Volume 2012, Issue 1 327434

Research Article

Open Access

Common Fixed Points for Asymptotic Pointwise Nonexpansive Mappings in Metric and Banach Spaces

P. Pasom,

P. Pasom

Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand cmu.ac.th

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B. Panyanak,

Corresponding Author

B. Panyanak

[email protected]

Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand cmu.ac.th

Centre of Excellence in Mathematics, CHE, Si Ayutthaya Road, Bangkok 10400, Thailand

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P. Pasom,

P. Pasom

Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand cmu.ac.th

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B. Panyanak,

Corresponding Author

B. Panyanak

[email protected]

Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand cmu.ac.th

Centre of Excellence in Mathematics, CHE, Si Ayutthaya Road, Bangkok 10400, Thailand

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First published: 20 February 2012

https://doi.org/10.1155/2012/327434

Citations: 1

Academic Editor: Rudong Chen

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Abstract

Let C be a nonempty bounded closed convex subset of a complete CAT(0) space X. We prove that the common fixed point set of any commuting family of asymptotic pointwise nonexpansive mappings on C is nonempty closed and convex. We also show that, under some suitable conditions, the sequence defined by , converges to a common fixed point of T₁, T₂, …, T_m where they are asymptotic pointwise nonexpansive mappings on C, are sequences in [0,1] for all i = 1,2, …, m, and {n_k} is an increasing sequence of natural numbers. The related results for uniformly convex Banach spaces are also included.

1. Introduction

A mapping T on a subset C of a Banach space X is said to be asymptotic pointwise nonexpansive if there exists a sequence of mappings α_n : C → [0, ∞) such that

(1.1)

where limsup _n→∞α_n(x) ≤ 1, for all x, y ∈ C. This class of mappings was introduced by Kirk and Xu [1], where it was shown that if C is a bounded closed convex subset of a uniformly convex Banach space X, then every asymptotic pointwise nonexpansive mapping T : C → C always has a fixed point. In 2009, Hussain and Khamsi [2] extended Kirk-Xu′s result to the case of metric spaces, specifically to the so-called CAT(0) spaces. Recently, Kozlowski [3] defined an iterative sequence for an asymptotic pointwise nonexpansive mapping T : C → C by x₁ ∈ C and

(1.2)

where {t_k} and {s_k} are sequences in [0,1] and {n_k} is an increasing sequence of natural numbers. He proved, under some suitable assumptions, that the sequence {x_k} defined by (1.2) converges weakly to a fixed point of T where X is a uniformly convex Banach space which satisfies the Opial condition and {x_k} converges strongly to a fixed point of T provided T^r is a compact mapping for some r ∈ ℕ. On the other hand, Khan et al. [4] studied the iterative process defined by

(1.3)

where T₁, …, T_m are asymptotically quasi-nonexpansive mappings on C and

are sequences in [0,1] for all i = 1,2, …, m.

In this paper, motivated by the results mentioned above, we ensure the existence of common fixed points for a family of asymptotic pointwise nonexpansive mappings in a CAT(0) space. Furthermore, we obtain ▵ and strong convergence theorems of a sequence defined by

(1.4)

where T₁, … , T_m are asymptotic pointwise nonexpansive mappings on a subset C of a complete CAT(0) space and

are sequences in [0,1] for all i = 1,2, … , m, and {n_k} is an increasing sequence of natural numbers. We also note that our method can be used to prove the analogous results for uniformly convex Banach spaces.

2. Preliminaries

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 [5]), ℝ-trees (see [6]), Euclidean buildings (see [7]), and the complex Hilbert ball with a hyperbolic metric (see [8]). For a thorough discussion of these spaces and of the fundamental role they play in geometry, we refer the reader to Bridson and Haefliger [5].

Fixed point theory in CAT(0) spaces was first studied by Kirk (see [9, 10]). He showed that every nonexpansive (single-valued) mapping defined on a bounded closed convex subset of a complete CAT(0) space always has a fixed point. Since then the fixed point theory for single-valued and multivalued mappings in CAT(0) spaces has been rapidly developed, and many papers have appeared (see, e.g., [2, 11–22] and the references therein). It is worth mentioning that fixed point theorems in CAT(0) spaces (specially in ℝ-trees) can be applied to graph theory, biology, and computer science (see, e.g., [6, 23–26]).

Let (X, d) be a metric space. A geodesic path joining x ∈ X to y ∈ X (or, more briefly, a geodesic from x to y) is a map c from a closed interval [0, l] ⊂ ℝ to X such that c(0) = x, c(l) = y, and d(c(t), c(t^′)) = |t − t^′| for all t, t^′ ∈ [0, l]. In particular, c is an isometry and d(x, y) = l. The image α of c is called a geodesic (or metric) segment joining x and y. When it is unique, this geodesic is denoted by [x, y]. The space (X, d) is said to be a geodesic space if every two points of X are joined by a geodesic, and X is said to be uniquely geodesic if there is exactly one geodesic joining x and y for each x, y ∈ X. A subset Y ⊂ X is said to be convex if Y includes every geodesic segment joining any two of its points.

A geodesic triangle ▵(x₁, x₂, x₃) in a geodesic space (X, d) consists of three points x₁, x₂, x₃ in X (the vertices of ▵) and a geodesic segment between each pair of vertices (the edges of ▵). A comparison triangle for geodesic triangle ▵(x₁, x₂, x₃) in (X, d) is a triangle in the Euclidean plane 𝔼² such that for i, j ∈ {1,2, 3}.

A geodesic space is said to be a CAT(0) space if all geodesic triangles satisfy the following comparison axiom.

CAT(0): Let ▵ be a geodesic triangle in X, and let

be a comparison triangle for ▵. Then, ▵ is said to satisfy the CAT(0) inequality if for all x, y ∈ ▵ and all comparison points

(2.1)

Let x, y ∈ X, by Lemma 2.1(iv) of [14] for each t ∈ [0,1], there exists a unique point z ∈ [x, y] such that

(2.2)

We will use the notation (1 − t)x ⊕ ty for the unique point z satisfying (2.2). We now collect some elementary facts about CAT(0) spaces.

Lemma 2.1. Let X be a complete CAT(0) space.

(i) [5, Proposition 2.4] If C is a nonempty closed convex subset of X, then, for every x ∈ X, there exists a unique point P(x) ∈ C such that d(x, P(x)) = inf {d(x, y) : y ∈ C}. Moreover, the map x ↦ P(x) is a nonexpansive retract from X onto C.
(ii) [14, Lemma 2.4] For x, y, z ∈ X and t ∈ [0,1], we have
(2.3)
(iii) [14, Lemma 2.5] For x, y, z ∈ X and t ∈ [0,1], we have
(2.4)

We now give the concept of Δ-convergence and collect some of its basic properties. Let {x_n} be a bounded sequence in a CAT(0) space X. For x ∈ X, we set

(2.5)

The asymptotic radius r({x_n}) of {x_n} is given by

(2.6)

and the asymptotic center A({x_n}) of {x_n} is the set

(2.7)

It is known from Proposition 7 of [27] that, in a CAT(0) space, A({x_n}) consists of exactly one point.

Definition 2.2 (see [28], [29].)A sequence {x_n} in a CAT(0) space X is said to Δ-converge to x ∈ X if x is the unique asymptotic center of {u_n} for every subsequence {u_n} of {x_n}. In this case, we write and call x the Δ-limit of {x_n}.

Lemma 2.3. Let X be a complete CAT(0) space.

(i)
[28, page 3690] Every bounded sequence in X has a Δ-convergent subsequence.
(ii)
[30, Proposition 2.1] If C is a closed convex subset of a complete CAT(0) space and if {x_n} is a bounded sequence in C, then the asymptotic center of {x_n} is in C.
(iii)
[14, Lemma 2.8] If {x_n} is a bounded sequence in a complete CAT(0) space with A({x_n}) = {x} and {u_n} is a subsequence of {x_n} with A({u_n}) = {u} and the sequence {d(x_n, u)} converges, then x = u.

Recall that a mapping T : X → X is said to be nonexpansive [31] if

(2.8)

where T is called asymptotically nonexpansive [32] if there is a sequence {k_n} of positive numbers with the property lim _n→∞ k_n = 1 and such that

(2.9)

where T is called an asymptotic pointwise nonexpansive mapping [1] if there exists a sequence of functions α_n : X → [0, ∞) such that

(2.10)

where lim sup _n→∞ α_n(x) ≤ 1. The following implications hold.

(2.11)

A point x ∈ X is called a fixed point of T if x = Tx. We shall denote by F(T) the set of fixed points of T. The existence of fixed points for asymptotic pointwise nonexpansive mappings in CAT(0) spaces was proved by Hussain and Khamsi [2] as the following result.

Theorem 2.4. Let C be a nonempty bounded closed convex subset of a complete CAT(0) space X. Suppose that T : C → C is an asymptotic pointwise nonexpansive mapping. Then, F(T) is nonempty closed and convex.

3. Existence Theorems

Let M be a metric space and ℱ a family of subsets of M. Then, we say that ℱ defines a convexity structure on M if it contains the closed balls and is stable by intersection.

Definition 3.1 (see [2].)Let ℱ be a convexity structure on M. We will say that ℱ is compact if any family of elements of ℱ has a nonempty intersection provided ⋂_α∈FA_α ≠ ∅ for any finite subset F ⊂ Γ.

Let X be a complete CAT(0) space. We denote by 𝒞(X) the family of all closed convex subsets of X. Then, 𝒞(X) is a compact convexity structure on X (see, e.g., [2]).

The following theorem is an extension of Theorem 4.3 in [33]. For an analog of this result in uniformly convex Banach spaces, see [34].

Theorem 3.2. Let C be a nonempty bounded closed and convex subset of a complete CAT(0) space X. Then, for any commuting family 𝒮 of asymptotic pointwise nonexpansive mappings on C, the set ℱ(𝒮) of common fixed points of 𝒮 is nonempty closed and convex.

Proof. Let 𝒯 be the family of all finite intersections of the fixed point sets of mappings in the commutative family 𝒮. We first show that 𝒯 has the finite intersection property. Let T₁, T₂, … , T_n ∈ 𝒮. By Theorem 2.4, F(T₁) is a nonempty closed and convex subset of C. We assume that is nonempty closed and convex for some k ∈ ℕ with 1 < k ≤ n. For x ∈ A and j ∈ ℕ with 1 ≤ j < k, we have

(3.1)

Thus, T_k(x) is a fixed point of T_j, which implies that T_k(x) ∈ A; therefore, A is invariant under T_k. Again, by Theorem 2.4, T_k has a fixed point in A, that is,

(3.2)

By induction,

. Hence, 𝒯 has the finite intersection property. Since 𝒞(X) is compact,

(3.3)

Obviously, the set is closed and convex.

As a consequence of Lemma 2.1(i) and Theorem 3.2, we obtain an analog of Bruck′s theorem [35].

Corollary 3.3. Let C be a nonempty bounded closed and convex subset of a complete CAT(0) space X. Then, for any commuting family 𝒮 of nonexpansive mappings on C, the set ℱ(𝒮) of common fixed points of 𝒮 is a nonempty nonexpansive retract of C.

4. Convergence Theorems

Throughout this section, X stands for a complete CAT(0) space. Let C be a closed convex subset of X. We shall denote by 𝒯(C) the class of all asymptotic pointwise nonexpansive mappings from C into C. Let T₁, …, T_m ∈ 𝒯(C), without loss of generality, we can assume that there exists a sequence of mappings α_n : C → [0, ∞) such that for all x, y ∈ C, i = 1, …, m, and n ∈ ℕ, we have

(4.1)

Let a_n(x) = max {α_n(x), 1}. Again, without loss of generality, we can assume that

(4.2)

for all x, y ∈ C, i = 1, … , m, and n ∈ ℕ. We define b_n(x) = a_n(x) − 1, then, for each x ∈ C, we have lim _n→∞ b_n(x) = 0.

The following definition is a mild modification of [3, Definition 2.3].

Definition 4.1. Define 𝒯_r(C) as a class of all T ∈ 𝒯(C) such that

(4.3)

a_n is a bounded function for every n ∈ ℕ.

Let T₁, … , T_m ∈ 𝒯_r(C), and let be bounded away from 0 and 1 for all i = 1,2, … , m, and {n_k} an increasing sequence of natural numbers. Let x₁ ∈ C, and define a sequence {x_k} in C as

(4.4)

We say that the sequence {x_k} in (4.4) is well defined if

. As in [3], we observe that lim _k→∞ a_k(x) = 1 for every x ∈ C. Hence, we can always choose a subsequence

which makes {x_k} well defined.

Lemma 4.2 (see [36], Lemma 2.2.)Let {a_n} and {u_n} be sequences of nonnegative real numbers satisfying

(4.5)

Then, (i) lim _n a_n exists, (ii) if lim inf _n a_n = 0, then lim _n a_n = 0.

Lemma 4.3 (see [37], [38].)Suppose {t_n} is a sequence in [b, c] for some b, c ∈ (0,1) and {u_n}, {v_n} are sequences in X such that lim sup _n d(u_n, w) ≤ r, lim sup _n d(v_n, w) ≤ r, and lim _n d((1 − t_n)u_n ⊕ t_nv_n, w) = r for some r ≥ 0. Then,

(4.6)

Lemma 4.4. Let C be a nonempty closed convex subset of X and T₁, … , T_m ∈ 𝒯_r(C). Let and {n_k} ⊂ ℕ be such that {x_k} in (4.4) is well defined. Assume that . Then,

(a)
there exists a sequence {v_k} in [0, ∞) such that and , for all p ∈ F and all k ∈ ℕ,
(b)
there exists a constant M > 0 such that d(x_k+l, p) ≤ Md(x_k, p), for all p ∈ F and k, l ∈ ℕ.

Proof. (a) Let p ∈ F and for all k ∈ ℕ. Since , we have . Now,

(4.7)

Suppose that

holds for some 1 ≤ j ≤ m − 2. Then,

(4.8)

By induction, we have

(4.9)

This implies

(4.10)

This completes the proof of (a).

(b) We observe that (1+α)ⁿ ≤ e^nα holds for all n ∈ ℕ and α ≥ 0. Thus, by (a), for k, l ∈ ℕ, we have

(4.11)

The proof is complete by setting

Theorem 4.5. Let C be a nonempty closed convex subset of X and T₁, …, T_m ∈ 𝒯_r(C). Let and {n_k} ⊂ ℕ be such that {x_k} in (4.4) is well defined. Assume that F ≠ ∅. Then, {x_k} converges to some point in F if and only if lim inf _k→∞ d(x_k, F) = 0, where d(x, F) = inf _p∈F d(x, p).

Proof. The necessity is obvious. Now, we prove the sufficiency. From Lemma 4.4(a), we have

(4.12)

This implies

(4.13)

Since

, then

. By Lemma 4.2(ii), we get lim _k→∞ d(x_k, F) = 0. Next, we show that {x_k} is a Cauchy sequence. From Lemma 4.4(b), there exists M > 0 such that

(4.14)

Since lim _k→∞ d(x_k, F) = 0, for each ɛ > 0, there exists k₁ ∈ ℕ such that

(4.15)

Hence, there exists z₁ ∈ F such that

(4.16)

By (4.14) and (4.16), for k ≥ k₁, we have

(4.17)

This shows that {x_k} is a Cauchy sequence and so converges to some q ∈ C. We next show that q ∈ F. Let L = sup {a₁(x) : x ∈ C}. Then, for each ϵ > 0, there exists k₂ ∈ ℕ such that

(4.18)

Since lim _k→∞ d(x_k, F) = 0, there exists k₃ ≥ k₂ such that

(4.19)

Thus, there exists z₂ ∈ F such that

(4.20)

By (4.18) and (4.20), for each i = 1,2, …, m, we have

(4.21)

Since ϵ is arbitrary, we have T_iq = q for all i = 1,2, …, m. Hence, q ∈ F.

As an immediate consequence of Theorem 4.5, we obtain the following.

Corollary 4.6. Let C be a nonempty closed convex subset of X and T₁, …, T_m ∈ 𝒯_r(C). Let and {n_k} ⊂ ℕ be such that {x_k} in (4.4) is well defined. Assume that F ≠ ∅. Then, {x_k} converges to a point p ∈ F if and only if there exists a subsequence of {x_k} which converges to p.

Definition 4.7. A strictly increasing sequence {n_k} ⊂ ℕ is called quasiperiodic [39] if the sequence {n_k+1 − n_k} is bounded or equivalently if there exists a number p ∈ ℕ such that any block of p consecutive natural numbers must contain a term of the sequence {n_k}. The smallest of such numbers p will be called a quasiperiod of {n_k}.

Lemma 4.8. Let C be a nonempty closed convex subset of X and T₁, …, T_m ∈ 𝒯_r(C). Let for some δ ∈ (0, 1/2) and {n_k} ⊂ ℕ be such that {x_k} in (4.4) is well defined. Then,

(i)
lim _k→∞ d(x_k, p) exists for all p ∈ F,
(ii)
, for all j = 1,2, …, m,
(iii)
if the set 𝒥 = {k ∈ ℕ : n_k+1 = 1 + n_k} is quasiperiodic, then lim _k→∞ d(x_k, T_jx_k) = 0, for all j = 1,2, …, m.

Proof. (i) Follows from Lemmas 4.2(i) and 4.4(a).

(ii) Let p ∈ F, then, by (i), we have lim _k→∞ d(x_k, p) exists. Let

(4.22)

By (4.9) and (4.22), we get that

(4.23)

Note that

(4.24)

Thus,

(4.25)

so that

(4.26)

From (4.23) and (4.26), we have

(4.27)

That is

(4.28)

for each j = 1,2, …, m − 1.

We also obtain from (4.23) that

(4.29)

By Lemma 4.3, we get that

(4.30)

For the case j = m, by (4.1), we have

(4.31)

But since lim _k→∞d(x_k, p) = c, then

(4.32)

Moreover,

(4.33)

Again, by Lemma 4.3, we get that

(4.34)

Thus, (4.30) and (4.34) imply that

(4.35)

(iii) For j = 1, from (ii), we have

(4.36)

If j = 2,3, …, m, then we have

(4.37)

By (ii) and

, we get

(4.38)

By (4.36) and (4.38), we have

(4.39)

By the construction of the sequence {x_k}, we have from (4.35) that

(4.40)

Next, we show that

(4.41)

It is enough to prove that d(T_jx_k, x_k) → 0 as k → ∞ though 𝒥. Indeed, let p be a quasiperiod of 𝒥, and let ɛ > 0 be given. Then, there exists N₁ ∈ ℕ such that

(4.42)

By the quasiperiodicity of 𝒥, for each l ∈ ℕ, there exists i_l ∈ 𝒥 such that |l − i_l | ≤ p. Without loss of generality, we can assume that l ≤ i_l ≤ l + p (the proof for the other case is identical). Let M = sup {a₁(x) : x ∈ C}. Then, M ≥ 1. Since lim _l→∞ d(x_l+1, x_l) = 0 by (4.40), there exists N₂ ∈ ℕ such that

(4.43)

This implies that

(4.44)

By the definition of T, we have

(4.45)

Let N = max {N₁, N₂}. Then, for l ≥ N, we have from (4.42), (4.44), and (4.45) that

(4.46)

To prove that d(T_jx_k, x_k) → 0 as k → ∞ though 𝒥. Since 𝒥 = {k ∈ ℕ : n_k+1 = n_k + 1} is quasiperiodic, for each k ∈ 𝒥, we have

(4.47)

From this, together with (4.39) and (4.40), we can obtain that d(T_jx_k, x_k) → 0 as k → ∞ through 𝒥.

The following lemmas can be found in [3] (see also [2]).

Lemma 4.9. Let C be a nonempty closed convex subset of X, and let T ∈ 𝒯_r(C). If lim _n→∞ d(x_n, Tx_n) = 0, then lim _n→∞ d(x_n, T^lx_n) = 0 for every l ∈ ℕ.

Lemma 4.10. Let C be a nonempty closed convex subset of X, and let T ∈ 𝒯_r(C). Suppose {x_n} is a bounded sequence in C such that lim _n d(x_n, Tx_n) = 0 and Δ-lim _n x_n = w. Then, Tw = w.

By using Lemmas 2.3 and 4.10, we can obtain the following result. We omit the proof because it is similar to the one given in [38].

Lemma 4.11. Let C be a closed convex subset of X, and let T : C → C be an asymptotic pointwise nonexpansive mapping. Suppose {x_n} is a bounded sequence in C such that lim _n d(x_n, T(x_n)) = 0 and d(x_n, v) converges for each v ∈ F(T), then ω_w(x_n) ⊂ F(T). Here, ω_w(x_n) = ⋃A({u_n}) where the union is taken over all subsequences {u_n} of {x_n}. Moreover, ω_w(x_n) consists of exactly one point.

Now, we are ready to prove our Δ-convergence theorem.

Theorem 4.12. Let C be a nonempty closed convex subset of X and T₁, …, T_m ∈ 𝒯_r(C). Let for some δ ∈ (0, 1/2) and {n_k} ⊂ ℕ be such that {x_k} in (4.4) is well defined. Suppose that and the set 𝒥 = {k ∈ ℕ : n_k+1 = 1 + n_k} is quasiperiodic. Then, {x_k}Δ-converges to a common fixed point of the family {T_i : i = 1,2, …, m}.

Proof. Let p ∈ F, by Lemma 4.8, lim _k→∞ d(x_k, p) existsm and hence {x_k} is bounded. Since lim _k→∞ d(x_k, T_jx_k) = 0 for all j = 1,2, …, m, then by Lemma 4.11 ω_w(x_k) ⊂ F(T_j) for all j = 1,2, …, m, and hence . Since ω_w(x_n) consists of exactly one point, then {x_k}Δ-converges to an element of F.

Before proving our strong convergence theorem, we recall that a mapping T : C → C is said to be semicompact if C is closed and, for any bounded sequence {x_n} in C with lim _n→∞ d(x_n, Tx_n) = 0, there exists a subsequence of {x_n} and x ∈ C such that .

Theorem 4.13. Let C be a nonempty closed convex subset of X and T₁, …, T_m ∈ 𝒯_r(C) such that is semicompact for some i ∈ {1, …, m} and l ∈ ℕ. Let for some δ ∈ (0, 1/2) and {n_k} ⊂ ℕ be such that {x_k} in (4.4) is well defined. Suppose that and the set 𝒥 = {k ∈ ℕ : n_k+1 = 1 + n_k} is quasiperiodic. Then, {x_k} converges to a common fixed point of the family {T_i : i = 1,2, …, m}.

Proof. By Lemma 4.8, we have

(4.48)

Let i ∈ {1, …, m} be such that

is semicompact. Thus, by Lemma 4.9,

(4.49)

We can also find a subsequence

of {x_k} such that

. Hence, from (4.48), we have

(4.50)

Thus, q ∈ F, and, by Corollary 4.6, {x_k} converges to q. This completes the proof.

5. Concluding Remarks

One may observe that our method can be used to obtain the analogous results for uniformly convex Banach spaces. Let C be a nonempty closed convex subset of a Banach space X and fix x₁ ∈ C. Define a sequence {x_k} in C as

(5.1)

where T₁, …, T_m ∈ 𝒯_r(C),

are sequences in [0,1] for all i = 1,2, …, m, and {n_k} is an increasing sequence of natural numbers.

Theorem 5.1. Let X be a uniformly convex Banach space with the Opial property, and let C be a nonempty closed convex subset of X. Let T₁, …, T_m ∈ 𝒯_r(C), for some δ ∈ (0, 1/2), and let {n_k} ⊂ ℕ be such that {x_k} in (5.1) is well defined. Suppose that and the set 𝒥 = {k ∈ ℕ : n_k+1 = 1 + n_k} is quasiperiodic. Then, {x_k} converges weakly to a common fixed point of the family {T_i : i = 1,2, …, m}.

Theorem 5.2. Let C be a nonempty closed convex subset of a uniformly convex Banach space X and T₁, …, T_m ∈ 𝒯_r(C) such that is semicompact for some i ∈ {1, …, m} and l ∈ ℕ. Let for some δ ∈ (0, 1/2), and let {n_k} ⊂ ℕ be such that {x_k} in (5.1) is well defined. Suppose that and the set 𝒥 = {k ∈ ℕ : n_k+1 = 1 + n_k} is quasiperiodic. Then, {x_k} converges strongly to a common fixed point of the family {T_i : i = 1,2, …, m}.

Acknowledgments

This research is (partially) supported by the Centre of Excellence in Mathematics, the Commission on Higher Education, Thailand. The first author also thanks the Graduate School of Chiang Mai University, Thailand.

References

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Common Fixed Points for Asymptotic Pointwise Nonexpansive Mappings in Metric and Banach Spaces

Abstract

1. Introduction

2. Preliminaries

3. Existence Theorems

4. Convergence Theorems

5. Concluding Remarks

Acknowledgments

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Common Fixed Points for Asymptotic Pointwise Nonexpansive Mappings in Metric and Banach Spaces

Abstract

1. Introduction

2. Preliminaries

3. Existence Theorems

4. Convergence Theorems

5. Concluding Remarks

Acknowledgments

References

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