Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2012, Article ID 395760, 24 pages
doi:10.1155/2012/395760
Research Article
The Modified Block Iterative Algorithms for
Asymptotically Relatively Nonexpansive
Mappings and the System of Generalized Mixed
Equilibrium Problems
Kriengsak Wattanawitoon1 and Poom Kumam2
1
Department of Mathematics and Statistics, Faculty of Science and Agricultural Technology, Rajamangala
University of Technology Lanna Tak, Tak 63000, Thailand
2
Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi
(KMUTT), Bangmod, Thrungkru, Bangkok 10140, Thailand
Correspondence should be addressed to Poom Kumam, poom.kum@kmutt.ac.th
Received 3 March 2012; Accepted 17 June 2012
Academic Editor: Hong-Kun Xu
Copyright q 2012 K. Wattanawitoon 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.
The propose of this paper is to present a modified block iterative algorithm for finding a common
element between the set of solutions of the fixed points of two countable families of asymptotically
relatively nonexpansive mappings and the set of solution of the system of generalized mixed
equilibrium problems in a uniformly smooth and uniformly convex Banach space. Our results
extend many known recent results in the literature.
1. Introduction
The equilibrium problem theory provides a novel and unified treatment of a wide class of
problems which arise in economics, finance, image reconstruction, ecology, transportation,
networks, elasticity, and optimization, and it has been extended and generalized in many
directions.
In the theory of equilibrium problems, the development of an efficient and implementable iterative algorithm is interesting and important. This theory combines theoretical
and algorithmic advances with novel domain of applications. Analysis of these problems
requires a blend of techniques from convex analysis, functional analysis, and numerical
analysis.
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Journal of Applied Mathematics
Let E be a Banach space with norm · , C be a nonempty closed convex subset of
E, and let E∗ denote the dual of E. Let fi : C × C → R be a bifunction, ψi : C → R be a
real-valued function, where R is denoted by the set of real numbers, and Ai : C → E∗ be a
nonlinear mapping. The goal of the system of generalized mixed equilibrium problem is to find
u ∈ C such that
f1 u, y A1 u, y − u ψ1 y − ψ1 u ≥ 0,
f2 u, y A2 u, y − u ψ2 y − ψ2 u ≥ 0,
∀y ∈ C,
∀y ∈ C,
1.1
..
.
fN u, y AN u, y − u ψN y − ψN u ≥ 0,
∀y ∈ C.
If fi f, Ai A, and ψi ψ, the problem 1.1 is reduced to the generalized mixed equilibrium
problem, denoted by GEMPf, A, ψ, to find u ∈ C such that
f u, y Au, y − u ψ y − ψu ≥ 0,
∀y ∈ C.
1.2
The set of solutions to 1.2 is denoted by Ω, that is,
Ω x ∈ C : f u, y Au, y − u ϕ y − ϕu ≥ 0, ∀y ∈ C .
1.3
If A 0, the problem 1.2 is reduced to the mixed equilibrium problem for f, denoted by
MEPf, ψ, to find u ∈ C such that
f u, y ψ y − ψu ≥ 0,
∀y ∈ C.
1.4
If f ≡ 0, the problem 1.2 is reduced to the mixed variational inequality of Browder type,
denoted by VIC, A, ψ, is to find u ∈ C such that
Au, y − u ψ y − ψu ≥ 0,
∀y ∈ C.
1.5
If A 0 and ψ 0, the problem 1.2 is reduced to the equilibrium problem for f, denoted by
EPf, to find u ∈ C such that
f u, y ≥ 0,
∀y ∈ C.
1.6
The above formulation 1.6 was shown in 1 to cover monotone inclusion problems,
saddle-point problems, variational inequality problems, minimization problems, vector
equilibrium problems, and Nash equilibria in noncooperative games. In addition, there are
several other problems, for example, the complementarity problem, fixed-point problem, and
optimization problem, which can also be written in the form of an EPf. In other words,
the EPf is a unifying model for several problems arising in physics, engineering, science,
economics, and so forth. In the last two decades, many papers have appeared in the literature
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3
on the existence of solutions to EPf; see, for example 1–4 and references therein. Some
solution methods have been proposed to solve the EPf; see, for example, 2, 4–15 and
references therein. In 2005, Combettes and Hirstoaga 5 introduced an iterative scheme of
finding the best approximation to the initial data when EPf is nonempty, and they also
proved a strong convergence theorem.
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. Let U {x ∈ E : x 1} be the unit sphere of E. Then the Banach
space E is said to be smooth, provided
x ty − x
lim
t→0
t
1.7
exists for each x, y ∈ U. It is also said to be uniformly smooth if the limit is attained uniformly
for x, y ∈ E. The modulus of convexity of E is the function δ : 0, 2 → 0, 1 defined by
x y
: x, y ∈ E, x y 1, x − y ≥ ε .
δε inf 1 − 2 1.8
A Banach space E is uniformly convex, if and only if δε > 0 for all ε ∈ 0, 2.
Let E be a Banach space, C be a closed convex subset of E, a mapping T : C → C is
said to be nonexpansive if
T x − T y ≤ x − y
1.9
for all x, y ∈ C. We denote by FT the set of fixed points of T . If C is a bounded closed
convex set and T is a nonexpansive mapping of C into itself, then FT is nonempty see
16. A point p in C is said to be an asymptotic fixed point of T 17 if C contains a sequence
{xn } which converges weakly to p such that limn → ∞ xn − T xn 0. The set of asymptotic
. A point p ∈ C is said to be a strong asymptotic fixed
fixed points of T will be denoted by FT
point of T , if there exists a sequence {xn } ⊂ C such that xn → p and xn − T xn → 0. The
. A mapping T from C into
set of strong asymptotic fixed points of T will be denoted by FT
FT and φp, T x ≤ φp, x for
itself is said to be relatively nonexpansive 18–20 if FT
all x ∈ C and p ∈ FT . The asymptotic behavior of a relatively nonexpansive mapping was
studied in 21, 22. T is said to be φ-nonexpansive, if φT x, T y ≤ φx, y for x, y ∈ C. T is
said to be quas-φ-nonexpansive if FT /
∅ and φp, T x ≤ φp, x for x ∈ C and p ∈ FT . A
mapping T is said to be asymptotically relatively nonexpansive, if FT /
∅, and there exists a real
sequence {kn } ⊂ 1, ∞ with kn → 1 such that φp, T n x ≤ kn φp, x, for all n ≥ 1, x ∈ C, and
p ∈ FT . {Tn }∞
n0 is said to be a countable family of weak relatively nonexpansive mappings 23 if
the following conditions are satisfied:
i F{Tn }∞
/ ∅;
n0 ii φu, Tn x ≤ φu, x, for all u ∈ FTn , x ∈ C, n ≥ 0;
∞
n }∞
iii F{T
n0 ∩n0 FTn .
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A mapping T : C → C is said to be uniformly L-Lipschitz continuous, if there exists a constant
L > 0 such that
n
T x − T n y ≤ Lx − y,
∀x, y ∈ C, ∀n ≥ 1.
1.10
A mapping T : C → C is said to be closed if for any sequence {xn } ⊂ C with xn → x and
∞
T xn → y, then T x y. Let {Ti }∞
i1 : C → C be a sequence of mappings. {Ti }i1 is said to be
∅,
a countable family of uniformly asymptotically relatively nonexpansive mappings, if ∩∞
n1 FTn /
and there exists a sequence {kn } ⊂ 1, ∞ with kn → 1 such that for each i > 1
φ p, Tin x ≤ kn φ p, x ,
∀p ∈
∞
FTn , x ∈ C, ∀n ≥ 1.
n1
1.11
In 2009, Petrot et al. 24, introduced a hybrid projection method for approximating
a common element of the set of solutions of fixed points of hemirelatively nonexpansive
or quasi-φ-nonexpansive mappings in a uniformly convex and uniformly smooth Banach
space:
x0 ∈ C,
C0 C,
yn J −1 αn Jxn 1 − αn JTn zn ,
zn J −1 βn Jxn 1 − βn JTn xn ,
Cn1 v ∈ Cn : φ v, yn ≤ φv, xn ,
1.12
xn1 ΠCn1 x0 .
They proved that the sequence {xn } converges strongly to p ∈ FT , where p ∈ ΠFT x and ΠC
is the generalized projection from E onto FT . Kumam and Wattanawitoon 25, introduced
a hybrid iterative scheme for finding a common element of the set of common fixed points of
two quasi-φ-nonexpansive mappings and the set of solutions of an equilibrium problem in
Banach spaces, by the following manner:
x0 ∈ C,
C0 C
yn J −1 αn Jxn 1 − αn JSzn ,
zn J −1 βn Jxn 1 − βn JT xn ,
1
y − un , Jun − Jyn ≥ 0,
un ∈ C such that f un , y rn
Cn1 z ∈ Cn : φz, un ≤ φz, xn ,
xn1 ΠCn1 x0 .
∀y ∈ C,
1.13
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5
They proved that the sequence {xn } converges strongly to p ∈ FT ∩ FS ∩ EPf, where
p ∈ ΠFT ∩FS∩EPf x under the assumptions C1 lim supn → ∞ αn < 1, C2 limn → ∞ βn < 1, and
C3 lim infn → ∞ 1 − αn βn 1 − βn > 0.
Recently, Chang et al. 26, introduced the modified block iterative method to propose
an algorithm for solving the convex feasibility problems for an infinite family of quasi-φasymptotically nonexpansive mappings,
C0 C,
x0 ∈ C chosen arbitrary,
∞
−1
n
αn,0 Jxn αn,i JSi xn ,
yn J
1.14
i1
Cn1
v ∈ Cn : φ v, yn ≤ φv, xn ξn ,
xn1 ΠCn1 x0 ,
∀n ≥ 0,
where ξn supu∈F kn − 1φu, xn . Then, they proved that under appropriate control
conditions the sequence {xn } converges strongly to Π∩∞n1 FSi x0 .
Very recently, Tan and Chang 27, introduced a new hybrid iterative scheme for
finding a common element between set of solutions for a system of generalized mixed
equilibrium problems, set of common fixed points of a family of quasi-φ-asymptotically
nonexpansive mappings which is more general than quasi-φ-nonexpansive mappings, and
null spaces of finite family of γ-inverse strongly monotone mappings in a 2-uniformly convex
and uniformly smooth real Banach space.
In this paper, motivated and inspired by Petrot et al. 24, Kumam and Wattanawitoon
25, Chang et al. 26, and Tan and Chang 27, we introduce the new hybrid block algorithm
for two countable families of closed and uniformly Lipschitz continuous and uniformly
asymptotically relatively nonexpansive mappings in a Banach space. Let {xn } be a sequence
defined by x0 ∈ C, C0 C and
∞
n βn,0 Jxn βn,i J Ti xn ,
yn J
−1
i1
zn J
−1
αn,0 Jxn ∞
αn,i J
i1
i
Cn1 ⎧
⎨
un Kfi ,ri Kfi−1 ,ri−1 · · · Kf1 ,r1 zn ,
i
z ∈ Cn : max φ z, un
⎩
i1,2,...,N
Sni yn
,
i 1, 2, . . . , N,
≤ φz, xn θn , φ z, yn ≤ φz, xn ξn
xn1 ΠCn1 x0 ,
1.15
⎫
⎬
,
⎭
∀n ≥ 0.
Under appropriate conditions, we will prove that the sequence {xn } generated by algorithms
1.15 converges strongly to the point Π∩Ni1 Ωi ∩∩∞i1 FTi ∩∩∞i1 FSi x0 . Our results extend many
known recent results in the literature.
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Journal of Applied Mathematics
2. Preliminaries
Let E be a real Banach space with norm · , and let J be the normalized duality mapping
∗
from E into 2E given by
Jx {x∗ ∈ E∗ : x, x∗ xx∗ , x x∗ }
2.1
for all x ∈ E, where E∗ denotes the dual space of E and ·, · the generalized duality pairing
between E and E∗ . It is also known that if E is uniformly smooth, then J is uniformly normto-norm continuous on each bounded subset of E.
We know the following see 28, 29:
i if E is smooth, then J is single valued;
ii if E is strictly convex, then J is one-to-one and x − y, x∗ − y∗ > 0 holds for all
x, x∗ , y, y∗ ∈ J with x / y;
iii if E is reflexive, then J is surjective;
iv if E is uniformly convex, then it is reflexive;
v if E is a reflexive and strictly convex, then J −1 is norm-weak-continuous;
vi E is uniformly smooth if and only if E∗ is uniformly convex;
vii if E∗ is uniformly convex, then J is uniformly norm-to-norm continuous on each
bounded subset of E;
viii each uniformly convex Banach space E has the Kadec-Klee property, that is, for any
sequence {xn } ⊂ E, if xn x ∈ E and xn → x, then xn → x.
Let E be a smooth, strictly convex, and reflexive Banach space, and let C be a nonempty
closed convex subset of E. Throughout this paper, we denote by φ the function defined by
2
φ x, y x2 − 2 x, Jy y ,
for x, y ∈ E.
2.2
Following Alber 30, the generalized projection ΠC : E → C is a map that assigns to an
arbitrary point x ∈ E the minimum point of the function φx, y, that is, ΠC x x, where x is
the solution to the minimization problem
φx, x inf φ y, x .
2.3
y∈C
Existence and uniqueness of the operator ΠC follows from the properties of the functional
φx, y and strict monotonicity of the mapping J. It is obvious from the definition of function
φ that see 30
y − x 2 ≤ φy, x ≤ y x 2 ,
∀x, y ∈ E.
2.4
If E is a Hilbert space, then φx, y x − y2 .
If E is a reflexive, strictly convex, and smooth Banach space, then for x, y ∈ E, φx, y 0 if and only if x y. It is sufficient to show that if φx, y 0, then x y. From 2.4, we
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have x y. This implies that x, Jy x2 Jy2 . From the definition of J, one has
Jx Jy. Therefore, we have x y; see 28, 29 for more details.
We also need the following lemmas for the proof of our main results.
Lemma 2.1 see Kamimura and Takahashi 31. Let E be a uniformly convex and smooth real
Banach space, and let {xn }, {yn } be two sequences of E. If φxn , yn → 0 and either {xn } or {yn } is
bounded, then xn − yn → 0.
Lemma 2.2 see Alber 30. Let C be a nonempty closed convex subset of a smooth Banach space E
and x ∈ E. Then, x0 ΠC x if and only if
x0 − y, Jx − Jx0 ≥ 0,
∀y ∈ C.
2.5
Lemma 2.3 see Alber 30. Let E be a reflexive, strictly convex, and smooth Banach space, let C
be a nonempty closed convex subset of E, and let x ∈ E. Then
φ y, ΠC x φΠC x, x ≤ φ y, x ,
∀y ∈ C.
2.6
Lemma 2.4 see Chang et al. 26. Let E be a uniformly convex Banach space, r > 0 a positive
number, and Br 0 a closed ball of E. Then, for any given sequence {xi }∞
i1 ⊂ Br 0 and for any given
∞
sequence {λi }∞
i1 of positive number with
n1 λn 1, there exists a continuous, strictly increasing,
and convex function g : 0, 2r → 0, ∞ with g0 0 such that for any positive integers i, j with
i < j,
2
∞
∞
λn xn 2 − λi λj g xi − xj .
λn xn ≤
n1
n1
2.7
Lemma 2.5 see Chang et al. 26. Let E be a real uniformly smooth and strictly convex Banach
space with Kadec-Klee property, and C be a nonempty closed convex subset of E. Let T : C → C be a
closed and asymptotically relatively nonexpansive mapping with a sequence {kn } ⊂ 1, ∞, kn → 1.
Then FT is closed and convex subset of C.
For solving the generalized mixed equilibrium problem or a system of generalized
mixed equilibrium problem, let us assume that the bifunction f : C × C → R and ψ : C → R
is convex and lower semicontinuous satisfies the following conditions:
A1 fx, x 0 for all x ∈ C;
A2 f is monotone, that is, fx, y fy, x ≤ 0 for all x, y ∈ C;
A3 for each x, y, z ∈ C,
lim sup f tz 1 − tx, y ≤ f x, y ;
t↓0
A4 for each x ∈ C, y → fx, y is convex and lower semicontinuous.
2.8
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Lemma 2.6 see Chang et al. 26. Let C be a closed convex subset of a smooth, strictly convex,
and reflexive Banach space E. Let A : C → E∗ be a continuous and monotone mapping, ψ : C → R
is convex and lower semicontinuous and f be a bifunction from C × C to R satisfying (A1)–(A4). For
r > 0 and x ∈ E, then there exists u ∈ C such that
1
f u, y Au, y − u ψ y − ψu y − u, Ju − Jx ≥ 0,
r
∀y ∈ C.
2.9
Define a mapping Kf,r : C → C as follows:
1
Kf,r x u ∈ C : f u, y Au, y − u ψ y − ψu y − u, Ju − Jx ≥ 0, ∀y ∈ C
r
2.10
for all x ∈ E. Then, the following hold:
i Kf,r is singlevalued;
ii Kf,r is firmly nonexpansive, that is, for all x, y ∈ E, Kf,r x − Kf,r y, JKf,r x − JKf,r y ≤
Kf,r x − Kf,r y, Jx − Jy;
iii FKf,r FK
f,r ;
iv u ∈ C is a solution of variational equation 2.9 if and only if u ∈ C is a fixed point of Kf,r ;
v FKf,r Ω;
vi Ω is closed and convex;
vii φp, Kf,r z φKf,r z, z ≤ φp, z, for all p ∈ FKf,r , z ∈ E.
3. Main Results
Theorem 3.1. Let E be a uniformly smooth and uniformly convex Banach space, let C be a nonempty,
closed, and convex subset of E. Let Ai : C → E∗ be a continuous and monotone mapping,
ψi : C → R be a lower semi-continuous and convex function, fi be a bifunction from C × C to
R satisfying (A1)–(A4), Kfi ,ri is the mapping defined by 2.10 where ri ≥ r > 0, and let {Ti }∞
i1 ,
{Si }∞
i1 be countable families of closed and uniformly Li , μi -Lipschitz continuous and asymptotically
relatively nonexpansive mapping with sequence {kn }, {ζn } ⊂ 1, ∞; kn → 1, ζn → 1 such that
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∞
∞
∅. Let {xn } be a sequence generated by x0 ∈ C and
F : ∩N
i1 Ωi ∩ ∩i1 FTi ∩ ∩i1 FSi /
C0 C,
∞
n βn,0 Jxn βn,i J Ti xn ,
yn J
−1
i1
∞
n αn,0 Jxn αn,i J Si yn ,
zn J
−1
i1
i
Cn1 un Kfi ,ri Kfi−1 ,ri−1 · · · Kf1 ,r1 zn ,
z ∈ Cn : max φ
i1,2,...,N
i
z, un
3.1
i 1, 2, . . . , N,
≤ φz, xn θn , φ z, yn ≤ φz, xn ξn ,
xn1 ΠCn1 x0 ,
∀n ≥ 0,
where ξn supp∈F kn − 1φp, xn , θn δn ξn ζn , and δn supp∈F ζn − 1φp, xn . The coefficient
sequences {αn,i } and {βn,i } ⊂ 0, 1 satisfy the following:
i ∞
i0 αn,i 1;
∞
ii i0 βn,i 1;
iii lim infn → ∞ αn,0 αn,i > 0, for all i ≥ 1;
iv lim infn → ∞ βn,0 βn,i > 0, for all i ≥ 1,
Ωi , i 1, 2, . . . , N is the set of solutions to the following generalized mixed equilibrium problem:
fi z, y Ai z, y − z ψi y − ψi z ≥ 0,
∀y ∈ C, i 1, 2, . . . , N.
3.2
Then the sequence {xn } converges strongly to ΠF x0 .
Proof. We first show that Cn , for all n ≥ 0 is closed and convex. Clearly C0 C is closed and
convex. Suppose that Ck is closed and convex for some k > 1. For each z ∈ Ck , we see that
i
φz, uk ≤ φz, xk is equivalent to
2
i i
2 z, xk − z, uk
≤ xk 2 − uk .
3.3
By the set of Ck1 , we have
Cn1
i
z ∈ Cn : max φ z, un ≤ φz, xn θn
i1,2,...,N
N
i
z ∈ C : φ z, un ≤ φz, xn θn .
i1
Hence, Cn1 is also closed and convex.
3.4
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j
By taking Θn Krj ,fi Krj−1 ,fj−1 · · · Kr1 ,f1 for any j ∈ {1, 2, . . . , i} and Θ0n I for all n ≥ 1.
i
We note that un Θin zn .
Next, we show that F ⊂ Cn , for all n ≥ 1. For n ≥ 1, we have F ⊂ C C1 . For any given
∞
∞
p ∈ F : ∩N
i1 Ωi ∩ ∩i1 FTi ∩ ∩i1 FSi . By 3.1 and Lemma 2.4, we have
φ p, yn
∞
n
βn,i JTi xn
φ p, J
−1
i0
2
∞
∞
2 n
n
p − βn,i 2 p, JTi xn βn,i JTi xn i0
i0
∞
∞
2 2
≤ p − βn,i 2 p, JTin xn βn,i JTin xn − βn,0 βn,i g JT0n xn − JTin xn i0
i0
∞
∞
2 2
p − βn,i 2 p, JTin xn βn,i Tin xn − βn,0 βn,i g Jxn − JTin xn i0
∞
i0
i0
βn,i φ p, Tin xn − βn,0 βn,i g Jxn − JTin xn ≤ kn φ p, xn − βn,0 βn,i g Jxn − JTin xn ≤ φ p, xn supkn − 1φ p, xn − βn,0 βn,i g Jxn − JTin xn p∈F
≤ φ p, xn ξn − βn,0 βn,i g Jxn − JTin xn ≤ φ p, xn ξn ,
3.5
where ξn supp∈F kn − 1φp, xn .
By 3.1 and 3.5, we note that
i
φ p, un φ p, Θin zn
≤ φ p, zn
≤ φ p, J
−1
αn,0 Jxn ∞
i1
JSni yn
2
∞
∞
2
p − 2 p, αn,0 Jxn JSni yn αn,0 Jxn JSni yn i1
i1
∞
∞ 2
2
≤ p − 2αn,0 p, Jxn − 2 αn,i p, JSni yn αn,0 xn 2 Sni yn i1
− αn,0 αn,i g Jxn − JSni yn i1
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11
∞
≤ αn,0 φ p, xn αn,i φ p, Sni yn − αn,0 αn,i g Jxn − JSni yn i1
∞
≤ αn,0 φ p, xn ζn αn,i φ p, yn − αn,0 αn,i g Jxn − JSni yn i1
∞
≤ αn,0 φ p, xn ζn αn,i φ p, xn ξn − αn,0 αn,i g Jxn − JSni yn i1
∞
∞
≤ αn,0 φ p, xn ζn αn,i φ p, xn ξn ζn αn,i − αn,0 αn,i g Jxn − JSni yn i1
≤ ζn φ p, xn ξn ζn
∞
i1
i1
αn,i − αn,0 αn,i g Jxn − JSni yn ∞
≤ φ p, xn supζn − 1φ p, xn ξn ζn αn,i − αn,0 αn,i g Jxn − JSni yn p∈F
i1
δn ξn ζn − αn,0 αn,i g Jxn − JSni yn ≤ φ p, xn
≤ φ p, xn θn ,
3.6
where δn supp∈F ζn − 1φp, xn , θn δn ξn ζn . By assumptions on {kn } and {ζn }, we have
ξn supkn − 1φ p, xn
p∈F
2
≤ supkn − 1 p M −→ 0 as n −→ ∞,
3.7
p∈F
δn supζn − 1φ p, xn
p∈F
2
≤ supζn − 1 p M −→ 0 as n −→ ∞,
3.8
p∈F
where M supn≥0 xn .
So, we have p ∈ Cn1 . This implies that F ∈ Cn , for all n ≥ 0 and also {xn } is well
defined.
From Lemma 2.2 and xn ΠCn x0 , we have
xn − z, Jx0 − Jxn ≥ 0,
xn − p, Jx0 − Jxn ≥ 0,
∀z ∈ Cn ,
∀p ∈ Cn .
3.9
From Lemma 2.3, one has
φxn , x0 φΠCn x0 , x0 ≤ φ p, x0 − φ p, xn ≤ φ p, x0
3.10
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Journal of Applied Mathematics
for all p ∈ F ⊂ Cn and n ≥ 1. Then, the sequence {φxn , x0 } is also bounded. Thus {xn } is
bounded. Since xn ΠCn x0 and xn1 ΠCn1 x0 ∈ Cn1 ⊂ Cn , we have
φxn , x0 ≤ φxn1 , x0 ,
∀n ∈ N.
3.11
Therefore, {φxn , x0 } is nondecreasing. Hence, the limit of {φxn , x0 } exists. By the
construction of Cn , one has that Cm ⊂ Cn and xm ΠCm x0 ∈ Cn for any positive integer
m ≥ n. It follows that
φxm , xn φxm , ΠCn x0 ≤ φxm , x0 − φΠCn x0 , x0 3.12
φxm , x0 − φxn , x0 .
Letting m, n → 0 in 3.12, we get φxm , xn → 0. It follows from Lemma 2.1, that xm −
xn → 0 as m, n → ∞. That is, {xn } is a Cauchy sequence.
Since {xn } is bounded and E is reflexive, there exists a subsequence {xni } ⊂ {xn } such
that xni u. Since Cn is closed and convex and Cn1 ⊂ Cn , this implies that Cn is weakly
closed and u ∈ Cn for each n ≥ 0. since xn ΠCn x0 , we have
φxni , x0 ≤ φu, x0 ,
∀ni ≥ 0.
3.13
Since
lim inf φxni , x0 lim inf xni 2 − 2xni , Jx0 x0 2
ni → ∞
ni → ∞
≤ u2 − 2u, Jx0 x0 2
3.14
φu, x0 .
We have
φu, x0 ≤ lim inf φxni , x0 ≤ lim sup φxni , x0 ≤ φu, x0 .
ni → ∞
ni → ∞
3.15
This implies that limni → ∞ φxni , x0 φu, x0 . That is, xni → u. Since xni u, by the
Kadec-klee property of E, we obtain that
lim xni u.
3.16
n→∞
If there exists some subsequence {xnj } ⊂ {xn } such that xnj → q, then we have
φ u, q lim
φ xni , xnj ≤
ni → ∞,nj → ∞
lim
ni → ∞,nj → ∞
lim
ni → ∞,nj → ∞
φxni , x0 − φ xnj x0 , x0
φxni , x0 − φ ΠCnj x0 , x0
3.17
0.
Journal of Applied Mathematics
13
Therefore, we have u q. This implies that
lim xn u.
3.18
n→∞
Since
φxn1 , xn φxn1 , ΠCn x0 ≤ φxn1 , x0 − φΠCn x0 , x0 φxn1 , x0 − φxn , x0 3.19
for all n ∈ N, we also have
lim φxn1 , xn 0.
3.20
n→∞
Since xn1 ΠCn1 x0 ∈ Cn1 and by the definition of Cn1 , for i 1, 2, . . . , N, we have
φ xn1 , uin ≤ φxn1 , xn θn .
3.21
Noticing that limn → ∞ φxn1 , xn 0, we obtain
lim φ xn1 , uin 0,
n→∞
for i 1, 2, . . . , N.
3.22
It then yields that limn → ∞ xn1 −uin 0, for all i 1, 2, . . . , N. Since limn → ∞ xn1 u,
we have
lim uin u, ∀i 1, 2, . . . , N.
3.23
n→∞
Hence,
lim Juin Ju,
n→∞
∀i 1, 2, . . . , N.
3.24
From Lemma 2.1 and 3.22, we have
lim xn1 − xn lim xn1 − uin 0,
n→∞
n→∞
∀i 1, 2, . . . , N.
3.25
By the triangle inequality, we get
lim xn − uin 0,
n→∞
∀i 1, 2, . . . , N.
3.26
Since J is uniformly norm-to-norm continuous on bounded sets, we note that
lim Jxn − Juin lim Jxn1 − Juin 0,
n→∞
n→∞
∀i 1, 2, . . . , N.
3.27
14
Journal of Applied Mathematics
∞
Now, we prove that u ∈ ∩∞
i1 FTi ∩ ∩i1 FSi . From the construction of Cn , we
obtain that
φ xn1 , yn ≤ φxn1 , xn ξn .
3.28
n→∞
lim φ xn1 , yn 0.
3.29
lim xn1 − yn 0.
3.30
From 3.7 and 3.20, we have
By Lemma 2.1, we also have
n→∞
Since J is uniformly norm-to-norm continuous on bounded sets, we note that
lim Jxn1 − Jyn 0.
n→∞
3.31
From 2.4 and 3.29, we have xn1 − yn 2 → 0. Since xn1 → u, it yields that
yn −→ u
as n −→ ∞.
3.32
Since J is uniformly norm-to-norm continuous on bounded sets, it follows that
Jyn −→ Ju
as n −→ ∞.
3.33
This implies that {Jyn } is bounded in E∗ . Since E is reflexive, there exists a subsequence
{Jyni } ⊂ {Jyn } such that Jyni r ∈ E∗ . Since E is reflexive, we see that JE E∗ . Hence,
there exists x ∈ E such that Jx r. We note that
2
φ xni 1 , yni xni 1 2 − 2 xni 1 , Jyni yni 2
xni 1 2 − 2 xni 1 , Jyni Jyni .
3.34
Taking the limit interior of both side and in view of weak lower semicontinuity of norm · ,
we have
0 ≥ u2 − 2u, r r2
u2 − 2u, Jx Jx2
3.35
u2 − 2u, Jx x2 φu, x,
that is, u x. This implies that r Ju and so Jyn Jp. It follows from limn → ∞ Jyn Ju, as n → ∞ and the Kadec-Klee property of E∗ that Jyni → Ju as n → ∞. Note
Journal of Applied Mathematics
15
that J −1 : E∗ → E is hemicontinuous, it yields that yni u. It follows from limn → ∞ un u, as n → ∞ and the Kadec-Klee property of E that limni → ∞ yni u.
By similar, we can prove that
lim yn u.
3.36
lim xn − yn 0.
3.37
n→∞
By 3.20 and 3.30, we obtain
n→∞
Since J is uniformly norm-to-norm continuous on bounded sets, we note that
lim Jxn − Jyn 0.
n→∞
3.38
So, from 3.27 and 3.31, by the triangle inequality, we get
lim Jyn − Juin 0,
n→∞
for i 1, 2, . . . , N.
3.39
Since J −1 is uniformly norm-to-norm continuous on bounded sets, we note that
lim yn − uin 0,
n→∞
for i 1, 2, . . . , N.
3.40
Since
2
φ p, xn − φ p, yn xn 2 − yn − 2 p, Jxn − Jyn
2
≤ xn 2 − yn 2pJxn − Jyn ≤ xn − yn xn yn 2pJxn − Jyn .
3.41
From 3.37 and 3.38, we obtain
φ p, xn − φ p, yn −→ 0,
n −→ ∞.
3.42
On the other hand, we observe that, for i 1, 2, . . . , N.
2
φ p, xn − φ p, uin xn 2 − uin − 2 p, Jxn − Juin
2
≤ xn 2 − uin 2pJxn − Juin ≤ xn − uin xn uin 2pJxn − Juin .
3.43
16
Journal of Applied Mathematics
From 3.22 and 3.27, we have
φ p, xn − φ p, uin −→ 0,
n −→ ∞, ∀i 1, 2, . . . , N.
3.44
∞
∞
For any p ∈ ∩N
i1 Ωi ∩ ∩i1 FTi ∩ ∩i1 FSi , it follows from 3.5 that
βn,0 βn,i g Jxn − JTin xn ≤ φ p, xn ξn − φ p, yn .
3.45
From condition, lim infn → ∞ βn,0 βn,i > 0, property of g, 3.7, and 3.42, we have that
Jxn − JT n xn −→ 0,
i
n −→ ∞, ∀i 1, 2, . . . , N.
3.46
Since xn → u and J is uniformly norm-to-norm continuous. It yields Jxn → Jp. Hence from
3.46, we have
xn − T n xn −→ 0,
i
n −→ ∞, ∀i 1, 2, . . . , N.
3.47
Since xn → u, this implies that limn → ∞ JTin xn → Ju as n → ∞. Since J −1 : E∗ → E is
hemicontinuous, it follows that
Tin xn u,
for each i ≥ 1.
3.48
On the other hand, for each i ≥ 1, we have
n T xn − u T n xn − u
i
i
≤ Tin xn − u −→ 0,
n −→ ∞.
3.49
from this, together with 3.48 and the Kadec-Klee property of E, we obtain
Tin xn −→ u,
for each i ≥ 1.
3.50
On the other hand, by the assumption that Ti is uniformly Li -Lipschitz continuous, we have
n1
Ti xn − Tin xn ≤ Tin1 xn − Tin1 xn1 Tin1 xn1 − xn1 xn1 − xn xn − Tin xn ≤ Li 1xn1 − xn Tin1 xn1 − xn1 xn − Tin xn .
3.51
Journal of Applied Mathematics
17
By 3.18 and 3.50, we obtain
lim Tin1 xn − Tin xn 0,
n→∞
∀i ≥ 1,
3.52
and limn → ∞ Tin1 xn u, that is, Ti T n xn → u, for all i ≥ 1. By the closeness of Ti , we have
Ti u u, for all i ≥ 1. This implies that u ∈ ∩∞
i1 FTi .
By the similar way, we can prove that for each i ≥ 1
Jxn − JSn yn −→ 0,
i
n −→ ∞.
3.53
Since xn → u and J is uniformly norm-to-norm continuous. it yields Jxn → Jp. Hence from
3.53, we have
xn − Sn yn −→ 0,
i
n −→ ∞.
3.54
Since xn → u, this implies that limn → ∞ JSni yn → Ju as n → ∞. Since J −1 : E∗ → E is
hemicontinuous, it follows that
Sni yn u,
for each i ≥ 1.
3.55
On the other hand, for each i ≥ 1, we have
n S yn − u Sn yn − u
i
i
≤ Sni yn − u −→ 0,
n −→ ∞.
3.56
From this, together with 3.54 and the Kadec-Klee property of E, we obtain
Sni yn −→ u,
for each i ≥ 1.
3.57
On the other hand, by the assumption that Si is uniformly μi -Lipschitz continuous, we have
n1
n1
n1
y
−
S
y
y
−
y
Si yn − Sni yn ≤ Sn1
S
n
n1
n1
n1 i
i
i
yn1 − yn yn − Sni yn ≤ μi 1 yn1 − yn Sn1
y
−
y
n1
n1
i
yn − Sni yn .
3.58
18
Journal of Applied Mathematics
By 3.36 and 3.57, we obtain
n
y
−
S
y
lim Sn1
0
n
n
i
i
3.59
n→∞
n
and limn → ∞ Sn1
i yn u, that is, Si T yn → u. By the closeness of Si , we have Si u u, for all i ≥
∞
∞
1. This implies that u ∈ ∩i1 FSi . Hence u ∈ ∩∞
i1 FTi ∩ ∩i1 FSi .
N
Next, we prove that u ∈ ∩i1 Ωi . For any p ∈ F, for each i 1, 2, . . . , N, we have
φ uin , zn φ Θin zn , zn
≤ φ p, zn − φ p, Θin zn
3.60
φ p, zn − φ p, uin
≤ φ p, xn θn − φ p, uin → 0,
as n → ∞.
It then yields that limn → ∞ uin − zn 0. Since limn → ∞ uin u, for all i ≥ 1, we have
lim zn u.
3.61
lim Jzn Ju.
3.62
n→∞
Hence,
n→∞
This together with limn → ∞ uin u show that for each i 1, 2, . . . , N,
i
i−1 lim uin − ui−1
n lim Jun − Jun 0,
n→∞
n→∞
3.63
where u0n zn . On the other hand, we have
uin Kfi ,ri ui−1
n ,
for each i 2, 3, . . . , N,
3.64
and uin is a solution of the following variational equation
1
y − uin , Juin − Jui−1
≥ 0,
fi uin , y Ai uin , y − uin ψi y − ψi uin n
ri
∀y ∈ C. 3.65
By condition A2, we note that
1
Ai uin , y − uin ψi y − ψi uin y − uin , Juin − Jui−1
n
ri
≥ −fi uin , y ≥ fi y, uin , ∀y ∈ C.
3.66
Journal of Applied Mathematics
19
By A4, 3.63, and uin → u for each i 2, 3, . . . , N, we have
Ai u, y − u ψi y − ψi u ≥ fi y, u ,
∀y ∈ C.
3.67
For 0 < t < 1 and y ∈ C, define yt ty 1 − tu. Noticing that y, u ∈ C, we obtain yt ∈ C,
which yields that
Ai u, yt − u ψi yt − ψi u ≥ fi yt , u .
3.68
In view of the convexity of φ it yields
t Ai u, y − u t ψi y − ψi u ≥ fi yt , u .
3.69
It follows from A1 and A4 that
0 fi yt , yt ≤ tfi yt , y 1 − tfi yt , u
!
≤ tfi yt , y 1 − tt Ai u, y − u ψi y − ψi u .
3.70
Let t → 0, from A3, we obtain the following:
fi u, y Ai u, y − u ψi y − ψi u ≥ 0,
∀y ∈ C, i 1, 2, . . . , N.
3.71
This implies that u is a solution of the system of generalized mixed equilibrium problem 3.2,
N
∞
∞
that is, u ∈ ∩N
i1 Ωi . Hence, u ∈ F : ∩i1 Ωi ∩ ∩i1 FTi ∩ ∩i1 FSi .
Finally, we show that xn → u ΠF x0 . Indeed from w ∈ F ⊂ Cn and xn ΠCn x0 , we
have the following:
φxn , x0 ≤ φw, x0 ,
∀n ≥ 0.
3.72
This implies that
φu, x0 lim φxn , x0 ≤ φw, x0 .
n→∞
3.73
From the definition of ΠF x0 and 3.73, we see that u w. This completes the proof.
Since every asymptotically relatively nonexpansive mappings is quasi-φ-nonexpansive mappings, hence we obtain the following corollary.
Corollary 3.2. Let E be a uniformly convex and uniformly smooth Banach space, let C be a
nonempty, closed, and convex subset of E. Let Ai : C → E∗ be a continuous and monotone
mapping, ψi : C → R be a lower semi-continuous and convex function, fi be a bifunction from
C × C to R satisfying (A1)–(A4), Kfi ,ri is the mapping defined by 2.10 where ri ≥ r > 0, and
∞
let {Ti }∞
i1 , {Si }i1 be countable families of closed and quasi-φ-nonexpansive mapping such that
20
Journal of Applied Mathematics
∞
∞
∅. Let {xn } be a sequence generated by x0 ∈ C and
F : ∩N
i1 Ωi ∩ ∩i1 FTi ∩ ∩i1 FSi /
C0 C, such that
∞
βn,0 Jxn βn,i JTi xn ,
yn J
−1
zn J
−1
i1
∞
αn,0 Jxn αn,i J Si yn ,
i1
i
Cn1 un Kfi ,ri Kfi−1 ,ri−1 · · · Kf1 ,r1 zn ,
i 1, 2, . . . , N,
3.74
i
z ∈ Cn : max φ z, un ≤ φz, xn , φ z, yn ≤ φz, xn ,
i1,2,...,N
xn1 ΠCn1 x0 ,
∀n ≥ 0,
where ΠC is the generalized projection from E onto C, J is the duality mapping on E. The coefficient
sequences {αn,i } and {βn,i } ⊂ 0, 1, satisfying:
i
ii
∞
i0
∞
i0
αn,i 1;
βn,i 1;
iii lim infn → ∞ αn,0 αn,i > 0, for all i ≥ 1;
iv lim infn → ∞ βn,0 βn,i > 0, for all i ≥ 1.
Ωi , i 1, 2, . . . , N is the set of solutions to the following generalized mixed equilibrium problem:
fi z, y Ai z, y − z ψi y − ψi z ≥ 0,
∀y ∈ C, i 1, 2, . . . , N.
3.75
Then the sequence {xn } converges strongly to ΠF x0 .
If Ai A, ψi ψ, and fi f for all i ≥ 1 in Theorem 3.1, we obtain the following
corollary.
Corollary 3.3. Let E be a uniformly smooth and uniformly convex Banach space, let C be a nonempty,
closed, and convex subset of E. Let A : C → E∗ be a continuous and monotone mapping,
ψ : C → R be a lower semicontinuous and convex function, f be a bifunction from C × C
to R satisfying (A1)–(A4), Kf,r be the mapping define by 2.10 where r > 0, and let {Ti }∞
i1 ,
{Si }∞
i1 be countable families of closed and uniformly Li , μi -Lipschitz continuous, and asymptotically
relatively nonexpansive mappings with sequence {kn }, {ζn } ⊂ 1, ∞; kn → 1, ζn → 1 such that
Journal of Applied Mathematics
21
∞
∅. Let {xn } be a sequence generated by x0 ∈ C and C0 C,
F : Ω ∩ ∩∞
i1 FTi ∩ ∩i1 FSi /
such that
∞
n βn,0 Jxn βn,i J Ti xn ,
yn J
−1
i1
zn J
−1
αn,0 Jxn ∞
αn,i J
i1
Sni yn
,
3.76
un Kf,r zn ,
Cn1 i
z ∈ Cn : max φ z, un ≤ φz, xn θn , φ z, yn ≤ φz, xn ξn ,
i1,2,...,N
xn1 ΠCn1 x0 ,
∀n ≥ 0,
where ξn supp∈F kn − 1φp, xn , θn δn ξn ζn , and δn supp∈F ζn − 1φp, xn . The coefficient
sequences {αn,i } and {βn,i } ⊂ 0, 1, satisfying:
i
ii
∞
i0
αn,i 1;
i0
βn,i 1;
∞
iii lim infn → ∞ αn,0 αn,i > 0, for all i ≥ 1;
iv lim infn → ∞ βn,0 βn,i > 0, for all i ≥ 1.
Then the sequence {xn } converges strongly to ΠF x0 .
If i 1 in Theorem 3.1, then we obtain the following corollary.
Corollary 3.4. Let E be a uniformly smooth and uniformly convex Banach space, let C be a nonempty,
closed, and convex subset of E. Let A : C → E∗ be a continuous and monotone mapping,
ψ : C → R be a lower semicontinuous and convex function, f be a bifunction from C × C to R
satisfying (A1)–(A4), Kf,r is the mapping define by 2.10 where r > 0, and let T, S are two closed
and uniformly L, μ-Lipschitz continuous and asymptotically relatively nonexpansive mappings with
∅. Let {xn } be a
sequence {kn }, {ζn } ⊂ 1, ∞; kn → 1, ζn → 1 such that F : Ω ∩ FT ∩ FS /
sequence generated by x0 ∈ C and C0 C, we have
yn J −1 βn Jxn 1 − βn JT n xn ,
zn J −1 αn Jxn 1 − αn JSn yn ,
un Kf,r zn ,
3.77
Cn1 z ∈ Cn : φz, un ≤ φz, xn θn , φ z, yn ≤ φz, xn ξn ,
xn1 ΠCn1 x0 ,
∀n ≥ 0,
where ξn supp∈F kn − 1φp, xn , θn δn ξn ζn , and δn supp∈F ζn − 1φp, xn . The coefficient
sequences {αn } and {βn } ⊂ 0, 1, satisfying
22
Journal of Applied Mathematics
D1 lim infn → ∞ αn 1 − αn > 0;
D2 lim infn → ∞ βn 1 − βn > 0.
Then the sequence {xn } converges strongly to ΠF x0 .
Remark 3.5. Theorem 3.1 and Corollary 3.3 improve and extend the corresponding results of
Petrot et al. 24, Kumam and Wattanawitoon 25, and Chang et al. 26 in the following
senses:
i for the mappings, we extend the mappings from nonexpansive mappings, hemirelatively nonexpansive mappings to two infinite family of closed asymptotically
relatively nonexpansive mappings;
ii from a solution of the classical equilibrium problem to a system of generalized
mixed equilibrium problems and the generalized mixed equilibrium problem with
an infinite family of closed relatively nonexpansive mappings.
Remark 3.6. Corollary 3.4 improves and extends the corresponding results of Theorem 3.1 in
Kumam and Wattanawitoon 25 and Corollary 3.3 in Saewan et al. 11 in the following
senses:
i the mapping in 11 and 25
ii the conditions D1 and D2 of the parameters {αn } and {βn } are weaker and
not complicated than the conditions C1–C3 in 25, Theorem 3.1 and 11,
Theorem 3.1 which are easy to compute.
Acknowledgments
This first author was supported by The Hands-On Research and Development, Rajamangala
University of Technology Lanna Grant no. UR2L003. Moreover, this work was supported
by the Higher Education Research Promotion and National Research University Project of
Thailand, Office of the Higher Education Commission NRU-CSEC Grant no. 55000613.
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