An approximation method for common fixed points of a finite family of asymptotic pointwise nonexpansive mappings

In this article, we consider an iterative scheme to approximate a common fixed point for a finite family of asymptotic pointwise nonexpansive mappings. We obtain weak and strong convergence theorems of the proposed iteration in uniformly convex Banach spaces. The related results for complete CAT(0) spaces are also included.

MSC: 47H09, 47H10.

It is well known that many of the most important nonlinear problems of applied mathematics reduce to solving a given equation which in turn may be reduced to finding the fixed points of a certain operator. It is important not only to know the fixed points exist, but also to be able to construct that fixed points. Lau is a great mathematician who has published many good papers concerning to the existence and the approximation of fixed points for various types of mappings (see, e.g., [1-11]).

The existence of fixed points for nonexpansive mappings was studied independently by three authors in 1965 (see Browder [12], Göhde [13], and Kirk [14]). Since then the iteration methods for approximating fixed points of nonexpansive mappings has rapidly been developed and many of papers have appeared (see, e.g., [15-21]). One of the popular classes of generalized nonexpansive mappings is the class of asymptotically nonexpansive mappings which was introduced by Goebel and Kirk [22] in 1972. Later on, Kirk and Xu [23] introduced the concept of asymptotic pointwise nonexpansive mappings which generalizes the concept of asymptotically nonexpansive mappings and proved the existence of fixed points for such maps in a uniformly convex Banach space. In 2011, Kozlowski [24] defined an iterative sequence for an asymptotic pointwise nonexpansive mapping T on a convex subset C of a Banach space X by and

where and are sequences in and is an increasing sequence of natural numbers. He proved, under some suitable assumptions, that the sequence defined by (1) converges weakly to a fixed point of T where X is a uniformly convex Banach space which satisfies the Opial condition and converges strongly to a fixed point of T provided is a compact mapping for some . Recently, Pasom and Panyanak [25] extended Kozlowski’s results to a finite family of asymptotic pointwise nonexpansive mappings . Precisely, they proved weak and strong convergence theorems of the iterative process defined by

where are sequences in for all , and be an increasing sequence of natural numbers. On the other hand, Kettapun et al.[26] studied the iterative process defined by

where are asymptotically quasi-nonexpansive mappings on C.

In this article, motivated by the results mentioned above, we obtain weak and strong convergence theorems of the iterative process defined by

where are asymptotic pointwise nonexpansive mappings on C, are sequences in for all , and be an increasing sequence of natural numbers.

Let C be a nonempty subset of a metric space and T be a mapping on C. A point x in C is called a fixed point of T if . We shall denote by the set of fixed points of T. The mapping is said to be

(i) nonexpansive if ,

(ii) asymptotically nonexpansive if there is a sequence of positive numbers with the property and such that

(iii) asymptotically quasi-nonexpansive if there is a sequence of positive numbers with the property and such that

(iv) asymptotic pointwise nonexpansive if there exists a sequence of functions such that and

The following implications hold.

The existence of fixed points for asymptotic pointwise nonexpansive mappings in uniformly convex Banach spaces was proved by Kirk and Xu [23] as the following result.

Theorem 2.1LetCbe a nonempty bounded closed and convex subset of a uniformly convex Banach spaceX. Then every asymptotic pointwise nonexpansive mappinghas a fixed point. Moreover, is closed and convex.

For common fixed points of a family of commuting mappings, Pasom and Panyanak [27] obtained the following result.

Theorem 2.2LetCbe a nonempty bounded closed convex subset of a uniformly convex Banach spaceX. Then every commuting familyof asymptotic pointwise nonexpansive mappings onChas a nonempty closed convex common fixed point set.

Let C be a nonempty subset of a metric space . We shall denote by the class of all asymptotic pointwise nonexpansive mappings from C into C. Let , without loss of generality, we can assume that there exists a sequence of mappings such that for all , , and ,

Let . Again, without loss of generality, we can assume that

for all , , and . We define , then for each we have .

Definition 2.3[24]

Define as a class of all such that

Let C be a nonempty subset of a Banach space X and . Let be bounded away from 0 and 1 for all and be an increasing sequence of natural numbers. Let and define a sequence in C as

We say that the sequence in (9) is well defined if . As in [24], we observe that for every . Hence, we can always choose a subsequence which makes well defined.

Definition 2.4 A strictly increasing sequence is called quasi-periodic if the sequence is bounded, or equivalently if there exists a number such that any block of p consecutive natural numbers must contain a term of the sequence . The smallest of such numbers p will be called a quasi-period of .

Recall that a mapping is called semi-compact if for any sequence in C such that

there exists a subsequence of and such that . A family of mapping on C is said to satisfy Condition () if there exists a nondecreasing function with and for all such that , for some for all , where .

Lemma 2.5[28], Lemma 2.2]

Letandbe sequences of nonnegative real numbers satisfy:

Then (i) exists (ii) if, then.

Lemma 2.6[29], Lemma 1]

Supposeis a bounded sequence of real numbers andis a doubly index sequence of real numbers which satisfy

for each. Thenfor some.

Lemma 2.7[30,31]

LetXbe a uniformly convex Banach space and letbe a sequence infor some. Suppose thatandare sequences inXsuch that

for some. Then.

Lemma 2.8[24], Lemma 3.1]

LetCbe a nonempty closed convex subset of a uniformly convex Banach spaceXand let. Ifthen for any, .

Lemma 2.9[24], Theorem 3.1]

LetXbe a uniformly convex Banach space with the Opial property and letCbe a nonempty closed convex subset ofX. Letand let, be such that weak-and. Then.

3.1 Results for bounded domains

Recall that a subset C of a metric space is said to be bounded if

Lemma 3.1LetCbe a nonempty closed convex subset of a Banach spaceXand. Letandbe such thatin (9) is well defined. Assume that. Then for each, there are sequences of nonnegative real numbersand (depending onp) such that, and the following statements hold:

(i) , for all;

(ii) , for all;

(iii) ;

(iv) ifCis bounded, thenexists.

Proof Let and for all . Then .

(i) For , we have

(ii) By (9), we obtain

We assume that holds for some . From part (i), we have

By mathematical induction, we obtain

(iii) By part (ii), we get

where . Since , then .

(iv) By part (iii), we have for all . Thus, for each ,

Since , . The conclusion follows from Lemma 2.6 by letting and . □

Lemma 3.2LetCbe a nonempty bounded closed convex subset of a uniformly convex Banach spaceXand. Letandbe such thatin (9) is well defined. Assume that. Then

(i) , for all;

(ii) , for all;

(iii) If the setis quasi-periodic, then, for all.

Proof (i) Let , then by Lemma 3.1(iv) we have exists. Let

By (10) and Lemma 3.1(ii), we get that

Note that

for all . So that

From (11) and (12), we have

That is,

By Lemma 3.1(i) and (13), we get that

By (11), (14), (15), and Lemma 2.7, we obtain

For the case , by Lemma 3.1(i), we have

This implies by (13) that

Moreover,

Again, by Lemma 2.7, we get that

Thus, (16) and (18) imply that

(ii) From (9), we have

By (19), we obtain

From

it follows by (20) that

From

it implies by (19) and (21) that

(iii) For , from (ii) we have

If , then

By (21), (22), and , we get

By (23) and (24), we have

From (9), we have

From (19) and (21),

The proof of the remaining part is identical to the proof of [25], Lemma 4.8(iii)] upon replacing with . □

By using Lemma 3.1 and the argument in the proof of [26], Theorem 3.2], we can obtain the following result.

Lemma 3.3LetCbe a nonempty bounded closed convex subset of a Banach spaceXand. Letandbe such thatin (9) is well defined. Assume that. Thenconverges strongly to a point inFif and only if.

Theorem 3.4LetXbe a uniformly convex Banach space with the Opial property andCbe a nonempty bounded closed convex subset ofX. Letbe such that. andbe such thatin (9) is well defined. If the setis quasi-periodic, then the sequenceconverges weakly to a common fixed point of the family.

Proof We have by Lemma 3.1 that exists for every . We shall prove that has a unique weak subsequential limit in F. For this, we suppose that there are subsequences and of which converge weakly to u and v, respectively. By Lemma 3.2(iii), for all . It follows from Lemma 2.9 that for all . That is . Finally, we prove that . Suppose not, then by the Opial property we get that

This is a contradiction. Therefore, the proof is complete. □

Theorem 3.5LetXbe a uniformly convex Banach space andCbe a nonempty bounded closed convex subset ofX. Letbe such thatis semi-compact for someand. andbe such thatin (9) is well defined. Suppose thatand the setis quasi-periodic. Thenconverges strongly to a common fixed point of the family.

Proof By Lemma 3.2, we have

Let be such that is semi-compact. Thus, by Lemma 2.8,

We can also find a subsequence of such that . Hence, from (27), we have

Thus . Therefore, converges strongly to . But since exists, must itself converges to q. This completes the proof. □

Theorem 3.6LetXbe a uniformly convex Banach space andCbe a nonempty bounded closed convex subset ofX. Letbe satisfy Condition (). Letandbe such thatin (9) is well defined. Suppose thatand the setis quasi-periodic. Thenconverges strongly to a common fixed point of the family.

Proof By Lemma 3.2, , for all . By using Condition (), there exists a nondecreasing function with , for such that

This implies that . The conclusion follows from Lemma 3.3. □

3.2 Results for unbounded domains

To relax the boundedness of the domains we have to add some condition on the sequence .

Lemma 3.7LetCbe a nonempty closed convex subset of a Banach spaceXandbe such that. Letandbe such thatin (9) is well defined. Assume that. Then for, we haveexists.

Proof Similar to the proof of Lemma 3.1, we can show that for all , where and . By assumption, we have for all . It follows that . By Lemma 2.5, we get that exists. □

By using Lemma 3.7 and the argument in Section 3.1 we can obtain the following results.

Lemma 3.8LetCbe a nonempty closed convex subset of a Banach spaceXandbe such that. Letandbe such thatin (9) is well defined. Assume that. Then

(i) , for all;

(ii) , for all;

(iii) If the setis quasi-periodic, then, for all.

Lemma 3.9LetCbe a nonempty closed convex subset of a Banach spaceXandbe such that. Letandbe such thatin (9) is well defined. Assume that. Thenconverges strongly to a point inFif and only if.

Theorem 3.10LetXbe a uniformly convex Banach space with the Opial property andCbe a nonempty closed convex subset ofX. Letbe such that. andbe such thatin (9) is well defined. Assume thatand the setis quasi-periodic. Then the sequenceconverges weakly to a common fixed point of the family.

Theorem 3.11LetXbe a uniformly convex Banach space andCbe a nonempty closed convex subset ofX. Letbe such thatis semi-compact for someand, andbe such thatin (9) is well defined. Suppose that, and the setis quasi-periodic. Thenconverges strongly to a common fixed point of the family.

Theorem 3.12LetXbe a uniformly convex Banach space andCbe a nonempty closed convex subset ofX. Letbe satisfy Condition (). Letandbe such thatin (9) is well defined. Suppose that, and the setis quasi-periodic. Thenconverges strongly to a common fixed point of the family.

A metric space X is a CAT(0) space if it is geodesically connected, and if every geodesic triangle in X is at least as thin as its comparison triangle in the Euclidean plane. It is well known that any complete, simply connected Riemannian manifold having nonpositive sectional curvature is a CAT(0) space. Other examples include Pre-Hilbert spaces (see [32]), -trees (see [33]), Euclidean buildings (see [34]), the complex Hilbert ball with a hyperbolic metric (see [35]), and many others. For a thorough discussion of these spaces and of the fundamental role they play in geometry, we refer the reader to Bridson and Haefliger [32].

Let , by Lemma 2.1(iv) of [36] for each , there exists a unique point such that

From now on, we will use the notation for the unique point z satisfying (28).

Let be a bounded sequence in a metric space . For , we set

The asymptotic radius of is given by

and the asymptotic center of is the set

It is known from Proposition 7 of [37] that in a CAT(0) space, consists of exactly one point. We now give the definition of Δ-convergence.

Definition 4.1[38,39]

A sequence in a metric space X is said to Δ-converge to if x is the unique asymptotic center of for every subsequence of . In this case we write and call x the Δ-limit of .

Let C be a nonempty closed convex subset of a CAT(0) space X and fix . Define a sequence in C as

where , are sequences in for all , and be an increasing sequence of natural numbers.

By using the argument in Section 3 together with the results in [25,36,40,41], we can also obtain the analogous results for CAT(0) spaces.

Theorem 4.2LetCbe a nonempty closed convex subset of a complete CAT(0) spaceX. Letbe such that, andbe such thatin (29) is well defined. Suppose that eitherCis bounded or. If the setis quasi-periodic, then the sequence Δ-converges to a common fixed point of the family.

Theorem 4.3LetCbe a nonempty closed convex subset of a complete CAT(0) spaceX. Letbe such thatandis semi-compact for someand. Letandbe such thatin (29) is well defined. Suppose that eitherCis bounded or. If the setis quasi-periodic, thenconverges strongly to a common fixed point of the family.

Theorem 4.4LetCbe a nonempty closed convex subset of a complete CAT(0) spaceX. Letbe satisfy Condition () and. Letandbe such thatin (29) is well defined. Suppose that eitherCis bounded or. If the setis quasi-periodic, thenconverges strongly to a common fixed point of the family.

The authors declare that they have no competing interests.

Both the authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.

This article is dedicated to Professor Anthony To-Ming Lau for celebrating his great achievements in the development of fixed point theory and applications. It was supported by the Centre of Excellence in Mathematics, the Commission on Higher Education, Thailand. Bancha Nanjaras also thanks the Graduate School of Chiang Mai University, Thailand.

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