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Generalized extragradient iterative method for systems of variational inequalities
Journal of Inequalities and Applications volume 2012, Article number: 88 (2012)
Abstract
The purpose of this article is to investigate the problem of finding a common element of the solution sets of two different systems of variational inequalities and the set of fixed points a strict pseudocontraction mapping defined in the setting of a real Hilbert space. Based on the well-known extragradient method, viscosity approximation method and Mann iterative method, we propose and analyze a generalized extra-gradient iterative method for computing a common element. Under very mild assumptions, we obtain a strong convergence theorem for three sequences generated by the proposed method. Our proposed method is quite general and flexible and includes the iterative methods considered in the earlier and recent literature as special cases. Our result represents the modification, supplement, extension and improvement of some corresponding results in the references.
Mathematics Subject Classification (2000): Primary 49J40; Secondary 65K05; 47H09.
1. Introduction
Let H be a real Hilbert space with inner product 〈·, ·〉 and norm ║ · ║. Let C be a nonempty closed convex subset of H and S : C → C be a self-mapping on C. We denote by Fix(S) the set of fixed points of S and by P C the metric projection of H onto C. Moreover, we also denote by R the set of all real numbers. For a given nonlinear mapping A : C → H, consider the following classical variational inequality problem of finding x* ∈ C such that
The set of solutions of problem (1.1) is denoted by VI(A, C). It is now well known that the variational inequalities are equivalent to the fixed-point problems, the origin of which can be traced back to Lions and Stampacchia [1]. This alternative formulation has been used to suggest and analyze Picard successive iterative method for solving variational inequalities under the conditions that the involved operator must be strongly monotone and Lipschitz continuous. Related to the variational inequalities, we have the problem of finding fixed points of nonexpansive mappings or strict pseudo-contractions, which is the current interest in functional analysis. Several authors considered some approaches to solve fixed point problems, optimization problems, variational inequality problems and equilibrium problems; see, for example, [2–32] and the references therein.
For finding an element of Fix(S) ∩ VI(A, C) under the assumption that a set C ⊂ H is nonempty, closed and convex, a mapping S : C → C is nonexpansive and a mapping A : C → H is α-inverse strongly monotone, Takahashi and Toyoda [20] introduced the following iterative algorithm:
where {α n } is a sequence in (0, 1), and {λ n } is a sequence in (0, 2α). It was proven in [20] that if then the sequence {x n } converges weakly to some z ∈ Fix(S) ∩ VI(A, C). Recently, Nadezhkina and Takahashi [19] and Zeng and Yao [32] proposed some so-called extragra-dient method motivated by the idea of Korpelevich [33] for finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of a variational inequality. Further, these iterative methods were extended in [27] to develop a general iterative method for finding a element of Fix(S) ∩ VI(A, C).
Let A1, A2 : C → H be two mappings. In this article, we consider the following problem of finding (x*, y*) ∈ C × C such that
which is called a general system of variational inequalities, where λ1 > 0 and λ2 > 0 are two constants. It was introduced and considered by Ceng et al. [7]. In particular, if A1 = A2 = A, then problem (1.2) reduces to the following problem of finding (x*, y*) ∈ C × C such that
which was defined by Verma [22] (see also [21]) and it is called a new system of variational inequalities. Further, if x* = y* additionally, then problem (1.3) reduces to the classical variational inequality problem (1.1). We remark that in [34], Ceng et al. proposed a hybrid extragradient method for finding a common element of the solution set of a variational inequality problem, the solution set of problem (1.2) and the fixed-point set of a strictly pseudocontractive mapping in a real Hilbert space. Recently, Ceng et al. [7] transformed problem (1.2) into a fixed point problem in the following way:
Lemma 1.1.[7]. For given is a solution of problem (1.2) if and only if is a fixed point of the mapping G: C → C defined by
where
In particular, if the mapping A i : C → H is -inverse strongly monotone for i = 1, 2, then the mapping G is nonexpansive provided fori = 1, 2.
Utilizing Lemma 1.1, they proposed and analyzed a relaxed extragradient method for solving problem (1.2). Throughout this article, the set of fixed points of the mapping G is denoted by Γ. Based on the extragradient method [33] and viscosity approximation method [23], Yao et al. [26] introduced and studied a relaxed extragradient iterative algorithm for finding a common solution of problem (1.2) and the fixed point problem of a strictly pseudocontraction in a real Hilbert space H.
Theorem 1.1. [[26], Theorem 3.2]. Let C be a nonempty bounded closed convex subset of a real Hilbert space H. Let the mapping A i : C → H be -inverse strongly monotone for i = 1, 2. Let S : C → C be a k-strict pseudocontraction mapping such that . Let Q : C → C be a ρ-contraction mapping with For given x0 ∈ C arbitrarily, let the sequences {x n }, {y n } and {z n } be generated iteratively by
where for i = 1, 2, and {α n }, {β n }, {γ n }, {δ n } are four sequences in [0, 1] such that
(i) β n + γ n + δ n = 1 and (γ n + δ n )k ≤ γ n <(1 - 2ρ)δ n for all n ≥ 0;
(ii) and
(iii) and
(iv)
Then the sequence {x n } generated by (1.5) converges strongly to x* = P Ω Qx* and (x*, y*) is a solution of the general system of variational inequalities (1.2), where y* = P C (x* - λ2A2x*).
Let B1, B2 : C → H be two mappings. In this article, we also consider another general system of variational inequalities, that is, finding (x*, y*) ∈ C × C such that
where μ1 > 0 and μ2 > 0 are two constants.
Utilizing Lemma 1.1, we know that for given is a solution of problem (1.6) if and only if is a fixed point of the mapping F : C → C defined by
where In particular, if the mapping B i : C → H is -inverse strongly monotone for i = 1, 2, then the mapping F is nonexpansive provided for i = 1, 2. Throughout this article, the set of fixed points of the mapping F is denoted by Γ0.
Assume that A i : C → H is -inverse strongly monotone and and B i : C → H is -inverse strongly monotone for i = 1, 2. Let S : C → C be a k-strict pseudocontraction mapping such that . Let Q : C → C be a ρ-contraction mapping with . Motivated and inspired by the research work going on in this area, we propose and analyze the following iterative scheme for computing a common element of the solution set Γ of one general system of variational inequalities (1.2), the solution set Γ0 of another general system of variational inequalities (1.6), and the fixed point set Fix(S) of the mapping S:
where and for i = 1, 2, and {α n },{β n }, {γ n }, {δ n } ⊂ [0, 1] such that β n + γ n + δ n = 1 for all n ≥ 0. Furthermore, it is proven that the sequences {x n }, {y n } and {z n } generated by (1.8) converge strongly to the same point x* = P Ω Qx* under very mild conditions, and (x*, y*) and are a solution of general system of variational inequalities (1.2) and a solution of general system of variational inequalities (1.6), respectively, where y* = P C (x* - λ2A2x*) and
Our result represents the modification, supplement, extension and improvement of the above Theorem 1.1 in the following aspects.
-
(a)
our problem of finding an element of Fix(S) ∩ Γ ∩ Γ0 is more general and more complex than the problem of finding an element of Fix(S) ∩ Γ in the above Theorem 1.1.
-
(b)
Algorithm (1.8) for finding an element of Fix(S)∩Γ∩Γ0 is also more general and more flexible than algorithm (1.5) for finding an element of Fix(S) ∩ Γ in the above Theorem 1.1. Indeed, whenever B1 = B2 = 0, we have
In this case, algorithm (1.8) reduces essentially to algorithm (1.5).
-
(c)
Algorithm (1.8) is very different from algorithm (1.5) in the above Theorem YLK because algorithm (1.8) is closely related to the viscosity approximation method with the ρ-contraction Q : C → C and involves the Picard successive iteration for the general system of variational inequalities (1.6).
-
(d)
The techniques of proving strong convergence in our result are very different from those in the above Theorem 1.1 because our techniques depend on the norm inequality in Lemma 2.2 and the inverse-strong monotonicity of mappings A i , B i : C → H for i = 1, 2, the demiclosed-ness principle for strict pseudocontractions, and the transformation of two general systems of variational inequalities (1.2) and (1.6) into the fixed-point problems of the nonexpansive self-mappings G: C → C and F: C → C (see the above Lemma 1.1, respectively.
2. Preliminaries
Let H be a real Hilbert space whose inner product and norm are 〈·, ·〉 and ║ · ║, respectively. Let C be a nonempty closed convex subset of H. We write → to indicate that the sequence {x n } converges strongly to x and ⇀ to indicate that the sequence {x n } converges weakly to x. Moreover, we use ω w (x n ) to denote the weak ω-limit set of the sequence {x n }, that is,
Recall that a mapping A : C → H is called α-inverse strongly monotone if there exists a constant α > 0 such that
It is obvious that any α-inverse strongly monotone mapping is Lipschitz continuous. A mapping S : C → C is called a strict pseudocontraction [35] if there exists a constant 0 ≤ k < 1 such that
In this case, we also say that S is a k-strict pseudocontraction. Meantime, observe that (2.1) is equivalent to the following
It is easy to see that if S is a k-strictly pseudocontractive mapping, then I - S is -inverse strongly monotone and hence -Lipschitz continuous; for further detail, we refer to [30] and the references therein. It is clear that the class of strict pseudocontractions strictly includes the one of nonexpansive mappings which are mappings S : C → C such that ║Sx - Sy║ ≤ ║x - y║ for all x, y ∈ C.
For every point x ∈ H, there exists a unique nearest point in C, denoted by P C x such that
The mapping P C is called the metric projection of H onto C. We know that P C is a firmly nonexpansive mapping of H onto C; that is, there holds the following relation
Consequently, P C is nonexpansive and monotone. It is also known that P C is characterized by the following properties: P C x ∈ C and
See [36] for more details.
In order to prove our main result in the next section, we need the following lemmas. The following lemma is an immediate consequence of an inner product.
Lemma 2.1. In a real Hilbert space H, there holds the inequality
Recall that S : C → C is called a quasi-strict pseudocontraction if the fixed point set of S, Fix(S), is nonempty and if there exists a constant 0 ≤ k < 1 such that
We also say that S is a k-quasi-strict pseudocontraction if condition (2.6) holds.
The following lemma was proved by Suzuki [37].
Lemma 2.2.[37]Let {x n } and {y n } be bounded sequences in a Banach space X and let {β n } be a sequence in [0, 1] with . Suppose x n+ 1 = (1 - β n )y n + β n x n for all integers n ≥ 0 and . Then, .
Lemma 2.3. [[17], Proposition 2.1] Assume C is a nonempty closed convex subset of a real Hilbert space H and let S: C → C be a self-mapping on C.
(a) If S is a k-strict pseudocontraction, then S satisfies the Lipschitz condition
(b) if S is a k-strict pseudocontraction, then the mapping I - S is demiclosed (at 0). That is, if {x n } is a sequence in C such that and (I - S)x n → 0, then , i.e.,
(c) if S is a k-quasi-strict pseudocontraction, then the fixed point set Fix(S) of S is closed and convex so that the projection PFix(S)is well defined.
Lemma 2.4.[24]Let {a n } be a sequence of nonnegative numbers satisfying the condition
where {δ n }, {σ n } are sequences of real numbers such that
(i) {δ n } ⊂ [0, 1] and , or equivalently,
(ii) or
(ii) is convergent.
Then .
3. Strong convergence theorems
We are now in a position to state and prove our main result.
Lemma 3.1. [[26], Lemma 3.1] Let C be a nonempty closed convex subset of a real Hilbert space H. Let S: C → C be a k-strict pseudocontraction mapping. Let γ and δ be two nonnegative real numbers. Assume (γ + δ)k ≤ γ. Then
Theorem 3.1. Let C be a nonempty bounded closed convex subset of a real Hilbert space H. Assume that for i = 1, 2, the mappings A i , B i : C → H are -inverse strongly monotone and -inverse strongly monotone, respectively. Let S : C → C be a k-strict pseudocontraction mapping such that . Let Q : C → C be a ρ-contraction mapping with For given x0 ∈ C arbitrarily, let the sequences {x n }, {y n } and {z n } be generated iteratively by
where and for i = 1, 2, and {α n }, {β n }, {γ n }, {δ n } are four sequences in [0, 1] such that:
(i) β n + γ n + δ n = 1 and (γ n + δ n )k ≤ γ n <(1 - 2ρ)δ n for all n ≥ 0;
(ii) and
(iii) and
(iv)
Then, the sequences {x n }, {y n }, {z n } generated by (3.2) converge strongly to the same point x* = P Ω Qx*, and (x*, y*) and are a solution of general system of variational inequalities (1.2) and a solution of general system of variational inequalities (1.6), respectively, where y* = P C (x* - λ2A2x*) and .
Proof. Let us show that the mappings I - λ i A i and I - μ i Bi are nonexpansive for i = 1, 2. Indeed, since for i = 1, 2, A i , B i are -inverse strongly monotone and -inverse strongly monotone, respectively, we have for all x, y ∈ C
and
This shows that both I - λ i A i and I - μ i Bi are nonexpansive for i = 1, 2.
We divide the rest of the proof into several steps.
Step 1. .
Indeed, first, we can write (3.2) as x n+ 1 = β n x n + (1 - β n )u n , ∀n ≥ 0, where . Set . It follows that
From Lemma 3.1 and (3.2), we get
Note that
and
Then it follows from (3.5) and (3.6) that
Therefore, from (3.3), (3.4) and (3.7), we have
This implies that
Hence by Lemma 2.2 we get limn → ∞‖u n - x n ‖ = 0. Consequently,
Step 2. .
Indeed, let x* ∈ Ω. Utilizing Lemma 1.1 we have x* = Sx*, x* = P C [P C (x* - λ2A2x*) - λ1A1P C (x* - λ2A2x*)] and
Put y* = P C (x* - λ2A2x*) and . Then x* = P C (y* - λ1A1y*) and . Thus it follows that
and
It follows from (3.2), (3.9) and (3.10) that
Utilizing the convexity of ║ · ║2, we have
where M > 0 is some appropriate constant. So, from (3.11) and (3.12) we have
Therefore,
Since , ║x n - x n+1 ║ → 0 and α n → 0, we have
Step 3. limn → ∞║z n - y n ║ = limn → ∞║x n - z n ║ = limn → ∞║Sy n - y n ║ = 0.
Indeed, set . Noting that P C is firmly nonexpansive, we have
and
due to (3.9). Thus, we have
and
It follows that
Utilizing (3.2), (3.10), (3.12) and (3.13), we have
It follows that
Note that ║x n+ 1 - x n ║ → 0, α n → 0 and ║A2z n - A2x*║ → 0. Then we immediately deduce that
In the meantime, utilizing (3.10), (3.12) and (3.14) we have
So, we obtain
Hence,
This together with ║y n - υ n ║ ≤ α n ║Qx n - υ n ║ → 0, implies that
Thus, from (3.15) and (3.16) we conclude that
On the other hand, by firm nonexpansiveness of P C , we have
and
Thus, we have
and
Consequently, from (3.10), (3.11), (3.12) and (3.17) we have
It follows that
Note that ║x n +1 - x n ║ → 0, α n → 0 and ║B2x n - B2x*║ → 0. Then we immediately deduce that
Furthermore, utilizing (3.11), (3.12) and (3.18) we have
So, we get
Hence,
Thus, from (3.19) and (3.20) we conclude that
Since
so we obtain that
Step 4. lim sup n→ ∞ 〈Qx* - x*, x n - x*〉 ≤ 0 where x* = P Ω Qx*.
Indeed, as H is reflexive and {x n } is bounded, we may assume, without loss of generality, that there exists a subsequence of {x n } such that and
From Step 3 it is known that ║x n - y n ║ → 0 as n → ∞. This together with , implies that . Again from Step 3 it is known that ║y n - Sy n ║ → 0 as n → ∞. Thus it is clear from
Lemma 2.3 (ii) that υ ∈ Fix(S). Next, we prove that υ ∈ Γ ∩ Γ 0 . As a matter of fact, observe that
and
where G and F are given in (1.4) and (1.7), respectively. According to Lemma 2.3 (ii) we obtain υ ∈ Γ ∩ Γ 0 . Therefore, υ ∈ Ω. Hence, it follows from (2.3) that
Step 5. lim n→∞ x n = x*.
Indeed, from (3.2) and the convexity of || · ||2, we have
By Lemma 3.1 and (3.21), we have
From (3.10), we note that ║z n - x*║ ≤ ║x n - x*║. Hence, according to 1 - β n = γ n + δ n we have
that is,
Note that . It follows that . It is obvious that
Therefore, all conditions of Lemma 2.4 are satisfied. Consequently, in terms of Lemma 2.4 we immediately deduce that x n → x*. This completes the proof. □
Next we present some applications of Theorem 3.1 in several special cases.
Corollary 3.1. Let C be a nonempty bounded closed convex subset of a real Hilbert space H. Let the mapping A i : C → H be -inverse strongly monotone for i = 1, 2. Let S : C → C be a k-strict pseudocontraction such that . Let Q : C → C be a ρ-contraction with . For given x0 ∈ C arbitrarily, let the sequences {x n }, {y n } be generated iteratively by
where for i= 1, 2, and {α n }, {β n }, {γ n }, {δ n } are four sequences in [0, 1] such that:
(i) β n + γ n + δ n = 1 and (γ n + δ n )k ≤ γ n <(1 - 2ρ)δ n for all n ≥ 0;
(ii) limn→∞α n = 0 and ;
(iii) 0 < lim infn→∞β n ≤ lim supn→∞β n < 1 and lim infn→∞δ n > 0;
(iv) .
Then the sequences {x n }, {y n } converge strongly to the same point x* = P Ω Qx*, and (x*, y*) is a solution of general system (1.2) of variational inequalities, where y* = P C (x* - λ2A2x*).
Proof. It is easy to see that if B i = 0 for i = 1, 2, then for any given , B i is -inverse strongly monotone. In Theorem 3.1, putting B i = 0 and taking for i = 1, 2 we have Ω := Fix(S) ∩ Γ n Γ 0 = Fix(S) ∩ Γ and
In this case, algorithm (3.2) reduces to the following algorithm
Therefore, in terms of Theorem 3.1 we immediately obtain the desired result. □
Remark 3.1. Compared with Theorem YLK (i.e., [[26], Theorem 3.2]), Corollary 3.1 coincides essentially with Theorem YLK. Therefore, Theorem 3.1 includes Theorem YLK as a special case.
Corollary 3.2. Let C be a nonempty bounded closed convex subset of a real Hilbert space H. Assume that for i = 1, 2, the mappings A i , B i : C → H are -inverse strongly monotone and -inverse strongly monotone, respectively. Let S : C → C be a k-strict pseudocontraction such that . For fixed u ∈ C and given x0 ∈ C arbitrarily, let the sequences {x n }, {y n } and {z n } be generated iteratively by
where and and {α n }, {β n }, {γ n }, {δ n } are four sequences in [0, 1] such that:
(i) β n + γ n + δ n = 1 and (γ n + δ n )k ≤ γ n < δ n for all n ≥ 0;
(ii) limn→∞α n = 0 and ;
(iii) 0 < lim infn→∞β n ≤ lim supn→∞β n < 1 and lim infn→∞δ n > 0;
(iv) .
Then the sequences {x n }, {y n }, {z n } converge strongly to the same point x* = P Ω u, and (x*, y*) and are a solution of general system (1.2) of variational inequalities and a solution of general system (1.6) of variational inequalities, respectively, where y* = P C (x* - λ2A2x*) and .
Corollary 3.3. Let C be a nonempty bounded closed convex subset of a real Hilbert space H. Assume that for i = 1, 2, the mappings A i , B i : C → H are -inverse strongly monotone and -inverse strongly monotone, respectively. Let S : C → C be a nonexpansive mapping such that . Let Q : C → C be a ρ-contraction with . For given x0 ∈ C arbitrarily, let the sequences {x n }, {y n } and {z n } be generated iteratively by
where and for i = 1, 2, and {α n }, {β n }, {γ n }, {δ n } are four sequences in [0, 1] such that:
(i) β n + γ n + δ n = 1 and γ n <(1 - 2ρ)δ n for all n ≥ 0;
(ii) limn→∞α n = 0 and ;
(iii) 0 < lim infn→∞β n ≤ lim supn→∞β n < 1 and lim infn→∞γ n > 0;
(iv) .
Then the sequences {x n }, {y n }, {z n } converge strongly to the same point x* = P Ω Qx*, and (x*, y*) and are a solution of general system (1.2) of variational inequalities and a solution of general system (1.6) of variational inequalities, respectively, where y* = P C (x* - λ2A2x*) and .
Corollary 3.4. Let C be a nonempty bounded closed convex subset of a real Hilbert space H. Assume that for i = 1, 2, the mappings A i , B i : C → H are -inverse strongly monotone and -inverse strongly monotone, respectively. Let S : C → C be a nonexpansive mapping such that . For fixed u ∈ C and given x0 ∈ C arbitrarily, let the sequences {x n }, {y n } and {z n } be generated iteratively by
where and for i = 1, 2, and {α n }, {β n }, {γ n }, {δ n } are four sequences in [0, 1] such that:
(i) β n + γ n + δ n = 1 and γ n < δ n for all n ≥ 0;
(ii) limn→∞α n = 0 and ;
(iii) 0 < lim infn→∞β n ≤ lim supn→∞β n < 1 and lim infn→∞γ n > 0;
(iv) .
Then the sequences {x n }, {y n }, {z n } converge strongly to the same point x* = P Ω u, and (x*, y*) and are a solution of general system(1.2) of variational inequalities and a solution of general system (1.6) of variational inequalities, respectively, where y* = P C (x* - λ2A2x*) and .
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Acknowledgements
In this research, the first author was partially supported by the National Science Foundation of China (11071169), Innovation Program of Shanghai Municipal Education Commission (09ZZ133), Leading Academic Discipline Project of Shanghai Normal University (DZL707), Ph.D. Program Foundation of Ministry of Education of China (20070270004), Science and Technology Commission of Shanghai Municipality Grant (075105118), and Shanghai Leading Academic Discipline Project (S30405). The second author was partially supported by the NSC100-2115-M-033-001. The third author gratefully acknowledge the financial support from the Deanship of Scientific Research (DSR) at King Abdulaziz University, Jeddah.
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Ceng, L.C., Wong, M.M. & Latif, A. Generalized extragradient iterative method for systems of variational inequalities. J Inequal Appl 2012, 88 (2012). https://doi.org/10.1186/1029-242X-2012-88
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DOI: https://doi.org/10.1186/1029-242X-2012-88