KYUNGPOOK Math. J. 2019; 59(4): 703-723
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
* Corresponding Author.
Received: December 30, 2017; Revised: December 2, 2018; Accepted: December 6, 2018; Published online: December 23, 2019.
© Kyungpook Mathematical Journal. All rights reserved.

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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 H be a real Hilbert space with inner product ⟨·, ·⟩ and norm ||·||, and let C be a nonempty closed convex subset of H. Let f : H × H → ℝ be a bifunction such that f(x, x) = 0 for all xC. The classical Ky Fan inequality [7] consists of finding a point x* in C such that

$f(x*,y)≥0, ∀y∈C.$

The set of solutions of problem (1.1) is denoted by Sol(f,C). In fact, the Ky Fan inequality can be formulated as an equilibrium problem. If f(x, y) = ⟨Ax, yx⟩, where A : CH is a operator, then problem (1.1) become the following variational inequality problem (shortly, VI(A,C)): find x*C such that

$⟨Ax*,y-x*⟩≥0, ∀y∈C.$

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 {xn} generated by the projected gradient algorithm

${x1∈C,xn+1=PC(xn-λAxn), ∀n∈ℕ,$

converges to the unique solution x* of VI(A,C) under the assumption that A is strongly monotone and Lipschitz continuous, the mapping PC(IλA) is strictly contractive over C. If A is monotone and Lipschitz, the projected gradient algorithm (1.3) may not be convergent. In order to deal with this situation, Korpelevich [15] introduced an extragradient algorithm:

${x1∈C,yn=PC(xn-λAxn),xn+1=PC(xn-λAyn), ∀n∈ℕ.$

He also proved that the sequences {xn} and {yn} converge to the same solution x* of VI(A,C) under the assumptions that A is L-Lipschitz and monotone, $λ∈(0,1L)$.

In 2008, the extragradient algorithm (1.4) has been extended to Ky Fan inequality problem by Muu et al. [17] as follows:

${x1∈C,yn=argminw∈C [λf(xn,w)+12‖w-xn‖2],xn+1=argminz∈C [λf(yn,z)+12‖z-xn‖2], ∀n∈ℕ.$

Under assumptions that f is pseudomonotone and Lipschitz-type continuous, the authors showed that the sequence {xn} converges to an element of Sol(f,C).

For obtaining a common element of set of solutions of Ky Fan inequality (1.1) and the set of fixed points of a nonexpansive mapping T in a real Hilbert space H, Anh [3] introduced an iterative algorithm by the modified viscosity approximation method. The sequence {xn} is defined by

${x1∈C,yn=argminw∈C [λnf(xn,w)+12‖w-xn‖2],zn=argminz∈C [λnf(yn,z)+12‖z-xn‖2],xn+1=αnh(xn)+βnxn+γn(μTxn+(1-μ)zn), ∀n∈ℕ,$

where C is a nonempty closed convex subset of H and h is a contractive mapping of C into itself. The author showed that under certain conditions, the sequence {xn} converges strongly to an element of Sol(f,C) ∩ F(T).

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 φ-inverse strongly monotone mappings in real Hilbert spaces. We obtain strong convergence theorems for the sequence generated by the proposed algorithm in a real Hilbert space. Our results generalize and improve a number of known results including the results of Anh [3] and Vahidi et al. [24].

2. Preliminaries and Useful Lemmas

In this section, we recall some definitions and results for further use. Let C be a nonempty closed convex subset of a real Hilbert space H. We denote the strong convergence and the weak convergence of the sequence {xn} to a point xH by xnx and xnx, respectively. It is also known in [19] that a Hilbert space H satisfies Opial’s condition, that is, for any sequence {xn} with xnx, the inequality

$lim supn→∞‖xn-x‖

holds for every yH with yx. Let PC be the metric projection of H onto C i.e., for xH, PCx satisfies the property

$‖x-PCx‖ =miny∈C‖x-y‖.$

Since C is nonempty closed and convex, PCx exists and is unique. It is also known that PC has the following characteristic properties, see [11, 23] for more details.

### Lemma 2.1

Let C be a nonempty closed convex subset of a real Hilbert space H and let PC : HC be the metric projection. Then

(i) for all xC, yH, $‖x-PCy‖2+‖PCy-y‖2≤ ‖x-y‖2;$

(ii) PCx = y if and only if there holds the inequality$⟨x-y,y-z⟩≥0, ∀z∈C.$

### Lemma 2.2.([23])

Let C be a nonempty closed convex subset of a Hilbert space H and let A be a mapping of C into H. Let uC. Then for η > 0,

$u=PC(I-ηA)u⇔u∈VI(A,C).$

### Definition 2.3.([13])

A mapping A : CH is called δ-inverse strongly monotone if there exists a positive real number δ such that

$⟨x-y,Ax-Ay⟩≥δ‖Ax-Ay‖2, ∀x,y∈C.$

We now give some concepts of the monotonicity of a bifunction.

### Definition 2.4

Let H be a real Hilbert space, C be a nonempty closed convex subset of H, and let f : H × H → ℝ be a bifunction. A bifunction f is said to be:

(i) strongly monotone on C if there exists a constant α > 0 such that $f(x,y)+f(y,x)≤-α‖x-y‖2, ∀x,y∈C;$

(ii) monotone on C if $f(x,y)+f(y,x)≤0, ∀x,y∈C;$

(iii) pseudomonotone on C if $f(x,y)≥0⇒f(y,x)≤0, ∀x,y∈C;$

(iv) Lipschitz-type continuous on C if there exist two positive constants c1 and c2 such that $f(x,y)+f(y,z)≥f(x,z)-c1‖x-y‖2-c2‖y-z‖2, ∀x,y∈C.$

From the definition above we obviously have the following implications: (1) It is clear that (i) ⇒ (ii) ⇒ (iii), (2) If f(x, y) = ⟨Φ(x), yx⟩ for a mapping Φ : HH. Then the notions of monotonicity of bifunction f collapse to the notions of monotonicity of mapping Φ, respectively. In addition, if mapping Φ is L-Lipschitz on C, i.e., ||Φ(x) − Φ(y)|| ≤ L||xy|| for all x, yC. Then, f is also Lipschitz-type continuous on C, for example, with constants $L1=L2ɛ,L2=Lɛ2$, for any ε > 0.

### Definition 2.5

Let H be a real Hilbert space, and let f : H × H → ℝ be a bifunction. For each zH, by ∂f(z, u) we denote the subdifferential of the function f(z, ·) at u, i.e.,

$∂f(z,u)={ξ∈H:f(z,t)-f(z,u)≥⟨ξ,t-u⟩,∀t∈H}.$

### Definition 2.6

Let H be a real Hilbert space and C be a nonempty closed convex subset of H. The normal cone of C at vC is defined by

$NC(v)={z∈H:⟨z,y-v⟩≤0,∀y∈C}.$

### Lemma 2.7.([6])

Let H be a real Hilbert space, C be a nonempty closed convex subset of H, and f : H × H → ℝ be a bifunction. For each zH, suppose that f(z, ·) is subdifferentiable on C. Then x*is a solution to the following convex problem:

$min{f(z,x):x∈C}$

if and only if 0 ∈ ∂f(z, x*) + NC(x*), where f(z, ·) denotes the subdifferential of f(z, ·) and NC(x*) is the normal cone of C at x*C.

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

Let H be a real Hilbert space, C be a nonempty closed convex subset of H, and let f : H × H → ℝ be a pseudomonotone and Lipschitz-type continuous bifunction. For each xC, let f(x, ·) be convex and subdifferentiable on C. Let {xn}, {zn}, and {wn} be the sequences generated by x1C and by

$wn=argminw∈C [λnf(xn,w)+12‖w-xn‖2],zn=argminz∈C [λnf(wn,z)+12‖z-xn‖2].$

Then for each x*Sol(f,C),

$‖zn-x*‖2≤ ‖xn-x*‖2-(1-2λnc1)‖xn-wn‖2-(1-2λnc2)‖wn-zn‖2, ∀n∈ℕ.$

A mapping h : CC is a contraction if there exists a constant η ∈ (0, 1) such that ||h(x) − h(y)||≤ η||xy|| for all x, yC. Let T : CC be a single-valued mapping. An element xC is said to be a fixed point of T if x = Tx. The fixed point set of T is denoted by F(T) = {xC : x = Tx}. A single-valued mapping T is called strictly pseudononspreading [20] if there exists k ∈ [0, 1) such that, for all x, yC,

$‖Tx-TY‖2≤ ‖x-y‖2+ k‖(I-T)x-(I-T)y‖2+ 2⟨x-Tx,y-Ty⟩,$

where I denotes the identity mapping. Note that if k = 0, a mapping T is called nonspreading [14]. As a generalization of the class of strictly pseudononspreading mappings, the class of demicontractive mappings was introduced by Hicks and Kubicek [12] in 1977.

Recall that a single-valued mapping T is said to be demicontractive if F(T) ≠ ∅︀ and there exists κ ∈ [0, 1) such that, for all xC and for all zF(T),

$‖Tx-z‖2≤ ‖x-z‖2+ κ‖x-Tx‖2.$

We call κ the contraction coefficient. Clearly, strictly pseudononspreading mapping with a nonempty fixed point set is demicontractive.

We now give two examples for the class of demicontractive mappings.

### Example 2.9

Let H be the real line and C = [0, 1]. Define a mapping T : CC by

$Tx={47x sin (1x),x≠0,0,x=0.$

Obviously, F(T) = {0}. Also, for all xC, we have $∣Tx-T0∣2= ∣Tx∣2 = ∣47x sin (1x)∣2≤ ∣4x7∣2≤ ∣x∣2 ≤ ∣x-0∣2+ k∣x-Tx∣2$ for all k ∈ [0, 1). Therefore, T is demicontractive.

### Example 2.10

Let H be the real line and C = [−1, 1]. Define a mapping T : CC by

$Tx={9-x10,x∈[-1,0),x+910,x∈[0,1].$

Obviously, F(T) = {1} and T is demicontractive.

The following lemma obtained by Suantai and Phuengrattana [22] is useful for our results.

### Lemma 2.11

Let H be a Hilbert space and C be a nonempty closed convex subset of H. Let T : CC be a demicontractive mapping with contraction coefficient κ. Then, the following hold:

(i) F(T) = F(PC(Iμ(IT))) for all μ > 0;

(ii) PC(Iμ(IT)) is quasi-nonexpansive, for all μ ∈ (0, 1 − κ].

The set C of H is called proximinal if for each xH there exists zC such that

$‖x-z‖ =inf{‖x-y‖:y∈C}=dist(x,C).$

It is clear that every nonempty closed convex subset of a real Hilbert space is proximinal. We denote by CB(C) and KC(C) the families of all nonempty closed bounded subsets, and nonempty compact convex subsets of C, respectively. The Pompeiu-Hausdorff metric ℋ on CB(C) is defined by

$H(A,B):=max{supx∈A dist(x,B),supy∈B dist(y,A)}, ∀A,B∈CB(C).$

Let S : CCB(C) be a multi-valued mapping. An element xC is said to be a fixed point of S if xSx. The fixed point set of S is denoted by F(S) = {xC : xSx}.

### Definition 2.12

A multi-valued mapping S : CCB(C) is said to

(i) be nonexpansive if ℋ(Sx,Sy) ≤ ||xy|| for all x, yC;

(ii) be quasi-nonexpansive if F(S) ≠ ∅︀ and ℋ(Sx,Sz) ≤ ||xz|| for all xC and zF(S);

(iii) satisfy condition (Eμ) if there exists μ ≥ 1 such that for each x, yC, $dist(x,Sy)≤μdist(x,Sx)+‖x-y‖.$

We say that S satisfies condition (E) whenever S satisfies (Eμ) for some μ ≥ 1.

From the above definitions, it is clear that:

(i) if S is nonexpansive, then T satisfies the condition (E1);

(ii) if C is compact, then S is hemicompact.

We now give an example for the class of quasi-nonexpansiveness multi-valued mapping satisfying the condition (E).

### Example 2.13

Let C = [0,∞) and S : CCB(C) be defined by

$Sx=[x4,x2] for all x∈C.$

Then S is quasi-nonexpansive and satisfies condition (E).

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 C = [0,∞) and S : CCB(C) be defined by

$Sx=[x,2x] for all x∈C.$

Then S satisfies condition (E) and is not quasi-nonexpansive.

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 C = [−1, 1] and S : CCB(C) be defined by

$Sx={{x1+∣x∣sin(1x)}ifx≠0;{0}ifx=0.$

Then S is quasi-nonexpansive and does not satisfy condition (E).

### Lemma 2.16.([16])

Let {tn} be a sequence of real numbers such that there exists a subsequence {ni} of {n} such that tni < tni+1for all i ∈ ℕ. Then there exists a nondecreasing sequence {τ (n)} ⊂ ℕ such that τ (n) → ∞, and the following properties are satisfied by all (sufficiently large) numbers n ∈ ℕ:

$tτ(n)≤tτ(n)+1, tn≤tτ(n)+1.$

In fact,

$τ(n)=max{k≤n:tk

### Lemma 2.17.([23])

In Hilbert space H, the following inequality holds:

$‖x+y‖2≤ ‖x‖2+ 2⟨y,x+y⟩, ∀x,y∈H.$

### Lemma 2.18.([28])

Let H be a Hilbert space. Let x1, x2,..., xNH and α1, α2,..., αN be real numbers in [0, 1] such that $∑i=1Nαi=1$. Then,

$‖∑i=1Nαixi‖2=∑i=1Nαi‖xi‖2-∑1≤i,j≤Nαiαj‖xi-xj‖2.$

### Lemma 2.19.([26])

Let {an} be a sequence of nonnegative real numbers, let {bn} be a sequence in (0, 1) with $∑n=1∞bn=∞$, let {dn} be a sequence of nonnegative real numbers with $∑n=1∞dn<∞$, and let {cn} be a sequence of real numbers with lim supn→∞cn ≤ 0. Suppose that the following inequality holds:

$an+1≤(1-bn)an+bncn+dn, ∀n∈ℕ.$

Then limn→∞an = 0.

3. Main Results

In this section, we show strong convergence theorems for the sequence generated by the hybrid algorithm (3.1) based on extragradient algorithm which solve the problem of finding of four sets, i.e., F(T), F(S), Sol(f,C), and VI(B,C).

Now, let C be a nonempty, closed and convex subset of a real Hilbert space H and f : H × H → ℝ be a bifunction such that f(x, x) = 0, for all xC. In order to find a point in F(T) ∩ F(S) ∩ Sol(f,C) ∩ VI(B,C) ≠ ∅︀, we make use of the following blanket assumptions:

Assumptions

(A1) f is monotone on C;

(A2) F is Lipschitz-type continuous on C with constants c1 > 0 and c2 > 0;

(A3) f(x, ·) is convex and subdifferentiable on C, for all xC;

(A4) f is jointly weakly continuous on C × C in the sense that, if x, yC and {xn}, {yn} ⊂ C converge weakly to x and y, respectively, then f(xn, yn) → f(x, y) as n→∞.

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

### Theorem 3.1

Let H be a real Hilbert space and C be a nonempty closed convex subset of H. Let f be a bifunction satisfying assumptionson C, T : CC be a demicontractive single-valued mapping with contraction coefficient κ, S : CKC(C) be a quasi-nonexpansive multi-valued mapping satisfying the condition (E), and B : CH be a δ-inverse strongly monotone mapping. Assume that ℱ = F(T)∩F(S)∩Sol(f,C)∩VI(B,C) ≠ ∅︀ and Sp = {p} for all p ∈ ℱ. Let h : CC be a k-contraction. For x1C, let {xn}, {yn}, {zn}, and {wn} be sequences generated by

${wn=argminw∈C [λnf(xn,w)+12‖w-xn‖2],zn=argminz∈C [λnf(wn,z)+12‖z-xn‖2],yn=αnzn+βnun+γnPC(I-μn(I-T))zn+ζnPC(I-ηnB)zn,xn+1=σnh(xn)+(1-σn)yn, ∀n∈ℕ,$

where unSzn and {αn}, {βn}, {γn}, {ζn}, {σn}, {μn}, {ηn}, and {λn} satisfy the following conditions:

(C1) {σn} ⊂ (0, 1), limn→∞σn = 0, $∑n=1∞σn=∞$;

(C2) ${λn}⊂[a,b]⊂(0,1L)$, where L = max{2c1, 2c2};

(C3) μn ∈ (0, 1 − κ] with limn→∞μn = 0;

(C4) ηn ∈ [d, e] for some d, e ∈ (0, 2δ) and for all n ∈ ℕ;

(C5) 0 < aαn, βn, γn, ζnb < 1 and αn + βn + γn + ζn = 1 for all n ∈ ℕ.

Then the sequence {xn} converges strongly to q ∈ ℱ, which solves the variational inequality

$⟨q-h(q),x-q⟩≥0, ∀x∈ℱ.$
Proof

Let Q = P and it easy to see that Qh is contraction. By the Banach contraction principle, there exists q ∈ ℱ such that q = (Qh)(q). Applying Lemma 2.8, we have

$‖zn-q‖2≤ ‖xn-q‖2- (1-2λnc1)‖xn-wn‖2- (1-2λnc2)‖wn-zn‖2.$

This implies that

$‖zn-q‖ ≤ ‖xn-q‖.$

Since S is quasi-nonexpansive and Sq = {q}, by (3.3), we have

$‖un-q‖ =dist(un,Sq)≤H(Szn,Sq)≤ ‖zn-q‖ ≤ ‖xn-q‖.$

By Lemma 2.11(ii), PC(Iμn(IT)) is quasi-nonexpansive for all n ∈ ℕ. It implies by PC(Iμn(IT))q = q and (3.3) that

$‖PC(I-μn(I-T))zn-q‖ ≤ ‖zn-q‖ ≤ ‖xn-q‖.$

Let x, yC. Since B is δ-inverse strongly monotone, by condition (C4), we have

$‖PC(I-ηnB)x-PC(I-ηnB)y‖2≤ ‖(I-ηnB)x-(I-ηnB)y‖2= ‖x-y‖2-2ηn⟨x-y,Bx-By⟩+ηn2‖Bx-By‖2≤ ‖x-y‖2-2ηnδ‖Bx-By‖2+ ηn2‖Bx-By‖2= ‖x-y‖2-ηn⟨2δ-ηn)‖Bx-By‖2≤ ‖x-y‖2-d(2δ-e)‖Bx-By‖2≤ ‖x-y‖2.$

This shows that PC(IηnB) is nonexpansive for all n ∈ ℕ. Thus, by PC(IηnB)q = q and (3.3), we have

$‖PC(I-ηnB)zn-q‖ ≤ ‖zn-q‖ ≤ ‖xn-q‖.$

From (3.3)–(3.6), we get that

$‖yn-q‖ = ‖αnzn+βnun+γnPC(I-μn(I-T))zn+ζnPC(I-ηnB)zn-q‖≤αn‖zn-q‖+βn‖un-q‖+γn‖PC(I-μn(I-T))zn-q‖+ζnPC(I-ηnB)zn-q‖≤(αn+βn+γn+ζn)‖xn-q‖=‖xn-q‖.$

Consequently,

$‖xn+1-q‖ = ‖σnh(xn)+(1-σn)yn-q‖≤σn‖h(xn)-q‖+ (1-σn)‖yn-q‖≤σn(‖h(xn)-h(q)‖+‖h(q)-q‖)+(1-σn)‖xn-q‖≤σn(‖h(xn)-h(q)‖+‖h(q)-q‖)+(1-σn)‖xn-q‖≤σnk‖xn-q‖+ σn‖h(q)-q‖+(1-σn)‖xn-q‖=(1-σn(1-k))‖xn-q‖+ σn‖h(q)-q‖≤max {‖xn-q‖,‖h(q)-q‖1-k}.$

By induction, we get

$‖xn-q‖ ≤max {‖x1-q‖,‖h(q)-q‖1-k}, ∀n∈ℕ.$

This implies that {xn} is bounded, and we also obtain that {un}, {zn}, {yn} and {h(xn)} are bounded.

By Lemma 2.18, (3.1), (3.2), and (3.3), we obtain that

$‖yn-q‖2≤ αn‖zn-q‖2+βn‖un-q‖2+ γn‖PC(I-μn(I-T))zn-q‖2 + ζn‖PC(I-ηnB)zn-q‖2- αnβn‖zn-un‖2 -αnγn‖zn-PC(I-μn(I-T))zn‖2 -αnζn‖zn-PC(I-ηnB)zn‖-βnγn‖un-PC(I-μn(I-T))zn‖2 -βnζn‖un-PC(I-ηnB)zn‖2 -γnζn‖PC(I-μn(I-T))zn-PC(I-ηnB)zn‖2≤ αn‖xn-q‖2+ βn‖xn-q‖2+ γn‖zn-q‖2 + ζn‖zn-q‖2- αnβn‖zn-un‖2 -αnγn‖zn-PC(I-μnI-T))zn‖2-αnζn‖zn-PC(I-ηnB)zn‖2 -βnγn‖un-PC(I-μn(I-T))zn‖2-βnζn‖un-PC(I-ηnB)zn‖2 -γnζn‖PC(I-μn(I-T))zn-PC(I-ηnB)zn‖2 -αn(1-2λnc1)‖xn-wn‖2- αn(1-2λnc2)‖wn-zn‖2≤ ‖xn-q‖2- αnβn‖zn-un‖2-αnγn‖zn-PC(I-μn(I-T))zn‖2 -αnζn‖zn-PC(I-ηnB)zn‖2-βnγn‖un-PC(I-μn(I-T))zn‖2 -βnζn‖un-PC(I-ηnB)zn‖2 -γnζn‖PC(I-μn(I-T))zn-PC(I-ηnB)zn‖2 -αn(1-2λnc1)‖xn-wn‖2-αn(1-2λnc2)‖wn-zn‖2.$

Consequently, utilizing (3.8), we conclude that

$‖xn+1-q‖2≤ σn‖h(xn)-q‖2+ (1-σn)‖yn-q‖2≤ σn‖h(xn)-q‖2+ (1-σn)‖xn-q‖2- (1-σn)αnβn‖zn-un‖2 -(1-σn)αnγn‖zn-PC(I-μn(I-T))zn‖2 -(1-σn)αnζn‖zn-PC(I-ηnB)zn‖2 -(1-σn)βnγn‖un-PC(I-μn(I-T))zn‖2 -(1-σn)βnζn‖un-PC(I-ηnB)zn‖2 -(1-σn)γnζn‖PC(I-μn(I-T))zn-PC(I-ηnB)zn‖2 -(1-σn)αn(1-2λnc1)‖xn-wn‖2 -(1-σn)αn(1-2λnc2)‖wn-zn‖2.$

Therefore, we have

$(1-σn)αnβn‖zn-un‖2≤ ‖xn-q‖2-‖xn+1-q‖2+ σn‖h(xn)-q‖2.$

In order to prove that xnq as n→∞, we have consider the following two cases.

Case 1

Suppose that there exists n0 such that {||xnq||} is nonincreasing, for all nn0. Boundedness of {||xnq||} implies that {||xnq||} is convergent. Since {h(xn)} is bounded and σn → 0 as n → ∞, from (3.10) and condition (C5), we obtain that

$limn→∞‖zn-un‖ =0.$

By (3.9), we have

$(1-σn)αnγn‖zn-PC(I-μn(I-T))zn‖2≤ ‖xn-q‖2-‖xn+1-q‖2+ σn‖h(xn)-q‖2.$

This implies by conditions (C1) and (C5) that

$limn→∞‖zn-PC(-μn(I-T))zn‖ =0.$

By similar argument we can obtain that lim

$limn→∞‖zn-PC(I-ηnB)zn‖ =0,$$limn→∞‖xn-wn‖ =0, limn→∞‖wn-zn‖ =0.$

Also, by (3.14), we have

$‖xn-zn‖ ≤ ‖xn-wn‖+‖wn-zn‖→0, as n→∞.$

Next, we will show that

$lim supn→∞⟨h(q)-q,xn-q⟩≤0$

where q = Qh(q). To show this inequality, take a subsequence {xni} of {xn} such that

$lim supn→∞⟨h(q)-q,xn-q⟩=limi→∞⟨h(q)-q,xni-q⟩.$

Without loss of generality, we may assume that xnix* as i→∞ where x*C. Since ||xnizni|| → 0 as i → ∞, we have znix*. We will show that x* = ℱ. Assume x*F(T). From Lemma 2.11(i), we have that x*F(PC(Iμni (IT))) for all i ∈ ℕ. That is x*PC(Iμni (IT))x*. By Opial’s property, condition (C3), and (3.12), we have

$lim infi→∞‖zni-x*‖ < lim infi→∞‖zni-PC(I-μni(I-T))x*‖=lim infi→∞(‖zni-PC(I-μni(I-T))zni‖ +‖PC(I-μni(I-T))zni-PC(I-μni(I-T))x*‖)≤ lim infi→∞(‖zni-PC(I-μni(I-T))zni‖ +‖zni-x*‖+μni‖(I-T)zni-(I-T)x*‖)≤ lim infi→∞‖zni-x*‖.$

This is a contradiction. Then x*F(T).

Since Sx* is compact and convex, for all i ∈ ℕ, we can choose qniSx* such that ||zniqni|| = dist(zni, Sx*) and the sequence {qni} has a convergent subsequence {qnk} with limk→∞qnk = qSx*. By condition (E), there exists μ ≥ 1 such that

$dist(zni,Sx*)≤μdist(zni,Szni)+‖zni-x*‖.$

Suppose that qx*. Since znix*, it follows by the Opial’s condition and (3.11) that

$lim supk→∞‖znk-x*‖

This is a contradiction. Then x*F(S).

Assume x*VI(B,C). From Lemma 2.2, we have that x*F(PC(IηnB)) for all n ∈ ℕ. That is x*PC(IηnB)x*. Now, since znix*, it follows by (3.13) and Opial’s property that

$lim infi→∞‖zni-x*‖

This is a contradiction. Then x*VI(B,C).

It follows from Lemma 2.7 and f(x, ·) is convex on C for each xC, we see that

$wn=argminy∈C[λnf(xn,y)+12‖y-xn‖2]$

if and only if

$0∈∂(λnf(xn,wn)+12‖wn-xn‖2)+NC(wn),$

where NC(wn) is the normal cone of C at wnC. Then there exists v∂f(xn, wn) and unNC(wn) such that

$0=λnv+wn-xn+un.$

Using successively the definition of the normal cone to C at wn and the subdifferential of the convex function f(xn, ·) at wn, we can write the following two inequalities

$⟨wn-xn,y-wn⟩≥λn⟨v,wn-y⟩, y∈C,$

and

$f(xn,y)-f(xn,wn)≥⟨v,y-wn⟩, y∈C$

Thus, we have

$λn(f(xn,y)-f(xn,wn))≥⟨wn-xn,wn-y⟩, y∈C.$

Hence

$f(xni,y)-f(xni,wni)≥1λni⟨wni-xni,wni-y⟩, y∈C.$

Since limi→∞ ||xniwni|| = 0, we have wnix*. Passing to the limit in the inequality (3.16) as i → ∞ and using the hypothesis (A4) and (C2), we obtain f(x*, y) ≥ 0 for all yC. This implies that x*Sol(f,C) and hence x* = ℱ. Since q = (Qh)(q) and x* = ℱ, it follows that

$limsupn→∞⟨h(q)-q,xn-q⟩=limi→∞⟨h(q)-q,xni-q⟩=⟨h(q)-q,x*-q⟩≤0.$

By using Lemma 2.17 and (3.7), we have

$‖xn+1-q‖2= ‖σnh(xn)+(1-σ)yn-q‖2≤ ‖(1-σn)(yn-q)‖2+2σn⟨h(xn)-q,xn+1-q⟩≤ (1-σn)2‖yn-q‖2+ 2σn⟨h(xn)-h(q),xn+1-q⟩ + 2σn⟨h(q)-q,xn+1-q⟩≤ (1-σn)2‖xn-q‖2+ 2σnk‖xn-q‖‖xn+1-q‖ +2σn⟨h(q)-q,xn+1-q⟩≤ (1-σn)2‖xn-q‖2+ σnk(‖xn-q‖2+‖xn+1-q‖2) +2σn⟨h(q)-q,xn+1-q⟩= ((1-σn)2+σnk)‖xn-q‖2+ σnk‖xn+1-q‖2 +2σn⟨h(q)-q,xn+1-q⟩= (1-σnk-2σn(1-k)+σn2)‖xn-q‖2+ σnk‖xn+1-q‖2 +2σn⟨h(q)-q,xn+1-q⟩.$

This implies that

$‖xn+1-q‖2≤(1-2(1-k)σn-σn21-σnk)‖xn-q‖2+2σn1-σnk⟨h(q)-q,xn+1-q⟩.$

Putting $bn=2(1-k)σn-σn21-σnk$ and $cn=21(1-k)-σn⟨h(q)-q,xn+1-q⟩$, we have $Σn=1∞bn=∞$ and lim supn→∞cn ≤ 0. Hence, by Lemma 2.19, we conclude the the sequence {xn} converge strongly to q.

Case 2

Assume that there exists a subsequence {xni} of {xn} such that

$‖xni-q‖ < ‖xni+1-q‖,$

for all i ∈ ℕ. In this case from Lemma 2.16, there exists a nondecreasing sequence {τ (n)} of ℕ for all nn0, for some n0 large enough, such that τ (n) → ∞ as n→∞, and the following inequalities hold for all nn0,

$‖xτ(n)-q‖ < ‖xτ(n)+1-q‖, ‖xn-q‖ < ‖xτ(n)+1-q‖.$

From (3.10), we have limn→∞ ||zτ(n)uτ(n)|| = 0, and similarly we obtain

$limn→∞‖zτ(n)-PC(I-μτ(n)(I-T))zτ(n)‖ =0,limn→∞‖zτ(n)-PC(I-ητ(n)B)zτ(n)‖ =0,limn→∞‖xτ(n)-wτ(n)‖ =0, limn→∞‖wτ(n)-zτ(n)‖ =0.$

Following an argument similar to that in Case 1, we have

$limn→∞‖xτ(n)-q‖ =0, limn→∞‖xτ(n)+1-q‖ =0.$

Thus, by Lemma 2.16, we have

$0≤ ‖xn-q‖ ≤max {‖xτ(n)-q‖,‖xn-q‖}≤ ‖xτ(n)+1-q‖.$

Therefore, the sequence {xn} converges strongly to q ∈ ℱ.

Recall that a multi-valued mapping S : CHCB(C) is said to satisfy Condition (A) if ||xp|| = dist(x, Sp) for all xH and pF(S); see [21]. We see that S satisfies Condition (A) if and only if Sp = {p} for all pF(S). Then the following result can be obtained from Theorem 3.1 immediately.

### Theorem 3.2

Let H be a real Hilbert space and C be a nonempty closed convex subset of H. Let f be a bifunction satisfying assumptionson C, T : CC be a demicontractive single-valued mapping with contraction coefficient κ, S : CKC(C) be a quasi-nonexpansive multi-valued mapping satisfying the condition (E), and B : CH be a δ-inverse strongly monotone mapping. Assume that ℱ = F(T)∩F(S)∩Sol(f,C)∩VI(B,C) ≠ ∅︀ and S satisfies Condition (A). Let h : CC be a k-contraction. For x1C, let {xn}, {yn}, {zn}, and {wn} be generated by (3.1), where {αn}, {βn}, {γn}, {ζn}, {σn}, {μn}, {ηn}, and {λn} satisfy the following conditions:

(C1) {σn} ⊂ (0, 1), limn→∞σn = 0, $∑n=1∞σn=∞$;

(C2) ${λn}⊂[a,b]⊂(0,1L)$, where L = max{2c1, 2c2};

(C3) μn ∈ (0, 1 − κ] with limn→∞μn = 0;

(C4) ηn ∈ [d, e] for some d, e ∈ (0, 2δ) and for all n ∈ ℕ;

(C5) 0 < aαn, βn, γn, ζnb < 1 and αn + βn + γn + ζn = 1 for all n ∈ ℕ.

Then the sequence {xn} converges strongly to q ∈ ℱ, which solves the variational inequality

$⟨q-h(q),x-q⟩≥0, ∀x∈ℱ.$

### 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 φ-inverse strongly monotone mappings in real Hilbert spaces.

(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 φ-inverse strongly monotone and monotone Lipschitz-type continuous mappings.

We consider the particular Ky Fan inequality, corresponding to the bifunction f, defined by f(x, y) = ⟨Ax, yx⟩ for all x, yC with A : CH. Then, the solution wn in algorithm (3.1) can be expressed as

$wn=argminw∈C [λnf(xn,w)+12‖w-xn‖2]=argminw∈C [λn⟨Axn,w-xn)+12‖w-xn‖2]=argminw∈C [12‖w-(xn-λnAxn)‖2-λn22‖Axn‖2]=argminw∈C [12‖w-(xn-λnAxn)‖2]=PC(xn-λnAxn).$

Also, the solution zn can be expressed as

$zn=argminz∈C [λnf(wn,z)+12‖z-xn‖2]=argminz∈C [λn⟨Awn,z-wn)+12‖z-xn‖2]=argminz∈C [12‖z-(xn-λnAwn)‖2-λn22‖Awn‖2-λn⟨Awn,wn-xn⟩]=argminz∈C [12‖z-(xn-λnAwn)‖2]=PC(xn-λnAwn).$

Let A be L-Lipschitz-type continuous on C, that is ||AxAy|| ≤ L||xy|| for all x, yC. Then, for x, y, zC, we have

$f(x,y)+f(y,z)-f(x,z)=-⟨Ay-Ax,y-z⟩≥-‖Ax-Ay‖ ‖y-z‖≥-L‖x-y‖ ‖y-z‖≥-L2‖x-y‖2-L2‖y-z‖2.$

Therefore, f is Lipschitz-type continuous on C with $c1=c2=L2$.

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

Let H be a real Hilbert space and C be a nonempty closed convex subset of H. Let A : CH be a monotone and L-Lipschitz-type continuous function, T : CC be a demicontractive single-valued mapping with contraction coefficient κ, S : CKC(C) be a quasi-nonexpansive multi-valued mapping satisfying the condition (E), and B : CH be a δ-inverse strongly monotone mapping. Assume that ℱ = F(T)∩F(S)∩VI(A,C)∩VI(B,C) ≠ ∅︀ and Sp = {p} for all p ∈ ℱ. Let h : CC be a k-contraction. For x1C, let {xn}, {yn}, {zn}, and {wn} be sequences generated by

${wn=PC(xn-λnAxn),zn=PC(xn-λnAwn),yn=αnzn+βnun+γnPC(I-μn(I-T))zn+ζnPC(I-ηnB)zn,xn+1=σnh(xn)+(1-σn)yn, ∀n∈ℕ,$

where unSzn and {αn}, {βn}, {γn}, {ζn}, {σn}, {μn}, {ηn}, and {λn} satisfy the following conditions:

(C1) {σn} ⊂ (0, 1), limn→∞σn = 0, $∑n=1∞σn=∞$;

(C2) ${λn}⊂[a,b]⊂(0,1L)$, where L = max{2c1, 2c2};

(C3) μn ∈ (0, 1 − κ] with limn→∞μn = 0;

(C4) ηn ∈ [d, e] for some d, e ∈ (0, 2δ) and for all n ∈ ℕ;

(C5) 0 < aαn, βn, γn, ζnb < 1 and αn + βn + γn + ζn = 1 for all n ∈ ℕ.

Then the sequence {xn} converges strongly to q ∈ ℱ, which solves the variational inequality

$⟨q-h(q),x-q⟩≥0, ∀x∈ℱ.$
5. Numerical Example

In this section, we give an example which shows numerical experiment for supporting our main results.

### Example 5.1

Let H be a real line with the Euclidean norm and C = [0, 10]. For all xC, we define mappings T,S,B, h on C as follows:

$Tx={47x sin (1x),x≠0,0,x=0, , Sx=[x4,x2], Bx=x15, hx=x2.$

For each x, yC, define the bifunction f by f(x, y) = ⟨Ax, yx⟩, where $Ax=x5$. Let {xn}, {yn}, {zn}, and {wn} be generated by (3.1), where $un=xn4,αn=2n5+n+1,βn=n10n+3, γn=3n50n+1, ζn=1-2n5n+1-n10n+3-3n50n+1, σn=1n+2, μn=1n+3$

, ηn = 4, and λn = 2. It can be observed that all the assumptions of Theorem 3.1 are satisfied and F(T) ∩ F(S) ∩ Sol(f,C) ∩ V I(B,C) = {0}. By using SciLab, we compute the iterates of (3.1) for the initial point x1 = 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 {xn}, {yn}, {zn}, and {wn} converge to a unique point 0, where {0} = F(T) ∩ F(S) ∩ Sol(f,C) ∩ V I(B,C).

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.

Tables
Table 1

Numerical results of Example 5.1 for the algorithm (3.1)

nxnwnznyn|xnxn−1|
19.00000005.40000006.84000005.4263101-
25.11754013.07052403.88933053.11467643.8825e+00
32.97569981.78541992.26153191.82044322.1418e+00
41.75392461.05235471.33298271.07707361.2218e+00
51.04372170.62623300.79322850.64273647.1020e-01
200.00057920.00034750.00044020.00035753.6920e-04
310.00000270.00000160.00000200.00000171.6776e-06
320.00000160.00000100.00000120.00000101.0298e-06
330.00000100.00000060.00000080.00000066.3427e-07
340.00000060.00000040.00000050.00000043.8953e-07
350.00000040.00000020.00000030.00000022.3928e-07
References
1. PN. Anh. An LQ regularization method for pseudomonotone equilibrium problems on polyhedra. Vietnam J Math., 36(2008), 209-228.
2. PN. Anh. Strong convergence theorems for nonexpansive mappings and Ky Fan inequalities. J Optim Theory Appl., 154(2012), 303-320.
3. PN. Anh. A hybrid extragradient method extended to fixed point problems and equilibrium problems. Optimization., 62(2013), 271-283.
4. E. Blum, and W. Oettli. From optimization and variational inequalities to equilibrium problems. Math Student., 63(1994), 123-145.
5. H. Brezis, L. Nirenberg, and G. Stampacchia. A remark on Ky Fan’s minimax principle. Boll Unione Mat Ital., 6(1972), 293-300.
6. F. Facchinei, and JS. Pang. Finite-dimensional variational inequalities and complementarity problems, , Springer-Verlag, New York, 2003.
7. K. Fan. A minimax inequality and applications. Inequality, III, Academic Press, New York, 1972:103-113.
8. J. Garcia-Falset, E. Lorens-Fuster, and T. Suzuki. Fixed point theory for a class of generalized nonexpansive mappings. J Math Anal Appl., 375(2011), 185-195.
9. F. Giannessi, and A. Maugeri. Variational inequalities and network equilibrium problems, , Plenum Press, New York, 1995.
10. F. Giannessi, A. Maugeri, and PM. Pardalos. Equilibrium problems: Nonsmooth optimization and variational inequality models, , Kluwer Academic Publ, Dordrecht, 2004.
11. K. Goebel, and S. Reich. Uniform convexity, hyperbolic geometry, and nonexpansive mappings, , Marcel Dekker, New York, 1984.
12. TL. Hicks, and JR. Kubicek. On the Mann iteration process in Hilbert spaces. J Math Anal Appl., 59(1977), 498-504.
13. H. Iiduka, and W. Takahashi. Weak convergence theorems by Cesáro means for non-expansive mappings and inverse-strongly monotone mappings. J Nonlinear Convex Anal., 7(2006), 105-113.
14. F. Kohsaka, and W. Takahashi. Fixed point theorems for a class of nonlinear mappings related to maximal monotone operators in Banach spaces. Arch Math (Basel)., 91(2008), 166-177.
15. GM. Korpelevich. An extragradient method for finding saddle points and for other problems. Ekonom i Mat Metody., 12(1976), 747-756.
16. PE. Mainge. Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization. Set-Valued Anal., 16(2008), 899-912.
17. LD. Muu, VH. Nguyen, and TD. Quoc. Extragradient algorithms extended to equilibrium problems. Optimization., 57(2008), 749-776.
18. LD. Muu, and W. Oettli. Convergence of an adaptive penalty scheme for finding constrained equilibria. Nonlinear Anal., 18(1992), 1159-1166.
19. Z. Opial. Weak convergence of the sequence of successive approximations for nonexpansive mappings. Bull Amer Math Soc., 73(1967), 591-597.
20. MO. Osilike, and FO. Isiogugu. Weak and strong convergence theorems for nonspreading-type mappings in Hilbert spaces. Nonlinear Anal., 74(2011), 1814-1822.
21. S. Suantai, P. Cholamjiak, YJ. Cho, and W. Cholamjiak. On solving split equilibrium problems and fixed point problems of nonspreading multi-valued mappings in Hilbert spaces. Fixed Point Theory Appl., (2016) Paper No. 35, 16 pp.
22. S. Suantai, and W. Phuengrattana. A hybrid shrinking projection method for common fixed points of a finite family of demicontractive mappings with variational inequality problems. Banach J Math Anal., 11(2017), 661-675.
23. W. Takahashi. Nonlinear functional analysis, , Yokohama Publishers, Yokohama, 2000.
24. J. Vahidi, A. Latif, and M. Eslamian. New iterative scheme with strict pseudo-contractions and multivalued nonexpansive mappings for fixed point problems and variational inequality problems. Fixed Point Theory Appl., 2013(2013), 13 pp.
25. HK. Xu. On weakly nonexpansive and *–nonexpansive multivalued mappings. Math Japon., 36(1991), 441-445.
26. HK. Xu. An iterative approach to quadratic optimization. J Optim Theory Appl., 116(2003), 659-678.
27. I. Yamada. The hybrid steepest descent method for the variational inequality problem over the intersection of fixed point sets of nonexpansive mappings. Stud Comput Math., 8(2001), 473-504.
28. H. Zegeye, and N. Shahzad. Convergence of Mann’s type iteration method for generalized asymptotically nonexpansive mappings. Comput Math Appl., 62(2011), 4007-4014.

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