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 (
The notation
Let
where dist(
A mapping
A point
A mapping
A multivalued mapping
A multivalued mapping
Let
A point
Then,
For every continuous mapping
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
for all
(i)
(ii)
Then the sequence {
In 2010, Sokhuma and Kaewkhao [15] proved the convergence theorem for a common fixed point in Banach spaces as follows.
Let
for all
In 2013, Sokhuma [14] proved the convergence theorem for a common fixed point in CAT(0) spaces as follows.
Let
for all
In 2013, Laowang and Panyanak proved the convergence theorem for a common fixed point in CAT(0) spaces as follows.
Theorem 1.1.([10])
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
In 2016, Uddin and Imdad [17] introduced the following iteration scheme:
Let
where
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
for all
2. Preliminaries
Relevant basic definitions followed previous research results and iterative methods were used frequently.
Let (
A geodesic triangle Δ(
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
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
Let
and
The number
If
The definition of Δ–convergence is presented below.
Definition 2.1.([12, 8])
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])
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])
Lemma 2.8.([11])
The following fact is well-known.
Lemma 2.9
The important property can be found in [18].
Lemma 2.10
3. Main Results
The following lemmas play very important roles in this section.
Lemma 3.1
(1)
(2)
(3)
(1) implies (2). Since
(2) implies (3). Since
(3) implies (1). Since
This implies that Fix(
Lemma 3.2
Let
By the convergence of
By condition
Lemma 3.3
Let
Notice that
Then,
By
Since
Lemma 3.4
Let
and hence
Therefore, by 0 <
Thus,
It follows that
Since
Recall that
Hence,
Thus,
Lemma 3.5
Consider,
Then,
Hence, by Lemmas 3.3 and 3.4,
Lemma 3.6
Consider,
It follows from Lemmas 3.2 – 3.4 that,
Theorem 3.7
Since {
By Corollary 1.6,
It follows that
Theorem 3.8
Since Lemma 3.6 guarantees that {
It follows that
a contradiction, and hence
To show that {
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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