Implicit and explicit iterative algorithms for hierarchical variational inequality in uniformly smooth Banach spaces.

The purpose of this paper is to solve the hierarchical variational inequality with the constraint of a general system of variational inequalities in a uniformly convex and 2-uniformly smooth Banach space. We introduce implicit and explicit iterative algorithms which converge strongly to a unique solution of the hierarchical variational inequality problem. Our results improve and extend the corresponding results announced by some authors.

Introduction

Let X be a real Banach space with its topological dual , and C be a nonempty closed convex subset of X. Let be a nonlinear mapping on C. We denote by the set of fixed points of T and by R the set of all real numbers. A mapping is called L-Lipschitz continuous if there exists a constant such that In particular, if then T is called a nonexpansive mapping; if then T is called a contraction.

The normalized dual mapping is defined as where denotes the generalized duality pairing; see, e.g., [1] for further details.

Let be the unit sphere of X. Then the space X is said to We note that if X is smooth, then the normalized duality mapping is single-valued; and if the norm of X is uniformly Gâteaux differentiable, then the normalized duality mapping is norm to weak star uniformly continuous on every bounded subset of X (see [1]). In the sequel, we shall denote by j the single-valued normalized duality mapping.

have a Gâteaux differentiable norm if the limit exists for each .

have a uniformly Gâteaux differentiable norm if the limit is attained uniformly for .

be strictly convex if and only if for with , we have

Let X be a smooth Banach space. Let be two nonlinear mappings and be two positive real numbers. The general system of variational inequalities (GSVI, for short) is to find such that

The equivalence between GSVI (2) and the fixed point problem of some nonexpansive mapping defined on a Banach space is established in Yao et al. [2]. The authors [2] introduced and analyzed implicit and explicit iterative algorithms for solving GSVI (2) by using this equivalence, and they proved the strong convergence of the sequences generated by the proposed algorithms. Subsequently, Ceng et al. [3] proposed and analyzed an implicit algorithm of Mann’s type and another explicit algorithm of Mann’s type for solving GSVI (2).

If X is a real Hilbert space, then GSVI (2) was introduced and studied by Ceng et al. [4]. In this case, for , it was considered by Verma [5] (see also [6]). Further, in this case, when , problem (2) reduces to the following classical variational inequality (in short, VI) of finding such that

This problem is a fundamental problem in the variational analysis, in particular, in the optimization theory and mechanics; see, e.g., [7-12] and the references therein. A large number of algorithms for solving this problem are essentially projection algorithms that employ projections onto the feasible set C of the VI, or onto some related set, so as to iteratively reach a solution. In particular, Korpelevich [13] proposed extragradient method for solving the VI in the Euclidean space. This method further has been improved by several researchers; see, e.g., [4, 14, 15] and the references therein.

In the case of a Banach space setting, that is, if and , the VI is defined as Aoyama et al. [16] proposed an iterative scheme to find the approximate solution of (4) and proved the weak convergence of the sequences generated by the proposed scheme. It is also well known [16] that, in a smooth Banach space, this problem is equivalent to a fixed-point equation, containing a sunny nonexpansive retraction from any point of the space onto the feasible set, which is usually assumed to be closed and convex. For the complexity of the feasible set, the sunny nonexpansive retraction is difficult to compute. To overcome this drawback in a Hilbert space, where the retraction is a metric projection, in Yamada [17], the feasible set is assumed to be the common fixed points of a finite family of nonexpansive mappings, and an explicit hybrid steepest-descent method is introduced. In this case, the variational inequality defined on such a feasible set is also called a hierarchical variational inequality (in short, HVI). Yamada’s method is then extended to solve more complex problems involving finite or infinite nonexpansive mappings (one can refer to, e.g., [18, 19] and the references therein). Zeng and Yao [19] introduced an implicit method that converges weakly to a solution of a variational inequality, containing a Lipschitz continuous and strongly monotone mapping in a Hilbert space H, where the feasible set is the common fixed points of a finite family of nonexpansive mappings on H. Ceng et al. [20] extended this result from nonexpansive mappings to Lipschitz pseudocontractive mappings and strictly pseudocontractive mappings on H. Recently, Buong and Anh [21] modified Yamada’s result and proposed a strongly convergent implicit method.

In this paper, we are going to solve the hierarchical variational inequality with the constraint of a general system of variational inequalities in a uniformly convex and 2-uniformly smooth Banach space. We introduce implicit and explicit iterative algorithms for finding a solution of the problem and derive the strong convergence of the proposed algorithms to a unique solution of the problem. Our results improve and extend the corresponding results announced by some others, e.g., Ceng et al. [3] and Buong and Phuong [18].

Preliminaries

Let X be a real Banach space with the dual space . For simplicity, the norms of X and are denoted by the symbol . Let X be a nonempty closed convex subset of a real Banach space X. We write (respectively, ) to indicate that the sequence converges weakly (respectively, strongly) to x. A mapping , defined by is called the normalized duality mapping of X. We know that for all and , and .

Let . A Banach space X is said to be uniformly convex if for each , there exists such that for any , . It is known that a uniformly convex Banach space is reflexive and strictly convex. Also, it is known that if a Banach space X is reflexive, then X is strictly convex if and only if is smooth as well as X is smooth if and only if is strictly convex.

Proposition 1

[22]

Let X be a smooth and uniformly convex Banach space, and let . Then there exists a strictly increasing, continuous and convex function , such that where .

Here 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 . For further details on the geometry of Banach spaces, we refer to [1, 23] and the references therein. Takahashi et al. [24] reminded us of the fact that no Banach space is q-uniformly smooth for . So, in this paper, we focus on only a 2-uniformly smooth Banach space as in [2].

Lemma 1

[25]

Let q be a given real number with , and let X be a q-uniformly smooth Banach space. Then where κ is the q-uniformly smooth constant of X and is the generalized duality mapping from X into defined by

Let D be a subset of C, and let Π be a mapping of C into D. Then Π is said to be sunny if whenever for and . A mapping Π of C into itself is called a retraction if . If a mapping Π of C into itself is a retraction, then for each , 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

[26]

Let C be a nonempty closed convex subset of a smooth Banach space X, D be a nonempty subset of C and Π be a retraction of C onto D. Then the following are equivalent:

Π is sunny and nonexpansive;

, , ;

, , .

It is well known that if X is a Hilbert space, then a sunny nonexpansive retraction coincides with the metric projection from X onto C. Let C be a nonempty closed convex subset of a uniformly convex and uniformly smooth Banach space X, and let T be a nonexpansive mapping of C into itself with the fixed point set . Then the set is a sunny nonexpansive retract of C; see, e.g., [2].

Lemma 3

[2]

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 mappings be α-inverse-strongly accretive and β-inverse-strongly accretive, respectively. For given , is a solution of GSVI (2) if and only if , where is the set of fixed points of the mapping and .

Proposition 2

[27]

Let C be a nonempty closed convex subset of a real 2-uniformly smooth Banach space X. Let the mappings be α-inverse-strongly accretive and β-inverse-strongly accretive, respectively. Then and In particular, if and , then and are nonexpansive.

Lemma 4

[2]

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 mappings be α-inverse-strongly accretive and β-inverse-strongly accretive, respectively. Let the mapping be defined as . If and , then is nonexpansive.

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 mappings be α-inverse-strongly accretive and β-inverse-strongly accretive, respectively. Let be δ-strongly accretive and ζ-strictly pseudocontractive with . Assume that and , where κ is the 2-uniformly smooth constant of X (see Lemma 2). Very recently, in order to solve GSVI (2), Ceng et al. [3] introduced an implicit algorithm of Mann’s type.

Algorithm 1

[3]

For each , choose a number arbitrarily. The net is generated by the implicit method where is a unique fixed point of the contraction

It was proven in [3] that the net converges in norm, as , to the unique solution to the following VI: provided . In the meantime, the authors [3] also proposed another explicit algorithm of Mann’s type.

Algorithm 2

[3]

For arbitrarily given , let the sequence be generated iteratively by where and are three sequences in such that , .

On the other hand, a mapping F with domain and range in X is called

accretive if for each , there exists such that where J is the normalized duality mapping;

δ-strongly accretive if for each , there exists such that

α-inverse-strongly accretive if for each , there exists such that

ζ-strictly pseudocontractive if for each , there exists such that

It is easy to see that (5) can be rewritten as (see [28]) where I denotes the identity mapping of X. Clearly, if F is ζ-strictly pseudocontractive with , then it is said to be pseudocontractive. It is not hard to find that every nonexpansive mapping is pseudocontractive.

Let C be a nonempty closed convex subset of a smooth Banach space X and be an infinite family of nonexpansive self-mappings on C. Then we set . In 2013, Buong and Phuong [18] considered the following HVI with : find such that In the case where , a Hilbert space, we have , and hence problem (7) reduces to the HVI: find such that Assume that is the set of common fixed points of a family of N nonexpansive mappings on H, and F is an L-Lipschitz continuous and η-strongly monotone mapping, i.e., and for all . Zeng and Yao [19] introduced the following implicit iteration: for an arbitrarily initial point , the sequence is generated as follows: where , for integer , with the mod function taking values in the set . They proved the following result.

Theorem 3

[19]

Let H be a real Hilbert space, and let be a mapping such that, for some positive constants L and η, F is L-Lipschitz continuous and η-strongly monotone. Let be N nonexpansive mappings on H such that . Let , , and satisfying the conditions , and let , for some . Then the sequence , defined by (9), converges weakly to .

Recently, for deriving the strong convergence and weakening the condition on , Buong and Anh [21] proposed the following implicit iteration method: where the sequence is defined by and proved that the nets , defined by (10)-(11), converge strongly to an element in (8). Under the assumptions that , X is a real reflexive and strictly convex Banach space with a uniformly Gâteaux differentiable norm, and T is a continuous pseudocontractive mapping, Ceng et al. [29] proved the following result.

Theorem 4

[29]

Let F be a δ-strongly accretive and ζ-strictly pseudocontractive mapping with , and let T be a continuous and pseudocontractive mapping on X, which is a real reflexive and strictly convex Banach space with a uniformly Gâteaux differentiable norm such that . For each , choose a number arbitrarily, and let be defined by Then, as , converges strongly to .

In [30], Takahashi introduced the following W-mapping which is generated by and real numbers as follows: Kikkawa and Takahashi [31] considered the following strongly convergent implicit method: However, the method (14) is very difficult to be grasped due to the limit mapping U.

Motivated by methods (10) and (12), Buong and Phuong [18] considered two implicit methods by introducing a mapping defined by where In both methods, the iteration sequence is defined, respectively, by and where and are the positive parameters satisfying some additional conditions. The strong convergence theorems for methods (17) and (18) are also established.

We will make use of the following well-known results.

Lemma 5

Let X be a real normed linear space. Then the following inequality holds:

Lemma 6

[32]

Let C be a nonempty closed convex subset of a uniformly convex Banach space X and be a nonexpansive mapping with . If is a sequence of C such that and , then . In particular, if , then .

Lemma 7

[2]

Let C be a nonempty closed convex subset of a real smooth Banach space X. Assume that the mapping is accretive and weakly continuous along segments (that is, as ). Then the variational inequality is equivalent to the following Minty type variational inequality:

Lemma 8

[3]

Let X be a real smooth Banach space and be a mapping.

If F is ζ-strictly pseudocontractive, then F is Lipschitz continuous with constant .

If F is δ-strongly accretive and ζ-strictly pseudocontractive with , then is contractive with constant .

If F is δ-strongly accretive and ζ-strictly pseudocontractive with , then for any fixed number , is contractive with constant .

Iterative algorithms and convergence criteria

In this section, we study iterative methods for computing approximate solutions of the HVI (for an infinite family of nonexpansive mappings) with a GSVI constraint. We introduce implicit and explicit iterative algorithms for solving such a problem. We show the strong convergence theorems for the sequences generated by the proposed algorithms.

The following lemmas will be used to prove our main results in the sequel.

Lemma 9

[18]

Let C be a nonempty closed convex subset of a strictly convex Banach space X, and let , be k nonexpansive self-mappings on C such that the set of common fixed points . Let and , be real numbers such that , and let be a mapping defined by (15) for all . Then .

Lemma 10

[18]

Let C be a nonempty closed convex subset of a Banach space X, and let be an infinite family of nonexpansive self-mappings on C such that the set of common fixed points . Let be a mapping defined by (15), and let satisfy (16). Then, for each and , exists.

Remark 1

We can define the mappings

It can be readily seen from the proof of Lemma 10 that if D is a nonempty and bounded subset of C, then the following holds: In particular, whenever , we have

Lemma 11

[18]

Let C be a nonempty closed convex subset of a strictly convex Banach space X, and let be an infinite family of nonexpansive self-mappings on C such that the set of common fixed points . Let satisfy the first condition in (16). Then .

Lemma 12

[33]

Let and be bounded sequences in a Banach space X, and let be a sequence in such that Suppose that , and Then .

Lemma 13

[34]

Assume that is a sequence of nonnegative real numbers such that where is a sequence in and is a sequence in R such that Then .

;

or .

Now, we are in a position to prove the following main results.

Theorem 5

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 mappings be α-inverse-strongly accretive and β-inverse-strongly accretive, respectively. Let be δ-strongly accretive and ζ-strictly pseudocontractive with . Assume that and where κ is the 2-uniformly smooth constant of X. Let be an infinite family of nonexpansive self-mappings on C such that . Let be defined by (15) and (16). Let be defined by where and are sequences in such that and as . Then converges strongly to a unique solution to the following VI:

Proof

Let the mapping be defined as , where and . In terms of Lemma 4 we know that is nonexpansive. Then the implicit iterative scheme can be rewritten as Consider the mapping . From Lemmas 4 and 8(c), it follows that for each , where (by using ). Due to , is a contraction of C into itself. Hence, by Banach’s contraction principle, there exists a unique element satisfying (20).

Next, we divide the rest of the proof into several steps.

Step 1. We show that is bounded. Indeed, take an arbitrarily given . Then we have and . Hence, by Lemma 8(c) we get Therefore, , which also leads to the boundedness of . So, the sequences , and , where , are also bounded. Since as , and the following relation holds we obtain from the boundedness of that as .

Step 2. We show that as , where for all . Indeed, for simplicity, put and . Then for all . From Lemma 2, we have and Substituting (22) for (23), we obtain From (21) and (24), we have which immediately yields So, from , and as , we deduce that Utilizing Proposition 1 and Lemma 2, we have which implies that In the same way, we derive which implies that Substituting (26) for (27), we get So, from (21) and (28) it follows that which hence leads to From (25), as , and the boundedness of and , we deduce that Utilizing the properties of and , we conclude that From (30), we get That is, This together with implies that

Step 3. We show that , where Indeed, we first claim that as . It is easy to see from Remark 1(ii) that if D is a nonempty and bounded subset of C, then, for , there exists such that for all Taking and , we have So, it follows that Noting that from (31), (33) and (34) we obtain that Also, noting that , from (32) and (33) we get In addition, observing that from (31) and (32) we get Since X is reflexive, there exists at least a weak convergence subsequence of , and hence . Take an arbitrary . Then there exists a subsequence of such that . Since is nonexpansive for all , V is a nonexpansive self-mapping on C. Also, since X is uniformly convex and V and G are two nonexpansive self-mappings on C, utilizing Lemma 6 we know from (36) and (37) that and (due to Lemma 11). Consequently, . This shows that .

Step 4. We show that , where Indeed, by Lemma 9, we have for any fixed , and hence Therefore, by Lemma 8(b) we get which immediately leads to where . Note that Since the uniform smoothness of X guarantees the uniform continuity of j on every nonempty bounded subset of X, we deduce from (32), and the boundedness of that

Now, take an arbitrary . Then there exists a subsequence of such that . In terms of Step 3, we know that . Thus, we can substitute for and p for z in (38) to get Consequently, the weak convergence of to p together with (39) actually implies that as , and hence . This shows that .

Step 5. We show that each solves the variational inequality (19). Indeed, take an arbitrary . Then there exists a subsequence of such that as . According to Steps 3 and 4, we know that . Replacing in (39) with , and noticing that , we have the Minty type variational inequality which is equivalent to the variational inequality (see Lemma 7) That is, is a solution of (19).

Step 6. We show that converges strongly to a unique solution in to VI (19). Indeed, we first claim that the solution set of (19) is a singleton. As a matter of fact, assume that is also a solution of (19). Then we have From (40), we have So, by the δ-strong accretiveness of F, we have Therefore, . In summary, we have shown that each cluster point of (as ) equals p. Consequently, as . □

Theorem 6

Let and μ be as in Theorem 5. Let be defined by (15) and (16). For arbitrarily given , let be defined by where and are three sequences in with , and satisfy the following conditions: Then converges strongly to a unique solution .

and ;

;

.

Let the mapping be defined as , where and . In terms of Lemma 4 we know that is nonexpansive. Then the explicit iterative scheme can be rewritten as

Next, we divide the rest of the proof into several steps.

Step 1. We show that is bounded. Indeed, take an arbitrarily given . Then we have and . Hence, by Lemma 8(c) we get By induction, we conclude that Therefore, is bounded. So, the sequences , and , where and , are also bounded.

Step 2. We show that and as . Indeed, observe that Set for all . Then . Note that Hence Since and are bounded, we have that and are bounded. So it follows from and conditions (i) and (ii) that Hence, by Lemma 12, we get as . Consequently, We also note that It follows that

Step 3. We show that and as . Indeed, for simplicity, put and . Then for all . Repeating the same arguments as those of (24), we obtain Observe that which together with (44) implies that So, it follows that Since , and as , we deduce from (42) and condition (iii) that Repeating the same arguments as those of (28), we get Combining (45) and (48), we have which immediately leads to Since as , and and are bounded, we deduce from (42), (47) and condition (iii) that Utilizing the properties of and , we conclude that From (49), we get That is, This together with (43) implies that

Step 4. We show that , where Indeed, repeating the same arguments as those of (36) and (37) in the proof of Theorem 5, we obtain that Utilizing Lemma 6 and Lemma 11 and the nonexpansivity of V and G, we conclude that .

Step 5. We show that , where for all and is the unique solution of VI (19). Indeed, we first take a subsequence of such that We may also assume that . Note that Combining (53) with (due to Step 4), we get . So, it follows from VI (19) that Since for all , according to Lemma 2(iii), we have From (54), we have which immediately yields

Step 6. We show that as . Indeed, from (45) and (55), we have Since , and , we get Taking into account , we can apply Lemma 13 to relation (56) and conclude that as . □