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,yC.

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,yC.

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

{x1C,xn+1=PC(xn-λAxn),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:

{x1C,yn=PC(xn-λAxn),xn+1=PC(xn-λAyn),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:

{x1C,yn=argminwC[λf(xn,w)+12w-xn2],xn+1=argminzC[λf(yn,z)+12z-xn2],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

{x1C,yn=argminwC[λnf(xn,w)+12w-xn2],zn=argminzC[λnf(yn,z)+12z-xn2],xn+1=αnh(xn)+βnxn+γn(μTxn+(1-μ)zn),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 supnxn-x<lim supnxn-y

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=minyCx-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-PCy2+PCy-y2x-y2;

(ii) PCx = y if and only if there holds the inequalityx-y,y-z0,zC.

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)uuVI(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-AyδAx-Ay2,x,yC.

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)-αx-y2,x,yC;

(ii) monotone on C if f(x,y)+f(y,x)0,x,yC;

(iii) pseudomonotone on C if f(x,y)0f(y,x)0,x,yC;

(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)-c1x-y2-c2y-z2,x,yC.

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)={ξH:f(z,t)-f(z,u)ξ,t-u,tH}.

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)={zH:z,y-v0,yC}.

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):xC}

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=argminwC[λnf(xn,w)+12w-xn2],zn=argminzC[λnf(wn,z)+12z-xn2].

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

zn-x*2xn-x*2-(1-2λnc1)xn-wn2-(1-2λnc2)wn-zn2,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-TY2x-y2+k(I-T)x-(I-T)y2+2x-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-z2x-z2+κx-Tx2.

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={47xsin (1x),x0,0,x=0.

Obviously, F(T) = {0}. Also, for all xC, we have Tx-T02=Tx2=47xsin (1x)24x72x2x-02+kx-Tx2 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:yC}=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{supxAdist(x,B),supyBdist(y,A)},A,BCB(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 xC.

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 xC.

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+xsin(1x)}ifx0;{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,         tntτ(n)+1.

In fact,

τ(n)=max{kn:tk<tk+1}.

Lemma 2.17.([23])

In Hilbert space H, the following inequality holds:

x+y2x2+2y,x+y,x,yH.

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=1Nαixi2=i=1Nαixi2-1i,jNαiαjxi-xj2.

Lemma 2.19.([26])

Let {an} be a sequence of nonnegative real numbers, let {bn} be a sequence in (0, 1) with n=1bn=, let {dn} be a sequence of nonnegative real numbers with n=1dn<, 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.

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=argminwC[λnf(xn,w)+12w-xn2],zn=argminzC[λnf(wn,z)+12z-xn2],yn=αnzn+βnun+γnPC(I-μn(I-T))zn+ζnPC(I-ηnB)zn,xn+1=σnh(xn)+(1-σn)yn,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=1σn=;

(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-q0,x.
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-q2xn-q2-(1-2λnc1)xn-wn2-(1-2λnc2)wn-zn2.

This implies that

zn-qxn-q.

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

un-q=dist(un,Sq)H(Szn,Sq)zn-qxn-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-qzn-qxn-q.

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

PC(I-ηnB)x-PC(I-ηnB)y2(I-ηnB)x-(I-ηnB)y2=x-y2-2ηnx-y,Bx-By+ηn2Bx-By2x-y2-2ηnδBx-By2+ηn2Bx-By2=x-y2-ηn2δ-ηn)Bx-By2x-y2-d(2δ-e)Bx-By2x-y2.

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-qzn-qxn-q.

From (3.3)–(3.6), we get that

yn-q=αnzn+βnun+γnPC(I-μn(I-T))zn+ζnPC(I-ηnB)zn-qαnzn-q+βnun-q+γnPC(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-qσnh(xn)-q+(1-σn)yn-qσn(h(xn)-h(q)+h(q)-q)+(1-σn)xn-qσn(h(xn)-h(q)+h(q)-q)+(1-σn)xn-qσnkxn-q+σnh(q)-q+(1-σn)xn-q=(1-σn(1-k))xn-q+σnh(q)-qmax {xn-q,h(q)-q1-k}.

By induction, we get

xn-qmax {x1-q,h(q)-q1-k},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-q2αnzn-q2+βnun-q2+γnPC(I-μn(I-T))zn-q2+ζnPC(I-ηnB)zn-q2-αnβnzn-un2-αnγnzn-PC(I-μn(I-T))zn2-αnζnzn-PC(I-ηnB)zn-βnγnun-PC(I-μn(I-T))zn2-βnζnun-PC(I-ηnB)zn2-γnζnPC(I-μn(I-T))zn-PC(I-ηnB)zn2αnxn-q2+βnxn-q2+γnzn-q2+ζnzn-q2-αnβnzn-un2-αnγnzn-PC(I-μnI-T))zn2-αnζnzn-PC(I-ηnB)zn2-βnγnun-PC(I-μn(I-T))zn2-βnζnun-PC(I-ηnB)zn2-γnζnPC(I-μn(I-T))zn-PC(I-ηnB)zn2-αn(1-2λnc1)xn-wn2-αn(1-2λnc2)wn-zn2xn-q2-αnβnzn-un2-αnγnzn-PC(I-μn(I-T))zn2-αnζnzn-PC(I-ηnB)zn2-βnγnun-PC(I-μn(I-T))zn2-βnζnun-PC(I-ηnB)zn2-γnζnPC(I-μn(I-T))zn-PC(I-ηnB)zn2-αn(1-2λnc1)xn-wn2-αn(1-2λnc2)wn-zn2.

Consequently, utilizing (3.8), we conclude that

xn+1-q2σnh(xn)-q2+(1-σn)yn-q2σnh(xn)-q2+(1-σn)xn-q2-(1-σn)αnβnzn-un2-(1-σn)αnγnzn-PC(I-μn(I-T))zn2-(1-σn)αnζnzn-PC(I-ηnB)zn2-(1-σn)βnγnun-PC(I-μn(I-T))zn2-(1-σn)βnζnun-PC(I-ηnB)zn2-(1-σn)γnζnPC(I-μn(I-T))zn-PC(I-ηnB)zn2-(1-σn)αn(1-2λnc1)xn-wn2-(1-σn)αn(1-2λnc2)wn-zn2.

Therefore, we have

(1-σn)αnβnzn-un2xn-q2-xn+1-q2+σnh(xn)-q2.

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

limnzn-un=0.

By (3.9), we have

(1-σn)αnγnzn-PC(I-μn(I-T))zn2xn-q2-xn+1-q2+σnh(xn)-q2.

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

limnzn-PC(-μn(I-T))zn=0.

By similar argument we can obtain that lim

limnzn-PC(I-ηnB)zn=0,limnxn-wn=0,limnwn-zn=0.

Also, by (3.14), we have

xn-znxn-wn+wn-zn0,   as n.

Next, we will show that

lim supnh(q)-q,xn-q0

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

lim supnh(q)-q,xn-q=limih(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 infizni-x*<lim infizni-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 infizni-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 supkznk-x*<lim supkznk-qlim supk(znk-qnk+qnk-q)=lim supk(dist(znk,Sx*)+qnk-q)lim supk(μdist(znk,Sznk)+znk-x*+qnk-q)lim supk(μznk-unk+znk-x*+qnk-q)=lim supkznk-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 infizni-x*<lim infizni-PC(I-ηniB)x*=lim infi(zni-PC(I-ηniB)zni+PC(I-ηniB)zni-PC(I-ηniB)x*)lim infi(zni-PC(I-ηniB)zni+zni-x*)=lim infiznk-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=argminyC[λnf(xn,y)+12y-xn2]

if and only if

0(λnf(xn,wn)+12wn-xn2)+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-wnλnv,wn-y,yC,

and

f(xn,y)-f(xn,wn)v,y-wn,yC

Thus, we have

λn(f(xn,y)-f(xn,wn))wn-xn,wn-y,yC.

Hence

f(xni,y)-f(xni,wni)1λniwni-xni,wni-y,yC.

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

limsupnh(q)-q,xn-q=limih(q)-q,xni-q=h(q)-q,x*-q0.

By using Lemma 2.17 and (3.7), we have

xn+1-q2=σnh(xn)+(1-σ)yn-q2(1-σn)(yn-q)2+2σnh(xn)-q,xn+1-q(1-σn)2yn-q2+2σnh(xn)-h(q),xn+1-q+2σnh(q)-q,xn+1-q(1-σn)2xn-q2+2σnkxn-qxn+1-q+2σnh(q)-q,xn+1-q(1-σn)2xn-q2+σnk(xn-q2+xn+1-q2)+2σnh(q)-q,xn+1-q=((1-σn)2+σnk)xn-q2+σnkxn+1-q2+2σnh(q)-q,xn+1-q=(1-σnk-2σn(1-k)+σn2)xn-q2+σnkxn+1-q2+2σnh(q)-q,xn+1-q.

This implies that

xn+1-q2(1-2(1-k)σn-σn21-σnk)xn-q2+2σn1-σnkh(q)-q,xn+1-q.

Putting bn=2(1-k)σn-σn21-σnk and cn=21(1-k)-σnh(q)-q,xn+1-q, we have Σn=1bn= 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

limnzτ(n)-PC(I-μτ(n)(I-T))zτ(n)=0,limnzτ(n)-PC(I-ητ(n)B)zτ(n)=0,limnxτ(n)-wτ(n)=0,limnwτ(n)-zτ(n)=0.

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

limnxτ(n)-q=0,limnxτ(n)+1-q=0.

Thus, by Lemma 2.16, we have

0xn-qmax{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=1σn=;

(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-q0,x.

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=argminwC[λnf(xn,w)+12w-xn2]=argminwC[λnAxn,w-xn)+12w-xn2]=argminwC[12w-(xn-λnAxn)2-λn22Axn2]=argminwC[12w-(xn-λnAxn)2]=PC(xn-λnAxn).

Also, the solution zn can be expressed as

zn=argminzC[λnf(wn,z)+12z-xn2]=argminzC[λnAwn,z-wn)+12z-xn2]=argminzC[12z-(xn-λnAwn)2-λn22Awn2-λnAwn,wn-xn]=argminzC[12z-(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-Ayy-z-Lx-yy-z-L2x-y2-L2y-z2.

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,

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

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

(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-q0,x.
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={47xsin(1x),x0,0,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
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