- Open Access
Hybrid viscosity approximation methods for general systems of variational inequalities in Banach spaces
© Latif et al.; licensee Springer. 2013
- Received: 24 June 2013
- Accepted: 3 September 2013
- Published: 7 November 2013
Let X be a uniformly convex and 2-uniformly smooth Banach space. In this paper, we propose an implicit iterative method and an explicit iterative method for solving a general system of variational inequalities (in short, GSVI) in X based on Korpelevich’s extragradient method and viscosity approximation method. We show that the proposed algorithms converge strongly to some solutions of the GSVI under consideration. When X is a 2-uniformly smooth Banach space with weakly sequentially continuous duality mapping, we also propose two methods, which were inspired and motivated by Korpelevich’s extragradient method and Mann’s iterative method. Furthermore, it is also proven that the proposed algorithms converge strongly to some solutions of the considered GSVI.
MSC:49J30, 47H09, 47J20.
- general system of variational inequalities
- iterative methods
- nonexpansive mapping
- sunny nonexpansive retraction
- fixed point
- weakly sequentially continuous duality map
- uniform smoothness
It is known that X is uniformly smooth if and only if . Let q be a fixed real number with . Then a Banach space X is said to be q-uniformly smooth if there exists a constant such that for all .
We use the notation ⇀ to indicate the weak convergence and the one → to indicate the strong convergence.
- (i)accretive if for each there exists such that
where J is the normalized duality mapping;
- (ii)α-strongly accretive if for each there exists such that
for some ;
- (iii)β-inverse-strongly accretive if for each , there exists such that
for some ;
- (iv)λ-strictly pseudocontractive if for each , there exists such that
for some .
Lemma 1.1 (See )
In this paper, we continue to study problem GSVI (1.1). We propose implicit and explicit algorithms based on Korpelevich’s extragradient method , viscosity approximation method  and Mann’s iterative method  to find approximate solutions of GSVI (1.1). Strong convergence results of these methods will be established under very mild conditions. We observe that some recent results in this direction have been obtained in, e.g., [6–10] and the references therein.
We need the following lemmas that will be used in the sequel.
Lemma 2.1 (See )
Lemma 2.2 (See )
for all , implies that ;
for any fixed positive integer N;
for all .
Lemma 2.3 (See )
Let . If , then there exists a subsequence of such that as .
We also need the following lemmas for the proofs of our main results.
Lemma 2.4 (See )
whenever for and . A mapping Π of C into itself is called a retraction if . If a mapping Π of C into itself is a retraction, then for every , where is the range of Π. A subset D of C is called a sunny nonexpansive retract of C if there exists a sunny nonexpansive retraction from C onto D. The following lemma concerns the sunny nonexpansive retraction.
Lemma 2.5 (See )
for all and .
It is well known that if a Hilbert space, then a sunny nonexpansive retraction is coincident with the metric projection from X onto C; that is, . Let C be a nonempty closed convex subset of a uniformly convex and uniformly smooth Banach space X, and let be a nonexpansive mapping with the fixed point set . Then the set is a sunny nonexpansive retract of C.
Lemma 2.6 (Demiclosedness principle; see )
Let X be a uniformly convex Banach space or a reflexive Banach space satisfying Opial’s condition, let C be a nonempty closed convex subset of X, and let be a nonexpansive mapping. Then the mapping is demiclosed on C, where I is the identity mapping; that is, if is a sequence of C such that and , then .
Lemma 2.7 (See )
Suppose that , and . Then .
Lemma 2.8 (See )
Lemma 2.9 (See )
for . In particular, if , then is nonexpansive for .
Lemma 2.10 (See )
Let C be a nonempty closed convex subset of a real 2-uniformly smooth Banach space X. Let be a sunny nonexpansive retraction from X onto C. Let the mapping be -inverse-strongly accretive for . Let be the mapping defined by (1.3). If , then is nonexpansive for .
In this section, we propose implicit iterative schemes and show the strong convergence theorems. First, we state the following obvious proposition.
If is α-strongly accretive and λ-strictly pseudocontractive with , then is nonexpansive, and F is Lipschitz-continuous with constant ;
If is α-strongly accretive and λ-strictly pseudocontractive with , then for any fixed , is a contraction with coefficient .
Lemma 3.1 Let C be a nonempty closed convex subset of a real smooth Banach space X. Let be a sunny nonexpansive retraction from X onto C, and let the mapping be -inverse-strongly accretive for . For given , is a solution of GSVI (1.1) if and only if , where .
the proof then follows from Lemma 2.4. □
which implies that is a fixed point of the mapping G. Throughout this paper, the set of fixed points of the mapping G is denoted by Ω.
where , .
We now state and prove our first main result.
and hence, (3.6) holds.
and there exists a subsequence which is still denoted by such that .
Since , this implies that . Therefore, as .
Hence, Q reduces to the sunny nonexpansive retraction from C to Ω. □
This means that is a contraction. Therefore, the Banach contraction principle guarantees that has a unique fixed point in C, which we denote by . This shows that the implicit scheme (3.20) is well defined.
This has proven the relative norm compactness of the net as .
Therefore, . In summary, we have shown that each (strong) cluster point of the net (as ) equals to . Therefore, as . This completes the proof. □
In this section, we propose explicit iterative schemes which are the discretization of the implicit iterative schemes, and show the strong convergence theorems.
where , and , are two positive numbers.
Then the sequence generated by scheme (4.1) converges strongly to , where is defined by (3.15).
- (i)the scheme (4.1) is rewritten as(4.3)
is nonexpansive by the similar argument to that of the nonexpansivity of in (3.5);
for all .
Taking the lim sup as in (4.10), and noticing the fact that the two limits are interchangeable due to the fact that the duality map J is norm-to-norm uniformly continuous on bounded sets of X, we obtain (4.8).
Finally, apply Lemma 2.2 to (4.11) to conclude that . □
where and are two sequences in and , are two positive numbers.
Then the sequence converges strongly to the unique solution of VIP (3.21).
where is the unique solution of VIP (3.21).
We apply Lemma 2.1 to the relation (4.18) and conclude that as . This completes the proof. □
This work was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University under grant No. HiCi/15-130-1433. The authors, therefore, acknowledge with thanks DSR technical and financial support. The authors are thankful to the referees for their valuable suggestion/comments.
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