Mathematical and Computer Modelling ( ) –

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Mathematical and Computer Modelling

journal homepage: www.elsevier.com/locate/mcm

Viscosity approximation methods based on generalizedcontraction mappings for a countable family of strictpseudo-contractions, a general system of variationalinequalities and a generalized mixed equilibrium problem inBanach spaces

Pongsakorn Sunthrayuth, Poom Kumam ∗

Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), Bangmod,Bangkok 10140, Thailand

a r t i c l e i n f o

Article history:Received 23 November 2011Received in revised form 29 October 2012Accepted 21 February 2013

Keywords:Strict pseudo-contractionsGeneralized mixed equilibrium problemGeneral system of variational inequalitiesVariational inequalitiesLipschitzian mappingsRelaxed cocoercive mappingsViscosity approximation method

a b s t r a c t

In this paper, we introduce viscosity approximation methods based on generalizedcontractionmappings for finding a set of common fixed points of a countable family of strictpseudo-contractionmappings, the commonelement of the set solutions of a general systemof variational inequalities with Lipschitzian and relaxed cocoercive mappings and the setof solutions of a generalizedmixed equilibrium problem. Furthermore, strong convergencetheorems of the purposed iterative process are established in the framework of Banachspaces. The results presented in this paper improve and extend many recent importantresults.

© 2013 Elsevier Ltd. All rights reserved.

1. Introduction

Throughout this paper, we assume that X and X∗ are real Banach spaces and the dual space of X , respectively. Let C be anonempty, closed and convex subset of X . The duality mapping J : X → 2X∗

is defined by

J(x) = f ∗∈ X∗

: ⟨x, f ∗⟩ = ∥x∥2, ∥f ∗

∥ = ∥x∥, ∀x ∈ X .

If X := H is a Hilbert space, then J = I , where I is the identity mapping. It is well known that if X is smooth, then J issingle-valued.

Let Θ : C × C → R be a bifunction, where R is the set of real numbers, Ψ : C → X∗ be a nonlinear mapping andϕ : C → R be a real-valued function. The generalized mixed equilibrium problem is to find x∗

∈ C such that

Θ(x∗, y)+ ⟨Ψ x∗, y − x∗⟩ + ϕ(y)− ϕ(x∗) ≥ 0, ∀y ∈ C . (1.1)

The set of solutions of (1.1) is denoted by GMEP(Θ, ϕ,Ψ ). As special cases of the problem (1.1), we have the followingresults:

The authors were supported by the Higher Education Research Promotion and National Research University Project of Thailand, Office of the HigherEducation Commission (NRU-CSEC No. 55000613 and No. 56000508).∗ Corresponding author. Tel.: +66 02 470 8998; fax: +66 02 428 4025.

E-mail addresses: sci_math@hotmail.com (P. Sunthrayuth), poom.kum@kmutt.ac.th, kumampoom@hotmail.com (P. Kumam).

0895-7177/$ – see front matter© 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.mcm.2013.02.010

2 P. Sunthrayuth, P. Kumam / Mathematical and Computer Modelling ( ) –

(1) If Ψ = 0, then problem (1.1) reduces to themixed equilibrium problem, which is to find x∗∈ C such that

Θ(x∗, y)+ ϕ(y)− ϕ(x∗) ≥ 0, ∀y ∈ C . (1.2)

The set of solutions of (1.2) is denotedMEP(Θ, ϕ).(2) If ϕ = 0, then problem (1.1) reduces to the generalized equilibrium problem, which is to find x∗

∈ C such that

Θ(x∗, y)+ ⟨Ψ x, y − x∗⟩ ≥ 0, ∀y ∈ C . (1.3)

The set of solutions of (1.3) is denoted by GEP(Θ,Ψ ).(3) If Ψ = ϕ = 0, then problem (1.1) reduces to the equilibrium problem, which is to find x∗

∈ C such that

Θ(x∗, y) ≥ 0, ∀y ∈ C . (1.4)

The set of solutions of (1.4) is denoted by EP(Θ).

The generalized mixed equilibrium problems include fixed point problems, variational inequality problems, optimizationproblems, Nash equilibrium problems, and equilibrium problems as special cases. Numerous problems in physics,optimization, and economics reduce to finding a solution of (1.4). Some iterative methods have been proposed to solvethe equilibrium problem and their generalizations in Hilbert spaces and Banach spaces (see, e.g., [1–20]).

Let C be a nonempty, closed and convex subset of X and T be a self-mapping of C . We denote the fixed points set of themapping T by Fix(T ) = x ∈ C : Tx = x and denote → and by strong and weak convergence, respectively.

Definition 1.1. A mapping T : C → C is said to be:

(1) λ-strictly pseudocontractive [21] if for all x, y ∈ C , there exists λ > 0 and j(x − y) ∈ J(x − y) such that

⟨Tx − Ty, j(x − y)⟩ ≤ ∥x − y∥2− λ∥(I − T )x − (I − T )y∥2,

or equivalently

⟨(I − T )x − (I − T )y, j(x − y)⟩ ≥ λ∥(I − T )x − (I − T )y∥2.

(2) L-Lipschitzian if for all x, y ∈ C , there exists a constant L > 0 such that

∥Tx − Ty∥ ≤ L∥x − y∥.

If 0 < L < 1, then T is a contraction and if L = 1, then T is a nonexpansive mapping. By the definition, we know thatevery λ-strictly pseudocontractive mapping is

1+λλ

-Lipschitzian (see [22]).

Remark 1.2. Let C be a nonempty subset of a real Hilbert spaceH and T : C → C be amapping. Then T is said to be k-strictlypseudocontractive [21] if for all x, y ∈ C , there exists k ∈ [0, 1) such that

∥Tx − Ty∥2≤ ∥x − y∥2

+ k∥(I − T )x − (I − T )y∥2. (1.5)

It is well known that (1.5) is equivalent to the following:

⟨Tx − Ty, x − y⟩ ≤ ∥x − y∥ −1 − k2

∥(I − T )x − (I − T )y∥2.

Recall the following definitions of a nonlinear mapping A : C → X , the following are mentioned.

Definition 1.3. Let C be a nonempty, closed and convex subset of X . The mapping A : C → X is said to be:

(1) accretive

⟨Ax − Ay, J(x − y)⟩ ≥ 0, ∀x, y ∈ C,

(2) strongly accretive if there exists a constant c > 0 such that

⟨Ax − Ay, J(x − y)⟩ ≥ c∥x − y∥2, ∀x, y ∈ C,

(3) inverse-strongly accretive or cocoercive if there exists a constant d > 0 such that

⟨Ax − Ay, J(x − y)⟩ ≥ d∥Ax − Ay∥2, ∀x, y ∈ C,

(4) relaxed cocoercive if there exists constants c, d > 0 such that

⟨Ax − Ay, J(x − y)⟩ ≥ −c∥Ax − Ay∥2+ d∥x − y∥2, ∀x, y ∈ C .

P. Sunthrayuth, P. Kumam / Mathematical and Computer Modelling ( ) – 3

Remark 1.4. (1) Every strongly accretive mapping is an accretive mapping.(2) Every strongly accretive mapping is a relaxed cocoercive mapping but the converse is not true in general. Then the

class of relaxed cocoercive operators is more general than the class of strongly accretive operators.(3) Evidently, the definition of the inverse-strongly accretive operator is based on that of the inverse-strongly monotone

operator in real Hilbert spaces (see, e.g., [23]).(4) The notion of the cocoercivity is applied in several directions, especially for solving variational inequality problems

using the auxiliary problem principle and projection methods [24]. Several classes of relaxed cocoercive variationalinequalities have been studied in [25,26].

Let C be a nonempty, closed and convex subset of a real Banach space X . Let Ai : C → X (i = 1, 2, 3) be a nonlinearmapping. We consider the following problem of finding (x∗, y∗, z∗) ∈ C × C × C such that

⟨ρ1A1y∗+ x∗

− y∗, J(x − x∗)⟩ ≥ 0, ∀x ∈ C,⟨ρ2A2z∗

+ y∗− z∗, J(x − y∗)⟩ ≥ 0, ∀x ∈ C,

⟨ρ3A3x∗+ z∗

− x∗, J(x − z∗)⟩ ≥ 0, ∀x ∈ C,(1.6)

which is called a general system of variational inequalities in Banach spaces, where ρi > 0 for all i = 1, 2, 3.As special cases of problem (1.6), we have the following results:

(1) If A3 = 0, z∗= x∗, then problem (1.6) reduces to finding (x∗, y∗) ∈ C × C such that

⟨ρ1A1y∗+ x∗

− y∗, J(x − x∗)⟩ ≥ 0, ∀x ∈ C,⟨ρ2A2x∗

+ y∗− x∗, J(x − y∗)⟩ ≥ 0, ∀x ∈ C, (1.7)

which was considered by Yao et al. [27].(2) If A3 = 0, z∗

= x∗ and ρi = 1 for all i = 1, 2, 3, then problem (1.6) reduces to finding (x∗, y∗) ∈ C × C such that⟨A1y∗

+ x∗− y∗, J(x − x∗)⟩ ≥ 0, ∀x ∈ C,

⟨A2x∗+ y∗

− x∗, J(x − y∗)⟩ ≥ 0, ∀x ∈ C, (1.8)

which was introduced by Yao et al. [28].(3) If X := H is a real Hilbert space, then problem (1.6) reduces to finding (x∗, y∗) ∈ C × C such that

⟨ρ1A1y∗+ x∗

− y∗, x − x∗⟩ ≥ 0, ∀x ∈ C,

⟨ρ2A2x∗+ y∗

− x∗, x − y∗⟩ ≥ 0, ∀x ∈ C, (1.9)

which was considered by Ceng et al. [29].In 2010, Qin et al. [9] considered the generalized equilibriumproblemand a strictly pseudocontractivemapping inHilbert

spaces to prove the following result.

Theorem QCC (See [9]). Let C be a nonempty, closed and convex subset of a Hilbert space H. Let Θ : C × C → R be a bifunctionwhich satisfies (A1)–(A4), Ψ : C → H be t-inverse-strongly monotone, A : C → H be a-inverse-strongly monotone andB : C → H be b-inverse-strongly monotone. Let S : C → C be a k-strict pseudo-contraction with a fixed point. Define a mappingSk by Skx := kx + (1 − k)Sx, ∀x ∈ C. Assume that Ω = EP(Θ,Φ) ∩ Fix(S) ∩ Fix

PC (I − λ1A)PC (I − λ2B)) = ∅. Let u, x1 ∈ C

and xn be a sequence defined byΘ(un, y)+ ⟨Ψ un, y − un⟩ +

1r⟨y − un, un − xn⟩ ≥ 0, ∀y ∈ C,

yn = PC (xn − ηBxn),vn = PC (yn − λAyn),xn+1 = αnu + βnxn + γn[µ1Skxn + µ2un + µ3vn], ∀n ≥ 1,

(1.10)

where µ1, µ2, µ3 ∈ [0, 1) such that µ1 + µ2 + µ3 = 1, λ ∈ (0, 2a], η ∈ (0, 2b] and r ∈ (0, 2t], and αn, βn and γn aresequences in [0, 1] which satisfy the following conditions:

(C1) αn + βn + γn = 1;(C2) limn→∞ αn = 0 and

∞

n=1 αn = ∞;(C3) 0 < lim infn→∞ βn ≤ lim supn→∞ βn < 1.

Then, the sequence xn defined by (1.10) converges strongly to x∗= PΩu and (x∗, y∗) is the solution of the problem (1.9),

where y∗= PC (x∗

− ηBx∗).Recently, Imnang and Suantai [30] considered the problem for finding the solutions of a general system of variational

inequality (1.6). To be more precise, they obtained the following results.

4 P. Sunthrayuth, P. Kumam / Mathematical and Computer Modelling ( ) –

Lemma 1.5 (See [30]). Let C be a nonempty, closed and convex subset of a 2-uniformly smooth Banach space X. Let QC bethe sunny nonexpansive retraction from X onto C. Let Ai : C → X (i = 1, 2, 3) be a nonlinear mapping. For given(x∗, y∗, z∗) ∈ C × C × C, where y∗

= QC (z∗−ρ2A2z∗) and z∗

= QC (x∗−ρ3A3x∗), (x∗, y∗, z∗) is the solution of problem (1.6) if

and only if x∗ is a fixed point of the mapping G : C → C defined by

Gx := QCQC

QC (x − ρ3A3x)− ρ2A2QC (x − ρ3A3x)

− ρ1A1QC

QC (x − ρ3A3x)− ρ2A2QC (x − ρ3A3x)

, ∀x ∈ C .

Theorem IS (See [30]). Let C be a nonempty, closed and convex subset of a uniformly convex and 2-uniformly smooth Banachspace X which admits a weakly sequentially continuous duality mapping. Let QC be the sunny nonexpansive retraction from Xonto C. Let the mapping Ai : C → X be αi-inverse strongly accretive with αi ≥ ρiK 2, for all i = 1, 2, 3 andΩ = Fix(G) = ∅. Forgiven x1, u ∈ C, let the sequence xn be defined byzn = QC (xn − ρ1A1xn),

yn = QC (zn − ρ2A2zn),xn+1 = αnu + βnxn + (1 − αn − βn)QC (yn − ρ3A3yn), ∀n ≥ 1.

(1.11)

Suppose αn, βn are sequences in (0, 1) satisfy the following conditions:

(C1) limn→∞ αn = 0 and

∞

n=1 αn = ∞;(C2) 0 < lim infn→∞ βn ≤ lim supn→∞ βn < 1.

Then, the sequence xn defined by (1.11) converges strongly to Qωu, where Qωu is a sunny nonexpansive retraction of C ontoΩ .

The following questions naturally arise in connection with above results:

Question I. Can we extend Theorem QCC to a general Banach space? Such as a uniformly convex and 2-uniformly smoothBanach space?

Question II. Can we extend the iterative process (1.10) to a general iterative process define over the set of fixed points of acountable family of strict pseudo-contractions?

Question III. Canwe extend the iterative process (1.11) for finding a common element of the set of solutions of a generalizedmixed equilibrium problem, the set of solutions of a general system of variational inequalities and the set of common fixedpoints of a countable family of strict pseudocontraction mappings?

Question IV. Can we remove the assumption ‘‘weakly sequentially continuous duality mapping’’ in Theorem IS?

Question V. We know that the generalized contraction mapping is more general than the contraction mappings. Whathappens if the contraction mapping is replaced by the generalized contraction mapping?

The purpose of this paper is to give an affirmative answer to these questionsmentioned above, motivated by Qin et al. [9]and Imnang and Suantai [30], we introduce viscosity approximation methods based on generalized contraction mappingsfor finding a set of common fixed points of a countable family of strict pseudo-contraction mappings, the common elementof the set solutions of a general system of variational inequalities with Lipschitzian and relaxed cocoercive mappings andthe set of solutions of generalized mixed equilibrium problems in Banach spaces. Furthermore, we prove that the purposediterative process converges strongly to a common element of the three aforementioned sets. The results presented in thispaper improve and extend the corresponding results announced by Qin et al. [9] and Imnang and Suantai [30] and manyauthors.

2. Preliminaries

Let S(X) = x ∈ X : ∥x∥ = 1. The norm of X is said to be Gâteaux differentiable if the limit

limt→0

∥x + ty∥ − ∥x∥t

(2.1)

exists for each x, y ∈ S(X). In this case X is smooth. The norm of X is said to be Fréchet differentiable if for each x ∈ S(X),the limit Eq. (2.1) is attained uniformly for y ∈ S(X). The norm of X is called uniformly Fréchet differentiable if the limitEq. (2.1) is attained uniformly for x, y ∈ S(X). It is well known that (uniform) Fréchet differentiability of the norm of Ximplies (uniform) Gâteaux differentiability of the norm of X . A Banach space X is said to be strictly convex if ∥x+y∥

2 < 1 for allx, y ∈ X with ∥x∥ = ∥y∥ = 1 and x = y. A Banach space X is called uniformly convex if for each ϵ > 0 there is a δ > 0 suchthat for x, y ∈ X with ∥x∥, ∥y∥ ≤ 1 and ∥x− y∥ ≥ ϵ, ∥x+ y∥ ≤ 2(1− δ) holds. The modulus of convexity of X is defined by

δX (ϵ) = inf1 −

12(x + y)

: ∥x∥, ∥y∥ ≤ 1, ∥x − y∥ ≥ ϵ

,

P. Sunthrayuth, P. Kumam / Mathematical and Computer Modelling ( ) – 5

for all ϵ ∈ [0, 2]. X is said to be uniformly convex if δX (0) = 0 and δX (ϵ) > 0 for all 0 < ϵ ≤ 2. A Hilbert space H is2-uniformly convex, while Lp is maxp, 2-uniformly convex for every p > 1.

Let ρX : [0,∞) → [0,∞) be the modulus of smoothness of X defined by

ρX (τ ) = sup12(∥x + y∥ + ∥x − y∥)− 1 : x ∈ S(X), ∥y∥ ≤ t

.

A Banach space X is said to be uniformly smooth if ρX (t)t → 0 as t → 0. A typical example of uniformly smooth Banachspaces is Lp, where p > 1. More precisely, Lp is minp, 2-uniformly smooth for every p > 1. Let q be a fixed real numberwith 1 < q ≤ 2. Then a Banach space X is said to be q-uniformly smooth if there exists a constant c > 0 such that ρX (t) ≤ ctqfor all t > 0. It is well known that every q-uniformly smooth Banach space is uniformly smooth and has a uniformly Gâteauxdifferentiable norm.

Let C be a nonempty, closed and convex subset of a real Banach space X , and D be a nonempty subset of C . A mappingQ : C → D is said to be sunny if

Q (Qx + t(x − Qx)) = Qx,

whenever Qx + t(x − Qx) ∈ C for x ∈ C and t ≥ 0. A mapping Q : C → D is called a retraction if Qx = x for all x ∈ D.A subset D of C is called a sunny nonexpansive retraction of C if there exists a sunny nonexpansive retraction from C ontoD. It is well known that if X := H is a Hilbert space, then a sunny nonexpansive retraction Q is coincident with the metricprojection from X onto C .

A subsetD of C is called a sunny nonexpansive retraction of C if there exists a sunny nonexpansive retraction from C ontoD. The following propositions concern the sunny nonexpansive retraction.

Proposition 2.1 (See [31]). Let C be a closed and convex subset of a smooth Banach space X. Let D be a nonempty subset of C.Let Q : C → D be a retraction and let J be the normalized duality mapping on X. Then the following are equivalent:

(a) Q is sunny and nonexpansive.(b) ∥Qx − Qy∥2

≤ ⟨x − y, J(Qx − Qy)⟩, ∀x, y ∈ C.(c) ⟨x − Qx, J(y − Qx)⟩ ≤ 0, ∀x ∈ C, y ∈ D.

Proposition 2.2 (See [32]). Let C be a nonempty, closed and convex subset of a uniformly convex and uniformly smooth Banachspace X. Let T : C → C be a nonexpansive mapping with Fix(T ) = ∅. Then, the set Fix(T ) is a sunny nonexpansive retract of C.

For solving the generalized mixed equilibrium problem, let us assume that the function ϕ : C → R is convex and lowersemi-continuous, the nonlinear mapping Ψ : C → X∗ is continuous and monotone, and the bifunction Θ : C × C → Rsatisfies the following conditions:

(A1) Θ(x, x) = 0,∀x ∈ C;(A2) Θ is monotone, i.e.,Θ(x, y)+Θ(y, x) ≤ 0,∀x, y ∈ C;(A3) limt→0Θ(x + t(z − x), y) ≤ Θ(x, y),∀x, y ∈ C;(A4) The function y → Θ(x, y) is convex and lower semicontinuous.

Lemma 2.3 (See [10]). Let C be nonempty, closed and convex subset of a uniformly smooth, strictly convex and reflexive Banachspace X. Let Ψ : C → X∗ be a continuous and monotone mapping, ϕ : C → R be a lower semicontinuous and convex functionandΘ : C × C → R be a bifunction satisfying (A1)–(A4). For r > 0 and x ∈ C, there exists u ∈ C such that

Θ(u, y)+ ⟨Ψ u, y − u⟩ + ϕ(y)− ϕ(u)+1r⟨y − u, Ju − Jx⟩ ≥ 0, ∀y ∈ C .

Define a mapping Kr : C → C by

Kr(x) =

u ∈ C : Θ(u, y)+ ⟨Ψ u, y − u⟩ + ϕ(y)− ϕ(u)+

1r⟨y − u, Ju − Jx⟩ ≥ 0,∀y ∈ C

, ∀x ∈ C,

then, the mapping Kr has the following properties:

(1) Kr is single-valued;(2) Kr is a firmly nonexpansive-type mapping, i.e., for all x, y ∈ C;

⟨Krx − Kry, JKrx − JKry⟩ ≤ ⟨Krx − Kry, Jx − Jy⟩;

(3) Fix(Kr) = GMEP(Θ, ϕ,Ψ );(4) GMEP(Θ, ϕ,Ψ ) is closed and convex.

Now, we denoted by N and R+ the set of all positive integers and all positive real numbers, respectively.A mapping ψ : R+

→ R+ is said to be an L-function if ψ(0) = 0, ψ(t) > 0 for each t > 0 and for every s > 0, thereexists u > s such that ψ(t) ≤ s for each t ∈ [s, u]. As a consequence, every L-function ψ satisfies ψ(t) < t for each t > 0.

6 P. Sunthrayuth, P. Kumam / Mathematical and Computer Modelling ( ) –

Definition 2.4. Let (X, d) be a metric space. A mapping f : X → X is said to be:

(1) a (ψ, L)-contraction if ψ : R+→ R+ is an L-function and d(f (x), f (y)) < ψ(d(x, y)),∀x, y ∈ X , with x = y,

(2) a Meir–Keeler type mapping if for each ϵ > 0, there exists δ = δ(ϵ) > 0 such that for each x, y ∈ X , with ϵ ≤ d(x, y)< ϵ + δ, we have d(f (x), f (y)) < ϵ.

Proposition 2.5 (See [33]). Let (X, d) be a metric space and f : X → X be a mapping. The following assertions are equivalent:

(i) f is a Meir–Keeler type mapping,(ii) there exists an L-function ψ : R+

→ R+ such that f is a (ψ, L)-contraction.

Proposition 2.6 (See [34]). Let C be a convex subset of a Banach space and X. Let f : C → C be a Meir–Keeler type mapping.Then, for each ϵ > 0 there exists r ∈ (0, 1) such that

∥x − y∥ ≥ ϵ implies ∥f (x)− f (y)∥ ≤ r∥x − y∥.

From now on, by a generalized contraction mapping, we mean a Meir–Keeler type mapping or a (ψ, L)-contraction. Inthe rest of paper we suppose that the L-function from the definition of (ψ, L)-contraction is continuous, strictly increasingand limt→∞ η(t) = ∞, where η(t) := t − ψ(t),∀t ∈ R+. As a consequence, we have that η is a bijection on R+.

Lemma 2.7 (See [35]). Let X be a 2-uniformly smooth Banach spacewith the best smoothness constant K > 0. Then the followinginequality holds:

∥x + y∥2≤ ∥x∥2

+ 2⟨y, Jx⟩ + 2∥Ky∥2, ∀x, y ∈ X .

Lemma 2.8 (See [36]). Let xn and ln be bounded sequences in a Banach space X and let βn be a sequence in [0, 1] with0 < lim infn→∞ βn ≤ lim supn→∞ βn < 1. Suppose xn+1 = (1 − βn)ln + βnxn for all integers n ≥ 0 and lim supn→∞

(∥ln+1 − ln∥ − ∥xn+1 − xn∥) ≤ 0. Then, limn→∞ ∥ln − xn∥ = 0.

Definition 2.9. Let C be a nonempty subset of a Banach space X and Tn be a sequence of mappings from C into X with∞

n=1 Fix(Tn) = ∅. Suppose that for any bounded subset B of C . We say that

(i) Tn satisfies the AKTT-condition (see [37]) if∞n=1

supω∈B

∥Tn+1ω − Tnω∥ < ∞. (2.2)

(ii) Tn satisfies the PU-condition (see [38]) if there exists a continuous and increasing function hB : R+→ R+, and for all

k, l ∈ N such that

hB(0) = 0 and limk,l→∞

supω∈B

hB(∥Tkω − Tlω∥) = 0. (2.3)

Remark 2.10. If Tn satisfies the AKTT -condition, then Tn satisfies the PU-condition (see [38, Remark 3.2]).

Lemma 2.11 (See [38]). Let Tn be a sequence of mappings from C into X. Suppose that for any bounded subset B of C, thereexists a continuous and increasing function hB : R+

→ R+ satisfying (2.3). Then the following hold:

(i) For each x ∈ C, Tn converges strongly to some point of C.(ii) If the mapping T : C → X be defined by Tx = limn→∞ Tnx for all x ∈ C.

Then, limn→∞ supω∈B hB(∥Tω − Tnω∥) = 0. Moreover, the properties of hB imply that limn→∞ supω∈B ∥Tω − Tnω∥ = 0.

Lemma 2.12 (See [39]). Let C be a closed and convex subset of a strictly convex Banach space X. Let Tn : n ∈ N be asequence of nonexpansive mappings on C. Suppose

∞

n=1 Fix(Tn) is a nonempty. Let µn be a sequence of positive numberswith

∞

n=1 µn = 1. Then the mapping S on C defined by Sx =

∞

n=1 µnTnx for all x ∈ C is well defined, nonexpansive such thatFix(S) =

∞

n=1 Fix(Tn).

Lemma 2.13 (See [40]). Let C be a nonempty subset of a 2-uniformly smooth Banach space X. Let T : C → C be aλ-strict pseudo-contraction mapping. For α ∈ (0, 1), we define Tαx := (1 − α)x + αTx. Then, as α ∈ (0, λ

K2

, Tα : C → C is nonexpansive such

that Fix(Tα) = Fix(T ).

Lemma 2.14 (See [41], Demiclosed Principle). Let C be a nonempty, closed and convex subset of a uniformly convex Banach space.Let T : C → C be a nonexpansive mapping. Then I − T is demiclosed at zero, i.e., xn x ∈ C and xn − Txn → 0 implies x = Tx.

P. Sunthrayuth, P. Kumam / Mathematical and Computer Modelling ( ) – 7

Lemma 2.15 (See [42]). Assume that an is a sequence of nonnegative real numbers such that

an+1 ≤ (1 − τn)an + σn,

where τn is a sequence in (0, 1) and σn is a sequence in R such that(i)

∞

n=0 τn = ∞;(ii) lim supn→∞

σnτn

≤ 0 or

∞

n=0 |σn| < ∞.

Then, limn→∞ an = 0.

Without loss of generality, we assume that c, d ∈ (0, 1) and L, K ∈ [1,∞).

Lemma 2.16. Let C be a nonempty, closed and convex subset of a 2-uniformly smooth Banach space X. Let A : C → X be anL-Lipschitzian and relaxed (c, d)-cocoercive mapping such that d > cL2. Then, we have

∥(I − ρA)x − (I − ρA)y∥2≤

1 + 2ρcL2 − 2ρd + 2ρ2K 2L2

∥x − y∥2,

where ρ > 0. In particular, if ρ ≤d−cL2

K2L2, then I − ρA is nonexpansive.

Proof. From Lemma 2.7, for all x, y ∈ C , we have

∥(I − ρA)x − (I − ρA)y∥2= ∥(x − y)− (ρAx − ρAy)∥2

≤ ∥x − y∥2− 2ρ⟨Ax − Ay, J(x − y)⟩ + 2ρ2K 2

∥Ax − Ay∥2

≤ ∥x − y∥2− 2ρ

−c∥Ax − Ay∥2

+ d∥x − y∥2+ 2ρ2K 2

∥Ax − Ay∥2

= ∥x − y∥2− 2ρd∥x − y∥2

+ 2ρc∥Ax − Ay∥2+ 2ρ2K 2

∥Ax − Ay∥2

≤1 + 2ρcL2 − 2ρd + 2ρ2K 2L2

∥x − y∥2.

It is clear that, if 0 < ρ ≤d−cL2

K2L2, then I − ρA is nonexpansive. This completes the proof.

3. Main results

In this section, we start our main results, before proof our main theorem we need the following lemmas.

Lemma 3.1. Let C be a nonempty, closed and convex subset of a uniformly smooth Banach space X. Let T : C → C be a nonex-pansive mapping such that Fix(T ) = ∅ and f : C → C be a generalized contraction mapping. Then xt defined by xt = tf (xt)+(1 − t)Txt for t ∈ (0, 1), converges strongly to x ∈ Fix(T ), which x solves the variational inequality:

⟨f (x)− x, J(z − x)⟩ ≤ 0, ∀z ∈ Fix(T ). (3.1)

Proof. First, we show that xt is bounded. Take p ∈ Fix(T ), we have

∥xt − p∥ = ∥t(f (xt)− p)+ (1 − t)(Txt − p)∥≤ t∥f (xt)− f (p)∥ + t∥f (p)− p∥ + (1 − t)∥Txt − p∥≤ tψ(∥xt − p∥)+ t∥f (p)− p∥ + (1 − t)∥xt − p∥,

which implies that ∥xt − p∥ − ψ(∥xt − p∥) ≤ ∥f (p)− p∥, or equivalently ∥xt − p∥ ≤ η−1(∥f (p)− p∥).Hence xt is bounded, so are Txt and f (xt). Assume tn ⊂ (0, 1) is a sequence such that tn → 0 as n → ∞. Set

xn := xtn . Define a mapping φ : C → R by

φ(x) = LIMn∥xn − x∥2, ∀x ∈ C,

where LIMn is a Banach limit on ℓ∞. Define a set

K =

x ∈ C : φ(x) = min

x∈CLIMn∥xn − x∥2

.

It easily see that K is a nonempty closed convex bounded subset of X . Since

∥xn − Txn∥ = tn∥f (xn)− Txn∥ → 0 as n → ∞. (3.2)

Then

φ(Tx) = LIMn∥xn − Tx∥2

= LIMn∥Txn − Tx∥2

≤ LIMn∥xn − x∥2

= φ(x).

8 P. Sunthrayuth, P. Kumam / Mathematical and Computer Modelling ( ) –

It follows that T (K) ⊂ K ; that is, K is invariant. Since a uniformly smooth Banach space has the fixed point property fornonexpansive mapping, T has a fixed point, say x ∈ K , it follows that, for t ∈ (0, 1) and x ∈ C ,

0 ≤φ(x + t(x − x))− φ(x)

t

= LIMn∥(xn − x)+ t(x − x)∥2

− ∥xn − x∥2

t≤ LIMn⟨x − x, J(xn − x + t(x − x))⟩. (3.3)

The uniform smoothness of X implies that the duality mapping J is norm-to-norm uniformly continuous on any boundedsubset of X (see [43, Lemma 1]). Taking t → 0 in (3.3), we have that the limits can be interchanged and obtain

LIMn⟨x − x, J(xn − x)⟩ ≤ 0, ∀x ∈ C . (3.4)

Since

∥xt − x∥2= t⟨f (xt)− x, J(xt − x)⟩ + (1 − t)⟨Txt − x, J(xt − x)⟩≤ t⟨f (xt)− x, J(xt − x)⟩ + (1 − t)∥xt − x∥2,

which implies that

∥xt − x∥2≤ ⟨f (xt)− x, J(xt − x)⟩

≤ ⟨f (xt)− x, J(xt − x)⟩ + ⟨x − x, J(xt − x)⟩. (3.5)

It follows from (3.5) that

LIMn∥xn − x∥2≤ LIMn⟨f (xn)− x, J(xn − x)⟩ + LIMn⟨x − x, J(xn − x)⟩≤ LIMn∥f (xn)− x∥∥xn − x∥.

In particular,

LIMn∥xn − x∥2≤ LIMn∥f (xn)− f (x)∥∥xn − x∥≤ LIMnψ(∥xn − x∥)∥xn − x∥,

which implies that

LIMn∥xn − x∥∥xn − x∥ − ψ(∥xn − x∥)

≤ 0.

Hence LIMn∥xn − x∥ = 0 and there exists a subsequence xni of xn such that xni → x as i → ∞. Now assume that thereexists another subsequence xnk of xn such that xnk → y ∈ Fix(T ) as k → ∞. Then from (3.5), we have

∥y − x∥2≤ ⟨f (y)− x, J(y − x)⟩.

Interchanging y and x, we have

∥x − y∥2≤ ⟨f (x)− y, J(x − y)⟩.

Adding up the last two inequalities, we obtain

2∥y − x∥2≤ ⟨f (y)− f (x)+ y − x, J(y − x)⟩≤ ψ(∥y − x∥)∥y − x∥ + ∥y − x∥2.

It follows that

∥y − x∥∥y − x∥ − ψ(∥y − x∥)

≤ 0,

which implies that y = x. Hence xt → x as t → 0.Next, we show that x solves the variational inequality (3.1). Since xt = tf (xt)+ (1 − t)Txt , we have

(I − f )xt = −1 − tt(I − T )xt .

Note that I − T is accretive (i.e., ⟨(I − T )x − (I − T )y, j(x − y)⟩ ≥ 0 for x, y ∈ C). Hence for z ∈ Fix(T ),

⟨(I − f )xt , J(xt − z)⟩ = −1 − tt

⟨(I − T )xt , J(xt − z)⟩

= −1 − tt

⟨(I − T )xt − (I − T )z, J(xt − z)⟩

≤ 0.

P. Sunthrayuth, P. Kumam / Mathematical and Computer Modelling ( ) – 9

Taking the limit as t → 0, we obtain ⟨(I − f )x, J(x − z)⟩ ≤ 0, that is x ∈ Fix(T ) is the solution of (3.1). This completes theproof.

Lemma 3.2. Let C be a nonempty, closed and convex subset of a uniformly smooth Banach space X. Let T : C → C be a nonex-pansivemapping such that Fix(T ) = ∅ and f : C → C be a generalized contractionmapping. Assume that xt = tf (xt)+(1−t)Txtfor t ∈ (0, 1), converges strongly to x ∈ Fix(T ) as t → 0. Suppose that xn is a bounded sequence such that xn − Tx → 0 asn → ∞. Then

lim supn→∞

⟨f (x)− x, J(xn − x)⟩ ≤ 0. (3.6)

Proof. Since

∥xt − xn∥2= t⟨f (xt)− xn, J(xt − xn)⟩ + (1 − t)⟨Txt − xn, J(xt − xn)⟩= t⟨f (xt)− xt , J(xt − xn)⟩ + t⟨xt − xn, J(xt − xn)⟩ + (1 − t)⟨Txt − Txn, J(xt − xn)⟩

+ (1 − t)⟨Txn − xn, J(xt − xn)⟩≤ t⟨f (xt)− xt , J(xt − xn)⟩ + t∥xt − xn∥2

+ (1 − t)∥xt − xn∥2+ (1 − t)∥Txn − xn∥∥xt − xn∥

≤ t⟨f (xt)− xt , J(xt − xn)⟩ + ∥xt − xn∥2+ ∥Txn − xn∥∥xt − xn∥,

which implies that

⟨f (xt)− xt , J(xn − xt)⟩ ≤∥Txn − xn∥

t∥xt − xn∥. (3.7)

Taking the upper limit as n → ∞ firstly, and then as t → 0 in (3.7), we have

lim supt→0

lim supn→∞

⟨f (xt)− xt , J(xn − xt)⟩ ≤ 0. (3.8)

Since X is a uniformly smooth Banach space, we have the duality mapping J is norm-to-norm uniformly on any boundedsubset of X (see [43, Lemma 1]), which ensures that the limits lim supn→∞ and lim supt→0 are interchangeable, we have

lim supn→∞

⟨f (x)− x, J(xn − x)⟩ ≤ 0.

This completes the proof.

Theorem 3.3. Let C be a nonempty, closed and convex subset of a uniformly convex and 2-uniformly smooth Banach space X.Let QC be a sunny nonexpansive retraction from X onto C. Let Ai : C → X (i = 1, 2, 3) be an Li-Lipschitzian and relaxed(ci, di)-cocoercive mapping such that di > ciL2i . Let Θ : C × C → R be a bifunction satisfying the conditions (A1)–(A4),Ψ : C → X∗ be a continuous and monotone mapping and ϕ : C → R be a lower semi-continuous and convex function andr ∈ (0,∞) be a constant. Let f : C → C be a generalized contraction mapping and Tn∞n=1 : C → C be a countable familyof λ-strict pseudocontraction mappings with constant λ ∈ (0, 1). Define a mapping Snx := (1 − α)x + αTnx for all x ∈ C andn ≥ 1. Assume that Ω :=

∞

n=1 Fix(Tn) ∩ Fix(G) ∩ GMEPΘ, ϕ,Ψ

= ∅, where G is defined as in Lemma 1.5. For x1 ∈ C, let

the sequence xn defined byΘ(un, y)+ ⟨Ψ un, y − un⟩ + ϕ(y)− ϕ(un)+

1r⟨y − un, Jun − Jxn⟩ ≥ 0, ∀y ∈ C,

zn = QC (xn − ρ3A3xn),yn = QC (zn − ρ2A2zn),vn = QC (yn − ρ1A1yn),xn+1 = αnf (xn)+ βnxn + γn[µ1Snxn + µ2un + µ3vn], ∀n ≥ 1,

(3.9)

where µ1, µ2, µ3 ∈ (0, 1) such that µ1 + µ2 + µ3 = 1, ρi ∈ (0, di−ciL2iK2L2i

] for all i = 1, 2, 3, α ∈ (0, λ

K2 ], and αn, βn and

γn are sequences in (0, 1) which satisfy the following conditions:

(C1) αn + βn + γn = 1;(C2) limn→∞ αn = 0 and

∞

n=1 αn = ∞;(C3) 0 < lim infn→∞ βn ≤ lim supn→∞ βn < 1.

Suppose that Tn satisfies the PU-condition. Let T : C → C be a mapping defined by Tx = limn→∞ Tnx for all x ∈ C andsuppose that Fix(T ) =

∞

n=1 Fix(Tn). Then, the sequence xn defined by (3.9) converges strongly to x = QΩ f (x), where Q isthe sunny nonexpansive retraction of C ontoΩ and (x, y, z) is the solution of the problem (1.6), where y = QC (z − ρ2A2z) andz = QC (x − ρ3A3x).

10 P. Sunthrayuth, P. Kumam / Mathematical and Computer Modelling ( ) –

Proof. First, we show xn is bounded. Note that un can be rewritten as un = Krxn. Take x∗∈ Ω , we have

∥un − x∗∥ = ∥Krxn − Krx∗

∥

≤ ∥xn − x∗∥.

Put y∗= QC (z∗

− ρ2A2z∗) and z∗= QC (x∗

− ρ3A3x∗). Then x∗= QC (y∗

− ρ1A1y∗). It follows from Lemma 1.5 that

x∗= QC

QC

QC (x∗

− ρ3A3x∗)− ρ2A2QC (x∗− ρ3A3x∗)

− ρ1A1QC

QC (x∗

− ρ3A3x∗)− ρ2A2QC (x∗− ρ3A3x∗)

.

We observe that

∥vn − x∗∥ = ∥Gxn − Gx∗

∥

≤ ∥xn − x∗∥.

Set θn := µ1Snxn +µ2un +µ3vn. By Lemma 2.13, we have Sn is nonexpansive such that Fix(Sn) = Fix(Tn) for all n ≥ 1. Then,we have

∥θn − x∗∥ = ∥µ1(Snxn − x∗)+ µ2(un − x∗)+ µ3(vn − x∗)∥

≤ µ1∥Snxn − x∗∥ + µ2∥un − x∗

∥ + µ3∥vn − x∗∥

≤ µ1∥xn − x∗∥ + µ2∥xn − x∗

∥ + µ3∥xn − x∗∥

= ∥xn − x∗∥. (3.10)

It follows that

∥xn+1 − x∗∥ = ∥αn(f (xn)− x∗)+ βn(xn − x∗)+ γn(θn − x∗)∥

≤ αn∥f (xn)− x∗∥ + βn∥xn − x∗

∥ + γn∥θn − x∗∥

≤ αn∥f (xn)− f (x∗)∥ + αn∥f (x∗)− x∗∥ + βn∥xn − x∗

∥ + γn∥θn − x∗∥

≤ αnψ(∥xn − x∗∥)+ (1 − αn)∥xn − x∗

∥ + αn∥f (x∗)− x∗∥

= (1 − αnη)∥xn − x∗∥ + αn∥f (x∗)− x∗

∥

≤ max∥xn − x∗

∥, η−1(∥f (x∗)− x∗∥)

.

By induction, we have

∥xn − x∗∥ ≤ max

∥x1 − x∗

∥, η−1(∥f (x∗)− x∗∥)

, ∀n ≥ 1.

Hence, xn is bounded, so are un and vn. Without loss of generality, we assume that there exists a subset B of C whichcontains xn, un and vn.

From the definition of Sn and for all k, l ∈ N, there exists a continuous and increasing function hB : R+→ R+, we note

that

supω∈B

hB(∥Skω − Slω∥) = supω∈B

hB(α∥Tkω − Tlω∥)

≤ supω∈B

hB(∥Tkω − Tlω∥).

By our assumption, that Tn satisfies the PU-condition, we obtain that

limk,l→∞

supω∈B

hB(∥Skω − Slω∥) = 0,

that is Sn satisfies the PU-condition.Next, we show that limn→∞ ∥xn+1 − xn∥ = 0. We observe that

∥vn+1 − vn∥ = ∥Gxn+1 − Gxn∥≤ ∥xn+1 − xn∥.

It follows that

∥θn+1 − θn∥ = ∥(µ1Sn+1xn+1 + µ2un+1 + µ3vn+1)− (µ1Snxn + µ2un + µ3vn)∥

= ∥µ1(Sn+1xn+1 − Snxn)+ µ2(un+1 − un)+ µ3(vn+1 − vn)∥

≤ µ1∥Sn+1xn+1 − Snxn∥ + µ2∥un+1 − un∥ + µ3∥vn+1 − vn∥

≤ µ1∥Sn+1xn+1 − Sn+1xn∥ + µ1∥Sn+1xn − Snxn∥ + µ2∥un+1 − un∥ + µ3∥vn+1 − vn∥

= µ1∥Sn+1xn+1 − Sn+1xn∥ + µ1∥Sn+1xn − Snxn∥ + µ2∥Krxn+1 − Krxn∥ + µ3∥vn+1 − vn∥

≤ µ1∥xn+1 − xn∥ + µ2∥xn+1 − xn∥ + µ3∥xn+1 − xn∥ + µ1∥Sn+1xn − Snxn∥= ∥xn+1 − xn∥ + µ1∥Sn+1xn − Snxn∥. (3.11)

P. Sunthrayuth, P. Kumam / Mathematical and Computer Modelling ( ) – 11

Let xn+1 = (1 − βn)ln + βnxn, for all n ≥ 1. Then, we have

ln+1 − ln =xn+2 − βn+1xn+1

1 − βn+1−

xn+1 − βnxn1 − βn

=αn+1f (xn+1)+ γn+1θn+1

1 − βn+1−αnf (xn)+ γnθn

1 − βn

=αn+1

1 − βn+1f (xn+1)+

1 − βn+1 − αn+1

1 − βn+1θn+1 −

αn

1 − βnf (xn)−

1 − βn − αn

1 − βnθn

=αn+1

1 − βn+1(f (xn+1)− θn+1)+

αn

1 − βn(θn − f (xn))+ θn+1 − θn. (3.12)

Combining (3.11) and (3.12), we have

∥ln+1 − ln∥ − ∥xn+1 − xn∥ ≤αn+1

1 − βn+1∥f (xn+1)− θn+1∥ +

αn

1 − βn∥θn − f (xn)∥ + µ1∥Sn+1xn − Snxn∥.

Now, we show that limn→∞ ∥Sn+1xn − Snxn∥ = 0. Since Sn satisfies the PU-condition, we can define S : C → C by

Sx = limn→∞

Snx = limn→∞

[(1 − α)x + αTnx] = (1 − α)x + αTx, ∀x ∈ C .

We observe that12∥Sn+1xn − Snxn∥ ≤

12∥Sn+1xn − Sxn∥ +

12∥Sxn − Snxn∥.

Since hB : R+→ R+ is a continuous, increasing and convex function, then, we have

hB

12∥Sn+1xn − Snxn∥

≤

12hB(∥Sn+1xn − Sxn∥)+

12hB(∥Sxn − Snxn∥)

≤12supω∈B

hB(∥Sn+1ω − Sω∥)+12supω∈B

hB(∥Sω − Snω∥).

By Lemma 2.11 and the continuity of hB, we obtain that limn→∞ hB(12∥Sn+1xn − Snxn∥) = 0. This implies that

limn→∞

∥Sn+1xn − Snxn∥ = 0. (3.13)

Consequently, it follows from the conditions (C2), (C3) and (3.13) that

lim supn→∞

∥ln+1 − ln∥ − ∥xn+1 − xn∥

≤ 0.

Hence, by Lemma 2.8, we obtain that limn→∞ ∥ln − xn∥ = 0. Consequently, we have

limn→∞

∥xn+1 − xn∥ = limn→∞

(1 − βn)∥ln − xn∥ = 0. (3.14)

On the other hand, from (3.9), we observe that

xn+1 − xn = αn(f (xn)− xn)+ (1 − αn − βn)(θn − xn).

It follows that

(1 − αn − βn)∥θn − xn∥ ≤ ∥xn+1 − xn∥ + αn∥f (xn)− xn∥.

From the conditions (C2), (C3) and (3.14), we obtain that

limn→∞

∥θn − xn∥ = 0. (3.15)

Next, we show that z ∈ Ω :=

∞

n=1 Fix(Tn) ∩ Fix(G) ∩ GMEPΘ, ϕ,Ψ

. Define a mapping Wn : C → C by

Wnx := µ1Snx + µ2Krx + µ3Gx, ∀x ∈ C, (3.16)

where G is defined as in Lemma 1.5. From (3.15), we have

limn→∞

∥Wnxn − xn∥ = 0. (3.17)

For all k, l ∈ N, there exists a continuous and increasing function hB : R+→ R+, we note that

supω∈B

hB(∥Wkω − Wlω∥) = supω∈B

hB(µ1∥Skω − Slω∥)

≤ supω∈B

hB(∥Skω − Slω∥).

12 P. Sunthrayuth, P. Kumam / Mathematical and Computer Modelling ( ) –

Since Sn satisfies the PU-condition, we obtain that

limk,l→∞

supω∈B

hB(∥Wkω − Wlω∥) = 0,

that is Wn is satisfies the PU-condition. Define a mappingW : C → C by

Wx = limn→∞

Wnx = limn→∞

[µ1Snx + µ2Krx + µ3Gx] = µ1Sx + µ2Krx + µ3Gx, ∀x ∈ C,

where G is defined as in Lemma 1.5. By Lemma 2.11, we obtain that

limn→∞

supω∈B

hB(∥Wnω − Wω∥) = 0. (3.18)

From Lemma 2.12, we see thatW is nonexpansive and

Fix(W ) = Fix(S) ∩ Fix(Kr) ∩ Fix(G)

=

∞n=1

Fix(Sn) ∩ Fix(Kr) ∩ Fix(G)

=

∞n=1

Fix(Tn) ∩ GMEPΘ, ϕ,Ψ

∩ Fix(G)

=

∞n=1

Fix(Wn).

On the other hand, we observe that

12∥xn − Wxn∥ ≤

12∥xn − Wnxn∥ +

12∥Wnxn − Wxn∥

≤12∥xn − Wnxn∥ +

12supω∈B

∥Wnω − Wω∥.

Since hB : R+→ R+ is continuous, increasing and convex, then, we have

hB

12∥xn − Wxn∥

≤

12hB(∥xn − Wnxn∥)+

12hB(∥Wnxn − Wxn∥)

≤12hB(∥xn − Wnxn∥)+

12supω∈B

hB(∥Wnω − Wω∥).

From Lemma 2.11, (3.17) and the continuity of hB, we obtain that limn→∞ hB(12∥xn − Wxn∥) = 0. This implies that

limn→∞

∥xn − Wxn∥ = 0. (3.19)

By reflexivity of a Banach space X and boundedness of xn, there exists a subsequence xni of xn such that xni z ∈ Cas i → ∞. From (3.19) and Lemma 2.14, we obtain that z ∈ Ω .

Next, we show that

lim supn→∞

⟨f (x)− x, J(xn − x)⟩ ≤ 0, (3.20)

where x = limt→0 xt and xt is the unique fixed point of the contraction mapping Tt : C → C given by

Ttx = tf (x)+ (1 − t)Wx, t ∈ (0, 1).

By Lemma 3.1, we have that x ∈ Fix(W ) = Ω solves the variational inequality

⟨f (x)− x, J(z − x)⟩ ≤ 0, ∀z ∈ Ω.

From (3.19) and Lemma 3.2, we obtain (3.20) holds.Finally, we show that xn → x as n → ∞. Assume that the sequence xn does not converge strongly to x ∈ Ω . Then there

exist ϵ > 0 and a subsequence xnj of xn such that ∥xnj − x∥ ≥ ϵ, for all j ∈ 0, 1, . . .. By Proposition 2.6, for this ϵ thereexists r ∈ (0, 1) such that ∥f (xnj)− f (x)∥ ≤ r∥xnj − x∥. From (3.10), we have

∥xnj+1 − x∥2= ⟨αnj(f (xnj)− x)+ βnj(xnj − x)+ γnj(θnj − x), J(xnj+1 − x)⟩

= αnj⟨f (xnj)− f (x), J(xnj+1 − x)⟩ + αnj⟨f (x)− x, J(xnj+1 − x)⟩

+βnj⟨xnj − x, J(xnj+1 − x)⟩ + γnj⟨θnj − x, J(xnj+1 − x)⟩

P. Sunthrayuth, P. Kumam / Mathematical and Computer Modelling ( ) – 13

≤ αnj∥f (xnj)− f (x)∥ ∥xnj+1 − x∥ + βnj∥xnj − x∥ ∥xnj+1 − x∥

+ γnj∥θnj − x∥ ∥xnj+1 − x∥ + αnj⟨f (x)− x, J(xnj+1 − x)⟩

≤ αnj r∥xnj − x∥ ∥xnj+1 − x∥ + βnj∥xnj − x∥ ∥xnj+1 − x∥

+ γnj∥xnj − x∥ ∥xnj+1 − x∥ + αnj⟨f (x)− x, J(xnj+1 − x)⟩

=1 − (1 − r)αnj

∥xnj − x∥ ∥xnj+1 − x∥ + αnj⟨f (x)− x, J(xnj+1 − x)⟩

≤1 − (1 − r)αnj

2

∥xnj − x∥2

+ ∥xnj+1 − x∥2+ αnj⟨f (x)− x, J(xnj+1 − x)⟩

≤1 − (1 − r)αnj

2∥xnj − x∥2

+12∥xnj+1 − x∥2

+ αnj⟨f (x)− x, J(xnj+1 − x)⟩,

which implies that

∥xnj+1 − x∥2≤

1 − (1 − r)αnj

∥xnj − x∥2

+ 2αnj⟨f (x)− x, J(xnj+1 − x)⟩. (3.21)

Put τnj = (1 − r)αnj and δnj =2

1−r ⟨f (x)− x, J(xnj+1 − x)⟩. Then, (3.21) reduces to formula

∥xnj+1 − x∥2≤ (1 − τnj)∥xnj − x∥2

+ τnjδnj .

It follows from the condition (C2) and (3.20) that

∞

j=1 τnj = ∞ and lim supj→∞ δnj ≤ 0. From Lemma 2.7, we obtain thatxnj → x as j → ∞. The contradiction permits us to conclude that xn converges strongly to x ∈ Ω . This completes theproof.

Remark 3.4. Theorem 3.3 improves and extends the main result of Qin et al. [9] in the following ways:

• From a generalized equilibrium problem in Hilbert spaces to a generalized mixed equilibrium problem in Banach spaces.• From a system of variational inequalities to a general system of variational inequalities.• From a single strict pseudocontraction mapping to a family of strict pseudocontraction mappings.• From a fixed element u to a generalized contraction mapping f .

Remark 3.5. Theorem 3.3 improves and extends the main result of Imnang and Suantai [30] in the following ways:

• From the class of inverse-strongly accretive mappings to the class of Lipschitzian and relaxed cocoercive mappings.• From the problemof finding an element of Fix(G) to problemof finding an element of

∞

n=1 Fix(Tn)∩Fix(G)which involvesthe fixed point problem of a countable family of strict pseudocontraction mappings Tn∞n=1.

• From a fixed element u to a generalized contraction mapping.

From Theorem 3.3, we obtain the following results:

Corollary 3.6. Let C be a nonempty, closed and convex subset of a uniformly convex and 2-uniformly smooth Banach space X.Let QC be a sunny nonexpansive retraction from X onto C. Let Ai : C → X (i = 1, 2, 3) be an Li-Lipschitzian and relaxed (ci, di)-cocoercive mapping such that di > ciL2i . Let Θ : C × C → R be a bifunction satisfying the conditions (A1)–(A4), Ψ : C → X∗

be a continuous and monotone mapping and ϕ : C → R be a lower semi-continuous and convex function and r ∈ (0,∞) bea constant. Let f : C → C be a generalized contraction mapping and Tn∞n=1 : C → C be a countable family of nonexpansivemappings. Assume that Ω :=

∞

n=1 Fix(Tn) ∩ Fix(G) ∩ GMEPΘ, ϕ,Ψ

= ∅, where G is defined as in Lemma 1.5. For x1 ∈ C,

let the sequence xn defined byΘ(un, y)+ ⟨Ψ un, y − un⟩ + ϕ(y)− ϕ(un)+

1r⟨y − un, Jun − Jxn⟩ ≥ 0, ∀y ∈ C,

zn = QC (xn − ρ3A3xn),yn = QC (zn − ρ2A2zn),vn = QC (yn − ρ1A1yn),xn+1 = αnf (xn)+ βnxn + γn[µ1Tnxn + µ2un + µ3vn], ∀n ≥ 1,

(3.22)

where µ1, µ2, µ3 ∈ (0, 1) such that µ1 + µ2 + µ3 = 1, ρi ∈ (0, di−ciL2iK2L2i

] for all i = 1, 2, 3, and αn, βn and γn are

sequences in (0, 1) which satisfy the following conditions:

(C1) αn + βn + γn = 1;(C2) limn→∞ αn = 0 and

∞

n=1 αn = ∞;(C3) 0 < lim infn→∞ βn ≤ lim supn→∞ βn < 1.

14 P. Sunthrayuth, P. Kumam / Mathematical and Computer Modelling ( ) –

Suppose that Tn satisfies the PU-condition. Let T : C → C be a mapping defined by Tx = limn→∞ Tnx for all x ∈ C andsuppose that Fix(T ) =

∞

n=1 Fix(Tn). Then, the sequence xn defined by (3.22) converges strongly to x = QΩ f (x), where Q isthe sunny nonexpansive retraction of C ontoΩ and (x, y, z) is the solution of the problem (1.6), where y = QC (z − ρ2A2z) andz = QC (x − ρ3A3x).

Proof. Taking Tn∞n=1 : C → C is a countable family of nonexpansive mappings in Theorem 3.3, we can get the desiredconclusion easily.

Corollary 3.7. Let C be a nonempty, closed and convex subset of a uniformly convex and 2-uniformly smooth Banach space X.Let QC be a sunny nonexpansive retraction from X onto C. Let Ai : C → X (i = 1, 2, 3) be an Li-Lipschitzian and relaxed(ci, di)-cocoercive mapping such that di > ciL2i . Let f : C → C be a generalized contraction mapping and Tn∞n=1 : C → C be acountable family of λ-strict pseudocontraction mappings with constant λ ∈ (0, 1). Define a mapping Snx := (1 − α)x + αTnxfor all x ∈ C and n ≥ 1. Assume that Ω :=

∞

n=1 Fix(Tn) ∩ Fix(G) = ∅, where G is defined as in Lemma 1.5. For x1 ∈ C, let thesequence xn defined by

zn = QC (xn − ρ3A3xn),yn = QC (zn − ρ2A2zn),vn = QC (yn − ρ1A1yn),xn+1 = αnf (xn)+ βnxn + γn[µ1Snxn + µ3vn], ∀n ≥ 1,

(3.23)

where µ1, µ3 ∈ (0, 1) such that µ1 + µ3 = 1, ρi ∈ (0, di−ciL2iK2L2i

] for all i = 1, 2, 3, α ∈ (0, λ

K2 ], and αn, βn and γn are

sequences in (0, 1] which satisfy the following conditions:

(C1) αn + βn + γn = 1;(C2) limn→∞ αn = 0 and

∞

n=1 αn = ∞;(C3) 0 < lim infn→∞ βn ≤ lim supn→∞ βn < 1.

Suppose that Tn satisfies the PU-condition. Let T : C → C be a mapping defined by Tx = limn→∞ Tnx for all x ∈ C andsuppose that Fix(T ) =

∞

n=1 Fix(Tn). Then, the sequence xn defined by (3.23) converges strongly to x = QΩ f (x) where Q isthe sunny nonexpansive retraction of C ontoΩ and (x, y, z) is the solution of the problem (1.6), where y = QC (z − ρ2A2z) andz = QC (x − ρ3A3x).

Proof. Taking Ψ = ϕ = 0, r = 1 and µ2 = 0 in Theorem 3.3, then un = QCxn = xn, we can get the desired conclusioneasily.

Acknowledgments

The authors wish to thank the referees for a careful reading and suggestions which improved the quality of this work.Also, the authors would like to thank the Higher Education Research Promotion and National Research University Project ofThailand’s Office of the Higher Education Commission for financial support.

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