Volume 2010, Article ID 264628, 13 pagesdoi:10.1155/2010/264628 Research Article Hybrid Viscosity Iterative Method for Fixed Point, Variational Inequality and Equilibrium Problems Yi-An
Trang 1Volume 2010, Article ID 264628, 13 pages
doi:10.1155/2010/264628
Research Article
Hybrid Viscosity Iterative Method for Fixed Point, Variational Inequality and Equilibrium Problems
Yi-An Chen and Yi-Ping Zhang
College of Mathematics and Statistics, Chongqing Technology and Business University,
Chongqing 400067, China
Correspondence should be addressed to Yi-An Chen,chenyian1969@sohu.com
Received 27 December 2009; Revised 3 May 2010; Accepted 1 June 2010
Academic Editor: Simeon Reich
Copyrightq 2010 Y.-A Chen and Y.-P Zhang 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
We introduce an iterative scheme by the viscosity iterative method for finding a common element
of the solution set of an equilibrium problem, the solution set of the variational inequality, and the fixed points set of infinitely many nonexpansive mappings in a Hilbert space Then we prove our main result under some suitable conditions
1 Introduction
LetH be a real Hilbert space with the inner product and the norm being denoted by ·, ·
and · , respectively Let C be a nonempty, closed, and convex subset of H and let F be a
bifunction ofC × C into R, where R denotes the real numbers The equilibrium problem for
F : C × C → R is to find x ∈ C such that
The solution set of1.1 is denoted by EPF.
Let A : C → H be a mapping The classical variational inequality, denoted by
VIA, C, is to find x∗∈ C such that
Ax∗, v − x∗ ≥ 0, ∀v ∈ C. 1.2
Trang 2The variational inequality has been extensively studied in the literaturesee, e.g., 1 3 The mappingA is called α-inverse-strongly monotone if
whereα is a positive real number.
A mappingT : C → C is called strictly pseudocontractive if there exists k with 0 ≤
k < 1 such that
Tx − Ty2≤x − y2 kI − Tx − I − Ty2, ∀x, y ∈ C. 1.4
It is easy to know thatI − T is 1 − k/2-inverse-strongly-monotone If k 0, then T is
nonexpansive We denote byFT the fixed points set of T.
In 2003, for x0 ∈ C, Takahashi and Toyoda 4 introduced the following iterative scheme:
x n1 α n x n 1 − α n SP C x n − λ n Ax n , n ≥ 0, 1.5
where{α n } is a sequence in 0, 1, A is an α-inverse-strongly monotone mapping, {λ n} is a sequence in0, 2α, and P C is the metric projection They proved that ifFSVIA, C / ∅,
then{x n } converges weakly to some z ∈ FSVIA, C
Recently, S Takahashi and W Takahashi5 introduced an iterative scheme for finding
a common element of the solution set of 1.1 and the fixed points set of a nonexpansive mapping in a Hilbert space IfF is bifunction which satisfies the following conditions:
A1 Fx, x 0 for all x ∈ C;
A2 F is monotone, that is, Fx, y Fy, x ≤ 0 for all x, y ∈ C;
A3 for each x, y, z ∈ C, lim t → 0 Ftz 1 − tx, y ≤ Fx, y;
A4 for each x ∈ C, y → Fx, y is convex and lower semicontinuous,
then they proved the following strong convergence theorem
Theorem A see 5 Let C be a closed and convex subset of a real Hilbert space H Let F : C×C →
R be a bifunction which satisfies conditions A1–A4.
Let T : C → H be a nonexpansive mapping such that FTEP F / ∅ and let f : H → H
be a contraction; that is, there is a constant k ∈ 0, 1 such that
and let {x n } and {u n } be sequences generated by x1∈ C and
Fu n , yr1
n
y − u n , u n − x n
≥ 0, ∀y ∈ C,
x n1 α n fx n 1 − α n Tu n , n ≥ 1,
1.7
Trang 3where {α n } ⊂ 0, 1 and {r n } ⊂ 0, ∞ satisfy lim n → ∞ α n 0,∞n1 α n ∞,∞n1 |α n1 − α n | <
∞, lim inf n → ∞ r n > 0, and∞
n1 |r n1 − r n | < ∞.
Then, {x n } and {u n } converge strongly to z ∈ FTEP F, where z P FTEP F fz Let {T n}∞n1 be a sequence of nonexpansive mappings of C into itself and {λ n}∞n1 a sequence
of nonnegative numbers in 0, 1 For each n ≥ 1, define a mapping W n of C into itself as follows:
U n,n1 I,
U n,n λ n T n U n,n1 1 − λ n I,
U n,n−1 λ n−1 T n−1 U n,n 1 − λ n−1 I,
.
U n,k λ k T k U n,k1 1 − λ k I,
U n,k−1 λ k−1 T k−1 U n,k 1 − λ k−1 I,
.
U n,2 λ2T2U n,3 1 − λ2I,
W n U n,1 λ1T1U n,2 1 − λ1I.
1.8
Such a mapping W n is called the W-mapping generated by T n , T n−1 , , T1and λ n , λ n−1 , , λ1(see [ 6 ]).
In this paper, we introduced a new iterative scheme generated byx1 ∈ C and find u n
such that
Fu n , yr1
n
y − u n , u n − x n≥ 0, ∀y ∈ C,
y n β n fx n 1− β nx n , n ≥ 1,
x n1 α n y n 1 − α n W n P C u n − δ n Au n ,
1.9
where {α n } and {β n } are sequences in 0, 1, {r n } and {δ n } are sequences in 0, ∞, f is a
fixed contractive mapping with contractive coefficient k ∈ 0, 1, A is an α-inverse-strongly monotone mapping ofC to H, F is a bifunction which satisfies conditions A1–A4, and
{W n} is generated by 1.8 Then we proved that the sequences {x n } and {u n} converge strongly tox∗∈∞
n1 FT nVIA, CEPF F, where x∗ P F fx∗
Trang 42 Preliminaries
LetH be a real Hilbert space and let C be a closed and convex subset of H P C is the metric projection fromH onto C, that is, for any x ∈ H, x − P C x ≤ x − y for all y ∈ C It is easy
to see thatP Cis nonexpansive and
u ∈ VIA, C ⇐⇒ u P C u − λAu, λ > 0. 2.1
IfA is an α-inverse-strongly monotone mapping of C to H, then it is obvious that A is
1/α-Lipschitz continuous We also have that for allx, y ∈ C and λ > 0,
I − λAx − I − λAy2x − y2− 2λx − y, Ax − Ay λ2Ax − Ay2
So, ifλ ≤ 2α, then I − λA is nonexpansive.
Lemma 2.1 see 7 Let {x n } and {z n } be bounded sequences in a Banach space E, and let {β n } be
a sequence in 0, 1 with 0 < lim inf n → ∞ β n≤ lim supn → ∞ β n < 1 Suppose x n1 1−β n z n β n x n for all n ≥ 1 and lim sup n → ∞ z n1 − z n − x n1 − x n ≤ 0 Then, lim n → ∞ z n − x n 0.
Lemma 2.2 see 8 Assume that {a n } is a sequence of nonnegative real numbers such that
where {α n } is a sequence in 0, 1 and {δ n } is a sequence in R such that
∞
n1
α n ∞; lim sup
n → ∞
δ n
α n ≤ 0 or
∞
n1
Then lim n → ∞ α n 0.
Lemma 2.3 see 9 Let C be a nonempty, closed, and convex subset of H and F a bifunction of
C × C into R that satisfies conditions A1–A4 Let r > 0 and x ∈ H Then, there exists z ∈ C such that
Fz, y1ry − z, z − x≥ 0, ∀y ∈ C. 2.5
Lemma 2.4 see 9 Assume that F : C × C → R satisfies conditions A1–A4 For r > 0 and
x ∈ H, define a mapping T r:H → C as follows:
T r x z ∈ C : Fz, y1ry − z, z − x≥ 0, ∀y ∈ C
Trang 5
Then, the following holds:
i T r is single-valued;
ii T r is firmly nonexpansive, that is,
T r x − T r y2≤T r x − T r y, x − y, ∀x, y ∈ H; 2.7
iii FT r EPF;
iv EPF is closed and convex.
Lemma 2.5 Opial’s theorem 10 Each Hilbert space H satisfies Opial’s condition; that is, for
any sequence {x n } ⊂ H with x n x, the inequality
lim inf
n → ∞ x n − x < lim inf n → ∞ x n − y 2.8
holds for each y ∈ H with x / y.
Let {T n}∞n1 be a sequence of nonexpansive self-mappings on C, where C is a nonempty, closed and convex subset of a real Hilbert space H Given a sequence {λ n}∞n1 in 0, 1, one defines a sequence {W n}∞
n1 of self-mappings on C generated by 1.8 Then one has the following results.
Lemma 2.6 see 6 Let C be a nonempty, closed, and convex subset of a real Hilbert space H Let {T n}∞n1 be a sequence of nonexpansive self-mappings on C such that∞n1 FT n / ∅ and {λ n } is a
sequence in 0, b for some b ∈ 0, 1 Then, for every x ∈ C and k ≥ 1 the limit lim n → ∞ U n,k x exists Remark 2.7 It can be shown fromLemma 2.6that ifD is a nonempty and bounded subset of
C, then for ε > 0 there exists n0≥ k such that sup x∈D U n,k x − U n−1,k x ≤ ε for all n > n0
Remark 2.8 UsingLemma 2.6, we can define a mappingW : C → C as follows:
Wx lim
n → ∞ W n x lim
for allx ∈ C Such a W is called the W-mapping generated by T1, T2, and λ1, λ2, Since
W nis nonexpansive,W : C → C is also nonexpansive Indeed, observe that for each x, y ∈ C,
Wx − Wy lim n → ∞ W n x − W n y ≤ x − y. 2.10
Let{x n } be a bounded sequence in C and D {x n:n ≥ 0} Then, it is clear fromRemark 2.7
that forε > 0 there exists N0≥ 1 such that for all n > N0,
W n x n − Wx n U n,1 x n − U1x n ≤ sup
x∈D U n,1 x − U1x ≤ ε. 2.11
This implies that limn → ∞ W n x n − Wx n 0.
Lemma 2.9 see 6 Let C be a nonempty, closed, and convex subset of a real Hilbert space H Let {T n}∞n1 be a sequence of nonexpansive self-mappings on C such that∞
n1 FT n / ∅ and {λ n } is a
sequence in 0, b for some b ∈ 0, 1 Then, FW ∞
n1 FT n .
Trang 63 Strong Convergence Theorem
Theorem 3.1 Let H be a Hilbert space Let C be a nonempty, closed, and convex subset of H Let
F : C × C → R be a bifunction which satisfies conditions A1–A4, A an α-inverse-strongly monotone mapping of C to H, f a contraction of C into itself, and {T n}∞n1 a sequence of nonexpansive self-mappings on C such that F / ∅ Suppose that {α n }, {β n }, and {λ n } are sequences in 0, 1, and {r n } and {δ n } are sequences in 0, ∞ which satisfies the following conditions:
i 0 < lim inf n → ∞ α n≤ lim supn → ∞ α n < 1;
ii limn → ∞ β n 0;∞
n1 β n ∞;
iii lim infn → ∞ r n > 0,∞n1 |r n1 − r n | < ∞;
iv δ n ∈ 0, b, b < 2α, lim n → ∞ δ n 0;
v λ n ∈ 0, c, c ∈ 0, 1.
Then {x n } and {u n } generated by 1.9 converge strongly to x∗∈ F, where x∗ P F fx∗.
Proof Let p ∈ F It follows fromLemma 2.4and1.9 that u n T r n x n, and hence,
u n − p T r n x n − T r n p ≤ x n − p, 3.1
for alln ∈ N Let z n P C u n − δ n Au n Since I − δ n A is nonexpansive and p P C p − δ n Ap,
we have
z n − p ≤ u n − δ n Au n−p − δ n Ap ≤ u n − p ≤ x n − p, 3.2
y n − p ≤ β n fx n − p 1 − β n x n − p
≤ β n fx n − fp β n fp − p 1 − β n x n − p
Thus,
x n1 − p α n y n 1 − α n W n z n − p
≤ α n y n − p 1 − α nzn − p
≤ α n
1− β n 1 − k n − p α n β n fp − p 1 − α nxn − p
1− α n β n 1 − k n − p α n β n 1 − k fp − p
1− k
≤ max
x n − p, fp − p
1− k
.
3.4
Hence{x n } is bounded So {u n }, {z n }, {W n x n }, {W n z n }, and {fx n} are also bounded
Trang 7Next, we claim that limn → ∞ x n1 −x n 0 Indeed, assume that x n1 ρ n x n 1−ρ n t n ,
whereρ n α n 1 − β n , n ≥ 0 Then,
t n1 − t n α n1 β n1 fx n1 1 − α n1 W n1 z n1
1− ρ n1 −α n β n fx n 1 − α n W n z n
1− ρ n
α n1 β n1 fx n1
1− ρ n1 −
α n β n fx n
1− ρ n
1− α n1
1− ρ n1 W n1 z n1 − W n1 z n
1− α n1
1− ρ n1 W n1 z n−1− α n
1− ρ n W n z n
≤ α n1 β n1 fx n1
1− ρ n1 −
α n β n fx n
1− ρ n
1− α n1
1− ρ n1 z n1 − z n
W n1 z n− α n1 β n1
1− ρ n1 W n1 z n − W n z n α n β n
1− ρ n W n z n ,
3.5
z n1 − z n ≤ u n1 − δ n1 Au n1 − u n − δ n Au n
≤ I − δ n1 Au n1 − I − δ n1 Au n I − δ n1 Au n − I − δ n Au n
≤ u n1 − u n δ n1 − δ n Au n .
3.6
Using1.8 and the nonexpansivity of T i, we deduce that
W n1 z n − W n z n λ1T1U n1,2 z n − λ1T1U n,2 z n
≤ λ1U n1,2 z n − U n,2 z n
≤ λ1λ2T2U n1,3 z n − λ2T2U n,3 z n
≤ λ1λ2U n1,3 z n − U n,3 z n
≤
n
i1
λ i
U n1,n1 z n − U n,n1 z n
≤ Mn
i1
λ i ,
3.7
for some constantM ≥ 0 On the other hand, from u n T r n x nandu n1 T r n1 x n1, we obtain
Fu n , yr1
n
y − u n , u n − x n
Fu n1 , yr1
n1
y − u n1 , u n1 − x n1
Trang 8Settingy u n1in3.8 and y u nin3.9, we get
Fu n , u n1 r1
n u n1 − u n , u n − x n ≥ 0,
Fu n1 , u n r1
n1 u n − u n1 , u n1 − x n1 ≥ 0.
3.10
FromA2, we have
u n1 − u n , u n − x n
r n −u n1 − x n1
r n1
and hence
u n1 − u n , u n − u n1 u n1 − x n−r r n
n1 u n1 − x n1
Without loss of generality, we may assume that there exists a real numberr such that r n > r >
0 for alln ≥ 0 Then
u n1 − u n2≤
u n1 − u n , x n1 − x n
1−r r n
n1
u n1 − x n1
≤ u n1 − u n
x n1 − x n
1 − r n
r n1
u n1 − x n1
,
3.13
and hence
u n1 − u n ≤ x n1 − x n
1 − r n
r n1
u n1 − x n1
≤ x n1− xn 1r |r n1 − r n |L,
3.14
whereL sup{u n − x n : n ≥ 0} It follows from 3.5, 3.6, 3.7, and 3.14 that
t n1 − t n − x n1 − x n ≤ α n1 β n1
1− ρ n1 fx n1 Wn1 z n α n β n
1− ρ n fx n Wn z n
1− α n1
1− ρ n1
x n1 − x n L r |r n1 − r n | |δ n1 − δ n |Au n
Mn
i1
λ i − x n1 − x n
Trang 9≤ α n1 β n1
1− ρ n1 fx n1 Wn1 z n α n β n
1− ρ n fx n Wn z n
1− α n1
1− ρ n1
L
r |r n1 − r n | |δ n1 − δ n |Au n
Mn
i1
λ i
3.15
Therefore, lim supn → ∞ t n1 − t n − x n1 − x n ≤ 0
Since 0< lim inf n → ∞ α n≤ lim supn → ∞ α n < 1 and lim n → ∞ β n 0, hence,
0< lim inf
n → ∞ ρ n≤ lim sup
Lemma 2.1yields that limn → ∞ t n − x n 0 Consequently, limn → ∞ x n1 − x n limn → ∞1 −
ρ n tn− x n 0.
Forp ∈ F, we obtain
u n − p2T r n x n − T r n p2
≤ T r n x n − T r n p, x n − p
u n − p, x n − p
1 2
u
n − p2x n − p2− x n − u n2
,
3.17
and hence
u n − p2≤x n − p2
This together with3.2 yields that
x n1 − p2 ≤ α n y n − p2
1 − α nzn − p2
≤ α n β n fx n − p 1 − β n x n − p2
1 − α nun − p2
≤ α n β n fx n − p2 α n
1− β n x n − p2
1 − α nx
n − p2
− u n − x n2
,
3.19
and hence,
1 − α n u n − x n2 ≤ α n β n fx n − p21− α n β n x n − p2−x n1 − p2
≤ α n β n fx
n − p2−x n − p2
x n1 − x nxn − p x n1 − p.
3.20
Trang 10Sou n − x n → 0 note that limn → ∞ β n 0 and limn → ∞ x n1 − x n 0 Since
W n u n − u n ≤ W n u n − W n x n W n x n − x n x n − u n
≤ 2x n − u n W n x n − x n ,
x n − W n x n ≤ x n − x n1 x n1 − W n x n
≤ x n − x n1 α n β n fx n − W n x n
α n1− β nx n − W n x n 1 − α n W n z n − W n x n
≤ x n − x n1 α n β n fx n − W n x n
α n1− β nx n − W n x n 1 − α n P C u n − δ n Au n − P C x n
≤ x n − x n1 α n β n fx n − W n x n α n
1− β n
x n − W n x n
1 − α n u n − x n 1 − α n δ n Au n ,
3.21
we obtain limn → ∞ x n − W n x n 0, and hence lim n → ∞ u n − W n u n 0 Thus, u n − Wu n ≤
u n − W n u n W n u n − Wu n → 0.
LetQ P F Then Qf is a contraction of H into itself In fact, there exists k ∈ 0, 1 such
thatfx − fy ≤ kx − y for all x, y ∈ H So
for all x, y ∈ H So Qf is a contraction by Banach contraction principle 11 Since H is a
complete space, there exists a unique elementx∗∈ C ⊂ H such that x∗ Qfx∗
Next we show that
lim sup
n → ∞ fx∗ − x∗, x n − x∗ ≤ 0, 3.23
wherex∗ Qfx∗ To show this inequality, we choose a subsequence {u n i} of {u n} such that
lim sup
n → ∞
fx∗ − x∗, u n − x∗
lim
n → ∞
fx∗ − x∗, u n i − x∗
Since{u n i } is bounded, there exists a subsequence of {u n i} which converges weakly to some
ω ∈ C, that is, u n i ω From Wu n − u n → 0, we obtain that Wu n i ω Now we will show
thatω ∈ FWVIA, CEPF First, we will show ω ∈ EPF From un T r n x n , we have
Fu n , yr1
n
y − u n , u n − x n
ByA2, we also have
1
r n
y − u n , u n − x n
≥ Fy, u n
... Trang 42 Preliminaries
LetH be a real Hilbert space and let C be a closed and convex... 1,
1.7
Trang 3where {α n } ⊂ 0, 1 and {r n } ⊂ 0, ∞ satisfy... n z n }, and {fx n} are also bounded
Trang 7Next, we claim that