Abstract
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.
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1 Introduction
Let X be a real Banach space whose dual space is denoted by . Let . A Banach space X is said to be uniformly convex if for each , there exists such that for all ,
It is known that a uniformly convex Banach space is reflexive and strict convex. A Banach space X is said to be smooth if
exists for all . X is said to be uniformly smooth if this limit is attained uniformly for . The norm of X is said to be the Frechet differential if for each , this limit is attained uniformly for . Also, we define a function called the modulus of smoothness of X as follows:
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 .
Let be the dual of X. The normalized duality mapping is defined by
where denotes the generalized duality pairing. It is an immediate consequence of the Hahn-Banach theorem that is nonempty for each . Moreover, it is known that J is single-valued if and only if X is smooth, whereas if X is uniformly smooth, then the mapping J is uniformly continuous on bounded subsets of X. Let C be a nonempty closed convex subset of a real Banach space X. A mapping is called nonexpansive if
We use the notation ⇀ to indicate the weak convergence and the one → to indicate the strong convergence.
Definition 1.1 Let be a mapping of C into X. Then A is said to be
-
(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 .
Very recently, Yao et al. [1] studied the following general system of variational inequalities (GSVI) in a real smooth Banach space X, which is to find such that
where C is a nonempty, closed and convex subset of X, are two nonlinear mappings, and and are two positive constants. The set of solutions of GSVI (1.1) is denoted by . In particular, if , a real Hilbert space, then GSVI (1.1) reduces to the following GSVI of finding such that
which was considered by Ceng et al. [2]. The set of solutions of problem (1.2) is also denoted by . In [2], problem (1.2) was transformed into a fixed point problem in the following way.
Lemma 1.1 (See [2])
For given , is a solution of problem (1.2) if and only if is a fixed point of the mapping defined by
where .
In this paper, we continue to study problem GSVI (1.1). We propose implicit and explicit algorithms based on Korpelevich’s extragradient method [3], viscosity approximation method [4] and Mann’s iterative method [5] 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.
2 Preliminaries
We need the following lemmas that will be used in the sequel.
Lemma 2.1 (See [11])
Let be a sequence of nonnegative real numbers satisfying
where , and satisfy the conditions:
-
(i)
, ;
-
(ii)
;
-
(iii)
(), .
Then .
Lemma 2.2 (See [11])
In a smooth Banach space X, the following inequality holds:
Let LIM be a continuous linear functional on and . We write instead of . LIM is said to be a Banach limit if LIM satisfies , and for all . It is well known that for the Banach limit LIM, the following hold:
-
(i)
for all , implies that ;
-
(ii)
for any fixed positive integer N;
-
(iii)
for all .
Lemma 2.3 (See [4])
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 [12])
Let q be a given real number with , and let X be a q-uniformly smooth Banach space. Then
where κ is the q-uniformly smooth constant of X, and is the generalized duality mapping from X into defined by
Let D be a subset of C, and let Π be a mapping of C into D. Then Π is said to be sunny if
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 [13])
Let C be a nonempty closed convex subset of a real smooth Banach space X, let D be a nonempty subset of C, and let Π be a retraction from C onto D. Then Π is sunny and nonexpansive if and only if
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 [14])
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 [15])
Let and be bounded sequences in a Banach space X, and let be a sequence in , which satisfies the following condition
Suppose that , and . Then .
Lemma 2.8 (See [1])
Let C be a nonempty closed convex subset of a real smooth Banach space X. Assume that the mapping is accretive and weakly continuous along segments (i.e., as ). Then the variational inequality
is equivalent to the dual variational inequality
Lemma 2.9 (See [1])
Let C be a nonempty closed convex subset of a 2-uniformly smooth Banach space X. Let the mapping be -inverse-strongly accretive for . Then we have
for . In particular, if , then is nonexpansive for .
Lemma 2.10 (See [1])
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 .
3 Implicit iterative schemes
In this section, we propose implicit iterative schemes and show the strong convergence theorems. First, we state the following obvious proposition.
Proposition 3.1 Let C be a nonempty closed convex subset of a real smooth Banach space X, and let be a mapping.
-
(i)
If is α-strongly accretive and λ-strictly pseudocontractive with , then is nonexpansive, and F is Lipschitz-continuous with constant ;
-
(ii)
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 .
Proof Rewriting GSVI (1.1) as
the proof then follows from Lemma 2.4. □
By Lemma 3.1, we observe that
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.
To solve GSVI (1.1), we first propose an implicit algorithm as follows. 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. As previously, let be the set of all contractions on C. Let the mapping be -inverse-strongly accretive for . Let with coefficient and be α-strongly accretive and λ-strictly pseudocontractive with . In what follows, we assume that for . For any given , we define a mapping by
where , .
Define another mapping :
Then is rewritten as
Let us show that is nonexpansive. As a matter of fact, since , by Proposition 3.1 we know that is nonexpansive; that is,
Hence, is nonexpansive. So, is nonexpansive. We note that by Lemma 2.9, is nonexpansive for . Thus, it follows from (3.3) that is nonexpansive. This together with (3.4) implies that for all ,
So, is a contraction. Therefore, the Banach contraction principle guarantees that has a unique fixed point in C, which we denote by ; that is,
We now state and prove our first main result.
Theorem 3.1 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 with coefficient , and let be α-strongly accretive and λ-strictly pseudocontractive with . Assume that for . Let be the unique solution in C to Equation (3.5), where , and . Then if and only if
and in this case, converges as strongly to an element of Ω. In addition, if we define by
then solves the variational inequality problem (VIP)
In particular, if is a constant, then (3.7) reduces to the sunny nonexpansive retraction of Reich from C onto Ω,
Proof If , we can take to derive from (3.5) that for ,
which implies that
Because , we deduce that
and hence, (3.6) holds.
Conversely, assume (3.6); that is, remains bounded when ; hence, and are bounded when , where G is defined by (1.3). Because in terms of (3.5),
we obtain
which hence yields
Now, assume that . Since remains bounded as . Set . Then is bounded. Now, define by
where LIM is a Banach limit on . Let
It is easily seen that K is nonempty closed convex bounded subset of X. Since (note that )
it follows that ; that is, K is invariant under G. Since a uniformly smooth Banach space has the fixed point property for nonexpansive mappings, G has a fixed point, say z, in K. Since z is also a minimizer of g over C, it follows that, for and ,
The uniform smoothness of X implies that the duality map J is norm-to-norm uniformly continuous on bounded sets of X. Letting , we find that the two limits above can be interchanged and obtain
Since
Hence,
So by (3.11), for ,
In particular,
Thus,
and there exists a subsequence which is still denoted by such that .
Now, assume that there exists another subsequence of such that . It follows from (3.12) that
Interchange and q to obtain
Adding up (3.13) and (3.14) yields
Since , this implies that . Therefore, as .
Define by
Since , we have
Hence, for ,
Because and as , taking the limit as in (3.17), we obtain that
If () is a constant, then
Hence, Q reduces to the sunny nonexpansive retraction from C to Ω. □
Theorem 3.2 Let C be a nonempty closed convex subset of a 2-uniformly smooth Banach space X with weakly sequentially continuous duality mapping J. Let be a sunny nonexpansive retraction from X onto C. Let the mapping be -inverse-strongly accretive with for . Let be α-strongly accretive and λ-strictly pseudocontractive with . Assume that . Let the net be defined by the implicit scheme
Then converges in norm, as , to the unique solution of the VIP
Proof For any given , consider the following mapping
By Proposition 3.1(ii) and Lemma 2.9, we know that is nonexpansive for , and is contractive with coefficient . Hence,
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.
Now, we show that is bounded. As a matter of fact, take arbitrarily. Then it follows from (3.20) and Lemma 3.1 that
where . Thus, it immediately follows that
Therefore, is bounded and so are the nets , . Furthermore, by Lemma 2.10, we know that is nonexpansive. Thus,
as . That is,
Furthermore, we show that is relatively norm-compact as . Assume that is such that as . Put . Then it is clear that
We can rewrite (3.20) as
For any , by Lemma 2.5, we have
According to this fact, we deduce that
It turns out that
In particular,
Since is bounded, we may assume, without loss of generality, that converges weakly to a point . Noticing (3.23), we can use Lemma 2.6 to get . Therefore, we can substitute for p in (3.25) to get
which together with the weakly sequential continuity of J implies that
This has proven the relative norm compactness of the net as .
We also show that solves the VIP (3.21). From (3.20), we have
For any , utilizing the nonexpansivity of G, we obtain that
and hence,
Therefore,
Since F is α-strongly accretive, we have
It follows that
Combining (3.27) and (3.28), we get
Now, replacing t in (3.29) with and letting , noticing that and as , we derive
which is equivalent to its dual variational inequality (see Lemma 2.8)
That is, is a solution of VIP (3.21). Now, we show that the solution set of VIP (3.21) is a singleton. As a matter of fact, we assume that is another solution of VIP (3.21). Then, we have
From (3.30), we have
So,
Therefore, . In summary, we have shown that each (strong) cluster point of the net (as ) equals to . Therefore, as . This completes the proof. □
4 Explicit iterative schemes
In this section, we propose explicit iterative schemes which are the discretization of the implicit iterative schemes, and show the strong convergence theorems.
Algorithm 4.1 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. Let be two nonlinear mappings. Let and be α-strongly accretive and λ-strictly pseudocontractive. For arbitrarily given , let the sequence be generated iteratively by
where , and , are two positive numbers.
In particular, if , then (4.1) reduces to the following:
Theorem 4.1 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 with coefficient , and let be α-strongly accretive and λ-strictly pseudocontractive with . Assume that for . Let , and assume that
-
(i)
and ;
-
(ii)
;
-
(iii)
or ;
-
(iv)
or .
Then the sequence generated by scheme (4.1) converges strongly to , where is defined by (3.15).
Proof For each , let be defined by
Then we know that
-
(i)
the scheme (4.1) is rewritten as
(4.3) -
(ii)
is nonexpansive by the similar argument to that of the nonexpansivity of in (3.5);
-
(iii)
for all .
Thus, we deduce that for ,
Because , we may assume without loss of generality that for all . Hence, from (4.4), we get
By induction, we conclude that
Therefore, is bounded, so are the sequences , , and . Also, from (4.1), we have
which together with and , implies that
Now, we note that
Thus, it follows that
where for some . So, utilizing Lemma 2.1, we obtain that
This together with (4.6) implies that
Let us show that
where . Indeed we can write
Putting
and using Lemma 2.2, we obtain
The last inequality implies that
Note that
where is a constant such that for all and . It follows from (4.9) that
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, we show that . Write
and apply Lemma 2.2 to get
It then follows that
Put
and
It follows that
Observe that
due to (4.8). It is easily seen from conditions (i), (ii) that
Finally, apply Lemma 2.2 to (4.11) to conclude that . □
Algorithm 4.2 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. Let be two nonlinear mappings. Let be α-strongly accretive and λ-strictly pseudocontractive. For arbitrarily given , let the sequence be generated iteratively by
where and are two sequences in and , are two positive numbers.
In particular, if , then (4.12) reduces to the following:
Theorem 4.2 Let C be a nonempty closed convex subset of a 2-uniformly smooth Banach space X with weakly sequentially continuous duality mapping J. Let be a sunny nonexpansive retraction from X onto C. Let the mapping be -inverse-strongly accretive with for . Let be α-strongly accretive and λ-strictly pseudocontractive with . Let , and let be the sequence generated by (4.12). Assume that the sequences and satisfy the following conditions:
-
(i)
and ;
-
(ii)
.
Then the sequence converges strongly to the unique solution of VIP (3.21).
Proof Take a fixed arbitrarily. Then due to Lemma 1.1. By Lemma 2.10, we have
Hence, it follows from Proposition 3.1(ii) that
where . By induction, we deduce that
Therefore, is bounded. Hence, and are also bounded. Now, set for all . Then for . Hence, it follows that
which together with and the boundedness of implies that
So, by Lemma 2.7 we get
Consequently,
At the same time, we note that
It follows from that
Since for all , by Lemma 2.10 we have
Next, we show that
where is the unique solution of VIP (3.21).
To see this, we choose a subsequence of such that
We may also assume that . Note that in terms of Lemma 2.6 and (4.14). Therefore, it follows from VIP (3.21) and the weakly sequential continuity of J that
Since , according to Lemma 2.5, we have
From (4.16), we have
It follows that
Finally, we prove that as . From (4.12) and (4.17),
We apply Lemma 2.1 to the relation (4.18) and conclude that as . This completes the proof. □
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Acknowledgements
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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Latif, A., Al-Mazrooei, A.E., Dehaish, B.A. et al. Hybrid viscosity approximation methods for general systems of variational inequalities in Banach spaces. Fixed Point Theory Appl 2013, 258 (2013). https://doi.org/10.1186/1687-1812-2013-258
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DOI: https://doi.org/10.1186/1687-1812-2013-258