Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2013, Article ID 107296, 13 pages
http://dx.doi.org/10.1155/2013/107296
Research Article
A New Hybrid Projection Algorithm for System of Equilibrium
Problems and Variational Inequality Problems and Two Finite
Families of Quasi-𝜙-Nonexpansive Mappings
Pongrus Phuangphoo1,2 and Poom Kumam1
1
Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), Bang Mod,
Thung Kru, Bangkok 10140, Thailand
2
Department of Mathematics, Faculty of Education, Bansomdejchaopraya Rajabhat University (BSRU), Thonburi,
Bangkok 10600, Thailand
Correspondence should be addressed to Poom Kumam; poom.kum@kmutt.ac.th
Received 28 July 2012; Revised 24 September 2012; Accepted 28 September 2012
Academic Editor: Yongfu Su
Copyright © 2013 P. Phuangphoo 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 modified Mann’s iterative procedure by using the hybrid projection method for solving the common solution of the
system of equilibrium problems for a finite family of bifunctions satisfying certain condition, the common solution of fixed point
problems for two finite families of quasi-𝜙-nonexpansive mappings, and the common solution of variational inequality problems
for a finite family of continuous monotone mappings in a uniformly smooth and strictly convex real Banach space. Then, we prove
a strong convergence theorem of the iterative procedure generated by some mild conditions. Our result presented in this paper
improves and generalizes some well-known results in the literature.
1. Introduction
Throughout this paper, we denote by R and N the set of all real
numbers and the set of all positive numbers, respectively. We
also assume that 𝐸 is a real Banach space and 𝐸∗ is the dual
space of 𝐸. Let 𝐶 be a nonempty, closed and convex subset of a
real Banach space 𝐸 with the dual 𝐸∗ . We recall the following
definitions.
A mapping 𝐴 : 𝐶 → 𝐶 is said to be nonexpansive if
󵄩
󵄩 󵄩
󵄩󵄩
󵄩󵄩𝐴𝑥 − 𝐴𝑦󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩 ,
∀𝑥, 𝑦 ∈ 𝐶.
(1)
A mapping 𝐴 : 𝐶 → 𝐸∗ is said to be monotone if for each
𝑥, 𝑦 ∈ 𝐶, such that
⟨𝑥 − 𝑦, 𝐴𝑥 − 𝐴𝑦⟩ ≥ 0.
(2)
A mapping 𝐴 : 𝐶 → 𝐸∗ is said to be 𝛿-strongly monotone
if there exists a constant 𝛿 > 0 such that
󵄩2
󵄩
⟨𝑥 − 𝑦, 𝐴𝑥 − 𝐴𝑦⟩ ≥ 𝛿󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩 ,
∀𝑥, 𝑦 ∈ 𝐶.
(3)
A mapping 𝐴 : 𝐶 → 𝐸∗ is said to be 𝛿-inverse strongly
monotone if there exists a constant 𝛿 > 0 such that
󵄩2
󵄩
⟨𝑥 − 𝑦, 𝐴𝑥 − 𝐴𝑦⟩ ≥ 𝛿󵄩󵄩󵄩𝐴𝑥 − 𝐴𝑦󵄩󵄩󵄩 ,
∀𝑥, 𝑦 ∈ 𝐶.
(4)
If 𝐴 is 𝛿-inverse strongly monotone, then it is Lipschitz
continuous with constant 1/𝛿, that is, for all 𝑥, 𝑦 ∈ 𝐶, ‖𝐴𝑥 −
𝐴𝑦‖ ≤ (1/𝛿)‖𝑥 − 𝑦‖.
Clearly, the class of monotone mappings include the class
of 𝛿-inverse strongly monotone mappings.
Let 𝐴 : 𝐶 → 𝐸∗ be a monotone mapping. The variational
inequality problem is to find a point 𝑧 ∈ 𝐶 such that
⟨𝑦 − 𝑧, 𝐴𝑧⟩ ≥ 0,
∀𝑦 ∈ 𝐶.
(5)
The set of the solution of the variational inequality
problem is denoted by VI(𝐶, 𝐴).
Let 𝐶 be a nonempty, closed and convex subset of a
smooth, strictly convex and reflexive Banach space 𝐸, let 𝑇 :
𝐶 → 𝐶 be a mapping, and 𝐹(𝑇) be the set of fixed points of
𝑇.
2
Abstract and Applied Analysis
A point 𝑥 ∈ 𝐶 is said to be a fixed point of 𝑇 if 𝑇𝑥 = 𝑥.
The set of the solution of the fixed point of 𝑇 is denoted by
𝐹(𝑇) := {𝑥 ∈ 𝐶 : 𝑇𝑥 = 𝑥}.
A point 𝑝 ∈ 𝐶 is said to be an asymptotic fixed point of
𝑇 if there exists a sequence {𝑥𝑛 } ⊂ 𝐶 such that 𝑥𝑛 ⇀ 𝑝 and
‖𝑥𝑛 − 𝑇𝑥𝑛 ‖ → 0. We denoted the set of all asymptotic fixed
̂
points of 𝑇 by 𝐹(𝑇).
A point 𝑝 ∈ 𝐶 is said to be a strong asymptotic fixed point
of 𝑇 if there exists a sequence {𝑥𝑛 } ⊂ 𝐶 such that 𝑥𝑛 → 𝑝 and
‖𝑥𝑛 − 𝑇𝑥𝑛 ‖ → 0. We denoted the set of all strong asymptotic
̃
fixed points of 𝑇 by 𝐹(𝑇).
Let 𝐹 : 𝐶 × 𝐶 → R be a bifunction. The equilibrium
problem is to find a point 𝑧 ∈ 𝐶 such that
𝐹 (𝑧, 𝑦) ≥ 0,
∀𝑦 ∈ 𝐶.
(6)
The set of the solution of equilibrium problem is denoted
by EP(𝐹). Numerous problems in sciences, mathematics,
optimizations, and economics reduced to find a solution
of equilibrium problems. The equilibrium problems include
variational inequality problems and fixed point problem,
and optimization problems as special cases (see, e.g., [1–
3]). Recently, many authors have considered the problem for
finding the common solution of fixed point problems, the
common solution of equilibrium problems, and the common
solution of variational inequality problems.
In 1953, Mann [4] introduced the iterative sequence
{𝑥𝑛 }𝑛∈N which is defined by
𝑥𝑛+1 = 𝛼𝑛 𝑥𝑛 + (1 − 𝛼𝑛 ) 𝑇𝑥𝑛 ,
(7)
where the initial element 𝑥0 ∈ 𝐶 is arbitrary, 𝑇 is a
nonexpansive mapping, and {𝛼𝑛 } is the sequence in [0, 1] such
that lim𝑛 → ∞ 𝛼𝑛 = 0 and ∑∞
𝑛=1 𝛼𝑛 = ∞. The sequence of (7) is
generally referred to as the Mann iteration.
In 2009, Takahashi and Zembayashi [5] introduced
the following iterative scheme by the shrinking projection
method, and they proved that {𝑥𝑛 }𝑛∈N converges strongly to
𝑞 = ΠEP(𝐹)∩𝐹(𝑆) , under appropriate conditions.
Theorem TZ. Let 𝐸 be a uniformly smooth and uniformly
convex real Banach space, and let 𝐶 be a nonempty, closed
and convex subset of 𝐸. Let 𝐹 be a bifunction from 𝐶 × 𝐶 to
R satisfying (A1)–(A4) and let 𝑇 be a relatively nonexpansive
mapping from 𝐶 into itself such that 𝐹(𝑆) ∩ EP(𝐹) ≠ 0. Let {𝑥𝑛 }
be a sequence generated by
for every 𝑛 ∈ N ∪ {0}, where 𝐽 is the duality mapping on 𝐸,
the sequence 𝛼𝑛 ⊂ [0, 1] satisfies lim inf𝑛 → ∞ 𝛼𝑛 (1 − 𝛼𝑛 ) > 0,
and {𝑟𝑛 }𝑛∈N ⊂ (0, ∞) satisfies lim inf𝑛 → ∞ 𝑟𝑛 > 0. Then,
the sequence {𝑥𝑛 } converges strongly to Π𝐹(𝑆)∩EP(𝑓) 𝑥0 , where
Π𝐹(𝑆)∩EP(𝐹) 𝑥0 is the generalized projection of 𝐸 onto 𝐹(𝑆) ∩
EP(𝐹).
In 2009, Qin et al. [6] extended the iterative process
(8) from a single relatively nonexpansive mapping to two
relatively quasi-nonexpansive mappings. In 2011, Zegeye and
Shahzad [7] introduced an iterative process for finding an
element in the common fixed point set of finite family
of closed relatively quasi-nonexpansive mappings, common
solutions of finite family of equilibrium problems, and common solutions of the finite family of variational inequality
problems for monotone mappings in Banach spaces.
Theorem ZS. Let 𝐶 be a nonempty, closed and convex subset
of a uniformly smooth and strictly convex real Banach space 𝐸
with the dual 𝐸∗ . Let 𝑓𝑘 : 𝐶 × 𝐶 → R, 𝑘 = 1, 2, 3, . . . , 𝐿, be a
finite family of bifunctions. Let 𝑆𝑗 : 𝐶 → 𝐶, 𝑗 = 1, 2, 3, . . . , 𝐷
be a finite family relatively quasi-nonexpansive mappings and
𝐴 𝑖 : 𝐶 → 𝐸∗ , 𝑖 = 1, 2, 3, . . . , 𝑁 be a finite family of continuous
monotone mappings.
For 𝑥 ∈ 𝐸, they define mappings 𝐹𝑟𝑛 , 𝑇𝑟𝑛 : 𝐸 → 𝐶 by
𝐹𝑟𝑛 𝑥 : = {𝑧 ∈ 𝐶 : ⟨𝑦 − 𝑧, 𝐴 𝑛 𝑧⟩
+
1
⟨𝑦 − 𝑧, 𝐽𝑧 − 𝐽𝑥⟩ ≥ 0, ∀𝑦 ∈ 𝐶} ,
𝑟𝑛
𝑇𝑟𝑛 𝑥 : = {𝑧 ∈ 𝐶 : 𝑓𝑛 (𝑧, 𝑦)
+
1
⟨𝑦 − 𝑧, 𝐽𝑧 − 𝐽𝑥⟩ ≥ 0, ∀𝑦 ∈ 𝐶} ,
𝑟𝑛
where 𝑆𝑛 ≡ 𝑆𝑛 (mod D), 𝐴 𝑛 ≡ 𝐴 𝑛 (mod N), 𝑓𝑛 (⋅, ⋅) ≡ 𝑓𝑛 (mod
L)(⋅, ⋅), and {𝑟𝑛 }𝑛∈N ⊂ [𝑐1 , ∞) for some 𝑐1 > 0. Assume that
𝑁
𝐿
F := (⋂𝐷
𝑗=1 𝐹(𝑆𝑗 )) ⋂(⋂𝑖=1 𝑉𝐼(𝐶, 𝐴 𝑖 )) ⋂(⋂𝑘=1 EP(𝑓𝑘 )) ≠ 0.
Let {𝑥𝑛 } be a sequence generated by
𝑥0 ∈ 𝐶0 = 𝐶 𝑐ℎ𝑜𝑠𝑒𝑛 𝑎𝑟𝑏𝑖𝑡𝑟𝑎𝑟𝑦,
𝑧𝑛 = 𝐹𝑟𝑛 𝑥𝑛 ,
𝑢𝑛 = 𝑇𝑟𝑛 𝑥𝑛 ,
𝑥0 ∈ 𝐶0 = 𝐶 𝑐ℎ𝑜𝑠𝑒𝑛 𝑎𝑟𝑏𝑖𝑡𝑟𝑎𝑟𝑦,
𝑦𝑛 = 𝐽−1 (𝛼0 𝐽𝑥𝑛 + 𝛼1 𝐽𝑧𝑛 + 𝛼2 𝐽𝑆𝑛 𝑢𝑛 ) ,
𝑦𝑛 = 𝐽−1 (𝛼𝑛 𝐽𝑥𝑛 + (1 − 𝛼𝑛 ) 𝐽𝑇𝑥𝑛 ) ,
𝐶𝑛+1 = {𝑧 ∈ 𝐶𝑛 : 𝜙 (𝑧, 𝑦𝑛 ) ≤ 𝜙 (𝑧, 𝑥𝑛 )} ,
𝑢𝑛 ∈ 𝐶 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡 𝐹 (𝑢𝑛 , 𝑦) +
1
⟨𝑦 − 𝑢𝑛 , 𝐽𝑢𝑛 − 𝐽𝑦𝑛 ⟩ ≥ 0,
𝑟𝑛
∀𝑦 ∈ 𝐶,
𝐶𝑛+1 = {𝑧 ∈ 𝐶𝑛 : 𝜙 (𝑧, 𝑢𝑛 ) ≤ 𝜙 (𝑧, 𝑥𝑛 )} ,
𝑥𝑛+1 = Π𝐶𝑛+1 (𝑥0 ) ,
∀𝑛 ≥ 0,
(8)
(9)
𝑥𝑛+1 = Π𝐶𝑛+1 (𝑥0 ) ,
(10)
∀𝑛 ≥ 0,
where the real numbers 𝛼0 , 𝛼1 , 𝛼2 ∈ (0, 1) such that 𝛼0 + 𝛼1 +
𝛼2 = 1. Then, {𝑥𝑛 }𝑛∈N converges strongly to an element of F.
In this paper, motivated and inspired by the previously
mentioned above results, we introduce a new iterative procedure for solving the common solution of system of equilibrium problems for a finite family of bifunctions satisfying
certain conditions and the common solution of fixed point
Abstract and Applied Analysis
3
problems for two countable families of quasi-𝜙-nonexpansive
mappings and the common solution of variational inequality
problems for a finite family of monotone mappings in a
uniformly smooth and strictly convex real Banach space.
Then, we prove a strong convergence theorem of the iterative
procedure generated by the conditions. The results obtained
in this paper extend and improve several recent results in this
area.
2. Preliminaries
A Banach space 𝐸 is said to be strictly convex if ‖𝑥 + 𝑦‖/2 <
1 for all 𝑥, 𝑦 ∈ 𝐸 with ‖𝑥‖ = ‖𝑦‖ = 1 and 𝑥 ≠ 𝑦. It is said
to be uniformly convex if lim𝑛 → ∞ ‖𝑥𝑛 − 𝑦𝑛 ‖ = 0 for any two
sequences {𝑥𝑛 } and {𝑦𝑛 } in 𝐸 such that ‖𝑥𝑛 ‖ ≤ 1, ‖𝑦𝑛 ‖ ≤ 1 and
󵄩󵄩 𝑥 + 𝑦𝑛 󵄩󵄩
󵄩󵄩 = 1.
lim 󵄩󵄩 𝑛
󵄩
2 󵄩󵄩
󵄩󵄩
𝑛→∞ 󵄩
(11)
Let 𝑈𝐸 = {𝑥 ∈ 𝐸 : ‖𝑥‖ = 1} be the unit sphere of 𝐸. Then
the Banach space 𝐸 is said to be smooth if
󵄩
󵄩󵄩
󵄩𝑥 + 𝑡𝑦󵄩󵄩󵄩 − ‖𝑥‖
lim 󵄩
𝑡→0
𝑡
(12)
exists for each 𝑥, 𝑦 ∈ 𝑈𝐸 . It is said to be uniformly smooth if
the limit (12) is attained uniformly for all 𝑥, 𝑦 ∈ 𝑈𝐸 .
Let 𝐸 be a Banach space. Then a function 𝜌𝐸 : R+ → R+
is said to be the modulus of smoothness of 𝐸 if
󵄩
󵄩 󵄩
󵄩󵄩
󵄩𝑥 + 𝑦󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩
󵄩 󵄩
𝜌𝐸 (𝑡) = sup { 󵄩
− 1 : ‖𝑥‖ = 1, 󵄩󵄩󵄩𝑦󵄩󵄩󵄩 = 𝑡} .
2
(13)
The space 𝐸 is said to be smooth if 𝜌𝐸 (𝑡) > 0, ∀𝑡 > 0 and is
said to be uniformly smooth if and only if lim𝑡 → 0+ 𝜌𝐸 (𝑡)/𝑡 = 0.
The modulus of convexity of 𝐸 is the function 𝛿𝐸 : [0, 2] →
[0, 1] defined by
󵄩󵄩 𝑥 + 𝑦 󵄩󵄩
󵄩󵄩 : ‖𝑥‖ ≤ 1, 󵄩󵄩󵄩𝑦󵄩󵄩󵄩 ≤ 1; 󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩 ≥ 𝜖} .
𝛿𝐸 (𝜖) = inf {1 − 󵄩󵄩󵄩󵄩
󵄩
󵄩
󵄩
󵄩 󵄩
󵄩 2 󵄩󵄩
(14)
A Banach space 𝐸 is said to be uniformly convex if and
only if 𝛿𝐸 (𝜖) > 0 for all 𝜖 ∈ (0, 2].
We recall the following definitions.
Definition 1. Let 𝐶 be a nonempty set.
(1) A mapping 𝑇 : 𝐶 → 𝐶 is said to be closed if for each
{𝑥𝑛 } ⊂ 𝐶, 𝑥𝑛 → 𝑥 and 𝑇𝑥𝑛 → 𝑦 imply 𝑇𝑥 = 𝑦.
(2) A mapping 𝑇 : 𝐶 → 𝐶 is said to be quasi-𝜙-nonexpansive (relatively quasi-nonexpansive) if 𝐹(𝑇) ≠ 0,
and
𝜙 (𝑝, 𝑇𝑥) ≤ 𝜙 (𝑝, 𝑥) ,
∀𝑥 ∈ 𝐶, 𝑝 ∈ 𝐹(𝑇) .
(15)
(3) A mapping 𝑇 : 𝐶 → 𝐶 is said to be relatively nonex̂
pansive [8, 9] if 𝐹(𝑇) ≠ 0, 𝐹(𝑇) = 𝐹(𝑇)
and
𝜙 (𝑝, 𝑇𝑥) ≤ 𝜙 (𝑝, 𝑥) ,
∀𝑥 ∈ 𝐶, 𝑝 ∈ 𝐹(𝑇) .
(16)
(4) A mapping 𝑇 : 𝐶 → 𝐶 is said to be weak relatively
̃
nonexpansive [10] if 𝐹(𝑇) ≠ 0, 𝐹(𝑇) = 𝐹(𝑇)
and
𝜙 (𝑝, 𝑇𝑥) ≤ 𝜙 (𝑝, 𝑥) ,
∀𝑥 ∈ 𝐶, 𝑝 ∈ 𝐹(𝑇) .
(17)
Remark 2. We here the following basic properties.
(1) Each relatively nonexpansive mapping is closed.
(2) The class of quasi-𝜙-nonexpansive mappings contains
properly the class of weak relatively nonexpansive
mappings as a subclass, but the converse may be not
true.
(3) The class of weak relatively nonexpansive mappings
contains properly the class of relatively nonexpansive
mappings as a subclass, but the converse may be not
true.
(4) The class of quasi-𝜙-nonexpansive mappings contains
properly the class of relatively nonexpansive mappings as a subclass, but the converse may be not true.
(5) If 𝐸 is a real uniformly smooth Banach space, then 𝐽 is
uniformly continuous on each bounded subset of 𝐸.
(6) If 𝐸 is a strictly convex reflexive Banach space, then
𝐽−1 is hemicontinuous, that is, 𝐽−1 is norm-to-weak∗
continuous.
(7) If 𝐸 is a smooth and strictly convex reflexive Banach
space, then 𝐽 is single-valued, one-to-one, and onto.
(8) A Banach space 𝐸 is uniformly smooth if and only if
𝐸∗ is uniformly convex.
(9) 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.
(10) A Banach space 𝐸 is strictly convex if and only if 𝐽 is
strictly monotone, that is,
⟨𝑥 − 𝑦, 𝑥∗ − 𝑦∗ ⟩ > 0,
whenever 𝑥, 𝑦 ∈ 𝐸,
𝑥 ≠ 𝑦, 𝑥∗ ∈ 𝐽𝑥, 𝑦∗ ∈ 𝐽𝑦.
(18)
(11) Both uniformly smooth Banach spaces and uniformly
convex Banach spaces are reflexive.
(12) 𝐸∗ is uniformly convex, then 𝐽 is uniformly norm-tonorm continuous on each bounded subset of 𝐸.
Let 𝐸 be a real Banach space and {𝑥𝑛 } be a sequence in 𝐸.
We denote by 𝑥𝑛 → 𝑥 and 𝑥𝑛 ⇀ 𝑥 the strong convergence
and weak convergence of {𝑥𝑛 }, respectively. The normalized
∗
duality mapping 𝐽 from 𝐸 to 2𝐸 is defined by
󵄩 󵄩2
𝐽𝑥 = {𝑓∗ ∈ 𝐸∗ : ⟨𝑥, 𝑓∗ ⟩ = ‖𝑥‖2 = 󵄩󵄩󵄩𝑓∗ 󵄩󵄩󵄩 } ,
∀𝑥 ∈ 𝐸, (19)
where ⟨⋅, ⋅⟩ denotes the duality pairing. It is well known that if
𝐸 is smooth, then 𝐽 is single-valued and demicontinuous, and
if 𝐸 is uniformly smooth, then 𝐽 is uniformly continuous on
bounded subset of 𝐸. Moreover, if 𝐸 is reflexive and strictly
convex Banach space with a strictly convex dual, then 𝐽−1
is single-valued, one-to-one, surjective, and it is the duality
4
Abstract and Applied Analysis
mapping from 𝐸∗ to 𝐸 and so 𝐽𝐽−1 = 𝐼𝐸∗ and 𝐽−1 𝐽 = 𝐼𝐸 (see
[11, 12]). We note that in a Hilbert space 𝐻, the mapping 𝐽 is
the identity operator.
Now, let 𝐸 be a smooth and strictly convex reflexive
Banach space. As Alber (see [13]) and Kamimura and Takahashi (see [14]) did, the Lyapunov functional 𝜙 : 𝐸 × 𝐸 → R+
is defined by
󵄩 󵄩2
𝜙 (𝑥, 𝑦) = ‖𝑥‖2 − 2 ⟨𝑥, 𝐽𝑦⟩ + 󵄩󵄩󵄩𝑦󵄩󵄩󵄩 ,
∀𝑥, 𝑦 ∈ 𝐸.
(20)
It follows from Kohsaka and Takahashi (see [15]) that
𝜙(𝑥, 𝑦) = 0 if and only if 𝑥 = 𝑦 and that
󵄩 󵄩2
󵄩 󵄩2
(‖𝑥‖ − 󵄩󵄩󵄩𝑦󵄩󵄩󵄩) ≤ 𝜙 (𝑥, 𝑦) ≤ (‖𝑥‖ + 󵄩󵄩󵄩𝑦󵄩󵄩󵄩) .
(21)
Further suppose that 𝐶 is nonempty, closed and convex
subset of 𝐸. The generalized, projection (Alber see [13]) Π𝐶 :
𝐸 → 𝐶 is defined by for each 𝑥 ∈ 𝐸,
Π𝐶 (𝑥) = arg min𝜙 (𝑥, 𝑦) .
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𝑦∈𝐶
Remark 3. If 𝐸 is a real Hilbert space 𝐻, then 𝜙(𝑥, 𝑦) = ‖𝑥 −
𝑦‖2 and Π𝐶 = 𝑃𝐶 (the metric projection of 𝐻 onto 𝐶).
Lemma 4 (Alber [13]). Let 𝐶 be a nonempty, closed and
convex subset of a smooth and strictly convex reflexive Banach
space 𝐸, and let 𝑥 ∈ 𝐸. Then
𝜙 (𝑥, Π𝐶 (𝑦)) + 𝜙 (Π𝐶 (𝑦) , 𝑦)
≤ 𝜙 (𝑥, 𝑦) ,
∀𝑥 ∈ 𝐶, ∀𝑦 ∈ 𝐸.
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Lemma 5 (Kamimura and Takahashi [14]). Let 𝐶 be a
nonempty, closed and convex subset of a smooth and strictly
convex reflexive Banach space 𝐸, and let 𝑥 ∈ 𝐸 and 𝑝̂ ∈ 𝐶.
Then,
̂ ≥ 0,
𝑝̂ = Π𝐶 (𝑥) ⇐⇒ ⟨𝑝̂ − 𝑦, 𝐽𝑥 − 𝐽𝑝⟩
∀𝑦 ∈ 𝐶.
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Lemma 6 (Qin et al. [6] and Kohsaka and Takahashi [16]).
Let 𝐸 be a smooth, strictly convex and reflexive Banach space,
and 𝐴 ⊂ 𝐸 × 𝐸∗ is a continuous monotone mapping with
𝐴−1 (0) ≠ 𝜙. Then, it is proved in [16] that the resolvent 𝐽𝑟 :=
(𝐽 + 𝑟𝐴)−1 𝐽, for 𝑟 > 0 is quasi-𝜙-nonexpansive. Moreover, if
𝑇 : 𝐸 → 𝐸 is quasi-𝜙-nonexpansive, then using the definition
of 𝜙 one can show that 𝐹(𝑇) is closed and convex (see [6]).
Lemma 7 (Kamimura and Takahashi [14]). Let 𝐸 be a
uniformly convex and smooth real Banach space and let {𝑥𝑛 }
and {𝑦𝑛 } be two sequences of 𝐸. If 𝜙(𝑥𝑛 , 𝑦𝑛 ) → 0 and either
{𝑥𝑛 } or {𝑦𝑛 } is bounded, then ‖𝑥𝑛 − 𝑦𝑛 ‖ → 0.
Lemma 8. Let 𝐸 be a uniformly smooth and strictly convex
Banach space with the Kadec-Klee property, let {𝑥𝑛 } and {𝑦𝑛 } be
two sequences of 𝐸, and 𝑝 ∈ 𝐸. If 𝑥𝑛 → 𝑝 and 𝜙(𝑥𝑛 , 𝑦𝑛 ) → 0,
then 𝑦𝑛 → 𝑝.
For solving the equilibrium problem, let us assume that
the bifunction 𝐹 : 𝐶 × 𝐶 → R satisfies the following
conditions:
(A1) 𝐹(𝑥, 𝑥) = 0, for all 𝑥 ∈ 𝐶;
(A2) 𝐹 is monotone, that is, 𝐹(𝑥, 𝑦) + 𝐹(𝑦, 𝑥) ≤ 0, ∀𝑥, 𝑦 ∈
𝐶;
(A3) lim𝑡 → 0 𝐹(𝑡𝑧 + (1 − 𝑡)𝑥, 𝑦) ≤ 𝐹(𝑥, 𝑦), ∀𝑥, 𝑦, 𝑧 ∈ 𝐶;
(A4) for any 𝑥 ∈ 𝐶, 𝑦 󳨃→ 𝐹(𝑥, 𝑦) is convex and lower
semicontinuous.
Lemma 9 (Blum and Oettli [1]). Let 𝐶 be a nonempty, closed
and convex subset of a smooth and strictly convex reflexive
Banach space 𝐸 and let 𝐹 : 𝐶 × 𝐶 → R be a bifunction
satisfying the following conditions (A1)–(A4). Let 𝑟 > 0 be any
given number and 𝑥 ∈ 𝐸 be any point. Then, there exists a point
𝑧 ∈ 𝐶 such that
𝐹 (𝑧, 𝑦) +
1
⟨𝑦 − 𝑧, 𝐽𝑧 − 𝐽𝑥⟩ ≥ 0,
𝑟
∀𝑦 ∈ 𝐶.
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Lemma 10 (Takahashi and Zembayashi [5]). Let 𝐶 be a
nonempty, closed and convex subset of a uniformly smooth and
strictly convex real Banach space 𝐸, and 𝐹 : 𝐶 × 𝐶 → R be
a bifunction satisfying the following conditions (A1)–(A4). Let
𝑟 > 0 be any given number and 𝑥 ∈ 𝐸 be any point defined
a mapping 𝑇𝑟 : 𝐸 → 𝐶 as follows. Then, there exists a point
𝑧 ∈ 𝐶 such that
𝑇𝑟 (𝑥) = {𝑧 ∈ 𝐶 : 𝐹 (𝑧, 𝑦)
1
+ ⟨𝑦 − 𝑧, 𝐽𝑧 − 𝐽𝑥⟩ ≥ 0, ∀𝑦 ∈ 𝐶} .
𝑟
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Then, the following conclusions hold:
(1) 𝑇𝑟 is single-valued;
(2) 𝑇𝑟 is a firmly nonexpansive type mapping, that is,
⟨𝑇𝑟 𝑥 − 𝑇𝑟 𝑦, 𝐽𝑇𝑟 𝑥 − 𝐽𝑇𝑟 𝑦⟩
≤ ⟨𝑇𝑟 𝑥 − 𝑇𝑟 𝑦, 𝐽𝑥 − 𝐽𝑦⟩ ,
∀𝑥, 𝑦 ∈ 𝐸;
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(3) 𝐹(𝑇𝑟 ) = EP(𝐹);
(4) EP(𝐹) is a closed and convex subset of 𝐶;
(5) 𝜙(𝑝, 𝑇𝑟 𝑥) + 𝜙(𝑇𝑟 𝑥, 𝑥) ≤ 𝜙(𝑝, 𝑥), ∀𝑝 ∈ 𝐹(𝑇𝑟 ).
Lemma 11 (Zegeye and Shahzad [7]). Let 𝐶 be a nonempty,
closed and convex subset of a uniformly smooth and strictly
convex real Banach space 𝐸, and 𝐴 : 𝐶 × 𝐸∗ → R be a
continuous monotone mapping. Let 𝑟 > 0 be any given number
and 𝑥 ∈ 𝐸 be any point defined a mapping 𝐾𝑟 : 𝐸 → 𝐶 as
follows:
𝐾𝑟 (𝑥) = {𝑧 ∈ 𝐶 : ⟨𝑦 − 𝑧, 𝐴𝑧⟩
1
+ ⟨𝑦 − 𝑧, 𝐽𝑧 − 𝐽𝑥⟩ ≥ 0, ∀𝑦 ∈ 𝐶} .
𝑟
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Abstract and Applied Analysis
5
𝐷
𝑀
Assume that Ω := (⋂𝐷
𝑖=1 𝐹(𝑇𝑖 )) ⋂(⋂𝑖=1 𝐹(𝑆𝑖 )) ⋂ (⋂𝑘=1
𝑁
SEP(𝐹𝑘 )) ⋂(⋂𝑛=1 VI(𝐶, 𝐴 𝑛 )) is a nonempty and bounded in 𝐶.
Let {𝑥𝑛 } be a sequence generated by
Then, the following conclusions hold:
(1) 𝐾𝑟 is single valued;
(2) 𝐹(𝐾𝑟 ) is a firmly nonexpansive type mapping, that is,
⟨𝐾𝑟 𝑥 − 𝐾𝑟 𝑦, 𝐽𝐾𝑟 𝑥 − 𝐽𝐾𝑟 𝑦⟩
≤ ⟨𝐾𝑟 𝑥 − 𝐾𝑟 𝑦, 𝐽𝑥 − 𝐽𝑦⟩ ,
∀𝑥, 𝑦 ∈ 𝐸;
𝑥𝑛+1 = Π𝐶𝑛+1 (𝑥0 ) ,
Lemma 12 (Xu [17]). Let 𝐸 be a uniformly convex Banach
space, let 𝐵𝑟 (0) be a closed ball of 𝐸, where 𝑟 > 0. Then, there
exists a continuous, strictly increasing, and convex function
𝑔 : [0, ∞) → [0, ∞) with 𝑔(0) = 0 such that
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In this section, we prove a strong convergence theorem which
solves the problem for finding a common solution of the
system of equilibrium problems and variational inequality
problems and fixed point problems in Banach spaces.
Theorem 13. Let 𝐶 be a nonempty, closed and convex subset
of a uniformly smooth and strictly convex real Banach space 𝐸
which has the Kadec-Klee property. Suppose that
(1) {𝐹𝑘 }𝑀
𝑘=1 : 𝐶 × 𝐶 → R is a finite family of bifunctions
satisfying conditions (A1)–(A4), where 𝑘 = 1, 2, 3, . . . ,
𝑀;
𝐷
(2) {𝑇𝑖 }𝐷
𝑖=1 and {𝑆𝑖 }𝑖=1 are two finite families of quasi-𝜙nonexpansive mappings from 𝐶 into 𝐸, where 𝑖 =
1, 2, 3, . . . , 𝐷;
∗
(3) {𝐴 𝑛 }𝑁
𝑛=1 : 𝐶 → 𝐸 is a finite family of continuous
monotone mappings, where 𝑛 = 1, 2, 3, . . . , 𝑁;
(4) For 𝑥 ∈ 𝐸, one defines the mappings 𝐾𝑟𝑛 , 𝑇𝑟𝑛 : 𝐸 → 𝐶
by
Proof. We will complete this proof by seven steps below.
󵄩 󵄩2 󵄩 󵄩2
2 ⟨𝑧, 𝐽𝑥𝑛 − 𝐽𝑦𝑛 ⟩ − 󵄩󵄩󵄩𝑥𝑛 󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑦𝑛 󵄩󵄩󵄩 ≤ 0.
Step 2. We will show that Ω ⊂ 𝐶𝑛 for each 𝑛 ≥ 0.
From the assumption, we see that 𝐹 ⊂ 𝐶0 = 𝐶. Suppose
that 𝐹 ⊂ 𝐶𝑘 for some 𝑘 ≥ 1. Now, for 𝑝 ∈ Ω, Since 𝑆𝑘 and
𝑇𝑘 are quasi-𝜙-nonexpansive and by Lemmas 10 and 11, we
compute
𝜙 (𝑝, 𝑦𝑘 ) = 𝜙 (𝑝, 𝐽−1 (𝛼𝐽𝑥𝑘 + 𝛽𝐽𝑇𝑘 𝑧𝑘 + 𝛾𝐽𝑆𝑘 𝑢𝑘 ))
󵄩 󵄩2
= 󵄩󵄩󵄩𝑝󵄩󵄩󵄩 − 2 ⟨𝑝, 𝛼𝐽𝑥𝑘 + 𝛽𝐽𝑇𝑘 𝑧𝑘 + 𝛾𝐽𝑆𝑘 𝑢𝑘 ⟩
󵄩2
󵄩
+ 󵄩󵄩󵄩𝛼𝐽𝑥𝑘 + 𝛽𝐽𝑇𝑘 𝑧𝑘 + 𝛾𝐽𝑆𝑘 𝑢𝑘 󵄩󵄩󵄩
󵄩 󵄩2
≤ 󵄩󵄩󵄩𝑝󵄩󵄩󵄩 − 2𝛼 ⟨𝑝, 𝐽𝑥𝑘 ⟩ − 2𝛽⟨𝑝, 𝐽𝑇𝑘 𝑧𝑘 ⟩
󵄩2
󵄩 󵄩2
󵄩
− 2𝛾 ⟨𝑝, 𝐽𝑆𝑘 𝑢𝑘 ⟩ + 𝛼󵄩󵄩󵄩𝐽𝑥𝑘 󵄩󵄩󵄩 + 𝛽󵄩󵄩󵄩𝐽𝑇𝑘 𝑧𝑘 󵄩󵄩󵄩
󵄩2
󵄩
+ 𝛾 󵄩󵄩󵄩𝐽𝑆𝑘 𝑢𝑘 󵄩󵄩󵄩
≤ 𝛼𝜙 (𝑝, 𝑥𝑘 ) + 𝛽𝜙 (𝑝, 𝑧𝑘 ) + 𝛾𝜙 (𝑝, 𝑢𝑘 )
(31)
(33)
It follows that 𝐶𝑛 is convex for each 𝑛 ≥ 0. Therefore, 𝐶𝑛 is
closed and convex for each 𝑛 ≥ 0.
= 𝛼𝜙 (𝑝, 𝑥𝑘 ) + 𝛽𝜙 (𝑝, 𝑇𝑘 𝑧𝑘 ) + 𝛾𝜙 (𝑝, 𝑆𝑘 𝑢𝑘 )
𝐾𝑟𝑛 (𝑥) : = {𝑧 ∈ 𝐶 : ⟨𝑦 − 𝑧, 𝐴 𝑛 𝑧⟩
1
⟨𝑦 − 𝑧, 𝐽𝑧 − 𝐽𝑥⟩ ≥ 0, ∀𝑦 ∈ 𝐶} .
𝑟𝑛
where 𝐴 𝑛 ≡ 𝐴 𝑛 (mod 𝑁), 𝐹𝑛 ≡ 𝐹𝑛 (mod 𝑀), 𝑇𝑛 ≡ 𝑇𝑛 (mod
𝐷), 𝑆𝑛 ≡ 𝑆𝑛 (mod 𝐷), {𝑟𝑛 } ⊂ [𝑎, ∞) for some 𝑎 > 0 and 𝑛 ∈
N and 𝛼, 𝛽, 𝛾 are real numbers in (0, 1) such that 𝛼 + 𝛽 + 𝛾 = 1.
̂
Then, the sequence {𝑥𝑛 }∞
𝑛=0 converges strongly to 𝑝 = ΠΩ (𝑥0 ).
𝜙 (𝑧, 𝑦𝑛 ) ≤ 𝜙 (𝑧, 𝑥𝑛 ) is equivalent to
3. Main Results
+
∀𝑛 ≥ 0,
Step 1. We will show that 𝐶𝑛 is closed and convex for each 𝑛 ≥ 0.
From the definition of 𝐶𝑛 , it is obvious that 𝐶𝑛 is closed.
Moreover, since
for all 𝑥, 𝑦 ∈ 𝐵𝑟 (0) and all 𝛼, 𝛽 ∈ [0, 1] with 𝛼 + 𝛽 = 1.
𝑇𝑟𝑛 (𝑥) : = {𝑧 ∈ 𝐶 : 𝐹𝑛 (𝑧, 𝑦)
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𝐶𝑛+1 = {𝑧 ∈ 𝐶𝑛 : 𝜙 (𝑧, 𝑦𝑛 ) ≤ 𝜙 (𝑧, 𝑥𝑛 )} ,
(5) 𝜙(𝑝, 𝐾𝑟 𝑥) + 𝜙(𝐾𝑟 𝑥, 𝑥) ≤ 𝜙(𝑝, 𝑥), ∀𝑝 ∈ 𝐹(𝐾𝑟 ).
1
⟨𝑦 − 𝑧, 𝐽𝑧 − 𝐽𝑥⟩ ≥ 0, ∀𝑦 ∈ 𝐶} ,
𝑟𝑛
𝑢𝑛 = 𝑇𝑟𝑛 𝑥𝑛 ,
𝑦𝑛 = 𝐽−1 (𝛼𝐽𝑥𝑛 + 𝛽𝐽𝑇𝑛 𝑧𝑛 + 𝛾𝐽𝑆𝑛 𝑢𝑛 ) ,
(4) VI(𝐶, 𝐴) is a closed and convex subset of 𝐶;
+
𝑧𝑛 = 𝐾𝑟𝑛 𝑥𝑛 ,
(29)
(3) 𝐹(𝐾𝑟 ) = VI(𝐶, 𝐴);
󵄩
󵄩2
󵄩 󵄩2
󵄩
󵄩󵄩
2
󵄩󵄩𝛼𝑥 + 𝛽𝑦󵄩󵄩󵄩 ≤ 𝛼‖𝑥‖ + 𝛽󵄩󵄩󵄩𝑦󵄩󵄩󵄩 − 𝛼𝛽𝑔 (󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩) ,
𝑥0 ∈ 𝐶0 = 𝐶, 𝑐ℎ𝑜𝑠𝑒𝑛 𝑎𝑟𝑏𝑖𝑡𝑟𝑎𝑟𝑦,
= 𝛼𝜙 (𝑝, 𝑥𝑘 ) + 𝛽𝜙 (𝑝, 𝐾𝑟𝑘 𝑥𝑘 ) + 𝛾𝜙 (𝑝, 𝑇𝑟𝑘 𝑥𝑘 )
≤ 𝛼𝜙 (𝑝, 𝑥𝑘 ) + 𝛽𝜙 (𝑝, 𝑥𝑘 ) + 𝛾𝜙 (𝑝, 𝑥𝑘 )
= (𝛼 + 𝛽 + 𝛾) 𝜙 (𝑝, 𝑥𝑘 )
= 𝜙 (𝑝, 𝑥𝑘 ) .
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6
Abstract and Applied Analysis
Therefore, 𝑝 ∈ 𝐶𝑘+1 . By the induction, this implies that Ω ⊂
𝐶𝑛 for each 𝑛 ≥ 0.
Hence, the sequence {𝑥𝑛 } is well defined.
Step 3. We will show that the sequence{𝑥𝑛 } is bounded.
Let 𝐺 := ⋂∞
𝑛=0 𝐶𝑛 . From Ω ⊂ 𝐶𝑛 for each 𝑛 ≥ 0 and {𝑥𝑛 }
is well defined. From the assumption of 𝐺, we see that 𝐺 is
closed and convex subset of 𝐶.
Let 𝑝̂ = Π𝐺(𝑥0 ), where 𝑝̂ is the unique element that
̂ 𝑥0 ). Now, we will show that
satisfies inf 𝑥∈𝐺 𝜙(𝑥, 𝑥0 ) = 𝜙(𝑝,
̂ → 0 as 𝑛 → ∞.
‖𝑥𝑛 − 𝑝‖
From the assumption of 𝐶𝑛 , we know that 𝐶 ⊃ 𝐶1 ⊃ 𝐶2 ⊃
𝐶3 ⊃ ⋅ ⋅ ⋅ and since 𝑥𝑛 = Π𝐶𝑛 (𝑥0 ) and 𝑥𝑛+1 = Π𝐶𝑛+1 (𝑥0 ) ∈
𝐶𝑛+1 ⊂ 𝐶𝑛 . Using Lemma 4, we get
̂ 𝑥0 ) ,
𝜙 (𝑥𝑛 , 𝑥0 ) ≤ 𝜙 (𝑥𝑛+1 , 𝑥0 ) ≤ ⋅ ⋅ ⋅ ≤ 𝜙 (𝑝,
(35)
for all 𝑝̂ ∈ Ω ⊂ 𝐶𝑛 , where 𝑛 ≥ 1. Then, the sequence
{𝜙(𝑥𝑛 , 𝑥0 )} is bounded. Hence, the sequence {𝑥𝑛 } is also
bounded.
̂
Step 4. We will show that there exists 𝑝̂ ∈ 𝐶 such that 𝑥𝑛 → 𝑝,
as 𝑛 → ∞.
Since 𝑥𝑛 = Π𝐶𝑛 (𝑥0 ) and 𝑥𝑛+1 = Π𝐶𝑛+1 𝑥0 ∈ 𝐶𝑛+1 ⊂ 𝐶𝑛 , we
have
𝜙 (𝑥𝑛 , 𝑥0 ) ≤ 𝜙 (𝑥𝑛+1 , 𝑥0 ) ,
∀𝑛 ≥ 0.
(36)
Therefore, the sequence {𝜙(𝑥𝑛 , 𝑥0 )} is nondecreasing. Hence
lim𝑛 → ∞ 𝜙(𝑥𝑛 , 𝑥0 ) exists. By the definition of 𝐶𝑛 , one has that
𝐶𝑚 ⊂ 𝐶𝑛 and 𝑥𝑚 = Π𝐶𝑚 𝑥0 ∈ 𝐶𝑛 for any positive integer
𝑚 ≥ 𝑛.
It follows that
(37)
= 𝜙 (𝑥𝑚 , 𝑥0 ) − 𝜙 (𝑥𝑛 , 𝑥0 ) .
Since lim𝑛 → ∞ 𝜙(𝑥𝑛 , 𝑥0 ) exists, by taking 𝑚, 𝑛 → ∞ in (37),
we have 𝜙(𝑥𝑚 , 𝑥𝑛 ) → 0.
From Lemma 7, it follows that ‖𝑥𝑚 − 𝑥𝑛 ‖ → 0 as 𝑚, 𝑛 →
∞. Thus {𝑥𝑛 } is a Cauchy sequence.
Without a loss of generalization, we can assume that 𝑥𝑛 ⇀
𝑝0 ∈ 𝐸. Since {𝑥𝑛 } is bounded and 𝐸 is reflexive. Since 𝐺 :=
⋂∞
𝑛=0 𝐶𝑛 is closed and convex, it follows that 𝑝0 ∈ 𝐺, ∀𝑛 ≥
0. Moreover, by using the weak lower semicontinuous of the
norm on 𝐸 and (35), we obtain
̂ 𝑥0 ) ≤ 𝜙 (𝑝0 , 𝑥0 )
𝜙 (𝑝,
≤ lim inf 𝜙 (𝑥𝑛 , 𝑥0 )
𝑛→∞
≤ lim sup 𝜙 (𝑥𝑛 , 𝑥0 )
𝑛→∞
≤ inf 𝜙 (𝑥, 𝑥0 )
𝑥∈𝐺
̂ 𝑥0 ) .
= 𝜙 (𝑝,
̂ 𝑥0 ) = 𝜙 (𝑝0 , 𝑥0 ) = inf 𝜙 (𝑥, 𝑥0 ) .
lim 𝜙 (𝑥𝑛 , 𝑥0 ) = 𝜙 (𝑝,
𝑛→∞
𝑥∈𝐺
(39)
By using Lemma 5, we have ⟨𝑝̂ − 𝑝0 , 𝐽𝑝̂ − 𝐽𝑝0 ⟩ = 0 and hence,
𝑝̂ = 𝑝0 . By the definition of 𝜙, we get
󵄩 󵄩2
󵄩 󵄩2
lim inf 𝜙 (𝑥𝑛 , 𝑥0 ) = lim inf (󵄩󵄩󵄩𝑥𝑛 󵄩󵄩󵄩 − 2 ⟨𝑥𝑛 , 𝐽𝑥0 ⟩ + 󵄩󵄩󵄩𝑥0 󵄩󵄩󵄩 )
𝑛→∞
𝑛→∞
󵄩 󵄩2
̂ 𝐽𝑥0 ⟩ + 󵄩󵄩󵄩󵄩𝑥0 󵄩󵄩󵄩󵄩2
≥ 󵄩󵄩󵄩𝑝̂󵄩󵄩󵄩 − 2 ⟨𝑝,
̂ 𝑥0 ) .
= 𝜙 (𝑝,
(40)
̂ Since 𝑥𝑛 ⇀ 𝑝,
̂ by the Kadec-Klee
Therefore, ‖𝑥𝑛 ‖ → ‖𝑝‖.
property of 𝐸, we obtain
lim 𝑥
𝑛→∞ 𝑛
(38)
̂
= 𝑝.
(41)
From 𝐽 is uniformly continuous, we also have
̂
lim 𝐽𝑥𝑛 = 𝐽𝑝.
𝑛→∞
(42)
̂ 𝑧𝑛 → 𝑝̂ and 𝑢𝑛 → 𝑝̂ as
Step 5. We will show that 𝑦𝑛 → 𝑝,
𝑛 → ∞.
Since 𝑥𝑛+1 ∈ 𝐶𝑛+1 , we have 𝜙(𝑥𝑛+1 , 𝑦𝑛 ) ≤ 𝜙(𝑥𝑛+1 , 𝑥𝑛 ) →
0, as 𝑛 → ∞. Thus, from (21), we obtain
󵄩 󵄩
󵄩󵄩 󵄩󵄩
󵄩󵄩𝑦𝑛 󵄩󵄩 󳨀→ 󵄩󵄩󵄩𝑝̂󵄩󵄩󵄩 ,
as 𝑛 󳨀→ ∞,
(43)
and so
󵄩󵄩 󵄩󵄩
󵄩 󵄩
󵄩󵄩𝐽𝑦𝑛 󵄩󵄩 󳨀→ 󵄩󵄩󵄩𝐽𝑝̂󵄩󵄩󵄩 ,
𝜙 (𝑥𝑚 , 𝑥𝑛 ) = 𝜙 (𝑥𝑚 , Π𝐶𝑛 𝑥0 )
≤ 𝜙 (𝑥𝑚 , 𝑥0 ) − 𝜙 (Π𝐶𝑛 𝑥0 , 𝑥0 )
This implies that
as 𝑛 󳨀→ ∞.
(44)
This implies that {𝐽𝑦𝑛 } is bounded. Note that reflexivity of 𝐸
implies reflexivity of 𝐸∗ . Thus, we assume that 𝐽𝑦𝑛 ⇀ 𝑦 ∈ 𝐸∗ .
Furthermore, the reflexivity of 𝐸 implies there exists 𝑥 ∈ 𝐸
such that 𝑦 = 𝐽𝑥. Then, it follows that
󵄩2
󵄩
󵄩 󵄩2
𝜙 (𝑥𝑛+1 , 𝑦𝑛 ) = 󵄩󵄩󵄩𝑥𝑛+1 󵄩󵄩󵄩 − 2 ⟨𝑥𝑛+1 , 𝐽𝑦𝑛 ⟩ + 󵄩󵄩󵄩𝑦𝑛 󵄩󵄩󵄩
󵄩2
󵄩
󵄩 󵄩2
= 󵄩󵄩󵄩𝑥𝑛+1 󵄩󵄩󵄩 − 2 ⟨𝑥𝑛+1 , 𝐽𝑦𝑛 ⟩ + 󵄩󵄩󵄩𝐽𝑦𝑛 󵄩󵄩󵄩 .
(45)
Taking the lim inf 𝑛 → ∞ on both sides of (45) and using weak
lower semicontinuous of norm to get that
󵄩 󵄩2
̂ 𝑦⟩ + 󵄩󵄩󵄩󵄩𝑦󵄩󵄩󵄩󵄩2
0 ≥ 󵄩󵄩󵄩𝑝̂󵄩󵄩󵄩 − 2 ⟨𝑝,
󵄩 󵄩2
̂ 𝐽𝑥⟩ + ‖𝐽𝑥‖2
= 󵄩󵄩󵄩𝑝̂󵄩󵄩󵄩 − 2 ⟨𝑝,
󵄩 󵄩2
̂ 𝐽𝑥⟩ + ‖𝑥‖2
= 󵄩󵄩󵄩𝑝̂󵄩󵄩󵄩 − 2 ⟨𝑝,
(46)
̂ 𝑥) .
= 𝜙 (𝑝,
̂ It follows that 𝐽𝑦𝑛 ⇀ 𝐽𝑝.
̂
Thus 𝑝̂ = 𝑥, and so 𝑦 = 𝐽𝑥 = 𝐽𝑝.
Now, from (44) and the Kadec-Klee property of 𝐸∗ , we obtain
̂
𝐽𝑦𝑛 󳨀→ 𝐽𝑝,
as 𝑛 󳨀→ ∞.
(47)
Abstract and Applied Analysis
7
̂ Now,
Thus the demicontinuity of 𝐽−1 implies that 𝑦𝑛 ⇀ 𝑝.
from (43) and the fact that 𝐸 has the Kadec-Klee property, we
obtain
lim 𝑦
𝑛→∞ 𝑛
̂
= 𝑝.
Taking the lim inf 𝑛 → ∞ on both sides of (58) and using the
continuity of 𝐽, we get that
̂ + 󵄩󵄩󵄩󵄩𝐽𝑝̂󵄩󵄩󵄩󵄩2
0 ≥ ‖𝑢‖2 − 2 ⟨𝑢, 𝐽𝑝⟩
(48)
In the fact that 𝑥𝑛 → 𝑝̂ and 𝑦𝑛 → 𝑝̂ as 𝑛 → ∞, we get
̂ .
lim 𝜙 (𝑝, 𝑧𝑛 ) = 𝜙 (𝑝, 𝑝)
(49)
𝑛→∞
̂
This implies that 𝑝̂ = 𝑢 and hence 𝑢𝑛 ⇀ 𝑝.
Now, from (57) and the Kadec-Klee property of 𝐸, we
obtain
Since 𝑧𝑛 = 𝐾𝑟𝑛 𝑥𝑛 , it follows from Lemma 11 that
lim 𝑢
𝑛→∞ 𝑛
𝜙 (𝑧𝑛 , 𝑥𝑛 ) = 𝜙 (𝐾𝑟𝑛 𝑥𝑛 , 𝑥𝑛 )
(59)
̂ .
= 𝜙 (𝑢, 𝑝)
̂
= 𝑝.
(60)
Step 6. We will show that 𝑝̂ ∈ Ω.
≤ 𝜙 (𝑝, 𝑥𝑛 ) − 𝜙 (𝑝, 𝐾𝑟𝑛 𝑥𝑛 )
≤ 𝜙 (𝑝, 𝑥𝑛 ) − 𝜙 (𝑝, 𝑧𝑛 ) 󳨀→ 0,
(50)
as 𝑛 󳨀→ ∞.
Substep 1. We will show that 𝑝̂ ∈ ⋂𝑁
𝑛=1 VI(𝐶, 𝐴 𝑛 ).
From the definition of 𝐾𝑟𝑛 of algorithm (32), we have
1
⟨𝑦 − 𝑧𝑛 , 𝐽𝑧𝑛 − 𝐽𝑥𝑛 ⟩ ≥ 0,
𝑟𝑛
⟨𝑦 − 𝑧𝑛 , 𝐴 𝑛 𝑧𝑛 ⟩ +
From (21), we obtain
󵄩 󵄩
󵄩󵄩 󵄩󵄩
󵄩󵄩𝑧𝑛 󵄩󵄩 󳨀→ 󵄩󵄩󵄩𝑝̂󵄩󵄩󵄩 ,
∀𝑦 ∈ 𝐶.
(51)
(61)
and so {𝑧𝑛 } is bounded. Since 𝐸 is reflexive, we assume that
𝑧𝑛 ⇀ 𝑧 ∈ 𝐸. It follows that
Let {𝑛𝑘 }𝑘∈N ⊂ N be such that 𝐴 𝑛𝑘 = 𝐴 1 , for all 𝑘 ∈ N. Then
from (61), we obtain
󵄩 󵄩2
󵄩 󵄩2
𝜙 (𝑧𝑛 , 𝑥𝑛 ) = 󵄩󵄩󵄩𝑧𝑛 󵄩󵄩󵄩 − 2 ⟨𝑧𝑛 , 𝐽𝑥𝑛 ⟩ + 󵄩󵄩󵄩𝐽𝑥𝑛 󵄩󵄩󵄩 .
(52)
Taking the lim inf 𝑛 → ∞ on both sides of (52) and using the
continuity of 𝐽, we get that
̂ + 󵄩󵄩󵄩󵄩𝐽𝑝̂󵄩󵄩󵄩󵄩2
0 ≥ ‖𝑧‖ − 2 ⟨𝑧, 𝐽𝑝⟩
2
(53)
̂ .
= 𝜙 (𝑧, 𝑝)
1
⟨𝑦 − 𝑧𝑛𝑘 , 𝐽𝑧𝑛𝑘 − 𝐽𝑥𝑛𝑘 ⟩ ≥ 0,
𝑟𝑛𝑘
(62)
and that is
⟨𝑦 − 𝑧𝑛𝑘 , 𝐴 1 𝑧𝑛𝑘 ⟩ + ⟨𝑦 − 𝑧𝑛𝑘 ,
𝐽𝑧𝑛𝑘 − 𝐽𝑥𝑛𝑘
𝑟𝑛𝑘
⟩ ≥ 0,
∀𝑦 ∈ 𝐶.
̂
= 𝑝.
(54)
̂ for all 𝑡 ∈ (0, 1] and 𝑣 ∈ 𝐶.
Now, we set 𝑣𝑡 = 𝑡𝑣 + (1 − 𝑡)𝑝,
Therefore, we get 𝑣𝑡 ∈ 𝐶. From (63), it follows that
⟨𝑣𝑡 − 𝑧𝑛𝑘 , 𝐴 1 𝑣𝑡 ⟩ ≥ ⟨𝑣𝑡 − 𝑧𝑛𝑘 , 𝐴 1 𝑣𝑡 ⟩ − ⟨𝑣𝑡 − 𝑧𝑛𝑘 , 𝐴 1 𝑧𝑛𝑘 ⟩
̂ 𝑦𝑛 → 𝑝̂ and 𝑧𝑛 → 𝑝̂ as 𝑛 → ∞,
In the fact that 𝑥𝑛 → 𝑝,
we get
̂ .
lim 𝜙 (𝑝, 𝑢𝑛 ) = 𝜙 (𝑝, 𝑝)
(55)
𝑛→∞
− ⟨𝑣𝑡 − 𝑧𝑛𝑘 ,
𝐽𝑧𝑛𝑘 − 𝐽𝑥𝑛𝑘
𝑟𝑛𝑘
− ⟨𝑣𝑡 − 𝑧𝑛𝑘 ,
𝐽𝑧𝑛𝑘 − 𝐽𝑥𝑛𝑘
𝑟𝑛𝑘
𝜙 (𝑢𝑛 , 𝑥𝑛 ) = 𝜙 (𝑇𝑟𝑛 𝑥𝑛 , 𝑥𝑛 )
⟩.
(64)
≤ 𝜙 (𝑝, 𝑥𝑛 ) − 𝜙 (𝑝, 𝑇𝑟𝑛 𝑥𝑛 )
≤ 𝜙 (𝑝, 𝑥𝑛 ) − 𝜙 (𝑝, 𝑢𝑛 ) 󳨀→ 0,
⟩
= ⟨𝑣𝑡 − 𝑧𝑛𝑘 , 𝐴 1 𝑣𝑡 − 𝐴 1 𝑧𝑛𝑘 ⟩
Since 𝑢𝑛 = 𝑇𝑟𝑛 𝑥𝑛 , it follows from Lemma 10 that
as 𝑛 󳨀→ ∞.
(56)
From (21), we obtain
󵄩 󵄩
󵄩󵄩 󵄩󵄩
󵄩󵄩𝑢𝑛 󵄩󵄩 󳨀→ 󵄩󵄩󵄩𝑝̂󵄩󵄩󵄩 ,
(57)
and so {𝑢𝑛 } is bounded. Since 𝐸 is reflexive, we assume that
𝑢𝑛 ⇀ 𝑢 ∈ 𝐸. It follows that
󵄩󵄩2
∀𝑦 ∈ 𝐶,
(63)
̂
This implies that 𝑝̂ = 𝑧 and hence 𝑧𝑛 ⇀ 𝑝.
Now, from (51) and the Kadec-Klee property of 𝐸, we
obtain
lim 𝑧
𝑛→∞ 𝑛
⟨𝑦 − 𝑧𝑛𝑘 , 𝐴 1 𝑧𝑛𝑘 ⟩ +
󵄩󵄩2
󵄩
󵄩
𝜙 (𝑢𝑛 , 𝑥𝑛 ) = 󵄩󵄩󵄩𝑢𝑛 󵄩󵄩 − 2 ⟨𝑢𝑛 , 𝐽𝑥𝑛 ⟩ + 󵄩󵄩󵄩𝐽𝑥𝑛 󵄩󵄩 .
(58)
̂
From the continuity of 𝐽 and (41) and (54), we have 𝑥𝑛𝑘 → 𝑝,
̂ as 𝑘 → ∞, we obtain
𝑧𝑛𝑘 → 𝑝,
𝐽𝑧𝑛𝑘 − 𝐽𝑥𝑛𝑘
𝑟𝑛𝑘
󳨀→ 0,
as 𝑘 󳨀→ ∞.
(65)
Since 𝐴 1 is a monotone mapping, we also have ⟨𝑣𝑡 −𝑧𝑛𝑘 , 𝐴 1 𝑣𝑡 −
𝐴 1 𝑧𝑛𝑘 ⟩ ≥ 0.
Thus, it follows that
0 ≤ lim ⟨𝑣𝑡 − 𝑧𝑛𝑘 , 𝐴 1 𝑣𝑡 ⟩
𝑛→∞
̂ 𝐴 1 𝑣𝑡 ⟩ ,
= ⟨𝑣𝑡 − 𝑝,
(66)
8
Abstract and Applied Analysis
and hence
From algorithm (32) and Lemma 12, we compute
̂ 𝐴 1 𝑣𝑡 ⟩ ≥ 0,
⟨𝑦 − 𝑝,
∀𝑦 ∈ 𝐶.
(67)
𝜙 (𝑝, 𝑦𝑛 ) = 𝜙 (𝑝, 𝐽−1 (𝛼𝐽𝑥𝑛 + 𝛽𝐽𝑇𝑛 𝑧𝑛 + 𝛾𝐽𝑆𝑛 𝑢𝑛 ))
󵄩 󵄩2
= 󵄩󵄩󵄩𝑝󵄩󵄩󵄩 − 2 ⟨𝑝, 𝛼𝐽𝑥𝑛 + 𝛽𝐽𝑇𝑛 𝑧𝑛 + 𝛾𝐽𝑆𝑛 𝑢𝑛 ⟩
󵄩2
󵄩
+ 󵄩󵄩󵄩𝛼𝐽𝑥𝑛 + 𝛽𝐽𝑇𝑛 𝑧𝑛 + 𝛾𝐽𝑆𝑛 𝑢𝑛 󵄩󵄩󵄩
󵄩 󵄩2
≤ 󵄩󵄩󵄩𝑝󵄩󵄩󵄩 − 2𝛼 ⟨𝑝, 𝐽𝑥𝑛 ⟩ − 2𝛽 ⟨𝑝, 𝐽𝑇𝑛 𝑧𝑛 ⟩
󵄩2
󵄩 󵄩2
󵄩
− 2𝛾 ⟨𝑝, 𝐽𝑆𝑛 𝑢𝑛 ⟩ + 𝛼󵄩󵄩󵄩𝑥𝑛 󵄩󵄩󵄩 + 𝛽󵄩󵄩󵄩𝑇𝑛 𝑧𝑛 󵄩󵄩󵄩
󵄩2
󵄩
󵄩
󵄩
+ 𝛾󵄩󵄩󵄩𝑆𝑛 𝑢𝑛 󵄩󵄩󵄩 − 𝛼𝛽𝑔 (󵄩󵄩󵄩𝐽𝑥𝑛 − 𝐽𝑇𝑛 𝑧𝑛 󵄩󵄩󵄩)
󵄩
󵄩
− 𝛼𝛾𝑔 (󵄩󵄩󵄩𝐽𝑥𝑛 − 𝐽𝑆𝑛 𝑢𝑛 󵄩󵄩󵄩)
If 𝑡 → 0, we obtain
̂ 𝐴 1 𝑝⟩
̂ ≥ 0,
⟨𝑦 − 𝑝,
∀𝑦 ∈ 𝐶.
(68)
This implies that 𝑝̂ ∈ VI(𝐶, 𝐴 1 ).
Similarly, let {𝑛𝑘 }𝑘∈N ⊂ N be such that 𝐴 𝑛𝑘 = 𝐴 2 , for all
𝑘 ∈ N.
Then, we have again that 𝑝̂ ∈ VI(𝐶, 𝐴 2 ).
Continuing in the same way, we obtain that 𝑝̂ ∈
VI(𝐶, 𝐴 𝑛 ), where 𝑛 = 3, 4, 5, . . . , 𝑁.
Hence, 𝑝̂ ∈ ⋂𝑁
𝑛=1 VI(𝐶, 𝐴 𝑛 ).
= 𝛼𝜙 (𝑝, 𝑥𝑛 ) + 𝛽𝜙 (𝑝, 𝑇𝑛 𝑧𝑛 ) + 𝛾𝜙 (𝑝, 𝑆𝑛 𝑢𝑛 )
Substep 2. We will show that 𝑝̂ ∈ ⋂𝑀
𝑘=1 SEP(𝐹𝑘 ).
From the definition of 𝑇𝑟𝑛 of algorithm (32) and (A2), we
have
1
⟨𝑦 − 𝑢𝑛 , 𝐽𝑢𝑛 − 𝐽𝑥𝑛 ⟩ ≥ −𝐹𝑛 (𝑢𝑛 , 𝑦) ≥ 𝐹 (𝑦, 𝑢𝑛 ) ,
𝑟𝑛
󵄩
󵄩
󵄩
󵄩
− 𝛼𝛽𝑔 (󵄩󵄩󵄩𝐽𝑥𝑛 − 𝐽𝑇𝑛 𝑧𝑛 󵄩󵄩󵄩) − 𝛼𝛾𝑔 (󵄩󵄩󵄩𝐽𝑥𝑛 − 𝐽𝑆𝑛 𝑢𝑛 󵄩󵄩󵄩)
≤ 𝛼𝜙 (𝑝, 𝑥𝑛 ) + 𝛽𝜙 (𝑝, 𝑧𝑛 ) + 𝛾𝜙 (𝑝, 𝑢𝑛 )
󵄩
󵄩
󵄩
󵄩
− 𝛼𝛽𝑔 (󵄩󵄩󵄩𝐽𝑥𝑛 − 𝐽𝑇𝑛 𝑧𝑛 󵄩󵄩󵄩) − 𝛼𝛾𝑔 (󵄩󵄩󵄩𝐽𝑥𝑛 − 𝐽𝑆𝑛 𝑢𝑛 󵄩󵄩󵄩)
∀𝑦 ∈ 𝐶.
(69)
= 𝛼𝜙 (𝑝, 𝑥𝑛 ) + 𝛽𝜙 (𝑝, 𝐾𝑟𝑛 𝑥𝑛 ) + 𝛾𝜙 (𝑝, 𝑇𝑟𝑛 𝑥𝑛 )
󵄩
󵄩
󵄩
󵄩
− 𝛼𝛽𝑔 (󵄩󵄩󵄩𝐽𝑥𝑛 − 𝐽𝑇𝑛 𝑧𝑛 󵄩󵄩󵄩) − 𝛼𝛾𝑔 (󵄩󵄩󵄩𝐽𝑥𝑛 − 𝐽𝑆𝑛 𝑢𝑛 󵄩󵄩󵄩)
Let {𝑛𝑘 }𝑘∈N ⊂ N be such that 𝐹𝑛𝑘 = 𝐹1 , for all 𝑘 ∈ N. Then
from (69), we obtain
≤ 𝛼𝜙 (𝑝, 𝑥𝑛 ) + 𝛽𝜙 (𝑝, 𝑥𝑛 ) + 𝛾𝜙 (𝑝, 𝑥𝑛 )
𝐽𝑢𝑛𝑘 − 𝐽𝑥𝑛𝑘
1
⟨𝑦 − 𝑢𝑛𝑘 , 𝐽𝑢𝑛𝑘 − 𝐽𝑥𝑛𝑘 ⟩ = ⟨𝑦 − 𝑢𝑛𝑘 ,
⟩
𝑟𝑛𝑘
𝑟𝑛𝑘
(70)
≥ 𝐹1 (𝑦, 𝑢𝑛𝑘 ) ,
󵄩
󵄩
󵄩
󵄩
− 𝛼𝛽𝑔 (󵄩󵄩󵄩𝐽𝑥𝑛 − 𝐽𝑇𝑛 𝑧𝑛 󵄩󵄩󵄩) − 𝛼𝛾𝑔 (󵄩󵄩󵄩𝐽𝑥𝑛 − 𝐽𝑆𝑛 𝑢𝑛 󵄩󵄩󵄩)
= (𝛼 + 𝛽 + 𝛾) 𝜙 (𝑝, 𝑥𝑛 )
∀𝑦 ∈ 𝐶.
󵄩
󵄩
− 𝛼𝛽𝑔 (󵄩󵄩󵄩𝐽𝑥𝑛 − 𝐽𝑇𝑛 𝑧𝑛 󵄩󵄩󵄩)
̂
From the continuity of 𝐽 and (41) and (60), we have 𝑥𝑛𝑘 → 𝑝,
̂ as 𝑘 → ∞ and we obtain
𝑢𝑛𝑘 → 𝑝,
𝐽𝑢𝑛𝑘 − 𝐽𝑥𝑛𝑘
𝑟𝑛𝑘
󳨀→ 0,
as 𝑘 󳨀→ ∞.
(71)
Therefore, 𝐹1 (𝑦, 𝑢𝑛𝑘 ) ≤ 0, ∀𝑦 ∈ 𝐶.
̂ for all 𝑡 ∈ (0, 1] and 𝑤 ∈ 𝐶.
Now, we set 𝑤𝑡 = 𝑡𝑤+(1−𝑡)𝑝,
̂ ≤ 0.
Consequently, we get 𝑤𝑡 ∈ 𝐶. And so 𝐹1 (𝑤𝑡 , 𝑝)
Therefore, from (A1), we obtain
0 = 𝐹1 (𝑤𝑡 , 𝑤𝑡 )
̂
≤ 𝑡𝐹1 (𝑤𝑡 , 𝑦) + (1 − 𝑡) 𝐹1 (𝑤𝑡 , 𝑝)
≤ 𝜙 (𝑝, 𝑥𝑛 ) .
(73)
From (73), we have
󵄩
󵄩
𝜙 (𝑝, 𝑦𝑛 ) ≤ 𝜙 (𝑝, 𝑥𝑛 ) − 𝛼𝛽𝑔 (󵄩󵄩󵄩𝐽𝑥𝑛 − 𝐽𝑇𝑛 𝑧𝑛 󵄩󵄩󵄩)
󵄩
󵄩
− 𝛼𝛾𝑔 (󵄩󵄩󵄩𝐽𝑥𝑛 − 𝐽𝑆𝑛 𝑢𝑛 󵄩󵄩󵄩)
(72)
≤ 𝑡𝐹1 (𝑤𝑡 , 𝑦) .
Thus, 𝐹1 (𝑤𝑡 , 𝑦) ≥ 0, ∀𝑦 ∈ 𝐶. From (A3), if 𝑡 → 0, then we
̂ 𝑦) ≥ 0, ∀𝑦 ∈ 𝐶. This implies that 𝑝̂ ∈ EP(𝐹1 ).
get 𝐹1 (𝑝,
Similarly, let {𝑛𝑘 }𝑘∈N ⊂ N be such that 𝐹𝑛𝑘 = 𝐹2 , for all
𝑘 ∈ N. Then, we have again that 𝑝̂ ∈ EP(𝐹2 ).
Continuing in the same way, we obtain that 𝑝̂ ∈ EP(𝐹𝑘 ),
where 𝑘 = 3, 4, 5, . . . , 𝑀. Hence, 𝑝̂ ∈ ⋂𝑀
𝑘=1 SEP(𝐹𝑘 ).
Substep 3. We will show that 𝑝̂ ∈ ⋂𝐷
𝑖=1 𝐹(𝑆𝑖 ).
󵄩
󵄩
− 𝛼𝛾𝑔 (󵄩󵄩󵄩𝐽𝑥𝑛 − 𝐽𝑆𝑛 𝑢𝑛 󵄩󵄩󵄩)
󵄩
󵄩
= 𝜙 (𝑝, 𝑥𝑛 ) − 𝛼𝛽𝑔 (󵄩󵄩󵄩𝐽𝑥𝑛 − 𝐽𝑇𝑛 𝑧𝑛 󵄩󵄩󵄩)
󵄩
󵄩
− 𝛼𝛾𝑔 (󵄩󵄩󵄩𝐽𝑥𝑛 − 𝐽𝑆𝑛 𝑢𝑛 󵄩󵄩󵄩)
(74)
󵄩
󵄩
≤ 𝜙 (𝑝, 𝑥𝑛 ) − 𝛼𝛾𝑔 (󵄩󵄩󵄩𝐽𝑥𝑛 − 𝐽𝑆𝑛 𝑢𝑛 󵄩󵄩󵄩) .
̂ 𝑦𝑛 → 𝑝,
̂ and 𝛼, 𝛾 > 0, we obtain
From 𝑥𝑛 → 𝑝,
󵄩
󵄩
𝛼𝛾𝑔 (󵄩󵄩󵄩𝐽𝑥𝑛 − 𝐽𝑆𝑛 𝑢𝑛 󵄩󵄩󵄩)
≤ 𝜙 (𝑝, 𝑥𝑛 ) − 𝜙 (𝑝, 𝑦𝑛 ) 󳨀→ 0,
as 𝑛 󳨀→ ∞.
(75)
Therefore,
󵄩
󵄩
𝑔 (󵄩󵄩󵄩𝐽𝑥𝑛 − 𝐽𝑆𝑛 𝑢𝑛 󵄩󵄩󵄩) 󳨀→ 0,
as 𝑛 󳨀→ ∞.
(76)
Abstract and Applied Analysis
9
It follows from the property of 𝑔 that
󵄩󵄩
󵄩
󵄩󵄩𝐽𝑥𝑛 − 𝐽𝑆𝑛 𝑢𝑛 󵄩󵄩󵄩 󳨀→ 0,
as 𝑛 󳨀→ ∞.
(77)
Moreover, the demicontinuity of 𝐽−1 implies that 𝑇𝑛 𝑧𝑛 ⇀ 𝑝̂
as 𝑛 → ∞. Thus, the Kadec-Klee property of 𝐸, we obtain
̂
𝑇𝑛 𝑧𝑛 󳨀→ 𝑝,
̂ as 𝑛 → ∞. Then,
From (42), we have 𝐽𝑥𝑛 → 𝐽𝑝,
̂
𝐽𝑆𝑛 𝑢𝑛 󳨀→ 𝐽𝑝,
as 𝑛 󳨀→ ∞,
(78)
and so
󵄩 󵄩
󵄩󵄩
󵄩
󵄩󵄩𝑆𝑛 𝑢𝑛 󵄩󵄩󵄩 󳨀→ 󵄩󵄩󵄩𝑝̂󵄩󵄩󵄩 ,
as 𝑛 󳨀→ ∞.
(79)
Moreover, the demicontinuity of 𝐽−1 implies that 𝑆𝑛 𝑢𝑛 ⇀ 𝑝̂
as 𝑛 → ∞. Thus, the Kadec-Klee property of 𝐸, we obtain
̂
𝑆𝑛 𝑢𝑛 󳨀→ 𝑝,
as 𝑛 󳨀→ ∞.
(80)
Let {𝑛𝑘 }𝑘∈N ⊂ N be such that 𝑆𝑛𝑘 = 𝑆1 , for all 𝑘 ∈ N.
̂ as 𝑛 → ∞. It follows
Then, from (60), we have 𝑢𝑛𝑘 → 𝑝,
from (80) and the closedness of 𝑆1 that
̂
𝑝̂ = lim 𝑆𝑛𝑘 𝑢𝑛𝑘 = lim 𝑆1 𝑢𝑛𝑘 = 𝑆1 𝑝.
𝑘→∞
𝑘→∞
(81)
This implies that 𝑝̂ ∈ 𝐹(𝑆1 ).
Similarly, let {𝑛𝑘 }𝑘∈N ⊂ N be such that 𝑆𝑛𝑘 = 𝑆2 , for all
𝑘 ∈ N. Then, we have again that 𝑝̂ ∈ 𝐹(𝑆2 ).
Continuing in the same way, we obtain that 𝑝̂ ∈ 𝐹(𝑆𝑖 ),
where 𝑖 = 3, 4, 5, . . . , 𝐷.
Hence, 𝑝̂ ∈ ⋂𝐷
𝑖=1 𝐹(𝑆𝑖 ).
Substep 4. We will show that 𝑝̂ ∈
From (73), we have
⋂𝐷
𝑖=1
𝐹(𝑇𝑖 ).
(82)
󵄩
󵄩
≤ 𝜙 (𝑝, 𝑥𝑛 ) − 𝛼𝛽𝑔 (󵄩󵄩󵄩𝐽𝑥𝑛 − 𝐽𝑇𝑛 𝑧𝑛 󵄩󵄩󵄩) .
󵄩
󵄩
𝛼𝛽𝑔 (󵄩󵄩󵄩𝐽𝑥𝑛 − 𝐽𝑇𝑛 𝑧𝑛 󵄩󵄩󵄩)
≤ 𝜙 (𝑝, 𝑥𝑛 ) − 𝜙 (𝑝, 𝑦𝑛 ) 󳨀→ 0,
as 𝑛 󳨀→ ∞.
̂
𝑝̂ = lim 𝑇𝑛𝑘 𝑧𝑛𝑘 = lim 𝑇1 𝑧𝑛𝑘 = 𝑇1 𝑝.
𝑘→∞
𝑘→∞
(89)
This implies that 𝑝̂ ∈ 𝐹(𝑇1 ).
Similarly, let {𝑛𝑘 }𝑘∈N ⊂ N be such that 𝑇𝑛𝑘 = 𝑇2 , for all
𝑘 ∈ N. Then, we have again that 𝑝̂ ∈ 𝐹(𝑇2 ).
Continuing in the same way, we obtain that 𝑝̂ ∈ 𝐹(𝑇𝑖 ),
where 𝑖 = 3, 4, 5, . . . , 𝐷.
Hence, 𝑝̂ ∈ ⋂𝐷
𝑖=1 𝐹(𝑇𝑖 ).
From Substeps (6.1)–(6.4), we can conclude that
𝐷
𝐷
𝑀
𝑖=1
𝑖=1
𝑘=1
𝑝̂ ∈ Ω := (⋂ 𝐹 (𝑇𝑖 )) ⋂ (⋂ 𝐹 (𝑆𝑖 )) ⋂ ( ⋂ SEP (𝐹𝑘 ))
𝑁
⋂ ( ⋂ VI (𝐶, 𝐴 𝑛 )) .
𝑛=1
(90)
Step 7. Finally, we will show that 𝑝̂ = ΠΩ (𝑥0 ).
From 𝑥𝑛 = Π𝐶𝑛 (𝑥0 ), we have
∀𝑝 ∈ Ω.
(91)
Taking 𝑛 → ∞ in (91), one has
̂ 𝑝̂ − 𝑝⟩ ≥ 0,
⟨𝐽𝑥0 − 𝐽𝑝,
̂ 𝑦𝑛 → 𝑝,
̂ and 𝛼, 𝛽 > 0, we obtain
From 𝑥𝑛 → 𝑝,
(88)
Let {𝑛𝑘 }𝑘∈N ⊂ N be such that 𝑇𝑛𝑘 = 𝑇1 , for all 𝑘 ∈ N.
̂ as 𝑛 → ∞. It follows
Then, from (54), we have 𝑧𝑛𝑘 → 𝑝,
from (88) and the closedness of 𝑇1 that
⟨𝐽𝑥0 − 𝐽𝑥𝑛 , 𝑥𝑛 − 𝑝⟩ ≥ 0,
󵄩
󵄩
𝜙 (𝑝, 𝑦𝑛 ) ≤ 𝜙 (𝑝, 𝑥𝑛 ) − 𝛼𝛽𝑔 (󵄩󵄩󵄩𝐽𝑥𝑛 − 𝐽𝑇𝑛 𝑧𝑛 󵄩󵄩󵄩)
󵄩
󵄩
− 𝛼𝛾𝑔 (󵄩󵄩󵄩𝐽𝑥𝑛 − 𝐽𝑆𝑛 𝑢𝑛 󵄩󵄩󵄩)
as 𝑛 󳨀→ ∞.
∀𝑝 ∈ Ω.
(92)
Now, we have 𝑝̂ ∈ Ω and by Lemma 5, we get
(83)
𝑝̂ = ΠΩ (𝑥0 ) .
(93)
This completes the proof of Theorem 13.
Therefore,
󵄩
󵄩
𝑔 (󵄩󵄩󵄩𝐽𝑥𝑛 − 𝐽𝑇𝑛 𝑧𝑛 󵄩󵄩󵄩) 󳨀→ 0,
as 𝑛 󳨀→ ∞.
(84)
If we set 𝑀 = 𝑁 = 𝐷 = 1 in Theorem 13, then we obtain
the following result.
(85)
Corollary 14. Let 𝐶 be a nonempty, closed and convex subset
of a uniformly smooth and strictly convex real Banach space 𝐸
which has the Kadec-Klee property. Suppose that
It follows from the property of 𝑔 that
󵄩󵄩
󵄩
󵄩󵄩𝐽𝑥𝑛 − 𝐽𝑇𝑛 𝑧𝑛 󵄩󵄩󵄩 󳨀→ 0,
as 𝑛 󳨀→ ∞.
̂ as 𝑛 → ∞. Then,
From (42), we have 𝐽𝑥𝑛 → 𝐽𝑝,
̂
𝐽𝑇𝑛 𝑧𝑛 󳨀→ 𝐽𝑝,
as 𝑛 󳨀→ ∞,
(86)
(2) 𝑇 and 𝑆 are two quasi-𝜙-nonexpansive mappings from
𝐶 into 𝐸;
and so
󵄩 󵄩
󵄩󵄩
󵄩
󵄩󵄩𝑇𝑛 𝑧𝑛 󵄩󵄩󵄩 󳨀→ 󵄩󵄩󵄩𝑝̂󵄩󵄩󵄩 ,
(1) 𝐹 : 𝐶 × 𝐶 → R is a bifunction satisfying conditions
(A1)–(A4);
as 𝑛 󳨀→ ∞.
(87)
(3) 𝐴 : 𝐶 → 𝐸∗ is a continuous monotone mapping.
10
Abstract and Applied Analysis
(4) For 𝑥 ∈ 𝐸, one defines the mappings 𝐾𝑟𝑛 , 𝑇𝑟𝑛 : 𝐸 → 𝐶
as follows:
(4) For 𝑥 ∈ 𝐸, one defines the mappings 𝐾𝑟𝑛 , 𝑇𝑟𝑛 : 𝐸 → 𝐶
by
𝐾𝑟𝑛 (𝑥) := {𝑧 ∈ 𝐶 : ⟨𝑦 − 𝑧, 𝐴 𝑛 𝑧⟩
𝐾𝑟𝑛 (𝑥) : = {𝑧 ∈ 𝐶 : ⟨𝑦 − 𝑧, 𝐴𝑧⟩
1
+ ⟨𝑦 − 𝑧, 𝐽𝑧 − 𝐽𝑥⟩ ≥ 0, ∀𝑦 ∈ 𝐶} ;
𝑟𝑛
+
(94)
𝑇𝑟𝑛 (𝑥) : = {𝑧 ∈ 𝐶 : 𝐹 (𝑧, 𝑦)
1
⟨𝑦 − 𝑧, 𝐽𝑧 − 𝐽𝑥⟩ ≥ 0, ∀𝑦 ∈ 𝐶} ;
𝑟𝑛
𝑇𝑟𝑛 (𝑥) := {𝑧 ∈ 𝐶 : 𝐹𝑛 (𝑧, 𝑦)
+
1
+ ⟨𝑦 − 𝑧, 𝐽𝑧 − 𝐽𝑥⟩ ≥ 0, ∀𝑦 ∈ 𝐶} .
𝑟𝑛
Assume that Ω := 𝐹(𝑇) ∩ 𝐹(𝑆) ∩ EP(𝐹) ∩ VI(𝐶, 𝐴) is
a nonempty and bounded in 𝐶 and let {𝑥𝑛 } be a sequence
generated by
(96)
1
⟨𝑦 − 𝑧, 𝐽𝑧 − 𝐽𝑥⟩ ≥ 0, ∀𝑦 ∈ 𝐶} .
𝑟𝑛
𝑀
𝑁
Assume that Ω := (⋂𝐷
𝑖=1 𝐹(𝑆𝑖 )) ⋂ (⋂𝑘=1 SEP(𝐹𝑘 )) ⋂(⋂𝑛=1
VI(𝐶, 𝐴 𝑛 )) is a nonempty and bounded in 𝐶 and let {𝑥𝑛 } be a
sequence generated by
𝑥0 ∈ 𝐶0 = 𝐶, 𝑐ℎ𝑜𝑠𝑒𝑛 𝑎𝑟𝑏𝑖𝑡𝑟𝑎𝑟𝑦,
𝑧𝑛 = 𝐾𝑟𝑛 𝑥𝑛 ,
𝑥0 ∈ 𝐶0 = 𝐶, 𝑐ℎ𝑜𝑠𝑒𝑛 𝑎𝑟𝑏𝑖𝑡𝑟𝑎𝑟𝑦,
𝑢𝑛 = 𝑇𝑟𝑛 𝑥𝑛 ,
𝑧𝑛 = 𝐾𝑟𝑛 𝑥𝑛 ,
𝑦𝑛 = 𝐽−1 (𝛼𝐽𝑥𝑛 + 𝛽𝐽𝑧𝑛 + 𝛾𝐽𝑆𝑛 𝑢𝑛 ) ,
𝑢𝑛 = 𝑇𝑟𝑛 𝑥𝑛 ,
−1
𝑦𝑛 = 𝐽 (𝛼𝐽𝑥𝑛 + 𝛽𝐽𝑇𝑧𝑛 + 𝛾𝐽𝑆𝑢𝑛 ) ,
(95)
𝐶𝑛+1 = {𝑧 ∈ 𝐶𝑛 : 𝜙 (𝑧, 𝑦𝑛 ) ≤ 𝜙 (𝑧, 𝑥𝑛 )} ,
𝑥𝑛+1 = Π𝐶𝑛+1 (𝑥0 ) ,
∀𝑛 ≥ 0,
where {𝑟𝑛 } ⊂ [𝑎, ∞) for some 𝑎 > 0 and 𝑛 ∈ N, and 𝛼, 𝛽, 𝛾 are
real numbers in (0, 1) such that 𝛼 + 𝛽 + 𝛾 = 1.
̂
Then, the sequence {𝑥𝑛 }∞
𝑛=0 converges strongly to 𝑝 =
ΠΩ (𝑥0 ).
If we set 𝑇𝑖 = 𝐼, for any 𝑖 = 1, 2, 3, . . . , 𝐷 in Theorem 13,
then we obtain the following result which extends and
improves the result’s Zegeye and Shahzad [7].
Corollary 15 (Zegeye and Shahzad [7]). Let 𝐶 be a nonempty,
closed and convex subset of a uniformly smooth and strictly
convex real Banach space 𝐸 which has Kadec-Klee property.
Suppose that
(1) {𝐹𝑘 }𝑀
𝑘=1 : 𝐶 × 𝐶 → R is a finite family of bifunctions satisfying conditions (A1)–(A4), where 𝑘 = 1, 2,
3, . . . , 𝑀;
(2) {𝑆𝑖 }𝐷
𝑖=1 : 𝐶 → 𝐸 is a finite family of quasi-𝜙-nonexpansive mappings, where 𝑖 = 1, 2, 3, . . . , 𝐷;
(3)
{𝐴 𝑛 }𝑁
𝑛=1
∗
: 𝐶 → 𝐸 is a finite family of continuous
monotone mappings, where 𝑛 = 1, 2, 3, . . . , 𝑁;
(97)
𝐶𝑛+1 = {𝑧 ∈ 𝐶𝑛 : 𝜙 (𝑧, 𝑦𝑛 ) ≤ 𝜙 (𝑧, 𝑥𝑛 )} ,
𝑥𝑛+1 = Π𝐶𝑛+1 (𝑥0 ) ,
∀𝑛 ≥ 0,
where 𝐴 𝑛 ≡ 𝐴 𝑛 (mod 𝑁), 𝐹𝑛 ≡ 𝐹𝑛 (mod 𝑀), 𝑆𝑛 ≡ 𝑆𝑛 (mod
𝐷), {𝑟𝑛 } ⊂ [𝑎, ∞) for some 𝑎 > 0 and 𝑛 ∈ N, and 𝛼, 𝛽, 𝛾 are
real numbers in (0,1) such that 𝛼 + 𝛽 + 𝛾 = 1.
̂
Then, the sequence {𝑥𝑛 }∞
𝑛=0 converges strongly to 𝑝 =
ΠΩ (𝑥0 ).
4. Deduced Theorems
If we set 𝐹𝑘 ≡ 0, for any 𝑘 = 1, 2, 3, . . . , 𝑀 in Theorem 13, then
we obtain the following result.
Corollary 16. Let 𝐶 be a nonempty, closed and convex subset
of a uniformly smooth and strictly convex real Banach space 𝐸
which has the Kadec-Klee property. Suppose that
𝐷
(1) {𝑇𝑖 }𝐷
𝑖=1 and {𝑆𝑖 }𝑖=1 are finite families of quasi-𝜙-nonexpansive mappings from 𝐶 into 𝐸, where 𝑖 =
1, 2, 3, . . . , 𝐷;
∗
(2) {𝐴 𝑛 }𝑁
𝑛=1 : 𝐶 → 𝐸 is a finite family of continuous
monotone mappings, where 𝑛 = 1, 2, 3, . . . , 𝑁;
(3) For 𝑥 ∈ 𝐸, we define the mappings 𝐾𝑟𝑛 : 𝐸 → 𝐶 by
𝐾𝑟𝑛 (𝑥) := {𝑧 ∈ 𝐶 : ⟨𝑦 − 𝑧, 𝐴 𝑛 𝑧⟩
1
+ ⟨𝑦 − 𝑧, 𝐽𝑧 − 𝐽𝑥⟩ ≥ 0, ∀𝑦 ∈ 𝐶} .
𝑟𝑛
(98)
Abstract and Applied Analysis
11
𝐷
𝑁
Assume that Ω := (⋂𝐷
𝑖=1 𝐹(𝑇𝑖 )) ⋂(⋂𝑖=1 𝐹(𝑆𝑖 )) ⋂ (⋂𝑛=1
VI(𝐶, 𝐴 𝑛 )) is a nonempty and bounded in 𝐶 and let {𝑥𝑛 } be
a sequence generated by
If we set 𝐹𝑘 ≡ 0, for any 𝑘 = 1, 2, 3, . . . , 𝑀 and 𝐴 𝑛 ≡ 0,
for any 𝑛 = 1, 2, 3, . . . , 𝑁 in Theorem 13, then we obtain the
following result.
𝑥0 ∈ 𝐶0 = 𝐶, 𝑐ℎ𝑜𝑠𝑒𝑛 𝑎𝑟𝑏𝑖𝑡𝑟𝑎𝑟𝑦,
Corollary 18. Let 𝐶 be a nonempty, closed and convex subset
of a uniformly smooth and strictly convex real Banach space 𝐸
which has the Kadec-Klee property. Suppose that {𝑇𝑖 }𝐷
𝑖=1 and
{𝑆𝑖 }𝐷
are
finite
families
of
quasi-𝜙-nonexpansive
mappings
𝑖=1
from 𝐶 into 𝐸, where 𝑖 = 1, 2, 3, . . . , 𝐷.
𝐷
Assume that Ω := (⋂𝐷
𝑖=1 𝐹(𝑇𝑖 )) ⋂(⋂𝑖=1 𝐹(𝑆𝑖 )) is a nonempty and bounded in 𝐶 and let {𝑥𝑛 } be a sequence generated
by
𝑧𝑛 = 𝐾𝑟𝑛 𝑥𝑛 ,
𝑦𝑛 = 𝐽−1 (𝛼𝐽𝑥𝑛 + 𝛽𝐽𝑇𝑛 𝑧𝑛 + 𝛾𝐽𝑆𝑛 𝑥𝑛 ) ,
(99)
𝐶𝑛+1 = {𝑧 ∈ 𝐶𝑛 : 𝜙 (𝑧, 𝑦𝑛 ) ≤ 𝜙 (𝑧, 𝑥𝑛 )} ,
𝑥𝑛+1 = Π𝐶𝑛+1 (𝑥0 ) ,
∀𝑛 ≥ 0,
where 𝐴 𝑛 ≡ 𝐴 𝑛 (mod 𝑁), 𝑇𝑛 ≡ 𝑇𝑛 (mod 𝐷), 𝑆𝑛 ≡ 𝑆𝑛 (mod
𝐷), {𝑟𝑛 } ⊂ [𝑎, ∞) for some 𝑎 > 0 and 𝑛 ∈ N, and 𝛼, 𝛽, 𝛾 are
real numbers in (0,1) such that 𝛼 + 𝛽 + 𝛾 = 1.
̂
Then, the sequence {𝑥𝑛 }∞
𝑛=0 converges strongly to 𝑝 =
ΠΩ (𝑥0 ).
If we set 𝐴 𝑛 ≡ 0, for any 𝑛 = 1, 2, 3, . . . , 𝑁 in Theorem 13,
then we obtain the following result.
Corollary 17. Let 𝐶 be a nonempty, closed and convex subset
of a uniformly smooth and strictly convex real Banach space 𝐸
which has the Kadec-Klee property. Suppose that
(1)
{𝐹𝑘 }𝑀
𝑘=1
: 𝐶 × 𝐶 → R is a finite family of bifunctions satisfying conditions (A1)–(A4), where 𝑘 = 1, 2,
3, . . . , 𝑀;
𝐷
(2) {𝑇𝑖 }𝐷
𝑖=1 and {𝑆𝑖 }𝑖=1 are finite families of quasi-𝜙-nonexpansive mappings from 𝐶 into 𝐸, where 𝑖 = 1, 2,
3, . . . , 𝐷;
(3) For 𝑥 ∈ 𝐸, one defines the mappings 𝑇𝑟𝑛 : 𝐸 → 𝐶 by
𝑥0 ∈ 𝐶0 = 𝐶, 𝑐ℎ𝑜𝑠𝑒𝑛 𝑎𝑟𝑏𝑖𝑡𝑟𝑎𝑟𝑦,
𝑦𝑛 = 𝐽−1 (𝛼𝐽𝑥𝑛 + 𝛽𝐽𝑇𝑛 𝑥𝑛 + 𝛾𝐽𝑆𝑛 𝑥𝑛 ) ,
𝐶𝑛+1 = {𝑧 ∈ 𝐶𝑛 : 𝜙 (𝑧, 𝑦𝑛 ) ≤ 𝜙 (𝑧, 𝑥𝑛 )} ,
𝑥𝑛+1 = Π𝐶𝑛+1 (𝑥0 ) ,
(102)
∀𝑛 ≥ 0,
where 𝑇𝑛 ≡ 𝑇𝑛 (mod 𝐷), 𝑆𝑛 ≡ 𝑆𝑛 (mod 𝐷), {𝑟𝑛 } ⊂ [𝑎, ∞) for
some 𝑎 > 0 and 𝑛 ∈ N, and 𝛼, 𝛽, 𝛾 are real numbers in (0,1)
such that 𝛼 + 𝛽 + 𝛾 = 1.
̂
Then, the sequence {𝑥𝑛 }∞
𝑛=0 converges strongly to 𝑝 =
ΠΩ (𝑥0 ).
5. Some Applications
(100)
5.1. Application to Weak Relatively Nonexpansive Mappings. If
we change the condition (2) in Theorem 13 as follows: {𝑇𝑖 }𝐷
𝑖=1
and {𝑆𝑖 }𝐷
𝑖=1 are finite families of weak relatively nonexpansive
mappings. From Remark 2(2), every weak relatively nonexpansive mappings is quasi-𝜙-nonexpansive mappings. Then,
we obtain the following result.
𝐷
𝑀
Assume that Ω := (⋂𝐷
𝑖=1 𝐹(𝑇𝑖 )) ⋂(⋂𝑖=1 𝐹(𝑆𝑖 )) ⋂ (⋂𝑘=1
SEP(𝐹𝑘 )) is a nonempty and bounded in 𝐶 and let {𝑥𝑛 } be a
sequence generated by
Corollary 19. Let 𝐶 be a nonempty, closed and convex subset
of a uniformly smooth and strictly convex real Banach space 𝐸
which has the Kadec-Klee property. Suppose that
𝑇𝑟𝑛 (𝑥) := {𝑧 ∈ 𝐶 : 𝐹𝑛 (𝑧, 𝑦)
1
+ ⟨𝑦 − 𝑧, 𝐽𝑧 − 𝐽𝑥⟩ ≥ 0, ∀𝑦 ∈ 𝐶} .
𝑟𝑛
𝑥0 ∈ 𝐶0 = 𝐶, 𝑐ℎ𝑜𝑠𝑒𝑛 𝑎𝑟𝑏𝑖𝑡𝑟𝑎𝑟𝑦,
𝑢𝑛 = 𝑇𝑟𝑛 𝑥𝑛 ,
𝑦𝑛 = 𝐽−1 (𝛼𝐽𝑥𝑛 + 𝛽𝐽𝑇𝑛 𝑥𝑛 + 𝛾𝐽𝑆𝑛 𝑢𝑛 ) ,
(101)
(1) {𝐹𝑘 }𝑀
𝑘=1 : 𝐶 × 𝐶 → R is a finite family of bifunctions satisfying conditions (A1)–(A4), where 𝑘 =
1, 2, 3, . . . , 𝑀;
𝐶𝑛+1 = {𝑧 ∈ 𝐶𝑛 : 𝜙 (𝑧, 𝑦𝑛 ) ≤ 𝜙 (𝑧, 𝑥𝑛 )} ,
𝑥𝑛+1 = Π𝐶𝑛+1 (𝑥0 ) ,
∀𝑛 ≥ 0,
where 𝐹𝑛 ≡ 𝐹𝑛 (mod 𝑀), 𝑇𝑛 ≡ 𝑇𝑛 (mod 𝐷), 𝑆𝑛 ≡ 𝑆𝑛 (mod
𝐷), {𝑟𝑛 } ⊂ [𝑎, ∞) for some 𝑎 > 0 and 𝑛 ∈ N, and 𝛼, 𝛽, 𝛾 are
real numbers in (0, 1) such that 𝛼 + 𝛽 + 𝛾 = 1.
̂
Then, the sequence {𝑥𝑛 }∞
𝑛=0 converges strongly to 𝑝 =
ΠΩ (𝑥0 ).
𝐷
(2) {𝑇𝑖 }𝐷
𝑖=1 and {𝑆𝑖 }𝑖=1 are finite families of weak relatively
nonexpansive mappings from 𝐶 into 𝐸, where 𝑖 =
1, 2, 3, . . . , 𝐷;
∗
(3) {𝐴 𝑛 }𝑁
𝑛=1 : 𝐶 → 𝐸 is a finite family of continuous
monotone mappings, where 𝑛 = 1, 2, 3, . . . , 𝑁;
12
Abstract and Applied Analysis
(4) For 𝑥 ∈ 𝐸, one defines the mappings 𝐾𝑟𝑛 , 𝑇𝑟𝑛 : 𝐸 → 𝐶
by
𝐾𝑟𝑛 (𝑥) := {𝑧 ∈ 𝐶 : ⟨𝑦 − 𝑧, 𝐴 𝑛 𝑧⟩
+
𝐾𝑟𝑛 (𝑥) := {𝑧 ∈ 𝐶 : ⟨𝑦 − 𝑧, 𝐴 𝑛 𝑧⟩
1
⟨𝑦 − 𝑧, 𝐽𝑧 − 𝐽𝑥⟩ ≥ 0, ∀𝑦 ∈ 𝐶} ;
𝑟𝑛
+
(103)
𝑇𝑟𝑛 (𝑥) := {𝑧 ∈ 𝐶 : 𝐹𝑛 (𝑧, 𝑦)
+
+
𝐷
𝑀
Assume that Ω := (⋂𝐷
𝑖=1 𝐹(𝑇𝑖 )) ⋂(⋂𝑖=1 𝐹(𝑆𝑖 )) ⋂ (⋂𝑘=1
𝑁
SEP(𝐹𝑘 )) ⋂(⋂𝑛=1 VI(𝐶, 𝐴 𝑛 )) is a nonempty and bounded in 𝐶
and let {𝑥𝑛 } be a sequence generated by
𝑥0 ∈ 𝐶0 = 𝐶, 𝑐ℎ𝑜𝑠𝑒𝑛 𝑎𝑟𝑏𝑖𝑡𝑟𝑎𝑟𝑦,
𝑧𝑛 = 𝐾𝑟𝑛 𝑥𝑛 ,
(105)
1
⟨𝑦 − 𝑧, 𝐽𝑧 − 𝐽𝑥⟩ ≥ 0, ∀𝑦 ∈ 𝐶} .
𝑟𝑛
𝐷
𝑀
Assume that Ω := (⋂𝐷
𝑖=1 𝐹(𝑇𝑖 )) ⋂(⋂𝑖=1 𝐹(𝑆𝑖 )) ⋂ (⋂𝑘=1
SEP(𝐹𝑘 )) ⋂(⋂𝑁
𝑛=1 VI(𝐶, 𝐴 𝑛 )) is a nonempty and bounded in 𝐶
and let {𝑥𝑛 } be a sequence generated by
𝑥0 ∈ 𝐶0 = 𝐶, 𝑐ℎ𝑜𝑠𝑒𝑛 𝑎𝑟𝑏𝑖𝑡𝑟𝑎𝑟𝑦,
𝑢𝑛 = 𝑇𝑟𝑛 𝑥𝑛 ,
𝑦𝑛 = 𝐽−1 (𝛼𝐽𝑥𝑛 + 𝛽𝐽𝑇𝑛 𝑧𝑛 + 𝛾𝐽𝑆𝑛 𝑢𝑛 ) ,
(104)
𝐶𝑛+1 = {𝑧 ∈ 𝐶𝑛 : 𝜙 (𝑧, 𝑦𝑛 ) ≤ 𝜙 (𝑧, 𝑥𝑛 )} ,
∀𝑛 ≥ 0,
where 𝐴 𝑛 ≡ 𝐴 𝑛 (mod 𝑁), 𝐹𝑛 ≡ 𝐹𝑛 (mod 𝑀), 𝑇𝑛 ≡ 𝑇𝑛 (mod
𝐷), 𝑆𝑛 ≡ 𝑆𝑛 (mod 𝐷), {𝑟𝑛 } ⊂ [𝑎, ∞) for some 𝑎 > 0 and 𝑛 ∈
N, and 𝛼, 𝛽, 𝛾 are real numbers in (0,1) such that 𝛼 + 𝛽 + 𝛾 = 1.
̂
Then, the sequence {𝑥𝑛 }∞
𝑛=0 converges strongly to 𝑝 =
ΠΩ (𝑥0 ).
5.2. Application to Relatively Nonexpansive Mappings. If we
change the condition (2) in Theorem 13 as follows: {𝑇𝑖 }𝐷
𝑖=1 and
{𝑆𝑖 }𝐷
are
finite
families
of
relatively
nonexpansive
mappings.
𝑖=1
From Remark 2(3) and (2) every relatively nonexpansive
mappings is weak relatively nonexpansive mappings and
every weak relatively nonexpansive mappings is quasi-𝜙nonexpansive mappings. Then, we obtain the following result.
Corollary 20. Let 𝐶 be a nonempty, closed and convex subset
of a uniformly smooth and strictly convex real Banach space 𝐸
which has the Kadec-Klee property. Suppose that
(1)
1
⟨𝑦 − 𝑧, 𝐽𝑧 − 𝐽𝑥⟩ ≥ 0, ∀𝑦 ∈ 𝐶} ;
𝑟𝑛
𝑇𝑟𝑛 (𝑥) := {𝑧 ∈ 𝐶 : 𝐹𝑛 (𝑧, 𝑦)
1
⟨𝑦 − 𝑧, 𝐽𝑧 − 𝐽𝑥⟩ ≥ 0, ∀𝑦 ∈ 𝐶} .
𝑟𝑛
𝑥𝑛+1 = Π𝐶𝑛+1 (𝑥0 ) ,
(4) For 𝑥 ∈ 𝐸, one defines the mappings 𝐾𝑟𝑛 , 𝑇𝑟𝑛 : 𝐸 → 𝐶
by
{𝐹𝑘 }𝑀
𝑘=1
: 𝐶 × 𝐶 → R is a finite family of bifunctions satisfying conditions (A1)–(A4), where 𝑘 =
1, 2, 3, . . . , 𝑀;
𝐷
(2) {𝑇𝑖 }𝐷
𝑖=1 and {𝑆𝑖 }𝑖=1 are finite families of relatively nonexpansive mappings from 𝐶 into 𝐸, where 𝑖 = 1, 2,
3, . . . , 𝐷;
∗
(3) {𝐴 𝑛 }𝑁
𝑛=1 : 𝐶 → 𝐸 is a finite family of continuous
monotone mappings, where 𝑛 = 1, 2, 3, . . . , 𝑁;
𝑧𝑛 = 𝐾𝑟𝑛 𝑥𝑛 ,
𝑢𝑛 = 𝑇𝑟𝑛 𝑥𝑛 ,
𝑦𝑛 = 𝐽−1 (𝛼𝐽𝑥𝑛 + 𝛽𝐽𝑇𝑛 𝑧𝑛 + 𝛾𝐽𝑆𝑛 𝑢𝑛 ) ,
(106)
𝐶𝑛+1 = {𝑧 ∈ 𝐶𝑛 : 𝜙 (𝑧, 𝑦𝑛 ) ≤ 𝜙 (𝑧, 𝑥𝑛 )} ,
𝑥𝑛+1 = Π𝐶𝑛+1 (𝑥0 ) ,
∀𝑛 ≥ 0,
where 𝐴 𝑛 ≡ 𝐴 𝑛 (mod 𝑁), 𝐹𝑛 ≡ 𝐹𝑛 (mod 𝑀), 𝑇𝑛 ≡ 𝑇𝑛 (mod
𝐷), 𝑆𝑛 ≡ 𝑆𝑛 (mod 𝐷), {𝑟𝑛 } ⊂ [𝑎, ∞) for some 𝑎 > 0 and 𝑛 ∈
N, and 𝛼, 𝛽, 𝛾 are real numbers in (0,1) such that 𝛼 + 𝛽 + 𝛾 = 1.
̂
Then, the sequence {𝑥𝑛 }∞
𝑛=0 converges strongly to 𝑝 =
ΠΩ (𝑥0 ).
5.3. Application to Hilbert Spaces. If 𝐸 = 𝐻, a real Hilbert
space, then 𝐸 is uniformly smooth and strictly convex real
Banach space. In this case, 𝐽 = 𝐼 and Π𝐶 = 𝑃𝐶. Then, we
obtain the following result.
Corollary 21. Let 𝐶 be a nonempty, closed and convex subset
of a real Hilbert space 𝐻. Suppose that
(1) {𝐹𝑘 }𝑀
𝑘=1 : 𝐶 × 𝐶 → R is a finite family of bifunctions satisfying conditions (A1)–(A4), where 𝑘 =
1, 2, 3, . . . , 𝑀;
𝐷
(2) {𝑇𝑖 }𝐷
𝑖=1 and {𝑆𝑖 }𝑖=1 are finite families of nonexpansive mappings from 𝐶 into 𝐶, where 𝑖 = 1, 2,
3, . . . , 𝐷;
(3) {𝐴 𝑛 }𝑁
𝑛=1 : 𝐶 → 𝐻 is a finite family of continuous
monotone mappings, where 𝑛 = 1, 2, 3, . . . , 𝑁;
Abstract and Applied Analysis
13
(4) For 𝑥 ∈ 𝐸, one defines the mappings 𝐾𝑟𝑛 , 𝑇𝑟𝑛 : 𝐻 → 𝐶
by
𝐾𝑟󸀠𝑛 (𝑥) := {𝑧 ∈ 𝐶 : ⟨𝑦 − 𝑧, 𝐴 𝑛 𝑧⟩
+
𝑇𝑟󸀠𝑛
1
⟨𝑦 − 𝑧, 𝑧 − 𝑥⟩ ≥ 0, ∀𝑦 ∈ 𝐶} ;
𝑟𝑛
(107)
(𝑥) := {𝑧 ∈ 𝐶 : 𝐹𝑛 (𝑧, 𝑦)
+
1
⟨𝑦 − 𝑧, 𝑧 − 𝑥⟩ ≥ 0, ∀𝑦 ∈ 𝐶} .
𝑟𝑛
𝐷
𝑀
Assume that Ω := (⋂𝐷
𝑖=1 𝐹(𝑇𝑖 )) ⋂(⋂𝑖=1 𝐹(𝑆𝑖 )) ⋂ (⋂𝑘=1
SEP(𝐹𝑘 )) ⋂(⋂𝑁
𝑛=1 VI(𝐶, 𝐴 𝑛 )) is a nonempty and bounded in 𝐶
and let {𝑥𝑛 } be a sequence generated by
𝑥0 ∈ 𝐶0 = 𝐶, 𝑐ℎ𝑜𝑠𝑒𝑛 𝑎𝑟𝑏𝑖𝑡𝑟𝑎𝑟𝑦,
𝑧𝑛 = 𝐾𝑟󸀠𝑛 𝑥𝑛 ,
𝑢𝑛 = 𝑇𝑟󸀠𝑛 𝑥𝑛 ,
𝑦𝑛 = 𝛼𝑥𝑛 + 𝛽𝑇𝑛 𝑧𝑛 + 𝛾𝑆𝑛 𝑢𝑛 ,
(108)
󵄩 󵄩
󵄩
󵄩
𝐶𝑛+1 = {𝑧 ∈ 𝐶𝑛 : 󵄩󵄩󵄩𝑧 − 𝑦𝑛 󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩𝑧 − 𝑥𝑛 󵄩󵄩󵄩} ,
𝑥𝑛+1 = 𝑃𝐶𝑛+1 (𝑥0 ) ,
∀𝑛 ≥ 0,
where 𝐴 𝑛 ≡ 𝐴 𝑛 (mod 𝑁), 𝐹𝑛 ≡ 𝐹𝑛 (mod 𝑀), 𝑇𝑛 ≡ 𝑇𝑛 (mod
𝐷), 𝑆𝑛 ≡ 𝑆𝑛 (mod 𝐷), {𝑟𝑛 } ⊂ [𝑎, ∞) for some 𝑎 > 0 and 𝑛 ∈
N, and 𝛼, 𝛽, 𝛾 are real numbers in (0,1) such that 𝛼 + 𝛽 + 𝛾 = 1.
Then, the sequence {𝑥𝑛 }∞
𝑛=0 converges strongly to an element
of Ω.
Remark 22. Our theorem extends and improves the corresponding results in [5–7] in the following aspect.
(a) For the mapping, we extend the mappings from nonexpansive mappings, relatively nonexpansive mappings, and weak relatively nonexpansive mappings to
more general than quasi-𝜙-nonexpansive mappings.
(b) For the common solution, we extend the common solution of a single finite family of quasi𝜙-nonexpansive mappings to two finite families of
quasi-𝜙-nonexpansive mappings.
Acknowledgments
This work was supported by the Higher Education Research
Promotion and National Research University Project of Thailand, Office of the Higher Education Commission. The first
author would like to thank the Bansomdejchaopraya Rajabhat
University (BSRU) for the financial support. Finally, the
authors would like to thank the referees for reading this paper
carefully, providing valuable suggestions and comments, and
pointing out a major error in the original version of this
paper.
References
[1] E. Blum and W. Oettli, “From optimization and variational
inequalities to equilibrium problems,” The Mathematics Student,
vol. 63, no. 1–4, pp. 123–145, 1994.
[2] A. Moudafi, “A partial complement method for approximating
solutions of a primal dual fixed-point problem,” Optimization
Letters, vol. 4, no. 3, pp. 449–456, 2010.
[3] P. M. Pardalos, T. M. Rassias, and A. A. Khan, Nonlinear Analysis
and Variational Problems, vol. 35 of Springer Optimization and
Its Applications, Springer, New York, NY, USA, 2010.
[4] W. R. Mann, “Mean value methods in iteration,” Proceedings of
the American Mathematical Society, vol. 4, pp. 506–510, 1953.
[5] W. Takahashi and K. Zembayashi, “Strong and weak convergence theorems for equilibrium problems and relatively
nonexpansive mappings in Banach spaces,” Nonlinear Analysis:
Theory, Methods & Applications, vol. 70, no. 1, pp. 45–57, 2009.
[6] X. Qin, Y. J. Cho, and S. M. Kang, “Convergence theorems of
common elements for equilibrium problems and fixed point
problems in Banach spaces,” Journal of Computational and
Applied Mathematics, vol. 225, no. 1, pp. 20–30, 2009.
[7] H. Zegeye and N. Shahzad, “A hybrid scheme for finite families
of equilibrium, variational inequality and fixed point problems,”
Nonlinear Analysis: Theory, Methods & Applications, vol. 74, no.
1, pp. 263–272, 2011.
[8] S.-y. Matsushita and W. Takahashi, “A strong convergence
theorem for relatively nonexpansive mappings in a Banach
space,” Journal of Approximation Theory, vol. 134, no. 2, pp. 257–
266, 2005.
[9] S. Plubtieng and K. Ungchittrakool, “Hybrid iterative methods
for convex feasibility problems and fixed point problems of
relatively nonexpansive mappings in Banach spaces,” Fixed
Point Theory and Applications, Article ID 583082, 19 pages,
2008.
[10] Y. Su, H.-k. Xu, and X. Zhang, “Strong convergence theorems
for two countable families of weak relatively nonexpansive mappings and applications,” Nonlinear Analysis: Theory, Methods &
Applications, vol. 73, no. 12, pp. 3890–3906, 2010.
[11] I. Cioranescu, Geometry of Banach Spaces, Duality Mappings
and Nonlinear Problems, vol. 62 of Mathematics and Its Applications, Kluwer Academic, Dordrecht, The Netherlands, 1990.
[12] W. Takahashi, Nonlinear Functional Analysis, Yokohama, Yokohama, Japan, 2000.
[13] Y. I. Alber, “Metric and generalized projection operators in
Banach spaces: properties and applications,” in Theory and
Applications of Nonlinear Operators of Accretive and Monotone
Type, vol. 178 of Lecture Notes in Pure and Applied Mathematics,
pp. 15–50, Dekker, New York, NY, USA, 1996.
[14] S. Kamimura and W. Takahashi, “Strong convergence of a
proximal-type algorithm in a Banach space,” SIAM Journal on
Optimization, vol. 13, no. 3, pp. 938–945, 2002.
[15] F. Kohsaka and W. Takahashi, “Existence and approximation of
fixed points of firmly nonexpansive-type mappings in Banach
spaces,” SIAM Journal on Optimization, vol. 19, no. 2, pp. 824–
835, 2008.
[16] F. Kohsaka and W. Takahashi, “Strong convergence of an
iterative sequence for maximal monotone operators in a Banach
space,” Abstract and Applied Analysis, vol. 2004, no. 3, pp. 239–
249, 2004.
[17] H. K. Xu, “Inequalities in Banach spaces with applications,”
Nonlinear Analysis: Theory, Methods & Applications, vol. 16, no.
12, pp. 1127–1138, 1991.
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