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System of variational inequalities and an accretive operator in Banach spaces
Fixed Point Theory and Applications volume 2013, Article number: 249 (2013)
Abstract
In this paper, we introduce composite Mann iteration methods for a general system of variational inequalities with solutions being also common fixed points of a countable family of nonexpansive mappings and zeros of an accretive operator in real smooth Banach spaces. Here, the composite Mann iteration methods are based on Korpelevich’s extragradient method, viscosity approximation method and the Mann iteration method. We first consider and analyze a composite Mann iterative algorithm in the setting of uniformly convex and 2-uniformly smooth Banach space, and then another composite Mann iterative algorithm in a uniformly convex Banach space having a uniformly Gâteaux differentiable norm. Under suitable assumptions, we derive some strong convergence theorems. The results presented in this paper improve, extend, supplement and develop the corresponding results announced in the earlier and very recent literature.
MSC:49J30, 47H09, 47J20.
1 Introduction
Let X be a real Banach space whose dual space is denoted by . 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 . Let denote the unit sphere of X. 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 the limit
exists for all ; in this case, X is also said to have a Gâteaux differentiable norm. X is said to have a uniformly Gâteaux differentiable norm if for each , the limit is attained uniformly for . Moreover, it 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 . Let C be a nonempty closed convex subset of X. A mapping is called nonexpansive if for every . The set of fixed points of T is denoted by . A mapping is said to be accretive if for each , there exists such that .
Recently, Yao et al. [1] combined the viscosity approximation method and the Mann iteration method, and gave the following hybrid viscosity approximation method:
Let C be a nonempty closed convex subset of a real uniformly smooth Banach space X, a nonexpansive mapping such that and with a contractive coefficient , where is the set of all contractive self-mappings on C. For an arbitrary , define in the following way:
where and are two sequences in . They proved under certain control conditions on the sequences and that converges strongly to a fixed point of T. Subsequently, Ceng and Yao [2] under the convergence of no parameter sequences to zero proved that the sequence generated by (YCY) converges strongly to a fixed point of T. Such a result includes [[1], Theorem 1] as a special case.
Theorem 1.1 (See [[2], Theorem 3.1])
Let C be a nonempty closed convex subset of a uniformly smooth Banach space X. Let be a nonexpansive mapping with and with contractive coefficient . Given sequences and in , the following control conditions are satisfied:
-
(i)
, for some integer ;
-
(ii)
;
-
(iii)
;
-
(iv)
.
For an arbitrary , let be generated by (YCY). Then
where solves the variational inequality problem (VIP):
On the other hand, Cai and Bu [3] considered the following general system of variational inequalities (GSVI) in a real smooth Banach space X, which involves finding such that
where C is a nonempty, closed and convex subset of X, are two nonlinear mappings and and are two positive constants. Here, the set of solutions of GSVI (1.1) is denoted by . In particular, if in a real Hilbert space, then GSVI (1.1) reduces to the following GSVI of finding such that
which and are two positive constants. The set of solutions of problem (1.2) is still denoted by . In particular, if , then problem (1.2) reduces to the new system of variational inequalities (NSVI), introduced and studied by Verma [4]. Further, if additionally, then the NSVI reduces to the classical variational inequality problem (VIP) of finding such that
The solution set of VIP (1.3) is denoted by . Variational inequality theory has been studied quite extensively and has emerged as an important tool in the study of a wide class of obstacle, unilateral, free, moving, equilibrium problems. 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 [5]. This alternative formulation has been used to suggest and analyze projection iterative method for solving variational inequalities under the conditions that the involved operator must be strongly monotone and Lipschitz continuous.
Recently, Ceng et al. [6] transformed problem (1.2) into a fixed point problem in the following way.
Lemma 1.1 (See [6])
For given , is a solution of problem (1.2) if and only if is a fixed point of the mapping defined by
where and is the projection of H onto C.
In particular, if the mappings is -inverse strongly monotone for , then the mapping G is nonexpansive provided for .
In 1976, Korpelevich [7] proposed an iterative algorithm for solving the VIP (1.3) in Euclidean space :
with a given number, which is known as the extragradient method. The literature on the VIP is vast, and Korpelevich’s extragradient method has received great attention given by many authors, who improved it in various ways; see, e.g., [3, 8–14] and the references therein, to name but a few.
In particular, whenever X is still a real smooth Banach space, and , then GSVI (1.1) reduces to the variational inequality problem (VIP) of finding such that
which was considered by Aoyama et al. [15]. Note that VIP (1.5) is connected with the fixed point problem for nonlinear mapping (see, e.g., [16, 17]), the problem of finding a zero point of a nonlinear operator (see, e.g., [18]) and so on. It is clear that VIP (1.5) extends VIP (1.3) from Hilbert spaces to Banach spaces.
In order to find a solution of VIP (1.5), Aoyama et al. [15] introduced the following Mann iterative scheme for an accretive operator A:
where is a sunny nonexpansive retraction from X onto C. Then they proved a weak convergence theorem.
Obviously, it is an interesting and valuable problem of constructing some algorithms with strong convergence for solving GSVI (1.1), which contains VIP (1.5) as a special case. Very recently, Cai and Bu [3] constructed an iterative algorithm for solving GSVI (1.1) and a common fixed point problem of a countable family of nonexpansive mappings in a uniformly convex and 2-uniformly smooth Banach space.
Theorem 1.2 (See [[3], 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 with for . Let f be a contraction of C into itself with coefficient . Let be a countable family of nonexpansive mappings of C into itself such that , where Ω is the fixed point set of the mapping on C. For arbitrarily given , let be the sequence generated by
Suppose that and are two sequences in satisfying the following conditions:
-
(i)
and ;
-
(ii)
.
Assume that for any bounded subset D of C, and let T be a mapping of C into X defined by for all and suppose that . Then converges strongly to , which solves the following VIP:
Furthermore, recall that a (possibly multivalued) operator with domain and range in a real Banach space X is accretive if, for each and (), there exists a such that . (Here J is the duality mapping.) An accretive operator A is said to satisfy the range condition if for all . An accretive operator A is m-accretive if for each . If A is an accretive operator which satisfies the range condition, then we can define, for each a mapping defined by , which is called the resolvent of A. We know that is nonexpansive and for all . Hence,
If , then the inclusion is solvable. The following resolvent identity is well known to us; see [19], where more details on accretive operators can be found.
Proposition 1.1 (Resolvent identity)
For , and ,
Recently, Aoyama et al. [20] studied the following iterative scheme in a uniformly convex Banach space having a uniformly Gâteaux differentiable norm: for resolvents of an accretive operator A such that and and
They proved that the sequence generated by (1.7) converges strongly to a zero of A under appropriate assumptions on and . Subsequently, Ceng et al. [21] introduced and analyzed the following composite iterative scheme in either a uniformly smooth Banach space or a reflexive Banach space having a weakly sequentially continuous duality mapping
where is an arbitrary (but fixed) element, under the following control conditions:
-
(H1)
;
-
(H2)
, or, equivalently, ;
-
(H3)
;
-
(H4)
, , for some and ;
-
(H5)
for some and .
Further, as the viscosity approximation method, Jung [22] purposed and analyzed the following composite iterative scheme for finding a zero of an accretive operator A: for resolvent of an accretive operator A such that and , ( denotes the set of all contractions on C) and ,
Theorem 1.3 (See [[22], Theorem 3.1])
Let X be a strictly convex and reflexive Banach space having a uniformly Gâteaux differentiable norm. Let C be a nonempty closed convex subset of X and an accretive operator in X such that and . Let and be sequences in which satisfy the conditions:
-
(i)
and ;
-
(ii)
for some for all .
Let and be chosen arbitrarily. Let be a sequence generated by (JS) for . If is asymptotically regular, i.e., , then converges strongly to , which is the unique solution of the variational inequality problem (VIP)
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 be a contraction with coefficient . Motivated and inspired by the research going on in this area, we introduce the composite Mann iteration methods for finding solutions of GSVI (1.1), which are also common fixed points of a countable family of nonexpansive mappings and zeros of an accretive operator such that . Here, the composite Mann iteration methods are based on Korpelevich’s extragradient method, viscosity approximation method and the Mann iteration method. We first consider and analyze a composite Mann iterative algorithm in the setting of uniformly convex and 2-uniformly smooth Banach space, and then another composite Mann iterative algorithm in a uniformly convex Banach space having a uniformly Gâteaux differentiable norm. Under suitable assumptions, we derive some strong convergence theorems. The results presented in this paper improve, extend, supplement and develop the corresponding results announced in the earlier and very recent literature; see, e.g., [2, 3, 6, 8, 22].
2 Preliminaries
Let X be a real Banach space. 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 . As pointed out in [23], no Banach space is q-uniformly smooth for . In addition, it is also known that J is single-valued if and only if X is smooth, whereas if X is uniformly smooth, then J is norm-to-norm uniformly continuous on bounded subsets of X. If X has a uniformly Gâteaux differentiable norm, then the duality mapping J is norm-to-weak∗ uniformly continuous on bounded subsets of X. We use the notation ⇀ to indicate the weak convergence and the one → to indicate the strong convergence.
Let C be a nonempty closed convex subset of X. Recall that a mapping is said to be
-
(i)
α-strongly accretive if for each , there exists such that
for some ;
-
(ii)
β-inverse-strongly-accretive if for each , there exists such that
for some ;
-
(iii)
λ-strictly pseudocontractive [24] if for each , there exists such that
for some .
It is worth emphasizing that the definition of the inverse strongly accretive mapping is based on that of the inverse strongly monotone mapping, which was studied by so many authors; see, e.g., [9, 25, 26].
Proposition 2.1 (See [27])
Let X be a 2-uniformly smooth Banach space. Then
where κ is the 2-uniformly smooth constant of X, and J is the normalized duality mapping from X into .
Proposition 2.2 (See [28])
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 .
Next, we list some lemmas that will be used in the sequel. Lemma 2.1 can be found in [29]. Lemma 2.2 is an immediate consequence of the subdifferential inequality of the function .
Lemma 2.1 Let be a sequence of nonnegative real numbers satisfying
where , and satisfy the conditions
-
(i)
and ;
-
(ii)
;
-
(iii)
, , and .
Then .
Lemma 2.2 In a real smooth Banach space X, the following inequality holds:
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 2.3 (See [30])
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. Then the following are equivalent:
-
(i)
Π is sunny and nonexpansive;
-
(ii)
, ;
-
(iii)
, , .
It is well known that if in 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.
Lemma 2.4 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 be nonlinear mappings. For given , is a solution of GSVI (1.1) if and only if , where .
Proof We can rewrite GSVI (1.1) as
which is obviously equivalent to
because of Lemma 2.3. This completes the proof. □
In terms of Lemma 2.4, 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.
Lemma 2.5 (See [27])
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 [31])
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 the set of all contractions on C. For and , let be the unique fixed point of the contraction on C; that is,
Let X be a uniformly smooth Banach space, or a reflexive and strictly convex Banach space with a uniformly Gâteaux 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 [33])
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 that is nonempty. Let be a sequence of positive numbers with . Then a mapping S on C defined by for is defined well, nonexpansive and holds.
Lemma 2.9 (See [15])
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 be an accretive mapping. Then for all ,
Lemma 2.10 (See [34])
Let and be bounded sequences in a Banach space X, and let be a sequence of nonnegative numbers in with . Suppose that for all integers and . Then .
Lemma 2.11 (See [35])
Let X be a uniformly convex Banach space and , . Then there exists a continuous, strictly increasing and convex function , such that
for all and all with .
3 Composite Mann iterative algorithms in uniformly convex and 2-uniformly smooth Banach spaces
In this section, we introduce our composite Mann iterative algorithms in uniformly convex and 2-uniformly smooth Banach spaces and show the strong convergence theorems. We will use some useful lemmas in the sequel.
Lemma 3.1 (See [[3], Lemma 2.8])
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 3.2 (See [[3], Lemma 2.9])
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.
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 be an accretive operator in X such that . Let be -inverse strongly accretive for . Let be a contraction with coefficient . Let be a countable family of nonexpansive mappings of C into itself such that , where Ω is the fixed point set of the mapping with for . For arbitrarily given , let be the sequence generated by
where , , and are the sequences in such that for all . Suppose that the following conditions hold:
-
(i)
and , for some integer ;
-
(ii)
and ;
-
(iii)
;
-
(iv)
and for all ;
-
(v)
.
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,
where solves the following VIP:
Proof First of all, let us show that the sequence is bounded. Indeed, take a fixed arbitrarily. Then we get , and for all . By Lemma 3.2, we know that G is nonexpansive. Then from (3.1), we have
and hence
By induction, we obtain
Thus, is bounded, and so are the sequences , and .
Let us show that
As a matter of fact, put , . Then it follows from (i) and (v) that
and hence
Define
Observe that
and hence
On the other hand, if , using the resolvent identity in Proposition 1.1,
we get
If , we derive in the similar way
Thus, combining the above cases, we obtain
where for some . Substituting (3.8) for (3.7), we have
which hence yields
where for some . So, from (3.9), conditions (iii), (iv) and the assumption on , it follows that
Consequently, by Lemma 2.10, we have
It follows from (3.5) and (3.6) that
From (3.1), we have
which hence implies that
Since and , we get
Next, we show that as .
Indeed, for simplicity, put , and . Then for all . From Lemma 3.1, we have
and
Substituting (3.13) for (3.14), we obtain
From (3.1) and (3.15), we have
which hence implies that
Since for , and is bounded, we obtain from (3.12), (3.17) and condition (ii) that
Utilizing Proposition 2.2 and Lemma 2.3, we have
which implies that
In the same way, we derive
which implies that
Substituting (3.19) for (3.20), we get
By Lemma 2.2, we have from (3.16) and (3.21)
which hence leads to
From (3.18), (3.22), condition (ii) and the boundedness of , , and , we deduce that
Utilizing the properties of and , we deduce that
From (3.23), we get
That is,
Next, let us show that
Indeed, utilizing Lemma 2.5 and (3.1), we have
which immediately implies that
So, from (3.12), the boundedness of , and conditions (ii), (v), it follows that
From the properties of , we have
Taking into account that
we have
From (3.12), (3.25) and condition (ii), it follows that
Note that
So, in terms of (3.25) and Lemma 2.6, we have
Also, note that
From (3.24) and (3.26), we have
Furthermore, we claim that for a fixed number r such that . In fact, taking into account the resolvent identity in Proposition 1.1, we have
Thus, we get from (3.28) and (3.29)
That is,
Define a mapping , where are two constants with . Then by Lemma 2.8, we have that . We observe that
From (3.24), (3.27) and (3.30), we obtain
Now, we claim that
where with being the fixed point of the contraction
Then solves the fixed point equation . Thus, we have
By Lemma 2.2, we conclude that
where
It follows from (3.33) that
Letting in (3.35) and noticing (3.34), we derive
where is a constant such that for all and . Taking in (3.36), we have
On the other hand, we have
It follows that
Taking into account that as , we have
Since X has a uniformly Frechet 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 (3.32) holds. From (3.4), we get . Noticing the norm-to-norm uniform continuity of J on bounded subsets of X, we deduce from (3.32) that
Finally, let us show that as . Utilizing Lemma 2.2, from (3.1) and the convexity of , we get
and
Applying Lemma 2.1 to (3.39), 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 be an accretive operator in X such that . Let be an α-strictly pseudocontractive mapping. Let be a contraction with coefficient . Let be a countable family of nonexpansive mappings of C into itself such that . For arbitrarily given , let be the sequence generated by
where , and , , and are the sequences in such that for all . Suppose that the following conditions hold:
-
(i)
and , for some integer ;
-
(ii)
and ;
-
(iii)
;
-
(iv)
and for all ;
-
(v)
.
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
where solves the following VIP:
Proof In Theorem 3.1, we put , and , where . Then GSVI (1.1) is equivalent to the VIP of finding such that
In this case, is α-inverse strongly accretive. It is not hard to see that . As a matter of fact, we have, for ,
Accordingly, we know that , and
So, the scheme (3.1) reduces to (3.40). Therefore, the desired result follows from Theorem 3.1. □
Remark 3.1 Theorem 3.1 improves, extends, supplements and develops Jung [[22], Theorem 3.1], Ceng and Yao [[2], Theorem 3.1] and Cai and Bu [[3], Theorem 3.1] in the following aspects.
(i) The problem of finding a point in our Theorem 3.1 is more general and more subtle than any of the problems of finding a point in [[22], Theorem 3.1], the problem of finding a point in [[2], Theorem 3.1], and the problem of finding a point in [[3], Theorem 3.1].
(ii) The iterative scheme in [[2], Theorem 3.1] is extended to develop the iterative scheme (3.1) of Theorem 3.1 by virtue of the iterative schemes of [[22], Theorem 3.1] and [[3], Theorem 3.1]. The iterative scheme (3.1) of Theorem 3.1 is more advantageous and more flexible than the iterative scheme of [[2], Theorem 3.1], because it can be applied to solving three problems (i.e., GSVI (1.1), fixed point problem and zero point problem) and involves several parameter sequences , , , and .
(iii) Our Theorem 3.1 extends and generalizes Ceng and Yao [[2], Theorem 3.1] from a nonexpansive mapping to a countable family of nonexpansive mappings, and Jung [[22], Theorems 3.1] to the setting of a countable family of nonexpansive mappings and GSVI (1.1) for two inverse-strongly accretive mappings. In the meantime, our Theorem 3.1 extends and generalizes Cai and Bu [[3], Theorem 3.1] to the setting of an accretive operator.
(iv) The iterative scheme (3.1) in Theorem 3.1 is very different from any in [[22], Theorem 3.1], [[2], Theorem 3.1] and [[3], Theorem 3.1], because the mapping G in [[3], Theorem 3.1] and the mapping in [[22], Theorem 3.1] are replaced by the same composite mapping in the iterative scheme (3.1) of our Theorem 3.1.
(v) Cai and Bu’s proof in [[3], Theorem 3.1] depends on the argument techniques in [6], the inequality in 2-uniformly smooth Banach spaces (see Proposition 2.1) and the inequality in smooth and uniform convex Banach spaces (see Proposition 2.2). Because the composite mapping appears in the iterative scheme (3.1) of our Theorem 3.1, the proof of our Theorem 3.1 depends on the argument techniques in [6], the inequality in 2-uniformly smooth Banach spaces (see Proposition 2.1), the inequality in smooth and uniform convex Banach spaces (see Proposition 2.2), the inequalities in uniform convex Banach spaces (see Lemmas 2.5 and 2.9 in Section 2 of this paper), and the resolvent identity for accretive operators (see Proposition 1.1).
(vi) It is worth emphasizing that the assumption of asymptotic regularity on in [[22], Theorem 3.1] is dropped by Theorem 3.1, and there is no assumption of the convergence of parameter sequences to zero in our Theorem 3.1.
4 Composite Mann iterative algorithms in uniformly convex Banach spaces having uniformly Gâteaux differentiable norms
In this section, we introduce our composite Mann iterative algorithms in uniformly convex Banach spaces having uniformly Gâteaux differentiable norms and show the strong convergence theorems. First, we give some useful lemmas whose proofs will be omitted because they can be obtained by standard argument.
Lemma 4.1 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 4.2 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 the mapping defined by
If , then is nonexpansive.
We now state and prove the main result of this section.
Theorem 4.1 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 be an accretive operator in X such that . Let be -strictly pseudocontractive and -strongly accretive with for . Let be a contraction with coefficient . Let be a countable family of nonexpansive mappings of C into itself such that , where Ω is the fixed point set of the mapping with for . For arbitrarily given , let be the sequence generated by
where , , , and are the sequences in such that for all . Suppose that the following conditions hold:
-
(i)
and ;
-
(ii)
for some ;
-
(iii)
;
-
(iv)
and for all ;
-
(v)
and .
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 First of all, take a fixed arbitrarily. Then we obtain , and for all . By Lemma 4.2, we get from (4.1)
and hence
By induction, we have
which implies that is bounded and so are the sequences , , .
Let us show that
As a matter of fact, observe that can be rewritten as follows
where . Observe that
On the other hand, repeating the same arguments as those of (3.8) in the proof of Theorem 3.1, we can deduce that for all ,
where for some . Taking into account , we may assume, without loss of generality, that . So, from (4.4) and (4.5), we have
where for some . In the meantime, observe that
This together with (4.6) implies that
where for some . Since , we obtain from conditions (i) and (v) that . Thus, applying Lemma 2.1 to (4.7), we deduce from conditions (iii), (iv) and the assumption on that
Next, we show that as .
Indeed, according to Lemma 2.2, we have from (4.1)
Utilizing Lemma 2.5, we get from (4.1) and (4.8)
which hence yields
Since and , from condition (v) and the boundedness of and , it follows that
Utilizing the properties of g, we have
which together with (4.1) and (4.3) implies that
That is,
Since
it immediately follows from (4.9) and (4.10) that
On the other hand, observe that can be rewritten as follows:
where and . Utilizing Lemma 2.11, we have
which hence implies that
Utilizing (4.10), conditions (i), (ii), (v) and the boundedness of , and , we get
From the properties of , we have
Utilizing Lemma 2.5 and the definition of , we have
which leads to
Since and are bounded, we deduce from (4.12) and condition (ii) that
From the properties of , we have
Furthermore, can also be rewritten as follows:
where and . Utilizing Lemma 2.11 and the convexity of , we have
which hence implies that
Utilizing (4.10), conditions (i), (ii), (v) and the boundedness of , and , we get
From the properties of , we have
Thus, from (4.13) and (4.14), we get
That is,
In terms of (4.14) and Lemma 2.6, we have
That is,
Furthermore, repeating the same arguments as those of (3.30) in the proof of Theorem 3.1, we can conclude that
for a fixed number r such that .
Define a mapping , where are two constants with . Then by Lemma 2.8, we have that . We observe that
From (4.11), (4.16) and (4.17), we obtain
Now, we claim that
where with being the fixed point of the contraction
Then solves the fixed point equation . Repeating the same arguments as those of (3.37) in the proof of Theorem 3.1, we can obtain 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 (4.19) holds. From (4.10), we get . Noticing the norm-to-weak∗ uniform continuity of J on bounded subsets of X, we deduce from (4.19) that
Finally, let us show that as . Indeed, observe that
and hence
Applying Lemma 2.1 to (4.21), we conclude from conditions (i), (v) and (4.20) that as . This completes the proof. □
Corollary 4.1 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 be an accretive operator in X such that . Let be a self-mapping such that is λ-strictly pseudocontractive and α-strongly accretive with . Let be a contraction with coefficient . Let be a countable family of nonexpansive mappings of C into itself such that . For arbitrarily given , let be the sequence generated by
where , and , , , and are the sequences in such that for all . Suppose that the following conditions hold:
-
(i)
and ;
-
(ii)
for some ;
-
(iii)
;
-
(iv)
and for all ;
-
(v)
and .
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 In Theorem 4.1, we put , and , where . Then GSVI (1.1) is equivalent to the VIP of finding such that
In this case, is λ-strictly pseudocontractive and α-strongly accretive. Repeating the same arguments as those in the proof of Corollary 3.1, we can infer that . Accordingly, , and
So, the scheme (4.1) reduces to (4.22). Therefore, the desired result follows from Theorem 4.1. □
Remark 4.1 Theorem 4.1 improves, extends, supplements and develops Jung [[22], Theorem 3.1], Ceng and Yao [[2], Theorem 3.1] and Cai and Bu [[3], Theorem 3.1] in the following aspects.
(i) The problem of finding a point in our Theorem 4.1 is more general and more subtle than any of the problems of finding a point in [[22], Theorem 3.1], the problem of finding a point in [[2], Theorem 3.1], and the problem of finding a point in [[3], Theorem 3.1].
(ii) The iterative scheme in [[22], Theorem 3.1] is extended to develop the iterative scheme (4.1) of Theorem 4.1 by virtue of the iterative schemes of [[2], Theorems 3.1] and [[3], Theorem 3.1]. The iterative scheme (4.1) of Theorem 4.1 is more advantageous and more flexible than the iterative scheme of [[2], Theorem 3.1], because it can be applied to solving three problems (i.e., GSVI (1.1), fixed point problem and zero point problem) and involves several parameter sequences , , , , and .
(iii) Theorem 4.1 extends and generalizes Ceng and Yao [[2], Theorem 3.1] from a nonexpansive mapping to a countable family of nonexpansive mappings, and Jung [[22], Theorem 3.1] to the setting of a countable family of nonexpansive mappings and GSVI (1.1) for two strictly pseudocontractive and strongly accretive mappings. In the meantime, Theorem 4.1 extends and generalizes Cai and Bu [[3], Theorem 3.1] to the setting of an accretive operator.
(iv) The iterative scheme (4.1) in Theorem 4.1 is very different from any in [[22], Theorem 3.1], [[2], Theorem 3.1] and [[3], Theorem 3.1] because the mapping in [[22], Theorem 3.1] and the mapping G in [[3], Theorem 3.1] are replaced by the same composite mapping in the iterative scheme (4.1) of Theorem 4.1.
(v) Cai and Bu’s proof in [[3], Theorem 3.1] depends on the argument techniques in [6], the inequality in 2-uniformly smooth Banach spaces (see Proposition 2.1) and the inequality in smooth and uniform convex Banach spaces (see Proposition 2.2). However, the proof of Theorem 4.1 does not depend on the argument techniques in [6], the inequality in 2-uniformly smooth Banach spaces (see Proposition 2.1), and the inequality in smooth and uniform convex Banach spaces (see Proposition 2.2). It depends on only the inequalities in uniform convex Banach spaces (see Lemmas 2.5 and 2.11 in Section 2 of this paper) and the resolvent identity for accretive operators (see Proposition 1.1).
(vi) The assumption of the uniformly convex and 2-uniformly smooth Banach space X in [[3], Theorem 3.1] is weakened to the one of the uniformly convex Banach space X having a uniformly Gâteaux differentiable norm in Theorem 4.1. Moreover, the assumption of the uniformly smooth Banach space X in [[2], Theorem 3.1] is replaced by the one of the uniformly convex Banach space X having a uniformly Gâteaux differentiable norm in Theorem 4.1. It is worth emphasizing that the assumption of asymptotic regularity on in [[22], Theorem 3.1] is dropped by Theorem 4.1.
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Acknowledgements
The first author was partially supported by the National Science Foundation of China (11071169), the Innovation Program of Shanghai Municipal Education Commission (09ZZ133) and the Ph.D. Program Foundation of Ministry of Education of China (20123127110002). The second author was partially supported by a grant from the NSC 102-2115-M-037-001.
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Ceng, LC., Wen, CF. System of variational inequalities and an accretive operator in Banach spaces. Fixed Point Theory Appl 2013, 249 (2013). https://doi.org/10.1186/1687-1812-2013-249
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DOI: https://doi.org/10.1186/1687-1812-2013-249