Convergence theorems for mixed type asymptotically nonexpansive mappings

In this paper, we introduce a new two-step iterative scheme of mixed type for two asymptotically nonexpansive self-mappings and two asymptotically nonexpansive nonself-mappings and prove strong and weak convergence theorems for the new two-step iterative scheme in uniformly convex Banach spaces.

Let K be a nonempty subset of a real normed linear space E. A mapping is said to be asymptotically nonexpansive if there exists a sequence with such that

for all and .

In 1972, Goebel and Kirk [1] introduced the class of asymptotically nonexpansive self-mappings, which is an important generalization of the class of nonexpansive self-mappings, and proved that if K is a nonempty closed convex subset of a real uniformly convex Banach space E and T is an asymptotically nonexpansive self-mapping of K, then T has a fixed point.

Since then, some authors proved weak and strong convergence theorems for asymptotically nonexpansive self-mappings in Banach spaces (see [2-16]), which extend and improve the result of Goebel and Kirk in several ways.

Recently, Chidume et al.[10] introduced the concept of asymptotically nonexpansive nonself-mappings, which is a generalization of an asymptotically nonexpansive self-mapping, as follows.

Definition 1.1[10]

Let K be a nonempty subset of a real normed linear space E. Let be a nonexpansive retraction of E onto K. A nonself-mapping is said to be asymptotically nonexpansive if there exists a sequence with as such that

for all and .

Let K be a nonempty closed convex subset of a real uniformly convex Banach space E.

In 2003, also, Chidume et al.[10] studied the following iteration scheme:

for each , where is a sequence in and P is a nonexpansive retraction of E onto K, and proved some strong and weak convergence theorems for an asymptotically nonexpansive nonself-mapping.

In 2006, Wang [11] generalized the iteration process (1.3) as follows:

for each , where are two asymptotically nonexpansive nonself-mappings and , are real sequences in , and proved some strong and weak convergence theorems for two asymptotically nonexpansive nonself-mappings. Recently, Guo and Guo [12] proved some new weak convergence theorems for the iteration process (1.4).

The purpose of this paper is to construct a new iteration scheme of mixed type for two asymptotically nonexpansive self-mappings and two asymptotically nonexpansive nonself-mappings and to prove some strong and weak convergence theorems for the new iteration scheme in uniformly convex Banach spaces.

Let E be a real Banach space, K be a nonempty closed convex subset of E and be a nonexpansive retraction of E onto K. Let be two asymptotically nonexpansive self-mappings and be two asymptotically nonexpansive nonself-mappings. Then we define the new iteration scheme of mixed type as follows:

for each , where , are two sequences in .

If and are the identity mappings, then the iterative scheme (2.1) reduces to the sequence (1.4).

We denote the set of common fixed points of , , and by and denote the distance between a point z and a set A in E by .

Now, we recall some well-known concepts and results.

Let E be a real Banach space, be the dual space of E and be the normalized duality mapping defined by

for all , where denotes duality pairing between E and . A single-valued normalized duality mapping is denoted by j.

A subset K of a real Banach space E is called a retract of E[10] if there exists a continuous mapping such that for all . Every closed convex subset of a uniformly convex Banach space is a retract. A mapping is called a retraction if . It follows that if a mapping P is a retraction, then for all y in the range of P.

A Banach space E is said to satisfy Opial’s condition[17] if, for any sequence of E, weakly as implies that

for all with .

A Banach space E is said to have a Fréchet differentiable norm[18] if, for all ,

exists and is attained uniformly in .

A Banach space E is said to have the Kadec-Klee property[19] if for every sequence in E, weakly and , it follows that strongly.

Let K be a nonempty closed subset of a real Banach space E. A nonself-mapping is said to be semi-compact[11] if, for any sequence in K such that as , there exists a subsequence of such that converges strongly to some .

Lemma 2.1[15]

Let, andbe three nonnegative sequences satisfying the following condition:

for each, whereis some nonnegative integer, and. Thenexists.

Lemma 2.2[8]

LetEbe a real uniformly convex Banach space andfor each. Also, suppose thatandare two sequences ofEsuch that

hold for some. Then.

Lemma 2.3[10]

LetEbe a real uniformly convex Banach space, Kbe a nonempty closed convex subset ofEandbe an asymptotically nonexpansive mapping with a sequenceandas. Thenis demiclosed at zero, i.e., ifweakly andstrongly, then, whereis the set of fixed points ofT.

Lemma 2.4[16]

LetXbe a uniformly convex Banach space andCbe a convex subset ofX. Then there exists a strictly increasing continuous convex functionwithsuch that, for each mappingwith a Lipschitz constant,

for alland.

Lemma 2.5[16]

LetXbe a uniformly convex Banach space such that its dual spacehas the Kadec-Klee property. Supposeis a bounded sequence andsuch that

exists for all, wheredenotes the set of all weak subsequential limits of . Then.

In this section, we prove strong convergence theorems for the iterative scheme given in (2.1) in uniformly convex Banach spaces.

Lemma 3.1LetEbe a real uniformly convex Banach space andKbe a nonempty closed convex subset ofE. Letbe two asymptotically nonexpansive self-mappings withandbe two asymptotically nonexpansive nonself-mappings withsuch thatandfor, respectively, and. Letbe the sequence defined by (2.1), whereandare two real sequences in. Then

(1) exists for any;

(2) exists.

Proof (1) Set . For any , it follows from (2.1) that

and so

Since and for , we have . It follows from Lemma 2.1 that exists.

(2) Taking the infimum over all in (3.2), we have

for each . It follows from and Lemma 2.1 that the conclusion (2) holds. This completes the proof. □

Lemma 3.2LetEbe a real uniformly convex Banach space andKbe a nonempty closed convex subset ofE. Letbe two asymptotically nonexpansive self-mappings withandbe two asymptotically nonexpansive nonself-mappings withsuch thatandfor, respectively, and. Letbe the sequence defined by (2.1) and the following conditions hold:

(a) andare two real sequences infor some;

(b) for alland.

Thenfor.

Proof Set . For any given , exists by Lemma 3.1. Now, we assume that . It follows from (3.2) and that

and

Taking lim sup on both sides in (3.1), we obtain and so

Using Lemma 2.2, we have

By the condition (b), it follows that

and so, from (3.3), we have

Since

Taking lim inf on both sides in the inequality above, we have

by (3.4) and so

Using (3.1), we have

In addition, we have

and

It follows from Lemma 2.2 that

Now, we prove that

Indeed, since by the condition (b). It follows from (3.5) that

Since and is a nonexpansive retraction of E onto K, we have

and so

Furthermore, we have

Thus it follows from (3.5), (3.6) and (3.7) that

Since by the condition (b) and

Using (3.3) and (3.8), we have

and

It follows from

and (3.3) that

In addition, we have

Using (3.3) and (3.11), we obtain that

Thus, using (3.9), (3.10) and the inequality

we have . It follows from (3.6) and the inequality

that

Since

from (3.8), (3.11) and (3.13), it follows that

Again, since , for and , are two asymptotically nonexpansive nonself-mappings, we have

for . It follows from (3.12), (3.14) and (3.15) that

for . Moreover, we have

Using (3.4), (3.8) and (3.12), we have

In addition, we have

for . Thus it follows from (3.6), (3.10), (3.16) and (3.17) that

Finally, we prove that

In fact, by the condition (b), we have

for . Thus it follows from (3.5), (3.6), (3.9) and (3.10) that

This completes the proof. □

Now, we find two mappings, and , satisfying the condition (b) in Lemma 3.2 as follows.

Example 3.1[20]

Let ℝ be the real line with the usual norm and let . Define two mappings by

and

Now, we show that T is nonexpansive. In fact, if or , then we have

If and or and , then we have

This implies that T is nonexpansive and so T is an asymptotically nonexpansive mapping with for each . Similarly, we can show that S is an asymptotically nonexpansive mapping with for each .

Next, we show that two mappings S, T satisfy the condition (b) in Lemma 3.2. For this, we consider the following cases:

Case 1. Let . Then we have

Case 2. Let . Then we have

Case 3. Let and . Then we have

Case 4. Let and . Then we have

Therefore, the condition (b) in Lemma 3.2 is satisfied.

Theorem 3.1Under the assumptions of Lemma 3.2, if one of, , andis completely continuous, then the sequencedefined by (2.1) converges strongly to a common fixed point of, , and.

Proof Without loss of generality, we can assume that is completely continuous. Since is bounded by Lemma 3.1, there exists a subsequence of such that converges strongly to some . Moreover, we know that

and

by Lemma 3.2, which imply that

as and so . Thus, by the continuity of , , and , we have

and

for . Thus it follows that . Furthermore, since exists by Lemma 3.1, we have . This completes the proof. □

Theorem 3.2Under the assumptions of Lemma 3.2, if one of, , andis semi-compact, then the sequencedefined by (2.1) converges strongly to a common fixed point of, , and.

Proof Since for by Lemma 3.2 and one of , , and is semi-compact, there exists a subsequence of such that converges strongly to some . Moreover, by the continuity of , , and , we have and for . Thus it follows that . Since exists by Lemma 3.1, we have . This completes the proof. □

Theorem 3.3Under the assumptions of Lemma 3.2, if there exists a nondecreasing functionwithandfor allsuch that

for all, where, then the sequencedefined by (2.1) converges strongly to a common fixed point of, , and.

Proof Since for by Lemma 3.2, we have . Since is a nondecreasing function satisfying , for all and exists by Lemma 3.1, we have .

Now, we show that is a Cauchy sequence in K. In fact, from (3.2), we have

for each , where and . For any m, n, , we have

where . Thus, for any , we have

Taking the infimum over all , we obtain

Thus it follows from that is a Cauchy sequence. Since K is a closed subset of E, the sequence converges strongly to some . It is easy to prove that , , and are all closed and so F is a closed subset of K. Since , , the sequence converges strongly to a common fixed point of , , and . This completes the proof. □

In this section, we prove weak convergence theorems for the iterative scheme defined by (2.1) in uniformly convex Banach spaces.

Lemma 4.1Under the assumptions of Lemma 3.1, for all, the limit

exists for all, whereis the sequence defined by (2.1).

Proof Set . Then and, from Lemma 3.1, exists. Thus it remains to prove Lemma 4.1 for any .

Define the mapping by

for all . It is easy to prove that

for all , where . Letting , it follows from and that . Setting

for each , from (4.1) and (4.2), it follows that

for all and , for any . Let

Then, using (4.3) and Lemma 2.4, we have

It follows from Lemma 3.1 and that uniformly for all m. Observe that

Thus we have , that is, exists for all . This completes the proof. □

Lemma 4.2Under the assumptions of Lemma 3.1, ifEhas a Fréchet differentiable norm, then, for all, the limit

exists, whereis the sequence defined by (2.1). Furthermore, ifdenotes the set of all weak subsequential limits of, thenfor alland.

Proof This follows basically as in the proof of Lemma 3.2 of [12] using Lemma 4.1 instead of Lemma 3.1 of [12]. □

Theorem 4.1Under the assumptions of Lemma 3.2, ifEhas a Fréchet differentiable norm, then the sequencedefined by (2.1) converges weakly to a common fixed point of, , and.

Proof Since E is a uniformly convex Banach space and the sequence is bounded by Lemma 3.1, there exists a subsequence of which converges weakly to some . By Lemma 3.2, we have

for . It follows from Lemma 2.3 that .

Now, we prove that the sequence converges weakly to q. Suppose that there exists a subsequence of such that converges weakly to some . Then, by the same method given above, we can also prove that . So, . It follows from Lemma 4.2 that

Therefore, , which shows that the sequence converges weakly to q. This completes the proof. □

Theorem 4.2Under the assumptions of Lemma 3.2, if the dual spaceofEhas the Kadec-Klee property, then the sequencedefined by (2.1) converges weakly to a common fixed point of, , and.

Proof Using the same method given in Theorem 4.1, we can prove that there exists a subsequence of which converges weakly to some .

Now, we prove that the sequence converges weakly to q. Suppose that there exists a subsequence of such that converges weakly to some . Then, as for q, we have . It follows from Lemma 4.1 that the limit

exists for all . Again, since , by Lemma 2.5. This shows that the sequence converges weakly to q. This completes the proof. □

Theorem 4.3Under the assumptions of Lemma 3.2, ifEsatisfies Opial’s condition, then the sequencedefined by (2.1) converges weakly to a common fixed point of, , and.

Proof Using the same method as given in Theorem 4.1, we can prove that there exists a subsequence of which converges weakly to some .

Now, we prove that the sequence converges weakly to q. Suppose that there exists a subsequence of such that converges weakly to some and . Then, as for q, we have . Using Lemma 3.1, we have the following two limits exist:

Thus, by Opial’s condition, we have

which is a contradiction and so . This shows that the sequence converges weakly to q. This completes the proof. □

The authors declare that they have no competing interests.

All authors read and approved the final manuscript.

The project was supported by the National Natural Science Foundation of China (Grant Number: 11271282) and the second author was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (Grant Number: 2012-0008170).

1. Goebel, K, Kirk, WA: A fixed point theorem for asymptotically nonexpansive mapping. Proc. Am. Math. Soc.. 35, 171–174 (1972). Publisher Full Text

2. Chang, SS, Cho, YJ, Zhou, HY: Demi-closed principle and weak convergence problems for asymptotically nonexpansive mappings. J. Korean Math. Soc.. 38, 1245–1260 (2001)

3. Chang, SS, Tan, KK, Lee, HWJ, Chan, CK: On the convergence of implicit iteration process with error for a finite family of asymptotically nonexpansive mappings. J. Math. Anal. Appl.. 313, 273–283 (2006). Publisher Full Text

4. Cho, YJ, Kang, JI, Zhou, HY: Approximating common fixed points of asymptotically nonexpansive mappings. Bull. Korean Math. Soc.. 42, 661–670 (2005)

5. Cho, YJ, Zhou, HY, Guo, GT: Weak and strong convergence theorems for three-step iterations with errors for asymptotically nonexpansive mappings. Comput. Math. Appl.. 47, 707–717 (2004). Publisher Full Text

6. Chen, W, Guo, W: Convergence theorems for two finite families of asymptotically nonexpansive mappings. Math. Comput. Model.. 54, 1311–1319 (2011). Publisher Full Text

7. Guo, W, Cho, YJ: On strong convergence of the implicit iterative processes with errors for a finite family of asymptotically nonexpansive mappings. Appl. Math. Lett.. 21, 1046–1052 (2008). Publisher Full Text

8. Schu, J: Weak and strong convergence of fixed points of asymptotically nonexpansive mappings. Bull. Aust. Math. Soc.. 43, 153–159 (1991). Publisher Full Text

9. Zhou, YY, Chang, SS: Convergence of implicit iterative process for a finite family of asymptotically nonexpansive mappings in Banach spaces. Numer. Funct. Anal. Optim.. 23, 911–921 (2002). Publisher Full Text

10. Chidume, CE, Ofoedu, EU, Zegeye, H: Strong and weak convergence theorems for asymptotically nonexpansive mappings. J. Math. Anal. Appl.. 280, 364–374 (2003). Publisher Full Text

11. Wang, L: Strong and weak convergence theorems for common fixed points of nonself asymptotically nonexpansive mappings. J. Math. Anal. Appl.. 323, 550–557 (2006). Publisher Full Text

12. Guo, W, Guo, W: Weak convergence theorems for asymptotically nonexpansive nonself-mappings. Appl. Math. Lett.. 24, 2181–2185 (2011). Publisher Full Text

13. Pathak, HK, Cho, YJ, Kang, SM: Strong and weak convergence theorems for nonself-asymptotically perturbed nonexpansive mappings. Nonlinear Anal.. 70, 1929–1938 (2009). Publisher Full Text

14. Zhou, HY, Cho, YJ, Kang, SM: A new iterative algorithm for approximating common fixed points for asymptotically nonexpansive mappings. Fixed Point Theory Appl.. 2007, Article ID 64974 (2007)

15. Sun, ZH: Strong convergence of an implicit iteration process for a finite family of asymptotically quasi-nonexpansive mappings. J. Math. Anal. Appl.. 286, 351–358 (2003). Publisher Full Text

16. Falset, JG, Kaczor, W, Kuczumow, T, Reich, S: Weak convergence theorems for asymptotically nonexpansive mappings and semigroups. Nonlinear Anal.. 43, 377–401 (2001). Publisher Full Text

17. Opial, Z: Weak convergence of successive approximations for nonexpansive mappings. Bull. Am. Math. Soc.. 73, 591–597 (1967). Publisher Full Text

18. Osilike, MO, Udomene, A: Demiclosedness principle and convergence theorems for strictly pseudocontractive mappings of Browder-Petryshyn type. J. Math. Anal. Appl.. 256, 431–445 (2001). Publisher Full Text

19. Goebel, K, Kirk, WA: Topics in Metric Fixed Point Theory, Cambridge University Press, Cambridge (1990)

20. Liu, Z, Feng, C, Ume, JS, Kang, SM: Weak and strong convergence for common fixed points of a pair of nonexpansive and asymptotically nonexpansive mappings. Taiwan. J. Math.. 11, 27–42 (2007)