Abstract
In this paper, we present relaxed and composite viscosity methods for computing a common solution of a general systems of variational inequalities, common fixed points of infinitely many nonexpansive mappings and zeros of accretive operators in real smooth and uniformly convex Banach spaces. The relaxed and composite viscosity methods are based on Korpelevich’s extragradient method, the viscosity approximation method and the Mann iteration method. Under suitable assumptions, we derive some strong convergence theorems for relaxed and composite viscosity algorithms not only in the setting of a uniformly convex and 2-uniformly smooth Banach space but also in a uniformly convex Banach space having a uniformly Gâteaux differentiable norm. The results presented in this paper improve, extend, supplement, and develop the corresponding results given in the literature.
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1 Introduction
The theory of variational inequalities is well established and a tool to solve many problems arising from science, engineering, social sciences, etc., see, for example, [1–4] and the references therein. One of the interesting directions, from the research view point, in the theory of variational inequalities is to develop some new iterative methods for computing the approximate solutions of different kinds of variational inequalities. In 1976, Korpelevich [5] proposed an iterative algorithm for solving variational inequalities (VI) in the finite dimensional space setting, It is now known as the extragradient method. Korpelevich’s extragradient method has received great attention by many authors, who improved it in various ways and in different directions, see, for example [6–16] and the references therein. In the recent past, several iterative methods for solving VI were proposed and analyzed in [17–24] in the setting of Banach spaces. In the last three decades, the system of variational inequalities is used as a tool to study the Nash equilibrium problem for a finite or infinite number of players, see, for example, [2, 3, 25, 26] and the references therein. Cai and Bu [20] considered a system of two variational inequalities (SVI) in the setting of real smooth Banach spaces. They proposed and analyzed an iterative method for computing the approximate solutions of system of variational inequalities. Such a solution is also a common fixed point of a family of nonexpansive mappings.
One of the most interesting problems in nonlinear analysis is to find a zero of an accretive operator. In 2007, Aoyama et al. [27] suggested a Halpern type iterative method for finding a common fixed point of a countable family of nonexpansive mappings and a zero of an accretive operator. They studied the strong convergence of the sequence generated by the proposed method in the setting of a uniformly convex Banach space having a uniformly Gâreaux differentiable norm. Ceng et al. [28] introduced and analyzed the composite iterative scheme to compute a zero of m-accretive operator A defined on a uniformly smooth Banach space or a reflexive Banach space having a weakly sequentially continuous duality mapping. It is shown that the iterative process in each case converges strongly to a zero of A. Subsequently, Jung [29] studied a viscosity approximation method, which generalizes the composite method in [28], to investigate the zero of an accretive operator.
During the last decade, several iterative methods have been proposed and analyzed to find a common solution of two different fixed point problems, a fixed point problem and a variational inequality problem, a fixed point problem for a family of nonexpansive mappings and a variational inequality problem or a fixed point problem and a system of variational inequalities, etc. See, for example, [8, 16, 20, 30, 31] and the references therein.
In the present paper, we mainly propose two different methods, namely, relaxed viscosity method and composite viscosity method, to find a common fixed point of an infinite family of nonexpansive mappings, a system of variational inequalities and zero of an accretive operator in the setting of a uniformly convex and 2-uniformly smooth Banach spaces. These methods are based on Korpelevich’s extragradient method, viscosity approximation method and Mann iteration method. Under suitable assumptions, we derive some strong convergence theorems for relaxed and composite viscosity algorithms not only in the setting of a uniformly convex and 2-uniformly smooth Banach space but also in the setting of uniformly convex Banach spaces having a uniformly Gâteaux differentiable norm. The results presented in this paper improve, extend, supplement, and develop the corresponding results in [10, 20, 24, 29, 30].
2 Preliminaries
Throughout the paper, unless otherwise specified, we adopt the following assumptions and notations.
Let X be a real Banach space whose dual space is denoted by . Let C be a nonempty closed convex subset of X. We denote by the set of all contractive mappings from C into itself.
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 in X. A Banach space X is said to be uniformly convex if for each , there exists such that for all ,
It is well 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 Gâteaux differentiable norm. X is said to have a uniformly Gâteaux differentiable norm if for each , the limit is attained uniformly for all . Moreover, it is said to be uniformly smooth if this limit is attained uniformly for all . The norm of X is said to be Fréchet differentiable if, for each , this limit is attained uniformly for all . A function defined by
is called the modulus of smoothness of X. It is well 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 [32], 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 Gâteaux differentiable norm then the duality mapping J is norm-to-weak∗ uniformly continuous on bounded subsets of X. For further details of the geometry of Banach spaces, we refer to [33–35].
Now, we present some lemmas which will be used in the sequel.
Lemma 2.1 [36]
Let X be a 2-uniformly smooth Banach space. Then
where κ is the 2-uniformly smooth constant of X.
The following lemma is an immediate consequence of the subdifferential inequality of the function .
Lemma 2.2 [37]
Let X be a real Banach space X. Then, for all ,
-
(a)
, ;
-
(b)
, .
Lemma 2.3 [36]
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 2.4 [38]
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 .
Proposition 2.1 [22]
Let X be a real smooth and uniform convex Banach space and . Then there exists a strictly increasing, continuous, and convex function , such that
where .
Lemma 2.5 [39]
Let C be a nonempty closed convex subset of a strictly convex Banach space X. Let be a sequence of nonexpansive mappings from C into itself such that is nonempty. Let be a sequence of positive numbers with . Then a mapping defined by , for all , is well defined and nonexpansive, and .
Lemma 2.6 [40]
Let and be bounded sequences in a Banach space X and be a sequence of nonnegative numbers in with . Suppose that for all integers and . Then .
Lemma 2.7 [41]
Let be a sequence of nonnegative real numbers satisfying
where , , and satisfy the conditions:
-
(i)
and ;
-
(ii)
;
-
(iii)
, , and .
Then .
A mapping is called nonexpansive if for every . The set of fixed points of T is denoted by . A mapping is said to be
-
(a)
accretive if for each , there exists such that
-
(b)
α-strongly accretive if for each , there exists such that
-
(c)
β-inverse strongly accretive if for each , there exists such that
-
(d)
λ-strictly pseudocontractive [18, 42] if for each , there exists such that
It is worth to emphasize that the definition of the inverse strongly accretive mapping is based on that of the inverse strongly monotone mapping [43].
Lemma 2.8 [[20], Lemma 2.8]
Let C be a nonempty closed convex subset of a real 2-uniformly smooth Banach space X and for each , be an -inverse strongly accretive mapping. Then, for each ,
where . In particular, if , then is nonexpansive for each .
Let C be a nonempty closed convex subset of a Banach space X and be a nonexpansive mapping with . For all and , let be a unique fixed point of the contraction on C, that is,
Let X be an 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, 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
Recall that a (possibly set-valued mapping) operator with domain and range in X is accretive if, for each and (), there exists a such that . 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 define a mapping by for each , which is called the resolvent of A. It is well known that is nonexpansive and for all . Therefore,
If , then the inclusion is solvable.
Proposition 2.2 (Resolvent Identity [46])
For , and ,
Let D be a subset of C. A mapping is said to be sunny if
whenever for all and . A mapping is called a retraction if . If a mapping 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.
Lemma 2.10 [23]
Let C be a nonempty closed convex subset of a real smooth Banach space X, D be a nonempty subset of C and Π be a retraction of C onto D. Then the following statements are equivalent:
-
(a)
Π is sunny and nonexpansive;
-
(b)
, ;
-
(c)
, , .
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 2.11 [[20], Lemma 2.9]
Let C be a nonempty closed convex subset of a real 2-uniformly smooth Banach space X and be a sunny nonexpansive retraction from X onto C. For each , let be an -inverse strongly accretive mapping and be defined by
If for each , then is nonexpansive.
Let with a contractive coefficient , be a sequence of nonexpansive self-mappings on C and be a sequence of nonnegative numbers in . For any , a self-mapping on C defined by
is called W-mapping [47] generated by and .
Lemma 2.12 [[37], 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 be a sequence of positive numbers in for some . Then, for every and , the limit exists.
B using Lemma 2.12, we define a W-mapping generated by the sequences and by
Throughout this paper, we assume that is a sequence of positive numbers in for some .
Lemma 2.13 [[37], 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 .
Let μ be a continuous linear functional on and . We write instead of . μ is called a Banach limit if μ satisfies and for all . If μ is a Banach limit, then the following implications hold:
-
(a)
for all , implies ;
-
(b)
for any fixed positive integer r;
-
(c)
for all .
Lemma 2.14 [48]
Let be a real number and a sequence satisfy the condition for all Banach limits μ. If , then .
In particular, if in Lemma 2.14, then we obtain the following corollary.
Corollary 2.1 [49]
Let be a real number and a sequence satisfy the condition for all Banach limits μ. If , then .
3 Formulations
Let C be a nonempty closed convex subset of a smooth Banach space X, be nonlinear mappings and and be two positive constants. The problem of system of variational inequalities (SVI) in the setting of a real smooth Banach space X is to find such that
The set of solutions of SVI (3.1) is denoted by . Very recently, Cai and Bu [20] constructed an iterative algorithm for solving SVI (3.1) and a common fixed point problem of an infinite family of nonexpansive mappings in a uniformly convex and 2-uniformly smooth Banach space. They studied the strong convergence of the proposed algorithm.
In particular, if , a real Hilbert space, then SVI (3.1) reduces to the following problem of SVI of finding such that
Further, if , where is an operator, and , then the SVI (3.2) reduces to the classical variational inequality problem (VIP) of finding such that
The solution set of the VIP (3.3) is denoted by . For details and applications of theory of variational inequalities, we refer to [1–4] and the references therein.
Recently, Ceng et al. [10] transformed problem (3.2) into a fixed point problem in the following way.
Lemma 3.1 [10]
For given is a solution of problem (3.2) 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 for each , is a -inverse strongly monotone mapping, then G is a nonexpansive mapping provided for each .
In particular, whenever X is a real smooth Banach space, and , then SVI (3.1) reduces to the variational inequality problem (VIP) of finding such that
which was considered by Aoyama et al. [17]. Note that VIP (3.5) is connected with the fixed point problem for nonlinear mapping [44], the problem of finding a zero point of a nonlinear operator [50] and so on. It is clear that VIP (3.5) extends VIP (3.3) from Hilbert spaces to Banach spaces. For further study on VIP in the setting of Banach spaces, we refer to [17, 21] and the references therein.
Define a mapping by
The fixed point set of G is denoted by Ω.
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 be nonlinear mappings. Then is a solution of SVI (3.1) if and only if , where .
Proof We rewrite SVI (3.1) as
which is obviously equivalent to
because of Lemma 2.10. This completes the proof. □
In terms of Lemma 3.2, we observe that
which implies that is a fixed point of the mapping G.
Motivated and inspired by the research going on in this area, we introduce some relaxed and composite viscosity methods for finding a zero of an accretive operator such that , solving SVI (3.1) and the common fixed point problem of an infinite family of nonexpansive self-mappings on C. Our methods are based on Korpelevich’s extragradient method, the viscosity approximation method, and Mann’s iteration method. Under suitable assumptions, we derive some strong convergence theorems for relaxed and composite viscosity algorithms not only in the setting of uniformly convex and 2-uniformly smooth Banach space but also in a uniformly convex Banach space having a uniformly Gâteaux differentiable norm. The results presented in this paper improve, extend, supplement, and develop the corresponding results given in [10, 20, 24, 29, 48].
4 Relaxed viscosity algorithms and convergence criteria
In this section, we introduce relaxed viscosity algorithms in the setting of real smooth uniformly convex Banach spaces and study the strong convergence of the sequences generated by the proposed algorithms.
Throughout this paper, we denote by Ω the fixed point set of the mapping .
Assumption 4.1 Let , , , , be 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 .
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 and be an accretive operator such that . For each , let be -inverse strongly accretive mapping and be a contraction with coefficient . Let be an infinite family of nonexpansive mappings from C into itself such that with for . Assume that Assumption 4.1 holds. For arbitrarily given , let be a sequence generated by
where is the W-mapping generated by (2.1). Then
-
(a)
;
-
(b)
the sequence converges strongly to some which is a unique solution of the following variational inequality problem (VIP):
provided for some fixed .
Proof We first claim that the sequence is bounded. Indeed, take a fixed arbitrarily. Then we get , , and for all . By Lemma 2.11, G is nonexpansive. Then, from (4.1), we have
and
By induction, we obtain
Hence, is bounded, and so are the sequences , , , and .
Next we show that
We note that can be rewritten as follows:
where . Observe that
On the other hand, if , using the resolvent identity in Proposition 2.2,
we get
If , then it is easy to see that
By combining the above cases, we obtain
Similarly, we have
Therefore, we obtain
where
and
for some . Since and are nonexpansive, from (2.1), we deduce that for each
By simple computations, we obtain
It follows from (4.6) that
Taking into account that , without loss of generality, we may assume that . Utilizing (4.5)-(4.8), we have
where for some . Thus, it follows from (4.9) and conditions (i), (iii), (iv) that
Since , by Lemma 2.6, we get
Consequently,
Now we show that as . Indeed, by Lemma 2.3 and (4.1), we get
By Lemma 2.2(a), (4.1), and (4.10), we obtain
and thus
Since and , from condition (v) and the boundedness of , it follows that
Utilizing the properties of g, we have
and thus,
For simplicity, we put , and . Then for all . From Lemma 2.8, we have
and
By combining (4.13) and (4.14), we obtain
By the convexity of , we have, from (4.1) and (4.15),
and thus
Since and for , and and are bounded, we obtain from condition (v) that
Utilizing Proposition 2.2 and Lemma 2.10, we have
which implies that
In the same way, we derive
and we get
Combining (4.17) and (4.18), we get
By the convexity of , we have, from (4.1) and (4.19),
and hence
From (4.12), (4.16), condition (v), and the boundedness of , , , and , we deduce
Utilizing the properties of and , we obtain
Hence,
that is,
Next, we show that
Indeed, observe that can be rewritten as
where and . Utilizing Lemma 2.4 and (4.22), we have
which implies that
Utilizing (4.4), conditions (i), (ii), (v), and the boundedness of and , we obtain
From the properties of , we have
Utilizing Lemma 2.3 and the definition of , we have
and thus
Since and are bounded and as , we deduce from condition (ii) that
From the properties of , we have
On the other hand, can also be rewritten as
where and . Utilizing Lemma 2.4 and the convexity of , we have
which implies that
From (4.4), conditions (i), (ii), (v), and the boundedness of and , we have
Utilizing the properties of , we have
which together with (4.12) and (4.24), implies that
that is,
We note that
Thus, from (4.12), (4.21), (4.23), and (4.24), it follows that
Now, we claim that for a fixed number r such that . In fact, using the resolvent identity in Proposition 2.2, we have
Thus, from (4.26) and (4.27), we get
that is,
Suppose that for some fixed such that for all . Define a mapping , where are two constants with . Then, by Lemmas 2.5 and 2.13, we have . For each , let be a unique element of C such that
From Lemma 2.9, we conclude that as . Observe that for every n, k
where . Since , we know that as .
From (4.29), we obtain
where as .
For any Banach limit μ, from (4.30), we have
In addition, note that
and
It is easy to see from (4.21) and (4.28) that
Utilizing (4.31) and (4.32), we have
Also, observe that
that is,
It follows from Lemma 2.2(ii) and (4.34) that
So by (4.33) and (4.35), we have
and hence,
This implies that
Since as , by the uniform Fréchet differentiability of the norm of X, we have
On the other hand, from (4.4) and the norm-to-norm uniform continuity of J on bounded subsets of X, we have
Utilizing Lemma 2.14, we deduce from (4.36) and (4.37) that
Finally, we show that as . It is easy to see from (4.1) that
Utilizing Lemma 2.2(a), from (4.1) and the convexity of we get
Applying Lemma 2.7 to (4.38), we obtain 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 and be a sunny nonexpansive retraction from X onto C. Let be an accretive operator such that . Let be α-strictly pseudocontractive mapping and be a contraction with coefficient . Let be an infinite family of nonexpansive mappings from C into itself such that . Suppose that Assumption 4.1 holds. For arbitrarily given , let be the sequence generated by
where , is the W-mapping generated by (2.1). Then
-
(a)
;
-
(b)
the sequence converges strongly to some which is a unique solution of the following variational inequality problem (VIP):
provided for some fixed .
Proof In Theorem 4.1, we put , and where . Then SVI (3.1) is equivalent to the VIP of finding such that
In this case, is α-inverse strongly accretive. It is not hard to see that . Indeed, for , we have
Accordingly, we have , and
Similarly, we get
So, the scheme (4.1) reduces to (4.39), and therefore, the desired result follows from Theorem 4.1. □
We give the following important lemmas which will be used in our next result.
Lemma 4.1 Let C be a nonempty closed convex subset of a smooth Banach space X and be -strictly pseudocontractive mappings and -strongly accretive with for . Then, for ,
for . In particular, if , then is nonexpansive for .
Proof Using the -strict pseudocontractivity of , we derive for every
which implies that
Hence,
Utilizing the -strong accretivity and -strict pseudocontractivity of , we get
So, we have
Therefore, for , we have
Since , it follows that
This implies that is nonexpansive for . □
Lemma 4.2 Let C be a nonempty closed convex subset of a smooth Banach space X and be a sunny nonexpansive retraction from X onto C. For each , let be -strictly pseudocontractive and -strongly accretive with . Let be the mapping defined by
If , then is nonexpansive.
Proof By Lemma 4.1, is nonexpansive for . Therefore, for all , we have
This shows that is nonexpansive. This completes the proof. □
Theorem 4.2 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 and be an accretive operator in X such that . For each , let be -strictly pseudocontractive and -strongly accretive with and be a contraction with coefficient . Let be an infinite family of nonexpansive mappings from C into itself such that with for . For arbitrarily given , let be the sequence generated by
where is the W-mapping generated by (2.1). Assume that Assumption 4.1 holds except condition (iii), which is replaced by the following condition:
-
(iii)
.
Then
-
(a)
;
-
(b)
the sequence converges strongly to some which is the unique solution of the following variational inequality problem (VIP):
provided for some fixed .
Proof Take a fixed arbitrarily. Then we obtain , and for all . By using Lemma 4.2 and the same argument as in the proof beginning of the proof of Theorem 4.1, we have , , , , are bounded sequences. Let us show that as . In fact, repeating the same argument as those in the proof of Theorem 4.1, we obtain
where
and
for some . By (4.40) and simple calculations, we have
It follows that
Repeating the same argument as in (4.7) in the proof of Theorem 4.1, we get
Considering condition (v), without loss of generality, we may assume that for some . From (4.40), it follows that can be rewritten as
where . Utilizing (4.3) and (4.42) we have
By simple calculations, (4.44) implies that
This together with (4.42) and (4.45) we have
where for some . By Lemma 2.7 and conditions (i), (iii), and (iv), we conclude that (noting that , )
Next we show that as . Indeed, utilizing Lemma 2.3, we get from (4.40)
By Lemma 2.2 (a), (4.40), and (4.47), we have
which yields
Since and , from condition (v) and the boundedness of , it follows that
Utilizing the properties of g, we have
On the other hand, observe that can be rewritten as
where and . By Lemma 2.4, (4.3), and (4.49), we have
which implies that
Utilizing (4.46), conditions (i), (ii), (v), and the boundedness of and , we get
From the properties of , we have
Utilizing Lemma 2.3 and the definition of , we have
which leads to
Since and are bounded, from (4.50) and condition (ii), we deduce
From the properties of , we have
Furthermore, can also be rewritten as
where and . Utilizing Lemma 2.4, the convexity of , and (4.3), we have
which implies that
From (4.46), conditions (i), (ii), (v), and the boundedness of and , we have
Utilizing the properties of , we have
Thus, from (4.51) and (4.52), we get
that is,
Therefore, from (4.40), (4.46), (4.52), (4.53), and , we have
that is,
Utilizing (4.40), (4.48), and (4.54), we obtain
that is,
In addition, from (4.52) and (4.54), we have
that is,
Note that
So, from (4.48) and (4.55), it follows that
Repeating the same argument as in (4.28) in the proof of Theorem 4.1, we get
for a fixed number r such that .
Suppose that for some fixed satisfying for all . Define a mapping , where are two constants with . Then, by Lemmas 2.5 and 2.13, we have . For each , let be a unique element of C such that
From Lemma 2.9, we conclude that as . Observe that for every n, k
where . Since , we know that as .
Repeating the same argument as in (4.31) and (4.32) in the proof of Theorem 4.1, we conclude that for any Banach limit μ,
and
Utilizing (4.60) and (4.61), we obtain
Repeating the same argument as in (4.36) in the proof of Theorem 4.1, we get
Since as , by the uniform Gâteaux differentiability of the norm of X, we have
On the other hand, from (4.4) and the norm-to-weak∗ uniform continuity of J on bounded subsets of X, it follows that
Using Lemma 2.14, we deduce from (4.63) and (4.64) that
Finally, we show that as . It is easy to see from (4.1) that
Utilizing Lemma 2.2(a), from (4.1) and the convexity of we get
Applying Lemma 2.7 to (4.65), we obtain as . This completes the proof. □
Corollary 4.2 Let C be a nonempty closed convex subset of a uniformly convex Banach space X which has an uniformly Gâteaux differentiable norm. Let be a sunny nonexpansive retraction from X onto C and be an accretive operator in X such that . Let be a mapping such that is ζ-strictly pseudocontractive and θ-strongly accretive with . Let be a contraction with coefficient and be an infinite family of nonexpansive mappings of C into itself such that . For arbitrarily given , let be the sequence generated by
where , is the W-mapping generated by (2.1). Assume that Assumption 4.1 holds except condition (iii), which is replaced by the following condition:
-
(iii)
.
Then
-
(a)
;
-
(b)
the sequence converges strongly to some which is a unique solution of the following variational inequality problem (VIP):
provided for some fixed .
Proof In Theorem 4.2, we put , and where . Then SVI (3.1) is equivalent to the VIP of finding such that
In this case, is ζ-strictly pseudocontractive and θ-strongly accretive. Repeating the same arguments as in the proof of Corollary 4.1, we can infer that . Accordingly, ,
So, the scheme (4.40) reduces to (4.66). Therefore, the desired result follows from Theorem 4.2. □
Remark 4.1 Theorems 4.1 and 4.2 improve and extend [[30], Theorem 3.2], [[20], Theorem 3.1] and [[29], Theorem 3.1] in the following aspects.
-
(a)
The problem of finding a point in Theorems 4.1 and 4.2 is more general and more subtle than the problem of finding a point in [[30], Theorem 3.2], the problem of finding a point in [[20], Theorem 3.1] and the problem of finding a point in [[29], Theorem 3.1].
-
(b)
Theorems 4.1 and 4.2 are proved without the assumption of the asymptotical regularity of in [[29], Theorem 3.1] (that is, ).
-
(c)
The iterative scheme in [[20], Theorem 3.1] is extended to develop the iterative schemes (4.1) and (4.40) in Theorems 4.1 and 4.2 by virtue of the iterative schemes of [[30], Theorem 3.2] and [[29], Theorem 3.1]. The iterative schemes (4.1) and (4.40) in Theorems 4.1 and 4.2 are more advantageous and more flexible than the iterative scheme in [[20], Theorem 3.1] because they involve several parameter sequences.
-
(d)
The iterative schemes (4.1) and (4.40) in Theorems 4.1 and 4.2 are different from the iterative schemes in [[30], Theorem 3.2], [[20], Theorem 3.1] and [[29], Theorem 3.1] because the mapping G in [[20], Theorem 3.1] and the mapping in [[29], Theorem 3.1] are replaced by the composite mapping in Theorems 4.1 and 4.2.
-
(e)
The proof of [[20], Theorem 3.1] depends on the argument techniques in [10], the inequality in 2-uniformly smooth Banach spaces, and the inequality in smooth and uniform convex Banach spaces. Because the composite mapping appears in the iterative scheme (4.1) of Theorem 4.1, the proof of Theorem 4.1 depends on the argument techniques in [10], the inequality in 2-uniformly smooth Banach spaces, the inequality in smooth and uniform convex Banach spaces, the inequalities in uniform convex Banach spaces, and the properties of the W-mapping and the Banach limit. However, the proof of our Theorem 4.2 does not depend on the argument techniques in [10], the inequality in 2-uniformly smooth Banach spaces, and the inequality in smooth and uniform convex Banach spaces. It depends on only the inequalities in uniform convex Banach spaces and the properties of the W-mapping and the Banach limit.
-
(f)
The assumption of the uniformly convex and 2-uniformly smooth Banach space X in [[20], Theorem 3.1] is weakened to the uniformly convex Banach space X having a uniformly Gâteaux differentiable norm in Theorem 4.2.
5 Composite viscosity algorithms and convergence criteria
In this section, we introduce composite viscosity algorithms in real smooth and uniformly convex Banach spaces and study the strong convergence theorems. We first state the following important and useful lemma which will be used in the sequel.
Lemma 5.1 [27]
Let C be a nonempty closed convex subset of a Banach space X and be a sequence of mappings of C into itself. Suppose that . Then, for each , converges strongly to some point in C. Moreover, let be a mapping defined by for all . Then .
Assumption 5.1 Let , , , , be 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 .
We now state and prove our first result on the composite implicit viscosity algorithm.
Theorem 5.1 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 and be an accretive operator on X such that . Let the mapping be -inverse strongly accretive for , and be a contraction with coefficient . Let be an infinite family of nonexpansive mappings of C into itself such that with for . Suppose that Assumption 5.1 holds. For arbitrarily given , let be the sequence generated by
Assume that for any bounded subset D of C, be a mapping defined by for all , and . Then the sequence converges strongly to , which 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 2.11, G is nonexpansive. Then, from (5.1), we have
which implies that
So, we have
By induction, we obtain
Hence, is bounded, and so are the sequences , , , and .
Let us show that
Observe that can be rewritten as
where . Note that
On the other hand, if , using the resolvent identity in Proposition 2.2,
we get
If , then it is easy to see that
Thus, combining the above cases, we obtain
In a similar way, we derive
Therefore, we have
for all , where
and
for some . Combining (5.6) and (5.5), we have
where for some . By simple calculations, we have
Taking into account condition (v), without loss of generality, we may assume that for some . Hence, from (5.7) and (5.8), we deduce
where for some . This leads to
Again by simple calculations, we have
This together with (5.6) and (5.9) implies that
where for some . Noting that for all , from condition (i), we know that . Utilizing Lemma 2.7, we conclude from conditions (iii), (iv), and the assumption on that
Next we show that as .
Indeed, according to Lemma 2.2(a), we have from (5.1)
which implies that
Utilizing Lemma 2.3, we get from (5.1) and (5.10)
and hence
Since and , from condition (v) and the boundedness of and , it follows that
Utilizing the properties of g, we have
Observe that
From (5.4) and (5.11), we have
For simplicity, put , and . Then for all . From Lemma 2.8, we have
and
Combining (5.13) and (5.14), we obtain
By Lemma 2.2(a), (5.1), and (5.15), we have
Thus, we have
Since for , from (5.12) and conditions (i), (ii), we obtain
Utilizing Proposition 2.2 and Lemma 2.10, we have
which implies that
In the same way, we derive
which implies that
Combining (5.17) and (5.18), we get
By Lemma 2.2(a), (5.1), and (5.19), we have
and hence
Utilizing conditions (i), (ii), from (5.12) and (5.16), we have
Utilizing the properties of and , we have
From (5.21), we get
that is,
Next, let us show that
Indeed, observe that can be rewritten as
where and . Utilizing Lemma 2.4 and (5.23), we have
which implies that
Utilizing (5.12), conditions (i), (ii), (v), and the boundedness of , , and , we get
From the properties of , we have
Utilizing Lemma 2.3 and the definition of , we have
which leads to
Since and are bounded and as , we deduce from condition (ii) that
From the properties of , we have
On the other hand, can also be rewritten as
where and . Utilizing Lemma 2.4 and the convexity of , we have
which implies that
From (5.12), conditions (i), (ii), (v), and the boundedness of , , and , we have
Utilizing the properties of , we have
By Lemma 5.1, we get
that is,
We note that
So, from (5.22), (5.24), and (5.25), it follows that
Furthermore, we claim that for a fixed number r such that . In fact, taking into account the resolvent identity in Proposition 2.2, we have
From (5.27) and (5.8), we get
that is,
Define a mapping , where are two constants with . Then by Lemma 2.5, we have . We observe that
From (5.22), (5.26), and (5.29), 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(a), we obtain
where
It follows from (5.32) that
Letting in (5.34) and noticing (5.33), we derive
where is a constant such that for all and . Taking in (5.35), we have
On the other hand, we have
It follows that
Taking into account that as , we have from (5.36)
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 (5.31) holds. From (5.12) we get . Noticing that J is norm-to-norm uniformly continuous on bounded subsets of X, we deduce from (5.31) that
Finally, let us show that as . We observe that
which implies that
From (5.1) and the convexity of , we get
Applying Lemma 2.7 to (5.39), we obtain as . This completes the proof. □
Corollary 5.1 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 and be an accretive operator on X such that . Let the mapping be -inverse strongly accretive for , and be a contraction with coefficient . Let be a nonexpansive mapping such that with for . For arbitrarily given , let be the sequence generated by
Suppose that Assumption 5.1 holds. Assume that for any bounded subset D of C, is a mapping defined by for all , and . Then the sequence converges strongly to , which solves the following VIP:
We now establish the following strong convergence result on the composite explicit viscosity algorithm.
Theorem 5.2 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 and be an accretive operator on X such that . For each , let be a -strictly pseudocontractive and -strongly accretive mapping with . Let be a contraction with coefficient and be an infinite family of nonexpansive mappings such that with for . Suppose that Assumption 5.1 holds. For arbitrarily given , let be the sequence generated by
Assume that for any bounded subset D of C, is a mapping defined by for all , and . Then converges strongly to , which solves the following VIP:
Proof Take a fixed arbitrarily. Then we obtain , and for all . Moreover, by Lemma 4.2, we have
and therefore
By induction, we get
which implies that is bounded and so are the sequences , , , .
Let us show that as . As a matter of fact, repeating the same arguments as those in the proof of Theorem 4.1, we obtain
where
and
for some . By (5.40) and simple calculations, we have
It follows that
Taking into account condition (v), without loss of generality we may assume that for some . From (5.40), can be rewritten as
where . Utilizing (5.42) and (5.43), we have
By simple calculations and (5.44), we get
This together with (5.43) and (5.45) implies that
where for some . So, in terms of Lemma 2.7 and conditions (i), (iii), and (iv), we conclude that
Next we show that as .
Indeed, utilizing Lemma 2.3 and (5.40), we get
According to Lemma 2.2, we have from (5.40) and (5.47)
which hence yields
Since and , from condition (v) and the boundedness of , it follows that
Utilizing the properties of g, we have
On the other hand, can be rewritten as
where and . Utilizing Lemma 2.4, from (5.41) and (5.49), we have
which hence implies that
Utilizing (5.46), conditions (i), (ii), (v), and the boundedness of and , we get
From the properties of , we have
Utilizing Lemma 2.3 and the definition of , we have
which leads to
Since and are bounded, we deduce from (5.50) and condition (ii) that
From the properties of , we have
Furthermore, can also be rewritten as
where and . Utilizing Lemma 2.4 and the convexity of , we have from (5.41)
which implies that
From (5.46), conditions (i), (ii), (v), and the boundedness of and , we have
Utilizing the properties of , we have
Thus, from (5.51) and (5.52), we get
that is,
Therefore, from (5.40), (5.46), (5.52), (5.53), and , it follows that
that is,
Utilizing (5.40), (5.48), and (5.54), we obtain
that is,
In addition, from (5.52) and (5.54), we have
that is,
In terms of (5.56) and Lemma 2.6, we have
that is,
We note that
So, from (5.53), (5.54), and (5.55), we obtain
Furthermore, repeating the same arguments as those of (5.29) in the proof of Theorem 4.1, we can derive
for a fixed number . Define a mapping , where are two constants with . Then by Lemma 2.5, we have . We observe that
From (5.55), (5.57), and (5.59), 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 (5.36) in the proof of Theorem 4.1, we derive
Repeating the same arguments as those of (5.37) in the proof of Theorem 4.1, we obtain
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 (5.61) holds. From (5.46), we get . Noticing the norm-to-weak∗ uniform continuity of J on bounded subsets of X, we deduce from (5.61) that
Finally, let us show that as . We observe that
and hence
Thus, we have
Since and , by Lemma 2.7, we conclude from (5.64) that as . This completes the proof. □
Corollary 5.2 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 and be an accretive operator on X such that . Let the mapping be -strictly pseudocontractive and -strongly accretive with for . Let be a contraction with coefficient and be a nonexpansive mapping such that with for . Suppose that Assumption 5.1 holds. For arbitrarily given , let be the sequence generated by
Then the sequence converges strongly to , which solves the following VIP:
Remark 5.1 Our Theorems 5.1 and 5.2 improve and extend [[30], Theorem 3.2], [[20], Theorem 3.1] and [[29], Theorem 3.1] in the following aspects.
-
(a)
The problem of finding a point in Theorems 5.1 and 5.2 is more general and more subtle than the problem of finding in [[30], Theorem 3.2], the problem of finding in [[20], Theorem 3.1] and the problem of finding in [[29], Theorem 3.1].
-
(b)
Theorems 5.1 and 5.2 are proved without the asymptotical regularity assumption of in [[29], Theorem 3.1] (that is, ).
-
(c)
The iterative scheme in [[20], Theorem 3.1] is extended to develop the iterative schemes (5.1) and (5.40) in Theorems 5.1 and 5.2 by virtue of the iterative schemes of [[30], Theorem 3.2] and [[29], Theorem 3.1]. The iterative schemes (5.1) and (5.40) in Theorems 5.1 and 5.2 are more advantageous and more flexible than the iterative scheme in [[20], Theorem 3.1] because they involves several parameter sequences.
-
(d)
The iterative schemes (5.1) and (5.40) in Theorems 5.1 and 5.2 are different from the one given in [[30], Theorem 3.2], [[20], Theorem 3.1] and [[29], Theorem 3.1] because the first iteration step in (5.1) is implicit and because the mapping G in [[20], Theorem 3.1] and the mapping in [[29], Theorem 3.1] are replaced by the same composite mapping in Theorems 5.1 and 5.2.
-
(e)
The proof of [[20], Theorem 3.1] depends on the argument techniques in [10], the inequality in 2-uniformly smooth Banach spaces and the inequality in smooth and uniform convex Banach spaces. Because the composite mapping appears in the iterative scheme (5.1) in Theorem 5.1, the proof of Theorem 5.1 depends on the argument techniques in [10], the inequality in 2-uniformly smooth Banach spaces, the inequality in smooth and uniform convex Banach spaces, and the inequalities in uniform convex Banach spaces. However, the proof of our Theorem 5.1 does not depend on the argument techniques in [10], the inequality in 2-uniformly smooth Banach spaces, and the inequality in smooth and uniform convex Banach spaces. It depends on only the inequalities in uniform convex Banach spaces.
-
(f)
The assumption of the uniformly convex and 2-uniformly smooth Banach space X in [[20], Theorem 3.1] is weakened to the one of the uniformly convex Banach space X having a uniformly Gâteaux differentiable norm in Theorem 5.2.
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Acknowledgements
This article was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. The authors, therefore, acknowledge with thanks DSR for the technical and financial support. This research was partially supported to 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).
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Ceng, LC., Al-Otaibi, A., Ansari, Q.H. et al. Relaxed and composite viscosity methods for variational inequalities, fixed points of nonexpansive mappings and zeros of accretive operators. Fixed Point Theory Appl 2014, 29 (2014). https://doi.org/10.1186/1687-1812-2014-29
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DOI: https://doi.org/10.1186/1687-1812-2014-29