### Article

KYUNGPOOK Math. J. 2019; 59(4): 665-678

**Published online** December 23, 2019

Copyright © Kyungpook Mathematical Journal.

### An Ishikawa Iteration Scheme for two Nonlinear Mappings in CAT(0) Spaces

Kritsana Sokhuma

Department of Mathematics, Faculty of Science and Technology, Phranakhon Rajabhat University, Bangkok 10220, Thailand

e-mail : k_sokhuma@yahoo.co.th

**Received**: December 9, 2017; **Revised**: December 10, 2018; **Accepted**: December 11, 2018

We construct an iteration scheme involving a hybrid pair of mappings, one a single-valued asymptotically nonexpansive mapping

**Keywords**: Ishikawa iteration, CAT(0) spaces, multivalued mapping, asymptotically nonexpansive mapping.

### 1. Introduction

Fixed point theory in CAT(0) spaces was first studied by Kirk [6, 8] who showed that every nonexpansive mapping defined on a bounded closed convex subset of a complete CAT(0) space always has a fixed point. Since then, the existence problem of fixed point and the Δ–convergence problem of iterative sequences to a fixed point for nonexpansive mappings and asymptotically nonexpansive mappings in a CAT(0) space have been extensively developed with many papers published.

Let (^{K}

The notation

Let

where dist(

A mapping

A point

A mapping _{n}

A multivalued mapping

A multivalued mapping

Let _{T}

A point

Then, _{n}_{n}_{→∞}_{n}_{n}

For every continuous mapping ^{K}

The notation Fix(

In 2009, Laokul and Panyanak [9] defined the iterative and proved the Δ–convergence for nonexpansive mapping in CAT(0) spaces as follows:

Let _{n}_{1} ∈

for all _{n}_{n}

(i) _{n}_{n}

(ii) _{n}_{n}

Then the sequence {_{n}

In 2010, Sokhuma and Kaewkhao [15] proved the convergence theorem for a common fixed point in Banach spaces as follows.

Let _{n}_{1} ∈

for all _{n}_{n}_{n}_{n}_{n}_{n}_{n}

In 2013, Sokhuma [14] proved the convergence theorem for a common fixed point in CAT(0) spaces as follows.

Let _{n}_{1} ∈

for all _{n}_{n}_{n}_{n}_{n}_{n}_{n}

In 2013, Laowang and Panyanak proved the convergence theorem for a common fixed point in CAT(0) spaces as follows.

### Theorem 1.1.([10])

_{λ}

In 2015, Akkasriworn and Sokhuma [1] proved the convergence theorem for a common fixed point in a complete CAT(0) space as follows.

### Theorem 1.2

_{n}

_{n}^{n}_{n}_{n}_{n}_{n}

In 2016, Uddin and Imdad [17] introduced the following iteration scheme:

Let _{T}_{n}

where _{0} ∈ _{n}_{T}_{n}_{n}_{n}_{n}

The Ishikawa iteration method was studied with respect to a pair of single-valued asymptotically nonexpansive mapping and a multivalued nonexpansive mapping. It also established the convergence theorem of a sequence from such process in a nonempty bounded closed convex subset of a complete CAT(0) space. A restricted condition (called end-point condition) in Akkasriworn and Sokhuma’s results was removed [1].

Here, an iteration method modifying the above was introduced and called the Ishikawa iteration method

### Definition 1.3

Let _{T}_{1} ∈ _{n}

for all _{n}_{T}^{n}_{n}_{n}_{n}

### 2. Preliminaries

Relevant basic definitions followed previous research results and iterative methods were used frequently.

Let (

A geodesic triangle Δ(_{1}, _{2}, _{3}) in a geodesic metric space (_{1}, _{2}, _{3} in _{1}, _{2}, _{3}) in (_{1}, _{2}, _{3}) := Δ(_{1}, _{2}, _{3}) in the Euclidean plane such that for

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

CAT(0): Let Δ be a geodesic triangle in

If _{1}, _{2} are points in a CAT(0) space and

then the CAT(0) inequality implies that

This is the (CN) inequality of Bruhat and Tits [3]. A geodesic space is a CAT(0) space if and only if it satisfies the (CN) inequality [2].

The following results and methods deal with the concept of asymptotic centres. Let _{n}_{n}

Let

and

The number _{n}

If _{n}_{n}_{n}_{n}

The definition of Δ–convergence is presented below.

### Definition 2.1.([12, 8])

_{n}_{n}_{n}_{n}_{n}

Some elementary facts about CAT(0) spaces which will be used in the proofs of the main results are stated. The following lemma can be found in [4, 5, 8].

### Lemma 2.2.([8])

### Lemma 2.3.([4])

_{n}_{n}

### Lemma 2.4.([5])

(i)

(ii)

The existence of fixed points for asymptotically nonexpansive mappings in CAT(0) spaces was proved by Kirk [7] as the following result.

### Theorem 2.5

### Theorem 2.6.([13])

### Corollary 2.7.([5])

_{n}

### Lemma 2.8.([11])

_{n}_{n}_{n}

The following fact is well-known.

### Lemma 2.9

_{n}

The important property can be found in [18].

### Lemma 2.10

_{n}_{n}

_{n}

### 3. Main Results

The following lemmas play very important roles in this section.

### Lemma 3.1

_{T}

(1)

(2) _{T}_{T}

(3) _{T}_{T}

_{T}

**Proof**

(1) implies (2). Since _{T}_{T}

(2) implies (3). Since _{T}_{T}_{T}

(3) implies (1). Since _{T}_{T}

This implies that Fix(_{T}

### Lemma 3.2

_{T}_{n}

**Proof**

Let _{1} ∈ _{T}

By the convergence of _{n}_{n}_{n}

By condition

### Lemma 3.3

_{T}_{n}

**Proof**

Let _{1} ∈ _{T}

Notice that

Then,

By _{n}_{n}_{n}

Since

### Lemma 3.4

_{T}_{n}

**Proof**

Let _{1} ∈ _{T}

and hence

Therefore, by 0 < _{n}

Thus,

It follows that

Since

Recall that

Hence,

Thus,

### Lemma 3.5

_{T}_{n}

**Proof**

Consider,

Then,

Hence, by Lemmas 3.3 and 3.4,

### Lemma 3.6

_{T}_{n}

**Proof**

Consider,

It follows from Lemmas 3.2 – 3.4 that,

### Theorem 3.7

_{T}_{n}_{n}

**Proof**

Since {_{n}

By Corollary 1.6,

It follows that _{T}

### Theorem 3.8

_{T}_{n}_{n}

**Proof**

Since Lemma 3.6 guarantees that {_{n}_{w}_{n}_{n}_{n}_{n}_{w}_{n}_{n}_{n}_{n}_{n}_{n}

It follows that _{T}

a contradiction, and hence

To show that {_{n}_{w}_{n}_{n}_{n}_{n}_{n}_{n}_{n}

It can complete the proof by showing that

a contradiction, and hence the conclusion follows.

### Acknowledgements

I would like to thank the Institute for Research and Development, Phranakhon Rajabhat University, for financial support.

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