝑚
= 𝜈𝑚
and 𝜇𝑚
= 𝜇𝑚
whenever
𝑛
𝑘
𝑛
𝑘
𝑘 ∈ K𝑖 for each 𝑖 ≥ 1. For example, by the definition of K1 ,
we have that K1 = {1, 2, 4, 7, 11, 16, . . .}, and 𝑖1 = 𝑖2 = 𝑖4 =
𝑖7 = 𝑖11 = 𝑖16 = ⋅ ⋅ ⋅ = 1. Then, we have that
𝑖→∞
󵄩 󵄩2
󵄩 󵄩2
≥ 󵄩󵄩󵄩𝑥∗ 󵄩󵄩󵄩 − 2 ⟨𝑥∗ , 𝐽𝑥1 ⟩ + 󵄩󵄩󵄩𝑥1 󵄩󵄩󵄩
= 𝑥∗ .
(IV) 𝑥∗ is some member of 𝐹.
(21)
𝑛→∞
𝜙 (𝑥𝑛𝑖 , 𝑥1 ) ≤ 𝜙 (𝑥∗ , 𝑥1 ) ,
that is, 𝑥∗ = 𝑦, and so
∗
= 𝜙 (𝑥 , 𝑥1 ) − 𝜙 (𝑥 , 𝑥1 ) = 0;
(26)
This, together with (35) and the Kadec-Klee property of 𝐸,
shows that
𝑚
lim 𝑇𝑖 𝑘 𝑥𝑘 = 𝑥∗ , ∀𝑖 ≥ 1.
(37)
K ∋𝑘 → ∞
𝑖
Abstract and Applied Analysis
5
On the other hand, by the assumptions that for each 𝑖 ≥ 1, 𝑇𝑖
is uniformly 𝐿 𝑖 -Lipschitz continuous, and noting again that
{𝑚𝑘 }𝑘∈K𝑖 = {𝑖, 𝑖 + 1, 𝑖 + 2, . . .}, that is, 𝑚𝑘+1 − 1 = 𝑚𝑘 for all
𝑘 ∈ K𝑖 , we then have
𝑚
𝑚
󵄩 󵄩 𝑚
󵄩
󵄩󵄩 𝑚𝑘+1
󵄩󵄩𝑇𝑖 𝑥𝑘 − 𝑇𝑖 𝑘 𝑥𝑘 󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩𝑇𝑖 𝑘+1 𝑥𝑘 − 𝑇𝑖 𝑘+1 𝑥𝑘+1 󵄩󵄩󵄩
󵄩 𝑚
󵄩
+ 󵄩󵄩󵄩𝑇𝑖 𝑘+1 𝑥𝑘+1 − 𝑥𝑘+1 󵄩󵄩󵄩
whose set of common solutions is denoted by Ω := ∩∞
𝑖=1 Ω𝑖 ,
where Ω𝑖 denotes the set of solutions to generalized mixed
equilibrium problem for 𝜃𝑖 , 𝐵𝑖 , and 𝜓𝑖 .
Define a countable family of mappings {𝑇𝑟,𝑖 }∞
𝑖=1 : 𝐸 → 𝐶
with 𝑟 > 0 as follows:
1
𝑇𝑟,𝑖 (𝑥) = {𝑧 ∈ 𝐶 : 𝜏𝑖 (𝑧, 𝑦)+ ⟨𝑦−𝑧, 𝐽𝑧−𝐽𝑥⟩ ≥ 0, ∀𝑦 ∈ 𝐶} ,
𝑟
𝑚
󵄩
󵄩 󵄩
󵄩
+ 󵄩󵄩󵄩𝑥𝑘+1 − 𝑥𝑘 󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑥𝑘 − 𝑇𝑖 𝑘 𝑥𝑘 󵄩󵄩󵄩
󵄩 󵄩 𝑚
󵄩
󵄩
≤ (𝐿 𝑖 +1) 󵄩󵄩󵄩𝑥𝑘+1 −𝑥𝑘 󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑇𝑖 𝑘+1 𝑥𝑘+1 −𝑥𝑘+1 󵄩󵄩󵄩
𝑚
󵄩
󵄩
+ 󵄩󵄩󵄩𝑥𝑘 − 𝑇𝑖 𝑘 𝑥𝑘 󵄩󵄩󵄩 .
(38)
𝑚
From (37) and 𝑥𝑘 → 𝑥∗ , we have that limK𝑖 ∋𝑘 → ∞ ‖𝑇𝑖 𝑘+1 𝑥𝑘 −
𝑚
𝑚
𝑇𝑖 𝑘 𝑥𝑘 ‖ = 0, and limK𝑖 ∋𝑘 → ∞ 𝑇𝑖 𝑘+1 𝑥𝑘 = 𝑥∗ ; that is,
𝑚 −1
limK𝑖 ∋𝑘 → ∞ 𝑇𝑖 (𝑇𝑖 𝑘+1 𝑥𝑘 ) = 𝑥∗ . It then follows that, for each
𝑖 ≥ 1,
𝑚
lim 𝑇𝑖 (𝑇𝑖 𝑘 𝑥𝑘 ) = 𝑥∗ .
K𝑖 ∋𝑘 → ∞
(39)
In view of the closeness of 𝑇𝑖 , it follows from (37) that 𝑇𝑖 𝑥∗ =
𝑥∗ , namely, for each 𝑖 ≥ 1, 𝑥∗ ∈ 𝐹(𝑇𝑖 ), and, hence, 𝑥∗ ∈ 𝐹.
(V) 𝑥∗ = Π𝐹 𝑥1 , and so 𝑥𝑛 → Π𝐹 𝑥1 as 𝑛 → ∞.
Put 𝑢 = Π𝐹 𝑥1 . Since 𝑢 ∈ 𝐹 ⊂ 𝐶𝑛 and 𝑥𝑛 = Π𝐶𝑛 𝑥1 , we have
that 𝜙(𝑥𝑛 , 𝑥1 ) ≤ 𝜙(𝑢, 𝑥1 ), for all 𝑛 ≥ 1. Then,
∗
𝜙 (𝑥 , 𝑥1 ) = lim 𝜙 (𝑥𝑛 , 𝑥1 ) ≤ 𝜙 (𝑢, 𝑥1 ) ,
𝑛→∞
(40)
∗
which implies that 𝑥 = 𝑢 since 𝑢 = Π𝐹 𝑥1 , and, hence, 𝑥𝑛 →
𝑥∗ = Π𝐹 𝑥1 .
This completes the proof.
4. Applications
Let 𝐸 be a smooth, strictly convex, and reflexive Banach
space, and let 𝐶 be a nonempty closed convex subset of 𝐸.
∗
Let {𝐵𝑖 }∞
𝑖=1 : 𝐶 → 𝐸 be a sequence of 𝛽𝑖 -inverse strongly
monotone mappings, {𝜓}∞
𝑖=1 : 𝐶 → R a sequence of lower
semicontinuous and convex functions, and {𝜃𝑖 }∞
𝑖=1 : 𝐶 ×
𝐶 → R a sequence of bifunctions satisfying the following
conditions:
(A1 ) 𝜃(𝑥, 𝑥) = 0;
(A3 ) lim sup𝑡↓0 𝜃(𝑥 + 𝑡(𝑧 − 𝑥), 𝑦) ≤ 𝜃(𝑥, 𝑦);
(A4 ) the mapping 𝑦 󳨃→ 𝜃(𝑥, 𝑦) is convex and lower semicontinuous.
A system of generalized mixed equilibrium problems
∞
∞
∗
(GMEPs), for {𝜃𝑖 }∞
𝑖=1 , {𝐵𝑖 }𝑖=1 , and {𝜓𝑖 }𝑖=1 is to find an 𝑥 ∈ 𝐶
such that
∀𝑦 ∈ 𝐶, 𝑖 ≥ 1,
where 𝜏𝑖 (𝑥, 𝑦) = 𝜃𝑖 (𝑥, 𝑦) + ⟨𝑦 − 𝑥, 𝐵𝑖 𝑥⟩ + 𝜓𝑖 (𝑦) − 𝜓𝑖 (𝑥), for all
𝑥, 𝑦 ∈ 𝐶, 𝑖 ≥ 1. It has been shown by Zhang [15] that
(1) {𝑇𝑟,𝑖 }∞
𝑖=1 is a sequence of single-valued mappings;
(2) {𝑇𝑟,𝑖 }∞
𝑖=1 is a sequence of closed quasi-𝜙-nonexpansive
mappings;
(3) ⋂∞
𝑖=1 𝐹(𝑇𝑟,𝑖 ) = Ω.
Now, we have the following result.
Theorem 9. Let 𝐸 be the same as that in Theorem 8, and let
C be a nonempty closed convex subset of E. Let {𝑇𝑟,𝑖 }∞
𝑖=1 :
𝐶 → 𝐶 be a sequence of mappings defined by (42) with
𝐹 := ⋂∞
𝑖=1 𝐹(𝑇𝑟,𝑖 ) ≠ 0. Let {𝛼𝑛 } be a sequence in [0, 𝜖] for some
𝜖 ∈ (0, 1), and let {𝛽𝑛 } be a sequence in [0, 1] satisfying 0 <
lim inf 𝑛 → ∞ 𝛽𝑛 (1 − 𝛽𝑛 ). Let {𝑥𝑛 } be the sequence generated by
𝑥1 ∈ 𝐶;
𝐶1 = 𝐶,
𝑦𝑛 = 𝐽−1 [𝛼𝑛 𝐽𝑥𝑛 + (1 − 𝛼𝑛 ) 𝐽𝑧𝑛 ] ,
𝑧𝑛 = 𝐽−1 [𝛽𝑛 𝐽𝑥𝑛 + (1 − 𝛽𝑛 ) 𝐽𝑇𝑟,𝑖𝑛 𝑥𝑛 ] ,
(43)
𝐶𝑛+1 = {𝑣 ∈ 𝐶𝑛 : 𝜙 (𝑣, 𝑦𝑛 ) ≤ 𝜙 (𝑣, 𝑥𝑛 )} ,
𝑥𝑛+1 = Π𝐶𝑛+1 𝑥1 ,
∀𝑛 ≥ 1,
where 𝑖𝑛 satisfies the positive integer equation: 𝑛 = 𝑖 + (𝑚 −
1)𝑚/2, and 𝑚 ≥ 𝑖 (𝑚 ≥ 𝑖, 𝑛 = 1, 2, . . .). Then, {𝑥𝑛 }
converges strongly to Π𝐹 𝑥1 which is some solution to the system
of generalized mixed equilibrium problems for {𝑇𝑟,𝑖 }∞
𝑖=1 .
Proof. Note that {𝑇𝑟,𝑖 }∞
𝑖=1 are quasi-𝜙-nonexpansive mappings; so, they are obviously generalized quasi-𝜙-asymptotically nonexpansive. Therefore, this conclusion can be
obtained immediately from Theorem 8.
Acknowledgments
(A2 ) 𝜃 is monotone; that is, 𝜃(𝑥, 𝑦) + 𝜃(𝑦, 𝑥) ≤ 0;
𝜃𝑖 (𝑥∗ , 𝑦) + ⟨𝑦 − 𝑥∗ , 𝐵𝑖 𝑥∗ ⟩ + 𝜓𝑖 (𝑦) − 𝜓𝑖 (𝑥∗ ) ≥ 0,
∀𝑖 ≥ 1,
(42)
(41)
The author is greatly grateful to the referees for their useful
suggestions by which the contents of this paper are improved.
This work was supported by the National Natural Science
Foundation of China (Grant no. 11061037).
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