Abstract
In this paper, considering the problem of solving a system of variational inequalities and a common fixed point problem of an infinite family of nonexpansive mappings in Banach spaces, we propose a two-step relaxed extragradient method which is based on Korpelevich’s extragradient method and viscosity approximation method. Strong convergence results are established.
MSC:49J30, 47H09, 47J20.
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1 Introduction
In the last three decades, the theory of variational inequalities has been used as a tool to study the Nash equilibrium problem for a finite or infinite number of players; see, for example, [1–6] and the references therein. There are two ways to study the Nash equilibrium problem by using variational inequality technique: (1) system of variational inequalities; (2) variational inequalities defined over the product of sets. If the number of players is finite, then the system of variational inequalities is equivalent to the variational inequality defined over the product of sets; see, for example, [7, 8] and the references therein.
Very recently, Cai and Bu [9] considered the following system of two variational inequalities in the setting of Banach spaces.
Let C be a nonempty, closed and convex subset of a real Banach space X, let be two nonlinear mappings and and be two positive constants. The problem of system of variational inequalities (SVI) [9] is to find such that
where J is the normalized duality mapping. The set of solutions of GSVI (1.1) is denoted by . This system could be useful to study the Nash equilibrium problem for two players. They proposed an iterative scheme to compute the approximate solutions of such a system.
In particular, if , a real Hilbert space, then GSVI (1.1) reduces to the following problem of a system of variational inequalities of finding such that
where and are two positive constants. The set of solutions of problem (1.2) is still denoted by .
In this paper, we introduce two-step relaxed extragradient method for solving SVI (1.1) and the common fixed point problem of an infinite family of nonexpansive mappings of C into itself. Here, the two-step relaxed extragradient method is based on Korpelevich’s extragradient method [10] and viscosity approximation method. We first suggest and analyze an implicit iterative algorithm by the two-step relaxed extragradient method in a uniformly convex and 2-uniformly smooth Banach space X, and then another explicit iterative algorithm in a uniformly convex Banach space X with a uniformly Gâteaux differentiable norm. On the other hand, we also propose and analyze a composite explicit iterative algorithm by the two-step relaxed extragradient method for solving SVI (1.1) and the common fixed point problem of in a uniformly convex and 2-uniformly smooth Banach space. The results presented in this paper improve, extend, supplement and develop the corresponding results that have appeared very recently in the literature.
2 Preliminaries
Let be the dual of X. The normalized duality mapping is defined by
where denotes the generalized duality pairing.
Let C be a nonempty closed convex subset of a real Banach space X. A mapping is said to be accretive if for each there exists such that
where J is the normalized duality mapping. A is said to be α-strongly accretive if for each there exists such that
for some . It is said to be β-inverse-strongly-accretive if for each there exists such that
for some ; and finally A is said to be λ-strictly pseudocontractive if for each there exists such that
for some .
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.
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, . If C is a nonempty closed convex subset of a strictly convex and uniformly smooth Banach space X and if is a nonexpansive mapping with the fixed point set , then the set is a sunny nonexpansive retract of C.
The following lemma concerns the sunny nonexpansive retraction.
Lemma 2.1 (see [11])
Let C be a nonempty closed convex subset of a real smooth Banach space X. Let D be a nonempty subset of C. Let Π be a retraction of C onto D and let J be a normalized duality mapping on X. Then the following are equivalent:
-
(i)
Π is sunny and nonexpansive;
-
(ii)
, ;
-
(iii)
, , .
Next, we present some more lemmas which are crucial for the proofs of our results.
Lemma 2.2 (see [12])
Let be a sequence of nonnegative real numbers satisfying
where , and satisfy the conditions:
-
(i)
and ;
-
(ii)
;
-
(iii)
, , and .
Then .
Lemma 2.3 (see [12])
In a smooth Banach space X, the following inequality holds:
Lemma 2.4 (see [13])
Let and be bounded sequences in a Banach space X and let be a sequence in which satisfies the following condition:
Suppose , and . Then .
Lemma 2.5 (see [14])
Given a number . A real Banach space X is uniformly convex if and only if there exists a continuous strictly increasing function , , such that
for all and such that and .
Lemma 2.6 (see [15])
Let C be a nonempty closed convex subset of a Banach space X. Let be a sequence of mappings of C into itself. Suppose that . Then, for each , converges strongly to some point of C. Moreover, let S be a mapping of C into itself defined by for all . Then .
Let C be a nonempty closed convex subset of a Banach space X and let be a nonexpansive mapping with . As previously, let be a set of all contractions on C. For and , let be a unique fixed point of the contraction on C; that is,
Lemma 2.7 (see [16])
Let X be a uniformly smooth Banach space, or a reflexive and strictly convex Banach space with a uniformly Gateaux differentiable norm. Let C be a nonempty closed convex subset of X, let be a nonexpansive mapping with , and . Then the net defined by converges strongly to a point in . If we define a mapping by , , then solves the VIP:
Lemma 2.8 (see [17])
Let C be a nonempty closed convex subset of a strictly convex Banach space X. Let be a sequence of nonexpansive mappings on C. Suppose is nonempty. Let be a sequence of positive numbers with . Then a mapping S on C defined by for is well defined, nonexpansive and holds.
Lemma 2.9 (see [12])
Let C be a nonempty closed convex subset of a smooth Banach space X and let the mapping be -strictly pseudocontractive and -strongly accretive with for . Then, for , we have
for . In particular, if , then is nonexpansive for .
Lemma 2.10 (see [9])
Let C be a nonempty closed convex subset of a smooth Banach space X. Let be a sunny nonexpansive retraction from X onto C and let the mapping be -strictly pseudocontractive and -strongly accretive with for . Let be a mapping defined by
If , then is nonexpansive.
Lemma 2.11 (see [9])
Let C be a nonempty closed convex subset of a real 2-uniformly smooth Banach space X. Let the mapping be -inverse-strongly accretive. Then we have
for , where . In particular, if , then is nonexpansive for .
Lemma 2.12 (see [9])
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
If for , then is nonexpansive.
Lemma 2.13 (see [12])
Let C be a nonempty closed convex subset of a smooth Banach space X. Let be a sunny nonexpansive retraction from X onto C and let the mapping be -strictly pseudocontractive and -strongly accretive for . For given , is a solution of GSVI (1.1) if and only if , where .
By Lemma 2.12, we observe that
which implies that is a fixed point of the mapping .
Proposition 2.1 (see [18])
Let X be a real smooth and uniform convex Banach space and let . Then there exists a strictly increasing, continuous and convex function , such that
where .
3 Two-step relaxed extragradient algorithms
In this section, we first suggest and analyze an implicit iterative algorithm by the two-step relaxed extragradient method in the setting of uniformly convex and 2-uniformly smooth Banach spaces, and then another explicit iterative algorithm in the setting of uniformly convex Banach spaces with a uniformly Gateaux differentiable norm.
Theorem 3.1 Let C be a nonempty closed convex subset of a uniformly convex and 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 a contraction with coefficient . Let be an infinite family of nonexpansive mappings of C into itself such that , where Ω is a fixed point set of the mapping G. For arbitrarily given , let be a sequence generated by
where for . Suppose that and are sequences in satisfying the following conditions:
-
(i)
and ;
-
(ii)
.
Assume that for any bounded subset D of C and let S be a mapping of C into itself defined by for all and suppose that . Then converges strongly to , which solves the following VIP:
Proof It is easy to see that scheme (3.1) can be rewritten as
Take a fixed arbitrarily. Then by Lemma 2.13 we know that . Moreover, by Lemma 2.12 we have
which hence implies that
Thus, from (3.2) we have
It immediately follows that is bounded, and so are the sequences , due to (3.3) and the nonexpansivity of G.
Let us show that as . As a matter of fact, from (3.2) we have
Simple calculations show that
It follows that
which hence yields
Thus we have from (3.4)
which implies that
From condition (i) and the assumption on , we have
It follows from Lemma 2.4 that
Hence we obtain
Next we show that as .
For simplicity, put , and . Then . From Lemma 2.11 we have
and
Substituting (3.7) into (3.8), we obtain
According to Lemma 2.3, we have from (3.2)
which hence yields
From this together with (3.9) and the convexity of we have
where for some . So, it follows that
Since for , from conditions (i), (ii) and (3.6) we obtain
Utilizing Proposition 2.1 and Lemma 2.1, we have
which implies that
In the same way, we derive
which implies that
Substituting (3.12) into (3.13), we get
From (3.10) and (3.14), we have
which implies that
Utilizing conditions (i), (ii), from (3.6) and (3.11) we have
Utilizing the properties of and , we deduce that
From (3.16) we obtain
That is,
On the other hand, we observe that
Since as , we have
We note that
From (3.5), (3.17) and (3.18), we obtain that
By (3.19) and Lemma 2.6, we have
In terms of (3.17) and (3.20), we have
Define a mapping , is a constant. Then by Lemma 2.8 we have that . We observe that
From (3.17) and (3.21), we obtain
Now, we claim that
where with being a fixed point of the contraction
Then solves the fixed point equation . Thus we have
By Lemma 2.3 we conclude that
where
It follows from (3.24) that
Letting in (3.26) and noticing (3.25), we derive
where is a constant such that for all and . Taking in (3.27), we have
On the other hand, we have
It follows that
Taking into account that as , we have
Since X has a uniformly Fréchet differentiable norm, the duality mapping J is norm-to-norm uniformly continuous on bounded subsets of X. Consequently, the two limits are interchangeable and hence (3.23) holds. From (3.17) and (3.18) we get . Noticing that J is norm-to-norm uniformly continuous on bounded subsets of X, we deduce from (3.23) that
Finally, let us show that as . We observe that
which implies that
By the convexity of and (3.2), we get
which together with (3.30) leads to
Applying Lemma 2.2 to (3.31), we obtain that as . This completes the proof. □
Corollary 3.1 Let C be a nonempty closed convex subset of a uniformly convex and 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 a contraction with coefficient . Let S be a nonexpansive mapping of C into itself such that , where Ω is the fixed point set of the mapping G. For arbitrarily given , let be a sequence generated by
where for . Suppose that and are sequences in satisfying the following conditions:
-
(i)
and ;
-
(ii)
.
Then converges strongly to , which solves the following VIP:
Theorem 3.2 Let C be a nonempty closed convex subset of a uniformly convex Banach space X which has a uniformly Gâteaux differentiable norm. Let be a sunny nonexpansive retraction from X onto C. Let the mapping be -strictly pseudocontractive and -strongly accretive with for . Let be a contraction with coefficient . Let be an infinite family of nonexpansive mappings of C into itself such that , where Ω is a fixed point set of the mapping G. For arbitrarily given , let be a sequence generated by
where for . Suppose that and are sequences in satisfying the following conditions:
-
(i)
;
-
(ii)
and ;
-
(iii)
or ;
-
(iv)
or .
Assume that for any bounded subset D of C and let S be a mapping of C into itself defined by for all and suppose that . Then converges strongly to , which solves the following VIP:
Proof It is easy to see that scheme (3.32) can be rewritten as
Take a fixed arbitrarily. Then by Lemma 2.13 we know that . Moreover, by Lemma 2.10 we have
From (3.34) we obtain
which implies that is bounded. By Lemma 2.10 we know from (3.34) that and both are bounded.
Let us show that and as . As a matter of fact, from (3.3) we have
Simple calculations show that
It follows that
Furthermore, from (3.33) we have
Also, simple calculations show that
It follows from (3.35) and (3.36) that
where for some . Utilizing Lemma 2.2, from conditions (ii)-(iv) and the assumption on , we deduce that
Since and both are bounded, by Lemma 2.5 there exists a continuous strictly increasing function , such that for
which together with (3.33) implies that
It immediately follows that
Since , and , we get and hence
Thus, from (3.33) and (3.40) it follows that
On the other hand, we observe that
Then we have
Since , and as , we get
In the meantime, since and as , we also get
We note that
From (3.40), (3.42) and (3.43), we obtain
By (3.44) and Lemma 2.6, we have
In terms of (3.40) and (3.45), we have
Define a mapping , is a constant. Then by Lemma 2.8 we have that . We observe that
From (3.40) and (3.46), we obtain
Now, we claim that
where with being the fixed point of the contraction . Then solves the fixed point equation . Utilizing the arguments similar to those of (3.28) in the proof of Theorem 3.1, we can deduce that
Since X has a uniformly Gâteaux differentiable norm, the duality mapping J is norm-to-weak∗ uniformly continuous on bounded subsets of X. Consequently, the two limits are interchangeable and hence the following holds:
From (3.38) we get . Noticing the norm-to-weak∗ uniform continuity of J on bounded subsets of X, we deduce from (3.48) that
Finally, let us show that as . We observe that
and
Since and , by Lemma 2.2 we conclude from (3.51) that as . This completes the proof. □
Corollary 3.2 Let C be a nonempty closed convex subset of a uniformly convex Banach space X which has a uniformly Gâteaux differentiable norm. Let be a sunny nonexpansive retraction from X onto C. Let the mapping be -strictly pseudocontractive and -strongly accretive with for . Let be a contraction with coefficient . Let S be a nonexpansive mapping of C into itself such that , where Ω is a fixed point set of the mapping G. For arbitrarily given , let be a sequence generated by
where for . Suppose that and are sequences in satisfying the following conditions:
-
(i)
;
-
(ii)
and ;
-
(iii)
or ;
-
(iv)
or .
Then converges strongly to , which solves the following VIP:
Remark 3.1 Theorems 3.1 and 3.2 improve, extend, supplement and develop [[14], Theorem 3.1] in the following aspects. Although the iterative algorithm in Theorem 3.1 is an implicit algorithm, we can derive the strong convergence of the proposed algorithm under the same conditions on the parameter sequences , as in [[14], Theorem 3.1]. The assumption of the uniformly convex and 2-uniformly smooth Banach space X in [[14], Theorem 3.1] is weakened to the one of the uniformly convex Banach space X having a uniformly Gâteaux differentiable norm in Theorem 3.2.
4 Relaxed extragradient composite algorithms
In this section, we propose and analyze a composite explicit iterative algorithm by the two-step relaxed extragradient method for solving GSVI (1.1) and the common fixed point problem of an infinite family of nonexpansive self-mappings in a 2-uniformly smooth and uniformly convex Banach space.
Theorem 4.1 Let C be a nonempty closed convex subset of a uniformly convex and 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 a contraction with coefficient . Let be an infinite family of nonexpansive mappings of C into itself such that , where Ω is a fixed point set of the mapping G. For arbitrarily given , let be a sequence generated by
where for . Suppose that and are sequences in satisfying the following conditions:
-
(i)
and ;
-
(ii)
for some ;
-
(iii)
or ;
-
(iv)
or .
Assume that for any bounded subset D of C and let S be a mapping of C into itself defined by for all and suppose that . Then converges strongly to , which solves the following VIP:
Proof It is easy to see that scheme (4.1) can be rewritten as
Take a fixed arbitrarily. Then by Lemma 2.13 we know that . Moreover, by Lemma 2.12 we have
From (4.3) we have
It immediately follows that is bounded, and so are the sequences , , due to (4.3) and the nonexpansivity of G.
Let us show that as . As a matter of fact, from (4.2) we have
Simple calculations show that
It follows that
So, we have from (4.4)
On the other hand, from (4.2) we have
Also, simple calculations show that
Thus, it follows from (4.4)-(4.6) that for all
where for some . Utilizing Lemma 2.2, we deduce from conditions (i), (iii), (iv) and the assumption on that
In terms of (4.4), we also have that as .
Let us show that and as . Indeed, since , we get
which together with implies that
Observe that
which together with condition (ii) implies that
Obviously, from (4.7) and (4.8) we know that as . This implies that
Also, from (4.8) and (4.9) we have
Let us show that as . Indeed, for simplicity, put , and . Then . From Lemma 2.11 we have
and
Substituting (4.11) into (4.12), we obtain
According to Lemma 2.2, we have from (4.2)
which together with (4.13) and the convexity of implies that
where for some . So, it follows that
Since for , from (4.7) and we obtain
Utilizing Proposition 2.1 and Lemma 2.1, we have
which implies that
In the same way, we derive
which implies that
Substituting (4.16) into (4.17), we get
From (4.14) and (4.18), we have
which hence implies that
Thus, from (4.7), (4.15) and we have
Utilizing the properties of and , we deduce that
From (4.20), we obtain
That is,
On the other hand, we observe that
So, it follows from (4.8), (4.9) and (4.21) that
By (4.23) and Lemma 2.6, we have
In terms of (4.21) and (4.24), we have
Define a mapping and is a constant. Then by Lemma 2.8 we have that . We observe that
From (4.21) and (4.25), we obtain
Utilizing the arguments similar to those of (3.29) in the proof of Theorem 3.1, we can deduce that
Finally, let us show that as . We observe that
By the convexity of and (4.2), we get
which together with (4.28) leads to
Applying Lemma 2.2 to (4.29), we obtain that as . This completes the proof. □
Corollary 4.1 Let C be a nonempty closed convex subset of a uniformly convex and 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 a contraction with coefficient . Let S be a nonexpansive mapping of C into itself such that , where Ω is a fixed point set of the mapping G. For arbitrarily given , let be a sequence generated by
where for . Suppose that and are sequences in satisfying the following conditions:
-
(i)
and ;
-
(ii)
for some ;
-
(iii)
or ;
-
(iv)
or .
Then converges strongly to , which solves the following VIP:
Remark 4.1 Theorem 4.1 improves, extends, supplements and develops [[14], Theorem 3.1] in the following aspects. The composite iterative algorithm in [[14], Theorem 3.1] is extended to develop the composite iterative algorithm in Theorem 4.1. Compared with the iterative algorithm in [[14], Theorem 3.1], each iteration step in the iterative algorithm of Theorem 4.1 is very different from the corresponding step in the iterative algorithm of [[14], Theorem 3.1] because each iteration step in the iterative algorithm of Theorem 4.1 involves the composite operator . In the proof of [[14], Theorem 3.1], Lemma 2.4 was used to derive . However, in the proof of Theorem 4.1, we only use Lemma 2.2 to derive . Thus, Theorem 4.1 drops the restriction .
Corollary 4.2 Let C be a nonempty closed convex subset of a real Hilbert space H. Let the mapping be -inverse-strongly monotone for . Let be a contraction with coefficient . Let be an infinite family of nonexpansive mappings of C into itself such that , where Ω is a fixed point set of the mapping G. For arbitrarily given , let be a sequence generated by
where for . Suppose that and are sequences in satisfying the following conditions:
-
(i)
and ;
-
(ii)
for some ;
-
(iii)
or ;
-
(iv)
or .
Assume that for any bounded subset D of C and let S be a mapping of C into itself defined by for all and suppose that . Then converges strongly to , which solves the following VIP:
Corollary 4.3 Let C be a nonempty closed convex subset of a real Hilbert space H. Let the mapping be -inverse-strongly monotone for . Let be a contraction with coefficient . Let S be a nonexpansive mapping of C into itself such that , where Ω is a fixed point set of the mapping G. For arbitrarily given , let be a sequence generated by
where for . Suppose that and are sequences in satisfying the following conditions:
-
(i)
and ;
-
(ii)
for some ;
-
(iii)
or ;
-
(iv)
or .
Then converges strongly to , which solves the following VIP:
Now, we say that a mapping has property (∗) if there exists a constant such that
Whenever , then T is nonexpansive. Put , where is a mapping having property (∗). Then A is -inverse-strongly monotone. Indeed, we have
Since H is a real Hilbert space, we have
and hence
Thus, if T is a mapping having property (∗), then T is Lipschitz continuous with constant , i.e., for all . We denote by a fixed point set of T. It is obvious that the class of mappings having property (∗) strictly includes the class of nonexpansive mappings.
Further, utilizing Corollary 4.3 we first derive a strong convergence result for finding a common fixed point of a nonexpansive mapping and a mapping having property (∗).
Corollary 4.4 Let C be a nonempty closed convex subset of a real Hilbert space H. Let be a mapping having property (∗) and let be a nonexpansive mapping such that . Let be a contraction with coefficient . For arbitrarily given , let be a sequence generated by
where . Suppose that and are sequences in satisfying the following conditions:
-
(i)
and ;
-
(ii)
for some ;
-
(iii)
or ;
-
(iv)
or .
Then converges strongly to , which solves the following VIP:
Proof In Corollary 4.3, we put , and . Then GSVI (1.1) is equivalent to the VIP of finding such that
In this case, is -inverse-strongly monotone. It is not hard to see that . As a matter of fact, we have, for ,
Accordingly, we know that ,
and
So, scheme (4.2) reduces to (4.30). Therefore, the desired result follows from Corollary 4.3. □
Utilizing Corollary 4.3, we also have the following result.
Corollary 4.5 Let H be a real Hilbert space. Let A be an α-inverse-strongly monotone mapping of H into itself and let S be a nonexpansive mapping of H into itself such that . Let be a contraction with coefficient . For arbitrarily given , let be a sequence generated by
where . Suppose that and are sequences in satisfying the following conditions:
-
(i)
and ;
-
(ii)
for some ;
-
(iii)
or ;
-
(iv)
or .
Then converges strongly to , which solves the following VIP:
Proof In Corollary 4.3, we put , , and . Then we know that and . Moreover, we know that ,
and
So, scheme (4.2) reduces to (4.31). Therefore, the desired result follows from Corollary 4.3. □
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Acknowledgements
This article was partially funded by the Deanship of Scientific Research (DSR), King Abdulaziz University. Therefore, the second and third authors gratefully acknowledge with thanks DSR for financial support. This research was partially supported to the first author by the National Science Foundation of China (11071169), Innovation Program of Shanghai Municipal Education Commission (09ZZ133) and Ph.D. Program Foundation of Ministry of Education of China (20123127110002). The authors thank the referees for their valuable comments and suggestions.
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Ceng, LC., Latif, A. & Yao, JC. On solutions of a system of variational inequalities and fixed point problems in Banach spaces. Fixed Point Theory Appl 2013, 176 (2013). https://doi.org/10.1186/1687-1812-2013-176
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DOI: https://doi.org/10.1186/1687-1812-2013-176