Article
KYUNGPOOK Math. J. 2019; 59(4): 703-723
Published online December 23, 2019
Copyright © Kyungpook Mathematical Journal.
Hybrid Algorithms for Ky Fan Inequalities and Common Fixed Points of Demicontractive Single-valued and Quasinonexpansive Multi-valued Mappings
Nawitcha Onjai-uea and Withun Phuengrattana∗
Department of Mathematics, Faculty of Science and Technology, Nakhon Pathom Rajabhat University, Nakhon Pathom 73000, Thailand
e-mail : nawitcha@hotmail.com and withun_ph@yahoo.com
Received: December 30, 2017; Revised: December 2, 2018; Accepted: December 6, 2018
Abstract
In this paper, we consider a common solution of three problems in real Hilbert spaces: the Ky Fan inequality problem, the variational inequality problem and the fixed point problem for demicontractive single-valued and quasi-nonexpansive multi-valued mappings. To find the solution we present a new iterative algorithm and prove a strong convergence theorem under mild conditions. Moreover, we provide a numerical example to illustrate the convergence behavior of the proposed iterative method.
Keywords: demicontractive mappings, quasi-nonexpansive mappings, Ky Fan inequality, variational inequality, Hilbert spaces.
1. Introduction
Let
The set of solutions of problem (
The equilibrium problem which was considered as the Ky Fan inequality is very general in the sense that it includes, as special cases, the optimization problem, the variational inequality problem, the complementarity problem, the saddle point problem, the Nash equilibrium problem in noncooperative games and the Kakutani fixed point problem, etc., see [1, 4, 5, 9, 10, 18] and the references therein. Recently, algorithms for solving the Ky Fan inequality have been studied extensively.
In 2001, Yamada [27] proved that the sequence {
converges to the unique solution
He also proved that the sequences {
In 2008, the extragradient algorithm (
Under assumptions that
For obtaining a common element of set of solutions of Ky Fan inequality (
where
Later in 2013, Vahidi et al. [24] introduced an iterative algorithm for finding a common element of the sets of fixed points for nonexpansive multi-valued mappings, strict pseudo-contractive single-valued mappings and the set of solutions of Ky Fan inequality for pseudomonotone and Lipschitz-type continuous bifunctions in Hilbert spaces.
In this paper, motivated by the research described above, we propose a new iterative algorithm for finding a common element of the sets of fixed points for demicontractive single-valued mappings, quasi-nonexpansive multi-valued mappings, the set of solutions of Ky Fan inequality for pseudomonotone and Lipschitz-type continuous bifunctions, and the set of solutions of variational inequality for
2. Preliminaries and Useful Lemmas
In this section, we recall some definitions and results for further use. Let
holds for every
Since
Lemma 2.1
(i)
(ii)
Lemma 2.2.([23])
Definition 2.3.([13])
A mapping
We now give some concepts of the monotonicity of a bifunction.
Definition 2.4
Let
(i)
(ii)
(iii)
(iv)
From the definition above we obviously have the following implications: (1) It is clear that (i) ⇒ (ii) ⇒ (iii), (2) If
Definition 2.5
Let
Definition 2.6
Let
Lemma 2.7.([6])
Lemma 2.8.([2, 17])
A mapping
where
Recall that a single-valued mapping
We call
We now give two examples for the class of demicontractive mappings.
Example 2.9
Let
Obviously,
Example 2.10
Let
Obviously,
The following lemma obtained by Suantai and Phuengrattana [22] is useful for our results.
Lemma 2.11
(i)
(ii)
The set
It is clear that every nonempty closed convex subset of a real Hilbert space is proximinal. We denote by
Let
Definition 2.12
A multi-valued mapping
(i) be
(ii) be
(iii) satisfy
We say that
From the above definitions, it is clear that:
(i) if
(ii) if
We now give an example for the class of quasi-nonexpansiveness multi-valued mapping satisfying the condition (E).
Example 2.13
Let
Then
Although the condition (E) implies the quasi-nonexpansiveness for single-valued mappings, but it is not true for multi-valued mappings as the following example.
Example 2.14.([25])
Let
Then
Notice also that the classes of (multi-valued) quasi-nonexpansive mappings and mappings satisfying condition (E) are different (see Examples 2.15).
Example 2.15.([8])
Let
Then
Lemma 2.16.([16])
Lemma 2.17.([23])
Lemma 2.18.([28])
Lemma 2.19.([26])
3. Main Results
In this section, we show strong convergence theorems for the sequence generated by the hybrid algorithm (
Now, let
(A1)
(A2)
(A3)
(A4)
We are now in a position to prove our main results.
Theorem 3.1
(C1) {
(C2)
(C3)
(C4)
(C5) 0 <
Let
This implies that
Since
By Lemma 2.11(ii),
Let
This shows that
From (
Consequently,
By induction, we get
This implies that {
By Lemma 2.18, (
Consequently, utilizing (
Therefore, we have
In order to prove that
Suppose that there exists
By (
This implies by conditions (
By similar argument we can obtain that lim
Also, by (
Next, we will show that
where
Without loss of generality, we may assume that
This is a contradiction. Then
Since
Suppose that
This is a contradiction. Then
Assume
This is a contradiction. Then
It follows from Lemma 2.7 and
if and only if
where
Using successively the definition of the normal cone to
and
Thus, we have
Hence
Since lim
By using Lemma 2.17 and (
This implies that
Putting
Assume that there exists a subsequence {
for all
From (
Following an argument similar to that in
Thus, by Lemma 2.16, we have
Therefore, the sequence {
Recall that a multi-valued mapping
Theorem 3.2
(C1) {
(C2)
(C3)
(C4)
(C5) 0 <
Remark 3.3
(1) Theorems 3.1 and 3.2 extends based on the work of Anh [3] and Vahidi et al. [24], that is, we present a hybrid algorithm for finding a common element of the sets of fixed points for demicontractive single-valued mappings, quasi-nonexpansive multi-valued mappings, the set of solutions of an equilibrium problem for a pseudomonotone, Lipschitz-type continuous bifunctions and variational inequality for
(2) It is know that the class of demicontractive single-valued mappings contains the classes of nonexpansive single-valued mappings, nonspreading singlevalued mappings, quasi-nonexpansive single-valued mappings, and strictly pseudononspreading single-valued mappings. Thus, Theorems 3.1 and 3.2 can be applied to these classes of mappings.
4. Application to Variational Inequalities
In this section, we discuss about an application of Theorem 3.1 to finding a common element of the set of fixed points for demicontractive single-valued mappings and quasi-nonexpansive multi-valued mappings and the set of solutions of variational inequalities for
We consider the particular Ky Fan inequality, corresponding to the bifunction
Also, the solution
Let
Therefore,
Now, using Theorem 3.1, we obtain the following strong convergence theorem for finding a common element of the set of common fixed points of a quasi-nonexpansive multi-valued mapping and a demicontractive single-valued mapping and the solution set of two variational inequalities.
Theorem 4.1
(C1) {
(C2)
(C3)
(C4)
(C5) 0 <
5. Numerical Example
In this section, we give an example which shows numerical experiment for supporting our main results.
Example 5.1
Let
For each
,
Remark 5.2
Table 1 shows that the sequences {
Acknowledgements
The authors are thankful to the referees for careful reading and the useful comments and suggestions. The first author would like to thank the Research and Development Institute, Nakhon Pathom Rajabhat University, Thailand for financial support.
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