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RESEARC H Open Access
A relaxed hybrid steepest descent method for
common solutions of generalized mixed
equilibrium problems and fixed point problems
Nawitcha Onjai-uea
1,3
, Chaichana Jaiboon
2,3*
and Poom Kumam
1,3
* Correspondence: chaichana.

2
Department of Mathematics,
Faculty of Liberal Arts, Rajamangala
University of Technology
Rattanakosin (Rmutr), Bangkok
10100, Thailand
Full list of author information is
available at the end of the article
Abstract
In the setting of Hilbert spaces, we introduce a relaxed hybrid steepest descent
method for finding a common element of the set of fixed points of a nonexp ansive
mapping, the set of solutions of a variational inequality for an inverse strongly
monotone mapping and the set of solutions of generalized mixed equilibrium
problems. We prove the strong convergence of the method to the unique solution
of a suitable variational inequality. The results obtained in this article improve and
extend the corresponding results.
AMS (2000) Subject Classification: 46C05; 47H09; 47H10.
Keywords: relaxed hybrid steepest descent method, inverse strongly monotone
mappings, nonexpansive mappings, generalized mixed equilibrium problem

1. Introduction
Let H be a real Hilbert space, C be a nonempty closed convex subset of H and let P
C
be the metric projection of H onto the closed convex subset C.LetS : C ® C be a
nonexpansive mapping, that is, ||Sx - Sy|| ≤ ||x - y|| for all x, y Î C.WedenotebyF
(S)thesetfixedpointofS.IfC ⊂ H is nonempty, bounded, closed and convex and S
is a nonexpansive mapping of C into itself, then F(S) is nonempty; see, for exa mple,
[1,2]. A mapping f : C ® C is a contraction on C if there exists a constant h Î (0, 1)
such that ||f(x)-f(y)|| ≤ h||x - y|| for all x, y Î C. In ad ditio n, let D : C ® H be a
nonlinear mapping,  : C ® ℝ ∪ {+∞} be a real-valued function and let F : C × C ® ℝ
be a bifunction such that C ∩ dom  ≠ ∅, where ℝ is the set of real numbers and dom
 ={x Î C : (x) <+∞}.
The generalized mixed equilibrium problem for finding x Î C such that
F
(
x, y
)
+ Dx, y − x + ϕ
(
y
)
− ϕ
(
x
)
≥ 0, ∀y ∈ C
.
(1:1)
The set of solutions of (1.1) is denoted by GMEP(F, , D), that is,
GMEP

(
F, ϕ, D
)
= {x ∈ C : F
(
x, y
)
+ Dx, y − x + ϕ
(
y
)
− ϕ
(
x
)
≥ 0, ∀y ∈ C}
.
We find that if x is a solution of a problem (1.1), then x Î dom .
If D = 0, then the problem (1.1) is reduced into the mixed equilibrium problem
which is denoted by MEP(F, ).
Onjai-uea et al. Fixed Point Theory and Applications 2011, 2011:32
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Attribution License (http://creativecommons.o rg/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
If  = 0, then the problem (1.1) is reduced into the generalized equilibrium proble m
which is denoted by GEP(F, D).
If D =0and = 0, then the problem (1.1) is reduced into the e quilibri um problem
which is denoted by EP(F).
If F = 0 and  = 0, then the problem (1.1) is reduced into th e variational inequality
problem which is denoted by VI( C, D).

The generalized mixed equilibrium problems include, as speci al cases, some optimi-
zation problems, fixed point problems, variational inequality problems, Nash equili-
brium problems in noncooperative games, equilibrium problem, Numerous problems
in physics, economics and others. Some methods have been proposed to solve the pro-
blem (1.1); see, for instance, [3,4] and the references therein.
Definition 1.1. Let B : C ® H be nonlinear mappings. Then, B is called
(1) monotone if 〈Bx - By, x - y〉 ≥ 0, ∀x, y Î C,
(2) b-inverse-strongly monotone if there exists a constant b > 0 such that
Bx − B
y
, x −
y
≥β|| Bx − B
y
||
2
, ∀x,
y
∈ C
.
(3) 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 called max-
imal if the graph G(Q)ofQ is not properly contained in the graph of any other
monotone mapping. It is well known that a monotone mapping Q is maximal if
and only if for (x, f)ÎH × H, 〈x - y, f - g〉 ≥ 0 for every ( y, g)ÎG(Q) implies f Î Qx.
A typical problem is to minimize a quadratic function over t he set of fixed points of

a nonexpansive mapping defined on a real Hilbert space H:
min
x∈F
1
2
Ax, x−x, b
,
where F is the fixed point set of a nonexpansive mapping S defi ned on H and b is a
given point in H.
A linear-bounded operator A is strongly positive if there exists a constant
¯
γ
>
0
with
the property
Ax, x≥ ¯
γ
||x||
2
, ∀x ∈ H
.
Recently, Marino and Xu [5] introduced a new iterative scheme by the viscosity
approximation method:
x
n+1
= ε
n
γ f
(

x
n
)
+
(
1 − ε
n
A
)
Sx
n
.
(1:2)
They proved that the sequences {x
n
} g enerated by (1.2) converges strongly to the
unique solution of the variational inequality
γ fz − Az, x − z≤0, ∀x ∈ F
(
S
),
Onjai-uea et al. Fixed Point Theory and Applications 2011, 2011:32
/>Page 2 of 20
which is the optimality condition for the minimization problem:
min
x∈F
(
S
)
1

2
Ax, x−h(x)
,
where h is a potential function for gf.
For finding a common element of the set of fixed poin ts of a nonexpansive mapping
and the set of solutions of variational inequalities for a ξ-inverse-strongly monotone
mapping, Takahashi and Toyoda [6] introduced the following iterative scheme:

x
0
∈ C chosen arbitrary,
x
n+1
= γ
n
x
n
+(1− γ
n
)SP
C
(x
n
− α
n
Bx
n
), ∀n ≥ 0,
(1:3)
where B is a ξ-inverse-strongly mo notone mapping, {g

n
} is a sequence in (0, 1), and
{a
n
}isasequencein(0,2ξ). They showed that if F(S) ∩ VI(C, B) is nonempty, then
the sequence {x
n
} generated by (1.3) converges weakly to some z Î F(S) ∩ VI(C, B).
The method of the steepest descent, also known as The Gradient Descent, is the
simplest of the gradient methods. By means of simple optimization algo rithm, this
popular method can find the local minimum of a function. It is a method that is widely
popular among mathematicians and physicists due to its easy concept.
For find ing a common element o f F(S) ∩ VI(C, B), let S : H ® H be nonexpansive
mappings, Yamada [7] introduced the following iterat ive scheme called the hybrid stee-
pest descent method:
x
n+1
=
S
x
n
− α
n
μ
B
S
x
n
, ∀n ≥ 1
,

(1:4)
where x
1
= x Î H,{a
n
} ⊂ (0, 1), B : H ® H is a strongly monotone and Lipschitz
continuous mapping and μ is a positi ve real number. He proved that the sequence {x
n
}
generated by (1.4) converged strongly to the unique solution of the F(S) ∩ VI(C, B).
On the other hand, for finding a n element of F( S) ∩ VI(C, B) ∩ EP(F), Su et al. [8]
introduced the following iterative scheme by the viscosity approximatio n method in
Hilbert spaces: x
1
Î H

F( u
n
, y)+
1
r
n
y − u
n
, u
n
− x
n
≥0, y ∈ C,
x

n+1
= α
n
f (x
n
)+(1− α
n
)SP
C
(u
n
− λ
n
Bu
n
), ∀n ≥ 1
,
(1:5)
where a
n
⊂ [0, 1) and r
n
⊂ (0, ∞) satisfy some appropriate conditions. Furthermore,
they prove {x
n
}and{u
n
} converge strongly to the same point z Î F(S) ∩ VI(C, B ) ∩ EP
(F), where z = P
F(S)∩VI(C,B) ∩ EP(F)

f(z).
For finding a common element of F(S) ∩ GEP(F, D), let C be a nonempty closed con-
vex subset of a real Hilbert space H. Let D be a b-inverse-strongly monotone mapping
of C into H, and let S be a nonexpansive mapping of C into itself, Takahashi and Taka-
hashi [9] introduced the following iterative scheme:
⎧
⎪
⎨
⎪
⎩
F( u
n
, y)+Dx
n
, y − u
n
 +
1
r
n
y − u
n
, u
n
− x
n
≥0, ∀y ∈ C
,
y
n

= α
n
x +(1− α
n
)u
n
,
x
n+1
= γ
n
x
n
+
(
1 − γ
n
)
Sy
n
, ∀n ≥ 1,
(1:6)
Onjai-uea et al. Fixed Point Theory and Applications 2011, 2011:32
/>Page 3 of 20
where {a
n
} ⊂ [0, 1], {g
n
} ⊂ [0, 1] and {r
n

} ⊂ [0, 2b] satisfy some parameters control-
ling conditions. They proved that the sequenc e {x
n
} defined by (1.6) converges strongly
to a common element of F(S) ∩ GEP(F, D).
Recently, Chantarangsi et al. [10] introduced a new iterative algorithm using a viscosity
hybrid steepest descent method for solving a common solution of a generalized mixed
equilibrium problem, the set of fixed points of a nonexpansive mapping and the set of
solutions of variational i nequality problem in a real H ilbert space. Jaiboon [11] suggests
and analyzes an iter ative scheme based on the hybrid steepest descent method for find-
ing a common element of the set of solutions of a system of equilibrium problems, the
set of fixed points of a nonexpansive mapping and the set of solutions of the vari ational
inequality problems for inverse strongly monotone mappings in Hilbert spaces.
In this article, motivated and inspired by the studies mentioned above, we introduce
an iterative scheme using a relaxed hybrid steepest descent method for finding a com-
mon element of the set of solutions of generalized mixed equilibrium problems, the set
of fixed points of a nonexpansive mapping and the set of solutions of variational inequal-
ity problems for inverse strongly monotone mapping in a real Hilbert space. Our results
improve and extend the corresponding results of Jung [12] and some others.
2. Preliminaries
Throughout this article, we always assume H to be a real Hilbert space, and let C be a
nonempty closed convex subset of H.Forasequence{x
n
}, the notation of x
n
⇀ x and
x
n
® x means that the sequence { x
n

} converges weakly and strongly to x, respectively.
For every point x Î H, there exists a unique nearest point in C,denotedbyP
C
x, such that
|
|x − P
C
x|| ≤ ||x −
y
||, ∀x ∈
C.
Such a mapping P
C
from H onto C is called the metric projection.
The following known lemmas will be used in the proof of our main results.
Lemma 2.1. Let H be a real Hilbert spaces H. Then, the following identities hold:
(i) for each x Î H and x* Î C, x*=P
C
x ⇔ 〈x - x*, y - x*〉 ≤ 0, ∀y Î C;
(ii) P
C
: H ® C is nonexpansive, that is,||P
C
x - P
C
y|| ≤ ||x - y||, ∀x, y Î H;
(iii) P
C
is firmly nonexpansive, that is,||P
C

x - P
C
y||
2
≤ 〈P
C
x - P
C
y, x - y〉, ∀x, y Î H;
(iv) ||tx +(1-t)y||
2
= t||x||
2
+(1-t)||y||
2
- t(1 - t)||x - y||
2
, ∀t Î [0, 1], ∀x, y Î H;
(v) ||x + y||
2
≤ ||x||
2
+2〈y, x + y〉.
Lemma 2.2.[2]Let H be a Hilbert space, let C be a nonempty closed convex subset of
H, and let B be a mapping of C into H. Let x* Î C. Then, for l >0,
x
∗
∈ VI
(
C, B

)
⇔ x
∗
= P
C
(
x
∗
− λBx
∗
),
where P
C
is the metric projection of H onto C.
Lemma 2.3.[2]Let H be a Hilbert space, and let C be a nonempty closed convex sub set
of H. Let b >0, and let A : C ® Hbeb-inv erse strongly mon otone. If 0 <ϱ ≤ 2b, then I
-ϱA is a nonexpansive mapping of C into H, where I is the identity mapping on H.
Lemma 2.4. Let H be a real Hilbert space, let C be a nonempty closed convex subset
of H, let S : C ® C be a nonexpansive mapping, and let B : C ® Hbeaξ-inverse
strongly monotone. If 0 < a
n
≤ 2ξ, then S - a
n
BS is a nonexpansive mapping in H.
Onjai-uea et al. Fixed Point Theory and Applications 2011, 2011:32
/>Page 4 of 20
Proof. For any x, y Î C and 0 < a
n
≤ 2ξ, we have



(S − α
n
BS)x − (S − α
n
BS)y


2
= ||(Sx − Sy) − α
n
(BSx − BSy)||
2
= ||Sx − Sy||
2
− 2α
n
Sx − Sy, BSx − BSy + α
2
n
||BSx − BSy||
2
≤||x − y||
2
− 2α
n
ξ||BSx − BSy|| + α
2
n
||BSx − BSy||

2
= ||x − y||
2
+ α
n
(α
n
− 2ξ)||BSx − BSy||
2
≤||x −
y
||
2
.
Hence, S - a
n
BS is a nonexpansive mapping of C into H. □
Lemma 2.5. [13]Let B be a monotone mapping of C into H and let N
C
w
1
be the nor-
mal cone to C at w
1
Î C, that is, N
C
w
1
={w Î H : 〈w
1

- w
2
, w〉 ≥ 0, ∀w
2
Î C} and
define a mapping Q on C by
Qw
1
=

Bw
1
+ N
C
w
1
, w
1
∈ C
;
∅, w
1
∈ C.
Then, Q is maximal monotone and 0 Î Qw
1
if and only if w
1
Î VI(C, B).
Lemma 2.6. [14]Each Hilbert space H satisfies Opial’ s condition, that is, for any
sequence {x

n
} ⊂ H with x
n
⇀ x, the inequality
lim in
f
n
→∞
||x
n
− x|| < lim in
f
n
→∞
||x
n
− y|
|
holds for each y Î H with y ≠ x.
Lemma 2.7.[5]Let C be a nonempty closed convex subset of H and let f be a contrac-
tion of H into itself with coefficient h Î (0, 1 ) and A be a strongly positive linear-
bounded operator on H with coefficient
¯
γ
>
0
. Then, for
0 <γ <
¯
γ

η
,

x − y,(A − γ f )x − (A − γ f )y

≥ ( ¯γ − ηγ)||x − y||
2
, x, y ∈ H
.
That is, A - g f is strongly monotone with coefficient
¯
γ
− η
γ
.
Lemma 2.8.[5]Assume A to be a strongly positive linear-bounded operator on H with
coefficient
¯
γ
>
0
and 0 < r ≤ ||A||
-1
. Then,
|
|I − ρ
A
|| ≤ 1 − ρ ¯
γ
.

For solving the generalized mixed equilibrium problem and the mixed equilibrium
problem, let us give the following assumptions for the bifunction F, the function  and
the set C:
(H1) F(x, x)=0,∀x Î C;
(H2) F is monotone, that is, F(x, y)+F(y, x) ≤ 0 ∀x, y Î C;
(H3) for each y Î C, x a F(x, y ) is weakly upper semicontinuous;
(H4) for each x Î C, y a F(x, y ) is convex;
(H5) for each x Î C, y a F(x, y ) is lower semicontinuous;
(B1) for each x Î H and l >0, there exist abounded subset G
x
⊆ C and y
x
Î C
such that for any z Î C \nG
x
,
F( z , y
x
)+ϕ(y
x
) − ϕ(z)+
1
λ
y
x
− z, z − x < 0
;
(2:1)
(B2) C is a bounded set.
Lemma 2.9. [15]Let C be a nonempty closed convex subset of H. Let F : C ×C ® ℝ be

a bifunction satisfies (H1)-(H5), and let  : C ® ℝ∪{+∞} be a proper lower semi contin-
uous and convex function. Assume that either (B1) or (B2) holds. For l >0and x Î H,
Onjai-uea et al. Fixed Point Theory and Applications 2011, 2011:32
/>Page 5 of 20
define a mapping
T
(
F,ϕ
)
λ
: H → C
as follows:
T
(F,ϕ)
λ
(x)=

z ∈ C : F(z, y)+ϕ(y) − ϕ(z)+
1
λ
y − z, z − x≥0, y ∈ C

, ∀z ∈ H
.
Then, the following properties hold:
(i) For each x Î H,
T
(
F,ϕ
)

λ
(x) =
∅
;
(ii)
T
(
F,ϕ
)
λ
is single-valued;
(iii)
T
(F,ϕ
)
λ
is firmly nonexpansive, that is, for any x, y Î H,
|
|T
(F,ϕ)
λ
x − T
(F,ϕ)
λ
y||
2
≤

T
(F,ϕ)

λ
x − T
(F,ϕ)
λ
y, x − y

;
(iv)
F( T
(
F,ϕ
)
λ
)=MEP(F, ϕ
)
;
(v) MEP(F, ) is closed and convex.
Lemma 2.10.[16]Assume {a
n
} to be a sequence of nonnegative real numbers such
that
a
n+1
≤
(
1 − b
n
)
a
n

+ c
n
, n ≥ 0
,
where {b
n
} is a sequence in (0, 1) and {c
n
} is a sequence in ℝ such that
(1)

∞
n
=1
b
n
=
∞
,
(2)
lim sup
n−∞
c
n
b
n
≤
0
or


∞
n
=1
|c
n
| <
∞
Then, lim
n ®∞
a
n
=0.
3. Main results
In this section, we are in a position to state and prove our main results.
Theorem 3.1. LetCbeanonemptyclosedconvexsubsetofarealHilbertspaceH.
Let F b e bifunction from C × Ctoℝ sati sfying (H1)-(H5), and let  : C ® ℝ ∪ {+∞} be
a proper lower semicontinuous and convex function with either (B1) or (B2). Let B, D
be two ξ, b-inverse strongly monotone mapping of C i nto H, respective ly, and let S : C
® C be a nonexpansive mapping. Let f : C ® C be a contraction mapping with h Î (0,
1), and let A be a strongly positive linear-bounded operator with
¯
γ
>
0
and
0 <γ <
¯γ
η
.
Assume that Θ := F (S) ∩ VI(C, B) ∩ GMEP(F, , D) ≠ ∅. Let {x

n
}, {y
n
} and {u
n
} be
sequences generated by the following iterative algorithm:
⎧
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎩
x
1
= x ∈ C chosen arbitrary,
u
n
= T
(F,ϕ)
λ
n
(x
n
− λ

n
Dx
n
),
y
n
= β
n
γ f (x
n
)+(I − β
n
A)P
C
(Su
n
− α
n
BSu
n
),
x
n+1
=
(
1 − δ
n
)
y
n

+ δ
n
P
C
(
Sy
n
− α
n
BSy
n
)
, ∀n ≥ 1
,
(3:1)
where {δn} and {b
n
} are two sequences in (0, 1) satisfying the following conditions:
Onjai-uea et al. Fixed Point Theory and Applications 2011, 2011:32
/>Page 6 of 20
(C1) lim
n ®∞
b
n
=0and

∞
n
=1
β

n
=
∞
,
(C2) {δ
n
} ⊂ [0, b], for some b Î (0, 1) and lim
n ®∞
|δ
n+1
- δ
n
|=0,
(C3) {l
n
} ⊂ [c, d] ⊂ (0, 2b) and lim
n ®∞
|l
n+1
- l
n
|=0,
(C4) {a
n
} ⊂ [e, g] ⊂ (0, 2ξ) and lim
n ®∞
|a
n+1
- a
n

|=0.
Then,{x
n
} converges strongly to z Î Θ, which is the unique solution of the variational
inequality

γ f (z) − Az, x − z

≤ 0, ∀x ∈ 
.
(3:2)
Proof. We may assume, in view of b
n
® 0asn ® ∞,thatb
n
Î (0, ||A||
-1
). By
Lemma 2.8, we obtain
|
|I − β
n
A
|| ≤ 1 − β
n
¯
γ
, ∀
n
Î N.

We divide the proof of Theorem 3.1 into six steps.
Step 1. We claim that the sequence {x
n
} is bounded.
Now, let p Î Θ. Then, it is clear that
p = Sp = P
C
(p − α
n
Bp)=T
(F,ϕ)
λ
n
(p − λ
n
Dp)
.
Let
u
n
= T
(
F,ϕ
)
λ
n
(x
n
− λ
n

Dx
n
) ∈ dom
ϕ
, D be b-inverse strongly monotone and 0 ≤ l
n
≤ 2b. Then, we have
|
|u
n
−
p
|| ≤ ||x
n
−
p
||
.
(3:3)
Let z
n
= PC( Su
n
- a
n
BSu
n
) and S - a
n
BS be a nonexpansive mapping. Then, we have

from Lemma 2.4 that
||
z
n
−
p||
≤
||
u
n
−
p||
≤
||
x
n
−
p||
(3:4)
and
||y
n
− p|| ≤ β
n
||γ f (x
n
) − Ap|| + ||1 − β
n
A||||z
n

− p||
≤ β
n
||γ f (x
n
) − Ap|| +(1− β
n
¯γ)||z
n
− p||
≤ β
n
γ ||f (x
n
) − f (p)|| + β
n
||γ f (p) − Ap|| +(1− β
n
¯γ)||x
n
− p|
|
≤ β
n
γη||x
n
− p|| + β
n
||γ f (p) − Ap|| +(1− β
n

¯γ)||x
n
− p||
=
(
1 −
(
¯γ − ηγ
)
β
n
)
||x
n
− p|| + β
n
||γ f
(
p
)
− Ap||.
Similarly, and let w
n
= P
C
(Sy
n
- a
n
BSy

n
) in (3.4). Then, we can prove that
|
|w
n
− p|| ≤ ||y
n
− p|| ≤
(
1 −
(
¯γ − ηγ
)
β
n
)
||x
n
− p|| + β
n
||γ f
(
p
)
− Ap||
,
(3:5)
which yields that
|
|x

n+1
− p|| ≤ (1 − δ
n
)||y
n
− p|| + δ
n
||w
n
− p||
≤ (1 − δ
n
)||y
n
− p|| + δ
n
||y
n
− p||
= ||y
n
− p|||
≤ (1 − ( ¯γ − ηγ)β
n
)||x
n
− p|| + β
n
||γ f (p) − Ap||
=(1− ( ¯γ − ηγ )β

n
)||x
n
− p|| +
( ¯γ − ηγ )β
n
( ¯γ − ηγ )
||γ f (p) − Ap|
|
≤ max

||x
n
− p||,
||γ f (p) − Ap||
( ¯γ − ηγ )

≤
≤ max

||x
1
− p||,
||γ f (p) − Ap||
(
¯γ − ηγ
)

, ∀n ≥ 1.
Onjai-uea et al. Fixed Point Theory and Applications 2011, 2011:32

/>Page 7 of 20
This shows that {x
n
} is bounded. Hence, {u
n
}, {z
n
}, {y
n
}, {w
n
}, {BSu
n
}, {BSy
n
}, {Az
n
} and
{f(x
n
)} are also bounded.
We can choose some appropriate constant M>0 such that
M ≥ max

sup
n≥1
{||BSu
n
||},sup
n≥1

{||BSy
n
||},sup
n≥1
{||γ f (x
n
) − Az
n
||}
,
sup
n
≥
1
{||u
n
− x
n
||},sup
n
≥
1
{||w
n
− y
n
||

.
(3:6)

Step 2. We claim that lim
n®∞
||x
n+1
- x
n
|| = 0.
It follows from Lemma 2.9 that
u
n−1
= T
(F,ϕ)
λ
n
−1
(x
n−1
− λ
n−1
Dx
n−1
)
and
u
n
= T
(F,ϕ)
λ
n
(x

n
− λ
n
Dx
n
)
for all n ≥ 1, and we get
F(u
n−1
, y)+ϕ(y)−ϕ(u
n−1
)+Dx
n−1
, y−u
n−1
+
1
λ
n
−1
y−u
n−1
, u
n−1
−x
n−1
≥0, ∀y ∈
C
(3:7)
and

F( u
n
, y)+ϕ(y) − ϕ(u
n
)+Dx
n
, y − u
n
 +
1
λ
n
y − u
n
, u
n
− x
n
≥0, ∀y ∈ C
.
(3:8)
Take y = u
n-1
in (3.8) and y = u
n
in (3.7), and then we have
F(u
n−1
, u
n

)+ϕ(u
n
)−ϕ(u
n−1
)+Dx
n−1
, u
n
−u
n−1
+
1
λ
n
−1
u
n
−u
n−1
, u
n−1
−x
n−1
≥
0
and
F(u
n
, u
n−1

)+ϕ(u
n−1
) − ϕ(u
n
)+Dx
n
, u
n−1
− u
n
 +
1
λ
n
u
n−1
− u
n
, u
n
− x
n
≥0
.
Adding the above two inequalities, the monotonicity of F implies that
Dx
n
− Dx
n−1
, u

n−1
− u
n
 +

u
n−1
− u
n
,
u
n
− x
n
λ
n
−
u
n−1
− x
n−1
λ
n−1

≥
0
and
0 ≤

u

n−1
− u
n
, λ
n−1
(Dx
n
− Dx
n−1
)+
λ
n−1
λ
n
(u
n
− x
n
) − (u
n−1
− x
n−1
)

=

u
n
− u
n−1

, u
n−1
− u
n
+

1 −
λ
n−1
λ
n

u
n
+(x
n
− λ
n−1
Dx
n
)
− (x
n−1
− λ
n−1
Dx
n−1
) − x
n
+

λ
n−1
λ
n
x
n

=

u
n
− u
n−1
, u
n−1
− u
n
+

1 −
λ
n−1
λ
n

(u
n
− x
n
)+(x

n
− λ
n−1
Dx
n
)
− (x
n−1
− λ
n−1
Dx
n−1
)

.
Without loss of generality, let us assume that there exists c Î ℝ such that l
n
>c>0,
∀n ≥ 1. Then, we have
|
|u
n
− u
n−1
||
2
≤||u
n
− u
n−1

||

||x
n
− x
n−1
|| +



1 −
λ
n−1
λ
n



||u
n
− x
n
||

Onjai-uea et al. Fixed Point Theory and Applications 2011, 2011:32
/>Page 8 of 20
and hence,
||u
n
− u

n−1
|| ≤ ||x
n
− x
n−1
|| +
1
λ
n
|λ
n
− λ
n−1
|||u
n
− x
n
|
|
≤||x
n
− x
n−1
|| +
1
c
|λ
n
− λ
n−1

|M.
(3:9)
Since S - a
n
BS is nonexpansive for each n ≥ 1, we have
||z
n
− z
n−1
|| = ||P
C
(Su
n
− α
n
BSu
n
) − P
C
(Su
n−1
− α
n−1
BSu
n−1
)||
≤||(Su
n
− α
n

BSu
n
) − (Su
n−1
− α
n−1
BSu
n−1
)||
= ||(Su
n
− α
n
BSu
n
) − (Su
n−1
− α
n
BSu
n−1
)+(α
n−1
− α
n
)BSu
n−1
||
≤||(Su
n

− α
n
BSu
n
) − (Su
n−1
− α
n
BSu
n−1
)|| + |α
n−1
− α
n
|||BSu
n−1
|
|
≤
||
u
n
− u
n−1
||
+
|
α
n−1
− α

n
|||
BSu
n−1
||
.
(3:10)
Substituting (3.9) into (3.10), we obtain
|
|z
n
− z
n−1
|| ≤ ||x
n
− x
n−1
|| +
1
c
|λ
n
− λ
n−1
|M + |α
n−1
− α
n
|||BSu
n−1

||
.
(3:11)
From (3.1), we have
|
|y
n
− y
n−1
|| = ||β
n
γ f (x
n
)+(I − β
n
A)z
n
− β
n−1
γ f (x
n−1
) − (I − β
n−1
A)z
n−1
|
|
= ||β
n
γ (f (x

n
) − f (x
n−1
)) + (β
n
− β
n−1
)γ f (x
n−1
)
+(I − β
n
A)(z
n
− z
n−1
) − (β
n
− β
n−1
)Az
n−1
||
= ||β
n
γ (f (x
n
) − f (x
n−1
)) + (β

n
− β
n−1
)(γ f (x
n−1
) − Az
n−1
)
+(I − β
n
A)(z
n
− z
n−1
)||
≤ β
n
γ ||f (x
n
) − f (x
n−1
)|| + |β
n
− β
n−1
|||γ f (x
n−1
) − Az
n−1
||

+(I − β
n
A)||z
n
− z
n−1
||
≤ β
n
γη||x
n
− x
n−1
|| + |β
n
− β
n−1
|||γ f (x
n−1
) − Az
n−1
||
+
(
1 − β
n
¯γ
)
||z
n

− z
n−1
||.
(3:12)
Substituting (3.11) into (3.12) yields
||y
n
− y
n−1
|| ≤ β
n
γη||x
n
− x
n−1
|| + |β
n
− β
n−1
|||γ f (x
n−1
) − Az
n−1
||
+(1− β
n
¯γ )

||x
n

− x
n−1
|| +
1
c
|λ
n
− λ
n−1
|M + |α
n−1
− α
n
|||BSu
n−1
||

=(1− ( ¯γ − γη)β
n
)||x
n
− x
n−1
|| + |β
n
− β
n−1
|||γ f (x
n−1
) − Az

n−1
||
+
(1 − β
n
¯γ )
c
|λ
n
− λ
n−1
|M +(1− β
n
¯γ )|α
n−1
− α
n
|||BSu
n−1
||.
(3:13)
Since w
n
= P
C
(Sy
n
- a
n
BSy

n
) and S - a
n
BS is nonexpansive mapping, we have
||w
n
− w
n−1
|| = ||P
C
(Sy
n
− α
n
BSy
n
) − P
C
(Sy
n−1
− α
n−1
BSy
n−1
)||
≤||(Sy
n
− α
n
BSy

n
) − (Sy
n−1
− α
n−1
BSy
n−1
)||
= ||( Sy
n
− α
n
BSy
n
) − (Sy
n−1
− α
n
BSy
n−1
)+(α
n−1
− α
n
)BSy
n−1
|
|
≤||
y

n
−
y
n−1
|| + |α
n−1
− α
n
|||BS
y
n−1
||.
(3:14)
Onjai-uea et al. Fixed Point Theory and Applications 2011, 2011:32
/>Page 9 of 20
Also, from (3.1) and (3.13), we have
||x
n+1
− x
n
|| = ||(1 − δ
n
)y
n
+ δ
n
w
n
−{(1 − δ
n−1

)y
n−1
+ δ
n−1
w
n−1
}||
= ||(1 − δ
n
)(y
n
− y
n−1
)+δ
n
(w
n
− w
n−1
)+(δ
n
− δ
n−1
)(w
n−1
− y
n−1
)||
≤ (1 − δ
n

)||y
n
− y
n−1
|| + δ
n
||w
n
− w
n−1
|| + |δ
n
− δ
n−1
|||w
n−1
− y
n−1
||
≤ (1 − δ
n
)||y
n
− y
n−1
|| + δ
n
{||y
n
− y

n−1
|| + |α
n−1
− α
n
|||BSy
n−1
||}
+ |δ
n
− δ
n−1
|||w
n−1
− y
n−1
||
= ||y
n
− y
n−1
|| + δ
n
|α
n−1
− α
n
|||BSy
n−1
|| + |δ

n
− δ
n−1
|||w
n−1
− y
n−1
||
≤ (1 − ( ¯γ − γη)β
n
)||x
n
− x
n−1
|| + |β
n
− β
n−1
|||γ f(x
n−1
) − Az
n−1
||
+
(1 − β
n
¯γ )
c
|λ
n

− λ
n−1
|M +(1− β
n
¯γ )|α
n−1
− α
n
|||BSu
n−1
||
+ δ
n
|α
n−1
− α
n
|||BSy
n−1
|| + |δ
n
− δ
n−1
|||w
n−1
− y
n−1
||
≤ (1 − ( ¯γ − γη)β
n

)||x
n
− x
n−1
|| +

|β
n
− β
n−1
| +
(1 − β
n
¯γ )
c
|λ
n
− λ
n−1
|
+(1 − β
n
¯γ + δ
n
)|α
n−1
− α
n
| + |δ
n

− δ
n−1
|

M.
(3:15)
Set
b
n
=
(
¯γ − γη
)
β
n
and
c
n
=

|β
n
− β
n−1
| +
(1−β
n
¯γ )
c
|λ

n
− λ
n−1
| +(1− β
n
¯γ + δ
n
)|α
n−1
− α
n
| + |δ
n
− δ
n−1
|

M
.
Then, we have
||x
n+1
− x
n
|| ≤
(
1 − b
n
)
||x

n
− x
n−1
|| + c
n
, ∀n ≥ 0
.
(3:16)
From the conditions (C1)-(C4), we find that
lim
n→∞
b
n
=0,
∞

n
=
0
b
n
= ∞ and limsup
n→∞
c
n
≤ 0
.
Therefore, applying Lemma 2.10 to (3.16), we have
lim
n

→
∞
||x
n+1
− x
n
|| =0
.
(3:17)
Step 3. We claim that lim
n®∞
||Sw
n
- w
n
|| = 0.
For any p Î Θ and Lemma 2.4, we obtain
||z
n
− p||
2
= ||P
C
(Su
n
− α
n
BSu
n
) − P

C
(p − α
n
Bp)||
2
≤||(Su
n
− α
n
BSu
n
) − (p − α
n
Bp)||
2
= ||(Su
n
− α
n
BSu
n
) − (Sp − α
n
BSp)||
2
≤||x
n
− p||
2
+(α

2
n
− 2α
n
ξ)||BSu
n
− Bp||
2
.
(3:18)
From (3.1) and (3.18), we have


y
n
− p


2
= ||β
n
(γ f (x
n
) − Ap)+(I − β
n
A)(z
n
− p)||
2
= || (I − β

n
A)(z
n
− p)||
2
+ β
2
n
||γ f (x
n
) − Ap||
2
+2β
n
(I − β
n
A)(z
n
− p), γ f (x
n
) − Ap
≤ (1 − β
n
¯γ )
2
||z
n
− p||
2
+ β

2
n
||γ f (x
n
) − Ap||
2
+2β
n
(I − β
n
A)(z
n
− p), γ f (x
n
) − Ap
≤ (1 − β
n
¯γ )
2

||x
n
− p||
2
+(α
2
n
− 2α
n
ξ)||BSu

n
− Bp||
2

+ β
2
n
||γ f (x
n
) − Ap||
2
+2β
n
(I − β
n
A)(z
n
− p), γ f (x
n
) − Ap
=(1− β
n
¯γ )
2
||x
n
− p||
2
+(1− β
n

¯γ )
2
(α
2
n
− 2α
n
ξ)||BSu
n
− Bp||
2
+ β
2
n
||γ f (x
n
) − Ap||
2
+2β
n
(I − β
n
A)(z
n
− p), γ f (x
n
) − Ap
≤||x
n
− p||

2
+(1− β
n
¯γ )
2
(α
2
n
− 2α
n
ξ)||BSu
n
− Bp||
2
+ β
2
n
||γ f (x
n
) − Ap||
2
+2β
n
(I − β
n
A)(z
n
− p), γ f (x
n
) − Ap.

(3:19)
Onjai-uea et al. Fixed Point Theory and Applications 2011, 2011:32
/>Page 10 of 20
From (3.1), (3.5), (3.19) and Lemma 2.1(iv), we have
||x
n+1
− p||
2
≤ (1 − δ
n
)||y
n
− p||
2
+ δ
n
||w
n
− p||
2
≤ (1 − δ
n
)||y
n
− p||
2
+ δ
n
||y
n

− p||
2
≤||y
n
− p||
2
≤||x
n
− p||
2
+(1− β
n
¯γ )
2
(α
2
n
− 2α
n
ξ)||BSu
n
− Bp||
2
+ β
2
n
||γ f (x
n
) − Ap||
2

+2β
n
(I − β
n
A)(z
n
− p), γ f (x
n
) − Ap
.
(3:20)
It follows that
(1 − β
n
¯γ )
2
(2gξ − e
2
)||BSu
n
− Bp||
2
≤ (1 − β
n
¯γ )
2
(2α
n
ξ − α
2

n
)||BSu
n
− Bp||
2
≤||x
n
− p||
2
−||x
n+1
− p||
2
+ β
2
n
||γ f(x
n
) − Ap||
2
+2β
n
(I − β
n
A)(z
n
− p), γ f (x
n
) − Ap
≤||x

n
− x
n+1
||(||x
n
− p|| + ||x
n+1
− p||)+β
2
n
||γ f(x
n
) − Ap||
2
+2β
n

(
I − β
n
A
)(
z
n
− p
)
, γ f
(
x
n

)
− Ap.
(3:21)
From condition (C1) and (3.17), we obtain
lim
n
→
∞
||BSu
n
− Bp|| =0
.
(3:22)
From w
n
= PC(Sy
n
- a
n
BSy
n
), (3.19) and Lemma 2.4, we have
||w
n
− p||
2
= ||P
C
(Sy
n

− α
n
BSy
n
) − P
C
(p − α
n
Bp)||
2
≤||(Sy
n
− α
n
BSy
n
) − (p − α
n
Bp)||
2
= ||( Sy
n
− α
n
BSy
n
) − (Sp − α
n
BSp)||
2

≤||y
n
− p||
2
+(α
2
n
− 2α
n
ξ)||BSy
n
− Bp||
2
≤

||x
n
− p||
2
+(1− β
n
¯γ)
2
(α
2
n
− 2α
n
ξ)||BSu
n

− Bp||
2
+β
2
n
||γ f (x
n
) − Ap||
2
+2β
n
(I − β
n
A)(z
n
− p), γ f (x
n
) − Ap

+(α
2
n
− 2α
n
ξ)||BSy
n
− Bp||.
2
(3:23)
Using (3.1), (3.19) and (3.23), we obtain

|
|x
n+1
− p||
2
≤ (1 − δ
n
)||y
n
− p||
2
+ δ
n
||w
n
− p||
2
≤ (1 − δ
n
)

||x
n
− p||
2
+(1− β
n
¯γ )
2
(α

2
n
− 2α
n
ξ)||BSu
n
− Bp||
2
+β
2
n
||γ f (x
n
) − Ap||
2
+2β
n
(I − β
n
A)(z
n
− p), γ f (x
n
) − Ap

+ δ
n

||x
n

− p||
2
+(1− β
n
¯γ )
2
(α
2
n
− 2α
n
ξ)||BSu
n
− Bp||
2
+ β
2
n
||γ f (x
n
) − Ap||
2
+2β
n
(I − β
n
A)(z
n
− p), γ f (x
n

) − Ap
]
+(α
2
n
− 2α
n
ξ)||BSy
n
− Bp||
2

= ||x
n
− p||
2
+(1− β
n
¯γ )
2
(α
2
n
− 2α
n
ξ)||BSu
n
− Bp||
2
+ β

2
n
||γ f (x
n
) − Ap||
2
+2β
n
(I − β
n
A)(z
n
− p), γ f (x
n
) − Ap
+(α
2
n
− 2α
n
ξ)δ
n
||BSy
n
− Bp||.
2
(3:24)
It follows that
(2gξ − e
2

)b||BSy
n
− Bp||
2
≤||x
n
− x
n+1
||(||x
n
− p|| + ||x
n+1
− p||)
+(1− β
n
¯γ )
2
(α
2
n
− 2α
n
ξ)||BSu
n
− Bp||
2
+ β
2
n
||γ f (x

n
) − Ap||
2
+2β
n

(
I − β
n
A
)(
z
n
− p
)
, γ f
(
x
n
)
− Ap.
(3:25)
Onjai-uea et al. Fixed Point Theory and Applications 2011, 2011:32
/>Page 11 of 20
From condition (C1), (3.17) and (3.22), we obtain
lim
n
→
∞
||BSy

n
− Bp|| =0
.
(3:26)
Since P
C
is firmly nonexpansive, we have
||w
n
− p||
2
= ||P
C
(Sy
n
− α
n
BSy
n
) − P
C
(p − α
n
Bp)||
2
≤(Sy
n
− α
n
BSy

n
) − (p − α
n
Bp), w
n
− p
=
1
2

||(Sy
n
− α
n
BSy
n
) − (p − α
n
Bp)||
2
+ ||w
n
− p||
2
−||(Sy
n
− α
n
BSy
n

) − (p − α
n
Bp) − (w
n
− p)||
2

≤
1
2

||y
n
− p||
2
+ ||w
n
− p||
2
−||(Sy
n
− w
n
) − α
n
(BSy
n
− Bp)||
2


≤
1
2
(||x
n
− p||
2
+(1− β
n
¯γ)
2
(α
2
n
− 2α
n
ξ)||BSu
n
− Bp||
2
+ β
2
n
||γ f (x
n
) − Ap||
2
+2β
n
(I − β

n
A)(z
n
− p), γ f (x
n
) − Ap)
+
1
2

||w
n
− p||
2
−||Sy
n
− w
n
||
2
−α
2
n
||BSy
n
− Bp||
2
+2α
n
Sy

n
− w
n
, BSy
n
− Bp

.
(3:27)
Hence, we have
||w
n
− p||
2
≤||x
n
− p||
2
−||Sy
n
− w
n
||
2
+(1− β
n
¯γ )
2
(α
2

n
− 2α
n
ξ)||BSu
n
− Bp||
2
+ β
2
n
||γ f (x
n
) − Ap||
2
+2β
n
(I − β
n
A)(z
n
− p), γ f (x
n
) − Ap
+2α
n
||S
y
n
− w
n

||||BS
y
n
− B
p
||.
(3:28)
Using (3.24) and (3.28), we have
||x
n+1
− p||
2
≤ (1 − δ
n
)||y
n
− p||
2
+ δ
n
||w
n
− p||
2
≤ (1 − δ
n
){||x
n
− p||
2

+(1− β
n
¯γ )
2
(α
2
n
− 2α
n
ξ)||BSu
n
− Bp||
2
+ β
2
n
||γ f (x
n
) − Ap||
2
+2β
n
(I − β
n
A)(z
n
− p), γ f (x
n
) − Ap}
+ δ

n
{||x
n
− p||
2
−||Sy
n
− w
n
||
2
+(1− β
n
¯γ )
2
(α
2
n
− 2α
n
ξ)||BSu
n
− Bp||
2
+2α
n
||Sy
n
− w
n

||||BSy
n
− Bp||
+ β
2
n
||γ f (x
n
) − Ap||
2
+2β
n
(I − β
n
A)(z
n
− p), γ f (x
n
) − Ap}
= ||x
n
− p||
2
− δ
n
||Sy
n
− w
n
||

2
+(1− β
n
¯γ )
2
(α
2
n
− 2α
n
ξ)||BSu
n
− Bp||
2
+2α
n
δ
n
||Sy
n
− w
n
||||BSy
n
− Bp|
|
+ β
2
n
||γ f (x

n
) − Ap||
2
+2β
n
(I − β
n
A)(z
n
− p), γ f (x
n
) − Ap.
(3:29)
It follows that
b
||Sy
n
− w
n
||
2
≤ δ
n
||Sy
n
− w
n
||
2
≤||x

n
− x
n+1
||(||x
n
− p|| + ||x
n+1
− p||)
+(1− β
n
¯γ )
2
(α
2
n
− 2α
n
ξ)||BSu
n
− Bp||
2
+2α
n
δ
n
||Sy
n
− w
n
||||BSy

n
− Bp|
|
+ β
2
n
||γ f (x
n
) − Ap||
2
+2β
n
(I − β
n
A)(z
n
− p), γ f (x
n
) − Ap.
(3:30)
From the condition (C1), (3.17), (3.22) and (3.26), we obtain
lim
n
→
∞
||Sy
n
− w
n
|| =0

.
(3:31)
Onjai-uea et al. Fixed Point Theory and Applications 2011, 2011:32
/>Page 12 of 20
Note that
||y
n
− p||
2
≤ (1 − β
n
¯γ )
2
||z
n
− p||
2
+ β
2
n
||γ f (x
n
) − Ap||
2
+2β
n
(I − β
n
A)(z
n

− p), γ f (x
n
) − Ap
≤ (1 − β
n
¯γ )
2
||u
n
− p||
2
+ β
2
n
||γ f (x
n
) − Ap||
2
+2β
n
(I − β
n
A)(z
n
− p), γ f (x
n
) − Ap

≤ (1 − β
n

¯γ )
2
{||x
n
− p||
2
+ λ
n
(λ
n
− 2β)||Dx
n
− Dp||
2
} + β
2
n
||γ f (x
n
) − Ap||
2
+2β
n
(I − β
n
A)(z
n
− p), γ f (x
n
) − Ap

≤||x
n
− p||
2
+(1− β
n
¯γ )
2
λ
n
(λ
n
− 2β)||Dx
n
− Dp||
2
+ β
2
n
||γ f (x
n
) − Ap||
2
+2β
n

(
I − β
n
A

)(
z
n
− p
)
, γ f
(
x
n
)
− Ap.
(3:32)
From (3.1) and (3.32), we can compute
||x
n+1
− p||
2
≤ (1 − δ
n
)||y
n
− p||
2
+ δ
n
||w
n
− p||
2
≤ (1 − δ

n
)||y
n
− p||
2
+ δ
n
||y
n
− p||
2
= ||y
n
− p||
2
≤||x
n
− p||
2
+(1− β
n
¯γ )
2
λ
n
(λ
n
− 2β)||Dx
n
− Dp||

2
+ β
2
n
||γ f (x
n
) − Ap||
2
+2β
n
(I − β
n
A)(z
n
− p), γ f (x
n
) − Ap
.
(3:33)
It follows that
(1 − β
n
¯γ )
2
d(2β − c)||Dx
n
− Dp||
2
≤||x
n

− x
n+1
||(||x
n
− p|| + ||x
n+1
− p||)+β
2
n
||γ f (x
n
) − Ap||
2
+2β
n

(
I − β
n
A
)(
z
n
− p
)
, γ f
(
x
n
)

− Ap,
(3:34)
which implies that
lim
n
→
∞
||Dx
n
− Dp|| =0
.
(3:35)
In addition, from the firmly nonexpansivity of
T
(
F,ϕ
)
λ
n
, we have
|
|u
n
− p||
2
= ||T
(
F,ϕ
)
λ

n
(x
n
− λ
n
Dx
n
) − T
(
F,ϕ
)
λ
n
(p − λ
n
Dp)||
2
≤(x
n
− λ
n
Dx
n
) − (p − λ
n
Dp), u
n
− p
=
1

2
{||(x
n
− λ
n
Dx
n
) − (p − λ
n
Dp)||
2
+ ||u
n
− p||
2
−||(x
n
− λ
n
Dx
n
) − (p − λ
n
Dp) − (u
n
− p)||
2
}
≤
1

2

||x
n
− p||
2
+ ||u
n
− p||
2
−||x
n
− u
n
− λ
n
(Dx
n
− Dp)||
2

=
1
2

||x
n
− p||
2
+ ||u

n
− p||
2
−||x
n
− u
n
||
2
+2λ
n
x
n
− u
n
, Dx
n
− Dp−λ
2
n
||Dx
n
− Dp||
2

.
Hence, we obtain
|
|u
n

−
p
||
2
≤||x
n
−
p
||
2
−||x
n
− u
n
||
2
+2λ
n
||x
n
− u
n
||||Dx
n
− D
p
||
.
(3:36)
Substituting (3.36) into (3.32) to get

||y
n
− p||
2
≤ (1 − β
n
¯γ )
2
||u
n
− p||
2
+ β
2
n
||γ f (x
n
) − Ap||
2
+2β
n
(I − β
n
A)(z
n
− p), γ f (x
n
) − Ap

≤ (1 − β

n
¯γ )
2

||x
n
− p||
2
−||x
n
− u
n
||
2
+2λ
n
||x
n
− u
n
||||Dx
n
− Dp||

+ β
2
n
||γ f (x
n
) − Ap||

2
+2β
n
(I − β
n
A)(z
n
− p), γ f (x
n
) − Ap
≤||x
n
− p||
2
− (1 − β
n
¯γ )
2
||x
n
− u
n
||
2
+2(1− β
n
¯γ )
2
λ
n

||x
n
− u
n
||||Dx
n
− Dp||
+ β
2
n
||γ f (x
n
) − Ap||
2
+2β
n
(I − β
n
A)(z
n
− p), γ f (x
n
) − Ap
(3:37)
Onjai-uea et al. Fixed Point Theory and Applications 2011, 2011:32
/>Page 13 of 20
and hence,
||x
n+1
− p||

2
≤||y
n
− p||
2
≤||x
n
− p||
2
− (1 − β
n
¯γ )
2
||x
n
− u
n
||
2
+2(1− β
n
¯γ )
2
λ
n
||x
n
− u
n
||||Dx

n
− Dp||
+ β
2
n
||γ f (x
n
) − Ap||
2
+2β
n
(I − β
n
A)(z
n
− p), γ f (x
n
) − Ap
.
(3:38)
It follows that
(1 − β
n
¯γ )
2
||x
n
− u
n
||

2
≤||x
n+1
− x
n
||(||x
n+1
− p|| + ||x
n
− p||)
+2(1− β
n
¯γ )
2
λ
n
||x
n
− u
n
||||Dx
n
− Dp|| + β
2
n
||γ f (x
n
) − Ap||
2
+2β

n

(
I − β
n
A
)(
z
n
− p
)
, γ f
(
x
n
)
− Ap.
(3:39)
This together w ith ||x
n+1
- x
n
|| ® 0, ||Dx
n
- D
p
|| ® 0, b
n
® 0asn ® ∞ and the
condition on l

n
implies that
lim
n
→∞
||x
n
− u
n
|| = 0 and lim
n
→∞
||x
n
−u
n
||
λ
n
=0
.
(3:40)
Consequently, from (3.17) and (3.40)
||
x
n+1
− u
n
||
≤

||
x
n+1
− x
n
||
+
||
x
n
− u
n
||
→ 0asn →∞
.
(3:41)
From (3.1) and condition (C1), we have
||y
n
− z
n
|| = ||β
n
γ f
(
x
n
)
+
(

1 − β
n
A
)
z
n
− z
n
|| ≤ β
n
||γ f
(
x
n
)
− Az
n
|| → 0asn →∞
.
(3:42)
Since S - a
n
BS is nonexpansive mapping(Lemma 2.4), we have
|
|w
n
− z
n
|| = ||P
C

(Sy
n
− α
n
BSy
n
) − P
C
(Su
n
− α
n
BSu
n
)|
|
≤||(S − α
n
BS)y
n
− (S − α
n
BS)u
n
||
≤||
y
n
− u
n

||.
(3:43)
Next, we will show that ||x
n
- y
n
|| ® 0asn ® ∞.
We consider x
n+1
- y
n
= δ
n
(w
n
- y
n
)=δn(w
n
- z
n
+ z
n
- y
n
).
From (3.43), we have
|
|x
n+1

− y
n
|| ≤ δ
n
(||w
n
− z
n
|| + ||z
n
− y
n
||)
≤ δ
n
(||y
n
− u
n
|| + ||z
n
− y
n
||)
≤ δ
n
(
||x
n+1
− y

n
|| + ||x
n+1
− u
n
|| + ||z
n
− y
n
||
).
(3:44)
From the condition (C2), (3.41) and (3.42), it follows that
||x
n+1
−y
n
|| ≤
δ
n
1 − δ
n
(||x
n+1
−u
n
||+||z
n
−y
n

||) ≤
b
1 − b
(||x
n+1
−u
n
||+||z
n
−y
n
||) → 0
.
(3:45)
From (3.17) and (3.45), we obtain
||
x
n
−
y
n
||
≤
||
x
n
− x
n+1
||
+

||
x
n+1
−
y
n
||
→ 0asn →∞
.
(3:46)
We observe that
|
|Sw
n
− w
n
|| ≤ ||Sw
n
− Sz
n
|| + ||Sz
n
− Sy
n
|| + ||Sy
n
− w
n
||
≤||w

n
− z
n
|| + ||z
n
− y
n
|| + ||Sy
n
− w
n
||
≤||y
n
− u
n
|| + ||z
n
− y
n
|| + ||Sy
n
− w
n
||
≤||
y
n
− x
n

|| + ||x
n
− u
n
|| + ||z
n
−
y
n
|| + ||S
y
n
− w
n
||.
(3:47)
Onjai-uea et al. Fixed Point Theory and Applications 2011, 2011:32
/>Page 14 of 20
Consequently, we obtain
lim
n
→
∞
||Sw
n
− w
n
|| =0
.
(3:48)

Step 4. We prove that the mapping P
Θ
(gf +(I - A)) has a unique fixed point.
Let f be a contraction of C into itself with coefficient h Î (0, 1). Then, we have


P

(γ f +(I − A))(x) − P

(γ f +(I − A))(y)


≤||(γ f +(I − A))(x) − (γ f +(I − A))(y)|
|
≤ γ ||f(x) − f (y)|| + ||I − A|| ||x − y||
≤ γη||x − y|| +(1−¯γ )||x − y||
=
(
1 −
(
¯γ − ηγ
))
||x − y||, ∀x, y ∈ C.
Since
0 < 1 −
(
¯γ − ηγ
)
<

1
, it follows that P
Θ
(gf +(I - A)) is a contraction of C into
itself. Therefore, by the Banach Contraction Mapping Principle, it has a unique fixed
point, say z Î C, that is,
z = P

(
γ f +
(
I − A
))(
z
).
Step 5. We claim that q Î F(S) ∩ VI(C, B) ∩ GMEP(F, , D).
First, we show that q Î F(S).
Assume q ∉ F(S). Since
w
n
i

q
and q ≠ Sq, based on Opial’s condition (Lemma 2.6),
it follows that
lim inf
i→∞
||w
n
i

− q|| < lim inf
i→∞
||w
n
i
− Sq||
≤ lim inf
i→∞
{||w
n
i
− Sw
n
i
|| + ||Sw
n
i
− Sq||
}
= lim inf
i→∞
||Sw
n
i
− Sq||
≤ lim inf
i
→
∞
||w

n
i
− q||.
This is a contradiction. Thus, we have q Î F(S).
Next, we prove that q Î GMEP(F, , D ).
From Lemma 2.9 that
u
n
= T
(F,ϕ)
λ
n
(x
n
− λ
n
Dx
n
)
for all n ≥ 1 is equivalent to
F( u
n
, y)+ϕ(y) − ϕ(u
n
)+Dx
n
, y − u
n
 +
1

λ
n
y − u
n
, u
n
− x
n
≥0, ∀y ∈ C
.
From (H2), we also have
ϕ(y) − ϕ(u
n
)+Dx
n
, y − u
n
 +
1
λ
n
y − u
n
, u
n
− x
n
≥−F(u
n
, y) ≥ F(y, u

n
)
.
Replacing n by n
i
, we obtain
ϕ(y) − ϕ(u
n
i
)+Dx
n
i
, y − u
n
i
 +

y − u
n
i
,
u
n
i
− x
n
i
λ
n
i


≥ F(y, u
n
i
)
.
(3:49)
Let y
t
= t
y
+(1-t)q for all t Î (0, 1] and y Î C. Since y Î C and q Î C, we obtain y
t
Î C. Hence, from (3.49), we have
Onjai-uea et al. Fixed Point Theory and Applications 2011, 2011:32
/>Page 15 of 20
y
t
− u
n
i
, Dy
t
≥y
t
− u
n
i
, Dy
t

−ϕ(y
t
)+ϕ(u
n
i
) −Dx
n
i
, y
t
− u
n
i

−

y
t
− u
n
i
,
u
n
i
− x
n
i
λ
n

i

+ F(y
t
, u
n
i
)
≥y
t
− u
n
i
, Dy
t
− Du
n
i
 + y
t
− u
n
i
, Du
n
i
− Dx
n
i
−ϕ(y

t
)
+ ϕ(u
n
i
) −{y
t
− u
n
i
,
u
n
i
− x
n
i
λ
n
i
} + F(y
t
, u
n
i
).
(3:50)
Since
||
u

n
i
− x
n
i
||
→ 0
, i ® ∞ we obt ain
||
Du
n
i
− Dx
n
i
||
→
0
. Furthermore, by the
monotonicity of D, we have

y
t
− u
n
i
, Dy
t
− Du
n

i

≥ 0.
Hence, from (H4), (H5) and the weak lower semicontinuity of ,
u
n
i
−x
n
i
λ
n
i
→
0
and
u
n
i
→
q
, we have
y
t
− q, Dy
t
≥−ϕ
(
y
t

)
+ ϕ
(
q
)
+ F
(
y
t
, q
)
as i →∞
.
(3:51)
From (H1), (H4) and (3.51), we also get
0=F(y
t
, y
t
)+ϕ(y
t
) − ϕ(y
t
)
≤ tF(y
t
, y)+(1− t)F ( y
t
, q)+tϕ(y)+(1− t)ϕ( q ) − ϕ(y
t

)
= t[F(y
t
, y)+ϕ(y) − ϕ(y
t
)] + (1 − t)[F(y
t
, q)+ϕ(q) − ϕ(y
t
)
]
≤ t[F(y
t
, y)+ϕ(y) − ϕ(y
t
)] + (1 − t)y
t
− q, Dy
t

= t[F
(
y
t
, y
)
+ ϕ
(
y
)

− ϕ
(
y
t
)
]+
(
1 − t
)
ty − q, Dy
t
.
Dividing by t, we get
F
(
y
t
, y
)
+ ϕ
(
y
)
− ϕ
(
y
t
)
+
(

1 − t
)
y − q, Dy
t
≥0
.
Letting t ® 0 in the above inequality, we arrive that, for each y Î C,
F
(
q, y
)
+ ϕ
(
y
)
− ϕ
(
q
)
+ y − q, Dq≥0
.
This implies that q Î GMEP(F, , D).
Finally, we prove that q Î VI(C, B).
We define the maximal monotone operator:
Qq
1
=

Bq
1

+ N
C
q
1
, q
1
∈ C
,
∅, q
1
∈ C.
Since B is ξ-inverse strongly monotone and by condition (C4), we have
Bx − B
y
, x −
y
≥ξ ||Bx −−B
y
||
2
≥ 0
.
Then, Q is maximal monotone. Let (q
1
, q
2
) Î G(Q). Since q
2
- Bq
1

Î N
C
q
1
and w
n
Î
C, we have 〈q
1
- w
n
, q
2
-Bq
1
〉 ≥ 0. On the other hand, from w
n
= P
C
(Sy
n
- a
n
BSy
n
), we
have
q
1
− w

n
, w
n
−
(
Sy
n
− α
n
BSy
n
)
≥0
,
that is,

q
1
− w
n
,
w
n
− Sy
n
α
n
+ BSy
n


≥ 0
.
Onjai-uea et al. Fixed Point Theory and Applications 2011, 2011:32
/>Page 16 of 20
Therefore, we obtain
q
1
− w
n
i
, q
2
≥q
1
− w
n
i
, Bq
1

≥q
1
− w
n
i
, Bq
1
−

q

1
− w
n
i
,
w
n
i
− Sy
n
i
α
n
i
+ BSy
n
i

=

q
1
− w
n
i
, Bq
1
− BSy
n
i

−
w
n
i
− Sy
n
i
α
n
i

= q
1
− w
n
i
, Bq
1
− Bw
n
i
 + q
1
− w
n
i
, Bw
n
i
− BSy

n
i

−

q
1
− w
n
i
,
w
n
i
− Sy
n
i
α
n
i

≥q
1
− w
n
i
, Bw
n
i
− BSy

n
i
−

q
1
− w
n
i
,
w
n
i
− Sy
n
i
α
n
i

.
(3:52)
Noting that
|
|w
n
i
− Sy
n
i

|| → 0
as i ® ∞, we obtain
q
1
− q, q
2
≥0.
Since Q is maximal monotone, we obtain that q Î Q
-1
0, and hence q Î VI(C, B).
This implies q Î Θ. Since z = P
Θ
(gf +(I - A))(z), we have
lim sup
n
→∞

γ f (z) − Az, x
n
− z

= lim
i→∞

γ f (z) − Az, x
n
i
− z

=


γ f (z) − Az, q − z

≤ 0
.
(3:53)
On the other hand, we have

γ f (z) − Az, y
n
− z

=

γ f (z) − Az, y
n
− x
n

+

γ f (z) − Az, x
n
− z

≤||γ f (z) − Az|| ||y
n
− x
n
|| +


γ f (z) − Az, x
n
− z

.
From (3.46) and (3.53), we obtain that
lim sup
n
→∞

γ f (z) − Az, y
n
− z

≤ 0
.
(3:54)
Step 6. Finally, we claim that x
n
® z, where z = P
Θ
(gf +(I - A))(z).
We note that


y
n
− z



2
= ||(I − β
n
A)(z
n
− z)+β
n
(γ f (x
n
) − Az)||
2
≤||(I − β
n
A)(z
n
− z)||
2
+2β
n
(γ f (x
n
) − Az), (I − β
n
A)(z
n
− z)+β
n
(γ f (x
n

) − Az)

= ||(I − β
n
A)(z
n
− z)||
2
+2β
n
(γ f (x
n
) − Az), y
n
− z
≤||I − β
n
A||
2
||z
n
− z||
2
+2β
n
γ f (x
n
) − f (z), y
n
− z +2β

n
γ f (z) − Az, y
n
− z
≤ (1 − β
n
¯γ )
2
||z
n
− z||
2
+2β
n
γη||x
n
− z|| ||y
n
− z|| +2β
n
γ f (z) − Az, y
n
− z
≤ (1 − β
n
¯γ )
2
||x
n
− z||

2
+ β
n
γη(||x
n
− z||
2
+ ||y
n
− z||
2
)+2β
n
γ f (z) − Az, y
n
− z
=(1− 2β
n
¯γ + β
2
n
¯γ
2
+ β
n
γη)||x
n
− z||
2
+ β

n
γη||y
n
− z||
2
+2β
n
γ f (z) − Az, y
n
− z
(3:55)
which implies that
|
|y
n
− z||
2
≤

1 −
(2 ¯γ − γη) β
n
1 − γηβ
n

||x
n
− z||
2
+

β
n
1 −
γ
ηβ
n

β
n
¯γ
2
||x
n
− z||
2
+2γ f (z) − Az, y
n
− z

.
(3:56)
Onjai-uea et al. Fixed Point Theory and Applications 2011, 2011:32
/>Page 17 of 20
On the other hand, we have

x
n+1
− z

2

≤||y
n
− z||
2
≤

1 −
(2 ¯γ − γη)β
n
1 − γηβ
n

||x
n
− z||
2
+
β
n
1 − γηβ
n

β
n
¯γ
2
||x
n
− z||
2

+2γ f (z) − Az, y
n
− z

≤

1 −
(2 ¯γ − γη)β
n
1 − γηβ
n

||x
n
− z||
2
+
β
n
1 −
γ
η
β
n

2γ f (z) − Az, y
n
− z + β
n
¯γ

2
K

,
(3:57)
where K is an appropriate constant such that K ≥ sup
n≥1
{||x
n
- z||
2
}.
Set
l
n
=
(2 ¯γ −γη)β
n
1−
γ
ηβ
n
and
e
n
=
β
n
1−
γ

ηβ
n

2γ f (z) − Az, y
n
− z + β
n
¯γ
2
K

. Then, we have
|
|x
n+1
− z||
2
≤
(
1 − b
n
)
||x
n
− z||
2
+ c
n
, ∀n ≥ 0
.

(3:58)
From the condition (C1) and (3.54), we see that
lim
n→∞
l
n
=0,
∞

n
=
0
l
n
= ∞ and lim sup
n→∞
e
n
≤ 0
.
Therefore, applying Lemma 2.10 to (3.58), we get that {x
n
} converges strongly to z Î
Θ.
This completes the proof. □
Corollary 3.2. Let C be a nonempty closed convex subset of a real Hilbert space H, let
Bbeξ-inverse-strongly monotone mapping of C into H, and let S : C ® Cbeanonex-
pansive mapping. Let f : C ® C be a contraction mapping with h Î (0, 1), and let A be
a strongly positive linear-bounded operator with
¯

γ
>
0
and
0 <γ <
¯γ
η
. Assume that Θ :
= F(S) ∩ VI(C, B) ≠ ∅. Let {x
n
} and {y
n
} be sequence generated by the following iterative
algorithm:
⎧
⎨
⎩
x
1
= x ∈ C chosen arbitrary,
y
n
= β
n
γ f (x
n
)+(I − β
n
A)P
C

(Sx
n
− α
n
BSx
n
),
x
n+1
=(1− δ
n
)y
n
+ δ
n
P
C
(Sy
n
− α
n
BSy
n
), ∀n ≥ 1
,
where {δ
n
} and {b
n
} are two sequences in (0, 1) satisfying the following conditions:

(C1) lim
n ® ∞
b
n
=0and

∞
n
=1
β
n
=
∞
,
(C2) {δ
n
} ⊂ [0, b], for some b Î (0, 1) and lim
n ® ∞
|δ
n+1
- δ
n
|=0,
(C3) {a
n
} ⊂ [e, g] ⊂ (0, 2ξ) and lim
n ® ∞
|a
n+1
- a

n
|=0.
Then,{x
n
} converges strongly to z Î Θ, which is the unique solution of the variational
inequality

γ f (z) − Az, x − z

≤ 0, ∀x ∈ 
.
(3:59)
Proof.PutF(x, y)= = D =0forallx, y Î C and l
n
=1foralln ≥ 1inTheorem
3.1, we get u
n
= x
n
. Hence, {x
n
} converges strongly to z Î Θ, which is the unique solu-
tion of the variational inequality (3.59). □
Onjai-uea et al. Fixed Point Theory and Applications 2011, 2011:32
/>Page 18 of 20
Corollary 3.3. [12]LetCbeanonemptyclosedconvexsubsetofarealHilbertspace
H and let F be bifunction from C × Ctoℝ satisfying (H1)-(H5). Let S : C ® Cbea
nonexpansive mapping and let f : C ® C be a contraction map ping with h Î (0, 1).
Assume that Θ := F(S) ∩ EP(F) ≠ ∅. Let {x
n

}, {y
n
} and {u
n
} be sequence generated by
the following iterative algorithm:
⎧
⎨
⎩
x
1
= x ∈ C chosen arbitrary,
y
n
= β
n
f (x
n
)+(1− β
n
)ST
F
λ
n
x
n
,
x
n+1
=(1− δ

n
)y
n
+ δ
n
Sy
n
, ∀n ≥ 1
,
(3:60)
where {δ
n
} and {b
n
} are two sequences in (0, 1) and {l
n
} ⊂ (0, ∞) satisfying the follow-
ing conditions:
(C1) lim
n ® ∞
b
n
=0and

∞
n
=1
β
n
=

∞
,
(C2) {δ
n
} ⊂ [0, b], for some b Î (0, 1) and lim
n ® ∞
|δ
n+1
- δ
n
|=0,
(C3) lim
n ® ∞
|l
n+1
- l
n
|=0.
Then,{x
n
} converges strongly to z Î Θ.
Proof. Put  = D =0,g =1,A = I and a
n
= 0 in Theorem 3.1. Then, we have P
C
(Su
n
)
= Su
n

and P
C
(Sy
n
)=Sy
n
. Hence, {x
n
} generated by ( 3.60) converges strongly to z Î Θ.
□
Acknowledgements
This research was partially supported by the Research Fund, Rajamangala University of Technology Rattanakosin. The
first author was supported by the ‘Centre of Excellence in Mathematics’, the Commission on High Education, Thailand
for Ph.D. program at King Mongkuts University of Technology Thonburi (KMUTT). The second author was supported
by Rajamangala University of Technology Rattanakosin Research and Development Institute, the Thailand Research
Fund and the Commission on Higher Education under Grant No. MRG5480206. The third author was supported by
the NRU-CSEC Project No. 54000267. Helpful comments by anonymous referees are also acknowledged.
Author details
1
Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (Kmutt), Bangkok
10140, Thailand
2
Department of Mathematics, Faculty of Liberal Arts, Rajamangala University of Technology
Rattanakosin (Rmutr), Bangkok 10100, Thailand
3
Centre of Excellence in Mathematics, Che, Si Ayuthaya Road, Bangkok
10400, Thailand
Authors’ contributions
All authors contribute equally and significantly in this research work. All authors read and approved the final
manuscript.

Competing interests
The authors declare that they have no competing interests.
Received: 13 January 2011 Accepted: 11 August 2011 Published: 11 August 2011
References
1. Goebeland, K, Kirk, WA: Topics in Metric Fixed Point Theory. Cambridge University Press, Cambridge (1990)
2. Takahashi, W: Nonlinear Functional Analysis. Yokohama Publishers, Yokohama (2000)
3. Peng, JW, Yao, JC: A new hybrid-extragradient method for generalized mixed equilibrium problems and fixed point
problems and variational inequality problems. Taiwanese J Math. 12, 1401–1433 (2008)
4. Peng, JW, Yao, JC: Two extragradient method for generalized mixed equilibrium problems, nonexpansive mappings
and monotone mappings. Comput Math Appl. 58, 1287–1301 (2009). doi:10.1016/j.camwa.2009.07.040
5. Marino, G, Xu, HK: A general iterative method for nonexpansive mapping in Hilbert spaces. J Math Anal Appl. 318,
43–52 (2006). doi:10.1016/j.jmaa.2005.05.028
6. Takahashi, W, Toyoda, M: Weak convergence theorems for nonexpansive mappings and monotone mappings. J Optim
Theory Appl. 118, 417–428 (2003). doi:10.1023/A:1025407607560
7. Yamada, I: The hybrid steepest descent method for the variational inequality problem of the intersection of fixed point
sets of nonexpansive mappings. In: Butnariu D, Censor Y, Reich S (eds.) Inherently Parallel Algorithm for Feasibility and
Optimization. pp. 473–504. Elsevier, Amsterdam (2001)
Onjai-uea et al. Fixed Point Theory and Applications 2011, 2011:32
/>Page 19 of 20
8. Su, Y, Shang, M, Qin, X: An iterative method of solution for equilibrium and optimization problems. Nonlinear Anal Ser
A Theory Methods Appl. 69, 2709–2719 (2008). doi:10.1016/j.na.2007.08.045
9. Takahashi, S, Takahashi, W: Strong convergence theorems for a generalized equilibrium problem and a nonexpansive
mappings in a Hilbert space. Nonlinear Anal Ser A Theory Methods Appl. 69, 1025–1033 (2008). doi:10.1016/j.
na.2008.02.042
10. Chantarangsi, W, Jaiboon, C, Kumam, P: A viscosity hybrid steepest descent method for generalized mixed equilibrium
problems and variational inequalities, for relaxed cocoercive mapping in Hilbert spaces. Abstr Appl Anal 2010,39
(2010). Article ID 390972
11. Jaiboon, C: The hybrid steepest descent method for addressing fixed point problems and system of equilibrium
problems. Thai J Math. 8(2):275–292 (2010)
12. Jung, JS: Strong convergence of composite iterative methods for equilibrium problems and fixed point problems. Appl

Math Comput. 213, 498–505 (2009). doi:10.1016/j.amc.2009.03.048
13. Rockafellar, RT: On the maximality of sums of nonlinear monotone operators. Trans Am Math Soc. 149,75–88 (1970).
doi:10.1090/S0002-9947-1970-0282272-5
14. Opial, Z: Weak convergence of the sequence of successive approximations for nonexpansive mappings. Bull Am Math
Soc. 73, 595–597 (1967)
15. Ceng, LC, Yao, JC: A hybrid iterative scheme for mixed equilibrium problems and fixed point problems. J Comput Appl
Math. 214, 186–201 (2008). doi:10.1016/j.cam.2007.02.022
16. Xu, HK: Viscosity approximation methods for nonexpansive mappings. J Math Anal Appl. 298, 279–291 (2004).
doi:10.1016/j.jmaa.2004.04.059
doi:10.1186/1687-1812-2011-32
Cite this article as: Onjai-uea et al.: A relaxed hybrid steepest descent method for common solutions of
generalized mixed equilibrium problems and fixed point problems. Fixed Point Theory and Applications 2011
2011:32.
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