On Strong Convergence by the Hybrid Method for Equilibrium and Fixed Point Problems for an Inifnite Family of Asymptotically Nonexpansive Mappings

Abstract

We introduce two modifications of the Mann iteration, by using the hybrid methods, for equilibrium and fixed point problems for an infinite family of asymptotically nonexpansive mappings in a Hilbert space. Then, we prove that such two sequences converge strongly to a common element of the set of solutions of an equilibrium problem and the set of common fixed points of an infinite family of asymptotically nonexpansive mappings. Our results improve and extend the results announced by many others.

1. Introduction

Let be a nonempty closed convex subset of a Hilbert space . A mapping is said to be nonexpansive if for all we have . It is said to be asymptotically nonexpansive [1] if there exists a sequence with and such that for all integers and for all . The set of fixed points of is denoted by .

Let be a bifunction, where is the set of real number. The equilibrium problem for the function is to find a point such that

(1.1)

The set of solutions of (1.1) is denoted by . In 2005, Combettes and Hirstoaga [2] introduced an iterative scheme of finding the best approximation to the initial data when is nonempty, and they also proved a strong convergence theorem.

For a bifunction and a nonlinear mapping , we consider the following equilibrium problem:

(1.2)

The set of such that is denoted by , that is,

(1.3)

In the case of , . In the case of , is denoted by . The problem (1.2) is very general in the sense that it includes, as special cases, some optimization problems, variational inequalities, minimax problems, the Nash equilibrium problem in noncooperative games, and others (see, e.g., [3, 4]).

Recall that a mapping is called monotone if

(1.4)

A mapping of into is called -inverse strongly monotone, see [5–7], if there exists a positive real number such that

(1.5)

for all . It is obvious that any inverse strongly monotone mapping is monotone and Lipschitz continuous.

Construction of fixed points of nonexpansive mappings and asymptotically nonexpansive mappings is an important subject in nonlinear operator theory and its applications, in particular, in image recovery and signal processing (see, e.g., [1, 8–10]). Fixed point iteration processes for nonexpansive mappings and asymptotically nonexpansive mappings in Hilbert spaces and Banach spaces including Mann [11] and Ishikawa [12] iteration processes have been studied extensively by many authors to solve nonlinear operator equations as well as variational inequalities; see, for example, [11–13]. However, Mann and Ishikawa iteration processes have only weak convergence even in Hilbert spaces (see, e.g., [11, 12]).

Some attempts to modify the Mann iteration method so that strong convergence is guaranteed have recently been made. In 2003, Nakajo and Takahashi [14] proposed the following modification of the Mann iteration method for a nonexpansive mapping in a Hilbert space :

(1.6)

where denotes the metric projection from onto a closed convex subset of . They proved that if the sequence bounded above from one, then defined by (1.6) converges strongly to .

Recently, Kim and Xu [15] adapted the iteration (1.6) to an asymptotically nonexpansive mapping in a Hilbert space :

(1.7)

where , as . They proved that if for all and for some , then the sequence generated by (1.7) converges strongly to .

Very recently, Inchan and Plubtieng [16] introduced the modified Ishikawa iteration process by the shrinking hybrid method [17] for two asymptotically nonexpansive mappings and , with a closed convex bounded subset of a Hilbert space . For and , define as follows:

(1.8)

where , as and and for all . They proved that the sequence generated by (1.8) converges strongly to a common fixed point of two asymptotically nonexpansive mappings and .

Zegeye and Shahzad [18] established the following hybrid iteration process for a finite family of asymptotically nonexpansive mappings in a Hilbert space :

(1.9)

where , as . Under suitable conditions strong convergence theorem is proved which extends and improves the corresponding results of Nakajo and Takahashi [14] and Kim and Xu [15].

On the other hand, for finding a common element of , Tada and Takahashi [19] introduced the following iterative scheme by the hybrid method in a Hilbert space: and let

(1.10)

for every , where for some and satisfies . Further, they proved that and converge strongly to , where .

Inspired and motivated by the above facts, it is the purpose of this paper to introduce the Mann iteration process for finding a common element of the set of common fixed points of an infinite family of asymptotically nonexpansive mappings and the set of solutions of an equilibrium problem. Then we prove some strong convergence theorems which extend and improve the corresponding results of Tada and Takahashi [19], Inchan and Plubtieng [16], Zegeye and Shahazad [18], and many others.

2. Preliminaries

We will use the following notations:

(1)"" for weak convergence and "" for strong convergence;

(2) denotes the weak -limit set of .

Let be a real Hilbert space. It is well known that

(2.1)

for all .

It is well known that satisfies Opial's condition [20], that is, for any sequence with , the inequality

(2.2)

holds for every with . Hilbert space satisfies the Kadec-Klee property [21, 22], that is, for any sequence with and together imply .

We need some facts and tools in a real Hilbert space which are listed as follows.

Lemma 2.1 ([23]).

Let be an asymptotically nonexpansive mapping defined on a nonempty bounded closed convex subset of a Hilbert space . If is a sequence in such that and , then .

Lemma 2.2 ([24]).

Let be a nonempty closed convex subset of and also give a real number . The set is convex and closed.

Lemma 2.3 ([22]).

Let be a nonempty closed convex subset of and let be the (metric or nearest) projection from onto i.e., is the only point in such that . Given and . Then if and only if it holds the relation:

(2.3)

For solving the equilibrium problem, let us assume that the bifunction satisfies the following conditions (see [3]):

(A1) for all ;

(A2) is monotone, that is, for any ;

(A3) is upper-hemicontinuous, that is, for each

(2.4)

(A4) is convex and weakly lower semicontinuous for each .

The following lemma appears implicity in [3].

Lemma 2.4 ([3]).

Let be a nonempty closed convex subset of and let be a bifunction of into satisfying (A1)–(A4). Let and . Then, there exists such that

(2.5)

The following lemma was also given in [2].

Lemma 2.5 ([2]).

Assume that satisfies (A1)–(A4). For and , define a mapping as follows:

(2.6)

for all . Then, the following holds

(1) is single-valued;

(2) is firmly nonexpansive, that is, for any , .

This implies that , that is, is a nonexpansive mapping:

(3);

(4) is a closed and convex set.

Definition 2.6 (see [25]).

Let be a nonempty closed convex subset of . Let be a family of asymptotically nonexpansive mappings of into itself, and let be a sequence of real numbers such that for every with . For any define a mapping as follows:

(2.7)

Such a mapping is called the modified -mapping generated by and .

Lemma 2.7 ([10, Lemma 4.1]).

Let be a nonempty closed convex subset of . Let be a family of asymptotically nonexpansive mappings of into itself with Lipschitz constants , that is, () such that and let be a sequence of real numbers with for all and for every and for some . Let be the modified -mapping generated by and . Let for every and . Then, the followings hold:

• (i) for all , and ;

(ii) if is bounded and , for every sequence in C,

(2.8)

(iii) if , and is closed convex.

Lemma 2.8 ([10, Lemma 4.4]).

Let be a nonempty closed convex subset of . Let be a family of asymptotically nonexpansive mappings of into itself with Lipschitz constants , that is, () such that . Let for every , where for every and with for every and for every and let for every . Then, the following holds:

(i) for all , and ;

• (ii) if is bounded and, for every sequence in C,

  

  (2.9)

• (iii) if , and is closed convex.

3. Main Results

In this section, we will introduce two iterative schemes by using hybrid approximation method for finding a common element of the set of common fixed points for a family of infinitely asymptotically nonexpansive mappings and the set of solutions of an equilibrium problem in Hilbert space. Then we show that the sequences converge strongly to a common element of the two sets.

Theorem 3.1.

Let be a nonempty bounded closed convex subset of a real Hilbert space , let be a bifunction satisfying the conditions (A1)–(A4), let be an -inverse strongly monotone mapping of into , let be a family of asymptotically nonexpansive mappings of into itself with Lipschitz constants , that is, () such that , where and let be a sequence of real numbers with for all and for every and for some . Let be the modified -mapping generated by and . Assume that for every and such that . Let and be sequences generated by the following algorithm:

(3.1)

where and and and . Then and converge strongly to .

Proof.

We show first that the sequences and are well defined.

We observe that is closed and convex by Lemma 2.2. Next we show that for all . we prove first that is nonexpansive. Let . Since is -inverse strongly monotone and , we have

(3.2)

Thus is nonexpansive.

Since

(3.3)

we obtain

(3.4)

By Lemma 2.5, we have , .

Let , it follows the definition of that

(3.5)

So,

(3.6)

Again by Lemma 2.5, we have ,.

Since and are nonexpansive, one has

(3.7)

Then using the convexity of and Lemma 2.7 we obtain that

(3.8)

So for all and hence for all . This implies that is well defined. From Lemma 2.4, we know that is also well defined.

Next, we prove that , , , , as .

It follows from that

(3.9)

So, for , we have

(3.10)

This implies that

(3.11)

and hence

(3.12)

Since is bounded, then and are bounded.

From and , we have

(3.13)

So,

(3.14)

This implies that

(3.15)

Hence, is nodecreasing, and so exists.

Next, we can show that . Indeed, From (2.1) and (3.13), we obtain

(3.16)

Since exists, we have

(3.17)

On the other hand, it follows from that

(3.18)

It follows that

(3.19)

Next, we claim that . Let , it follows from (3.8) that

(3.20)

This implies that

(3.21)

It follows from (3.19) that

(3.22)

From Lemma 2.5, one has

(3.23)

This implies that

(3.24)

By (3.8), we have

(3.25)

Substituting (3.24) into (3.25), we obtain

(3.26)

which implies that

(3.27)

Noticing that and (3.19), it follows from (3.27) that

(3.28)

From (3.17) and (3.28), we have

(3.29)

Similarly, from (3.19) and (3.28), one has

(3.30)

Noticing that the condition , it follows that

(3.31)

which implies that

(3.32)

Next, we prove that there exists a subsequence of which converges weakly to , where .

Since is bounded and is closed, there exists a subsequence of which converges weakly to , where . From (3.28), we have . Noticing (3.29) and (3.32), it follows from Lemma 2.7 that . Next we prove that . Since , for any we have

(3.33)

From (A2), one has

(3.34)

Replacing by , we obtain

(3.35)

Put for all and . Then, we have . So we have

(3.36)

Since , we have . Further, from monotonicity of , we have . So, from (A4) we have

(3.37)

as . From (A1) and (A4), we also have

(3.38)

and hence

(3.39)

Letting , we have, for each ,

(3.40)

This implies that . Therefore .

Finally we show that ,, where .

Putting and consider the sequence . Then we have and by the weak lower semicontinuity of the norm and by the fact that for all which is implied by the fact that , we obtain

(3.41)

This implies that (hence by the uniqueness of the nearest point projection of onto ) and that

(3.42)

It follows that , and hence . Since is an arbitrary (weakly convergent) subsequence of , we conclude that . From (3.28), we know that also. This completes the proof of Theorem 3.1.

Theorem 3.2.

Let be a nonempty bounded closed convex subset of a real Hilbert space , let be a bifunction satisfying the conditions (A1)–(A4), let be an -inverse strongly monotone mapping of into , and let be a family of asymptotically nonexpansive mappings of into itself with Lipschitz constants , that is, () such that , where . Let for every , where for every and with for each and for every and assume that for every such that . Let and be sequences generated by

(3.43)

where and and . Then and converge strongly to .

Proof.

We divide the proof of Theorem 3.2 into four steps.

(i)We show first that the sequences and are well defined.

From the definition of and , it is obvious that is closed and is closed and convex for each . We prove that is convex. Since

(3.44)

is equivalent to

(3.45)

it follows that is convex. So, is a closed convex subset of for any .

Next, we show that . Indeed, let and let be a sequence of mappings defined as in Lemma 2.5. Similar to the proof of Theorem 3.1, we have

(3.46)

By virtue of the convexity of norm , (3.46), and Lemma 2.8, we have

(3.47)

Therefore, for all .

Next, we prove that , . For , we have . Assume that . Since is the projection of onto , by Lemma 2.3, we have

(3.48)

In particular, we have

(3.49)

for each and hence . Hence , . Therefore, we obtain that

(3.50)

This implies that is well defined. From Lemma 2.4, we know that is also well defined.

(ii)We prove that , , , , as .

Since is a nonempty closed convex subset of , there exists a unique such that .

From , we have

(3.51)

Since , we have

(3.52)

Since is bounded, we have , and are bounded. From the definition of , we have , which together with the fact that implies that

(3.53)

This shows that the sequence is nondecreasing. So, exists.

It follows from (2.1) and (3.53) that

(3.54)

Noticing that exists, this implies that

(3.55)

Since , we have

(3.56)

So, we have . It follows that

(3.57)

Similar to the proof of Theorem 3.1, we have

(3.58)

From (3.55) and (3.58), we have

(3.59)

Similarly, from (3.57) and (3.58), one has

(3.60)

Noticing the condition , it follows that

(3.61)

which implies that

(3.62)

• (iii) We prove that there exists a subsequence of which converges weakly to , where .

Since is bounded and is closed, there exists a subsequence of which converges weakly to , where . From (3.58), we have . Noticing (3.59) and (3.62), it follows from Lemma 2.8 that . By using the same method as in the proof of Theorem 3.1, we easily obtain that .

• (iv) Finally we show that , , where .

Since and , we have

(3.63)

It follows from and the weak lower-semicontinuity of the norm that

(3.64)

Thus, we obtain that . Using the Kadec-Klee property of , we obtain that

(3.65)

Since is an arbitrary subsequence of , we conclude that converges strongly to . By (3.58), we have also. This completes the proof of Theorem 3.2.

Corollary 3.3.

Let be a nonempty bounded closed convex subset of a real Hilbert space , let be a bifunction satisfying the conditions (A1)–(A4), let be a family of asymptotically nonexpansive mappings of into itself with Lipschitz constants , that is, () such that , where and let be a sequence of real numbers with for all and for every and for some . Let be the modified -mapping generated by and . Assume that for every and such that . Let and be sequences generated by the following algorithm:

(3.66)

where and and and such that . Then and converge strongly to .

Proof.

Putting , the conclusion of Corollary 3.3 can be obtained as in the proof of Theorem 3.1.

Remark 3.4.

Corollary 3.3 extends the Theorem of Tada and Takahashi [19] in the following senses:

(1)from one nonexpansive mapping to a family of infinitely asymptotically nonexpansive mappings;

• (2) from computation point of view, the algorithm in Corollary 3.3 is also simpler and, more convenient to compute than the one given in [19].

Corollary 3.5.

Let be a nonempty bounded closed convex subset of a real Hilbert space , let be a family of asymptotically nonexpansive mappings of into itself with Lipschitz constants , that is, () such that and let be a sequence of real numbers with for all and for every and for some . Let be the modified -mapping generated by and . Assume that for every and such that . Let be a sequence generated by the following algorithm:

(3.67)

where and . Then converges strongly to .

Proof.

Putting , , and , for all in Theorem 3.1, we have , therefore . The conclusion of Corollary 3.5 can be obtained from Theorem 3.1 immediately.

Remark 3.6.

Corollary 3.5 extends Theorem 3.1 of Inchan and Plubtieng [16] from two asymptotically nonexpansive mappings to an infinite family of asymptotically nonexpansive mappings.

Corollary 3.7.

Let be a nonempty bounded closed convex subset of a real Hilbert space , and let be a family of asymptotically nonexpansive mappings of into itself with Lipschitz constants , that is, () such that . Let for every , where for every and with for each and for every and assume that for every such that . Let be a sequence generated by

(3.68)

where and . Then converges strongly to .

Proof.

Putting , , and , for all in Theorem 3.2, we have , therefore . The conclusion of Corollary 3.7 can be obtained from Theorem 3.2.

Remark 3.8.

Corollary 3.7 extends Theorem 3.1 of Zegeye and Shahzad [18] from a finite family of asymptotically nonexpansive mappings to an infinite family of asymptotically nonexpansive mappings.