Báo cáo hóa học: "A general composite iterative method for generalized mixed equilibrium problems, variational inequality problems and optimization problems" pot

RESEARC H Open Access
A general composite iterative method for
generalized mixed equilibrium problems,
variational inequality problems and optimization
problems
Jong Soo Jung
Correspondence: jungjs@mail.
donga.ac.kr
Department of Mathematics,
Dong-A University, Busan, 604-714,
Korea
Abstract
In this article, we introduce a new general composite iterative scheme for finding a
common element of the set of solutions of a generalized mixed equilibrium
problem, the set of fixed points of an infinite family of nonexpansive mappings and
the set of solutions of a variational inequality problem for an inverse-strongly
monotone mapping in Hilbert spaces. It is shown that the sequence generated by
the proposed iterative scheme converges strongly to a common element of the
above three sets under suitable control conditions, which solves a certain
optimization problem. The results of this article substantially improve, develop, and
complement the previous well-known results in this area.
2010 Mathematics Subject Classifications: 49J30; 49J40; 47H09; 47H10 ; 47J20;
47J25; 47J05; 49M05.
Keywords: generalized mixed equilibrium problem, fixed point, nonexpansive map-
ping; inverse-strongly monotone mapping, variational inequality; optimization pro-
blem, metric projection, strongly positive bounded linear operator
1 Introduction
Let H be a real Hilbert space with inner product 〈·, ·〉 and induced norm || · ||. Let C
beanonemptyclosedconvexsubsetofH and S : C ® C be a sel f-ma pping on C.Let
us denote by F(S)thesetoffixedpointsofS and by P
C

the metric projection of H
onto C .
Let B : C ® H be a nonlinear mapping and  : C ® ℝ be a function, and Θ be a
bifunction of C × C into ℝ, where ℝ is the set of real numbers.
Then, we consider the following generalized mixed equilibrium problem of finding
x Î C such that

(
x, y
)
+ Bx, y − x + ϕ
(
y
)
− ϕ
(
x
)
≥ 0, ∀y ∈ C
,
(1:1)
which was recently introduced by Peng and Yao [1]. The set of solutions of the pro-
blem (1.1) is denoted by GMEP(Θ, , B). Here, some special cases of the problem (1.1)
are stated as follows:
Jung Journal of Inequalities and Applications 2011, 2011:51
/>© 2011 Jung; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attrib ution
License (http://creat ivecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium,
provided the original work is properly cited.
If B = 0, then the problem (1.1) reduces the following mixed equilibrium problem of
finding x Î C such that


(
x, y
)
+ ϕ
(
y
)
− ϕ
(
x
)
≥ 0, ∀y ∈ C,
(1:2)
which was studied by Ceng and Yao [2] (see also [3]). The set of solutions of the
problem (1.2) is denoted by MEP (Θ ,).
If  =0andB = 0, then the problem (1.1) reduces the following equilibrium pro-
blem of finding x Î C such that

(
x, y
)
≥ 0, ∀y ∈ C
.
(1:3)
The set of solutions of the problem (1.3) is denoted by EP(Θ).
If  = 0 and Θ(x, y) = 0 for all x, y Î C, then the problem (1.1) reduces the following
variational inequality problem of finding x Î C such that
Bx,
y

− x≥0, ∀
y
∈
C.
(1:4)
The set of solutions of the problem (1.4) is denoted by VI(C, B).
The problem (1.1) is very general in the sense that it i ncludes, as special cases, fixed
point problems, optimization problems, variational inequality problems, minmax pro-
blems, Nash equilibrium problems in noncooperative games, and others; see [2,4-6].
Recently, in order to study the problem (1.3) coupled with the fixed point problem,
many authors have introduced some iterative schemes for finding a common element
of the set of the solutions of the problem (1.3) and the set of fixed points of a counta-
ble family of nonexpansive mappings; see [7-16] and the references therein.
In 2008, Su et al. [17] gave an iterative scheme for the problem (1.3), the problem
(1.4) for an inverse-strongly monotone mapping, and fixed point problems of non-
expansive mappings. In 2009, Yao et al. [18] considered an iterative scheme for the
problem (1.2), the problem (1.4) for a Lipschitz and relaxed-cocoercive mapping and
fixed point problems of nonexpansive mappings, and in 2008, Peng and Yao [1] stu-
died an iterative scheme for the problem (1.1), the problem (1.4) for a monotone, and
Lipschitz continuous mapping and fixed point problems of nonexpansive mappings.
In particular, in 2010, Jung [9] introduced the following new composite iterative
scheme for finding a com mon element of the set of solutions of the problem (1.3) and
the set of fixed points of a nonexpansive mapping: x
1
Î C and
⎧
⎪
⎨
⎪
⎩

(u
n
, y)+
1
r
n
y − u
n
, u
n
− x
n
≥0, ∀y ∈ C
,
y
n
= α
n
f (x
n
)+(1− α
n
)Tu
n
,
x
n+1
=
(
1 − β

n
)
y
n
+ β
n
Ty
n
, n ≥ 1,
(1:5)
where T is a nonexpansive mapping, f is a contra ction with constant k Î (0, 1), {a
n
},
{b
n
}⊂ [0, 1], and {r
n
} ⊂ (0, ∞). He showed that the sequences {x
n
}and{u
n
} generated
by (1.5) converge strongly to a point in F(T ) ∩ EP (Θ) under suitable conditions.
On the other hand, the followin g optimizatio n problem has been studied extensively
by many authors:
min
x∈
μ
2
Ax, x +

1
2

x − u

2
− h(x),
Jung Journal of Inequalities and Applications 2011, 2011:51
/>Page 2 of 23
where
 =

∞
n=1
C
n
, C
1
, C
2
, ··
·
are infinitely many closed convex subsets of H such
that

∞
n
=1
C
n

= ∅
,
u Î H, μ ≥ 0 is a real number, A is a strongly positive bounded linear
operator on H (i.e., there is a constant
¯
γ
>
0
such that
Ax, x≥ ¯
γ

x

2
, ∀x Î H) and h
is a potential function for g f (i.e., h’ (x)=g f(x) for all x Î H). For this kind of optimi-
zation problems, see, for examp le, Bauschke and Borwein [19], Combettes [20],
Deutsch and Yamada [21], Jung [22], and Xu [23] when

=

N
i
=1
C
i
;andh(x)=〈x, b〉
for a given point b in H.
In 2009, Yao et al. [3] considered the following iterative scheme for the problem (1.2)

and optimization problems:
⎧
⎨
⎩
(y
n
, y)+ϕ(y ) − ϕ(y
n
)+
1
r
K

(y
n
) − K

(x
n
), y − y
n
≥0, ∀y ∈ H,
x
n+1
= α
n
(u + γ f (x
n
)) + β
n

x
n
+ ((1 − β
n
)I − α
n
(I + μA))W
n
y
n
, n ≥ 1
,
(1:6)
where u Î H;{a
n
}and{b
n
}aretwosequencesin(0,1),μ >0,r>0, g >0; K’(x)isthe
Fréchet derivative of a functional K : H ® ℝ at x;andW
n
is the so-called W-mapping
related to a sequence {T
n
} of nonexpansive mappings. They showed that under appro-
priate conditions, the sequences {x
n
} and {y
n
} generated by (1.6) converge strongly to a
solution of the optimization problem:

min
x∈

∞
n
=1
F(T
n
) ∩ MEP(,ϕ)
μ
2
Ax, x +
1
2
||x − u||
2
− h(x)
.
In 2010, using the method of Yao et al. [3], Jaiboon and Kumam [24] also introduced
a general iterative method for finding a common element of the set of solutions of the
problem (1.2), the set of fixed points of a sequence {T
n
} of nonexpansive mappings,
and the set of solutions of the problem (1.4) for a a-inverse-strongly monotone map-
ping. We point out that in the main result s of [3,24], the condition of the sequentially
continuity from the weak topology to the strong topology for the derivative K’ of the
function K : C ® ℝ is very strong. E ven if
K(x)=

x


2
2
,thenK’ (x)=x is not sequen-
tially continuous from the weak topology to the strong topology.
In this article, inspired and motivated by above mentioned results, we introduce a
new iterative method for finding a common element of the set of solutions of a gener-
alized mixed equilibrium problem (1.1), the set of fixed points of a countable family of
nonexpansive mappings, and the set of solutions of the variational inequality probl em
(1.4) for an inverse-strongly monotone mapping in a Hilbert space. We show that
under suitable conditions, the sequence generated by the proposed iterative scheme
converges strongly to a common element of the above three sets, which is a solution
of a certain optimization problem. The results of this article can be viewed as an
improvement and complement of the recent results in this direction.
2 Preliminaries and lemmas
Let H be a real Hilbert space, and let C beanonemptyclosedconvexsubsetofH.In
thefollowing,wewritex
n
⇀ x to indicate that the sequence {x
n
}convergesweaklyto
x. x
n
® x implies that {x
n
} converges strongly to x.
Jung Journal of Inequalities and Applications 2011, 2011:51
/>Page 3 of 23
First, we know that a mapping f : C ® C is a contraction on C if there exists a co n-
stant k Î (0, 1) such that ||f(x)-f(y)|| ≤ k||x - y||, x, y Î C.AmappingT : C ® C is

called nonexpansive if ||Tx - Ty|| ≤ ||x - y||, x, y Î C .
In a real Hilbert space H, we have


λx +(1− λ)y


2
= λ

x

2
+(1− λ)


y


2
− λ(1 − λ)


x − y


2
for all x, y Î H and l Î ℝ . For every point x Î H, there exists the unique nearest
point in C, denoted by P
C

(x), such that
|
|x − P
C
(
x
)
|| ≤ ||x − y|
|
for all y Î C. P
C
is called the metric projection of H onto C. It is well known that P
C
is nonexpansive and P
C
satisfies
x − y, P
C
(x) − P
C
(y)≥


P
C
(x) − P
C
(y)



2
(2:1)
for every x, y Î H. Moreover, P
C
(x) is characterized by the properties:


x − y


2
≥


x − P
C
(x)


2
+


y − P
C
(x)


2
and

u
= P
C
(
x
)
⇔x − u, u − y ≥0forallx ∈ H, y ∈ C
.
In the context of the variational inequality problem fo r a nonlinear mapping F,this
implies that
u
∈ VI
(
C, F
)
⇔ u = P
C
(
u − λFu
)
,foranyλ>0
.
(2:2)
It is also well known that H satisfies the Opial condition, that is, for any sequence
{x
n
} with x
n
⇀ x, the inequality
lim inf

n
→∞

x
n
− x

< lim inf
n
→∞


x
n
− y


holds for every y Î H with y ≠ x.
A mapping F of C into H is called a-inve rse-s trongly monotone if there exists a con-
stant a >0 such that
x − y, Fx − Fy≥α


Fx − Fy


2
, ∀x, y ∈ C
.
We know that if F = I - T, where T is a nonexpansive mapping of C into itself and I

is the identity mapping of H,thenF is
1
2
-inverse-strongly monotone a nd VI(C, F )=
F(T ). A mapping F of C into H is called strongly monotone if there exists a positive
real number h such that
x − y, Fx − Fy≥η


x − y


2
, ∀x, y ∈ C
.
In such a case, we say F is h-strongly monotone. If F is h-strongly monotone and
-Lipschitz continuous,thatis,||Fx - Fy|| ≤ ||x - y|| for all x, y Î C,thenF is
η
κ
2
-inverse-strongly monotone. If F is an a-inverse-strongly monotone mapping of C
into H, then it is obvious that F is
1
α
-Lipschi tz continuous. We also have that for all x,
y Î C and l >0,
Jung Journal of Inequalities and Applications 2011, 2011:51
/>Page 4 of 23



(I − λF)x − (I − λF)y


2
=


(x − y) − λ(Fx − Fy)


2
=


x − y


2
− 2λx − y, Fx − Fy + λ
2


Fx − Fy


2
≤


x − y



2
+ λ(λ − 2α)


Fx − Fy


2
.
Hence, if l ≤ 2a,thenI - lF is a nonexpansive mapping of C into H. The following
result for the existence of solutions of the variational inequality problem for inverse-
strongly monotone mappings was given in Takahashi and Toyoda [25].
Proposition Let C be a bounded closed convex subset of a real Hilbert space, and F
be an a-inverse-strongly monotone mapping of C into H. Then, V I(C, F) is nonempty.
A set-valued mapping Q : H ® 2
H
is called monotone if for all x, y Î H, f Î Qx and
g Î Qy imply 〈x - y, f - g 〉 ≥ 0. A monotone mapping Q : H ® 2
H
is maximal if the
graph G(Q)ofQ is not properly contained in the graph of any other monotone map-
ping. It is known that a monotone mapping Q is maximal if and only if for (x, f ) Î H
× H, 〈x - y, f - g〉 ≥ 0forevery(y, g) Î G(Q) implies f Î Qx.LetF be an inverse-
strongly monotone mapping of C into H,andletN
C
v be the normal cone to C at v,
that is, N
C

v ={w Î H : 〈v - u, w〉 ≥ 0, for all u Î C}, and define
Qv =

Fv + N
C
v, v ∈ C
∅ v ∈ C
.
Then, Q is maximal monotone and 0 Î Qv if and only if v Î VI(C, F ); see [26,27].
For solving the equilib rium problem for a bifunction Θ : C × C ® ℝ, let us assume
that Θ and  satisfy the following conditions:
(A1) Θ(x, x) = 0 for all x Î C;
(A2) Θ is monotone, that is, Θ(x, y)+Θ (y, x) ≤ 0 for all x, y Î C;
(A3) for each x, y, z Î C,
lim
t
↓
0
(tz +(1− t)x, y) ≤ (x, y)
;
(A4) for each x Î C, y a Θ (x, y) is convex and lower semicontinuous;
(A5) For each y Î C, x a Θ (x, y) is weakly upper semicontinuous;
(B1) For each x Î H and r>0, there exists a bounded subset D
x
⊆ C and y
x
Î C
such that for any z Î C \D
x
,

(z, y
x
)+ϕ(y
x
) − ϕ(z)+
1
r
y
x
− z, z − x < 0
;
(B2) C is a bounded set;
The following lemmas were given in [1,4].
Lemma 2.1 ([4]) Let C be a nonempty closed convex subset of H, and Θ be a bifunc-
tion of C × Cintoℝ satisfying (A1)-(A4). Let r >0 and x Î H. Then, there exists z Î C
such that
(z, y)+
1
r
y − z, z − x≥0, ∀y ∈ C
.
Lemma 2.2 ([1]) LetCbeanonemptyclosedconvexsubsetofH.LetΘ be a bifunc-
tion form C × Ctoℝ satisfying (A1)-(A5) and  : C ® ℝ be a proper lower semicontin-
uous and convex function. For r >0 and x Î H, define a mapping S
r
: H ® C as follows:
Jung Journal of Inequalities and Applications 2011, 2011:51
/>Page 5 of 23
S
r

(x)={z ∈ C : (z, y)+ϕ(y) − ϕ(z)+
1
r
y − z, z − x≥0, ∀y ∈ C
}
for all z Î H. Assume that either (B1) or (B2) holds. Then, the following hold:
(1) For each x Î H, S
r
(x) ≠ ∅;
(2) S
r
is single-valued;
(3) S
r
is firmly nonexpansive, that is, for any x, y Î H,
||S
r
x − S
r
y
||
2
≤S
r
x − S
r
y
, x −
y


;
(4) F(S
r
)=MEP (Θ, );
(5) MEP (Θ, ) is closed and convex.
We also need the following lemmas for the proof of our main results.
Lemma 2.3 ([23]) Let {s
n
} be a sequence of non-negative real numbers satisfying
s
n+1
≤
(
1 − λ
n
)
s
n
+ β
n
, n ≥ 1
,
where { l
n
} and {b
n
} satisfy the following conditions:
(i) {l
n
} ⊂ [0, 1] and


∞
n
=1
λ
n
=
∞
or, equivalently,

∞
n
=1
(1 − λ
n
)=
0
,
(ii)
lim sup
n→∞
β
n
λ
n
≤ 0
or

∞
n

=1
|β
n
| <
∞
,
Then, lim
n®∞
s
n
=0.
Lemma 2.4 In a Hilbert space, there holds the inequality
|
|x +
y
||
2
≤||x||
2
+2
y
, x +
y
, ∀x,
y
∈ H
.
Lemma 2.5 (Aoyamaetal.[28])Let C be a nonemp ty closed convex subset of H and
{T
n

} be a sequence of nonexpansive mappings of C into itself. Suppose that
∞

n
=1
sup{||T
n+1
z − T
n
z|| : z ∈ C} < ∞
.
Then , for each y Î C,{T
n
y} converges strongly to some point of C. Moreo ver, let T be
a mapping of C into itself defined by Ty = lim
n®∞
T
n
y for all y Î C. Then,lim
n®∞
sup
{||Tz - T
n
z|| : z Î C}=0.
The following lemma can be found in [3](see also Lemma 2.1 in [22]).
Lemma 2.6 Let C be a nonempty closed convex subset of a real Hilbert space H and
g : C ® ℝ ∪{∞} be a proper lower semicontinuous differentiable convex function. If x* is
a solution to the minimization problem
g
(x

∗
)=inf
x
∈
C
g(x)
,
then
g

(
x
)
, x − x
∗
≥0, x ∈ C
.
In particular, if x* solves the optimization problem
min
x∈C
μ
2
Ax, x +
1
2
||x − u||
2
− h(x)
,
Jung Journal of Inequalities and Applications 2011, 2011:51

/>Page 6 of 23
then
u +
(
γ f −
(
I + μA
))
x
∗
, x − x
∗
≤0, x ∈ C
,
where h is a potential function for g f.
3 Main results
In this sect ion, we introduce a new composite iterati ve scheme for finding a common
point of the set of solutions of th e problem (1.1), the set of fixed points of a countable
fam ily of nonexpansive mappings, and the set of solut ions of the problem (1.4) for an
inverse-strongly monotone mapping.
Theorem 3.1 Let C be a nonempty closed convex sub set of a real Hilbert space H such
that C ± C ⊂ C. Let Θ be a bifunction from C × Ctoℝ satisfying (A1)-(A5) and  : C ®
ℝ be a lower semicontinuous and convex function. Let F, B be two a, b-inverse-strongly
monotone mappings of C into H, respectively. Let {T
n
} be a sequence of nonexpansive
mappings of C into itself such that

1
:=


∞
n
=1
F( T
n
) ∩ VI(C, F) ∩ GMEP(, ϕ, B) =
∅
.
Let μ >0and g >0be real numbers. Let f be a contraction of C into itself with constant k
Î (0, 1) and A be a strongly positive bounded linearoperatoronCwithconstant
¯γ ∈
(
0, 1
)
such that
0 <γ <
(1+μ) ¯γ
k
. Assume that either (B1) or (B2) holds. Let u Î C,
and let {x
n
},{y
n
}, and {u
n
} be sequences generated by x
1
Î C and
⎧

⎪
⎨
⎪
⎩
(u
n
, y)+Bx
n
, y − u
n
 + ϕ(y) − ϕ(u
n
)+
1
r
n
y − u
n
, u
n
− x
n
≥0, ∀y ∈ C,
y
n
= α
n
(u + γ f (x
n
)) + (I − α

n
(I + μA))T
n
P
C
(u
n
− λ
n
Fu
n
),
x
n+1
=
(
1 − β
n
)
y
n
+ β
n
T
n
P
C
(
y
n

− λ
n
Fy
n
)
, n ≥ 1,
(IS
)
where { a
n
}, {b
n
} ⊂ [0, 1], l
n
Î [a, b] ⊂ (0, 2a) and r
n
Î [c, d] ⊂ (0, 2b). Let {a
n
}, {l
n
}
and { b
n
} satisfy the following conditions:
(C1) a
n
® 0(n ® ∞);

∞
n

=1
α
n
= ∞
;
(C2) b
n
⊂ [0, a) for all n ≥ 0 and for some a Î (0, 1);
(C3)

∞
n
=1
|α
n+1
− α
n
| <
∞
,

∞
n
=1
|β
n+1
− β
n
| <
∞

,

∞
n
=1
|λ
n+1
− λ
n
| < ∞
,

∞
n
=1
|r
n+1
− r
n
| <
∞
.
Suppose that

∞
n
=1
sup{||T
n+1
z − T

n
z|| : z ∈ D} <
∞
for any bounded subset D of C.
Let T b e a mapping of C into itself defined by Tz = lim
n®∞
T
n
zforallzÎ Candsup-
pose that
F( T)=

∞
n
=1
F( T
n
)
. Then {x
n
} and {u
n
} converge strongly to q Î Ω
1
, which is
a solution of the optimization problem:
min
x∈
1
μ

2
Ax, x +
1
2
||x − u||
2
− h(x), (OP1
)
where h is a potential function for g f.
Proof First, from a
n
® 0(n ® ∞) in the condi tion (C1), we assume, without loss of
generality, that a
n
≤ (1 + μ||A ||)
-1
and
2
((
1+μ
)
¯γ − γ k
)
α
n
<
1
for n ≥ 1. We know
that if A is bounded linear self-adjoint operator on H, then
|

|A|| =su
p
{|Au, u| : u ∈ H, ||u|| =1}
.
Jung Journal of Inequalities and Applications 2011, 2011:51
/>Page 7 of 23
Observe that
(I − α
n
(I + μA))u, u =1− α
n
− α
n
μAu, u

≥ 1 − α
n
− α
n
μ||A||
≥
0,
which is to say I - a
n
(I + μA) is positive. It follows that
||I − α
n
(I + μA)|| =sup{(I − α
n
(I + μA))u, u : u ∈ H, ||u|| =1

}
=sup{1 − α
n
− α
n
μAu, u : u ∈ H, ||u|| =1}
≤ 1 − α
n
(1 + μ ¯γ )
< 1 − α
n
(
1+μ
)
¯γ .
Let us divide the proof into several step s. From now on, we put z
n
= P
C
(u
n
-l
n
Fu
n
)
and w
n
= P
C

(y
n
- l
n
Fy
n
).
Step 1: We show that {x
n
}isbounded.Tothisend,let
p ∈ 
1
:=

∞
n
=1
F( T
n
) ∩ VI(C, F) ∩ GMEP(, ϕ, B
)
and
{S
r
n
}
be a sequence of map-
pings defined as in Lemma 2.2. Then
p = T
n

p = S
r
n
(p − r
n
Bp
)
and p = P
C
(p - l
n
Fp)
from (2.2). From z
n
= P
C
(u
n
- l
n
Fu
n
)andthefactthatP
C
and I - l
n
F are nonexpan-
sive, it follows that
|
|z

n
− p|| ≤ ||
(
I − λ
n
F
)
u
n
−
(
I − λ
n
F
)
p|| ≤ ||u
n
− p||
.
Also, by
u
n
= S
r
n
(
x
n
− r
n

Bx
n
)
∈
C
and the b-inverse-strongly monotonicity of B,we
have with r
n
Î (0, 2b),
|
|u
n
− p||
2
≤||x
n
− r
n
Bx
n
− (p − r
n
Bp)||
2
≤||x
n
− p||
2
− 2r
n

x
n
− p, Bx
n
− Bp + r
2
n
||Bx
n
− Bp||
2
≤||x
n
− p||
2
+ r
n
(r
n
− 2β)β||Bx
n
− Bp||
2
≤||x
n
−
p
||
2
,

that is, ||u
n
- p|| ≤ ||x
n
- p||, and so
||
z
n
−
p||
≤
||
x
n
−
p||.
(3:1)
Similarly, we have
||
w
n
−
p||
≤
||y
n
−
p||.
(3:2)
Now, set

¯
A =
(
I + μA
)
. Then, from (IS) and (3.1), we obtain
||y
n
− p||
≤ (1 − (1 + μ ¯γ )α
n
)||z
n
− p|| + α
n
||u||
+ α
n
γ ||f (x
n
) − f (p)|| + α
n
||γ f (p) −
¯
Ap||
≤ (1 − (1 + μ ¯γ )α
n
)||z
n
− p|| + α

n
||u|| + α
n
γ k||x
n
− p|| + α
n
||γ f (p) −
¯
Ap||
≤ (1 − ((1 + μ) ¯γ − γ k)α
n
)||x
n
− p|| + α
n
(||γ f (p) −
¯
Ap|| + ||u||)
=(1− ((1 + μ) ¯γ − γ k)α
n
)||x
n
− p|| + ((1 + μ) ¯γ − γ k)α
n
||γ f (p) −
¯
Ap|| + ||u||
(
1+μ

)
¯γ − γ k
.
(3:3)
Jung Journal of Inequalities and Applications 2011, 2011:51
/>Page 8 of 23
From (3.2) and (3.3), it follows that
|
|x
n+1
− p|| ≤ (1 − β
n
)||y
n
− p|| + β
n
||w
n
− p||
≤ (1 − β
n
)||y
n
− p|| + β
n
||y
n
− p||
= ||y
n

− p||
≤ max

||x
n
− p||,
||γ f (p) −
¯
Ap|| + ||u||
(1 + μ) ¯γ − γ k

.
(3:4)
By induction, it follows from (3.4) that
||x
n
− p|| ≤ max

||x
1
− p||,
||γ f (p) −
¯
Ap|| + ||u||
(1 + μ) ¯γ − γ k

, n ≥ 1
.
Therefore, {x
n

} is bounded. Hence {u
n
}, {y
n
}, {z
n
}, {w
n
}, {f(x
n
)}, {Fu
n
}, {Fy
n
}, and
{
¯
AT
n
z
n
}
are bounded. Moreover, since ||T
n
z
n
- p|| ≤ ||x
n
- p|| and ||T
n

w
n
- p|| ≤ ||y
n
-
p||, {T
n
z
n
}and{T
n
w
n
} are also bounded, and since a
n
® 0 in the condition (C1), we
have
||y
n
− T
n
z
n
|| = α
n
||
(
u + γ f
(
x

n
))
−
¯
AT
n
z
n
|| → 0
(
as n →∞
).
(3:5)
Step 2: We show that lim
n®∞
||x
n+1
- x
n
|| = 0. Indeed, since I - l
n
F and P
C
are non-
expansive, we have
|
|z
n
− z
n−1

|| = ||P
C
(u
n
− λ
n
Fu
n
) − P
C
(u
n−1
− λ
n−1
Fu
n−1
)||
≤||(I − λ
n
F) u
n
− (I − λ
n
F) u
n−1
+(λ
n
− λ
n−1
)Fu

n−1
|
|
≤
||
u
n
− u
n−1
||
+
|
λ
n
− λ
n−1
|||
Fu
n−1
||
.
(3:6)
Similarly, we get
||
w
n
− w
n−1
||
≤

||y
n
−
y
n−1
||
+
|
λ
n
− λ
n−1
|||
F
y
n−1
||.
(3:7)
On the other hand, from
u
n−1
= S
r
n
−1
(x
n−1
− r
n
Bx

n−1
)
and
u
n
= S
r
n
(x
n
− r
n
Bx
n
)
,itfol-
lows that
(u
n−1
, y)+Bx
n−1
, y − u
n−1

+ ϕ(y) − ϕ(u
n−1
)+
1
r
n

−1
y − u
n−1
, u
n−1
− x
n−1
≥0, ∀y ∈ C
,
(3:8)
and
(u
n
, y)+Bx
n
, y − u
n

+ ϕ(y) − ϕ(u
n
)+
1
r
n
y − u
n
, u
n
− x
n

≥0, ∀y ∈ C
.
(3:9)
Substituting y = u
n
into (3.8) and y = u
n -1
into (3.9), we obtain
(u
n−1
, u
n
)+Bx
n−1
, u
n
−u
n−1
+ϕ(u
n
)−ϕ(u
n−1
)+
1
r
n
−1
u
n
−u

n−1
, u
n−1
−x
n−1
≥
0
and
(u
n
, u
n−1
)+Bx
n
, u
n−1
− u
n
 + ϕ(u
n−1
) − ϕ(u
n
)+
1
r
n
u
n−1
− u
n

, u
n
− x
n
≥0
.
Jung Journal of Inequalities and Applications 2011, 2011:51
/>Page 9 of 23
From (A2), we have
u
n
− u
n−1
, Bx
n−1
− Bx
n
+
u
n−1
− x
n−1
r
n
−1
−
u
n
− x
n

r
n
≥0
,
and then
u
n
− u
n−1
, r
n−1
(Bx
n−1
− Bx
n
)+u
n−1
− x
n−1
−
r
n−1
r
n
(u
n
− x
n
)≥0
.

Hence, it follows that
u
n
− u
n−1
,(I−r
n−l
B)x
n
− (I − r
n−1
B)x
n−1
+ u
n−1
− u
n
+ u
n
− x
n
−
r
n−1
r
n
(u
n
− x
n

)≥0
.
(3:10)
Without loss of generality, let us assume that there exists a real number c such that
r
n
>c >0foralln ≥ 1. Then, by (3.10) and the fact that (I - r
n-1
B) is nonex pansive, we
have
||u
n
− u
n−1
||
2
≤u
n
− u
n−1
,(I − r
n−1
B)x
n
− (I − r
n−1
B)x
n−1
+(1−
r

n−1
r
n
)(u
n
− x
n
)
≤||u
n
− u
n−1
||

||(I − r
n−1
B)x
n
− (I − r
n−1
B)x
n−1
|| +




1 −
r
n−1

r
n




||u
n
− x
n
||

≤||u
n
− u
n−1
||

||x
n
− x
n−1
|| +




1 −
r
n−1

r
n




||u
n
− x
n
||

,
which implies that
|
|u
n
− u
n−1
|| ≤ ||x
n
− x
n−1
|| +
1
r
n
|r
n
− r

n−1
|||u
n
− x
n
|
|
≤||x
n
− x
n−1
|| +
M
1
c
|r
n
− r
n−1
|,
(3:11)
where M
1
= sup {||u
n
- x
n
|| : n ≥ 1}. Substituting (3.11) into (3.6), we have
|
|z

n
− z
n−1
|| ≤ ||x
n
− x
n−1
|| +
M
1
c
|r
n
− r
n−1
| + |λ
n
− λ
n−1
|||Fu
n−1
||
.
(3:12)
Simple calculations show that
y
n
− y
n−1
=(α

n
− α
n−1
)(u + γ f (x
n−1
) −
¯
AT
n−1
z
n−1
)+α
n
γ (f (x
n
) − f( x
n−1
)
)
+
(
I − α
n
¯
A
)(
T
n
z
n

− T
n
z
n−1
)
+
(
I − α
n
¯
A
)(
T
n
z
n−1
− T
n−1
z
n−1
)
.
Jung Journal of Inequalities and Applications 2011, 2011:51
/>Page 10 of 23
Hence, by (3.11) and (3.12), we obtain
|
|y
n
− y
n−1

|| ≤ |α
n
− α
n−1
|(||u|| + γ ||f (x
n−1
)|| + ||
¯
A|| ||T
n−1
z
n−1
||)
+ α
n
γ k||x
n
− x
n−1
|| +(1− (1 + μ) ¯γα
n
)||z
n
− z
n−1
||
+(1− (1 + μ) ¯γα
n
)||T
n

z
n−1
− T
n−1
z
n−1
||
≤|α
n
− α
n−1
|(||u|| + γ ||f (x
n−1
)|| + ||
¯
A|| ||T
n−1
z
n−1
||)
+ α
n
γ k||x
n
− x
n−1
|| +(1− (1 + μ) ¯γα
n
)||x
n

− x
n−1
||
+
M
1
c
|r
n
− r
n−1
| + |λ
n
− λ
n−1
|||Fu
n−1
|| +sup
z∈D
1
||T
n
z − T
n−1
z||
,
(3:13)
where D
1
is a bounded subset of C containing {z

n
}. Also observe that
x
n+1
− x
n
=(1− β
n
)(y
n
− y
n−1
)+(β
n
− β
n−1
)(T
n−1
w
n−1
− y
n−1
)
+ β
n
(
T
n
w
n

− T
n
w
n−1
)
+ β
n
(
T
n
w
n−1
− T
n−1
w
n−1
)
.
(3:14)
By (3.7), (3.13) and (3.14), we have
||x
n+1
− x
n
||
≤ (1 − β
n
)||y
n
− y

n−1
|| + |β
n
− β
n−1
|(||T
n−1
x
n−1
|| + ||y
n−1
||)
+ β
n
||w
n
− w
n−1
|| + β
n
||T
n
w
n−1
− T
n−1
w
n−1
||
≤ (1 − β

n
)||y
n
− y
n−1
|| + β
n
||y
n
− y
n−1
|| + β
n
|λ
n
− λ
n−1
|||Fy
n−1
||
+|β
n
− β
n−1
|(||T
n−1
w
n−1
|| + ||y
n−1

||)+β
n
||T
n
w
n−1
− T
n−1
w
n−1
|
|
≤||y
n
− y
n−1
|| + |λ
n
− λ
n−1
|||Fy
n−1
||
+|β
n
− β
n−1
|(||T
n−1
w

n−1
|| + ||y
n−1
||)+||T
n
w
n−1
− T
n−1
w
n−1
||
≤ (1 − ((1 + μ) ¯γ − γ k)α
n
)||x
n
− x
n−1
||
+|(α
n
− α
n−1
|(||u|| + γ ||f (x
n−1
)|| + ||
¯
A|| ||T
n−1
z

n−1
||)
+|λ
n
− λ
n−1
|(||Fy
n−1
|| + ||Fu
n−1
||)+
M
1
c
|r
n
− r
n−1
|
+|β
n
− β
n−1
|(||T
n−1
w
n−1
|| + ||y
n−1
||)+2sup

z∈D
2
{||T
n
z − T
n−1
z||
≤ (1 − ((1 + μ) ¯γ − γ k)α
n
)||x
n
− x
n−1
|| +
M
1
c
|r
n
− r
n−1
|
+ M
2
|α
n
− α
n−1
| + M
3

|λ
n
− λ
n−1
| + M
4
|β
n
− β
n−1
|
+2sup
z∈D
2
||T
n
z − T
n−1
z||,
(3:15)
where D
2
is a bounded subset of C containing {z
n
}and{w
n
},
M
2
=sup{||u|| + γ ||f (x

n
)|| + ||
¯
A|| ||T
n
z
n
|| : n ≥ 1}
, M
3
=sup{||Fy
n
|| + ||Fu
n
|| : n ≥ 1},
and M
4
=sup{||T
n
w
n
|| + ||y
n
|| : n ≥ 1}. From the conditions (C1) and (C3) and the
condition

∞
n
=1
sup{||T

n+1
z − T
z
|| : z ∈ D} < ∞
for any bounded subset D of C,itis
easy to see that
lim
n→∞
((1 + μ) ¯γ − γ k)α
n
=0,
∞

n
=1
((1 + μ) ¯γ − γ k)α
n
= ∞
,
Jung Journal of Inequalities and Applications 2011, 2011:51
/>Page 11 of 23
and
∞

n=2

M
1
c
|r

n
− r
n−1
| + M
2
| α
n
− α
n−1
| + M
3
|λ
n
− λ
n−1
|
+ M
4
|β
n
− β
n−1
| +2sup
z∈D
2
||T
n
z − T
n−1
z||


< ∞
.
Applying Lemma 2.3 to (3.15), we obtain
lim
n
→
∞
||x
n+1
− x
n
|| =0
.
Moreover, from (3.11), it follows that
lim
n
→
∞
||u
n+1
− u
n
|| =0
.
From (3.12) and (3.13), we also have
lim
n
→
∞

||z
n+1
− z
n
|| = 0 and lim
n
→
∞
||y
n+1
− y
n
|| =0
.
Step 3: We show that lim
n-∞
||x
n
- u
n
|| = 0. To this end, let p Î Ω
1
.Since
S
r
n
is
firmly nonexpansive and
u
n

= S
r
n
(x
n
− r
n
Bx
n
)
, we have
|
|u
n
− p||
2
≤S
r
n
(x
n
− r
n
Bx
n
) − S
r
n
(p − r
n

Bp), x
n
− r
n
Bx
n
− (p − r
n
Bp)
=
1
2
{||u
n
− p||
2
+ ||x
n
− r
n
Bx
n
− (p − r
n
Bp)||
2
}
−
1
2

{||x
n
− r
n
Bx
n
− (p − r
n
Bp) − (u
n
− p)||
2
}
≤
1
2
{||u
n
− p||
2
+ ||x
n
− p||
2
−||x
n
− u
n
− r
n

(Bx
n
− Bp)||
2
}
≤
1
2
{||u
n
− p||
2
+ ||x
n
− p||
2
−||x
n
− u
n
||
2
+2r
n
Bx
n
− Bp, x
n
− u
n

−r
2
n
||Bx
n
− Bp||
2
}.
Hence,
||u
n
− p||
2
≤||x
n
− p||
2
−||x
n
− u
n
||
2
+2r
n
Bx
n
− Bp, x
n
− u

n
−r
2
n
||Bx
n
− Bp||
2
≤||x
n
− p||
2
−||x
n
− u
n
||
2
+2r
n
Bx
n
− Bp, x
n
− u
n

≤||x
n
−

p
||
2
−||x
n
− u
n
||
2
+2r
n
||Bx
n
− B
p
|| ||x
n
− u
n
||.
(3:16)
On the other hand, since z
n
= P
C
(u
n
- l
n
Fu

n
), we get
|
|z
n
− p||
2
≤||(I − λ
n
F) u
n
− (I − λ
n
F) p||
2
≤||u
n
−
p
||
2
.
(3:17)
Jung Journal of Inequalities and Applications 2011, 2011:51
/>Page 12 of 23
From (3.16), (3.17), and the convexity of || ||
2
, we obtain
||y
n

− p||
2
≤ α
n
||u + γ f (x
n
)+(I −
¯
A)T
n
z
n
− p||
2
+(1− α
n
)||T
n
z
n
− p||
2
≤ α
n
||u + γ f (x
n
)+(I −
¯
A)T
n

z
n
− p||
2
+(1− α
n
)||z
n
− p||
2
≤ α
n
||u + γ f (x
n
)+(I −
¯
A)T
n
z
n
− p||
2
+(1− α
n
)||u
n
− p||
2
≤ α
n

||u + γ f (x
n
)+(I −
¯
A)T
n
z
n
− p||
2
+(1− α
n
){||x
n
− p||
2
−||x
n
− u
n
||
2
+2r
n
||x
n
− u
n
|| ||Bx
n

− Bp||
}
≤ α
n
||u + γ f (x
n
)+(I −
¯
A)T
n
z
n
− p||
2
+ ||x
n
− p||
2
−
(
1 − α
n
)
||x
n
− u
n
||
2
+2

(
1 − α
n
)
r
n
||x
n
− u
n
|| ||Bx
n
− Bp||.
(3:18)
On the another hand, we note that
|
|u
n
− p||
2
≤||(x
n
− r
n
Bx
n
) − (p − r
n
Bp)||
2

≤||x
n
− p||
2
− 2r
n
x
n
− p, Bx
n
− Bp + r
2
n
||Bx
n
− Bp||
2
≤||x
n
− p||
2
− 2r
n
β||Bx
n
− Bp||
2
+ r
2
n

||Bx
n
− Bp||
2
.
(3:19)
Using the convexity of || ||
2
, (3.2), (3.18), and (3.19), we have
|
|x
n+1
− p||
2
≤||y
n
− p||
2
≤ α
n
||u + γ f (x
n
)+(I −
¯
A)T
n
z
n
− p||
2

+(1− α
n
)||u
n
− p||
2
≤ α
n
||u + γ f (x
n
)+(I −
¯
A)T
n
z
n
− p||
2
+(1− α
n
){||x
n
− p||
2
− 2r
n
β||Bx
n
− Bp||
2

+ r
2
n
||Bx
n
− Bp||
2
}
≤ α
n
||u + γ f (x
n
)+(I −
¯
A)T
n
z
n
− p||
2
+ ||x
n
− p||
2
+
(
1 − α
n
)
r

n
(
r
n
− 2β
)
||Bx
n
− Bp||
2
.
(3:20)
Hence, we have
(1 − α
n
)c(2β − d)||Bx
n
− Bp||
2
≤ (1 − α
n
)r
n
(2β − r
n
)||Bx
n
− Bp||
2
≤ α

n
||u + γ f (x
n
)+(I −
¯
A)T
n
z
n
− p||
2
+(||x
n
− p||
2
−||x
n+1
− p||
2
)
≤ α
n
||u + γ f
(
x
n
)
+
(
I −

¯
A
)
T
n
z
n
− p||
2
+ ||x
n
− x
n+1
||
(
||x
n
− p|| + ||x
n+1
− p||
).
From the condition (C1), {r
n
} ⊂ [c, d] ⊂ (0, 2b) and Step 2, it follows that
lim
n
→
∞
||Bx
n

− Bp||
2
=0
.
(3:21)
Also, by (3.16) and (3.20), we have
||x
n+1
− p||
2
≤||y
n
− p||
2
≤ α
n
||u + γ f (x
n
)+(I −
¯
A)T
n
z
n
− p||
2
+(1− α
n
)||u
n

− p||
2
≤ α
n
||u + γ f (x
n
)+(I −
¯
A)T
n
z
n
− p||
2
+(1− α
n
){||x
n
− p||
2
−||x
n
− u
n
||
2
+2r
n
||x
n

− u
n
|| ||Bx
n
− Bp||
}
≤ α
n
||u + γ f (x
n
)+(I −
¯
A)T
n
z
n
− p||
2
+ ||x
n
− p||
2
−
(
1 − α
n
)
||x
n
− u

n
||
2
+2r
n
(
1 − α
n
)
||x
n
− u
n
|| ||Bx
n
− Bp||,
Jung Journal of Inequalities and Applications 2011, 2011:51
/>Page 13 of 23
and so
(1 − α
n
)||x
n
− u
n
||
2
≤ α
n
||u + γ f (x

n
)+(I −
¯
A)T
n
z
n
− p||
2
+ ||x
n
− p||
2
−||x
n+1
− p||
2
+2r
n
(1 − α
n
)||x
n
− u
n
|| ||Bx
n
− Bp|
|
≤ α

n
M
5
+ ||x
n
− x
n+1
||(||x
n
− p|| + ||x
n+1
− p||)
+2r
n
(
1 − α
n
)
||x
n
− u
n
|| ||Bx
n
− Bp||,
where
M
5
=sup{||u + γ f
(

x
n
)
+
(
I −
¯
A
)
T
n
z
n
− p||
2
: n ≥ 1
}
. Since ||x
n+1
- x
n
|| ® 0, a
n
® 0, and ||Bx
n
- Bp|| ® 0asn ® ∞, we obtain
lim
n
→
∞

||x
n
− u
n
|| =0
.
(3:22)
Moreover, since lim inf
n®∞
r
n
> 0, we also have
lim
n→∞
||x
n
− u
n
||
r
n
= lim
n→∞
1
r
n
||x
n
− u
n

|| =0
.
(3:23)
Step 4: We show that lim
n®∞
||x
n
- T
n
z
n
|| = 0 and lim
n-∞
||x
n
- y
n
|| = 0. Indeed,
since z
n
= P
C
(u
n
- l
n
Fu
n
) and w
n

= P
C
(y
n
- l
n
Fy
n
), we obtain with the condition (C2)
|
|x
n+1
− y
n
|| ≤ β
n
(||T
n
z
n
− T
n
w
n
|| + ||y
n
− T
n
z
n

||)
≤ a(||z
n
− w
n
|| + ||y
n
− T
n
z
n
||)
≤ a(||u
n
− y
n
|| + ||y
n
− T
n
z
n
||)
≤ a
(
||u
n
− x
n
|| + ||x

n
− x
n+1
|| + ||x
n+1
− y
n
|| + ||y
n
− T
n
z
n
||
),
which implies that
||x
n+1
− y
n
|| ≤
a
1 −
a
(||u
n
− x
n
|| + ||x
n

− x
n+1
|| + ||y
n
− T
n
z
n
||)
.
Thus, from (3.5), Step 2, and Step 3, we have
||x
n+1
−
y
n
|| → 0asn →∞
.
Also we have
|
|x
n
−
y
n
|| ≤ ||x
n
− x
n+1
|| + ||x

n+1
−
y
n
|| → 0asn →∞
.
Since lim
n®∞
||y
n
- T
n
z
n
|| = 0 by (3.5) in Step 1, we obtain
||
x
n
− T
n
z
n
||
≤
||
x
n
−
y
n

||
+
||y
n
− T
n
z
n
||
→ 0asn →∞
.
Step 5: We show that lim
n®∞
||T
n
z
n
- z
n
|| = 0. Let p Î Ω
1
. Using the convexity of ||
||
2
, we compute
Jung Journal of Inequalities and Applications 2011, 2011:51
/>Page 14 of 23
||y
n
− p||

2
≤ α
n
||u + γ f (x
n
)+(I −
¯
A)T
n
z
n
− p||
2
+(1− α
n
)||T
n
z
n
− p||
2
≤ α
n
||u + γ f (x
n
)+(I −
¯
A)T
n
z

n
− p||
2
+(1− α
n
)||z
n
− p||
2
= α
n
||u + γ f (x
n
)+(I −
¯
A)T
n
z
n
− p||
2
+(1− α
n
)||P
C
(u
n
− λ
n
Fu

n
) − P
C
(p − λ
n
Fp)||
2
≤ α
n
||u + γ f (x
n
)+(I −
¯
A)T
n
z
n
− p||
2
+(1− α
n
)||(u
n
− p) − λ
n
(Fu
n
− Fp)||
2
= α

n
||u + γ f (x
n
)+(I −
¯
A)T
n
z
n
− p||
2
+(1− α
n
){||u
n
− p||
2
− 2λ
n
u
n
− p, Fu
n
− Fp + λ
2
n
||Fu
n
− Fp||
2

}
≤ α
n
||u + γ f (x
n
)+(I −
¯
A)T
n
z
n
− p||
2
+(1− α
n
){||u
n
− p||
2
− 2λ
n
α||Fu
n
− Fp||
2
+ λ
2
n
||Fu
n

− Fp||
2
}
≤ α
n
||u + γ f (x
n
)+(I −
¯
A)T
n
z
n
− p||
2
+ ||x
n
− p||
2
+
(
1 − α
n
)
λ
n
(
λ
n
− 2α

)
||Fu
n
− Fp||
2
.
(3:24)
Using (3.24), we obtain
|
|x
n+1
− p||
2
≤||y
n
− p||
2
≤ α
n
||u + γ f (x
n
)+(I −
¯
A)T
n
z
n
− p||
2
+ ||x

n
− p||
2
+
(
1 − α
n
)
λ
n
(
λ
n
− 2α
)
||Fu
n
− Fp||
2
.
Hence, we have
(1 − α
n
)a(2α − b)||Fu
n
− Fp||
2
≤ (1 − α
n
)λ

n
(2α − λ
n
)||Fu
n
− Fp||
2
≤ α
n
||u + γ f (x
n
)+(I −
¯
A)T
n
z
n
− p||
2
+(||x
n
− p|| − ||x
n+1
− p||)(||x
n
− p|| + ||x
n+1
− p||
)
≤ α

n
M
5
+ ||x
n
− x
n+1
||
(
||x
n
− p|| + ||x
n+1
− p||
)
.
where
M
5
=sup{||u + γ f
(
x
n
)
+
(
I −
¯
A
)

T
n
z
n
− p||
2
: n ≥ 1
}
. From the condition (C1),
l
n
Î [a, b] ⊂ (0, 2a), and Step 2, it follows that
lim
n
→
∞
||Fu
n
− Fp|| =0
.
(3:25)
On the other hand, using z
n
= P
C
(u
n
- l
n
Fu

n
) and (2.1), we observe that
|
|z
n
− p||
2
≤(u
n
− λ
n
Fu
n
) − (p − λ
n
Fp), z
n
− p
≤
1
2
{||u
n
− p||
2
+ ||z
n
− p||
2
−||(u

n
− z
n
) − λ
n
(Fu
n
− Fp)||
2
}
≤
1
2
{||x
n
− p||
2
+ ||z
n
− p||
2
−||u
n
− z
n
||
2
+2λ
n
u

n
− z
n
, Fu
n
− Fp−λ
2
n
||Fu
n
− Fp||
2
},
and so
||z
n
− p||
2
≤||x
n
− p||
2
−||u
n
− z
n
||
2
+2λ
n

u
n
− z
n
, Fu
n
− Fp−λ
2
n
||Fu
n
− Fp||
2
(3:26)
Jung Journal of Inequalities and Applications 2011, 2011:51
/>Page 15 of 23
Thus, from (3.26), we have
|
|y
n
− p||
2
≤ α
n
||u + γ f (x
n
)+(I −
¯
A)T
n

z
n
− p||
2
+(1− α
n
)||T
n
z
n
− p||
2
≤ α
n
||u + γ f (x
n
)+(I −
¯
A)T
n
z
n
− p||
2
+(1− α
n
)||z
n
− p||
2

≤ α
n
||u + γ f (x
n
)+(I −
¯
A)T
n
z
n
− p||
2
+(1− α
n
){||x
n
− p||
2
−||u
n
− z
n
||
2
+2λ
n
u
n
− z
n

, Fu
n
− Fp−λ
2
n
||Fu
n
− Fp||
2
}
≤ α
n
||u + γ f (x
n
)+(I −
¯
A)T
n
z
n
− p||
2
+ ||x
n
− p||
2
− (1 − α
n
)||u
n

− z
n
||
2
+2λ
n
(1 − α
n
)u
n
− z
n
, Fu
n
− Fp
− λ
2
n
||Fu
n
− Fp||
2
.
Hence, we obtain
|
|x
n+1
− p||
2
≤||y

n
− p||
2
≤ α
n
||u + γ f (x
n
)+(I −
¯
A)T
n
z
n
− p||
2
+ ||x
n
− p||
2
− (1 − α
n
)||u
n
− z
n
||
2
+2λ
n
(1 − α

n
)u
n
− z
n
, Fu
n
− Fp

− λ
2
n
||Fu
n
− Fp||
2
,
(3:27)
which implies that
(1 − α
n
)||u
n
− z
n
||
2
≤ α
n
||u + γ f (x

n
)+(I −
¯
A)T
n
z
n
− p||
2
+ ||x
n
− p||
2
−||x
n+1
− p||
2
+2λ
n
(1 − α
n
)||u
n
− z
n
|| ||Fu
n
− Fp||
≤ α
n

||u + γ f (x
n
)+(I −
¯
A)T
n
z
n
− p||
2
+ ||x
n+1
− x
n
||(||x
n
− p|| + ||x
n+1
− p||
)
+2λ
n
(
1 − α
n
)
||u
n
− z
n

|| ||Fu
n
− Fp||.
From ||x
n+1
- x
n
|| ® 0, a
n
® 0 and ||Fu
n
- Fp|| ® 0asn ® ∞, it follows that
lim
n
→
∞
||u
n
− z
n
|| =0
.
(3:28)
Since ||T
n
z
n
- z
n
|| ≤ ||T

n
z
n
- x
n
|| + ||x
n
- u
n
|| + || u
n
- z
n
||, from Step 3, Step 4, and
(3.28), we conclude that
lim
n
→
∞
||T
n
z
n
− z
n
|| =0
.
(3:29)
We notice that by the assumption on T, (3.29) and Lemma 2.5,
lim

n→∞
||Tz
n
− z
n
|| ≤ lim
n→∞
(||Tz
n
− T
n
z
n
|| + ||T
n
z
n
− z
n
||)
≤ lim
n
→
∞
sup{||Ty − T
n
y|| : y ∈ C} + lim
n
→
∞

||T
n
z
n
− z
n
|| =0
.
(3:30)
Step 6: We show that
lim sup
n
→∞
u +(γ f − (I + μA))q, y
n
− q = lim sup
n
→∞
u +(γ f −
¯
A)q, y
n
− q≤0
,
where q is a solution of the optimization problem (OP1). To this end, first we prove
that
lim sup
n
→∞
u +(γ f −

¯
A)q, Tz
n
− q≤0
.
Jung Journal of Inequalities and Applications 2011, 2011:51
/>Page 16 of 23
Since {z
n
} is bounded, we can choose a subsequence
{z
n
i
}
of {z
n
} such that
lim sup
n
→∞
u +(γ f −
¯
A)q, Tz
n
− q = lim
i→∞
u +(γ f −
¯
A)q, Tz
n

i
− q
.
(3:31)
Without loss of generality, we may assume that
{z
n
i
}
converges weakly to z Î C.
From ||Tz
n
- z
n
|| ® 0 in (3.30), it follows that
Tz
n
i
 z
.
Now, we will show that z Î Ω
1
.First,weshowthat
z
∈

∞
n
=1
F( T

n
)=F(T
)
. Assume
that z ∉ F(T). Since
z
n
i
 z
and Tz ≠ z, by the Opial condition, we obtain
lim inf
i→∞
||z
n
i
− z|| < lim inf
i→∞
||z
n
i
− Tz||
≤ lim inf
i→∞
(||z
n
i
− Tz
n
i
|| + ||Tz

n
i
− Tz||
)
≤ lim inf
i
→∞
||z
n
i
− z||,
which is a contradiction. Thus we have
z
∈

∞
n
=1
F( T
n
)=F(T
)
.
Next, we prove that z Î VI(C, F ). Let
Qv =

Fv + N
C
v, v ∈ C
∅, v ∈ C

,
where N
C
v is normal cone to C at v. We already know that, in this case, the mapping
Q is maximal monotone, and 0 Î Qv if and only if v Î VI(C, F). Let (v, w) Î G(Q).
Since w - Fv Î N
C
v and z
n
Î C, we have

v − z
n
, w − Fv

≥ 0
.
On the other hand, from z
n
= P
C
(u
n
- l
n
Fu
n
), we have
v − z
n

, z
n
−
(
u
n
− λ
n
Fu
n
)
≥0
,
that is,
v − z
n
,
z
n
− u
n
λ
n
+ Fu
n
≥0
.
Thus, we obtain
v − z
n

i
, w≥v − z
n
i
, Fv
≥v − z
n
i
, Fv−v − z
n
i
,
z
n
i
− u
n
i
λ
n
i
+ Fu
n
i

= v − z
n
i
, Fv − Fz
n

i
 + v − z
n
i
, Fz
n
i
− Fu
n
i

−v − z
n
i
,
z
n
i
− u
n
i
λ
n
i

≥v − z
n
i
, Fz
n

i
− Fu
n
i
−v − z
n
i
,
z
n
i
− u
n
i
λ
n
i

.
(3:32)
Since ||z
n
- u
n
|| ® 0 in (3.28) and F is a-inverse-strongly monotone, it follows from
(3.32) that

v − z, w

≥ 0, as i →∞

.
Since Q is maximal monotone, we have z Î Q
−1
0 and hence z Î VI(C, F).
Jung Journal of Inequalities and Applications 2011, 2011:51
/>Page 17 of 23
Finally, we show that z Î GMEP(Θ, ,B). By
u
n
= S
r
n
(x
n
− r
n
Bx
n
)
; we know that
(u
n
, y)+Bx
n
, y − u
n
 + ϕ(y) − ϕ(u
n
)+
1

r
n
y − u
n
, u
n
− x
n
≥0, ∀y ∈ C
.
It follows from (A2) that
Bx
n
, y − u
n
 + ϕ(y) − ϕ(u
n
)+
1
r
n
y − u
n
, u
n
− x
n
≥(y, u
n
), ∀y ∈ C

.
Hence
Bx
n
i
, y − u
n
i
 +(ϕ(y) − (ϕ( u
n
i
)+
1
r
n
i
y − u
n
i
, u
n
i
− x
n
i
≥(y, u
n
i
), ∀y ∈ C
.

(3:33)
For t with 0 <t≤ 1 and y Î C, let y
t
= ty +(1-t)z. Since y Î C and z Î C, we have
y
t
Î C and hence Θ(y
t,
z) ≤ 0. Hence, from (3.33), we have
y
t
− u
n
i
, By
t
≥y
t
− u
n
i
, By
t
−ϕ(y
t
)+ϕ(u
n
i
) −y
t

− u
n
i
, Bx
n
i

−y
t
− u
n
i
,
u
n
i
− x
n
i
r
n
i
 + (y
t
, u
n
i
)
= y
t

− u
n
i
, By
t
− Bu
n
i
 + y
t
− u
n
i
, Bu
n
i
− Bx
n
i

− ϕ(y
t
)+ϕ(u
n
i
) −y
t
− u
n
i

,
u
n
i
− x
n
i
r
n
i
 + (y
t
, u
n
i
)
Since
||u
n
i
− x
n
i
|| →
0
from Step 3, we have
||Bu
n
i
− Bx

n
i
|| →
0
and
u
n
i
−
x
n
i
r
n
i
→
0
.
Also by
||
u
n
− z
n
||
→
0
in (3.28), we have
u
n

i

z
. Moreover, from the inverse-strongly
monotonicity of B,wehave

y
t
− u
n
i
, By
t
− Bu
n
i

≥
0
. Hence, from (A4) and the weak
lower semicontinuity of , if follows that
y
t
− z, By
t
≥−ϕ
(
y
t
)

+ ϕ
(
z
)
+ 
(
y
t
, z
)
as i →∞
.
(3:34)
From (A1), (A4), and (3.34), we also obtain
0=(y
t
, y
t
)+ϕ(y
t
) − ϕ(y
t
)
≤ t(y
t
, y)+(1− t)(y
t
, z)+tϕ(y
t
)+(1− t)ϕ(z ) − ϕ( y

t
)
≤ t[(y
t
, y)+ϕ(y) − ϕ(y
t
)] + (1 − t)y
t
− z, By
t

= t[
(
y
t
, y
)
+ ϕ
(
y
)
− ϕ
(
y
t
)
]+
(
1 − t
)

ty − z, By
t
,
and hence
0 ≤ 
(
y
t
, y
)
+ ϕ
(
y
)
− ϕ
(
y
t
)
+
(
1 − t
)
y − z, By
t

.
(3:35)
Letting t ® 0 in (3.35), we have for each y Î C


(
z, y
)
+ Bz, y − z + ϕ
(
y
)
− ϕ
(
z
)
≥ 0
.
This implies that z Î GMEP(Θ, , B). Therefore z Î Ω
1
.
Now, from Lemma 2.6 and (3.31), we obtain
lim sup
n→∞
u +(γ f −
¯
A)q, Tz
n
− q = lim
i→∞
u +(γ f −
¯
A)q, Tz
n
i

− q

= u +(γ f −
¯
A)q, z − q
≤
0.
(3:36)
Jung Journal of Inequalities and Applications 2011, 2011:51
/>Page 18 of 23
Since lim
n®∞
||x
n
- Tz
n
|| ≤ ||x
n
- T
n
z
n
|| + ||T
n
z
n
- Tz
n
|| ® 0, from Step 4 and
Lemma 2.5, and from (3.36), we conclude that

lim sup
n→∞
u +(γ f −
¯
A)q, y
n
− q
≤ lim sup
n→∞
u +(γ f −
¯
A)q, y
n
− Tz
n
 + lim sup
n→∞
u +(γ f −
¯
A)q, Tz
n
− q
≤ lim sup
n→∞
||u +(γ f −
¯
A)q|| ||y
n
− Tz
n

|| + lim sup
n→∞
u +(γ f −
¯
A)q, Tz
n
− q

≤
0.
Ste p 7: We show that lim
n®∞
||x
n
- q|| = 0 and lim
n®∞
||u
n
- q|| = 0, where q is a
solution of the optimization problem (OP1). Indeed, from (IS) and Lemma 2.4, we have
||x
n+1
− q||
2
≤||y
n
− q||
2
≤||(I − α
n

¯
A)(T
n
z
n
− q)||
2
+2α
n
u + γ f (x
n
) −
¯
Aq, y
n
− q
≤ (1 − (1 + μ) ¯γα
n
)
2
||z
n
− q||
2
+2α
n
γ f (x
n
) − f( q ), y
n

− q
+2α
n
u + γ f (q) −
¯
Aq, y
n
− q)
≤ (1 − (1 + μ) ¯γα
n
)
2
||x
n
− q||
2
+2α
n
γ k||x
n
− q|| ||y
n
− q||
+2α
n
u +(γ f −
¯
A)q, y
n
− q

≤ (1 − (1 + μ) ¯γα
n
)
2
||x
n
− q||
2
+2α
n
γ k||x
n
− q||(||y
n
− x
n
|| + ||x
n
− q||
)
+2α
n
u +(γ f −
¯
A)q, y
n
− q
=(1− 2((1 + μ) ¯γ − γ k)α
n
)||x

n
− q||
2
+α
2
n
((1 + μ) ¯γ )
2
||x
n
− q||
2
+2α
n
γ k||x
n
− q|| ||y
n
− x
n
||
+2α
n
u +
(
γ f −
¯
A
)
q, y

n
− q,
that is,
|
|x
n+1
− q||
2
≤ (1 − 2((1 + μ) ¯γ − γ k)α
n
)||x
n
− q||
2
+ α
2
n
((1 + μ) ¯γ )
2
M
6
+2α
n
γ k||y
n
− x
n
||M
6
+2α

n
u +(γ f −
¯
A)q, y
n
− q
=
(
1 − α
n
)
||x
n
− q||
2
+ β
n
,
where
M
6
=sup{||x
n
− q|| : n ≥ 1}, α
n
=2
((
1+μ
)
¯γ − γ k

)
α
n
and
β
n
= α
n
[α
n
((1 + μ) ¯γ )
2
M
2
6
+2γ k||y
n
− x
n
||M
6
+2u +(γ f −
¯
A)q, y
n
− q]
.
From the condition (C1), ||y
n
- x

n
|| ® 0 in Step 4, and Step 6, it is easily seen that

∞
n
=1
α
n
= ∞
,

∞
n
=1
α
n
=
∞
, and
lim sup
n→∞
β
n
α
n
≤
0
. Hence, by Lemma 2.3, we conclude
x
n

® q as n ® ∞. Moreover, from Step 3, we obtain that u
n
® q as n ® ∞. This com-
pletes the proof. □
Corollary 3.2 Let H, C, Θ, ,F,A,f,μ, g,
¯
γ
and k be as in Theorem 3.1. Let {T
n
} be
a sequence of nonexpansive mappings of C into i tself such that

2
:=

∞
n
=1
F( T
n
) ∩ VI(C, F) ∩ MEP(, ϕ) =
0
. Assume that either (B1) or (B2) holds.
Let u Î C and let {x
n
} and {u
n
} be sequences generated by x
1
Î C and

⎧
⎪
⎪
⎨
⎪
⎪
⎩
(u
n
, y)+ϕ(y) − ϕ(u
n
)+
1
r
n
y − u
n
, u
n
− x
n
≥0, ∀y ∈ C
,
y
n
= α
n
(u + γ f (x
n
)) + (I − α

n
(I + μA))T
n
P
C
(u
n
− λ
n
Fu
n
),
x
n+1
=(1− β
n
)y
n
+ β
n
T
n
P
C
(y
n
− λ
n
Fy
n

), n ≥ 1,
Jung Journal of Inequalities and Applications 2011, 2011:51
/>Page 19 of 23
where {a
n
}, {b
n
} ⊂ [0, 1], l
n
Î [a, b] ⊂ (0, 2a), and r
n
Î [c, d] ⊂ (0, 2b). Let {a
n
},
{b
n
}, {l
n
}, and {r
n
} satisfy the conditions (C1), (C2), and (C3) in Theorem 3.1. Suppose
that

∞
n
=1
sup{||T
n+1
z − T
n

z|| : z ∈ D} <
∞
for any bounded subset D of C. Let T be a
mapping of C into itself defined by Tz =lim
n®∞
T
n
zforallzÎ C and suppose that
F( T)=

∞
n
=1
F( T
n
)
. Then {x
n
} and {u
n
} converge strongly to q Î Ω
2
, which is a solution
of the optimization problem:
min
x∈
2
μ
2
Ax, x +

1
2
||x − u||
2
− h(x), (OP2
)
where h is a potential function for gf.
Proof Putting B = 0 in Theorem 3.1, we obtain the required result. □
As direct consequences of Theorem 3.1, we can also obtain the following new strong
convergence theorems for the problem (1.1) and fixed point problem.
Corol lary 3.3 Let H, C, Θ, ,B,A,f,μ, g,
¯
γ
, and k be as in Theorem 3.1. Let T be a
nonexpansive mapping of C into itself such that Ω
3
:= F(T) ∩ GMEP( Θ, , B) ≠ Ø.
Assume that either (B1) or (B2) holds. Let u Î Candlet{x
n
} and {u
n
} be sequences
generated by x
1
Î C and
⎧
⎨
⎩
(u
n

, y)+Bx
n
, y − u
n
 + ϕ(y) − ϕ(u
n
)+
1
r
n
y − u
n
, u
n
− x
n
≥0, ∀y ∈ C
,
x
n+1
= α
n
(u + γ f (x
n
)) + (I − α
n
(I + μA))Tu
n
, n ≥ 1,
where {a

n
} ⊂ [0, 1], and r
n
Î [c, d] ⊂ (0; 2b). Let {a
n
} and {r
n
} satisfy the following
conditions:
(i) a
n
® 0(n ® ∞);

∞
n
=1
α
n
= ∞
;
(ii)

∞
n
=1
|α
n+1
− α
n
| <

∞
;

∞
n
=1
|r
n+1
− r
n
| <
∞
Then {x
n
} and {u
n
} converge strongly to q Î Ω
3
, which i s a solut ion of the optimiza -
tion problem:
min
x∈
3
μ
2
Ax, x +
1
2
||x − u||
2

− h(x), (OP3
)
where h is a potential function for gf.
Proof Putting F =0,b
n
=0,andT
n
= T for n ≥ 1 in Theorem 3.1, we obtain the
required result. □
A mapping S : C ® C is called strictly pseudocontractive if there exists r with 0 ≤ r
<1 such that
||Sx − Sy||
2
≤||x − y||
2
+ r||
(
I − S
)
x −
(
I − S
)
y||
2
for all x, y ∈ C
.
Such a mapping S is called r-strictly pseudocontractive. Putting F = I - S,weknow
that
x − y, Fx − Fy≥

1 − r
2
||Fx − Fy||
2
for all x, y ∈ C
;
see, for instance, [12,29]. Hence, we have the following result.
Corolla ry 3.4 Let H, C, Θ, , B, A, f, μ, g,
¯
γ
, and k be as in Theorem 3.1. Let S be a
r-strictly pseudocontractive mappings of C into itself. Let T be a nonexpansive mapping
of C into itself such that Ω
4
:= F(T) ∩ F(S) ∩ GMEP(Θ,, B) ≠ Ø. Assume that either
Jung Journal of Inequalities and Applications 2011, 2011:51
/>Page 20 of 23
(B1) or (B2) holds. Let u Î Candlet{x
n
} and {u
n
} be sequences generated by x
1
Î C
and
⎧
⎨
⎩
(u
n

, y)+Bx
n
, y − u
n
 + ϕ(y) − ϕ(u
n
)+
1
r
n
y − u
n
, u
n
− x
n
≥0, ∀y ∈ C
,
x
n+1
= α
n
(u + γ f (x
n
)) + (I − α
n
(I + μA))T(u
n
− λ
n

(u
n
− Su
n
)), n ≥ 1,
where {a
n
},{l
n
} ⊂ [0, 1], l
n
Î [a, b] ⊂ (0, 1 -r), and r
n
Î [c, d] ⊂ (0, 2b). Let {a
n
},
{l
n
}, and {r
n
} satisfy the following conditions:
(i) a
n
® 0(n ® ∞);

∞
n=1
α
n
= ∞

;
(ii)

∞
n
=1
|α
n+1
− α
n
| <
∞
;

∞
n
=1
|λ
n+1
− λ
n
| <
∞
;

∞
n
=1
|r
n+1

− r
n
| <
∞
.
Then {x
n
} and {u
n
} converge strongly to q Î Ω
4
, which i s a solut ion of the optimiza -
tion problem
min
x∈
4
μ
2
Ax, x +
1
2
||x − u||
2
− h(x), (OP4
)
where h is a potential function for gf.
Proof Putting F = I-S, b
n
=0,andT
n

= T for n ≥ 1, from Theorem 3.1, we obtain
the required result. □
Remark 3.1
(1) Theorem 3.1, Corollary 3.2, Corollary 3.3, and Corollary 3.4 improve and
develop the corresponding results, which were obtained recently by many authors
in various references, for example, see [2,3,7-9,11-18,24,30]. In particular, we men-
tion that the iterative scheme (3.1) in Theorem 3.1 of [24] is not well defined with-
out the assumption C ± C ⊂ C on a nonempty closed convex subset C of H.
(2) The condition

∞
n
=1
|α
n+1
− α
n
| <
∞
imposed on {a
n
} can be replaced by the
perturbed control condition |a
n+1
- a
n
|<o(a
n+1
)+s
n

,

∞
n
=1
σ
n
<
∞
or the condi-
tion a
n
Î (0, 1], n ≥ 1, and lim
n®∞
a
n
/a
n+1
=1.
(3) Some special cases of the generalized mixed equilibrium problem (1.1) are
known as follows:
(i) If  = 0, then the problem (1.1) reduced the following generalized equili-
brium problem (GEP ) of finding x Î C such that

(
x, y
)
+ Bx, y − x≥0, ∀y ∈ C
,
(3:31a)

which was studied by Takahashi and Takahashi [30].
(ii) If Θ(x, y)=0forallx, y Î C, then the problem (1.1) reduces the following
generalized variational inequality problem (GVI) of finding x Î C such that
Bx, y − x + ϕ
(
y
)
− ϕ
(
x
)
≥ 0, ∀y ∈ C
.
(3:32a)
(iii) If B = 0 and Θ(x, y) = 0 for all x, y Î C, then the problem (1.1) reduces the
following minimization problem of finding x Î C such that
Jung Journal of Inequalities and Applications 2011, 2011:51
/>Page 21 of 23
ϕ
(
y
)
− ϕ
(
x
)
≥ 0, ∀y ∈ C
.
(3:33a)
Applying Theorem 3.1 together with the one in the following assumptions instead of

(B1), we can also establish the new corresponding results for the above mentioned
problems:
(B2) C is a bounded set;
(B3) For each x Î H and r>0, there exist a bounded subset D
x
⊆ C and
y
x
Î C such that for any z Î C\D
x
,
ϕ(y
x
) − ϕ(z)+
1
r
y
x
− z, z − x < 0
;
(B4) For each x Î H and r>0, there exist a bounded subset D
x
⊆ C and y
x
Î C such
that for any z Î C\D
x
,
(z, y
x

)+
1
r
y
x
− z, z − x < 0
.
Acknowledgements
The author would like to thank the anonymous referees for their valuable comments and suggestions, which
improved the presentation of this manuscript. This research was supported by the Basic Science Research Program
through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and
Technology (2010-0017007).
Competing interests
The author declares that they have no competing interests.
Received: 21 February 2011 Accepted: 8 September 2011 Published: 8 September 2011
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doi:10.1186/1029-242X-2011-51
Cite this article as: Jung: A general composite iterative method for generalized mixed equilibrium problems,
variational inequality problems and optimization problems. Journal of Inequalities and Applications 2011 2011:51.
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