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Three-step Mann iterations for a general system of variational inequalities and an infinite family of nonexpansive mappings in Banach spaces
Journal of Inequalities and Applications volume 2013, Article number: 539 (2013)
Abstract
In this paper, let X be a uniformly convex Banach space which either is uniformly smooth or has a weakly continuous duality map. We introduce and consider three-step Mann iterations for finding a common solution of a general system of variational inequalities (GSVI) and a fixed point problem (FPP) of an infinite family of nonexpansive mappings in X. Here three-step Mann iterations are based on Korpelevich’s extragradient method, the viscosity approximation method and the Mann iteration method. We prove the strong convergence of this method to a common solution of the GSVI and the FPP, which solves a variational inequality on their common solution set. We also give a weak convergence theorem for three-step Mann iterations involving the GSVI and the FPP in a Hilbert space. 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 unite 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 strictly 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 Gateaux differentiable norm. X is said to have a uniformly Gateaux 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, and let be a nonlinear mapping. Denote by the set of fixed points of T, i.e., . Recall that T is nonexpansive if for all . A mapping is said to be a contraction on C if there exists a constant ρ in such that for all . 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, let be a nonexpansive mapping such that and with a contractive coefficient , where is the collection 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 a 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 VIP
Let C be a nonempty closed convex subset of a real Banach space X, and with a contractive coefficient . Let be an infinite family of nonexpansive self-mappings on C, and let be a sequence of nonnegative numbers in . For any , define a self-mapping on C as follows:
Such a mapping is called the W-mapping generated by and ; see [3].
In 2012, Ceng et al. [4] introduced and analyzed the following hybrid viscosity approximation method for finding a common fixed point of an infinite family of nonexpansive mappings in a strictly convex and reflexive Banach space, which either is uniformly smooth or has a weakly continuous duality map with gauge φ.
Theorem 1.2 (see [[4], Theorem 3.3])
Let C be a nonempty closed convex subset of a reflexive and strictly convex Banach space X. Assume, in addition, that X either is uniformly smooth or has a weakly continuous duality map with gauge φ. Let be an infinite family of nonexpansive self-mappings on C such that the common fixed point set and with a contractive coefficient . Given sequences , and in , the following conditions are satisfied:
-
(i)
, for some , and ;
-
(ii)
;
-
(iii)
;
-
(iv)
, for some constant .
For an arbitrary , let be generated by
where is the W-mapping generated by and . Then
In this case,
-
(i)
if X is uniformly smooth, then solves the VIP
-
(ii)
if X has a weakly continuous duality map with gauge φ, then solves the VIP
On the other hand, Cai and Bu [5] 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 , a real Hilbert space, then GSVI (1.1) reduces to the following GSVI of finding such that
which is studied in Ceng et al. [6]. 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 [7]. 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 [8]. This alternative formulation has been used to suggest and analyze the projection iterative method for solving variational inequalities under the conditions that the involved operator must be strongly monotone and Lipschitz continuous.
In 1976, Korpelevich [9] proposed an iterative algorithm for solving VIP (1.3) in Euclidean space :
with a given number, which is known as the extragradient method (see also [10, 11]). The literature on the VIP is vast and Korpelevich’s extragradient method has received great attention from many authors, who improved it in various ways; see, e.g., [5, 8, 12–29] and 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. [30]. Note that VIP (1.5) is connected with the fixed point problem for a nonlinear mapping (see, e.g., [31]), the problem of finding a zero point of a nonlinear operator (see, e.g., [32]) 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. [30] 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. For related work, please see [33] and the references therein.
Beyond doubt, 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 [5] constructed an iterative algorithm for solving GSVI (1.1) and a fixed point problem of an infinite family of nonexpansive mappings in a uniformly convex and 2-uniformly smooth Banach space. They proved the strong convergence of the proposed algorithm by virtue of the following inequality in a 2-uniformly smooth Banach space X.
Lemma 1.1 (see [34])
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 .
Theorem 1.3 (see [[5], 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 a 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 . For arbitrarily given , let be a 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:
For the convenience of implementing the argument techniques in [6], the authors [5] used the following inequality in a real smooth and uniform convex Banach space X.
Proposition 1.1 (see [35])
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 .
In this paper, let X be a uniformly convex Banach space which either is uniformly smooth or has a weakly continuous duality map. Let C be a nonempty closed convex subset of X, be a sunny nonexpansive retraction from X onto C and with a contractive coefficient . Motivated and inspired by the research going on in this area, we introduce and analyze three-step Mann iterations for finding a common solution of GSVI (1.1) and a fixed point problem (FPP) of an infinite family of nonexpansive self-mappings on C. Here, three-step Mann iterations are based on Korpelevich’s extragradient method, the viscosity approximation method and the Mann iteration method. We prove the strong convergence of this method to a common solution of GSVI (1.1) and the FPP, which solves a variational inequality on their common solution set. We also give a weak convergence theorem for three-step Mann iterations involving GSVI (1.2) and the FPP in the case of , a Hilbert space. 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, 4–6, 29].
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 [36], 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 the mapping J is norm-to-norm uniformly continuous on bounded subsets of X. If X has a uniformly Gateaux 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 → 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 [37] (see also [38]) 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., [17, 39, 40].
We list some lemmas that will be used in the sequel. Lemma 2.1 can be found in [41]. 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 the condition
where and are sequences of real numbers such that
-
(i)
and , or equivalently,
-
(ii)
, or .
Then .
Lemma 2.2 (see [42])
Let X be a real Banach space and J be the normalized duality map on X. Then, for any given , 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 [43])
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 , 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 [44])
Let X be a uniformly smooth Banach space, C be a nonempty closed convex subset of X, 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
In particular, if is a constant, then the map is reduced to the sunny nonexpansive retraction of Reich type from C onto , i.e.,
Recall that a gauge is a continuous strictly increasing function such that and as . Associated to the gauge φ is the duality map defined by
We say that a Banach space X has a weakly continuous duality map if there exists a gauge φ for which the duality map is single-valued and weak-to-weak∗ sequentially continuous. It is known that has a weakly continuous duality map with gauge for all . Set
Then for all , where ∂ denotes the subdifferential in the sense of convex analysis; see [42] for more details.
The first part of the following lemma is an immediate consequence of the subdifferential inequality, and the proof of the second part can be found in [45].
Lemma 2.6 Assume that X has a weakly continuous duality map with gauge φ.
-
(i)
For all , the following inequality holds:
-
(ii)
Assume that a sequence in X is weakly convergent to a point x. Then the following identity holds:
Lemma 2.7 ([[31], Theorem 3.1 and its proof])
Let X be a reflexive Banach space and have a weakly continuous duality map with gauge φ, let C be a nonempty closed convex subset of X, let be a nonexpansive mapping with , and let . Then defined by , , converges strongly to a point in as . Define by . Then solves the variational inequality
In particular, if is a constant, then the map is reduced to the sunny nonexpansive retraction of Reich type from C onto , i.e.,
Recall that X satisfies Opial’s property [46] provided, for each sequence in X, the condition implies
It is known in [46] that each () enjoys this property, while does not unless . It is known in [47] that every separable Banach space can be equivalently renormed so that it satisfies Opial’s property. We denote by the weak ω-limit set of , i.e.,
Also, recall that in a Hilbert space H, the following equality holds:
Lemma 2.8 (see [48])
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 [30])
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 [[49], Lemma 3.2])
Let C be a nonempty closed convex subset of a strictly convex Banach space X. Let be a sequence of nonexpansive self-mappings on C such that , and let be a sequence of positive numbers in for some . Then, for every and , the limit exists.
Using Lemma 2.10, one can define a mapping as follows:
Such W is called the W-mapping generated by the sequences and . Throughout this paper, we always assume that is a sequence of positive numbers in for some .
Lemma 2.11 (see [[49], Lemma 3.3])
Let C be a nonempty closed convex subset of a strictly convex Banach space X. Let be a sequence of nonexpansive self-mappings on C such that , and let be a sequence of positive numbers in for some . Then .
Lemma 2.12 (see [[50], Lemma 2])
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 2.13 (see [34])
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 .
We will also use the following elementary lemmas in the sequel.
Lemma 2.14 (see [51])
Let and be the sequences of nonnegative real numbers such that and for all . Then exists.
Lemma 2.15 (Demiclosedness principle [42])
Assume that T is a nonexpansive self-mapping of a nonempty closed convex subset C of a Hilbert space H. If T has a fixed point, then is demiclosed. That is, whenever in C and in H, it follows that . Here, I is the identity operator of H.
3 Main results
In this section, in order to prove our main results, we will use the following useful lemmas whose proofs will be omitted since they can be proved by standard arguments.
Lemma 3.1 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 3.2 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 paper.
Theorem 3.1 Let C be a nonempty closed convex subset of a uniformly convex Banach space X. Assume, in addition, that X either is uniformly smooth or has a weakly continuous duality map with gauge φ. Let be a sunny nonexpansive retraction from X onto C. Let the mapping be -strictly pseudocontractive and -strongly accretive with for . Let with a contractive coefficient . Let be a sequence of positive numbers in for some and be an infinite family of nonexpansive self-mappings on C such that , where Ω is the fixed point set of the mapping with for . For an arbitrary , let be generated by
where is the W-mapping generated by and , and , and are sequences in . Suppose that the following conditions hold:
-
(i)
, for some , and ;
-
(ii)
and ;
-
(iii)
;
-
(iv)
.
Then
In this case,
-
(i)
if X is uniformly smooth, then solves the VIP
-
(ii)
if X has a weakly continuous duality map with gauge φ, then solves the VIP
Proof First of all, let us show that is bounded. Indeed, taking an element arbitrarily, we obtain that and for all . By Lemma 3.2, we know that G is nonexpansive. It follows from the nonexpansivity of G and that
and
By induction, we have
Hence is bounded, and so are the sequences , , , and .
Suppose that as . Then and for all . From (3.1) it follows that
and
that is, . Again from (3.1) we obtain that
Conversely, suppose that (). Put for each . Then it follows from conditions (i) and (iii) that
and hence
Define by
Observe that
It follows that
In the meantime, simple calculations show that
which hence yields
Taking into account the nonexpansivity of and , from (CWY) we have
where for some . Thus, from (3.4), (3.5) and (3.6), we get
where for some . Then it immediately follows that
From condition (ii) and , , we deduce that
Hence by Lemma 2.12 we have
It follows from (3.2) and (3.3) that
From (3.1) we have
This implies that
Since and , we get
Observe that
It follows from condition (iii), (3.7) and (3.8) that
Also, utilizing Lemma 2.13, we obtain from (3.1) that for
and hence
Thus, we get
From (3.7), conditions (iii), (iv) and the boundedness of and , it follows that
Utilizing the properties of g, we have
This immediately implies that
Note that
which together with (3.9) and (3.11) implies that
Also, note that
From [[52], Remark 2.2] (see also [[53], Remark 3.1]), we have
It follows that
In terms of (2.3) and Lemma 2.11, is a nonexpansive mapping such that . Define a mapping , where θ is a constant in . Then by Lemma 2.8, we have that . Moreover, from (3.10) and (3.12), we get
that is,
In the following, we discuss two cases.
(i) Firstly, suppose that X is uniformly smooth. Let be the unique fixed point of the contraction mapping given by
By Lemma 2.5, we can define , and solves the VIP
Let us show that
Note that
Applying Lemma 2.2, we derive
where
The last inequality implies
It follows that
where is a constant such that for all and small enough . Taking the lim sup as in (3.15) and noticing the fact that the two limits are interchangeable due to the fact that the duality map J is uniformly norm-to-norm continuous on any bounded subset of X, we get (3.14).
Now, let us show that as .
Indeed, utilizing Lemma 2.2, we obtain from (3.1) that
and hence
Therefore, applying Lemma 2.1 to (3.17), we conclude from (3.14) and condition (i) that as .
(ii) Secondly, suppose that X has a weakly continuous duality map with gauge φ. Let be the unique fixed point of the contraction mapping given by
By Lemma 2.7, we can define , and solves the VIP
Let us show that
We take a subsequence of such that
Since X is reflexive and is bounded, we may further assume that for some . Since is weakly continuous, utilizing Lemma 2.6(ii), we have
Put , . It follows that
From (3.13), we have
Furthermore, observe that
Combining (3.21) with (3.22), we obtain
Hence and . Thus, from (3.18) and (3.20), it is easy to see that
Therefore, we deduce that (3.19) holds.
Next, let us show that as . Indeed, utilizing Lemma 2.6(i), we obtain from (3.1) that
and hence
Applying Lemma 2.1 to (3.23), we conclude from (3.19) and condition (i) that
which implies that (), i.e., (). This completes the proof. □
Corollary 3.1 The conclusion in Theorem 3.1 still holds, provided the conditions (i)-(iv) are replaced by the following:
-
(i)
, for some ;
-
(ii)
and ;
-
(iii)
and ;
-
(iv)
and .
Proof Observe that
Since and , it follows that
Consequently, all the conditions of Theorem 3.1 are satisfied. So, utilizing Theorem 3.1, we obtain the desired result. □
Corollary 3.2 Let C be a nonempty closed convex subset of a uniformly convex Banach space X. Assume, in addition, that X either is uniformly smooth or has a weakly continuous duality map with gauge φ. Let be a sunny nonexpansive retraction from X onto C. Let the mapping be -strictly pseudocontractive and -strongly accretive with for . Let with a contractive coefficient . Let be a sequence of positive numbers in for some and be an infinite family of nonexpansive self-mappings on C such that , where Ω is the fixed point set of the mapping with for . Suppose that , and are sequences in satisfying the following conditions:
-
(i)
and ;
-
(ii)
;
-
(iii)
;
-
(iv)
.
Then, for an arbitrary but fixed , the sequence defined by (3.1) converges strongly to some . Moreover,
-
(i)
if X is uniformly smooth, then solves the VIP
-
(ii)
if X has a weakly continuous duality map with gauge φ, then solves the VIP
Proof Repeating the same arguments as those in the proof of Theorem 3.1, we know that is bounded, and so are the sequences , , , and . Since , it is easy to see that the following hold:
-
(i)
();
-
(ii)
, for some integer ;
-
(iii)
.
Therefore, all the conditions of Theorem 3.1 are satisfied. So, utilizing Theorem 3.1, we derive the desired result. □
To end this paper, we give a weak convergence theorem for three-step Mann iterations (3.1) involving GSVI (1.2) and an infinite family of nonexpansive mappings in a Hilbert space H.
Theorem 3.2 Let C be a nonempty closed convex subset of a Hilbert space H. Let the mapping be -inverse strongly monotone for . Let with a contractive coefficient . Let be a sequence of positive numbers in for some , and let be an infinite family of nonexpansive self-mappings on C such that , where Ω is the fixed point set of the mapping with for . Suppose that , and are sequences in satisfying the following conditions:
-
(i)
;
-
(ii)
;
-
(iii)
.
Then, for an arbitrary but fixed , the sequence defined by (3.1) converges weakly to a point in F.
Proof First of all, by Lemma 1.1, we know that is nonexpansive. Take an arbitrary . Repeating the same arguments as those in the proof of Theorem 3.1, we know that is bounded, and so are the sequences , , , and .
It follows from (2.2) and (3.1) that
and hence
Since and is bounded, we obtain . Utilizing Lemma 2.14, we conclude that exists. Furthermore, it follows from (3.24) that for all ,
Since , and
we deduce from (3.25) that
It immediately follows that
Note that
Thus, from (3.26) and (3.27) we have
Also, observe that
From [[52], Remark 2.2] (see also [[53], Remark 3.1]), we have
This implies immediately that
Now, let us show that (see (2.1)). Indeed, let . Then there exists a subsequence of such that . Since and , by Lemma 2.15, we know that and . Thus, according to Lemmas 1.1 and 2.11.
Finally, let us show that is a singleton. Indeed, let be another subsequence of such that . Then also lies in F. If , by Opial’s property of H, we reach the following contradiction:
This implies that is a singleton. Consequently, converges weakly to a point in F. □
Remark 3.1 Our Theorem 3.1 improves, extends, supplements and develops Ceng et al. [[4], Theorem 3.3], Cai and Bu [[5], Theorem 3.1] and Ceng and Yao [[2], 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 the problem of finding a point in [[4], Theorem 3.3], and the problem of finding a point in [[2], Theorem 3.1].
-
(ii)
The iterative scheme in [[4], Theorems 3.3] is extended to develop the iterative scheme (3.1) of our Theorem 3.1 by virtue of the iterative schemes of [[5], Theorem 3.1] and [[2], Theorem 3.1]. The iterative scheme (3.1) of our Theorem 3.1 is more advantageous and more flexible than the iterative schemes of [[2], Theorem 3.1] and [[4], Theorem 3.3] because it can be applied to solving two problems (i.e., GSVI (1.1), fixed point problem of infinitely many nonexpansive mappings) 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 Ceng and Yao [[4], Theorems 3.3] to the setting of infinitely many nonexpansive mappings and GSVI (1.1) for two strictly pseudocontractive and strongly accretive mappings. In the meantime, our Theorem 3.1 drops the following restrictions in Cai and Bu [[5], Theorem 3.1]:
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 .
-
(iv)
The iterative scheme (3.1) in our Theorem 3.1 is very different from every one in [[4], Theorem 3.3], [[5], Theorem 3.1] and [[2], Theorem 3.1] because the mapping in [[5], Theorem 3.1] and the mapping T in [[2], Theorem 3.1] are replaced by the same W-mapping in the iterative scheme (3.1) of our Theorem 3.1, and (3.1) in our Theorem 3.1 is three-step Mann iterations for finding a point in comparison with two-step Mann iterations for finding a point in [[4], Theorem 3.3].
-
(v)
Cai and Bu’s proof in [[5], Theorem 3.1] depends on the argument techniques in [6], the inequality in 2-uniformly smooth Banach spaces (see Lemma 1.1) and the inequality in smooth and uniform convex Banach spaces (see Proposition 1.1). However, the proof of our Theorem 3.1 does not depend on the argument techniques in [6], the inequality in 2-uniformly smooth Banach spaces (see Lemma 1.1), and the inequality in smooth and uniform convex Banach spaces (see Proposition 1.1). It depends only on the inequality in uniform convex Banach spaces (see Lemma 2.13 in Section 2 of this paper), the properties of reflexive Banach space having a weakly continuous duality map with gauge φ, and the properties of the W-mapping (see Lemmas 2.6-2.7 and 2.10-2.11 in Section 2 of this paper).
-
(vi)
The assumption of the uniformly convex and 2-uniformly smooth Banach space X in [[5], Theorem 3.1] is weakened to the one of the uniformly convex Banach space X which either has a weakly continuous duality map with gauge φ or is uniformly smooth in our Theorem 3.1. Moreover, our Theorem 3.1 shows that the assumption of the uniformly smooth Banach space X in [[2], Theorem 3.1] can be replaced by the assumption of the uniformly convex Banach space X having a weakly continuous duality map with gauge φ in our Theorem 3.1. It is worth emphasizing that there is no assumption on the convergence of parameter sequences , and to the zero in our Theorem 3.1.
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Acknowledgements
The first author was partially supported 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 second author was partially supported by a grant from NSC 102-2115-M-037-001.
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Ceng, LC., Wen, CF. Three-step Mann iterations for a general system of variational inequalities and an infinite family of nonexpansive mappings in Banach spaces. J Inequal Appl 2013, 539 (2013). https://doi.org/10.1186/1029-242X-2013-539
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DOI: https://doi.org/10.1186/1029-242X-2013-539