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Approach to common elements of variational inequality problems and fixed point problems via a relaxed extragradient method
Computers and Mathematics with Applications 59 (2010) 3472–3480
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Computers and Mathematics with Applications journal homepage: www.elsevier.com/locate/camwa
Approach to common elements of variational inequality problems and fixed point problems via a relaxed extragradient method Yonghong Yao a , Yeong-Cheng Liou b , Shin Min Kang c,∗ a
Department of Mathematics, Tianjin Polytechnic University, Tianjin 300160, People’s Republic of China
b
Department of Information Management, Cheng Shiu University, Kaohsiung 833, Taiwan
c
Department of Mathematics and the RINS, Gyeongsang National University, Jinju 660-701, Republic of Korea
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abstract
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Article history: Received 30 April 2009 Accepted 22 March 2010 Keywords: Variational inequality Fixed point Monotone mapping Strictly pseudocontractive mapping
In this paper, we introduce an iterative method based on the extragradient method for finding a common element of the set of a general system of variational inequalities and the set of fixed points of a strictly pseudocontractive mapping in a real Hilbert space. Furthermore, we prove that the studied iterative method strongly converges to a common element of the set of a general system of variational inequalities and the set of fixed points of a strictly pseudocontractive mapping under some mild conditions imposed on algorithm parameters. © 2010 Elsevier Ltd. All rights reserved.
1. Introduction Let C be a nonempty closed convex subset of a real Hilbert space H. For a given nonlinear operator A : C → H, we consider the following variational inequality problem of finding x∗ ∈ C such that
hAx∗ , x − x∗ i ≥ 0,
∀x ∈ C .
(1.1)
The set of solutions of the variational inequality (1.1) is denoted by VI (A, C ). Variational inequality theory has emerged as an important tool in studying a wide class of obstacle, unilateral, free, moving, equilibrium problems arising in several branches of pure and applied sciences in a unified and general framework. The variational inequality problem has been extensively studied in the literature. See, e.g. [1–7] and the references therein. For finding an element of Fix(S ) ∩ VI (A, C ) under the assumption that a set C ⊂ H is closed and convex, a mapping S of C into itself is nonexpansive and a mapping A of C into H is α -inverse strongly monotone, Takahashi and Toyoda [8] introduced the following iterative scheme: xn+1 = αn xn + (1 − αn )SPC (xn − λn Axn ),
∀n ≥ 0,
where PC is the metric projection of H onto C , x0 = x ∈ C , {αn } is a sequence in (0, 1), and {λn } is a sequence in (0, 2α). They showed that, if Fix(S ) ∩ VI (A, C ) is nonempty, then the sequence {xn } converges weakly to some z ∈ Fix(S ) ∩ VI (A, C ). Recently, Nadezhkina and Takahashi [9] and Zeng and Yao [10] proposed some so-called extragradient method motivated by the idea of Korpelevich [11] for finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of a variational inequality problem. Further, these iterative methods are extended in [12] to develop a new iterative method for finding elements in Fix(S ) ∩ VI (A, C ).
∗
Corresponding author. Fax: +82 55 755 1917. E-mail addresses: [email protected] (Y. Yao), [email protected] (Y.-C. Liou), [email protected] (S.M. Kang).
0898-1221/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.camwa.2010.03.036
Y. Yao et al. / Computers and Mathematics with Applications 59 (2010) 3472–3480
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Let A, B : C → H be two mappings. Now we concern the following problem of finding (x∗ , y∗ ) ∈ C × C such that
hλAy∗ + x∗ − y∗ , x − x∗ i ≥ 0, hµBx∗ + y∗ − x∗ , x − y∗ i ≥ 0,
∀x ∈ C , ∀x ∈ C ,
(1.2)
which is called a general system of variational inequalities where λ > 0 and µ > 0 are two constants. The set of solutions of (1.2) is denoted by GSVI(A, B, C ). In particular, if A = B, then problem (1.2) reduces to finding (x∗ , y∗ ) ∈ C × C such that
hλAy∗ + x∗ − y∗ , x − x∗ i ≥ 0, hµAx∗ + y∗ − x∗ , x − y∗ i ≥ 0,
∀x ∈ C , ∀x ∈ C ,
(1.3)
which is defined by Verma [13] (see also [14]) and is called the new system of variational inequalities. Further, if we add up the requirement that x∗ = y∗ , then problem (1.3) reduces to the classical variational inequality problem (1.1). For solving problem (1.2), recently, Ceng et al. [15] introduced and studied a relaxed extragradient method. It is clear that their results unified and extended many results in the literature. Motivated and inspired by the above works, in this paper, we introduce an iterative method based on the extragradient method for finding a common element of the set of a general system of variational inequalities and the set of fixed points of a strictly pseudocontractive mapping in a real Hilbert space. Furthermore, we prove that the studied iterative method strongly converges to a common element of the set of a general system of variational inequalities and the set of fixed points of a strictly pseudocontractive mapping under some mild conditions imposed on algorithm parameters. 2. Preliminaries In this section, we collect some notations and lemmas. Let C be a nonempty closed convex subset of a real Hilbert space H. Recall that a mapping A : C → H is called α -inverse strongly monotone if there exists a real number α > 0 such that
hAx − Ay, x − yi ≥ αkAx − Ayk2 ,
∀ x, y ∈ C .
A mapping S : C → C is said to be a strictly pseudo-contractive if there exists a constant 0 ≤ k < 1 such that
kSx − Syk2 ≤ kx − yk2 + kk(I − S )x − (I − S )yk2 ,
∀x, y ∈ C .
(2.1)
For such case, we also say that S is a k-strict pseudo-contraction. It is clear that, in a real Hilbert space H, (2.1) is equivalent to the following
hSx − Sy, x − yi ≤ kx − yk2 −
1−k
k(I − S )x − (I − S )yk2 , ∀x, y ∈ C . (2.2) 2 From [16], we know that if S is a k-strictly pseudocontractive mapping, then S satisfies Lipschitz condition kSx − Syk ≤ 1 +k kx − yk for all x, y ∈ C . We use Fix(S ) to denote the set of fixed points of S. It is well-known that the class of strict 1 −k pseudo-contractions strictly includes the class of nonexpansive mappings which are mappings S : C → C such that kSx − Syk ≤ kx − yk, ∀x, y ∈ C . A mapping Q : C → C is called contraction if there exists a constant ρ ∈ [0, 1) such that kQx − Qyk ≤ ρkx − yk for all x, y ∈ C . For every point x ∈ H, there exists a unique nearest point in C , denoted by PC x such that kx − PC xk ≤ kx − yk,
∀y ∈ C .
The mapping PC is called the metric projection of H onto C . It is well known that PC is a nonexpansive mapping and satisfies
hx − y, PC x − PC yi ≥ kPC x − PC yk2 ,
∀ x, y ∈ H .
It is known that PC x is characterized by the following property:
hx − PC x, y − PC xi ≤ 0,
∀x ∈ H , y ∈ C .
(2.3)
In order to prove our main results in the next section, we need the following lemmas. Lemma 2.1 ([15]). For given x∗ , y∗ ∈ C , (x∗ , y∗ ) is a solution of problem (1.2) if and only if x∗ is a fixed point of the mapping G : C → C defined by G(x) = PC [PC (x − µBx) − λAPC (x − µBx)],
∀x ∈ C ,
where y = PC (x − µBx ). In particular, if the mappings A, B : C → H are α -inverse strongly monotone and β -inverse strongly monotone, respectively, then the mapping G is nonexpansive provided λ ∈ (0, 2α) and µ ∈ (0, 2β). ∗
∗
∗
Throughout this paper, the set of fixed points of the mapping G is denoted by Γ . Lemma 2.2 ([17]). Let {xn } and {yn } be bounded sequences in a Banach space X and let {βn } be a sequence in [0, 1] with 0 < lim infn→∞ βn ≤ lim supn→∞ βn < 1. Suppose xn+1 = (1 − βn )yn + βn xn for all integers n ≥ 0 and lim supn→∞ (kyn+1 − yn k − kxn+1 − xn k) ≤ 0. Then, limn→∞ kyn − xn k = 0.
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Lemma 2.3 ([17]). Let C be a closed convex subset of a real Hilbert space H and let T : C → C be a k-strictly pseudo-contractive mapping. Then, the mapping I − T is demiclosed. That is, if {xn } is a sequence in C such that xn → x∗ weakly and (I − T )xn → y strongly, then (I − T )x∗ = y. Lemma 2.4 ([18]). Assume {an } is a sequence of nonnegative real numbers such that an+1 ≤ (1 − γn )an + δn where {γn } is a sequence in (0, 1) and {δn } is a sequence such that P∞ (a) n=1 γn = ∞; P∞ (b) lim supn→∞ δn /γn ≤ 0 or n=1 |δn | < ∞. Then limn→∞ an = 0. 3. Main results Now we state and prove our main results. Lemma 3.1. Let C be a nonempty closed convex subset of a real Hilbert space H. Let S : C → C be a k-strictly pseudocontractive mapping. Let γ and δ be two nonnegative real numbers. Assume (γ + δ)k ≤ γ . Then
kγ (x − y) + δ(Sx − Sy)k ≤ (γ + δ)kx − yk,
∀ x, y ∈ C .
Proof. From (2.1) and (2.2), we have
kγ (x − y) + δ(Sx − Sy)k2 = γ 2 kx − yk2 + δ 2 kSx − Syk2 + 2γ δhSx − Sy, x − yi ≤ γ 2 kx − yk2 + δ 2 [kx − yk2 + kk(I − S )x − (I − S )yk2 ] 1−k 2 2 + 2γ δ kx − yk − k(I − S )x − (I − S )yk 2
= (γ + δ) kx − yk + δ[(γ + δ)k − γ ]k(I − S )x − (I − S )yk2 ≤ (γ + δ)2 kx − yk2 , 2
2
which implies that
kγ (x − y) + δ(Sx − Sy)k ≤ (γ + δ)kx − yk.
(3.1)
Theorem 3.2. Let C be a nonempty bounded closed convex subset of a real Hilbert space H. Let the mappings A, B : C → H be α -inverse strongly monotone and β -inverse strongly monotone, respectively. Let S : C → C be a k-strictly pseudocontractive mapping such that Ω := Fix(S ) ∩ Γ 6= ∅. Let Q : C → C be a ρ -contraction with ρ ∈ 0, 12 . For given x0 ∈ C arbitrarily, let the sequences {xn }, {yn } and {zn } be generated iteratively by zn = PC (xn − µBxn ), yn = αn Qxn + (1 − αn )PC (zn − λAzn ), xn+1 = βn xn + γn PC (zn − λAzn ) + δn Syn ,
(
(3.2)
∀n ≥ 0,
where λ ∈ (0, 2α), µ ∈ (0, 2β) and {αn }, {βn }, {γn }, {δn } are four sequences in [0, 1] such that: (i) βn + γn + δn = 1 and n + δn )k ≤ γn < (1 − 2ρ)δn for all n ≥ 0; P(γ ∞ (ii) limn→∞ αn = 0 and n=0 αn = ∞; (iii) 0 < lim inf n→∞ βn ≤ limsupn→∞ βn < 1 and lim infn→∞ δn > 0; (iv) limn→∞
γn+1 1−βn+1
−
γn
1−βn
= 0.
Then the sequence {xn } generated by (3.2) converges strongly to x∗ = PΩ · Qx∗ and (x∗ , y∗ ) is a solution of the general system of variational inequalities (1.2), where y∗ = PC (x∗ − µBx∗ ). Proof. We divide the proof into several steps. Step 1. limn→∞ kxn+1 − xn k = 0. First, we can write (3.2) as xn+1 = βn xn + (1 − βn )un , n ≥ 0, where un = un+1 − un =
xn+2 − βn+1 xn+1
−
xn+1 −βn xn . 1−βn
It follows that
xn+1 − βn xn
1 − βn+1 1 − βn γn+1 PC (zn+1 − λAzn+1 ) + δn+1 Syn+1 γn PC (zn − λAzn ) + δn Syn = − 1 − βn+1 1 − βn γn+1 [PC (zn+1 − λAzn+1 ) − PC (zn − λAzn )] + δn+1 (Syn+1 − Syn ) = 1 − βn+1 γn+1 γn δn+1 δn + − PC (zn − λAzn ) + − Syn . 1 − βn+1 1 − βn 1 − βn+1 1 − βn
(3.3)
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From Lemma 3.1 and (3.2), we get
kγn+1 [PC (zn+1 − λAzn+1 ) − PC (zn − λAzn )] + δn+1 (Syn+1 − Syn )k ≤ kγn+1 (yn+1 − yn ) + δn+1 (Syn+1 − Syn )k + γn+1 k[PC (zn+1 − λAzn+1 ) − yn+1 ] + [yn − PC (zn − λAzn )]k ≤ (γn+1 + δn+1 )kyn+1 − yn k + γn+1 αn+1 kQxn+1 − PC (zn+1 − λAzn+1 )k + γn+1 αn kQxn − PC (zn − λAzn )k.
(3.4)
Since A, B are α -inverse strongly monotone mapping and β -inverse strongly monotone mapping, respectively, then we have
k(I − λA)x − (I − λA)yk2 = kx − yk2 − 2λhAx − Ay, x − yi + λ2 kAx − Ayk2 ≤ kx − yk2 + λ(λ − 2α)kAx − Ayk2 ,
(3.5)
k(I − µB)x − (I − µB)yk2 ≤ kx − yk2 + µ(µ − 2β)kBx − Byk2 .
(3.6)
and
It is clear that if 0 ≤ λ ≤ 2α and 0 ≤ µ ≤ 2β , then (I − λA) and (I − µB) are nonexpansive. It follows that
kPC (zn+1 − λAzn+1 ) − PC (zn − λAzn )k ≤ ≤ = ≤ ≤
kzn+1 − λAzn+1 − (zn − λAzn )k kzn+1 − zn k kPC (xn+1 − µBxn+1 ) − PC (xn − µBxn )k k(xn+1 − µBxn+1 ) − (xn − µBxn )k kxn+1 − xn k.
Then,
kyn+1 − yn k ≤ kPC (zn+1 − λAzn+1 ) − PC (zn − λAzn )k + αn+1 kQxn+1 − PC (zn+1 − λAzn+1 )k + αn kQxn − PC (zn − λAzn )k ≤ kxn+1 − xn k + αn kQxn − PC (zn − λAzn )k + αn+1 kQxn+1 − PC (zn+1 − λAzn+1 )k.
(3.7)
Therefore, from (3.3), (3.4) and (3.7), we have
γn+1 kun+1 − un k ≤ kxn+1 − xn k + 1 + αn kQxn − PC (zn − λAzn )k 1 − βn+1 γn+1 + 1+ αn+1 kQxn+1 − PC (zn+1 − λAzn+1 )k 1 − βn+1 γn+1 γn (kPC (zn − λAzn )k + kSyn k). + − 1 − βn+1 1 − βn This implies that lim sup(kun+1 − un k − kxn+1 − xn k) ≤ 0. n→∞
Hence by Lemma 2.2 we get limn→∞ kun − xn k = 0. Consequently, lim kxn+1 − xn k = lim (1 − βn )kun − xn k = 0.
n→∞
n→∞
(3.8)
Step 2. limn→∞ kAzn − Ay∗ k = 0 and limn→∞ kBxn − Bx∗ k = 0. Let x∗ ∈ Ω . From Lemma 2.1, we have x∗ = Sx∗ and x∗ = PC [PC (x∗ − µBx∗ ) − λAPC (x∗ − µBx∗ )]. Put y∗ = PC (x∗ − µBx∗ ). Then x∗ = PC (y∗ − λAy∗ ). From (3.5) and (3.6), we have
kPC (zn − λAzn ) − PC (y∗ − λAy∗ )k2 ≤ k(zn − λAzn ) − (y∗ − λAy∗ )k2 ≤ kzn − y∗ k2 + λ(λ − 2α)kAzn − Ay∗ k2 ,
(3.9)
and
kzn − y∗ k2 = kPC (xn − µBxn ) − PC (x∗ − µBx∗ )k2 ≤ k(xn − µBxn ) − (x∗ − µBx∗ )k2 ≤ kxn − x∗ k2 + µ(µ − 2β)kBxn − Bx∗ k2 .
(3.10)
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It follows from (3.2), (3.9) and (3.10) that
kyn − x∗ k2 ≤ (1 − αn )kPC (zn − λAzn ) − PC (y∗ − λAy∗ )k2 + αn kQxn − x∗ k2 ≤ αn kQxn − x∗ k2 + kzn − y∗ k2 + λ(λ − 2α)kAzn − Ay∗ k2 ≤ αn kQxn − x∗ k2 + kxn − x∗ k2 + µ(µ − 2β)kBxn − Bx∗ k2 + λ(λ − 2α)kAzn − Ay∗ k2 .
(3.11)
Using the convexity of k · k, we have
kxn+1 − x∗ k2 = kβn (xn − x∗ ) + (1 − βn )
1 1 − βn
[γn (PC (zn − λAzn ) − x∗ ) + δn (Syn − x∗ )]k2
2
γn
δn ∗ ∗
≤ βn kxn − x k + (1 − βn ) (PC (zn − λAzn ) − x ) + (Syn − x ) 1 − βn 1 − βn αn γn (yn − x∗ ) + δn (Syn − x∗ ) + (PC (zn − λAzn ) − Qxn )k2 = βn kxn − x∗ k2 + (1 − βn )k 1 − βn 1 − βn
γn (yn − x∗ ) + δn (Syn − x∗ ) 2
+ M αn ≤ βn kxn − x∗ k2 + (1 − βn )
1−β ∗ 2
n
≤ βn kxn − x k + (1 − βn )kyn − x k + M αn , ∗ 2
∗ 2
(3.12)
where M > 0 is some appropriate constant. So, by (3.11) and (3.12), we have
kxn+1 − x∗ k2 ≤ kxn − x∗ k2 + µ(µ − 2β)(1 − βn )kBxn − Bx∗ k2 + λ(λ − 2α)(1 − βn )kAzn − Ay∗ k2 + (M + kQxn − x∗ k2 )αn . Therefore,
λ(2α − λ)(1 − βn )kAzn − Ay∗ k2 + µ(2β − µ)(1 − βn )kBxn − Bx∗ k2 ≤ kxn − x∗ k2 − kxn+1 − x∗ k2 + (M + kQxn − x∗ k2 )αn ≤ (kxn − x∗ k + kxn+1 − x∗ k)kxn − xn+1 k + (M + kQxn − x∗ k2 )αn . Since lim infn→∞ λ(2α − λ)(1 − βn ) > 0, lim infn→∞ µ(2β − µ)(1 − βn ) > 0, kxn − xn+1 k → 0 and αn → 0, we have lim kAzn − Ay∗ k = 0
n→∞
and
lim kBxn − Bx∗ k = 0.
n→∞
Step 3. limn→∞ kSyn − yn k = 0. Set vn = PC (zn − λAzn ). Noting that PC is firmly nonexpansive, then we have
kzn − y∗ k2 = kPC (xn − µBxn ) − PC (x∗ − µBx∗ )k2 ≤ h(xn − µBxn ) − (x∗ − µBx∗ ), zn − y∗ i = ≤ =
1 2 1 2 1 2
(kxn − x∗ − µ(Bxn − Bx∗ )k2 + kzn − y∗ k2 − k(xn − x∗ ) − µ(Bxn − Bx∗ ) − (zn − y∗ )k2 ) (kxn − x∗ k2 + kzn − y∗ k2 − k(xn − zn ) − µ(Bxn − Bx∗ ) − (x∗ − y∗ )k2 ) (kxn − x∗ k2 + kzn − y∗ k2 − kxn − zn − (x∗ − y∗ )k2
+ 2µhxn − zn − (x∗ − y∗ ), Bxn − Bx∗ i − µ2 kBxn − Bx∗ k2 ), and
kvn − x∗ k = kPC (zn − λAzn ) − PC (y∗ − λAy∗ )k2 ≤ hzn − λAzn − (y∗ − λAy∗ ), vn − x∗ i = ≤
1 2 1 2
kzn − λAzn − (y∗ − λAy∗ )k2 + kvn − x∗ k2 − kzn − λAzn − (y∗ − λAy∗ ) − (vn − x∗ )k2
(kzn − y∗ k2 + kvn − x∗ k2 − kzn − vn + (x∗ − y∗ )k2
+ 2λhAzn − Ay∗ , zn − vn + (x∗ − y∗ )i − λ2 kAzn − Ay∗ k2 ) ≤
1 2
(kxn − x∗ k2 + kvn − x∗ k2 − kzn − vn + (x∗ − y∗ )k2 + 2λhAzn − Ay∗ , zn − vn + (x∗ − y∗ )i).
Thus, we have
kzn − y∗ k2 ≤ kxn − x∗ k2 − kxn − zn − (x∗ − y∗ )k2 + 2µhxn − zn − (x∗ − y∗ ), Bxn − Bx∗ i − µ2 kBxn − Bx∗ k2 , (3.13)
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and
kvn − x∗ k2 ≤ kxn − x∗ k2 − kzn − vn + (x∗ − y∗ )k2 + 2λkAzn − Ay∗ k kzn − vn + (x∗ − y∗ )k. It follows that
kyn − x∗ k2 ≤ αn kQxn − x∗ k2 + (1 − αn )kvn − x∗ k2 ≤ αn kQxn − x∗ k2 + kvn − x∗ k2 ≤ αn kQxn − x∗ k2 + kxn − x∗ k2 − kzn − vn + (x∗ − y∗ )k2 + 2λkAzn − Ay∗ k kzn − vn + (x∗ − y∗ )k. (3.14) By (3.11), (3.12) and (3.13), we have
kxn+1 − x∗ k2 ≤ βn kxn − x∗ k2 + (1 − βn )αn kQxn − x∗ k2 + (1 − βn )kzn − y∗ k2 + M αn ≤ kxn − x∗ k2 − (1 − βn )kxn − zn − (x∗ − y∗ )k2 + 2(1 − βn )µkxn − zn − (x∗ − y∗ )k kBxn − Bx∗ k + (M + (1 − βn )αn )αn . It follows that
(1 − βn )kxn − zn − (x∗ − y∗ )k2 ≤ (kxn − x∗ k + kxn+1 − x∗ k)kxn+1 − xn k + (M + (1 − βn )αn )αn + 2(1 − βn )µkxn − zn − (x∗ − y∗ )k kBxn − Bx∗ k. Note that kxn+1 − xn k → 0, αn → 0 and kBxn − Bx∗ k → 0. Then we immediately deduce lim kxn − zn + (x∗ − y∗ )k = 0.
n→∞
(3.15)
By (3.12) and (3.14), we have
kxn+1 − x∗ k2 ≤ kxn − x∗ k2 − (1 − βn )kzn − vn + (x∗ − y∗ )k2 + 2λ(1 − βn )kAzn − Ay∗ k kzn − vn + (x∗ − y∗ )k + (M + kQxn − x∗ k2 )αn . So, we obtain
(1 − βn )kzn − vn + (x∗ − y∗ )k2 ≤ kxn − x∗ k2 − kxn+1 − x∗ k2 + 2λ(1 − βn )kAzn − Ay∗ k × kzn − vn + (x∗ − y∗ )k + (M + kQxn − x∗ k2 )αn . Hence, lim kzn − vn − (x∗ − y∗ )k = 0.
n→∞
This together with kyn − vn k ≤ αn kQxn − vn k → 0 imply that lim kzn − yn − (x∗ − y∗ )k = 0.
n→∞
(3.16)
Thus, from (3.15) and (3.16), we deduce that lim kxn − yn k = 0.
n→∞
Since
kδn (Syn − xn )k ≤ kxn+1 − xn k + γn kPC (zn − λAzn ) − xn k ≤ kxn+1 − xn k + γn kyn − xn k + γn αn kQxn − xn k. Therefore, lim kSyn − xn k = 0
n→∞
and
lim kSyn − yn k = 0.
n→∞
Step 4. lim supn→∞ hQx∗ − x∗ , xn − x∗ i ≤ 0 where x∗ = PΩ · Qx∗ . As {yn } is bounded, there exists a subsequence {yni } of {yn } such that yni → v weakly. First, it is clear from Lemma 2.3 that v ∈ Fix(S ). Next, we prove that v ∈ Γ . We note that
kyn − G(yn )k ≤ = ≤ →
αn kQxn − G(yn )k + (1 − αn )kPC [PC (xn − µBxn ) − λAPC (xn − µBxn )] − G(yn )k αn kQxn − G(yn )k + (1 − αn )kG(xn ) − G(yn )k αn kQxn − G(yn )k + (1 − αn )kxn − yn k 0.
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According to Lemma 2.3 we obtain v ∈ Γ . Therefore, v ∈ Ω . Hence, it follows from (2.3) that lim suphQx∗ − x∗ , xn − x∗ i = lim hQx∗ − x∗ , xni − x∗ i i→∞
n→∞
= hQx∗ − x∗ , v − x∗ i ≤ 0. Step 5. limn→∞ xn = x∗ . From (3.2) and the convexity of k · k, we have
kxn+1 − x∗ k2 = kβn (xn − x∗ ) + γn (yn − x∗ ) + δn (Syn − x∗ ) + γn αn (PC (zn − λAzn ) − Qxn )k2 ≤ kβn (xn − x∗ ) + γn (yn − x∗ ) + δn (Syn − x∗ )k2 + 2γn αn hPC (zn − λAzn ) − Qxn , xn+1 − x∗ i
2
1 ∗ ∗ [γ ( y − x ) + δ ( Sy − x )] ≤ βn kxn − x∗ k2 + (1 − βn ) n n
1 − β n n n
+ 2γn αn hPC (zn − λAzn ) − x , xn+1 − x∗ i + 2γn αn hx∗ − Qxn , xn+1 − x∗ i. ∗
(3.17)
By Lemma 3.1 and (3.17), we have
kxn+1 − x∗ k2 ≤ βn kxn − x∗ k2 + (1 − βn )kyn − x∗ k2 + 2γn αn kPC (zn − λAzn ) − x∗ k kxn+1 − x∗ k + 2γn αn hx∗ − Qxn , xn+1 − x∗ i ≤ βn kxn − x∗ k2 + (1 − βn )[(1 − αn )kzn − y∗ k2 + 2αn hQxn − x∗ , yn − x∗ i] + 2γn αn kzn − y∗ k kxn+1 − x∗ k + 2γn αn hx∗ − Qxn , xn+1 − x∗ i. From (3.10), we note that kzn − y∗ k ≤ kxn − x∗ k. Hence, we have
kxn+1 − x∗ k2 ≤ βn kxn − x∗ k2 + (1 − βn )(1 − αn )kxn − x∗ k2 + 2αn (1 − βn )hQxn − x∗ , yn − x∗ i + 2γn αn kxn − x∗ k kxn+1 − x∗ k + 2γn αn hx∗ − Qxn , xn+1 − x∗ i ≤ [1 − (1 − βn )αn ]kxn − x∗ k2 + 2αn γn hQxn − x∗ , yn − xn+1 i + 2αn δn hQxn − x∗ , yn − x∗ i + 2αn γn kxn − x∗ k kxn+1 − x∗ k ≤ [1 − (1 − βn )αn ]kxn − x∗ k2 + 2αn γn kQxn − x∗ k kyn − xn+1 k + 2αn δn hQxn − x∗ , xn − x∗ i + 2αn δn hQxn − x∗ , yn − xn i + 2αn γn kxn − x∗ k kxn+1 − x∗ k ≤ [1 − (1 − βn )αn ]kxn − x∗ k2 + 2αn γn kQxn − x∗ k kyn − xn+1 k + 2αn δn ρkxn − x∗ k2 + 2αn δn hQx∗ − x∗ , xn − x∗ i + 2αn δn kQxn − x∗ k kyn − xn k + 2αn γn kxn − x∗ k kxn+1 − x∗ k ≤ [1 − (1 − βn )αn ]kxn − x∗ k2 + 2αn γn kQxn − x∗ k kyn − xn+1 k + 2αn δn ρkxn − x∗ k2 + 2αn δn hQx∗ − x∗ , xn − x∗ i + 2αn δn kQxn − x∗ k kyn − xn k + αn γn (kxn − x∗ k2 + kxn+1 − x∗ k2 ), that is,
[(1 − 2ρ)δn − γn ]αn (1 − 2ρ)δn − γn kxn+1 − x∗ k2 ≤ 1 − αn kxn − x∗ k2 + 1 − αn γn 1 − αn γn ( 2γn kQxn − x∗ k kyn − xn+1 k × (1 − 2ρ)δn − γn + Note that lim infn→∞
2δn
(1 − 2ρ)δn − γn
(1−2ρ)δn −γn 1−αn γn
2δn
∗
kQxn − x k kyn − xn k +
> 0. It follows that
P∞
n =0
(1 − 2ρ)δn − γn
(1−2ρ)δn −γn αn 1−αn γn
) hQx − x , xn − x i . ∗
∗
∗
= ∞. It is clear that
(
2γn 2δn kQxn − x∗ k kyn − xn+1 k + kQxn − x∗ k kyn − xn k (1 − 2ρ)δn − γn (1 − 2ρ)δn − γn ) 2δn ∗ ∗ ∗ + hQx − x , xn − x i ≤ 0. (1 − 2ρ)δn − γn
lim sup n→∞
Therefore, all conditions of Lemma 2.4 are satisfied. Therefore, we immediately deduce that xn → x∗ . This completes the proof.
Y. Yao et al. / Computers and Mathematics with Applications 59 (2010) 3472–3480
3479
Corollary 3.3. Let C be a nonempty bounded closed convex subset of a real Hilbert space H. Let the mappings A, B : C → H be α -inverse strongly monotone and β -inverse strongly monotone, respectively. Let S : C → C be a k-strictly pseudocontractive mapping such that Ω := Fix(S ) ∩ Γ 6= ∅. For fixed u ∈ C and given x0 ∈ C arbitrarily, let the sequences {xn }, {yn } and {zn } be generated iteratively by zn = PC (xn − µBxn ), yn = αn u + (1 − αn )PC (zn − λAzn ), xn+1 = βn xn + γn PC (zn − λAzn ) + δn Syn ,
(
∀n ≥ 0,
where λ ∈ (0, 2α), µ ∈ (0, 2β) and {αn }, {βn }, {γn }, {δn } are four sequences in [0, 1] such that: (i) βn + γn + δn = 1 and n + δn )k ≤ γn < δn for all n ≥ 0; P(γ ∞ (ii) limn→∞ αn = 0 and n=0 αn = ∞; (iii) 0 < lim inf n→∞ βn ≤ limsupn→∞ βn < 1 and lim infn→∞ δn > 0; (iv) limn→∞
γn+1 1−βn+1
−
γn
1−βn
= 0.
Then the sequence {xn } converges strongly to x∗ = PΩ u and (x∗ , y∗ ) is a solution of the general system of variational inequalities (1.2), where y∗ = PC (x∗ − µBx∗ ). Corollary 3.4. Let C be a nonempty bounded closed convex subset of a real Hilbert space H. Let the mappings A, B : C → H be α -inverse strongly monotone and β -inverse strongly monotone, respectively. Let S : C → C be a nonexpansive mapping such that Ω := Fix(S ) ∩ Γ 6= ∅. Let Q : C → C be a ρ -contraction with ρ ∈ 0, 12 . For given x0 ∈ C arbitrarily, let the sequences {xn }, {yn } and {zn } be generated iteratively by zn = PC (xn − µBxn ), yn = αn Qxn + (1 − αn )PC (zn − λAzn ), xn+1 = βn xn + γn PC (zn − λAzn ) + δn Syn ,
(
∀n ≥ 0 ,
where λ ∈ (0, 2α), µ ∈ (0, 2β) and {αn }, {βn }, {γn }, {δn } are four sequences in [0, 1] such that: (i) βn + γn + δn = 1 and Pγ∞n < (1 − 2ρ)δn for all n ≥ 0; (ii) limn→∞ αn = 0 and n=0 αn = ∞; (iii) 0 < lim inf n→∞ βn ≤ limsupn→∞ βn < 1 and lim infn→∞ γn > 0; (iv) limn→∞
γn+1 1−βn+1
−
γn
1−βn
= 0.
Then the sequence {xn } converges strongly to x∗ = PΩ · Qx∗ and (x∗ , y∗ ) is a solution of the general system of variational inequalities (1.2), where y∗ = PC (x∗ − µBx∗ ). Corollary 3.5. Let C be a nonempty bounded closed convex subset of a real Hilbert space H. Let the mappings A, B : C → H be α -inverse strongly monotone and β -inverse strongly monotone, respectively. Let S : C → C be a nonexpansive mapping such that Ω := Fix(S ) ∩ Γ 6= ∅. For fixed u ∈ C and given x0 ∈ C arbitrarily, let the sequences {xn }, {yn } and {zn } be generated iteratively by zn = PC (xn − µBxn ), yn = αn u + (1 − αn )PC (zn − λAzn ), xn+1 = βn xn + γn PC (zn − λAzn ) + δn Syn ,
(
∀n ≥ 0,
where λ ∈ (0, 2α), µ ∈ (0, 2β) and {αn }, {βn }, {γn }, {δn } are four sequences in [0, 1] such that: (i) βn + γn + δn = 1 and Pγ∞n < (1 − 2ρ)δn for all n ≥ 0; (ii) limn→∞ αn = 0 and n=0 αn = ∞; (iii) 0 < lim inf n→∞ βn ≤ limsupn→∞ βn < 1 and lim infn→∞ γn > 0; (iv) limn→∞
γn+1 1−βn+1
−
γn
1−βn
= 0.
Then the sequence {xn } converges strongly to x∗ = PΩ · Qx∗ and (x∗ , y∗ ) is a solution of the general system of variational inequalities (1.2), where y∗ = PC (x∗ − µBx∗ ). Acknowledgements The first author was partially supported by the National Natural Science Foundation of China, Grant 10771050. The second author was partially supported by Grant 98-2622-E-230-006-CC3 and NSC 98-2923-E-110-003-MY3. References [1] [2] [3] [4]
F.E. Browder, W.V. Petryshyn, Construction of fixed points of nonlinear mappings in Hilbert space, J. Math. Anal. Appl. 20 (1967) 197–228. F. Liu, M.Z. Nashed, Regularization of nonlinear ill-posed variational inequalities and convergence rates, Set-Valued Anal. 6 (1998) 313–344. J.C. Yao, Variational inequalities with generalized monotone operators, Math. Oper. Res. 19 (1994) 691–705. L.C. Zeng, S. Schaible, J.C. Yao, Iterative algorithm for generalized set-valued strongly nonlinear mixed variational-like inequalities, J. Optim. Theory Appl. 124 (2005) 725–738.
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[5] L.C. Ceng, J.C. Yao, An extragradient-like approximation method for variational inequality problems and fixed point problems, Appl. Math. Comput. 190 (2007) 205–215. [6] M.Aslam Noor, Some developments in general variational inequalities, Appl. Math. Comput. (2004) 191–277. [7] Y. Censor, A.N. Iusem, S.A. Zenios, An interior point method with Bregman functions for the variational inequality problem with paramonotone operators, Math. Program. 81 (1998) 373–400. [8] W. Takahashi, M. Toyoda, Weak convergence theorems for nonexpansive mappings and monotone mappings, J. Optim. Theory Appl. 118 (2003) 417–428. [9] N. Nadezhkina, W. Takahashi, Weak convergence theorem by an extragradient method for nonexpansive mappings and monotone mappings, J. Optim. Theory Appl. 128 (2006) 191–201. [10] L.C. Zeng, J.C. Yao, Strong convergence theorem by an extragradient method for fixed point problems and variational inequality problems, Taiwanese J. Math. 10 (2006) 1293–1303. [11] G.M. Korpelevich, An extragradient method for finding saddle points and for other problems, Ekonom. i Mat. Metody 12 (1976) 747–756. [12] Y. Yao, J.C. Yao, On modified iterative method for nonexpansive mappings and monotone mappings, Appl. Math. Comput. 186 (2007) 1551–1558. [13] R.U. Verma, On a new system of nonlinear variational inequalities and associated iterative algorithms, Math. Sci. Res. Hot-Line 3 (1999) 65–68. [14] R.U. Verma, Iterative algorithms and a new system of nonlinear quasivariational inequalities, Adv. Nonlinear Var. Inequal. 4 (2001) 117–124. [15] L.C. Ceng, C. Wang, J.C. Yao, Strong convergence theorems by a relaxed extragradient method for a general system of variational inequalities, Math. Methods Oper. Res. 67 (2008) 375–390. [16] G.L. Acedo, H.K Xu, Iterative methods for strict pseudo-contractions in Hilbert spaces, Nonlinear Anal. 67 (2007) 2258–2271. [17] T. Suzuki, Strong convergence of Krasnoselskii and Mann’s type sequences for one-parameter nonexpansive semigroups without Bochner integrals, J. Math. Anal. Appl. 305 (2005) 227–239. [18] H.K. Xu, Viscosity approximation methods for nonexpansive mappings, J. Math. Anal. Appl. 298 (2004) 279–291.
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Computers and Mathematics with Applications journal homepage: www.elsevier.com/locate/camwa
Approach to common elements of variational inequality problems and fixed point problems via a relaxed extragradient method Yonghong Yao a , Yeong-Cheng Liou b , Shin Min Kang c,∗ a
Department of Mathematics, Tianjin Polytechnic University, Tianjin 300160, People’s Republic of China
b
Department of Information Management, Cheng Shiu University, Kaohsiung 833, Taiwan
c
Department of Mathematics and the RINS, Gyeongsang National University, Jinju 660-701, Republic of Korea
article
abstract
info
Article history: Received 30 April 2009 Accepted 22 March 2010 Keywords: Variational inequality Fixed point Monotone mapping Strictly pseudocontractive mapping
In this paper, we introduce an iterative method based on the extragradient method for finding a common element of the set of a general system of variational inequalities and the set of fixed points of a strictly pseudocontractive mapping in a real Hilbert space. Furthermore, we prove that the studied iterative method strongly converges to a common element of the set of a general system of variational inequalities and the set of fixed points of a strictly pseudocontractive mapping under some mild conditions imposed on algorithm parameters. © 2010 Elsevier Ltd. All rights reserved.
1. Introduction Let C be a nonempty closed convex subset of a real Hilbert space H. For a given nonlinear operator A : C → H, we consider the following variational inequality problem of finding x∗ ∈ C such that
hAx∗ , x − x∗ i ≥ 0,
∀x ∈ C .
(1.1)
The set of solutions of the variational inequality (1.1) is denoted by VI (A, C ). Variational inequality theory has emerged as an important tool in studying a wide class of obstacle, unilateral, free, moving, equilibrium problems arising in several branches of pure and applied sciences in a unified and general framework. The variational inequality problem has been extensively studied in the literature. See, e.g. [1–7] and the references therein. For finding an element of Fix(S ) ∩ VI (A, C ) under the assumption that a set C ⊂ H is closed and convex, a mapping S of C into itself is nonexpansive and a mapping A of C into H is α -inverse strongly monotone, Takahashi and Toyoda [8] introduced the following iterative scheme: xn+1 = αn xn + (1 − αn )SPC (xn − λn Axn ),
∀n ≥ 0,
where PC is the metric projection of H onto C , x0 = x ∈ C , {αn } is a sequence in (0, 1), and {λn } is a sequence in (0, 2α). They showed that, if Fix(S ) ∩ VI (A, C ) is nonempty, then the sequence {xn } converges weakly to some z ∈ Fix(S ) ∩ VI (A, C ). Recently, Nadezhkina and Takahashi [9] and Zeng and Yao [10] proposed some so-called extragradient method motivated by the idea of Korpelevich [11] for finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of a variational inequality problem. Further, these iterative methods are extended in [12] to develop a new iterative method for finding elements in Fix(S ) ∩ VI (A, C ).
∗
Corresponding author. Fax: +82 55 755 1917. E-mail addresses: [email protected] (Y. Yao), [email protected] (Y.-C. Liou), [email protected] (S.M. Kang).
0898-1221/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.camwa.2010.03.036
Y. Yao et al. / Computers and Mathematics with Applications 59 (2010) 3472–3480
3473
Let A, B : C → H be two mappings. Now we concern the following problem of finding (x∗ , y∗ ) ∈ C × C such that
hλAy∗ + x∗ − y∗ , x − x∗ i ≥ 0, hµBx∗ + y∗ − x∗ , x − y∗ i ≥ 0,
∀x ∈ C , ∀x ∈ C ,
(1.2)
which is called a general system of variational inequalities where λ > 0 and µ > 0 are two constants. The set of solutions of (1.2) is denoted by GSVI(A, B, C ). In particular, if A = B, then problem (1.2) reduces to finding (x∗ , y∗ ) ∈ C × C such that
hλAy∗ + x∗ − y∗ , x − x∗ i ≥ 0, hµAx∗ + y∗ − x∗ , x − y∗ i ≥ 0,
∀x ∈ C , ∀x ∈ C ,
(1.3)
which is defined by Verma [13] (see also [14]) and is called the new system of variational inequalities. Further, if we add up the requirement that x∗ = y∗ , then problem (1.3) reduces to the classical variational inequality problem (1.1). For solving problem (1.2), recently, Ceng et al. [15] introduced and studied a relaxed extragradient method. It is clear that their results unified and extended many results in the literature. Motivated and inspired by the above works, in this paper, we introduce an iterative method based on the extragradient method for finding a common element of the set of a general system of variational inequalities and the set of fixed points of a strictly pseudocontractive mapping in a real Hilbert space. Furthermore, we prove that the studied iterative method strongly converges to a common element of the set of a general system of variational inequalities and the set of fixed points of a strictly pseudocontractive mapping under some mild conditions imposed on algorithm parameters. 2. Preliminaries In this section, we collect some notations and lemmas. Let C be a nonempty closed convex subset of a real Hilbert space H. Recall that a mapping A : C → H is called α -inverse strongly monotone if there exists a real number α > 0 such that
hAx − Ay, x − yi ≥ αkAx − Ayk2 ,
∀ x, y ∈ C .
A mapping S : C → C is said to be a strictly pseudo-contractive if there exists a constant 0 ≤ k < 1 such that
kSx − Syk2 ≤ kx − yk2 + kk(I − S )x − (I − S )yk2 ,
∀x, y ∈ C .
(2.1)
For such case, we also say that S is a k-strict pseudo-contraction. It is clear that, in a real Hilbert space H, (2.1) is equivalent to the following
hSx − Sy, x − yi ≤ kx − yk2 −
1−k
k(I − S )x − (I − S )yk2 , ∀x, y ∈ C . (2.2) 2 From [16], we know that if S is a k-strictly pseudocontractive mapping, then S satisfies Lipschitz condition kSx − Syk ≤ 1 +k kx − yk for all x, y ∈ C . We use Fix(S ) to denote the set of fixed points of S. It is well-known that the class of strict 1 −k pseudo-contractions strictly includes the class of nonexpansive mappings which are mappings S : C → C such that kSx − Syk ≤ kx − yk, ∀x, y ∈ C . A mapping Q : C → C is called contraction if there exists a constant ρ ∈ [0, 1) such that kQx − Qyk ≤ ρkx − yk for all x, y ∈ C . For every point x ∈ H, there exists a unique nearest point in C , denoted by PC x such that kx − PC xk ≤ kx − yk,
∀y ∈ C .
The mapping PC is called the metric projection of H onto C . It is well known that PC is a nonexpansive mapping and satisfies
hx − y, PC x − PC yi ≥ kPC x − PC yk2 ,
∀ x, y ∈ H .
It is known that PC x is characterized by the following property:
hx − PC x, y − PC xi ≤ 0,
∀x ∈ H , y ∈ C .
(2.3)
In order to prove our main results in the next section, we need the following lemmas. Lemma 2.1 ([15]). For given x∗ , y∗ ∈ C , (x∗ , y∗ ) is a solution of problem (1.2) if and only if x∗ is a fixed point of the mapping G : C → C defined by G(x) = PC [PC (x − µBx) − λAPC (x − µBx)],
∀x ∈ C ,
where y = PC (x − µBx ). In particular, if the mappings A, B : C → H are α -inverse strongly monotone and β -inverse strongly monotone, respectively, then the mapping G is nonexpansive provided λ ∈ (0, 2α) and µ ∈ (0, 2β). ∗
∗
∗
Throughout this paper, the set of fixed points of the mapping G is denoted by Γ . Lemma 2.2 ([17]). Let {xn } and {yn } be bounded sequences in a Banach space X and let {βn } be a sequence in [0, 1] with 0 < lim infn→∞ βn ≤ lim supn→∞ βn < 1. Suppose xn+1 = (1 − βn )yn + βn xn for all integers n ≥ 0 and lim supn→∞ (kyn+1 − yn k − kxn+1 − xn k) ≤ 0. Then, limn→∞ kyn − xn k = 0.
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Y. Yao et al. / Computers and Mathematics with Applications 59 (2010) 3472–3480
Lemma 2.3 ([17]). Let C be a closed convex subset of a real Hilbert space H and let T : C → C be a k-strictly pseudo-contractive mapping. Then, the mapping I − T is demiclosed. That is, if {xn } is a sequence in C such that xn → x∗ weakly and (I − T )xn → y strongly, then (I − T )x∗ = y. Lemma 2.4 ([18]). Assume {an } is a sequence of nonnegative real numbers such that an+1 ≤ (1 − γn )an + δn where {γn } is a sequence in (0, 1) and {δn } is a sequence such that P∞ (a) n=1 γn = ∞; P∞ (b) lim supn→∞ δn /γn ≤ 0 or n=1 |δn | < ∞. Then limn→∞ an = 0. 3. Main results Now we state and prove our main results. Lemma 3.1. Let C be a nonempty closed convex subset of a real Hilbert space H. Let S : C → C be a k-strictly pseudocontractive mapping. Let γ and δ be two nonnegative real numbers. Assume (γ + δ)k ≤ γ . Then
kγ (x − y) + δ(Sx − Sy)k ≤ (γ + δ)kx − yk,
∀ x, y ∈ C .
Proof. From (2.1) and (2.2), we have
kγ (x − y) + δ(Sx − Sy)k2 = γ 2 kx − yk2 + δ 2 kSx − Syk2 + 2γ δhSx − Sy, x − yi ≤ γ 2 kx − yk2 + δ 2 [kx − yk2 + kk(I − S )x − (I − S )yk2 ] 1−k 2 2 + 2γ δ kx − yk − k(I − S )x − (I − S )yk 2
= (γ + δ) kx − yk + δ[(γ + δ)k − γ ]k(I − S )x − (I − S )yk2 ≤ (γ + δ)2 kx − yk2 , 2
2
which implies that
kγ (x − y) + δ(Sx − Sy)k ≤ (γ + δ)kx − yk.
(3.1)
Theorem 3.2. Let C be a nonempty bounded closed convex subset of a real Hilbert space H. Let the mappings A, B : C → H be α -inverse strongly monotone and β -inverse strongly monotone, respectively. Let S : C → C be a k-strictly pseudocontractive mapping such that Ω := Fix(S ) ∩ Γ 6= ∅. Let Q : C → C be a ρ -contraction with ρ ∈ 0, 12 . For given x0 ∈ C arbitrarily, let the sequences {xn }, {yn } and {zn } be generated iteratively by zn = PC (xn − µBxn ), yn = αn Qxn + (1 − αn )PC (zn − λAzn ), xn+1 = βn xn + γn PC (zn − λAzn ) + δn Syn ,
(
(3.2)
∀n ≥ 0,
where λ ∈ (0, 2α), µ ∈ (0, 2β) and {αn }, {βn }, {γn }, {δn } are four sequences in [0, 1] such that: (i) βn + γn + δn = 1 and n + δn )k ≤ γn < (1 − 2ρ)δn for all n ≥ 0; P(γ ∞ (ii) limn→∞ αn = 0 and n=0 αn = ∞; (iii) 0 < lim inf n→∞ βn ≤ limsupn→∞ βn < 1 and lim infn→∞ δn > 0; (iv) limn→∞
γn+1 1−βn+1
−
γn
1−βn
= 0.
Then the sequence {xn } generated by (3.2) converges strongly to x∗ = PΩ · Qx∗ and (x∗ , y∗ ) is a solution of the general system of variational inequalities (1.2), where y∗ = PC (x∗ − µBx∗ ). Proof. We divide the proof into several steps. Step 1. limn→∞ kxn+1 − xn k = 0. First, we can write (3.2) as xn+1 = βn xn + (1 − βn )un , n ≥ 0, where un = un+1 − un =
xn+2 − βn+1 xn+1
−
xn+1 −βn xn . 1−βn
It follows that
xn+1 − βn xn
1 − βn+1 1 − βn γn+1 PC (zn+1 − λAzn+1 ) + δn+1 Syn+1 γn PC (zn − λAzn ) + δn Syn = − 1 − βn+1 1 − βn γn+1 [PC (zn+1 − λAzn+1 ) − PC (zn − λAzn )] + δn+1 (Syn+1 − Syn ) = 1 − βn+1 γn+1 γn δn+1 δn + − PC (zn − λAzn ) + − Syn . 1 − βn+1 1 − βn 1 − βn+1 1 − βn
(3.3)
Y. Yao et al. / Computers and Mathematics with Applications 59 (2010) 3472–3480
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From Lemma 3.1 and (3.2), we get
kγn+1 [PC (zn+1 − λAzn+1 ) − PC (zn − λAzn )] + δn+1 (Syn+1 − Syn )k ≤ kγn+1 (yn+1 − yn ) + δn+1 (Syn+1 − Syn )k + γn+1 k[PC (zn+1 − λAzn+1 ) − yn+1 ] + [yn − PC (zn − λAzn )]k ≤ (γn+1 + δn+1 )kyn+1 − yn k + γn+1 αn+1 kQxn+1 − PC (zn+1 − λAzn+1 )k + γn+1 αn kQxn − PC (zn − λAzn )k.
(3.4)
Since A, B are α -inverse strongly monotone mapping and β -inverse strongly monotone mapping, respectively, then we have
k(I − λA)x − (I − λA)yk2 = kx − yk2 − 2λhAx − Ay, x − yi + λ2 kAx − Ayk2 ≤ kx − yk2 + λ(λ − 2α)kAx − Ayk2 ,
(3.5)
k(I − µB)x − (I − µB)yk2 ≤ kx − yk2 + µ(µ − 2β)kBx − Byk2 .
(3.6)
and
It is clear that if 0 ≤ λ ≤ 2α and 0 ≤ µ ≤ 2β , then (I − λA) and (I − µB) are nonexpansive. It follows that
kPC (zn+1 − λAzn+1 ) − PC (zn − λAzn )k ≤ ≤ = ≤ ≤
kzn+1 − λAzn+1 − (zn − λAzn )k kzn+1 − zn k kPC (xn+1 − µBxn+1 ) − PC (xn − µBxn )k k(xn+1 − µBxn+1 ) − (xn − µBxn )k kxn+1 − xn k.
Then,
kyn+1 − yn k ≤ kPC (zn+1 − λAzn+1 ) − PC (zn − λAzn )k + αn+1 kQxn+1 − PC (zn+1 − λAzn+1 )k + αn kQxn − PC (zn − λAzn )k ≤ kxn+1 − xn k + αn kQxn − PC (zn − λAzn )k + αn+1 kQxn+1 − PC (zn+1 − λAzn+1 )k.
(3.7)
Therefore, from (3.3), (3.4) and (3.7), we have
γn+1 kun+1 − un k ≤ kxn+1 − xn k + 1 + αn kQxn − PC (zn − λAzn )k 1 − βn+1 γn+1 + 1+ αn+1 kQxn+1 − PC (zn+1 − λAzn+1 )k 1 − βn+1 γn+1 γn (kPC (zn − λAzn )k + kSyn k). + − 1 − βn+1 1 − βn This implies that lim sup(kun+1 − un k − kxn+1 − xn k) ≤ 0. n→∞
Hence by Lemma 2.2 we get limn→∞ kun − xn k = 0. Consequently, lim kxn+1 − xn k = lim (1 − βn )kun − xn k = 0.
n→∞
n→∞
(3.8)
Step 2. limn→∞ kAzn − Ay∗ k = 0 and limn→∞ kBxn − Bx∗ k = 0. Let x∗ ∈ Ω . From Lemma 2.1, we have x∗ = Sx∗ and x∗ = PC [PC (x∗ − µBx∗ ) − λAPC (x∗ − µBx∗ )]. Put y∗ = PC (x∗ − µBx∗ ). Then x∗ = PC (y∗ − λAy∗ ). From (3.5) and (3.6), we have
kPC (zn − λAzn ) − PC (y∗ − λAy∗ )k2 ≤ k(zn − λAzn ) − (y∗ − λAy∗ )k2 ≤ kzn − y∗ k2 + λ(λ − 2α)kAzn − Ay∗ k2 ,
(3.9)
and
kzn − y∗ k2 = kPC (xn − µBxn ) − PC (x∗ − µBx∗ )k2 ≤ k(xn − µBxn ) − (x∗ − µBx∗ )k2 ≤ kxn − x∗ k2 + µ(µ − 2β)kBxn − Bx∗ k2 .
(3.10)
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It follows from (3.2), (3.9) and (3.10) that
kyn − x∗ k2 ≤ (1 − αn )kPC (zn − λAzn ) − PC (y∗ − λAy∗ )k2 + αn kQxn − x∗ k2 ≤ αn kQxn − x∗ k2 + kzn − y∗ k2 + λ(λ − 2α)kAzn − Ay∗ k2 ≤ αn kQxn − x∗ k2 + kxn − x∗ k2 + µ(µ − 2β)kBxn − Bx∗ k2 + λ(λ − 2α)kAzn − Ay∗ k2 .
(3.11)
Using the convexity of k · k, we have
kxn+1 − x∗ k2 = kβn (xn − x∗ ) + (1 − βn )
1 1 − βn
[γn (PC (zn − λAzn ) − x∗ ) + δn (Syn − x∗ )]k2
2
γn
δn ∗ ∗
≤ βn kxn − x k + (1 − βn ) (PC (zn − λAzn ) − x ) + (Syn − x ) 1 − βn 1 − βn αn γn (yn − x∗ ) + δn (Syn − x∗ ) + (PC (zn − λAzn ) − Qxn )k2 = βn kxn − x∗ k2 + (1 − βn )k 1 − βn 1 − βn
γn (yn − x∗ ) + δn (Syn − x∗ ) 2
+ M αn ≤ βn kxn − x∗ k2 + (1 − βn )
1−β ∗ 2
n
≤ βn kxn − x k + (1 − βn )kyn − x k + M αn , ∗ 2
∗ 2
(3.12)
where M > 0 is some appropriate constant. So, by (3.11) and (3.12), we have
kxn+1 − x∗ k2 ≤ kxn − x∗ k2 + µ(µ − 2β)(1 − βn )kBxn − Bx∗ k2 + λ(λ − 2α)(1 − βn )kAzn − Ay∗ k2 + (M + kQxn − x∗ k2 )αn . Therefore,
λ(2α − λ)(1 − βn )kAzn − Ay∗ k2 + µ(2β − µ)(1 − βn )kBxn − Bx∗ k2 ≤ kxn − x∗ k2 − kxn+1 − x∗ k2 + (M + kQxn − x∗ k2 )αn ≤ (kxn − x∗ k + kxn+1 − x∗ k)kxn − xn+1 k + (M + kQxn − x∗ k2 )αn . Since lim infn→∞ λ(2α − λ)(1 − βn ) > 0, lim infn→∞ µ(2β − µ)(1 − βn ) > 0, kxn − xn+1 k → 0 and αn → 0, we have lim kAzn − Ay∗ k = 0
n→∞
and
lim kBxn − Bx∗ k = 0.
n→∞
Step 3. limn→∞ kSyn − yn k = 0. Set vn = PC (zn − λAzn ). Noting that PC is firmly nonexpansive, then we have
kzn − y∗ k2 = kPC (xn − µBxn ) − PC (x∗ − µBx∗ )k2 ≤ h(xn − µBxn ) − (x∗ − µBx∗ ), zn − y∗ i = ≤ =
1 2 1 2 1 2
(kxn − x∗ − µ(Bxn − Bx∗ )k2 + kzn − y∗ k2 − k(xn − x∗ ) − µ(Bxn − Bx∗ ) − (zn − y∗ )k2 ) (kxn − x∗ k2 + kzn − y∗ k2 − k(xn − zn ) − µ(Bxn − Bx∗ ) − (x∗ − y∗ )k2 ) (kxn − x∗ k2 + kzn − y∗ k2 − kxn − zn − (x∗ − y∗ )k2
+ 2µhxn − zn − (x∗ − y∗ ), Bxn − Bx∗ i − µ2 kBxn − Bx∗ k2 ), and
kvn − x∗ k = kPC (zn − λAzn ) − PC (y∗ − λAy∗ )k2 ≤ hzn − λAzn − (y∗ − λAy∗ ), vn − x∗ i = ≤
1 2 1 2
kzn − λAzn − (y∗ − λAy∗ )k2 + kvn − x∗ k2 − kzn − λAzn − (y∗ − λAy∗ ) − (vn − x∗ )k2
(kzn − y∗ k2 + kvn − x∗ k2 − kzn − vn + (x∗ − y∗ )k2
+ 2λhAzn − Ay∗ , zn − vn + (x∗ − y∗ )i − λ2 kAzn − Ay∗ k2 ) ≤
1 2
(kxn − x∗ k2 + kvn − x∗ k2 − kzn − vn + (x∗ − y∗ )k2 + 2λhAzn − Ay∗ , zn − vn + (x∗ − y∗ )i).
Thus, we have
kzn − y∗ k2 ≤ kxn − x∗ k2 − kxn − zn − (x∗ − y∗ )k2 + 2µhxn − zn − (x∗ − y∗ ), Bxn − Bx∗ i − µ2 kBxn − Bx∗ k2 , (3.13)
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and
kvn − x∗ k2 ≤ kxn − x∗ k2 − kzn − vn + (x∗ − y∗ )k2 + 2λkAzn − Ay∗ k kzn − vn + (x∗ − y∗ )k. It follows that
kyn − x∗ k2 ≤ αn kQxn − x∗ k2 + (1 − αn )kvn − x∗ k2 ≤ αn kQxn − x∗ k2 + kvn − x∗ k2 ≤ αn kQxn − x∗ k2 + kxn − x∗ k2 − kzn − vn + (x∗ − y∗ )k2 + 2λkAzn − Ay∗ k kzn − vn + (x∗ − y∗ )k. (3.14) By (3.11), (3.12) and (3.13), we have
kxn+1 − x∗ k2 ≤ βn kxn − x∗ k2 + (1 − βn )αn kQxn − x∗ k2 + (1 − βn )kzn − y∗ k2 + M αn ≤ kxn − x∗ k2 − (1 − βn )kxn − zn − (x∗ − y∗ )k2 + 2(1 − βn )µkxn − zn − (x∗ − y∗ )k kBxn − Bx∗ k + (M + (1 − βn )αn )αn . It follows that
(1 − βn )kxn − zn − (x∗ − y∗ )k2 ≤ (kxn − x∗ k + kxn+1 − x∗ k)kxn+1 − xn k + (M + (1 − βn )αn )αn + 2(1 − βn )µkxn − zn − (x∗ − y∗ )k kBxn − Bx∗ k. Note that kxn+1 − xn k → 0, αn → 0 and kBxn − Bx∗ k → 0. Then we immediately deduce lim kxn − zn + (x∗ − y∗ )k = 0.
n→∞
(3.15)
By (3.12) and (3.14), we have
kxn+1 − x∗ k2 ≤ kxn − x∗ k2 − (1 − βn )kzn − vn + (x∗ − y∗ )k2 + 2λ(1 − βn )kAzn − Ay∗ k kzn − vn + (x∗ − y∗ )k + (M + kQxn − x∗ k2 )αn . So, we obtain
(1 − βn )kzn − vn + (x∗ − y∗ )k2 ≤ kxn − x∗ k2 − kxn+1 − x∗ k2 + 2λ(1 − βn )kAzn − Ay∗ k × kzn − vn + (x∗ − y∗ )k + (M + kQxn − x∗ k2 )αn . Hence, lim kzn − vn − (x∗ − y∗ )k = 0.
n→∞
This together with kyn − vn k ≤ αn kQxn − vn k → 0 imply that lim kzn − yn − (x∗ − y∗ )k = 0.
n→∞
(3.16)
Thus, from (3.15) and (3.16), we deduce that lim kxn − yn k = 0.
n→∞
Since
kδn (Syn − xn )k ≤ kxn+1 − xn k + γn kPC (zn − λAzn ) − xn k ≤ kxn+1 − xn k + γn kyn − xn k + γn αn kQxn − xn k. Therefore, lim kSyn − xn k = 0
n→∞
and
lim kSyn − yn k = 0.
n→∞
Step 4. lim supn→∞ hQx∗ − x∗ , xn − x∗ i ≤ 0 where x∗ = PΩ · Qx∗ . As {yn } is bounded, there exists a subsequence {yni } of {yn } such that yni → v weakly. First, it is clear from Lemma 2.3 that v ∈ Fix(S ). Next, we prove that v ∈ Γ . We note that
kyn − G(yn )k ≤ = ≤ →
αn kQxn − G(yn )k + (1 − αn )kPC [PC (xn − µBxn ) − λAPC (xn − µBxn )] − G(yn )k αn kQxn − G(yn )k + (1 − αn )kG(xn ) − G(yn )k αn kQxn − G(yn )k + (1 − αn )kxn − yn k 0.
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According to Lemma 2.3 we obtain v ∈ Γ . Therefore, v ∈ Ω . Hence, it follows from (2.3) that lim suphQx∗ − x∗ , xn − x∗ i = lim hQx∗ − x∗ , xni − x∗ i i→∞
n→∞
= hQx∗ − x∗ , v − x∗ i ≤ 0. Step 5. limn→∞ xn = x∗ . From (3.2) and the convexity of k · k, we have
kxn+1 − x∗ k2 = kβn (xn − x∗ ) + γn (yn − x∗ ) + δn (Syn − x∗ ) + γn αn (PC (zn − λAzn ) − Qxn )k2 ≤ kβn (xn − x∗ ) + γn (yn − x∗ ) + δn (Syn − x∗ )k2 + 2γn αn hPC (zn − λAzn ) − Qxn , xn+1 − x∗ i
2
1 ∗ ∗ [γ ( y − x ) + δ ( Sy − x )] ≤ βn kxn − x∗ k2 + (1 − βn ) n n
1 − β n n n
+ 2γn αn hPC (zn − λAzn ) − x , xn+1 − x∗ i + 2γn αn hx∗ − Qxn , xn+1 − x∗ i. ∗
(3.17)
By Lemma 3.1 and (3.17), we have
kxn+1 − x∗ k2 ≤ βn kxn − x∗ k2 + (1 − βn )kyn − x∗ k2 + 2γn αn kPC (zn − λAzn ) − x∗ k kxn+1 − x∗ k + 2γn αn hx∗ − Qxn , xn+1 − x∗ i ≤ βn kxn − x∗ k2 + (1 − βn )[(1 − αn )kzn − y∗ k2 + 2αn hQxn − x∗ , yn − x∗ i] + 2γn αn kzn − y∗ k kxn+1 − x∗ k + 2γn αn hx∗ − Qxn , xn+1 − x∗ i. From (3.10), we note that kzn − y∗ k ≤ kxn − x∗ k. Hence, we have
kxn+1 − x∗ k2 ≤ βn kxn − x∗ k2 + (1 − βn )(1 − αn )kxn − x∗ k2 + 2αn (1 − βn )hQxn − x∗ , yn − x∗ i + 2γn αn kxn − x∗ k kxn+1 − x∗ k + 2γn αn hx∗ − Qxn , xn+1 − x∗ i ≤ [1 − (1 − βn )αn ]kxn − x∗ k2 + 2αn γn hQxn − x∗ , yn − xn+1 i + 2αn δn hQxn − x∗ , yn − x∗ i + 2αn γn kxn − x∗ k kxn+1 − x∗ k ≤ [1 − (1 − βn )αn ]kxn − x∗ k2 + 2αn γn kQxn − x∗ k kyn − xn+1 k + 2αn δn hQxn − x∗ , xn − x∗ i + 2αn δn hQxn − x∗ , yn − xn i + 2αn γn kxn − x∗ k kxn+1 − x∗ k ≤ [1 − (1 − βn )αn ]kxn − x∗ k2 + 2αn γn kQxn − x∗ k kyn − xn+1 k + 2αn δn ρkxn − x∗ k2 + 2αn δn hQx∗ − x∗ , xn − x∗ i + 2αn δn kQxn − x∗ k kyn − xn k + 2αn γn kxn − x∗ k kxn+1 − x∗ k ≤ [1 − (1 − βn )αn ]kxn − x∗ k2 + 2αn γn kQxn − x∗ k kyn − xn+1 k + 2αn δn ρkxn − x∗ k2 + 2αn δn hQx∗ − x∗ , xn − x∗ i + 2αn δn kQxn − x∗ k kyn − xn k + αn γn (kxn − x∗ k2 + kxn+1 − x∗ k2 ), that is,
[(1 − 2ρ)δn − γn ]αn (1 − 2ρ)δn − γn kxn+1 − x∗ k2 ≤ 1 − αn kxn − x∗ k2 + 1 − αn γn 1 − αn γn ( 2γn kQxn − x∗ k kyn − xn+1 k × (1 − 2ρ)δn − γn + Note that lim infn→∞
2δn
(1 − 2ρ)δn − γn
(1−2ρ)δn −γn 1−αn γn
2δn
∗
kQxn − x k kyn − xn k +
> 0. It follows that
P∞
n =0
(1 − 2ρ)δn − γn
(1−2ρ)δn −γn αn 1−αn γn
) hQx − x , xn − x i . ∗
∗
∗
= ∞. It is clear that
(
2γn 2δn kQxn − x∗ k kyn − xn+1 k + kQxn − x∗ k kyn − xn k (1 − 2ρ)δn − γn (1 − 2ρ)δn − γn ) 2δn ∗ ∗ ∗ + hQx − x , xn − x i ≤ 0. (1 − 2ρ)δn − γn
lim sup n→∞
Therefore, all conditions of Lemma 2.4 are satisfied. Therefore, we immediately deduce that xn → x∗ . This completes the proof.
Y. Yao et al. / Computers and Mathematics with Applications 59 (2010) 3472–3480
3479
Corollary 3.3. Let C be a nonempty bounded closed convex subset of a real Hilbert space H. Let the mappings A, B : C → H be α -inverse strongly monotone and β -inverse strongly monotone, respectively. Let S : C → C be a k-strictly pseudocontractive mapping such that Ω := Fix(S ) ∩ Γ 6= ∅. For fixed u ∈ C and given x0 ∈ C arbitrarily, let the sequences {xn }, {yn } and {zn } be generated iteratively by zn = PC (xn − µBxn ), yn = αn u + (1 − αn )PC (zn − λAzn ), xn+1 = βn xn + γn PC (zn − λAzn ) + δn Syn ,
(
∀n ≥ 0,
where λ ∈ (0, 2α), µ ∈ (0, 2β) and {αn }, {βn }, {γn }, {δn } are four sequences in [0, 1] such that: (i) βn + γn + δn = 1 and n + δn )k ≤ γn < δn for all n ≥ 0; P(γ ∞ (ii) limn→∞ αn = 0 and n=0 αn = ∞; (iii) 0 < lim inf n→∞ βn ≤ limsupn→∞ βn < 1 and lim infn→∞ δn > 0; (iv) limn→∞
γn+1 1−βn+1
−
γn
1−βn
= 0.
Then the sequence {xn } converges strongly to x∗ = PΩ u and (x∗ , y∗ ) is a solution of the general system of variational inequalities (1.2), where y∗ = PC (x∗ − µBx∗ ). Corollary 3.4. Let C be a nonempty bounded closed convex subset of a real Hilbert space H. Let the mappings A, B : C → H be α -inverse strongly monotone and β -inverse strongly monotone, respectively. Let S : C → C be a nonexpansive mapping such that Ω := Fix(S ) ∩ Γ 6= ∅. Let Q : C → C be a ρ -contraction with ρ ∈ 0, 12 . For given x0 ∈ C arbitrarily, let the sequences {xn }, {yn } and {zn } be generated iteratively by zn = PC (xn − µBxn ), yn = αn Qxn + (1 − αn )PC (zn − λAzn ), xn+1 = βn xn + γn PC (zn − λAzn ) + δn Syn ,
(
∀n ≥ 0 ,
where λ ∈ (0, 2α), µ ∈ (0, 2β) and {αn }, {βn }, {γn }, {δn } are four sequences in [0, 1] such that: (i) βn + γn + δn = 1 and Pγ∞n < (1 − 2ρ)δn for all n ≥ 0; (ii) limn→∞ αn = 0 and n=0 αn = ∞; (iii) 0 < lim inf n→∞ βn ≤ limsupn→∞ βn < 1 and lim infn→∞ γn > 0; (iv) limn→∞
γn+1 1−βn+1
−
γn
1−βn
= 0.
Then the sequence {xn } converges strongly to x∗ = PΩ · Qx∗ and (x∗ , y∗ ) is a solution of the general system of variational inequalities (1.2), where y∗ = PC (x∗ − µBx∗ ). Corollary 3.5. Let C be a nonempty bounded closed convex subset of a real Hilbert space H. Let the mappings A, B : C → H be α -inverse strongly monotone and β -inverse strongly monotone, respectively. Let S : C → C be a nonexpansive mapping such that Ω := Fix(S ) ∩ Γ 6= ∅. For fixed u ∈ C and given x0 ∈ C arbitrarily, let the sequences {xn }, {yn } and {zn } be generated iteratively by zn = PC (xn − µBxn ), yn = αn u + (1 − αn )PC (zn − λAzn ), xn+1 = βn xn + γn PC (zn − λAzn ) + δn Syn ,
(
∀n ≥ 0,
where λ ∈ (0, 2α), µ ∈ (0, 2β) and {αn }, {βn }, {γn }, {δn } are four sequences in [0, 1] such that: (i) βn + γn + δn = 1 and Pγ∞n < (1 − 2ρ)δn for all n ≥ 0; (ii) limn→∞ αn = 0 and n=0 αn = ∞; (iii) 0 < lim inf n→∞ βn ≤ limsupn→∞ βn < 1 and lim infn→∞ γn > 0; (iv) limn→∞
γn+1 1−βn+1
−
γn
1−βn
= 0.
Then the sequence {xn } converges strongly to x∗ = PΩ · Qx∗ and (x∗ , y∗ ) is a solution of the general system of variational inequalities (1.2), where y∗ = PC (x∗ − µBx∗ ). Acknowledgements The first author was partially supported by the National Natural Science Foundation of China, Grant 10771050. The second author was partially supported by Grant 98-2622-E-230-006-CC3 and NSC 98-2923-E-110-003-MY3. References [1] [2] [3] [4]
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