Thai Journal of Mathematics
Volume 11 (2013) Number 2 : 313–317
http://thaijmath.in.cmu.ac.th
ISSN 1686-0209
Approximation Method for Fixed Point
Problem and Variational Inequality Problem
without Assumption on the Set of Fixed Pint
and the Set of Variational Inequality
Atid Kangtunyakarn
Department of Mathematics, Faculty of Science
King Mongkut, s Institute of Technology
Ladkrabang, Bangkok 10520, Thailand
e-mail : [email protected]
Abstract : The purpose of this paper is to prove a necessary and sufficient
condition to obtain strong convergence theorem for finding a common element of
the set of fixed point problem of nonexpansive mapping
and the set of variational
T
inequality problem without assumption F (T ) V I(C, A) 6= ∅ in framework of
Hilbert space.
Keywords : nonexpansive mappings; strongly positive operator; fixed point.
2010 Mathematics Subject Classification : 47H09; 47H10.
1
Introduction
Let H be a real Hilbert space and let C be a nonempty closed convex subset of
H. Let A : C → H be a nonlinear mapping. A mapping T of H into itself is called
nonexpansive if kT x − T yk ≤ kx − yk for all x, y ∈ H. We denote by F (T ) the set
of fixed points of T (i.e., F (T ) = {x ∈ H : T x = x}). It is known that F (T ) is
always closed convex and also nonempty provided T has a bounded trajectory.
A bounded linear operator A on H is called strongly positive with coefficient
c 2013 by the Mathematical Association of Thailand.
Copyright All rights reserved.
314
T hai J. M ath.
11 (2013)/ A. Kangtunyakarn
γ̄ if there is a constant γ̄ > 0 with the property
hAx, xi ≥ γ̄kxk2 .
The variational inequality problem is to find a point u ∈ C such that
hv − u, Aui ≥ 0
for all v ∈ C.
(1.1)
The set of solutions of the variational inequality is denoted by V I(C, A).
Variational inequalities were introduced and studied by Stampacchia [1] in
1964. It is now well-known that a wide classes of problems arising in various
branches of pure and applied sciences can be reduce to find element of (1.1). Many
author has been studied strong convergence theorem for finding a common element
of the set of fixed point
T problem and the set of variational inequality problem by
using condition F (T ) V I(C, A) 6= ∅. (see, e.g., [2, 3])
In this paper, we prove strong convergence theorem for finding a common
element of the set of fixed point problem of nonexpansiveTmapping and the set of
variational inequality problem without assumption F (T ) V I(C, A) 6= ∅.
2
Preliminaries
In this section, we provide the well-known lemmas are needed to prove our
main result.
Let C be closed convex subset of a real Hilbert space H, let PC be the metric
projection of H onto C i.e., for x ∈ H, PC x satisfies the property
kx − PC xk = min kx − yk.
y∈C
Lemma 2.1 (See [4]). Let H be a Hilbert space, let C be nonempty closed convex
subset of H and let A be a mapping of C into H. Let u ∈ C. Then for λ > 0,
u ∈ V I(C, A) ⇔ u = PC (I − λA)u
where PC is the metric projection of H onto C.
Lemma 2.2 (See [5]). Let E be a uniformly convex Banach space, C be a nonempty
closed convex subset of E and S : C → C be a nonexpansive mapping. Then I − S
is demi-closed at zero.
Lemma 2.3 (See [6]). Let A be a strongly positive linear bounded operator on a
Hilbert space H with coefficient γ and 0 < ρ ≤ kAk−1 . Then kI − ρAk ≤ 1 − ργ.
3
Main Results
Theorem 3.1. Let C be a nonempty closed convex subset of a Hilbert space H
and let T : C → C be a nonexpansive mapping, A : C → H a strongly positive
315
Approximation Method for Fixed Point Problem ...
linear bounded operator with coefficient γ > 0. Let {xn } be the sequence generated
by x0 ∈ C and
xn+1 = αT xn + (1 − α)PC (I − pA)xn , ∀n = 0, 1, 2, . . .
where α ∈ (0, 1), 0 < p ≤
(3.1)
1
kAk
and pγ < 1. Then the following are equivalent.
T
(i) The sequence {xn } defined by (3.1) converges strongly to x∗ ∈ F (T ) V I(C, A);
(ii) limn→∞ kxn − T xn k = 0.
Proof. (i) ⇒ (ii) Let condition (i) holds. Since x∗ ∈ F (T )
∗
T
V I(C, A), we have
∗
kxn − T xn k ≤ kxn − x k + kT x − T xn k ≤ 2kxn − x∗ k.
It implies that limn→∞ kxn − T xn k = 0.
Conversely, (ii) ⇒ (i), let condition (ii) holds. First we show that I − pA is
contraction mapping. Let x, y ∈ C. Since A is a strongly positive linear bounded
operator and Lemma 2.3, we have
k(I − pA)x − (I − pA)yk = k(I − pA)(x − y)k
≤ (1 − pγ)kx − yk.
Since 0 < pγ < 1, we have I − pA is a contraction mapping with coefficient 1 − pγ.
For every n ∈ N, we have
kxn+1 − xn k ≤ αkT xn − T xn−1 k + (1 − α)kPC (I − pA)xn − PC (I − pA)xn−1 k
≤ αkxn − xn−1 k + (1 − α)k(I − pA)xn − (I − pA)xn−1 k
= αkxn − xn−1 k + (1 − α)k(I − pA)(xn − xn−1 )k
≤ αkxn − xn−1 k + (1 − α)(1 − pγ)kxn − xn−1 k
≤ 1 − pγ(1 − α) kxn − xn−1 k
= akxn − xn−1 k
≤ a akxn−1 − xn−2 k
≤ a2 kxn−1 − xn−2 k
..
.
≤ an kx1 − x0 k
where a = 1 − pγ(1 − α) ∈ (0, 1).
For any number n, k ∈ N, k > 0 and (3.2), we have
kxn+k − xn k ≤
n+k−1
X
(3.2)
kxj+1 − xj k
j=n
≤
n+k−1
X
aj kx1 − x0 k
j=n
n
≤
a
kx1 − x0 k.
1−a
(3.3)
316
T hai J. M ath.
11 (2013)/ A. Kangtunyakarn
Since limn→∞ an = 0 and (3.3), we have {xn } is a Cauchy sequence. Since H is a
Hilbert space, we get {xn } converges to x∗ i.e.,
lim xn = x∗ .
n→∞
(3.4)
Since C is closed, so we get x∗ ∈ C. By limn→∞ kxn − T xnk = 0, (3.4) and Lemma
2.2, we have x∗ ∈ F (T ). Since limn→∞ xn = x∗ and definition of xn , we have
x∗ = αT x∗ + (1 − α)PC (I − pA)x∗
= αx∗ + (1 − α)PC (I − pA)x∗ ,
(3.5)
it implies that x∗ ∈ F (PC (I − pA)). From Lemma 2.1, weThave x∗ ∈ V I(C, A).
Hence, the sequence {xn } converges strongly to x∗ ∈ F (T ) V I(C, A). Complete
the proof.
The following result can be obtain from Theorem 3.1, then we, therefore, omit
the prove.
Corollary 3.2. Let C be a nonempty closed convex subset of a Hilbert space H
and let A : C → H be a strongly positive linear bounded operator with coefficient
γ > 0. Let {xn } be the sequence generated by x0 ∈ C and
xn+1 = αxn + (1 − α)PC (I − pA)xn , ∀n = 0, 1, 2, . . .
(3.6)
1
and pγ < 1. Then the sequence {xn } defined by
where α ∈ (0, 1), 0 < p ≤ kAk
∗
(3.6) converges strongly to x ∈ V I(C, A).
Acknowledgement : This research was supported by the Research Administration Division of King Mongkut’s Institute of Technology Ladkrabang.
References
[1] G. Stampacchia, Formes bilineaires coercitives sur les ensembles convexes,
C.R. Acad. Sci. Paris 258 (1964) 4413–4416.
[2] H. Iiduka, W. Takahashi, Strong convergence theorems for nonexpansive mappings and inverse-strongly monotone mappings, Nonlinear Anal. 61 (2005)
341–350.
[3] L.C. Zeng, J.C. Yao, Strong convergence theorem by an extragradient method
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Math. 10 (2006) 1293–1303.
[4] W. Takahashi, Nolinear and Convex Analysis, Yokohama Publishers, 2009.
[5] F.E. Browder, Nonlinear operators and nonlinear equations of evolution in
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Approximation Method for Fixed Point Problem ...
317
[6] G. Marino, H.K. Xu, A general iterative method for nonexpansive mappings
in Hilbert spaces, J. Math. Anal. Appl. 318 (1) (2006) 43–52.
(Received 11 April 2012)
(Accepted 19 September 2012)
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