Volume 2011, Article ID 173430, 15 pagesdoi:10.1155/2011/173430 Research Article An Application of Hybrid Steepest Descent Methods for Equilibrium Problems and Strict Pseudocontractions
Trang 1Volume 2011, Article ID 173430, 15 pages
doi:10.1155/2011/173430
Research Article
An Application of Hybrid Steepest Descent
Methods for Equilibrium Problems and Strict
Pseudocontractions in Hilbert Spaces
Ming Tian
College of Science, Civil Aviation University of China, Tianjin 300300, China
Correspondence should be addressed to Ming Tian,tianming1963@126.com
Received 9 December 2010; Accepted 13 February 2011
Academic Editor: Shusen Ding
Copyrightq 2011 Ming Tian 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 use the hybrid steepest descent methods for finding a common element of the set of solutions
of an equilibrium problem and the set of fixed points of a strict pseudocontraction mapping in the setting of real Hilbert spaces We proved strong convergence theorems of the sequence generated
by our proposed schemes
1 Introduction
LetH be a real Hilbert space and C a closed convex subset of H, and let φ be a bifunction of
C × C into R, where R is the set of real numbers The equilibrium problem for φ : C × C → R
is to findx ∈ C such that
EP :φx, y≥ 0 ∀y ∈ C 1.1
denoted the set of solution by EPφ Given a mapping T : C → H, let φx, y Tx, y − x for allx, y ∈ C, then z ∈ EPφ if and only if Tz, y − z ≥ 0 for all y ∈ C, that is, z
is a solution of the variational inequality Numerous problems in physics, optimizations, and economics reduce to find a solution of1.1 Some methods have been proposed to solve the equilibrium problem, see, for instance,1,2
A mappingT of C into itself is nonexpansive if Tx−Ty ≤ x−y, for all x, y ∈ C The
set of fixed points ofT is denoted by FT In 2007, Plubtieng and Punpaeng 3, S Takahashi and W Takahashi4, and Tada and W Takahashi 5 considered iterative methods for finding
an element of EPφ ∩ FT
Trang 2Recall that an operatorA is strongly positive if there exists a constant γ > 0 with the
property
Ax, x ≥ γx2, ∀x ∈ H. 1.2
In 2006, Marino and Xu6 introduced the general iterative method and proved that for a givenx0 ∈ H, the sequence {x n} is generated by the algorithm
x n1 α n γfx n I − α n ATx n , n ≥ 0, 1.3
whereT is a self-nonexpansive mapping on H, f is a contraction of H into itself with β ∈ 0, 1
and {α n } ⊂ 0, 1 satisfies certain conditions, and A is a strongly positive bounded linear
operator onH and converges strongly to a fixed-point x∗ofT which is the unique solution
to the following variational inequality:
γf − Ax∗, x − x∗ ≤ 0, for x ∈ FT, and is also the optimality condition for some
minimization problem A mappingS : C → H is said to be k-strictly pseudocontractive if
there exists a constantk ∈ 0, 1 such that
Sx − Sy2
≤x − y2 kI − Sx − I − Sy2, ∀x, y ∈ C. 1.4
Note that the class ofk-strict pseudo-contraction strictly includes the class of
nonex-pansive mapping, that is,S is nonexpansive if and only if S is 0-srictly pseudocontractive; it
is also said to be pseudocontractive ifk 1 Clearly, the class of k-strict pseudo-contractions
falls into the one between classes of nonexpansive mappings and pseudo-contractions The set of fixed points ofS is denoted by FS Very recently, by using the general
approximation method, Qin et al.7 obtained a strong convergence theorem for finding an element ofFS On the other hand, Ceng et al 8 proposed an iterative scheme for finding
an element of EPφ ∩ FS and then obtained some weak and strong convergence theorems Based on the above work, Y Liu9 introduced two iteration schemes by the general iterative method for finding an element of EPφ ∩ FS
In 2001, Yamada10 introduced the following hybrid iterative method for solving the variational inequality:
x n1 Tx n − μλ n FTx n , n ≥ 0, 1.5
whereF is k-Lipschitzian and η-strongly monotone operator with k > 0, η > 0, 0 < μ < 2η/k2, then he proved that if {λ n } satisfyies appropriate conditions, the {x n} generated by 1.5 converges strongly to the unique solution of variational inequality
F x, x − x ≥ 0, ∀x ∈ F ix T, x ∈ F ix T. 1.6
Motivated and inspired by these facts, in this paper, we introduced two iteration methods by the hybrid iterative method for finding an element of EPφ ∩ FS, where S : C → H is
ak-strictly pseudocontractive non-self mapping, and then obtained two strong convergence
theorems
Trang 32 Preliminaries
Throughout this paper, we always assume thatC is a nonempty closed convex subset of a
Hilbert spaceH We write x n x to indicate that the sequence {x n} converges weakly to
x x n → x implies that {x n } converges strongly to x For any x ∈ H, there exists a unique
nearest point inC, denoted by P C x, such that
x − P C x ≤ x − y, ∀y ∈ C. 2.1
Such aP C x is called the metric projection of H onto C It is known that P Cis nonexpansive Furthermore, forx ∈ H and u ∈ C, u p c x, ⇔ x − u, u − y ≥ 0, for all y ∈ C.
It is widely known thatH satisfies Opial’s condition 11, that is, for any sequence
{x n } with x n x, the inequality
lim inf
n → ∞ x n − x < lim inf n → ∞ x n − y, 2.2
holds for everyy ∈ H with y / x In order to solve the equilibrium problem for a bifunction
φ : C × C → R, let us assume that φ satisfies the following conditions:
A1 φx, x 0, for all x ∈ C,
A2 φ is monotone, that is, φx, y φy, x ≤ 0, for all x, y ∈ C,
A3 For all x, y, z ∈ C.
lim
A4 For each fixed x ∈ C, the function y → φx, y is convex and lower semicontinuous.
Let us recall the following lemmas which will be useful for our paper
Lemma 2.1 see 12 Let φ be a bifunction from C × C into R satisfying (A1), (A2),(A3) and (A4)
then, for any r > 0 and x ∈ H, there exists z ∈ C such that
φz, y1r y − z, z − x ≥ 0, ∀y ∈ C. 2.4
Further, if T r x {z ∈ C : φz, y 1/ry − z, z − x ≥ 0, ∀y ∈ C}, then the following hold:
1 T r is single-valued,
2 T r is firmly nonexpansive, that is,
T r x − T r y2 ≤T r x − T r y, x − y, ∀x, y ∈ H; 2.5
3 FT r EPφ,
4 EPφ is nonempty, closed and convex.
Trang 4Lemma 2.2 see 13 If S : C → H is a k-strict pseudo-contraction, then the fixed-point set FS
is closed convex, so that the projection P FS is well difened.
Lemma 2.3 see 14 Let S : C → H be a k-strict pseudo-contraction Define T : C → H by
Tx λx 1 − λSx for each x ∈ C, then, as λ ∈ k, 1, T is nonexpansive mapping such that FT FS.
Lemma 2.4 see 15 In a Hilbert space H, there holds the inequality
x y2
≤ x2 2y,x y, ∀x, y ∈ H. 2.6
Lemma 2.5 see 16 Assume that {a n } is a sequence of nonnegative real numbers such that
where {γ n } is a sequence in (0,1) and {δ n } is a sequence inÊ, such that
i∞
n1 γ n ∞,
ii lim supn → ∞ δ n ≤ 0 or∞
n1 |δ n γ n | < ∞.
Then lim n → ∞ a n 0.
3 Main Results
Throughout the rest of this paper, we always assume thatF is a L-lipschitzian continuous and η-strongly monotone operator with L, η > 0 and assume that 0 < μ < 2η/L2.τ μη −μL2/2.
Let {T λ n} be mappings defined asLemma 2.1 Define a mappingS n : C → H by S n x
β n x 1 − β n Sx, for all x ∈ C, where β n ∈ k, 1, then, byLemma 2.3,S nis nonexpansive We consider the mappingG nonH defined by
G n x I − α n μFS n T λ n x, x ∈ H, n ∈ N, 3.1 whereα n ∈ 0, 1 By Lemmas2.1and 2.3, we have
G n x − G n y ≤ 1 − α n τ T λ n x − T λ n y
≤ 1 − α n τ x − y. 3.2
It is easy to see thatG nis a contraction Therefore, by the Banach contraction principle,
G nhas a unique fixed-pointx F ∈ H such that
x F
n I − α n μFS n T λ n x F
Trang 5For simplicity, we will writex nforx F
n provided no confusion occurs Next, we prove that the sequence{x n } converges strongly to a q ∈ FS ∩ EPφ which solves the variational
inequality
Equivalently,q P FS∩EPφ I − μFq.
Theorem 3.1 Let C be a nonempty closed convex subset of a real Hilbert space H and φ a
bifunction from C × C into R satisfying (A1), (A2), (A3), and (A4) Let S : C → H be a k-strictly pseudocontractive nonself mapping such that FS ∩ EPφ / φ Let F : H → H be an L-Lipschitzian continuous and η-strongly monotone operator on H with L, η > 0 and 0 < μ < 2η/L2,
τ μη − μL2/2 Let {x n } be asequence generated by
φu n , yλ1
n y − u n , u n − x n ≥ 0, ∀y ∈ C,
y n β n u n1− β nSu n ,
x nI − α n μFy n , ∀n ∈ N,
3.5
where u n T λ n x n , y n S n u n , and {λ n } ⊂ 0, ∞ satisfy lim inf n → ∞ λ n > 0 if {α n } and {β n}
satisfy the following conditions:
i {α n } ⊂ 0, 1, lim n → ∞ α n 0,
ii 0 ≤ k ≤ β n ≤ λ < 1 and lim n → ∞ β n λ,
then {x n } converges strongly to a point q ∈ FS ∩ EPφ which solves the variational inequality
3.4.
Proof First, take p ∈ FS ∩ EPφ Since u n T λ n x nandp T λ n p, fromLemma 2.1, for any
n ∈ N, we have
u n − p T λ n x n − T λ n p ≤ x n − p. 3.6 Then, sinceS n p p, we obtain that
y n − p S n u n − S n p ≤ u n − p ≤ x n − p. 3.7
Further, we have
x n − p −α n μFp I − μα n Fy n−I − μα n Fp
≤ α n −μFp 1 − α n τ y n − p. 3.8
It follows thatx n − p ≤ μFp/τ.
Trang 6Hence,{x n } is bounded, and we also obtain that {u n } and {y n} are bounded Notice that
u n − y n ≤ u n − x n x n − y n
u n − x n α n −μFy n . 3.9
ByLemma 2.1, we have
u n − p2T λ n x n − T λ n p2≤x n − p, u n − p
1 2
x
n − p2u n − p2
− u n − x n2 . 3.10
It follows that
u n − p2
≤x n − p2− x n − u n2. 3.11 Thus, fromLemma 2.4,3.7, and 3.11, we obtain that
x n − p2α n
−μFpI − μα n Fy n−I − μα n Fp2
≤ 1 − α n τ2y n − p2 2α n−μFp, x n − p
≤ 1 − α n τ2u n − p2 2α n
−μFp, x n − p
≤ 1 − α n τ2 x
n − p2− x n − u n2 2α n −μFpx n − p
1− 2α n τ α n τ2 x n − p2
− 1 − α n τ2x n − u n2 2α n − μFpx n − p
≤x n − p2 α n τ2x n − p2− 1 − α n τ2x n − u n2 2α n −μFpx n − p.
3.12
It follows that
1 − α n τ2x n − u n2 ≤ α n τ2x n − p2 2α n μFpx n − p. 3.13 Sinceα n → 0, therefore
lim
From3.9, we derive that
lim
Trang 7DefineT : C → H by Tx λx 1 − λSx, then T is nonexpansive with FT FS
byLemma 2.3 We note that
Tu n − u n ≤Tu n − y n y n − u n ≤ λ − β n u n − Su n y n − u n . 3.16
So by3.15 and β n → λ, we obtain that
lim
Since{u n } is bounded, so there exists a subsequence {u n i} which converges weakly to
q Next, we show that q ∈ FS ∩ EPφ Since C is closed and convex, C is weakly closed So
we haveq ∈ C Let us show that q ∈ FS Assume that q ∈ FT, Since u n i q and q / Tq, it
follows from the Opial’s condition that
lim inf
n → ∞ u n i − q < liminf
n → ∞ u n i − Tq
≤ lim inf
n → ∞
u n i − Tu n i Tu n i − Tq
≤ lim inf
n → ∞ u n i − q.
3.18
This is a contradiction So, we getq ∈ FT and q ∈ FS.
Next, we show thatq ∈ EPφ Since u n T λ n x n, for anyy ∈ C, we obtain
φu n , y 1
λ n
FromA2, we have
1
λ n
Replacingn by n i, we have
y − u n i , u n i − xni
y, u n i
Sinceu n i − x n i /λ n i → 0 and u n i q, it follows from A4 that 0 ≥ φy, q, for all
y ∈ C Let z t ty 1 − tq for all t ∈ 0, 1 and y ∈ C, then we have z t ∈ C and hence
φz t , q ≤ 0 Thus, from A1 and A4, we have
0 φz t , z t ≤ tφz t , y 1 − tφz t , q≤ tφz t , y, 3.22
Trang 8and hence 0 ≤ φz t , y From A3, we have 0 ≤ φq, y for all y ∈ C and hence q ∈ EPφ.
Therefore,q ∈ FS ∩ EPφ On the other hand, we note that
x n − q −α n μFq I − μα n Fy n−I − μα n Fq. 3.23 Hence, we obtain
x n − q2−α n μFq, x n − qI − μα n Fy n−I − μα n Fq, x n − q
≤ α n
−μFq, x n − q 1 − α n τ x n − q2. 3.24
It follows that
x n − q2
≤ τ1−μFq, x n − q. 3.25 This implies that
x n − q2≤
−μFq, x n − q
In particular,
x n i − q2≤
−μFq, x n i − q
Sincex n i q, it follows from 3.27 that x n i → q as i → ∞ Next, we show that q
solves the variational inequality3.4
As a matter of fact, we have
x n I − α n μFy n
and we have
μFx n −α1
n
I − S n T λ n x n − μα n Fx n − FS n T λ n x n. 3.29 Hence, forp ∈ FS ∩ EPφ,
μFx n , x n − p −α1
n
I − S n T λ n x n − μα n Fx n − FS n T λ n x n, x n − p
− 1
α n
I − S n T λ n x n − I − S n T λ n p, x n − p μFx n − FS n T λ n x n , x n − p.
3.30
Trang 9SinceI − S n T λ n is monotonei.e., x − y, I − S n T λ n x − I − S n T λ n y ≥ 0, for all x, y ∈ H This
is due to the nonexpansivity ofS n T λ n
Now replacingn in 3.30 with n iand lettingi → ∞, we obtain
μFq, q − p lim
i → ∞
μFx n i , x n i − p
≤ lim
i → ∞ μFx n i − FS n T λ n x n i , x n i − p 0. 3.31
That is,q ∈ FS∩EPφ is a solution of 3.4 To show that the sequence {x n} converges strongly toq, we assume that x n k → x Similiary to the proof above, we derive x ∈ FS ∩
EPφ Moreover, it follows from the inequality 3.31 that
Interchangeq and x to obtain
μFx, x − q≤ 0. 3.33 Adding up3.32 and 3.33 yields
μη q − x2≤q − x,μFq −μFx≤ 0. 3.34 Hence,q x, and therefore x n → q as n → ∞,
I − μFq − q, q − p≥ 0, ∀p ∈ FS ∩ EPφ. 3.35 This is equivalent to the fixed-point equation
P FS∩EPφ
Theorem 3.2 Let C be a nonempty closed convex subset of a real Hilbert space H and φ a bifunction
from C × C into R satisfying (A1), (A2), (A3) and (A4) Let S : C → H be a k-strictly pseudocontractive nonself mapping such that FS∩EPφ / φ Let F : H → H be an L-Lipschitzian continuous and η-strongly monotone operator on H with L, η > 0 Suppose that 0 < μ < 2η/L2,
τ μη − μL2/2 Let {x n } and {u n } be sequences generated by x1∈ H and
φu n , y λ1
n
y − u n , u n − x n
≥ 0, ∀y ∈ C,
y n β n u n1− β nSu n ,
x n1I − α n μFy n , ∀n ∈ N,
3.37
Trang 10where u n T λ n x n , y n S n u n if {α n },{β n }, and {λ n } satisfy the following conditions:
i {α n } ⊂ 0, 1, lim n → ∞ α n 0,∞
n1 α n ∞,∞
n1 |α n1 − α n | < ∞,
ii 0 ≤ k ≤ β n ≤ λ < 1 and lim n → ∞ β n λ,∞
n1 |β n1 − β n | < ∞,
iii {λ n } ∈ 0, ∞, lim n → ∞ λ n > 0 and∞
n1 |λ n1 − λ n | < ∞,
then {x n } and {u n } converge strongly to a point q ∈ FS∩EPφ which solves the variational
inequality3.4.
Proof We first show that {x n } is bounded Indeed, pick any p ∈ FS ∩ EPφ to derive that
x n1 − p −α n μFp I − μα n Fy n−I − μα n Fp
≤ α n −μFp 1 − α n τ x n − p
≤ 1 − α n τ x n − p α n −μFp. 3.38
By induction, we have
x n − p ≤ max
x1− p,1τ − μFp
and hence {x n} is bounded From 3.6 and 3.7, we also derive that {u n } and {y n} are bounded Next, we show thatx n1 − x n → 0 We have
x n1 − x n I − α n μFy n−I − α n−1 μFy n−1
I − α n μFy n−I − α n μFy n−1I − α n μFy n−1−I − α n−1 μFy n−1
≤ 1 − α n τ y n − y n−1 |α n − α n−1|μFyn−1
≤ 1 − α n τ y n − y n−1 K|α n − α n−1 |,
3.40 where
K sup μFy n:n ∈ N
On the other hand, we have
y n − y n−1 S n u n − S n−1 u n−1
≤ S n u n − S n u n−1 Sn u n−1 − S n−1 u n−1
≤ u n − u n−1 S n u n−1 − S n−1 u n−1 .
3.42
... Trang 8and hence ≤ φz t , y From A3, we have ≤ φq, y for all y ∈ C and hence... p.
3.30
Trang 9SinceI − S n T λ n... C × C into R satisfying (A1), (A2), (A3) and (A4) Let S : C → H be a k-strictly pseudocontractive nonself mapping such that FS∩EPφ / φ Let F : H → H be an L-Lipschitzian continuous and η-strongly