We establish theorems of strong convergence, for the Ishikawa-type (or two step; cf. Ishikawa, 1974) iteration scheme, to a fixed point of a uniformly L-Lipschitzian asymptotically demicontractive mapping and a uniformly L-Lipschitzian hemicontractive mapping in CAT(0) space. Moreover, we will propose some open problems.

1. Introduction

Let (X, d) be a metric space. One of the most interesting aspects of metric fixed point theory is to extend a linear version of known result to the nonlinear case in metric spaces. To achieve this, Takahashi [1] introduced a convex structure in a metric space (X, d). A mapping W : X × X × [0,1] → X is a convex structure in X if

()

for all x, y ∈ X and λ ∈ [0,1]. A metric space together with a convex structure W is known as a convex metric space. A nonempty subset K of a convex metric space is said to be convex if

()

for all x, y ∈ K and λ ∈ [0,1]. In fact, every normed space and its convex subsets are convex metric spaces but the converse is not true, in general (see, [1]).

Example 1 (see [2].)Let X = {(x₁, x₂) ∈ ℝ² : x₁ > 0, x₂ > 0}, for all x = (x₁, x₂), y = (y₁, y₂) ∈ X, and λ ∈ [0,1]. We define a mapping W : X × X × [0,1] → X by

()

and define a metric d : X × X → [0, ∞) by

()

Then we can show that (X, d, W) is a convex metric space, but it is not a normed linear space.

A metric space X is a CAT(0) space (the term is due to Gromov [3] and it is an acronym for E. Cartan, A. D. Aleksandrov, and V. A. Toponogov) 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 (see, e.g., [4], page 159). It is well known that any complete, simply connected Riemannian manifold nonpositive sectional curvature is a CAT(0) space. The precise definition is given below. For a thorough discussion of these spaces and of the fundamental role they play in various branches of mathematics, see Bridson and Haefliger [4] or Burago et al. [5].

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 mapping 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 Δ(x₁, x₂, x₃) is a geodesic metric space (X, d) that consists of three points x₁, x₂, x₃ ∈ X (the vertices of Δ) and is a geodesic segment between each pair of vertices (the edges of Δ). A comparison triangle for the geodesic triangle Δ(x₁, x₂, x₃) in (X, d) is a triangle in ℝ² such that for i, j ∈ {1,2, 3}. Such a triangle always exists (see, [4]).

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

Let Δ be a geodesic triangle in X and let

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

()

Complete CAT(0) spaces are often called Hadamard spaces (see, [6]). If x, y₁, y₂ are points of a CAT(0) space and if y₀ is the midpoint of the segment [y₁, y₂], which we will denote by y₁ ⊕ y₂/2, then the CAT(0) inequality implies

()

This inequality is the (CN) inequality of Bruhat and Tits [7]. In fact, a geodesic space is a CAT(0) space if and only if it satisfies the (CN) inequality (cf. [4], page 163). The previous inequality has been extended by Khamsi and Kirk [8] as

for any α ∈ [0,1] and x, y, z ∈ X. The inequality (CN*) also appeared in [9].

Let us recall that a geodesic metric space is a CAT(0) space if and only if it satisfies the (CN) inequality (see, [4], page 163). Moreover, if X is a CAT(0) metric space and x, y ∈ X, then for any α ∈ [0,1], there exists a unique point αx ⊕ (1 − α)y ∈ [x, y] such that

()

for any z ∈ X and [x, y] = {αx ⊕ (1 − α)y : α ∈ [0,1]}. In view of the previous inequality, CAT(0) space has Takahashi′s convex structure W(x, y, α) = αx ⊕ (1 − α)y. It is easy to see that for any x, y ∈ X and λ ∈ [0,1],

()

As a consequence,

()

Moreover, a subset K of CAT(0) space X is convex if for any x, y ∈ K, we have [x, y] ⊂ K.

Definition 2. Let C be a nonempty subset of a metric space (X, d). Let F(T) denote the fixed point set of T. Let F(T) ≠ ∅.

(1)
A mapping T : C → C is said to be k-strict asymptotically pseudocontractive with sequence {a_n} if lim _n→∞a_n = 1 for some constant k, 0 ≤ k < 1 and
()
for all x, y ∈ C, n ∈ ℕ.
If k = 0, then T is said to be asymptotically nonexpansive with sequence {a_n}, that is,
()
(2)
A mapping T : C → C is said to be asymptotically demicontractive with sequence {a_n} if lim _n→∞a_n = 1 for some constant k, 0 ≤ k < 1, and
()
for all x ∈ C, n ∈ ℕ.
If k = 0, then T is said to be asymptotically quasi-nonexpansive with sequence {a_n}, that is,
()
(3)
A mapping T : C → C is said to be asymptotically pseudocontractive with sequence {a_n} if lim _n→∞a_n = 1 and
()
for all x, y ∈ C, n ∈ ℕ.
(4)
A mapping T : C → C is said to be asymptotically hemicontractive with sequence {a_n} if lim _n→∞a_n = 1 and
()
for all x ∈ C, n ∈ ℕ.
(5)
A mapping T : C → C is said to be uniformly L-Lipschitzian if for some constant L > 0,
()
for all n ∈ ℕ.

Liu [10] has proved the convergence of Mann and Ishikawa iterative sequence for uniformly L-Lipschitzian asymptotically demicontractive and hemicontractive mappings in Hilbert space (cf. [11]). The existence of (common) fixed points of one mapping (or two mappings or family of mappings) is not known in many situations. So the approximation of fixed points of one or more nonexpansive, asymptotically nonexpansive, or asymptotically quasi-nonexpansive mappings by various iterations have been extensively studied in Banach spaces, convex metric spaces, CAT(0) spaces, and so on (see, [2, 6, 8, 9, 12–27]).

In this paper, we establish theorems of strong convergence for the Ishikawa-type (or two step, cf. [28]) iteration scheme to a fixed point of a uniformly L-Lipschitzian asymptotically demicontractive mapping and a uniformly L-Lipschitzian asymptotically hemicontractive mapping in CAT(0) space. Moreover, we will propose some open problems.

2. Preliminaries

We introduce the following iteration process.

Let C be a nonempty convex subset of a CAT(0) space (X, d) and let T : C → C be a given mapping. Let x₁ ∈ C be a given point.

Algorithm 3. The sequences {x_n} and {y_n} defined by the iterative process

()

is called an Ishikawa-type iterative sequence (cf. [28]).

If β_n ≡ 0, then Algorithm 3 reduces to the following

Algorithm 4. The sequence {x_n} defined by the iterative process

()

is called a Mann-type iterative sequence (cf. [29]).

Lemma 5 (see [10].)Let sequences {a_n}, {b_n} satisfy that

()

a_n ≥ 0, for all n ∈ ℕ,

is convergent, and {a_n} has a subsequence

converging to 0. Then, we must have

()

3. Convergence Theorems

Lemma 6. Let (X, d) be a CAT(0) space and let C be a nonempty convex subset of X. Let T : C → C be an uniformly L-Lipschitzian mapping and let {α_n}, {β_n} be sequence in [0,1]. Define the iteration scheme {x_n} as Algorithm 3. Then

()

for all n ≥ 1.

Proof. Let C_n = d(x_n, Tⁿx_n). We have

()

From (22), we get

()

From (22) and (23), we get

()

This completes the proof of Lemma 6.

Theorem 7. Let (X, d) be a complete CAT(0) space, let C be a nonempty bounded closed convex subset of X, and let T : C → C be a completely continuous and uniformly L-Lipschitzian and asymptotically demicontractive with sequence {a_n}, a_n ∈ [1, ∞), , ε ≤ α_n ≤ 1 − k − ε, for all n ∈ ℕ and some ε > 0. Given x₀ ∈ C, define the iteration scheme {x_n} by

()

Then {x_n} converges strongly to some fixed point of T.

Proof. Since T is a completely continuous mapping in a bounded closed convex subset C of complete metric space, from Schauder′s theorem, F(T) is nonempty. It follows from (CN*) inequality that

()

for all p ∈ F(T). Since T is a asymptotically demicontractive, we get

()

Since 0 < ε ≤ α_n ≤ 1 − k − ε, we have 1 − k − α_n ≥ ε. Thus,

()

From (27), we have

()

for all p ∈ F(T). Since C is bounded and T is self-mapping in C, there exist some M > 0 so that d²(x_n, p) ≤ M, for all n ∈ ℕ. Since 0 ≤ α_n ≤ 1, it follows from (29) that

()

Therefore,

()

for all m ∈ ℕ. Since

, we get

()

Therefore,

()

Since T is a uniformly L-Lipschitzian, it follows from Lemma 6 that

()

Since {x_n} is a bounded sequence and T is completely continuous, there exist a convergent subsequence

of {Tx_n}. Therefore, from (35), {x_n} has a convergent subsequence

. Let

. It follows from the continuity of T and (35), we have q = Tq. Therefore, {x_n} has a subsequence which converges to the fixed point q of T. Let p = q in the inequality (30). Since

and

, from (30) and Lemma 5, we have

()

Therefore,

()

This completes the proof of Theorem 7.

Corollary 8. Let (X, d) be a complete CAT(0) space, let C be a nonempty bounded closed convex subset of X, and let T : C → C be a completely continuous and uniformly L-Lipschitzian and k-strict asymptotically pseudocontractive with sequence {a_n}, a_n ∈ [1, ∞), , and ε ≤ α_n ≤ 1 − k − ε, for all n ∈ ℕ and some ε > 0. Given x₀ ∈ C, define the iteration scheme {x_n} by

()

Then {x_n} converges strongly to some fixed point of T.

Proof. By Definition 2, T is k-strict asymptotically pseudocontractive; then T must be asymptotically demicontractive. Therefore, Corollary 8 can be proved by using Theorem 7.

Lemma 9. Let (X, d) be a CAT(0) space and let C be a nonempty convex subset of X. Let T : C → C be an uniformly L-Lipschitzian and asymptotically hemicontractive with sequence {a_n}⊂[1, ∞), for all n ∈ ℕ, and F(T) is nonempty. Define the iteration scheme {x_n} as follows:

()

Then the following inequality holds:

()

for all p ∈ F(T).

Proof. It follows from (CN*) inequality that

()

for all p ∈ F(T). Since T is asymptotically hemicontractive, we get

()

From (42) and (44), we have

()

From (CN*) inequality, we have

()

Substituting (45) and (46) into (43), we get

()

From (41) and (47), we obtain

()

Since T is uniformly L-Lipschitzian, we have

()

Substituting (49) into (48), we obtain

()

This completes the proof of Lemma 9.

Lemma 10. Let (X, d) be a CAT(0) space and let C be a nonempty bounded convex subset of X. Let T : C → C be a uniformly L-Lipschitzian and asymptotically hemicontractive with sequence {a_n}⊂[1, ∞), for all n ∈ ℕ and . Let F(T) be nonempty. Given x₁ ∈ C, define the iteration scheme {x_n} by

()

If ε ≤ α_n ≤ β_n ≤ b for some ε > 0 and

, then

()

Proof. First, we will prove lim _n→∞d(x_n, Tⁿx_n) = 0. From Lemma 9 and 0 ≤ α_n ≤ β_n, we have

()

Thus

()

Since

, we have lim _n→∞(a_n − 1) = 0. Hence, {a_n} is bounded. By boundedness of C and 0 ≤ α_n ≤ β_n ≤ 1, we obtain that {α_n(1 + a_nβ_n)d²(x_n, p)} is bounded. Therefore, there exists a constant M > 0 such that

()

From (54) and (55), we get

()

Let D = 1 − 2b − L²b² > 0. Since lim _n→∞a_n = 1, there exists N ∈ ℕ such that

()

for all n ≥ N. Suppose that lim _n→∞d(x_n, Tⁿx_n) ≠ 0, then there exist a ε₀ > 0 and a subsequence

of {x_n} such that

()

Without loss of generality, we let n₁ ≥ N. From (56), we have

()

From (57)–(60) and ε ≤ α_n ≤ β_n, we obtain

()

Since

and the boundedness of C, the right side of (61) is bounded. However, if we have i → ∞, then the left side of (61) is unbounded. This is a contradiction. Therefore,

()

Since T is a uniformly L-Lipschitzian, from Lemma 6, we get

()

This completes the proof of Lemma 10.

Theorem 11. Let (X, d) be a complete CAT(0) space, let C be a nonempty bounded closed convex subset of X, and let T : C → C be a completely continuous and uniformly L-Lipschitzian and asymptotically hemicontractive with sequence {a_n}⊂[1, ∞) satisfying , for all n ∈ ℕ. Given x₁ ∈ C, define the iterative scheme {x_n} by

()

If {α_n}, {β_n}⊂[0,1] with ε ≤ α_n ≤ β_n ≤ b for some ε > 0 and

, then {x_n} converges strongly to some fixed point of T.

Proof. Since T is a completely continuous mapping in a bounded closed convex subset C of complete metric space, from Schauder′s theorem, F(T) is nonempty. Since T is completely continuous, there exist a convergent subset of {Tx_n}. Let

()

Since lim _n→∞d(x_n, Tx_n) = 0, from Lemma 10, we have

()

On the other hand, from the continuity of T, (66), and Lemma 10, we have

()

This means that p is a fixed point of T. From (55), (57), and α_n ≤ β_n, we obtain Lemma 9 that

()

From (66), there exists a subsequence

of {d²(x_n, p)} which converges to 0. Therefore, from Lemma 5 and (68),

()

Hence,

()

This completes the proof of Theorem 11.

Corollary 12. Let (X, d) be a complete CAT(0) space, let C be a nonempty bounded closed convex subset of X, and let T : C → C be a completely continuous and uniformly L-Lipschitzian and asymptotically pseudocontractive with sequence {a_n}⊂[1, ∞) satisfying , for all n ∈ ℕ. Given x₁ ∈ C, define the iterative scheme {x_n} by

()

If {α_n}, {β_n}⊂[0,1] with ε ≤ α_n ≤ β_n ≤ b for some ε > 0 and

, then {x_n} converges strongly to some fixed point of T.

Proof. By Definition 2, T is an asymptotically pseudocontractive mapping, then T is an asymptotically hemicontractive mapping. Since a_n ∈ [1, ∞), we have . Obviously, . Therefore, Corollary 12 can be proved by using Theorem 11.

4. Some Remarks and Open Problems

Let S be a semigroup. We denote by B(S) the space of all bounded real-valued functions defined on S with supremum norm. For each s ∈ S, we define the left and right translation operators l_s and r_s on B(S) by

()

for each t ∈ S and f ∈ B(S), respectively. Let X be a subspace of B(S) containing 1. An element μ in the dual space X^* of X is said to be a mean on X if ∥μ∥ = μ(1) = 1. For s ∈ S, we can define a point evaluation δ_s by δ_s(f) = f(s) for each f ∈ X. It is well known that μ is mean on X if and only if

()

for each f ∈ X. Each mean on X is the weak*-limit of convex combination of point evaluations.

Let X be a translation invariant subspace of B(S) (i.e., l_sX ⊂ X and r_sX ⊂ X for each s ∈ S) containing 1. Then a mean μ on X is said to be left invariant (resp., right invariant) if

()

for each s ∈ S and f ∈ X. A mean μ on X is said to be invariant if μ is both left and right invariant ([30–34]). X is said to be left (resp., right) amenable if X has a left (resp., right) invariant mean. X is amenable if X is left and right amenable. In this case, we say that the semigroup S is an amenable semigroup (see [35, 36]). Moreover, B(S) is amenable when S is a commutative semigroup or a solvable group. However, the free group or semigroup of two generators is not left or right amenable.

A net {μ_α} of means on X is said to be asymptotically left (resp., right) invariant if

()

for each f ∈ X and s ∈ S, and it is said to be left (resp., right) strongly asymptotically invariant (or strong regular) if

()

for each s ∈ S, where

and

are the adjoint operators of l_s and r_s, respectively. Such nets were first studied by Day in [35] where they were called weak* invariant and norm invariant, respectively.

It is easy to see that if a semigroup S is left (resp., right) amenable, then the semigroup S^′ = S ∪ {e}, where es^′ = s^′e = s^′ for all s^′ ∈ S is also left (resp., right) amenable and conversely.

A semigroup S is called left reversible if any two right ideals of S have nonvoid intersection, that is, aS∩bS ≠ ∅ for a, b ∈ S. In this case, (S, ⪯) is a directed system when the binary relation “⪯” on S is defined by a⪯b if and only if {a} ∪ aS⊇{b} ∪ bS for a, b ∈ S. It is easy to see that t⪯ts for all t, s ∈ S. Further, if t⪯s, then pt⪯ps for all p ∈ S. The class of left reversible semigroup includes all groups and commutative semigroups. If a semigroup S is left amenable, then S is left reversible. But the converse is not true ([31, 37–41]).

Let S be a semigroup and F(T) denote the fixed point set of T. Then ℑ = {T_s : s ∈ S} is called a representation of S if T_e = I and T_st = T_sT_t for each s, t ∈ S. We denote by F(ℑ) the set of common fixed points of {T_s : s ∈ S}, that is,

()

Open Problem 1. It will be interesting to obtain a generalization of both Theorems 7 and 11 to commutative, amenable, and reversible semigroups as in the case of Hilbert spaces or some Banach spaces (cf. [8, 30, 32, 42–45]).

For a real number κ, a CAT(κ) space is defined by a geodesic metric space whose geodesic triangle is sufficiently thinner than the corresponding triangle in a model space with curvature κ.

For κ = 0, the 2-dimensional model space is the Euclidean space ℝ² with the metric induced from the Euclidean norm. For κ > 0, is the 2-dimensional sphere whose metric is length of a minimal great arc joining each two points. For κ < 0, is the 2-dimensional hyperbolic space with the metric defined by a usual hyperbolic distance. For more details about the properties of CAT(κ) spaces, see [4, 46–48].

Open Problem 2. It will be interesting to obtain a generalization of both Theorems 7 and 11 to CAT(κ) space.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The author would like to thank Professor A. T.-M Lau and Professor J. K. Kim for their helpful suggestions. Also, special thanks are due to the referees for their deep insight which improved the presentation of this paper. This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2012R1A1A4A01010526).

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Some Convergence Theorems for Contractive Type Mappings in CAT(0) Spaces

Abstract

1. Introduction

2. Preliminaries

3. Convergence Theorems

4. Some Remarks and Open Problems

Conflict of Interests

Acknowledgments

References

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Some Convergence Theorems for Contractive Type Mappings in CAT(0) Spaces

Abstract

1. Introduction

2. Preliminaries

3. Convergence Theorems

4. Some Remarks and Open Problems

Conflict of Interests

Acknowledgments

References

References

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