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RESEARC H Open Access
Strong convergence theorem for amenable
semigroups of nonexpansive mappings and
variational inequalities
Hossein Piri
*
and Ali Haji Badali
* Correspondence: h.

Department of Mathematics,
University of Bonab, Bonab 55517-
61167, Iran
Abstract
In this paper, using strongly monotone and lipschitzian operator, we introduce a
general iterative process for finding a common fixed point of a semigroup of
nonexpansive mappings, with respect to strongly left regular sequence of means
defined on an appropriate space of bounded real-valued functions of the
semigroups and the set of solutions of variational inequality for b-inverse strongly
monotone mapping in a real Hilbert space. Under suitable conditions, we prove the
strong convergence theorem for approximating a common element of the above
two sets.
Mathematics Subject Classification 2000: 47H09, 47H10, 43A07, 47J25
Keywords: projection, common fixed point, amenable semigroup, iterative process,
strong convergence, variational inequality
1 Introduction
Throughout this paper, we assume that H is a real Hilbert space with inner product
and norm are denoted by 〈.,.〉 and || . ||, respectively, and let C be a nonempty closed
convex subset of H.AmappingT of C into itself is called nonexpansive if || Tx - Ty
||≤|| x - y ||, for all x, y Î H.ByFix(T), we denote the set of fixed points of T (i.e., Fix
(T)={x Î H : Tx = x}), it is well known that Fix(T) is closed and convex. Recall that a
self-mapping f : C ® C is a contraction on C if there exists a constant a Î [0 , 1) such

that || f(x)-f(y)||≤ a || x - y || for all x, y Î C.
Let B : C ® H be a mapping. The variational inequality problem, denoted by VI(C,
B), is to fined x Î C such that

Bx, y − x

≥ 0
,
(1)
for all y Î C. The variational inequality problem has been extensively studied in lit-
erature. See, for example, [1,2] and the references therein.
Definition 1.1 Let B : C ® H be a mapping. Then B
(1) is called h-strongly monotone if there exists a positive constant h such that

Bx − By, x − y

≥ η  x − y 
2
, ∀x, y ∈ C
,
Piri and Badali Fixed Point Theory and Applications 2011, 2011:55
/>© 2011 Piri and Badali; licens ee Springer. This is an Open Access articl e distributed under the terms of the Creative Commons
Attribution License (http://crea tivecommons.org/licens es/by/2.0), which permits unrestricted use, distribution, and re prod uction in
any medium, provide d the origin al work is properly cited.
(2) is called k-Lipschitzian if there exist a positive constant k such that
 Bx −−B
y
≤ k  x −
y
, ∀x,

y
∈ C
,
(3) is called b-inverse strongly monotone if there exists a positive real number b >0
such that

Bx − By, x − y

≥ β  Bx − By 
2
, ∀x, y ∈ C
.
It is obvious that any b-inverse strongly monotone mapping B is
1
β
-Lipschitzian.
Moudafi [3] introduced the viscosity approximation method for fixed point of nonex-
pansive mappings (see [4] for further developments in both Hilbert and Banach
spaces). Starting with an arbitrary initial x
0
Î H, define a sequence {x
n
} recursively by
x
n+1
=
(
1 − α
n
)

Tx
n
+ α
n
f
(
x
n
)
, n ≥ 0
,
(2)
where a
n
is sequence in (0, 1). Xu [4,5] proved that under certain appropriate condi-
tions on {a
n
}, the sequences {x
n
} generated by (2) strongly converges to the unique
solution x* in Fix(T) of the variational inequality:

(
f − I
)
x
∗
, x − x
∗
≤0, ∀x ∈ Fix

(
T
).
Let A is strongly positive operator on C. That is, there is a constant
¯
γ
>
0
with the
property that
Ax, x≥ ¯
γ
 x 
2
, ∀x ∈ C
.
In [5], it is proved that the sequence {x
n
} generated by the iterative method bellow
with initial guess x
0
Î H chosen arbitrarily,
x
n+1
=
(
I − α
n
A
)

Tx
n
+ α
n
u, n ≥ 0
,
(3)
converges strongly to the unique solution of the minimization problem
min
x∈Fix
(
T
)
1
2

Ax, x

−

x, b

,
where b is a given point in H.
Combining the iterati ve method (2) and (3), M arino and Xu [6] consider the fo llow-
ing iterative method:
x
n+1
=
(

I − α
n
A
)
Tx
n
+ α
n
γ f
(
x
n
)
, n ≥ 0
,
(4)
it is proved that if the sequence {a
n
} of parameters satisfies the following conditions:
(C
1
) a
n
® 0,
(C
2
)
∞

n=0

α
n
=
∞
,
C
3
) either
∞

n
=
0
| α
n+1
− α
n
| <
∞
or
lim
n
→∞
α
n+1
α
n
=
1
.

then, the sequence {x
n
} generated by (4) converges strongly, as n ® ∞, to the unique
solution of the variational inequality:

(γ f − A)x
∗
, x − x
∗

≤ 0, ∀x ∈ Fix(T)
,
Piri and Badali Fixed Point Theory and Applications 2011, 2011:55
/>Page 2 of 16
which is the optimality condition for minimization problem
min
x∈Fix
(
T
)
1
2

Ax, x

− h(x)
,
where h is a potential function for gf (i.e., h’(x)=gf(x), for all x Î H). Some people
also study the application of the iterative method (4) [7,8].
Yamada [9] introduce the following hybrid iterative method for solving the varia-

tional inequality:
x
n+1
= Tx
n
− μα
n
F
(
Tx
n
)
, n ∈ N
,
(5)
where F is k-Lipschitzian and h-strongly monotone ope rator with k>0, h >0,
0 <μ<
2η
k
2
, then he proved that if {a
n
} satisfying appropriate conditions, then {x
n
}
generated by (5) converges strongly to the unique solution of the variational inequality:

Fx
∗
, x − x

∗

≥ 0, ∀x ∈ Fix(T)
.
In 2010, Tian [ 10] combined the iterative (4) with the iterative method (5) and con-
sidered the iterative methods:
x
n+1
=
(
I − μα
n
F
)
Tx
n
+ α
n
γ f
(
x
n
)
, n ≥ 0
,
(6)
andheprovethatifthesequence{a
n
} of parameters satisfies the con ditions (C
1

),
(C
2
), and (C
3
), then the sequences {x
n
} generated by (6) converges strongly to the
unique solution x* Î Fix(T) of the variational inequality:

(
μF − γ f
)
x
∗
, x − x
∗
≥0, ∀x ∈ Fix
(
T
).
In this paper motivated and inspired by Atsushiba and Takahashi [11], Ceng and Yao
[12], Kim [13], Lau et al. [14], Lau et al [15], Marino and Xu [6], Piri and Vaezi [16],
Tian [10], Xu [5] and Yamada [9], we introduce the following general iterative algo-
rithm: Let x
0
Î C and

y
n

= β
n
x
n
+(1− β
n
)P
C
(x
n
− δ
n
Bx
n
),
x
n+1
= α
n
γ f (x
n
)+(I − α
n
μF)T
μ
n
y
n
, n ≥ 0
.

(7)
where P
C
isametricprojectionofHontoC,Bisb-inverse strongly monotone,  =
{T
t
: t Î S} is a nonexpansive semigroup on H such that the set
F = Fix
(
ϕ
)
∩ VI
(
C, B
)
= ∅
,
, X isasubspaceofB(S)suchthat1Î X and the mapping t
® 〈T
t
x, y〉 is an element of X for each x, y Î H, and {μ
n
} is a sequence of means on X.
Our purpose in this paper is to introduce this general iterative algorithm for approxi-
mating a common element of the set of co mmon fixed point of a semigroup of nonex-
pansive mappings and the set of solutions of variational inequality for b-inverse
strongly monotone mapping which solves some variatio nal inequality. We will prove
that if {μ
n
} is left regular and the sequences {a

n
}, {b
n
}, and {δ
n
} of parameters satisfies
appropriate conditions, then the sequences {x
n
}and{y
n
} generated by (7) conve rges
strongly to the unique solution
x
∗
∈
F
of the variational inequalities:


(μF − γ f )x
∗
, x − x
∗

≥ 0, ∀x ∈
F
,

Bx
∗

, y − x
∗

≥ 0 ∀y ∈ C.
Piri and Badali Fixed Point Theory and Applications 2011, 2011:55
/>Page 3 of 16
2 Preliminaries
Let S be a semigroup and let B(S) be the space of all bounded real-valued functions
defined on S with supremum norm. For s Î S and f Î B(S), we define elements l
s
f and
r
s
f in B(S)by
(
l
s
f
)(
t
)
= f
(
st
)
,
(
r
s
f

)(
t
)
= f
(
ts
)
, ∀t ∈ S
.
Let X beasubspaceofB(S)containing1andletX* be its topological dual. An ele-
ment μ of X* issaidtobeameanonX if || μ || = μ(1) = 1. We often write μ
t
(f(t))
instead of μ(f)forμ Î X*andf Î X.LetX be left invariant (resp. right invariant), i.e.,
l
s
(X) ⊂ X (resp. r
s
(X) ⊂ X) for each s Î S.Ameanμ on X is said to be left invariant
(resp. right invariant) if μ(l
s
f)=μ(f)(resp.μ(r
s
f)=μ(f)) for each s Î S and f Î X. X is
said to be left (resp. right) amenable if X has a left (resp. right) invariant mean. X is
amenable if X is both left and right amenable. As is well known, B(S)isamenable
when S is a co mmutative semigroup, see [ 15]. A net {μ
a
} of means on X is said to be
strongly left regular if

lim
α
 l
s
∗
μ
α
− μ
α
=0,
for each s Î S, where
l
∗
s
is the adjoint operator of l
s
.
Let S be a se migroup and let C be a nonempty closed and conve x subset of a reflex-
ive Banach space E.Afamily ={T
t
: t Î S}ofmappingfromC into itself is said to
be a nonexpansive semigroup on C if T
t
is nonexpansive and T
ts
= T
t
T
s
for each t, s Î

S.ByFix(), we denote the set of common fixed points of , i.e.,
Fix(ϕ)=

t∈S
{x ∈ C : T
t
x = x}
.
Lemma 2.1 [15]Let S be a semigroup and C be a nonempty closed convex subset of a
reflexive Banach s pace E. Let  ={T
t
: t Î S} be a nonexp ansive semigroup on H such
that {T
t
x : t Î S} is bounded for some x Î C, let X be a s ubspace of B(S) such that 1 Î
X and the mapping t ® 〈T
t
x, y*〉 is an element of X for each x Î C and y* Î E*, and μ
is a mean on X. If we write T
μ
x instead of

T
t
xdμ(t
)
, then the followings hold .
(i) T
μ
is non-expansive mapping from C into C.

(ii) T
μ
x = x for each x Î Fix().
(iii)
T
μ
x ∈ co{T
t
x : t ∈ S
}
for each x Î C.
Let C be a nonempty subset of a Hilbert space H and T : C ® H a mapping. Then T
is said to be demiclosed at v Î H if, for any sequ ence {x
n
}inC, the following implica-
tion holds:
x
n
→ u ∈ C, Tx
n
→ vim
p
l
y
Tu = v
,
where ® (resp. ⇀) denotes strong (resp. weak) convergence.
Lemma 2.2 [17]LetCbeanonemptyclosedconvexsubsetofaHilbertspaceHand
suppose that T : C ® H is nonexpansive. Then, the mapping I - T is demiclosed at zero.
Lemma 2.3 [18]For a given x Î H, y Î C,

y = P
C
x ⇔

y − x, z − y

≥ 0, ∀z ∈ C
.
Piri and Badali Fixed Point Theory and Applications 2011, 2011:55
/>Page 4 of 16
It is well known that P
C
is a firmly nonexpansive mapping of H onto C and satisfies
 P
C
x − P
C
y 
2
≤

P
C
x − P
C
y, x − y

, ∀x, y ∈
H
(8)

Moreover, P
C
is characterized by the following properties: P
C
x Î C and for all x Î H,
y Î C,

x − P
C
x, y − P
C
x

≤ 0
.
(9)
It is easy to see that (9) is equivalent to the following inequality
 x −
y

2
≥ x − P
C
x 
2
+ 
y
− P
C
x 

2
.
(10)
Using Lemma 2.3, one can see that the variational inequality (24) is equivalent to a
fixed point problem.
It is easy to see that the following is true:
u
∈ VI
(
C, B
)
⇔ u = P
C
(
u − λBu
)
, λ>0
.
(11)
A set-valued mapping U : H ® 2
H
is call ed monotone if for all x, y Î H, f Î Ux and
g Î Uy imply 〈x-y, f - g〉 ≥ 0. A monotone mapping U : H ® 2
H
is maximal if the
graph of G(U)ofU is not properly contained in the graph of any other monotone
mapping. It is known that a monotone mapping U is maximal if and only if for (x, f) Î
H×H, 〈x - y, f - g〉 ≥ 0forevery(y, g) Î G(U) implies that f Î Ux.LetB beamono-
tone mapping of C into H and let N
C

x be the normal cone to C at x Î C, that is, N
C
x
={y Î H : 〈z - x, y〉 ≤ 0, ∀z Î C} and define
Ux =

Bx + N
C
x, x ∈ C
,
∅ x /∈ C
.
(12)
Then U is the maximal monotone and 0 Î Ux if and only if x Î VI(C, B); see [19].
The following lemma is well known.
Lemma 2.4 Let H be a real Hilbert space. Then, for all x, y Î H
 x − y 
2
≤x 
2
+2

y, x + y

,
.
Lemma 2.5 [5]Let {a
n
} be a sequence of nonnegative real numbers such that
a

n+1
≤
(
1 − b
n
)
a
n
+ b
n
c
n
, n ≥ 0
,
where {b
n
} and {c
n
} are sequences of real numbers satisfying the following conditions:
(i) {b
n
} ⊂ (0, 1),
∞

n
=
0
b
n
=

∞
,
(ii) either
lim sup
n
→∞
c
n
≤ 0
or
∞

n
=
0
| b
n
c
n
| <
∞
.
Then,
lim
n
→∞
a
n
=0
.

As far as we know, the following lemma has been used implicitly in some papers; for
the sake of completeness, we include its proof.
Lemma 2.6 LetHbearealHilbertspaceandFbeak-Lipschitzianandh-strongly
monotone operator with k >0, h >0. Let
0 <μ<
2η
k
2
and
τ
= μ(η −
μk
2
2
)
. Then for
t ∈ (0, min{1,
1
τ
}
)
, I - t
μ
F is contraction with constant 1-tτ.
Piri and Badali Fixed Point Theory and Applications 2011, 2011:55
/>Page 5 of 16
Proof. Notice that
 (I − tμF)x − (I − tμF)y 
2
=


(I − tμF)x − (I − tμF)y,(I − tμF)x − (I − tμF)y

=  x − y 
2
+ t
2
μ
2
 Fx − Fy 
2
− 2tμ

x − y, Fx − Fy

≤x − y 
2
+ t
2
μ
2
k
2
 x − y 
2
− 2tμη  x − y 
2
≤x − y 
2
+ tμ

2
k
2
 x − y 
2
− 2tμη  x − y 
2
=

1 − 2tμ

η −
μk
2
2

 x − y 
2
=(1− 2tτ )  x − y 
2
≤
(
1 − tτ
)
2
 x − y 
2
.
It follows that


(
I − tμF
)
x −
(
I − tμF
)
y ≤
(
1 − tτ
)
 x − y 
,
and hence I - tμF is contractive due to 1 - tτ Î (0, 1). □
Notation Throughout the rest of this paper, F will denote a k-Lipschitzian and h-
strongly monotone operator of C into H with k>0, h >0, f is a contraction on C with
coefficient 0 < a <1. We will al so always use g to mean a number in
(0,
τ
α
)
,where
τ = μ(η −
μk
2
2
)
and
0 <μ<
2

η
k
2
. The open ball of radius r centered at 0 is denoted by
B
r
and for a subset D of H,by
co
D
, we denote the closed convex hull of D. For ε >0
and a mapping T : D ® H,weletF
ε
(T; D) be the set of ε-approximate fixed points of
T, i.e., F
ε
(T; D)={x Î D :||x - T
x
|| ≤ ε }. Weak convergence is denoted by ⇀ and
strong convergence is denoted by ®.
3 Main results
Theorem 3.1 Let S be a semigroup, C a nonempty closed convex subset of real Hilbert
space H an d B : C ® Hbeab-inverse strongly monotone. Let  ={T
t
: t Î S} be a
nonexpansive semigroup of C into itself such that
F = Fix
(
ϕ
)
∩ VI

(
C, B
)
= ∅
,
, Xaleft
invariant subspace of B(S) such that 1 Î X, and the function t ® 〈T
t
x, y〉 is an element
of X fo r each x Î CandyÎ H,{μ
n
} a left regular sequence of means on X such that

∞
n
=1
 μ
n+1
− μ
n
<
∞
. Let {a
n
} and {b
n
} be sequences in (0, 1) and {δ
n
} be a
sequence in [a, b], where 0 <a<b<2b. Suppose the following conditions are satisfied.

(B
1
) lim
n®∞
a
n
= 0, lim
n®∞
b
n
=0,
(B
2
)

∞
n
=1
α
n
=
∞
,
(B
3
)

∞
n
=1

| α
n+1
− α
n
| <
∞
,

∞
n
=1
| β
n+1
− β
n
| <
∞
,

∞
n
=1
| δ
n+1
− δ
n
| <
∞
.
If {x

n
} and {y
n
} be generated by x
0
Î C and

y
n
= β
n
x
n
+(1− β
n
)P
C
(x
n
− δ
n
Bx
n
),
x
n+1
= α
n
γ f (x
n

)+(I − α
n
μF)T
μ
n
y
n
, n ≥ 0
.
Then,{x
n
} and {y
n
} converge strongly, as n ® ∞, to
x
∗
∈
F
, which is a unique solution
of the variational inequalities:
Piri and Badali Fixed Point Theory and Applications 2011, 2011:55
/>Page 6 of 16


(μF − γ f )x
∗
, x − x
∗

≥ 0, ∀x ∈

F
,

Bx
∗
, y − x
∗

≥ 0 ∀y ∈ C.
Proof.Since{a
n
} satisfies in condition (B
1
), we may assume, with no loss of general-
ity, that
α
n
< min{1,
1
τ
}
.SinceB is b-inverse strongly monotone and δ
n
<2b,forany
x, y Î C, we have
 (I − δ
n
B)x − (I − δ
n
B)y 

2
=  (x − y) − δ
n
(Bx − By) 
2
=  x − y 
2
− 2δ
n

x − y, Bx − By

+ δ
2
n
 Bx − By 
2
≤x − y 
2
− 2δ
n
β  Bx − By 
2
+ δ
2
n
 Bx − By 
2
=  x − y 
2

+ δ
n
(δ
n
− 2β)  Bx − By 
2
≤x −
y

2
.
It follows that

(
I − δ
n
B
)
x −
(
I − δ
n
B
)
y ≤  x − y 
.
(13)
Let
p
∈

F
, in the context of the variational inequality problem, the characterization
of projection (11) implies that p = P
C
(p - δ
n
B
p
). Using (13), we get
 y
n
− p  = β
n
x
n
+(1− β
n
)P
C
(x
n
− δ
n
Bx
n
) − p 
=  β
n
[x
n

− p]+(1− β
n
)[P
C
(x
n
− δ
n
Bx
n
) − P
C
(p − δ
n
Bp)] 
≤ β
n
 x
n
− p  +(1− β
n
)  P
C
(x
n
− δ
n
Bx
n
) − P

C
(p − δ
n
Bp)

≤ β
n
 x
n
− p  +
(
1 − β
n
)
 x
n
− p  =  x
n
− p  .
(14)
We claim that {x
n
} is bounded. Let
p
∈
F
, using Lemma 2.6 and (14), we have
 x
n+1
− p  =  α

n
γ f (x
n
)+(I − α
n
μF)T
μ
n
y
n
− p 
=  α
n
γ f (x
n
)+(I − α
n
μF)T
μ
n
y
n
− (I − α
n
μF)p − α
n
μFp 
≤ α
n
 γ f (x

n
) − μFp  +  (I − α
n
μF)T
μ
n
y
n
− (I − α
n
μF)p 
≤ α
n
 γ f (x
n
) − γ f (p) 
+ α
n
 γ f (p) − μFp  +(1− α
n
τ )  T
μ
n
y
n
− p 
≤ α
n
γα  x
n

− p  + α
n
 γ f (p) − μFp  +(1− α
n
τ )  y
n
− p 
≤ α
n
γα  x
n
− p  + α
n
 γ f (p) − μFp  +(1− α
n
τ )  x
n
− p

=(1− α
n
(τ − γα))  x
n
− p  + α
n
 γ f (p) − μFp 
≤ max{ x
n
− p ,
(

τ − γα
)
−1
 γ f
(
p
)
− μFp }.
By induction we have,
 x
n
− p ≤ max{
(
τ − γα
)
−1
 γ f
(
p
)
− μFp ,  x
0
− p } = M
0
.
Hence, {x
n
} is bounded and also {y
n
}and{f(x

n
)} are bounded. Set D ={y Î H :||y -
p||≤ =M
0
}. We remark that D is -invariant bounded closed convex set and {x
n
}, {y
n
}
⊂ D. Now we claim that
lim sup
n→∞
sup
y
∈D
 T
μ
n
y − T
t
T
μ
n
y  =0, ∀t ∈ S
.
(15)
Let ε >0. By [[20], Theorem 1.2], there exists δ >0 such that
coF
δ
(

T
t
; D
)
+ B
δ
⊂ F
ε
(
T
t
; D
)
, ∀t ∈ S
.
(16)
Piri and Badali Fixed Point Theory and Applications 2011, 2011:55
/>Page 7 of 16
Also by [[20], Corollary 1.1], there exists a natural number N such that





1
N +1
N

i=0
T

t
i
s
y − T
t

1
N +1
N

i=0
T
t
i
s
y






≤ δ
,
(17)
for all t, s Î S and y Î D. Let t Î S. Since {μ
n
} is strongly left regular, there exists N
0
Î N such that

 μ
n
− l
∗
t
i
μ
n
≤
δ
(M
0
+p)
for n ≥ N
0
and i = 0, 1, 2, , N. Then we have
sup
y∈D





T
μ
n
y −

1
N +1

N

i=0
T
t
i
s
ydμ
n
s





=sup
y∈D
sup
z=1





T
μ
n
y, z−



1
N +1
N

i=0
T
t
i
s
ydμ
n
s, z






=sup
y∈D
sup
z=1





1
N +1
N


i=0
(μ
n
)
s
T
s
y, z−
1
N +1
N

i=0
(μ
n
)
s
T
t
i
s
y, z





≤
1

N +1
N

i=0
sup
y∈D
sup
z=1
| (μ
n
)
s
T
s
y, z−(l
∗
t
i
μ
n
)
s
T
s
y, z|
≤ max
i=1
,
2
,

N
 μ
n
− l
∗
t
i
μ
n
 (M
0
+  p ) ≤ δ, ∀n ≥ N
0
.
(18)
By Lemma 2.1, we have

1
N +1
N

i=0
T
t
i
s
ydμ
n
s ∈ co


1
N +1
N

i=0
T
t
i
(T
s
y):s ∈ s

.
(19)
It follows from (16), (17), (18), and (19) that
T
μ
n
(y) ∈ co

1
N +1
N

i=0
T
t
i
s
(y):s ∈ S


+ B
δ
⊂ coF
δ
(
T
t
; D
)
+ B
δ
⊂ F
ε
(
T
t
; D
)
,
for all y Î D and n ≥ N
0
. Therefore,
lim sup
n→∞
sup
y
∈D
 T
t

(T
μ
n
y) − T
μ
n
y ≤ε
.
Since ε > 0 is arbitrary, we get (15). In this stage, we will show
lim
n
→
∞
 x
n
− T
t
x
n
 =0, ∀t ∈ S
.
(20)
Let t Î S and ε > 0. Then, there exists δ > 0, which satisiies (16). From lim
n®∞
a
n
=
0 and (15) there exists N
1
Î N such that

α
n
≤
δ
(
τ +μk
)
M
0
and
T
μ
n
y
n
∈ F
δ
(T
t
; D
)
,foralln
≥ N
1
. By Lemma 2.6 and (14), we have
α
n
γ f (x
n
) − μFT

μ
n
y
n

≤ α
n
(γ  f (x
n
) − f (p)  +  γ f (p) − μFp  +  μFp − μFT
μ
n
y
n

)
≤ α
n
(γα  x
n
− p  +  γ f (p) − μFp  +μk  y
n
− p )
≤ α
n
(γαM
0
+(τ − γα)M
0
+ μkM

0
)
≤ α
n
(
τ + μk
)
M
0
≤ δ,
Piri and Badali Fixed Point Theory and Applications 2011, 2011:55
/>Page 8 of 16
for all n ≥ N
1
. Therefore, we have
x
n+1
= T
μ
n
y
n
+ α
n
[γ f (x
n
)+μF(T
μ
n
y

n
)
]
∈ F
δ
(
T
t
; D
)
+ B
δ
⊂ F
ε
(
T
t
; D
)
,
for all n ≥ N
1
. This shows that

x
n
− T
t
x
n


≤ ε, ∀n ≥ N
1
.
Since ε > 0 is arbitrary, we get (20).
Let
Q
= P
F
. Then Q(I - μF + g f) is a contraction of H into itself. In fact, we see that
 Q(I − μF + γ f )x − Q(I − μF + γ f)y 
≤ (I − μF + γ f )x − (I − μF + γ f )y 
≤ (I − μF)x − (I − μF)y  + γ  f (x) − f(y) 
= lim
n→∞





I −

1 −
1
n

μF

x −


I −

1 −
1
n

μF

y




+ γ  f (x) − f(y)

≤ lim
n→∞
(1 − (1 −
1
n
)τ )  x − y  + γα  x − y 
=
(
1 − τ
)
 x − y  +γα  x − y ,
and hence Q(I - μF + g f) is a contraction due to (1 - (τ -ga)) Î (0, 1).
Therefore, by Banachs contraction principal,
P
F

(
I − μF + γ f
)
has a unique fixed
point x*. Then using (9), x* is the unique solution of the variational inequality:

(
γ f − μF
)
x
∗
, x − x
∗
≤0, ∀x ∈ F
.
(21)
We show that
lim sup
n
→∞
γ f (x
∗
) − μFx
∗
, x
n
− x
∗
≤0
.

(22)
Indeed, we can choose a subsequence
{
x
n
i
}
of {x
n
} such that
lim sup
n
→∞
γ f (x
∗
) − μFx
∗
, x
n
− x
∗
 = lim
i→∞
γ f (x
∗
) − μFx
∗
, x
n
i

− x
∗

.
(23)
Because
{x
n
i
}
is bounded, we may assume that
x
n
i
→ z
. In terms of Lemma 2.2 and
(20), we conclude that z Î Fix ().
Now, let us show that z Î VI (C, B). Let w
n
= P
C
(x
n
- δ
n
Bx
n
), it follows from the
definition of {y
n

} that
y
n+1
− y
n

=  β
n+1
x
n+1
+(1− β
n+1
)P
C
(x
n+1
− δ
n+1
Bx
n+1
) − β
n
x
n
− (1 − β
n
)P
C
(x
n

− δ
n
Bx
n
) 
=  β
n+1
(x
n+1
− x
n
)+(β
n+1
− β
n
)x
n
+(1− β
n+1
)P
C
(x
n+1
− δ
n+1
Bx
n+1
)
− (1 − β
n+1

)P
C
(x
n
− δ
n+1
Bx
n
)+(1− β
n+1
)P
C
(x
n
− δ
n+1
Bx
n
) − (1 − β
n
)P
C
(x
n
− δ
n
Bx
n
)


≤ β
n+1
 x
n+1
− x
n
 + | β
n+1
− β
n
| x
n

+(1− β
n+1
)  P
C
(x
n+1
− δ
n+1
Bx
n+1
) − P
C
(x
n
− δ
n+1
Bx

n
) 
+  P
C
(x
n
− δ
n+1
Bx
n
) − P
C
(x
n
− δ
n
Bx
n
) 
+  β
n
P
C
(x
n
− δ
n
Bx
n
) − β

n+1
P
C
(x
n
− δ
n+1
Bx
n
) ]
≤ β
n+1
 x
n+1
− x
n
 + | β
n+1
− β
n
| x
n
 +(1− β
n+1
)  x
n+1
− x
n

+ | δ

n+1
− δ
n
| Bx
n
 +  β
n
P
C
(x
n
− δ
n
Bx
n
) − β
n
P
C
(x
n
− δ
n+1
Bx
n
)
+ β
n
P
C

(x
n
− δ
n+1
Bx
n
) − β
n+1
P
C
(x
n
− δ
n+1
Bx
n
) 
≤ β
n+1
 x
n+1
− x
n
 + | β
n+1
− β
n
| x
n
 +(1− β

n+1
)  x
n+1
− x
n

+ | δ
n+1
− δ
n
| Bx
n
 +β
n
| δ
n+1
− δ
n
| Bx
n
 + | β
n+1
− β
n
| P
C
(x
n
− δ
n+1

Bx
n
) 
= x
n+1
− x
n
 + | β
n+1
− β
n
| x
n
 +(1+β
n
) | δ
n+1
− δ
n
| Bx
n

+ | β
n+1
− β
n
| P
C
(
x

n
− δ
n+1
Bx
n
)
 .
Piri and Badali Fixed Point Theory and Applications 2011, 2011:55
/>Page 9 of 16
Using the last inequality, we get

x
n+1
− x
n

=  α
n
γ f(x
n
)+(I − α
n
μF)T
μ
n
y
n
− α
n−1
γ f(x

n−1
) − (I − α
n−1
μF)T
μ
n−1
y
n−1

=  α
n
γ f(x
n
) − α
n
γ f(x
n−1
)+(α
n
− α
n−1
)γ f(x
n−1
)
+(I − α
n
μF)T
μ
n
y

n
− (I − α
n
μF)T
μ
n−1
y
n−1
+(I − α
n
μF)T
μ
n−1
y
n−1
− (I − α
n−1
μF)T
μ
n−1
y
n−1

≤ α
n
γα  x
n
− x
n−1
 + | α

n
− α
n−1
| γ  f (x
n−1
) 
+(1− α
n
τ )  T
μ
n
y
n
− T
μ
n−1
y
n−1
 + | α
n
− α
n−1
| μ  FT
μ
n−1
y
n−1

≤ α
n

γα  x
n
− x
n−1
 + | α
n
− α
n−1
| γ  f (x
n−1
)  +(1 − α
n
τ )  y
n
− y
n−1

+(1− α
n
τ )  T
μ
n
y
n−1
− T
μ
n−1
y
n−1
 + | α

n
− α
n−1
| μ  FT
μ
n−1
y
n−1

≤ α
n
γα  x
n
− x
n−1
 + | α
n
− α
n−1
| γ  f (x
n−1
)  +(1 − α
n
τ )  x
n
− x
n−1

+(1− α
n

τ ) | β
n
− β
n−1
| x
n−1
 +(1 − α
n
τ )(1 + β
n−1
) | δ
n
− δ
n−1
| Bx
n−1

+(1− α
n
τ ) | β
n
− β
n−1
| P
C
(x
n−1
− δ
n
Bx

n−1
) 
+(1− α
n
τ )  T
μ
n
y
n−1
− T
μ
n−1
y
n−1
 + | α
n
− α
n−1
| μ  FT
μ
n−1
y
n−1
 .
Thus, for some large enough constant M > 0, we have
 x
n+1
− x
n
≤(1 − α

n
(τ − γα))  x
n
− x
n−1

+
[
| α
n
− α
n−1
| + | β
n
− β
n−1
| + | δ
n
− δ
n−1
| +  μ
n
− μ
n−1

]
M
.
Therefore, using condition B
3

and Lemma 2.5, we get
lim
n
→
∞
 x
n+1
− x
n
 =0
.
(24)
Let
p
∈
F
, from (11) and deiinition of {y
n
}, we have
y
n
− p
2
=  β
n
x
n
+(1− β
n
)P

C
(x
n
− δ
n
Bx
n
) − p
2
=  β
n
(x
n
− p)+(1− β
n
)(P
C
(x
n
− δ
n
Bx
n
) − P
C
(p − δ
n
Bp))
2
≤ β

n
 x
n
− p
2
+(1− β
n
)  (x
n
− p) − δ
n
(Bx
n
− Bp))
2
= β
n
 x
n
− p
2
+(1− β
n
)  x
n
− p
2
+ δ
2
n

(1 − β
n
)  Bx
n
− Bp
2
− 2δ
n
(1 − β
n
)x
n
− p, Bx
n
− Bp
≤x
n
− p
2
+ δ
2
n
(1 − β
n
)  Bx
n
− Bp
2
− 2δ
n

(1 − β
n
)β  Bx
n
− Bp
2
=  x
n
− p
2
+ δ
n
(
1 − β
n
)(
δ
n
− 2β
)
 Bx
n
− Bp
2
.
(25)
Using (25), we have
x
n+1
− p

2
=  α
n
γ f (x
n
)+(I − α
n
μF)T
μ
n
y
n
− p
2
=  α
n
(γ f (x
n
) − μFT
μ
n
y
n
)+(T
μ
n
y
n
− p)
2

= α
2
n
 γ f (x
n
) − μFT
μ
n
y
n

2
+  T
μ
n
y
n
− p 
+2α
n
γ f (x
n
) − μFT
μ
n
y
n
, T
μ
n

y
n
− p
≤ α
2
n
 γ f (x
n
) − μFT
μ
n
y
n

2
+  y
n
− p
2
+2α
n
γ f (x
n
) − μFT
μ
n
y
n
, T
μ

n
y
n
− p
2
≤ α
2
n
 γ f (x
n
) − μFT
μ
n
y
n

2
+  x
n
− p
2
+ δ
n
(1 − β
n
)(δ
n
− 2β)  Bx
n
− Bp

2
+2α
n
γ f (x
n
) − μFT
μ
n
y
n
, T
μ
n
y
n
− p
= α
2
n
 γ f (x
n
) − μFT
μ
n
y
n

2
+  x
n

− p
2
+ δ
n
(δ
n
− 2β
n
)  Bx
n
− Bp
2
− δ
n
β
n
(δ
n
− 2β
n
)  Bx
n
− Bp
2
+2α
n
γ f (x
n
) − μFT
μ

n
y
n
, T
μ
n
y
n
− p.
(26)
Piri and Badali Fixed Point Theory and Applications 2011, 2011:55
/>Page 10 of 16
Therefore,
−δ
n
(δ
n
− 2β
n
)  Bx
n
− Bp
2
≤ α
2
n
 γ f (x
n
) − μFT
μ

n
y
n

2
+[ x
n
− p  +  x
n+1
− p ]  x
n+1
− x
n

− δ
n
β
n
(δ
n
− 2β
n
)  Bx
n
− Bp
2
+2α
n
γ f (x
n

) − μFT
μ
n
y
n
, T
μ
n
y
n
− p
.
Hence, using condition B
1
and (24), we get
lim
n
→
∞
 Bx
n
− Bp  =0
.
(27)
From (8), we have
w
n
− p
2
=  P

C
(x
n
− δ
n
Bx
n
) − P
C
(p − δ
n
Bp)
2
≤(x
n
− δ
n
Bx
n
) − (p − δ
n
Bp), w
n
− p
=
1
2
[ (x
n
− δ

n
Bx
n
) − (p − δ
n
Bp)
2
+  w
n
− p
2
−(x
n
− δ
n
Bx
n
) − (p − δ
n
Bp) − (w
n
− p)
2
]
≤
1
2
[ x
n
− p

2
+  w
n
− p
2
−(x
n
− δ
n
Bx
n
) − (p − δ
n
Bp) − (w
n
− p)
2
]
=
1
2
[ x
n
− p
2
+  w
n
− p
2
−x

n
− w
n

2
+2δ
n
x
n
− w
n
, Bx
n
− Bp−δ
2
n
 Bx
n
− Bp
2
].
So we obtain
 w
n
− p
2
≤x
n
− p
2

−x
n
− w
n

2
+2δ
n
x
n
− w
n
, Bx
n
− Bp−δ
2
n
 Bx
n
− Bp
2
.
(28)
It follows from (26) and (28) that
x
n+
− p
2
≤ α
2

n
 γ f(x
n
) − μFT
μ
n
y
n

2
+  y
n
− p
2
+2α
n
γ f (x
n
) − μFT
μ
n
y
n
, T
μ
n
y
n
− p
≤ α

2
n
 γ f(x
n
) − μFT
μ
n
y
n

2
+  β
n
x
n
+(1− β
n
)P
C
(x
n
− δ
n
Bx
n
) − p
2
+2α
n
γ f (x

n
) − μFT
μ
n
y
n
, T
μ
n
y
n
− p
≤ α
2
n
 γ f(x
n
) − μFT
μ
n
y
n

2
+ β
n
 x
n
− p
2

+(1− β
n
)  w
n
− p
2
+2α
n
γ f (x
n
) − μFT
μ
n
y
n
, T
μ
n
y
n
− p
≤ α
2
n
 γ f(x
n
) − μFT
μ
n
y

n

2
+ β
n
 x
n
− p
2
+(1− β
n
)  x
n
− p
2
− (1 − β
n
)  x
n
− w
n

2
+2δ
n
(1 − β
n
)x
n
− w

n
, Bx
n
− Bp−δ
2
n
(1 − β
n
)  Bx
n
− Bp
2
+2α
n
γ f (x
n
) − μFT
μ
n
y
n
, T
μ
n
y
n
− p.
Which implies that
 x
n

− w
n

2
≤ α
2
n
 γ f (x
n
) − μFT
μ
n
y
n

2
+[ x
n
− p  +  x
n+1
− p ]  x
n+1
− x
n

+ β
n
 x
n
− w

n

2
+2δ
n
(1 − β
n
)  x
n
− w
n
 Bx
n
− Bp 
− δ
2
n
(1 − β
n
)  Bx
n
− Bp
2
+2α
n
γ f (x
n
) − μFT
μ
n

y
n
, T
μ
n
y
n
− p.
Piri and Badali Fixed Point Theory and Applications 2011, 2011:55
/>Page 11 of 16
Therefore, using condition B
1
, (24) and (27), we get
lim
n
→
∞
 x
n
− w
n
 =0
.
(29)
Let U : H ®2
H
be a set-valued mapping is defined by
Ux =

Ax + N

C
x, x ∈ C
,
∅, x /∈ C
.
where N
C
x is the normal cone to C at x Î C.SinceB is relaxed, b-inverse strongly
mono tone. Thus, U is maximal monotone see [19]. Let (x, y) Î G (U), hence y - Bx Î
N
C
x.Sincew
n
= P
C
( x
n
- ζ
n
Bx
n
)therefore,〈x - w
n
, y - Bx〉 ≥ 0. On the other hand
from w
n
= P
C
(x
n

- ζ
n
Bx
n
), we have
x − w
n
, w
n
−
(
x
n
− δ
n
Bx
n
)
≥0,
that is

x − w
n
,
w
n
− x
n
δ
n

+ Bx
n

≥ 0
.
Therefore, we have

x − w
n
i
, y

≥x − w
n
i
, Bx
≥x − w
n
i
, Bx−

x − w
n
i
,
w
n
i
− x
n

i
δ
n
i
+ Bx
n
i
)

=

x − w
n
i
, Bx −
w
n
i
− x
n
i
δ
n
i
− Bx
n
i

= x − w
n

i
, Bx − Bw
n
i
 + x − w
n
i
, Bw
n
i
− Bx
n
i
−

x − w
n
i
,
w
n
i
− x
n
i
δ
n
i

≥x − w

n
i
, Bw
n
i
− Bx
n
i
−

x − w
n
i
,
w
n
i
− x
n
i
δ
n
i

≥x − w
n
i
, Bw
n
i

− Bx
n
i
−x − w
n
i





w
n
i
− x
n
i
δ
n
i




.
Noting that
lim
i→∞
 w
n

i
− x
n
i
 =0,x
n
i
→
z
,
x
n
i
→ z
and B is
1
β
-lipschitzian, we
obtain

x − z,
y
≥ 0
.
Since U is maximal monotone, we have z Î U
-1
0, and hence z Î VI(C, B). Therefore,
z
∈
F

.
Since
x
n
i
→ z
from (21) and (23), we have
lim sup
n
→∞
γ f (x
∗
) − μFx
∗
, x
n
− x
∗
≤0
.
Piri and Badali Fixed Point Theory and Applications 2011, 2011:55
/>Page 12 of 16
Finally, we prove that x
n
® x*asn ® ∞. By Lemmas 2.4, 2.6, and (14), we have
x
n+1
− x
∗


2
=  α
n
γ f (x
n
)+(I − α
n
μF)T
μ
n
y
n
− x
∗

2
=  α
n
γ f (x
n
) − α
n
μFx
∗
+(I − α
n
μF)T
μ
n
y

n
− (I − α
n
μF)x
∗

2
≤(I − α
n
μF)T
μ
n
y
n
− (I − α
n
μF)x
∗

2
+2α
n
γ f (x
n
) − μFx
∗
, x
n+1
− x
∗


≤ (1 − α
n
τ )
2
 y
n
− x
∗

2
+2α
n
γ f (x
n
) − μFx
∗
, x
n+1
− x
∗

≤ (1 − α
n
τ )
2
 y
n
− x
∗


2
+2α
n
γ f (x
n
) − γ f (x
∗
), x
n+1
− x
∗

+2α
n
γ f (x
∗
) − μFx
∗
, x
n+1
− x
∗
.
≤ (1 − α
n
τ )
2
 y
n

− x
∗

2
+ α
n
γα[ x
n
− x
∗

2
+  x
n+1
− x
∗

2
]
+2α
n
γ f (x
∗
) − μFx
∗
, x
n+1
− x
∗
.

≤ (1 − α
n
τ )
2
 x
n
− x
∗

2
+ α
n
γα[ x
n
− x
∗

2
+  x
n+1
− x
∗

2
]
+2α
n
γ f
(
x

∗
)
− μFx
∗
, x
n+1
− x
∗
.
(30)
So from (30), we reach the following
 x
n+1
− x
∗

2
≤
1+α
2
τ
2
− 2α
n
τ + α
n
γα
1 − α
n
γα

 x
n
− x
∗

2
+
2α
n
1 − α
n
γα
γ f (x
∗
) − μFx
∗
, x
n+1
− x
∗

≤ (1 − α
n
2(τ − γα) − α
n
τ
2
1 − α
n
γα

)  x
n
− x
∗

2
+ α
n
2(τ − γα) − α
n
τ
2
1 − α
n
γα
2
2
(
τ − γα
)
− α
n
τ
2
γ f (x
∗
) − μFx
∗
, x
n+1

− x
∗

It follows that
 x
n+1
− x
∗

2
≤
(
1 − b
n
)
 x
n
− x
∗

2
+ b
n
c
n
,
(31)
where
b
n

= α
n
2(τ − γα) − α
n
τ
2
1 − α
n
γα
, c
n
=
2
2
(
τ − γα
)
− α
n
τ
2
γ f (x
∗
) − μFx
∗
, x
n+1
− x
∗


Since a
n
® oand
∞

n
=
0
α
n
=
∞
,wehave
∞

n
=
0
b
n
=
∞
and by (22 ), we get lim sup
n®∞
c
n
≤ 0. Consequently, applying Lemma 2.5, to (31), we conclude that x
n
® x* . Since || y
n

- x*||≤ || x
n
- x* ||, we have y
n
®x*. □
Corollary 3.2 Let {a
n
}, {b
n
}, {δ
n
} and B be as in Theorem 3.1. Let T a nonexpansive
mapping of C into C such that
F = Fix
(
T
)
∩ VI
(
C, B
)
=
∅
. Suppose x
0
Î Hand{x
n
}
and {y
n

} be generated by the iteration algorithm

y
n
= β
n
x
n
+(1− β
n
)P
C
(x
n
− δ
n
Bx
n
),
x
n+1
= α
n
γ f (x
n
)+(I − α
n
μF)
n
n

−1

n − 1
n
0
T(t)y
n
dt, n ≥ 0
.
Then {x
n
} and {y
n
} convergence strongly to x* which is the unique solution of the sys-
tems of variational inequalities:
Piri and Badali Fixed Point Theory and Applications 2011, 2011:55
/>Page 13 of 16

(μF − γ f)x
∗
, x − x
∗
≥0, ∀x ∈ F
,
Bx
∗
, y − x
∗
≥0 ∀y ∈ C,
Proof.Take

λ
n
=
n−1
n
,forn Î N,wedefine
μ
n
(f )=
1
λ
n

λ
n
0
f (t)d
t
for each f Î C(ℝ
+
),
where C(ℝ
+
) denotes the space of all real-valued bounded continuous functions on R
+
with supremum norm. Then, { μ
n
} is regular sequence of means on C(ℝ
+
) such that

 μ
n+1
− μ
n
≤2

1 −
λ
n
λ
n+1

for more details, see [21]. Further, for each y Î C, we have
T
μ
n
y =
1
λ
n
λ
n

0
T(t)ydt
.
On the other hand
∞

n=1


μ
n+1
− μ
n

≤ 2
∞

n=1

λ
n+1
− λ
n
λ
n+1

=2
∞

n=1

n
n+1
−
n−1
n
n
n+1


=2
∞

n
=1
1
n
2
< ∞
Now, apply Theorem 3.1 to conclude the result. □
Corollary 3.3 Let S, ,X,{μ
n
},
F
,{a
n
}, {b
n
}, an d {δ
n
} be as in Theorem 3.1. Let A be
a strongly positive bounded linear operator with coefficient
¯
γ
>
0
, ζ a number in
(0,
¯τ

α
)
,
where
¯τ = ¯μ( ¯γ −
¯μA
2
2
)
and
0 < ¯μ<
2 ¯γ

A

2
. If {x
n
} and {y
n
} are generated by x
0
Î C and

y
n
= β
n
x
n

+(1− β
n
)P
C
(x
n
− δ
n
Ax
n
),
x
n+1
= α
n
γ f (x
n
)+(I − α
n
¯μA)T
μ
n
y
n
, n ≥ 0
.
Then,{x
n
} and {y
n

} converge strongly, as n ® ∞, to
x
∗
∈
F
, which is a unique solution
of the variational inequalities:

(μF − γ f)x
∗
, x − x
∗
≥0, ∀x ∈ F
,
Ax
∗
, y − x
∗
≥0 ∀y ∈ C.
Proof. Because A is strongly positive bounded linear operator on H with coefficient
¯
γ
,
we have
Ax − A
y
, x −
y
≥ ¯γ  x −
y


2
.
Therefore, A is
¯
γ
-strongly monotone.
On the other hand

A
x −
Ay
≤
A
 x −
y

.
Therefore,
¯γ

A

2
 Ax − Ay
2
≤Ax − Ay, x − y
.
Piri and Badali Fixed Point Theory and Applications 2011, 2011:55
/>Page 14 of 16

Hence, A is
¯γ

A

2
-inverse strongly monotone. Now apply Theorem 3.1 to conclude the
result. □
Corol lary 3.4 Let {a
n
}, {b
n
} and B be as in Theorem 3.1. Le t u, x
0
Î Cand{x
n
} and
{y
n
} be generated by the iterative algorithm

y
n
= β
n
x
n
+(1− β
n
)P

C
(x
n
− δ
n
Bx
n
),
x
n+1
= α
n
u +(I − α
n
μF)y
n
, n ≥ 0
.
Then {x
n
} and {y
n
} convergence strongly to x* which is the unique solution of the sys-
tems of variational inequalities:

(μF − γ f )x
∗
, x − x
∗
≥0, ∀x ∈ F

,
Bx
∗
, y − x
∗
≥0 ∀y ∈ C.
Proof. It is sufficient to take
f =
1
γ
u
and  ={I} in Theorem 3.1. □
Acknowledgements
The authors are extremely grateful to the reviewers for careful reading, valuable comment and suggestions that
improved the content of this paper. This work is supported by University of Bonab under Research Projection 100-22.
Authors’ contributions
The authors have equitably contributed in obtaining the new results presented in this article. All authors read and
approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Received: 25 November 2010 Accepted: 19 September 2011 Published: 19 September 2011
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Cite this article as: Piri and Badali: Strong convergence theorem for amenable semigroups of nonexpansive
mappings and variational inequalities. Fixed Point Theory and Applications 2011 2011:55.
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