Hybrid Proximal-Type Algorithms for Generalized Equilibrium Problems, Maximal Monotone Operators, and Relatively Nonexpansive Mappings

Abstract

The purpose of this paper is to introduce and consider new hybrid proximal-type algorithms for finding a common element of the set of solutions of a generalized equilibrium problem, the set of fixed points of a relatively nonexpansive mapping , and the set of zeros of a maximal monotone operator in a uniformly smooth and uniformly convex Banach space. Strong convergence theorems for these hybrid proximal-type algorithms are established; that is, under appropriate conditions, the sequences generated by these various algorithms converge strongly to the same point in . These new results represent the improvement, generalization, and development of the previously known ones in the literature.

1. Introduction

Let be a real Banach space with the dual and be a nonempty closed convex subset of . We denote by and the sets of positive integers and real numbers, respectively. Also, we denote by the normalized duality mapping from to defined by

(1.1)

where denotes the generalized duality pairing. Recall that if is smooth, then is single valued and if is uniformly smooth, then is uniformly norm-to-norm continuous on bounded subsets of . We will still denote by the single valued duality mapping. Let be a bifunction and be a nonlinear mapping. We consider the following generalized equilibrium problem:

(1.2)

The set of such is denoted by , that is,

(1.3)

Whenever a Hilbert space, problem (1.2) was introduced and studied by S. Takahashi and W. Takahashi [1]. Similar problems have been studied extensively recently. See, for example, [2–11]. In the case of is denoted by . In the case of , is also denoted by . The problem (1.2) is very general in the sense that it includes, as special cases, optimization problems, variational inequalities, minimax problems, the Nash equilibrium problem in noncooperative games and others; see, for example, [12–14]. A mapping is called nonexpansive if for all . Denote by the set of fixed points of , that is, . A mapping is called α-inverse-strongly monotone, if there exists an such that

(1.4)

It is easy to see that if is an α-inverse-strongly monotone mapping, then it is -Lipschitzian.

Let be a real Banach space with the dual . A multivalued operator with domain is called monotone if for each and , . A monotone operator is called maximal if its graph is not properly contained in the graph of any other monotone operator. A method for solving the inclusion is the proximal point algorithm. Denote by the identity operator on a Hilbert space. The proximal point algorithm generates, for any initial point , a sequence in , by the iterative scheme

(1.5)

where is a sequence in the interval . Note that this iteration is equivalent to

(1.6)

This algorithm was first introduced by Martinet [12] and generally studied by Rockafellar [15] in the framework of a Hilbert space. Later many authors studied its convergence in a Hilbert space or a Banach space. See, for instance, [16–21] and the references therein.

Let be a reflexive, strictly convex, and smooth Banach space with the dual and be a nonempty closed convex subset of . Let be a maximal monotone operator with domain and be a relatively nonexpansive mapping. Let be an α-inverse-strongly monotone mapping and be a bifunction satisfying (A1)–(A4): (A1) , ; (A2) is monotone, that is, , ; (A3) , ; (A4) the function is convex and lower semicontinuous. The purpose of this paper is to introduce and investigate two new hybrid proximal-type Algorithms 1.1 and 1.2 for finding an element of .

Algorithm 1.1.

(1.7)

where is a sequence in and , are sequences in .

Algorithm 1.2.

(1.8)

where is a sequence in and is a sequence in .

In this paper, strong convergence results on these two hybrid proximal-type algorithms are established; that is, under appropriate conditions, the sequence generated by Algorithm 1.1 and the sequence generated by Algorithm 1.2, converge strongly to the same point . These new results represent the improvement, generalization and development of the previously known ones in the literature including Solodov and Svaiter [22], Kamimura and Takahashi [23], Qin and Su [24], and Ceng et al. [25].

Throughout this paper the symbol ⇀ stands for weak convergence and → stands for strong convergence.

2. Preliminaries

Let be a real Banach space with the dual . We denote by the normalized duality mapping from to defined by

(2.1)

where denotes the generalized duality pairing. A Banach space is called strictly convex if for all with and . It is said to be uniformly convex if for any two sequences such that and . Let be a unit sphere of , then the Banach space is called smooth if

(2.2)

exists for each . If is smooth, then is single valued. We still denote the single valued duality mapping by .

It is also said to be uniformly smooth if the limit is attained uniformly for . Recall also that if is uniformly smooth, then is uniformly norm-to-norm continuous on bounded subsets of . A Banach space is said to have the Kadec-Klee property if for any sequence , whenever and , we have . It is known that if is uniformly convex, then has the Kadec-Klee property; see [26, 27] for more details.

Let be a nonempty closed convex subset of a real Hilbert space and be the metric projection of onto C, then is nonexpansive. This fact actually characterizes Hilbert spaces and hence, it is not available in more general Banach spaces. Nevertheless, Alber [28] recently introduced a generalized projection operator in a Banach space which is an analogue of the metric projection in Hilbert spaces.

Next, we assume that is a smooth Banach space. Consider the functional defined as in [28, 29] by

(2.3)

It is clear that in a Hilbert space , (2.3) reduces to , for all .

The generalized projection is a mapping that assigns to an arbitrary point the minimum point of the functional ; that is, , where is the solution to the minimization problem

(2.4)

The existence and uniqueness of the operator follows from the properties of the functional and strict monotonicity of the mapping (see, e.g., [30]). In a Hilbert space , . From [28], in uniformly smooth and uniformly convex Banach spaces, we have

(2.5)

Let be a nonempty closed convex subset of , and let be a mapping from into itself. A point is called an asymptotically fixed point of [31] if contains a sequence which converges weakly to such that . The set of asymptotical fixed points of will be denoted by . A mapping from into itself is called relatively nonexpansive [32–34] if and for all and .

We remark that if is a reflexive, strictly convex and smooth Banach space, then for any if and only if . It is sufficient to show that if then . From (2.5), we have . This implies that . From the definition of , we have . Therefore, we have ; see [26, 27] for more details.

We need the following lemmas for the proof of our main results.

Lemma 2.1 (see [23]).

Let be a smooth and uniformly convex Banach space and let and be two sequences of . If and either or is bounded, then .

Lemma 2.2 (see [23, 28]).

Let be a nonempty closed convex subset of a smooth, strictly convex and reflexive Banach space , let and let , then

(2.6)

Lemma 2.3 (see [23, 28]).

Let be a nonempty closed convex subset of a smooth, strictly convex and reflexive Banach space , then

(2.7)

Lemma 2.4 (see [35]).

Let be a nonempty closed convex subset of a reflexive, strictly convex and smooth Banach space , and let be a relatively nonexpansive mapping, then is closed and convex.

The following result is according to Blum and Oettli [36].

Lemma 2.5 (see [36]).

Let be a nonempty closed convex subset of a smooth, strictly convex and reflexive Banach space , let be a bifunction from to satisfying (A1)–(A4), and let and , then, there exists such that

(2.8)

Motivated by Combettes and Hirstoaga [37] in a Hilbert space, Takahashi and Zembayashi [38] established the following lemma.

Lemma 2.6 (see [38]).

Let be a nonempty closed convex subset of a uniformly smooth, strictly convex and reflexive Banach space , and let be a bifunction from to satisfying (A1)–(A4). For and , define a mapping as follows:

(2.9)

for all , then, the following hold:

(i) is single valued;

(ii) is a firmly nonexpansive-type mapping, that is, for all ,

(2.10)

(iii);

(iv) is closed and convex.

Using Lemma 2.6, one has the following result.

Lemma 2.7 (see [38]).

Let be a nonempty closed convex subset of a smooth, strictly convex and reflexive Banach space , let be a bifunction from to satisfying (A1)–(A4), and let , then, for and ,

(2.11)

Utilizing Lemmas 2.5, 2.6 and 2.7 as above, Chang [39] derived the following result.

Proposition 2.8 (see [39, Lemma 2.5]).

Let be a smooth, strictly convex and reflexive Banach space and be a nonempty closed convex subset of . Let be an α-inverse-strongly monotone mapping, let be a bifunction from to satisfying (A1)–(A4), and let , then there hold the following:

(I)for , there exists such that

(2.12)

(II)if is additionally uniformly smooth and is defined as

(2.13)

then the mapping has the following properties:

(i) is single valued,

(ii) is a firmly nonexpansive-type mapping, that is,

(2.14)

(iii),

(iv) is a closed convex subset of ,

(v), for all .

Proof.

Define a bifunction as follows:

(2.15)

Then it is easy to verify that satisfies the conditions (A1)–(A4). Therefore, The conclusions (I) and (II) of Proposition 2.8 follow immediately from Lemmas 2.5, 2.6 and 2.7.

Lemma 2.9 (see [13, 14]).

Let be a reflexive, strictly convex and smooth Banach space, and let be a maximal monotone operator with , then,

(2.16)

3. Main Results

Throughout this section, unless otherwise stated, we assume that is a maximal monotone operator with domain , is a relatively nonexpansive mapping, is an α-inverse-strongly monotone mapping and is a bifunction satisfying (A1)–(A4), where is a nonempty closed convex subset of a reflexive, strictly convex, and smooth Banach space . In this section, we study the following algorithm.

Algorithm 3.1.

(3.1)

where is a sequence in and , are sequences in .

First we investigate the condition under which the Algorithm 3.1 is well defined. Rockafellar [40] proved the following result.

Lemma 3.2 (Rockafellar [40]).

Let be a reflexive, strictly convex, and smooth Banach space and let be a multivalued operator, then there hold the following:

(i) is closed and convex if is maximal monotone such that ;

(ii) is maximal monotone if and only if is monotone with for all .

Utilizing this result, we can show the following lemma.

Lemma 3.3.

Let be a reflexive, strictly convex, and smooth Banach space. If , then the sequence generated by Algorithm 3.1 is well defined.

Proof.

For each , define two sets and as follows:

(3.2)

It is obvious that is closed and are closed convex sets for each . Let us show that is convex. For and , put . It is sufficient to show that . Indeed, observe that

(3.3)

is equivalent to

(3.4)

Note that there hold the following:

(3.5)

Thus we have

(3.6)

This implies that . Therefore, is convex and hence is closed and convex.

On the other hand, let be arbitrarily chosen, then and . From Algorithm 3.1, it follows that

(3.7)

So for all . Now, from Lemma 3.2 it follows that there exists such that and . Since is monotone, it follows that , which implies that and hence . Furthermore, it is clear that , then , and therefore is well defined. Suppose that and is well defined for some . Again by Lemma 3.2, we deduce that such that and , then from the monotonicity of we conclude that , which implies that and hence . It follows from Lemma 2.4 that

(3.8)

which implies that . Consequently, and so . Therefore is well defined, then, by induction, the sequence generated by Algorithm 3.1, is well defined for each integer .

Remark 3.4.

From the above proof, we obtain that

(3.9)

for each integer .

We are now in a position to prove the main theorem.

Theorem 3.5.

Let be a uniformly smooth and uniformly convex Banach space. Let be a sequence in and be sequences in such that

(3.10)

Let . If is uniformly continuous, then the sequence generated by Algorithm 3.1 converges strongly to .

Proof.

First of all, if follows from the definition of that . Since , we have

(3.11)

Thus is nondecreasing. Also from and Lemma 2.3, we have that

(3.12)

for each and for each . Consequently, is bounded. Moreover, according to the inequality

(3.13)

we conclude that is bounded. Thus, we have that exists. From Lemma 2.3, we derive the following:

(3.14)

for all . This implies that . So it follows from Lemma 2.1 that . Since , from the definition of , we also have

(3.15)

Observe that

(3.16)

At the same time,

(3.17)

Since and , it follows that

(3.18)

and hence that . Further, from , we have , which yields

(3.19)

Then it follows from that . Hence it follows from Lemma 2.1 that . Since from (3.15) we derive

(3.20)

we have

(3.21)

Thus, from , , and , we know that . Consequently from (3.16), , and it follows that

(3.22)

So it follows from (3.15), , and that . Utilizing Lemma 2.1 we deduce that

(3.23)

Furthermore, for arbitrarily fixed, it follows from Proposition 2.8 that

(3.24)

Since is uniformly norm-to-norm continuous on bounded subsets of , it follows from (3.23) that and , which hence yield . Utilizing Lemma 2.1, we get . Observe that

(3.25)

due to (3.23). Since is uniformly norm-to-norm continuous on bounded subsets of , we have that

(3.26)

On the other hand, we have

(3.27)

Noticing that

(3.28)

we have

(3.29)

From (3.26) and , we obtain

(3.30)

Since is also uniformly norm-to-norm continuous on bounded subsets of , we obtain

(3.31)

Observe that

(3.32)

Since is uniformly continuous, it follows from (3.27), (3.31) and that .

Now let us show that , where

(3.33)

Indeed, since is bounded and is reflexive, we know that . Take arbitrarily, then there exists a subsequence of such that . Hence . Let us show that . Since , we have that . Moreover, since is uniformly norm-to-norm continuous on bounded subsets of and , we obtain

(3.34)

It follows from and the monotonicity of that

(3.35)

for all and . This implies that

(3.36)

for all and . Thus from the maximality of , we infer that . Therefore, . Further, let us show that . Since and , from we obtain that and .

Since is uniformly norm-to-norm continuous on bounded subsets of , from we derive

(3.37)

From , it follows that

(3.38)

By the definition of , we have

(3.39)

where

(3.40)

Replacing by , we have from (A2) that

(3.41)

Since is convex and lower semicontinuous, it is also weakly lower semicontinuous. Letting in the last inequality, from (3.38) and (A4) we have

(3.42)

For , with , and , let . Since and , then and hence . So, from (A1) we have

(3.43)

Dividing by , we have

(3.44)

Letting , from (A3) it follows that

(3.45)

So, . Therefore, we obtain that by the arbitrariness of .

Next, let us show that and .

Indeed, put . From and , we have . Now from weakly lower semicontinuity of the norm, we derive for each

(3.46)

It follows from the definition of that and hence

(3.47)

So we have . Utilizing the Kadec-Klee property of , we conclude that converges strongly to . Since is an arbitrary weakly convergent subsequence of , we know that converges strongly to . This completes the proof.

Theorem 3.5 covers [25, Theorem 3.1] by taking and . Also Theorem 3.5 covers [24, Theorem 2.1] by taking , and .

Theorem 3.6.

Let be a nonempty closed convex subset of a uniformly smooth and uniformly convex Banach space . Let be a maximal monotone operator with domain , be a relatively nonexpansive mapping, be an α-inverse-strongly monotone mapping and be a bifunction satisfying (A1)–(A4). Assume that is a sequence in satisfying and that is a sequences in satisfying .

Define a sequence .

Algorithm 3.7.

(3.48)

where is the single valued duality mapping on . Let . If is uniformly continuous, then converges strongly to .

Proof.

For each , define two sets and as follows:

(3.49)

It is obvious that is closed and are closed convex sets for each . Let us show that is convex and so is closed and convex. Similarly to the proof of Lemma 3.3, since

(3.50)

is equivalent to

(3.51)

we know that is convex and so is . Next, let us show that for each . Indeed, utilizing Proposition 2.8, we have, for each ,

(3.52)

So for all and . As in the proof of Lemma 3.3, we can obtain and hence . It follows from Lemma 2.4 that

(3.53)

which implies that . Consequently, and so for all . Therefore, the sequence generated by Algorithm 3.7 is well defined. As in the proof of Theorem 3.5, we can obtain . Since , from the definition of we also have

(3.54)

As in the proof of Theorem 3.5, we can deduce not only from that but also from , and that

(3.55)

Since , from the definition of , we also have

(3.56)

It follows from (3.55) and that

(3.57)

Utilizing Lemma 2.1 we have

(3.58)

Furthermore, for arbitrarily fixed, it follows from Proposition 2.8 that

(3.59)

Since is uniformly norm-to-norm continuous on bounded subsets of , it follows from (3.58) that and , which together with , yield . Utilizing Lemma 2.1, we get . Observe that

(3.60)

due to (3.58). Since is uniformly norm-to-norm continuous on bounded subsets of , we have

(3.61)

Note that

(3.62)

Therefore, from we get

(3.63)

Since is also uniformly norm-to-norm continuous on bounded subsets of , we obtain

(3.64)

It follows that

(3.65)

Since is uniformly continuous, it follows from (3.58) and (3.64) that .

Finally, we prove that . Indeed, for arbitrarily fixed, there exists a subsequence of such that , then . Now let us show that . Since , we have that . Moreover, since is uniformly norm-to-norm continuous on bounded subsets of , and , we obtain that . It follows from and the monotonicity of that for all and . This implies that for all and . Thus from the maximality of , we infer that . Further, let us show that . Since and , from we obtain that and .

Since is uniformly norm-to-norm continuous on bounded subsets of , from we derive . From it follows that

(3.66)

By the definition of , we have

(3.67)

where . Replacing by , we have from (A2) that

(3.68)

Since is convex and lower semicontinuous, it is also weakly lower semicontinuous. Letting in the last inequality, from (3.66) and (A4) we have , for all . For , with , and , let . Since and , then and hence . So, from (A1) we have

(3.69)

Dividing by , we have , for all . Letting , from (A3) it follows that , for all . So, . Therefore, we obtain that by the arbitrariness of .

Next, let us show that and .

Indeed, put . From and , we have . Now from weakly lower semicontinuity of the norm, we derive for each

(3.70)

It follows from the definition of that and hence . So we have . Utilizing the Kadec-Klee property of , we know that . Since is an arbitrary weakly convergent subsequence of , we know that . This completes the proof.

Theorem 3.6 covers [25, Theorem 3.2] by taking and . Also Theorem 3.6 covers [24, Theorem 2.2] by taking and .