Hindawi Publishing Corporation
Fixed Point Theory and Applications
Volume 2011, Article ID 620284, 17 pages
doi:10.1155/2011/620284
Research Article
Iterative Methods for Variational Inequalities
over the Intersection of the Fixed Points Set of
a Nonexpansive Semigroup in Banach Spaces
Issa Mohamadi
Department of Mathematics, Islamic Azad University, Sanandaj Branch, Sanandaj 418, Kurdistan, Iran
Correspondence should be addressed to Issa Mohamadi, [email protected]
Received 8 November 2010; Accepted 19 November 2010
Academic Editor: Qamrul Hasan Ansari
Copyright q 2011 Issa Mohamadi. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
This paper presents a framework of iterative methods for finding specific common fixed points of a
nonexpansive self-mappings semigroup in a Banach space. We prove, with appropriate conditions,
the strong convergence to the solution of some variational inequalities.
1. Introduction
Let C be a nonempty closed convex subset of a Hilbert space H,andletF : C → H be a
nonlinear map. The classical variational inequality which is denoted by VIF, C is formulated
as finding x
∗
∈ C such that
Fx
∗
,x− x
∗
VIF, C has a unique solution. An important problem is how to find a solution of VIF, C.It
is known that
x
∗
∈ VI
F, C
⇐⇒ x
∗
P
C
x
∗
− λFx
∗
, 1.4
where λ>0 is an arbitrarily fixed constant and P
C
is the projection of H onto C.This
alternative equivalence has been used to study the existence theory of the solution and to
develop several iterative type algorithms for solving variational inequalities. But the fixed
point formulation in 1.4 involves the projection P
C
, which may not be easy to compute, due
to the complexity of the convex set C. So, projection methods and their variant forms can be
implemented for solving variational inequalities.
In order to reduce the complexity probably caused by the projection P
∞
n1
λ
n
∞,
C3 lim
n →∞
λ
n
− λ
n1
/λ
2
n1
0.
Starting with an arbitrary initial guess x
0
∈ H, generate a sequence {x
n
} by the
following algorithm:
x
n1
: Tx
n
− λ
n1
μF
Tx
t>0} of self-mappings of X satisfies the following conditions:
i T0x x for x ∈ X,
ii Tt sx TtTsx for t, s > 0andx ∈ X,
Fixed Point Theory and Applications 3
iii lim
t → 0
Ttx x for x ∈ X,
iv for each t>0,Tt is nonexpansive. that is,
T
t
x − T
t
y
≤
x − y
, ∀x, y ∈ X. 1.7
The problem is to find some fixed point in C
n
, 1.8
x
n1
: λ
n
x
n
1 − λ
n
T
t
n
x
n
− λ
n
μFx
n
, 1.9
x
n1
: λ
n
y
− F
x
n
,
1.10
for n ≥ 1. With some appropriate assumptions, we prove the strong convergence of 1.8,
1.9,and1.10 to the unique solution of the variational inequality Fx
∗
,Jx − x
∗
≥0inC,
where J is the single-valued normalized duality mapping from X into 2
X
∗
.
Our main purpose is to improve some of the conditions and results in the mentioned
papers, especially those of Song and Xu 11.
2. Preliminaries
Let S : {x ∈ X : x 1} be the unit sphere of the Banach space X. The space X is said to
have Gateaux differentiable norm or X is said to be smooth, if the limit
lim
t → 0
x ty
−
2
f
∗
2
, ∀x ∈ X. 2.2
where ·, · denotes the generalized duality pairing. It is well known if X is smooth then any
duality mapping on X is single valued, and if X has a uniformly Gateaux differentiable norm,
then the duality mapping is norm to weak
∗
uniformly continuous on bounded sets.
4 Fixed Point Theory and Applications
Recall that a Banach space X is said to be strictly convex if x y 1andx
/
y
implies x y/2 < 1. In a strictly convex Banach space X, we have that if λx 1 − λy 1
for λ ∈ 0, 1 and x, y ∈ X, then x y.
Now, we recall the concept of uniformly asymptotically regular semigroup. A
continuous operator semigroup {Tt : t>0} on X is said to be uniformly asymptotically
regular on X if for all h>0 and any bounded subset D of X, we have
lim
t →∞
sup
x∈D
inf
{
a
n
: n ∈ N
}
≤ μ
a
≤ sup
{
a
n
: n ∈ N
}
, 2.4
for every a a
1
,a
2
, ∈ l
∞
. Sometimes, we use μ
n
a
n
instead of μa. A mean μ on N is
called a Banach limit if μ
n
μ
n
a
n
μ
a
c. 2.6
A discussion on these and related concepts can be found in 12.
We make use of the following well-known results throughout the paper.
Lemma 2.1 see 12, Lemma 4.5.4. Let D be a nonempty closed convex subset of a Banach space
X with a uniformly Gateaux differentiable norm, and let {y
n
} be a bounded sequence in X.Ifz
0
∈ D,
then
μ
n
y
n
− z
0
n
}, {c
n
}⊂R
, {a
n
}⊂0, 1, and let {b
n
}⊂R be sequences such that
s
n1
≤
1 − a
n
s
n
b
n
c
n
, 2.9
Fixed Point Theory and Applications 5
for all n ≥ 0. Assume also that
n≥0
|c
n
x y
2
≤
x
2
2
y, j
x y
.
2.11
In order to reduce any possible complexity in writing, we set C
t>0
FixTt for a
nonexpansive semigroup {Tt : t>0} and
D
x
n
x ∈ X : g
x − y
≥ η
x − y
2
,
3.1
for a constant η>0.
The following lemma will be be used to show the convergence of 1.8 and 1.9.
Lemma 3.1. Let X be a Banach space, and let J be the single-valued normalized duality mapping from
X into 2
X
∗
. Assume also that F : X → X is η-strongly monotone and κ-Lipschitzian on X. Then,
ψ
x
I
x
− μF
x
x − y
μ
Fy − Fx
2
≤
x − y
2
2
μ
Fy − Fx
,J
x − y
μ
Fy − Fx
Fx − Fy,J
x − y
2μ
2
Fy − Fx
J
Fy − Fx
≤
x − y
2
− 2μη
x − y
x − y
2
≤
1 − 2μη 2μ
2
κ
2
x − y
2
,
3.3
Thus, we obtain
ψx − ψy
≤
1 − 2μ
/
∅. 3.5
Assume that F : X → X is η-strongly monotone and κ-Lipschitzian. Assume also that {t
n
} is a
sequence of positive numbers that lim
n →∞
t
n
∞ and {λ
n
}⊂0, 1.Ifμ ∈ 0,η/κ
2
,then{x
n
}
converges strongly to some fixed point x
∗
∈ C, which is the unique solution in C to the variational
inequality VI
∗
F, C, that is
Fx
∗
,J
x − x
∗
y
∗
− x
∗
≤ 0.
3.7
By adding up the last two inequalities, we obtain
η
x
∗
− y
∗
2
≤
Fx
∗
− Fy
∗
,J
x
∗
− y
∗
T
t
n
x
n
− λ
n
μFx
n
− x
∗
≤ λ
n
x
n
− μFx
n
− x
∗
λ
n
I − μF
x
∗
− x
∗
1 − λ
n
x
n
− x
∗
≤ λ
n
1 − 2μ
n
1 −
1 − 2μ
η − μκ
2
x
n
− x
∗
λ
n
μ
Fx
∗
.
3.9
Taking γ 1 −
1 − 2μη − μκ
2
and by using induction, we obtain
}.
Step 3. The sequence {x
n
} is sequentially compact. To prove this, we assume that the set
Dx
n
contains some x
∗
such that Ttx
∗
x
∗
for an arbitrary t>0. So, by using Lemma 2.1 ,
we can obtain
μ
n
x − x
∗
,J
x
n
− x
∗
≤ 0, ∀x ∈ X. 3.11
On the other hand, for any q ∈ C, we have
− μλ
n
Fx
n
,J
x
n
− q
≤ λ
n
I − μF
x
n
−
I − μF
q
x
− T
t
n
q, J
x
n
− q
≤
1 − λ
n
γ
x
n
− q
2
λ
n
−μF
x
n
− q
.
3.13
Also, we have
x
n
− q
2
≤ λ
n
I − μF
x
n
− q, J
x
n
− q
1 − λ
n
x
n
− q
2
.
3.14
It follows that
x
n
− q
2
≤
I − μF
x
n
∗
,J
x
n
− x
∗
≤ 0.
3.16
This yields μ
n
x
n
− x
∗
0. Hence, there exists a subsequence of {x
n
} such as {x
n
k
} that
converges strongly to x
∗
;thatis,{x
n
} is sequentially compact.
Step 4. We claim that x
∗
n
− x
λ
n
−μFx
∗
,J
x
n
− x
1 − λ
n
T
t
n
x
n
− T
x
n
− x
∗
λ
n
−μFx
∗
,J
x
n
− x
x
n
− x
2
.
3.17
Hence,
Fx
,J
x
n
k
− x
−→
Fx
∗
,J
x
∗
− x
, 3.19
and by taking limit as n
k
→∞in two sides of 3.18,weobtain
Fx
∗
,J
x
∗
∗
2
≤
I − μF
x
∗
− y
∗
,J
x
∗
− y
∗
,
y
∗
− x
∗
≤
I − μF
x
∗
−
I − μF
y
∗
x
∗
− y
∗
,J
x
∗
− y
∗
≤
2 − γ
n
},μ, and {x
n
} be as those in Theorem 3.2.Ifx
∗
lim
n →∞
x
n
, and there exists a bounded sequence {y
n
} such that
lim
n →∞
T
t
y
n
− y
n
0, ∀t>0.
4.1
Then,
− y
n
λ
n
−→ 0,
4.3
as n →∞.Letx
∗
λ
n
be the fixed point of the contraction
ϕ
∗
λ
n
x
λ
n
x
1 − λ
n
T
t
n
T
t
n
x
∗
λ
n
− y
n
. 4.5
Now, by using Lemma 2.3, we have
x
∗
λ
n
− y
n
2
x
∗
λ
n
− y
n
,J
x
∗
λ
n
− y
n
≤
1 − λ
n
2
T
t
n
2λ
n
x
∗
λ
n
− y
n
2
2λ
n
−μFx
∗
λ
n
,J
x
∗
λ
n
− y
n
− y
n
2
x
∗
λ
n
− y
n
T
t
n
y
n
− y
n
∗
λ
n
− y
n
≤
λ
n
2
x
∗
λ
n
− y
n
2
T
t
n
y
n
− y
n
.
4.7
Because {y
n
}, {Tt
n
y
n
} and {x
∗
λ
n
} are bounded, from 4.3 and 4.7, we conclude that
lim sup
n →∞
μFx
∗
λ
n
,J
I − μF
x
∗
λ
n
,J
x
∗
− y
n
x
∗
−
I − μF
x
∗
λ
n
,J
x
∗
4.9
By Theorem 3.2, x
∗
λ
n
→ x
∗
,asn →∞. So, using the boundedness of {y
n
},weget
x
∗
λ
n
− x
∗
,J
x
∗
λ
n
− y
n
−→ 0,n−→ ∞ . 4.10
On the other hand, noticing that the sequence {x
∗
λ
x
∗
− y
n
−→ 0,n−→ ∞ . 4.11
Fixed Point Theory and Applications 11
Therefore, from 4.8 and 4.9,weobtain
lim sup
n →∞
Fx
∗
,J
x
∗
− y
n
≤ 0.
4.12
This completes the proof.
Next, we prove the strong convergence of explicit iteration scheme 1.9.
Theorem 4.2. Let X be a real Banach space with a uniformly Gateaux differentiable norm, and let
{Tt : t>0} be a nonexpansive semigroup from X into itself. Let also {x
n
} defined by 1.9 satisfies
the following conditions:
2
,then{x
n
} converges strongly to some fixed point x
∗
∈ C, which is the unique
solution in C for the following variational inequality:
Fx
∗
,J
x − x
∗
≥ 0, ∀x ∈ C. 4.14
Proof. Existence and uniqueness of the solution of VI
∗
F, C is attained from Theorem 3.2.
Now, we claim that {x
n
} is bounded. Indeed, taking a fixed x
∗
∈ C, we have
x
n1
− x
∗
x
n
− μFx
n
− x
∗
1 − λ
n
T
t
n
x
n
− x
∗
≤ λ
n
x
n
− x
∗
≤ λ
n
1 − 2μ
η − μκ
2
x
n
− x
∗
1 − λ
n
x
n
− x
μ
Fx
∗
.
4.15
Taking a
n
λ
n
1 −
1 − 2μη − μκ
2
, b
n
λ
n
μFx
∗
, c
n
0, and using Lemma 2.2,
we conclude that x
n
− x
∗
is bounded and so is x
n
,J
x
n1
− x
∗
1 − λ
n
T
t
n
x
n
− x
∗
,J
x
n1
− x
∗
n1
− x
∗
1 − λ
n
T
t
n
x
n
− x
∗
x
n1
− x
∗
≤ μλ
n
−Fx
1 − λ
n
x
n
− x
∗
x
n1
− x
∗
≤ μλ
n
Fx
∗
,J
x
∗
− x
n1
λ
n
∗
2
x
n1
− x
∗
2
2
≤
1 − λ
n
γ
x
n
− x
∗
2
2μλ
n
Fx
∗
n1
− x
∗
2
0, that is, x
n
→ x
∗
in norm. This completes the
proof.
Corollary 4.3. Let X be a real reflexive strictly convex Banach space with a uniformly Gateaux
differentiable norm. Let also {Tt : t>0} be a nonexpansive semigroup from X into itself such that
C
t>0
FixTt
/
∅. Assume that {x
n
} defined by 1.9 satisfies condition ii in Theorem 4.2,
then condition i holds.
Proof. Clearly, gxμ
n
x
n
− x
2
is a convex and continuous function. Because X is a
reflexive Banach space, according to 12, Theorem 1.3.11, Dx
x
2
μ
n
x
n
− T
t
x
n
T
t
x
n
− Tx
2
≤ μ
x∈D
x
n
u − x
.
4.18
On the other hand, Ttu u for all t>0, Ttx
∗
∈ Dx
n
and Tt is nonexpansive, so we get
u − T
t
x
∗
T
t
u − T
} is defined by 1.9,whereλ
n
satisfies
C1, then condition ii in Theorem 4.2 holds.
Proof. From 1.9, C1, and the boundedness of {x
n
}, we conclude that
x
n1
− T
t
n
x
n
λ
n
x
n
− T
t
n
x
t
n
x
n
≤ lim
n →∞
sup
x∈S
T
t
T
t
n
x
− T
t
n
x
n
x
n
− T
t
T
t
n
x
n
T
t
T
t
n
x
n
x
n
− T
t
n
x
n
.
4.22
So, from 4.20 , 4.21,and4.22,weget
lim
n →∞
x
n
− T
t
x
n
0, ∀t>0,
4.23
and it completes the proof.
1 − λ
n
T
t
n
x
n
.
4.24
Remark 4.6. In the same way and with the same conditions mentioned in Theorem 4.2,it’s
easy to see that the sequence {x
n
} defined by
x
n1
: T
t
n
x
n
− λ
n1
μF
I
x
− μ
F I − T
x
5.1
is a contraction on X.
Proof. Considering the inequality
x y
2
≤
x
2
2
y, J
x y
2
x − y
μ
F I − T
y −
F I − T
x
2
≤
x − y
2
x − y
2
2μ
F I − T
y −
F I − T
x, J
x − y
2μ
2
F I − T
y
−
F I − T
x, J
2μ
2
F I − T
y −
F I − T
x
J
F I − T
y −
F I − T
x
≤
F I − T
y −
F I − T
x
J
F I − T
y −
F I − T
x
.
5.3
Noticing that
I − T
2
2μ
2
F I − T
y −
F I − T
x
2
≤
x − y
2
− 2μη
x − y
ϕx − ϕy
≤
1 − 2μ
η − μσ
2
x − y
.
5.6
Note that for μ ∈ 0,η/σ
2
, we conclude
1 − 2μη − μσ
2
∈ 0, 1.Thatis,ϕ is a contraction
and the proof is complete.
Theorem 5.2. Let X be a real Banach space with a uniformly Gateaux differentiable norm and {Tt :
t>0} a nonexpansive semigroup from X into itself. Let also {x
n
} defined by 1.10 satisfies the
∞
n1
λ
n
∞,
C
2
{μ
n
} does not take 0 as it’s limit point.
Then, {x
n
} converges strongly to some fixed point x
∗
∈ C, which is the unique solution in C
for the variational inequality VI
∗
F, C.
Proof. Existence and uniqueness of the solution of VI
∗
F, C is obtained from Theorem 3.2.We
claim that {x
n
} is bounded. Indeed, taking a fixed x
∗
∈ C, we have
x
I − μ
n
F I − T
t
n
x
n
− x
∗
1 − λ
n
T
t
n
x
n
x
∗
λ
n
I − μ
n
F I − T
t
n
x
∗
− x
∗
1 − λ
n
− x
∗
λ
n
μ
n
Fx
∗
≤
1 − λ
n
1 −
1 − 2μ
n
η − μ
n
σ
2
x
n
and by assumption that there exists >0sothatμ
n
>, for all n ∈ N. Thus, we get
x
n1
− x
∗
≤
1 − λ
n
μ
2
n
x
n
− x
∗
1
λ
n
μ
n
n
F I − T
t
n
x
n
− x
∗
,J
x
n1
− x
∗
1 − λ
n
T
t
n
n
F I − T
t
n
x
∗
,J
x
n1
− x
∗
λ
n
−μ
n
F I − T
t
n
x
≤ μ
n
λ
n
−Fx
∗
,J
x
n1
− x
∗
λ
n
1 − 2μ
n
η − μ
n
σ
2
x
n
− x
,J
x
∗
− x
n1
λ
n
1 − 2μ
n
η − μ
n
σ
2
x
n
− x
∗
2
x
n1
1 −
1 − 2μ
n
η − μ
n
σ
2
x
n
− x
∗
2
2λ
n
μ
n
Fx
∗
,J
x
∗
x
∗
− x
n1
.
5.10
Taking a
n
λ
n
μ
2
n
, b
n
2λ
n
μ
2
n
/Fx
∗
,Jx
∗
− x
n1
,andc
σ
2
, and therefore we can
remove C
2
,alsoC
1
turns to
∞
n1
λ
n
μ
n
∞.
Fixed Point Theory and Applications 17
Remark 5.4. We can easily see that under some restrictions all the strongly monotone and
Lipschitzian nonlinear operators used in this paper are replaceable by strongly accretive and
strictly pseudocontractive ones see 15.
References
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intersection of fixed point sets of nonexpansive mappings,” in Inherently Parallel Algorithm for
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York, NY, USA, 2001.
2 F. Deutsch and I. Yamada, “Minimizing certain convex functions over the intersection of the fixed
point sets of nonexpansive mappings,” Numerical Functional Analysis and Optimization, vol. 19, no.
1-2, pp. 33–56, 1998.
3 H. K. Xu and T. H. Kim, “Convergence of hybrid steepest-descent methods for variational
inequalities,” Journal of Optimization Theory and Applications, vol. 119, no. 1, pp. 185–201, 2003.
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