### 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 ^{*} in

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 {_{n}

converges to the unique solution ^{*} of _{C}

He also proved that the sequences {_{n}_{n}^{*} of

In 2008, the extragradient algorithm (

Under assumptions that _{n}

For obtaining a common element of set of solutions of Ky Fan inequality (_{n}

where _{n}

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

holds for every _{C}_{C}

Since _{C}_{C}

### Lemma 2.1

_{C}

(i)

(ii) _{C}

### 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) _{1} and _{2} such that

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])

^{*}

^{*}) + _{C}^{*}), _{C}^{*}) ^{*} ∈

### Lemma 2.8.([2, 17])

_{n}_{n}_{n}_{1} ∈

^{*} ∈

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) _{C}

(ii) _{C}

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 _{μ}

We say that _{μ}

From the above definitions, it is clear that:

(i) if _{1});

(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])

_{n}_{i}_{ni} < _{ni+1}

### Lemma 2.17.([23])

### Lemma 2.18.([28])

_{1}, _{2},..., _{N}_{1}, _{2}_{N}

### Lemma 2.19.([26])

_{n}_{n}_{n}_{n}_{n}_{→∞}_{n}

_{n}_{→∞}_{n}

### 3. Main Results

In this section, we show strong convergence theorems for the sequence generated by the hybrid algorithm (

Now, let

(A1)

(A2) _{1} > 0 and _{2} > 0;

(A3)

(A4) _{n}_{n}_{n}_{n}

We are now in a position to prove our main results.

### Theorem 3.1

_{1} ∈ _{n}_{n}_{n}_{n}

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

(C1) {_{n}_{n}_{→∞}_{n}

(C2) _{1}, 2_{2}}

(C3) _{n}_{n}_{→∞}_{n}

(C4) _{n}

(C5) 0 < _{n}_{n}_{n}_{n}_{n}_{n}_{n}_{n}

_{n}

**Proof**

Let _{ℱ} and it easy to see that

This implies that

Since

By Lemma 2.11(ii), _{C}_{n}_{C}_{n}

Let

This shows that _{C}_{n}_{C}_{n}

From (

Consequently,

By induction, we get

This implies that {_{n}_{n}_{n}_{n}_{n}

By Lemma 2.18, (

Consequently, utilizing (

Therefore, we have

In order to prove that _{n}

**Case 1**

Suppose that there exists _{0} such that {||_{n}_{0}. Boundedness of {||_{n}_{n}_{n}_{n}

By (

This implies by conditions (

By similar argument we can obtain that lim

Also, by (

Next, we will show that

where _{ni}} of {_{n}

Without loss of generality, we may assume that _{ni} ⇀ ^{*} as ^{*} ∈ _{ni} − _{ni}|| → 0 as _{ni} ⇀ ^{*}. We will show that ^{*} = ℱ. Assume ^{*} ∉ ^{*} ∈ _{C}_{ni} (^{*} ≠ _{C}_{ni} (^{*}. By Opial’s property, condition (

This is a contradiction. Then ^{*} ∈

Since ^{*} is compact and convex, for all _{ni} ∈ ^{*} such that ||_{ni} − _{ni}|| = dist(_{ni}, ^{*}) and the sequence {_{ni}} has a convergent subsequence {_{nk}} with lim_{k}_{→∞}_{nk} = ^{*}. By condition (E), there exists

Suppose that ^{*}. Since _{ni} ⇀ ^{*}, it follows by the Opial’s condition and (

This is a contradiction. Then ^{*} ∈

Assume ^{*} ∉ ^{*} ∉ _{C}_{n}^{*} ≠ _{C}_{n}^{*}. Now, since _{ni} ⇀ ^{*}, it follows by (

This is a contradiction. Then ^{*} ∈

It follows from Lemma 2.7 and

if and only if

where _{C}_{n}_{n}_{n}_{n}_{n}_{C}_{n}

Using successively the definition of the normal cone to _{n}_{n}_{n}

and

Thus, we have

Hence

Since lim_{i}_{→∞} ||_{ni} − _{ni}|| = 0, we have _{ni} ⇀ ^{*}. Passing to the limit in the inequality (^{*}, ^{*} ∈ ^{*} = ℱ. Since ^{*} = ℱ, it follows that

By using Lemma 2.17 and (

This implies that

Putting _{n}_{→∞}_{n}_{n}

**Case 2**

Assume that there exists a subsequence {_{ni}} of {_{n}

for all _{0}, for some _{0} large enough, such that _{0},

From (_{n}_{→∞} ||_{τ}_{(}_{n}_{)} − _{τ}_{(}_{n}_{)}|| = 0, and similarly we obtain

Following an argument similar to that in

Thus, by Lemma 2.16, we have

Therefore, the sequence {_{n}

Recall that a multi-valued mapping

### Theorem 3.2

_{1} ∈ _{n}_{n}_{n}_{n}_{n}_{n}_{n}_{n}_{n}_{n}_{n}_{n}

(C1) {_{n}_{n}_{→∞}_{n}

(C2) _{1}, 2_{2}}

(C3) _{n}_{n}_{→∞}_{n}

(C4) _{n}

(C5) 0 < _{n}_{n}_{n}_{n}_{n}_{n}_{n}_{n}

_{n}

### 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 _{n}

Also, the solution _{n}

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

_{1} ∈ _{n}_{n}_{n}_{n}

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

(C1) {_{n}_{n}_{→∞}_{n}

(C2) _{1}, 2_{2}}

(C3) _{n}_{n}_{→∞}_{n}

(C4) _{n}

(C5) 0 < _{n}_{n}_{n}_{n}_{n}_{n}_{n}_{n}

_{n}

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

, _{n}_{n}_{1} = 9. The numerical experiment’s results of our iteration for approximating the point 0 are given in Table 1.

### Remark 5.2

Table 1 shows that the sequences {_{n}_{n}_{n}_{n}

### 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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