Abstract and Applied Analysis

Volume 2013 (2013), Article ID 894926, 27 pages

http://dx.doi.org/10.1155/2013/894926

Research Article

Hybrid Extragradient Methods for Finding Zeros of Accretive Operators and Solving Variational Inequality and Fixed Point Problems in Banach Spaces

Lu-Chuan Ceng1,2 and Ching-Feng Wen³

¹Department of Mathematics, Shanghai Normal University, Shanghai 200234, China

²Scientific Computing Key Laboratory of Shanghai Universities, Shanghai 200234, China

³Center for Fundamental Science, Kaohsiung Medical University, Kaohsiung 807, Taiwan

Received 8 July 2013; Accepted 11 August 2013

Academic Editor: Jen-Chih Yao

Copyright © 2013 Lu-Chuan Ceng and Ching-Feng Wen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We introduce and analyze hybrid implicit and explicit extragradient methods for finding a zero of an accretive operator and solving a general system of variational inequalities and a fixed point problem of an infinite family of nonexpansive self-mappings in a uniformly convex Banach space which has a uniformly Gateaux differentiable norm. We establish some strong convergence theorems for hybrid implicit and explicit extra-gradient algorithms under suitable assumptions. Furthermore, we derive the strong convergence of hybrid implicit and explicit extragradient algorithms for finding a common element of the set of zeros of an accretive operator and the common fixed point set of an infinite family of nonexpansive self-mappings and a self-mapping whose complement is strictly pseudocontractive and strongly accretive in . The results presented in this paper improve, extend, supplement, and develop the corresponding results announced in the earlier and very recent literature.

1. Introduction

Let be a real Banach space whose dual space is denoted by . Let denote the unite sphere of . A Banach space is said to be uniformly convex if, for each , there exists such that for all ,

It is known that a uniformly convex Banach space is reflexive and strict convex. 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 . Moreover, it is known that is single-valued if and only if is smooth, whereas if is uniformly smooth, then is norm-to-norm uniformly continuous on bounded subsets of . If has a uniformly Gateaux differentiable norm, then the duality mapping is norm-to-weak* uniformly continuous on bounded subsets of ; see for example [1].

Let be a nonempty closed convex subset of a real Banach space . A mapping is called nonexpansive if

The set of fixed points of is denoted by . We use the notation to indicate the weak convergence and to indicate the strong convergence.

Recall that (possibly multivalued) operator with domain and range in is accretive if, for each and (), there exists a such that (here is the duality mapping). An accretive operator is said to satisfy the range condition if for all . An accretive operator is -accretive if for each . If is an accretive operator which satisfies the range condition, then we can define, for each , a mapping defined by , which is called the resolvent of . We know that is nonexpansive and for all . Hence

If , then the inclusion is solvable.

The following resolvent identity is well known to us; see [2], where more details on accretive operators can be found.

Proposition 1 (resolvent identity). For , and ,

Recently, Aoyama et al. [3] studied the following iterative scheme in a uniformly convex Banach space having a uniformly Gareaux differentiable norm: for resolvents of an accretive operator such that and and ,

They proved that the sequence generated by (6) converges strongly to a zero of under appropriate assumptions on and . Subsequently, Ceng et al. [4] introduced and analyzed the following composite iterative scheme in either a uniformly smooth Banach space or a reflexive Banach space having a weakly sequentially continuous duality mapping,

where is an arbitrary (but fixed) element, under the following control conditions:

(H1);(H2), or, equivalently, ;(H3);(H4), for all , for some and ;(H5) for some and .

Furthermore, as the viscosity approximation method, Jung [5] purposed and analyzed the following composite iterative scheme for finding a zero of an accretive operator : for resolvent of an accretive operator such that and , ( denotes the set of all contractions on ) and ,

He established the strong convergence of the sequence generated by (8) to a zero of under certain appropriate conditions.

Theorem 2 (see [5, Theorem 3.1]). Let be a strictly convex and reflexive Banach space having a uniformly Gateaux differentiable norm. Let be a nonempty closed convex subset of and an accretive operator in such that and . Let and be sequences in which satisfy the following conditions:

(C1) and ;(B1) for some for all .

Let and be chosen arbitrarily. Let be a sequence generated by (8) for . If is asymptotically regular, that is, , then converges strongly to , which is the unique solution of the variational inequality problem (VIP)

On the other hand, we first recall the following concepts.

Definition 3. Let be a nonempty closed convex subset of a real Banach space and let be a mapping of into . Then is said to be:

(i) accretive if, for each , there exists such that

where is the normalized duality mapping;

(ii) -strongly accretive if, for each , there exists such that

for some ;

(iii) -inverse-strongly-accretive if, for each , there exists such that

for some ;

(iv) -strictly pseudocontractive [6] 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, for example, [7–9].

Very recently, Cai and Bu [10] considered the following general system of variational inequalities (GSVI) in a real smooth Banach space , which involves finding such that

where is a nonempty, closed, and convex subset of , are two nonlinear mappings, and and are two positive constants. Here the set of solutions of GSVI (14) is denoted by . In particular, if , a real Hilbert space, then GSVI (14) reduces to the following GSVI of finding such that

where and are two positive constants. The set of solutions of problem (15) is still denoted by . In particular, if , then problem (15) reduces to the new system of variational inequalities (NSVI), introduced and studied by Verma [11]. Furthermore, if additionally, then the NSVI reduces to the classical variational inequality problem (VIP) of finding such that

The solution set of the VIP (16) 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 [12]. This alternative formulation has been used to suggest and analyze projection iterative method for solving variational inequalities under the conditions that the involved operator must be strongly monotone and Lipschitz continuous.

Recently, Ceng et al. [13] transformed problem (15) into a fixed point problem in the following way.

Lemma 4 (see [13]). For a given is a solution of problem (15) if and only if is a fixed point of the mapping defined by

where and is the projection of onto .

In particular, if the mapping is -inverse strongly monotone for , then the mapping is nonexpansive provided for .

In 1976, Korpelevič [14] proposed an iterative algorithm for solving the VIP (16) in Euclidean space as follows:

with a given number, which is known as the extragradient method (see also [15]). The literature on the VIP is vast and Korpelevich's extragradient method has received great attention given by many authors, who improved it in various ways; see, for example, [10, 13, 16–23] the and references therein, to name but a few.

In particular, whenever is still a real smooth Banach space, , and , then GSVI (17) reduces to the variational inequality problem (VIP) of finding such that

which was considered by Aoyama et al. [24]. Note that VIP (19) is connected with the fixed point problem for nonlinear mapping (see, e.g., [15, 25]), the problem of finding a zero point of a nonlinear operator (see, e.g., [1, 26]), and so on. It is clear that VIP (19) extends VIP (16) from Hilbert spaces to Banach spaces.

In order to find a solution of VIP (19), Aoyama et al. [24] introduced the following iterative scheme for an accretive operator :

where is a sunny nonexpansive retraction from onto . Then they proved a weak convergence theorem.

Beyond doubt, it is an interesting and valuable problem of constructing some algorithms with strong convergence for solving GSVI (14) which contains VIP (19) as a special case. Very recently, Cai and Bu [10] constructed an iterative algorithm for solving GSVI (14) and a common fixed point problem of an infinite family of nonexpansive mappings in a uniformly convex and -uniformly smooth Banach space. They proved the strong convergence of the proposed algorithm by virtue of the following inequality in a -uniformly smooth Banach space .

Lemma 5 (see [27]). Let be a -uniformly smooth Banach space. Then

where is the -uniformly smooth constant of and is the normalized duality mapping from into .

Define the mapping as follows:

The fixed point set of is denoted by . Then their strong convergence theorem on the proposed method is stated as follows.

Theorem 6 (see [10, Theorem 3.1]). Let be a nonempty closed convex subset of a uniformly convex and -uniformly smooth Banach space . Let be a sunny nonexpansive retraction from onto . Let the mapping be -inverse strongly accretive with for . Let be a contraction of into itself with coefficient . Let be an infinite family of nonexpansive mappings of into itself such that , where is the fixed point set of the mapping defined by (22). For arbitrarily given , let be the sequence generated by

Suppose that and are two sequences in satisfying the following conditions:

(i) and ;(ii).

Assume that for any bounded subset of and let be a mapping of into defined by for all and suppose that . Then converges strongly to , which solves the following VIP:

Corollary 7 (see [10, Corollary 3.2]). Let be a nonempty closed convex subset of a uniformly convex and -uniformly smooth Banach space . Let be a sunny nonexpansive retraction from onto . Let the mapping be -inverse strongly accretive with for . Let be a contraction of into itself with coefficient . Let be a nonexpansive mapping of into itself such that , where is the fixed point set of the mapping defined by (22). For arbitrarily given , let be the sequence generated by

Suppose that and are two sequences in satisfying the following conditions:

(i) and ;(ii).

Then converges strongly to , which solves the following VIP:

We remark that in Theorem 6, the Banach space is both uniformly convex and -uniformly smooth. According to Lemma 5, the -uniform smoothness of guarantees the nonexpansivity of the mapping for -inverse-strongly accretive mapping with for , and hence the composite mapping is nonexpansive where . In the meantime, for the convenience of implementing the argument techniques in [13], they have applied the following inequality in a real smooth and uniform convex Banach space .

Proposition 8 (see [28]). Let be a real smooth and uniform convex Banach space and let . Then there exists a strictly increasing, continuous, and convex function , such that

where .

Let be a nonempty closed convex subset of a uniformly convex Banach space which has a uniformly Gateaux differentiable norm. Let be a sunny nonexpansive retraction from onto . Motivated and inspired by the research going on this area, we introduce and analyze hybrid implicit and explicit extragradient methods for finding a zero of an accretive operator such that and solving GSVI (14) and a fixed point problem of an infinite family of nonexpansive self-mappings on . We establish some strong convergence theorems for hybrid implicit and explicit extragradient algorithms under suitable assumptions. Furthermore, we derive the strong convergence of hybrid implicit and explicit extragradient algorithms for finding a common element of the set of zeros of an accretive operator and the common fixed point set of an infinite family of nonexpansive self-mappings on and a self-mapping whose complement is strictly pseudocontractive and strongly accretive on . The results presented in this paper improve, extend, supplement, and develop the corresponding results announced in the earlier and very recent literature; see, for example, [5, 10, 13, 16].

2. Preliminaries

Let be a real Banach space. is said to be smooth if the limit

exists for all ; in this case, is also said to have a Gateaux differentiable norm. 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 is said to be the Frechet differential if, for each , this limit is attained uniformly for . In addition, we define a function called the modulus of smoothness of as follows:

It is known that is uniformly smooth if and only if . Let be a fixed real number with . Then a Banach space is said to be -uniformly smooth if there exists a constant such that for all . As pointed out in [29], no Banach space is -uniformly smooth for .

We list some lemmas that will be used in the sequel. Lemma 9 can be found in [30]. Lemma 10 is an immediate consequence of the subdifferential inequality of the function .

Lemma 9. Let be a sequence of nonnegative real numbers satisfying

where , , and satisfy the following conditions:

(i) and ;(ii);(iii), for all , and .

Then .

Lemma 10. In a smooth Banach space , there holds the inequality

Lemma 11 (see [6]). Let be a nonempty closed convex subset of a real smooth Banach space and let be a -strictly pseudocontractive mapping. Then is Lipschitz continuous with constant .

Proof. Since is a -strictly pseudocontractive mapping, we have for all

and hence

This yields

Therefore,

Let be a subset of and let be a mapping of into . Then is said to be sunny if

whenever for and . A mapping of into itself is called a retraction if . If a mapping of into itself is a retraction, then for every where is the range of . A subset of is called a sunny nonexpansive retract of if there exists a sunny nonexpansive retraction from onto . The following lemma concerns the sunny nonexpansive retraction.

Lemma 12 (see [31]). Let be a nonempty closed convex subset of a real smooth Banach space . Let be a nonempty subset of . Let be a retraction of onto . Then the following are equivalent:

(i) is sunny and nonexpansive;(ii), for all ;(iii), for all .

It is well known that if a Hilbert space, then a sunny nonexpansive retraction is coincident with the metric projection from onto ; that is, . If is a nonempty closed convex subset of a strictly convex and uniformly smooth Banach space and if is a nonexpansive mapping with the fixed point set , then the set is a sunny nonexpansive retract of .

Lemma 13 (see [32]). Let be a uniformly convex Banach space and . Then there exists a continuous, strictly increasing, and convex function such that

for all and all with .

Lemma 14 (see [33]). Let be a nonempty closed convex subset of a Banach space . Let be a sequence of mappings of into itself. Suppose that . Then for each , converges strongly to some point of . Moreover, let be a mapping of into itself defined by for all . Then .

Let be a nonempty closed convex subset of a Banach space and let be a nonexpansive mapping with . As previously mentioned, let be the set of all contractions on . For and , let be the unique fixed point of the contraction on ; that is,

Lemma 15 (see [15]). Let be a reflexive and strictly convex Banach space with a uniformly Gateaux differentiable norm. Let a nonempty closed convex subset of a nonexpansive mapping with , and . Then the net defined by converges strongly to a point in . If we define a mapping by , for all , then solves the following VIP:

Lemma 16 (see [34]). Let be a nonempty closed convex subset of a strictly convex Banach space . Let be a sequence of nonexpansive mappings on . Suppose is nonempty. Let be a sequence of positive numbers with . Then a mapping on defined by for is well defined, nonexpansive and holds.

Lemma 17 (see [27]). Given a number . A real Banach space is uniformly convex if and only if there exists a continuous strictly increasing function , , such that

for all and such that and .

Lemma 18 (see [24]). Let be a nonempty closed convex subset of a smooth Banach space . Let be a sunny nonexpansive retraction from onto and let be an accretive operator of into . Then, for all ,

Lemma 19. Let be a nonempty closed convex subset of a smooth Banach space and let the mapping be strictly pseudocontractive and strongly accretive with for . Then, for , we have

for . In particular, if , then is nonexpansive for .

Proof. Taking into account the -strict pseudocontractivity of , by Lemma 11 we derive for every

Utilizing the -strong accretivity and -strict pseudocontractivity of , we get

So, we have

Therefore, for , we have

Since , it follows immediately that

This implies that is nonexpansive for .

Lemma 20. Let be a nonempty closed convex subset of a smooth Banach space . Let be a sunny nonexpansive retraction from onto and let the mapping be strictly pseudocontractive and strongly accretive with for . Let be the mapping defined by

If , then is nonexpansive.

Proof. According to Lemma 19, we know that is nonexpansive for . Hence, for all , we have

This shows that is nonexpansive. This completes the proof.

Lemma 21. Let be a nonempty closed convex subset of a smooth Banach space . Let be a sunny nonexpansive retraction from onto and let be two nonlinear mappings. For a given is a solution of GSVI (14) if and only if where .

Proof. We can rewrite GSVI (14) as

which is obviously equivalent to

because of Lemma 12. This completes the proof.

Remark 22. By Lemma 21, we observe that

which implies that is a fixed point of the mapping . Throughout this paper, the set of fixed points of the mapping is denoted by .

3. Hybrid Implicit Extragradient Algorithm

In this section, let be a nonempty closed convex subset of a uniformly convex Banach space which has a uniformly Gateaux differentiable norm. We suggest and analyze a hybrid implicit extragradient algorithm for finding a zero of an accretive operator in with and solving a general system of variational inequalities and a common fixed point problem of an infinite family of nonexpansive self-mappings in .

Theorem 23. Let be a nonempty closed convex subset of a uniformly convex Banach space which has a uniformly Gateaux differentiable norm. Let be a sunny nonexpansive retraction from onto . Let be an accretive operator in such that . Let be strictly pseudocontractive and strongly accretive with for each . Define the mapping by where for . Let be a contraction with coefficient . Let be an infinite family of nonexpansive mappings of into itself such that where . For arbitrarily given , let be the sequence generated by

Suppose that , , and the following conditions hold:

(i) and ;(ii) for some ;(iii);(iv) and , for all for some ;(v) and .

Assume that for any bounded subset of and let be a mapping of into itself defined by for all and suppose that . Then converges strongly to , which solves the following VIP:

Proof. It is easy to see that (53) can be rewritten as follows:

where . By Lemma 20 we know that is a nonexpansive self-mapping on .

Now, let us show that the sequence is bounded. Indeed, take a fixed arbitrarily. Then from (55), we have

which hence implies that

So, we have

By induction, we obtain

Hence is bounded and so are , , , , and .

Let us show that

As a matter of fact, observe that can be rewritten as follows:

where . Observe that

On the other hand, if , using the resolvent identity in Proposition 1,

we get

If , we derive in a similar way

Thus, combining the above cases we obtain

where for some . Substituting (66) into (62), we have

where for some . In the meantime, simple calculations show that

Taking into account condition (iv), we may assume, without loss of generality, that for some . Hence, from (67) and (68) we deduce that

where for some . Thus, from condition (ii) we immediately get

Also, from (55) we have

Taking into account condition (v), we may assume, without loss of generality, that for some . This together with (70) implies that

where for some . Since and , we know that . So, applying Lemma 9 to (72), we obtain from conditions (iii), (iv), and the assumption on that

Next we show that as .

Indeed, according to Lemma 10 we have from (55)

which hence implies that

Utilizing Lemma 17, we get from (55) and (75)

which hence yields

Since and , from condition (iv) and the boundedness of and , it follows that

Utilizing the properties of , we have

Observe that

and hence

That is,

Next, let us show that

Indeed, observe that can be rewritten as follows:

where and . Utilizing Lemma 13 and (84), we have

which hence implies that

Utilizing (82), conditions (i), (ii), and (v), and the boundedness of , , and , we get

From the properties of , we have

Utilizing Lemma 17 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 follows:

where and . Utilizing Lemma 13 and the convexity of , we have

which hence implies that

From (82), conditions (i), (ii), and (v), and the boundedness of , , and , we have

Utilizing the properties of , we have

Note that

From (82), (92), and (97), we get

In terms of (99) and Lemma 14, we have

That is,

Furthermore, we claim that for a fixed number such that . In fact, from the resolvent identity in Proposition 1, we have

So, we get from (102)

Thus, from (97) it follows that

Define a mapping , where are two constants with . Then, by Lemma 16, we have that . We observe that

From (82), (101), and (104), 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 10, we conclude that

where

It follows from (110) that

Letting in (112) and noticing (111), we derive

where is a constant such that for all and . Taking in (113), we have

On the other hand, we have

It follows that

Taking into account that as , we have

Since has a uniformly Gateaux differentiable norm, the duality mapping is norm-to-weak* uniformly continuous on bounded subsets of . Consequently, the two limits are interchangeable and hence (107) holds. From (82), we get . Noticing the norm-to-weak* uniform continuity of on bounded subsets of , we deduce from (107) that

Finally, let us show that as . Indeed, observe that

which implies that

Thus, we have

Applying Lemma 9 to (121), we conclude from condition (i) and (118) that as . This completes the proof.

The following results can be obtained from Theorem 23. We, therefore, omit the proof.

Corollary 24. Let be a nonempty closed convex subset of a uniformly convex Banach space which has a uniformly Gateaux differentiable norm. Let be a sunny nonexpansive retraction from onto . Let be an accretive operator in such that . Let be strictly pseudocontractive and strongly accretive with for each . Define the mapping by where for . Let be a contraction with coefficient . Let be a nonexpansive mapping such that where . For arbitrarily given , let be the sequence generated by

Suppose that , , , and the following conditions hold:

(i) and ;(ii) for some ;(iii);(iv) and , for all for some ;(v) and .

Then converges strongly to , which solves the following VIP:

Corollary 25. Let be a nonempty closed convex subset of a uniformly convex Banach space which has a uniformly Gateaux differentiable norm. Let be a sunny nonexpansive retraction from onto . Let be an accretive operator in such that . Let be a contraction with coefficient . Let be a self-mapping such that is strictly pseudocontractive and strongly accretive with . Let be an infinite family of nonexpansive mappings of into itself such that . For arbitrarily given , let be the sequence generated by

where . Suppose that , , , and the following conditions hold:

(i) and ;(ii) for some ;(iii);(iv) and , for all for some ;(v) and .

Assume that for any bounded subset of and let be a mapping of into itself defined by for all and suppose that . Then converges strongly to , which solves the following VIP:

Proof. In Theorem 23, we put , and where . Then GSVI (14) is equivalent to the VIP of finding such that

In this case, is strictly pseudocontractive and strongly accretive. It is not hard to see that . As a matter of fact, we have, for ,

Accordingly, we know that , and

So, the scheme (53) reduces to (124). Therefore, the desired result follows from Theorem 23.

Corollary 26. Let be a nonempty closed convex subset of a uniformly convex Banach space which has a uniformly Gateaux differentiable norm. Let be a sunny nonexpansive retraction from onto . Let be an accretive operator in such that . Let be a contraction with coefficient . Let be a self-mapping such that is strictly pseudocontractive and strongly accretive with . Let be a nonexpansive mapping such that . For arbitrarily given , let be the sequence generated by

where . Suppose that , , , and the following conditions hold:

(i) and ;(ii) for some ;(iii);(iv) and , for all for some ;(v) and .

Then converges strongly to , which solves the following VIP:

Remark 27. Our Theorem 23 improves, extends, supplements, and develops Cai and Bu [10, Theorem 3.1 and Corollary 3.2] and Jung [5, Theorems 3.1] in the following aspects.

(i) The problem of finding a point in our Theorem 23 is more general and more subtle than every one of both the problem of finding a point in Cai and Bu [10, Theorem 3.1] and the problem of finding a point in Jung [5, Theorem 3.1].

(ii) Our Theorem 23 drops the assumption of the asymptotical regularity of in [5, Theorems 3.1] (i.e., ).

(iii) Cai and Bu's proof in [10, Theorem 3.1] depends on the argument techniques in [13], inequality (21) in -uniformly smooth Banach spaces, and inequality (27) in smooth and uniform convex Banach spaces. Jung's proof in [5, Theorem 3.1] depends on the resolvent identity in Proposition 1. It is worth emphasizing that the proof of our Theorem 23 does not depend on the argument techniques in [13], inequality (21) in -uniformly smooth Banach spaces, and inequality (27) in smooth and uniform convex Banach spaces. However, it depends on the resolvent identity in Proposition 1 and the inequalities in uniform convex Banach spaces; see Lemmas 13 and 17 in Section 2 of this paper.

(iv) The assumption of the uniformly convex and -uniformly smooth Banach space in [10, Theorem 3.1] is weakened to the one of the uniformly convex Banach space having a uniformly Gateaux differentiable norm in our Theorem 23.

(v) The iterative scheme in our Theorem 23 is very different from every one in both [10, Theorem 3.1] and [5, Theorem 3.1] because the first iteration step in our iterative scheme is implicit.

(vi) The problem of finding a point in [10, Corollary 3.2] is extended to develop the problem of finding a point in our Corollary 24.

4. Hybrid Explicit Extragradient Algorithm

In this section, let be a nonempty closed convex subset of a uniformly convex Banach space which has a uniformly Gateaux differentiable norm. We suggest and analyze a hybrid explicit extragradient algorithm for finding a zero of an accretive operator in with and solving a general system of variational inequalities and a common fixed point problem of an infinite family of nonexpansive self-mappings in .

Theorem 28. Let be a nonempty closed convex subset of a uniformly convex Banach space which has a uniformly Gateaux differentiable norm. Let be a sunny nonexpansive retraction from onto . Let be an accretive operator in such that . Let be strictly pseudocontractive and strongly accretive with for each . Define the mapping by where for . Let be a contraction with coefficient . Let be an infinite family of nonexpansive mappings of into itself such that where . For arbitrarily given , let be the sequence generated by

Suppose that , , , and the following conditions hold:

(i) and ;(ii) for some ;(iii);(iv) and , for all for some ;(v) and .

Assume that for any bounded subset of and let be a mapping of into itself defined by for all and suppose that . Then converges strongly to , which solves the following VIP:

Proof. It is easy to see that (131) can be rewritten as follows

where . By Lemma 20 we know that is a nonexpansive self-mapping on .

Now, let us show that the sequence is bounded. Indeed, take a fixed arbitrarily. Then from (133), we have

and hence

By induction, we obtain

It immediately follows that is bounded and so are , , , , and .

Let us show that

As a matter of fact, observe that can be rewritten as follows:

where . Observe that

On the other hand, repeating the same arguments as those of (66) in the proof of Theorem 23, we can derive

where for some . Substituting (140) into (139), we have

where for some . Also, from (133) we have

Taking into account condition (v), we may assume, without loss of generality, that for some . This together with (141) implies that

where for some . Since and , we know that . So, applying Lemma 9 to (143), we obtain from conditions (iii) and (iv) and the assumption on that

Next we show that as .

Indeed, according to Lemma 10 we have from (133)

Utilizing Lemma 17 we get from (133) and (145)

which hence yields

Since and , from condition (v) and the boundedness of and , it follows that

Utilizing the properties of , we have

Observe that

and hence

That is,

Next, let us show that

Indeed, observe that can be rewritten as follows:

where and . Utilizing Lemma 13 and (154), we have

which hence implies that

Utilizing (152), conditions (i), (ii), and (v) and the boundedness of , , and , we get

From the properties of , we have

Utilizing Lemma 17 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 follows:

where and . Utilizing Lemma 13 and the convexity of , we have

which hence implies that

From (152), conditions (i), (ii), and (v), and the boundedness of , , and , we have

Utilizing the properties of , we have

Note that

From (162) and (167), we get

In terms of (169) and Lemma 14, we have

That is,

Furthermore, repeating the same arguments as those of (104) in the proof of Theorem 23, we can conclude that

for a fixed number such that .

Define a mapping , where are two constants with . Then by Lemma 16, we have that . We observe that

From (152), (171), and (172), we obtain

Repeating the same arguments as those of (107) in the proof of Theorem 23, we can deduce that

Since is norm-to-weak* uniformly continuous over bounded subsets of , we obtain from that

Finally, let us show that as . Indeed, observe that

and hence

Applying Lemma 9 to (178), we conclude from condition (i) and (178) that as . This completes the proof.

The following results can be obtained from Theorem 28. We, therefore, omit the proof.

Corollary 29. Under the same conditions as those in Corollary 24, let be the sequence generated from any given by

Then converges strongly to , which solves the following VIP:

Corollary 30. Under the same conditions as those in Corollary 25, let be the sequence generated from any given by

Then converges strongly to    , which solves the following VIP:

Corollary 31. Under the same conditions as those in Corollary 26, let be the sequence generated from any given by

Then converges strongly to , which solves the following VIP:

Remark 32. Our Theorem 28 improves, extends, supplements, and develops Cai and Bu [10, Theorem 3.1 and Corollary 3.2] and Jung [5, Theorems 3.1] in the following aspects.

(i) The problem of finding a point in our Theorem 28 is more general and more subtle than every one of both the problem of finding a point in Cai and Bu [10, Theorem 3.1] and the problem of finding a point in Jung [5, Theorem 3.1].

(ii) Our Theorem 28 drops the assumption of the asymptotical regularity of in [5, Theorems 3.1] (i.e., ).

(iii) Cai and Bu's proof in [10, Theorem 3.1] depends on the argument techniques in [13], inequality (21) in -uniformly smooth Banach spaces, and inequality (27) in smooth and uniform convex Banach spaces. Jung's proof in [5, Theorem 3.1] depends on the resolvent identity in Proposition 1. It is worth emphasizing that the proof of our Theorem 28 does not depend on the argument techniques in [13], inequality (21) in -uniformly smooth Banach spaces, and inequality (27) in smooth and uniform convex Banach spaces. However, it depends on the resolvent identity in Proposition 1 and the inequalities in uniform convex Banach spaces; see Lemmas 13 and 17 in Section 2 of this paper.

(iv) The assumption of the uniformly convex and -uniformly smooth Banach space in [10, Theorem 3.1] is weakened to the one of the uniformly convex Banach space having a uniformly Gateaux differentiable norm in our Theorem 28.

(v) The iterative scheme in our Theorem 28 is very different from every one in both [10, Theorem 3.1] and [5, Theorem 3.1] because the first iteration step in the iterative scheme of [10, Theorem 3.1] is given by and the first iteration step in the iterative scheme of [5, Theorem 3.1] is given by . In the meantime, it is clear that the second iteration steps for in three iterative schemes are completely different.

(vi) The problem of finding a point in [10, Corollary 3.2] is extended to develop the problem of finding a point in our Corollary 29.

Acknowledgments

This research 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). This research was partially supported by a Grant from NSC 102-2115-M-037-001.

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