A New General Iterative Method for Solution of a New General System of Variational Inclusions for Nonexpansive Semigroups in Banach Spaces

Abstract

We introduce a new general system of variational inclusions in Banach spaces and propose a new iterative scheme for finding common element of the set of solutions of the variational inclusion with set-valued maximal monotone mapping and Lipschitzian relaxed cocoercive mapping and the set of fixed point of nonexpansive semigroups in a uniformly convex and 2-uniformly smooth Banach space. Furthermore, strong convergence theorems are established under some certain control conditions. As applications, finding a common solution for a system of variational inequality problems and minimization problems is given.

1. Introduction

In the theory of variational inequalities and variational inclusions, the development of an efficient and implementable iterative algorithm is interesting and important. The important generalization of variational inequalities called variational inclusions, have been extensively studied and generalized in different directions to study a wide class of problems arising in optimization, nonlinear programming, finance, economics, and applied sciences.

Variational inequalities are being used as a mathematical programming tool in modeling a wide class of problems arising in several branches of pure and applied mathematics. Several numerical techniques for solving variational inequalities and the related optimization problem have been considered by many authors.

Throughout this paper, we denoted by ℕ and ℝ⁺ the set of all positive integers and all positive real numbers, respectively. Let 𝑋 be a real Banach space and 𝑋^∗ be its dual space. Let 𝑈={𝑥∈𝑋∶‖𝑥‖=1} denote the unit sphere of 𝑋. 𝑋 is said to be uniformly convex if for each 𝜖∈(0,2], there exists a constant 𝛿>0 such that for all 𝑥,𝑦∈𝑈,𝑥−𝑦≥𝜖implies𝑥+𝑦2≤1−𝛿. (1.1) The norm on 𝑋 is said to be Gâteaux differentiable if the limitlim_𝑡→0𝑥+𝑡𝑦−‖𝑥‖𝑡 (1.2) exists for each 𝑥,𝑦∈𝑈, and in this case 𝑋 is smooth. Moreover, we say that the norm 𝑋 is said to have a uniformly Gâteaux differentiable if the above limit is attained uniformly for all 𝑥,𝑦∈𝑈 and in this case 𝑋 is said to be uniformly smooth. We define a function 𝜌∶[0,∞)→[0,∞), called the modulus of smoothness of 𝑋, as follows:𝜌(𝜏)1=sup2𝑥+𝑦+𝑥−𝑦−1∶𝑥,𝑦∈𝑋,‖𝑥‖=1,𝑦=𝜏. (1.3) It is know that 𝑋 is uniformly smooth if and only if lim_𝜏→0𝜌(𝜏)/𝜏=0. Let 𝑞 be a fixed real number 1<𝑞≤2. A Banach space 𝑋 is said to be 𝑞-uniformly smooth if there exists a constant 𝑐>0 such that 𝜌(𝜏)≤𝑐𝜏^𝑞 for all 𝜏>0. From [1], we know the following property.

Let 𝑞 be a real number with 1<𝑞≤2 and let 𝑋 be a Banach space. Then, 𝑋 is 𝑞-uniformly smooth if and only if there exists a constant 𝐾≥1 such that𝑥+𝑦^𝑞+𝑥−𝑦^𝑞≤2‖𝑥‖^𝑞+𝐾𝑦^𝑞,∀𝑥,𝑦∈𝑋. (1.4) The best constant 𝐾 in the above inequality is called the 𝑞-uniformly smoothness constant of 𝑋 (see [1] for more details).

Let 𝑋 be a real Banach space and 𝑋^∗ the dual space of 𝑋. Let ⟨⋅,⋅⟩ denote the pairing between 𝑋 and 𝑋^∗. For 𝑞>1, the generalized duality mapping𝐽_𝑞(𝑥)=𝑓∈𝑋^∗∶𝑥,𝑓=‖𝑥‖^𝑞,𝑓=‖𝑥‖^𝑞−1,∀𝑥∈𝑋. (1.5) In particular, if 𝑞=2, the mapping 𝐽₂ is called the normalized duality mapping and usually, we write 𝐽₂=𝐽. If 𝑋 is a Hilbert space, then 𝐽=𝐼 is the identity. Further, we have the following properties of the generalized duality mapping 𝐽_𝑞: (1)𝐽_𝑞(𝑥)=‖𝑥‖^𝑞−2𝐽₂(𝑥) for all 𝑥∈𝑋 with 𝑥≠0, (2)𝐽_𝑞(𝑡𝑥)=𝑡^𝑞−1𝐽_𝑞(𝑥) for all 𝑥∈𝑋 and 𝑡∈[0,∞), (3)𝐽_𝑞(−𝑥)=−𝐽_𝑞(𝑥) for all 𝑥∈𝑋.

It is know that if 𝑋 is smooth, then 𝐽 is single-valued, which is denoted by 𝑗.

Definition 1.1. Let 𝐶 be a nonempty closed convex subset of 𝑋. A mapping 𝑇∶𝐶→𝑋 is said to be (i)nonexpansive if 𝑇𝑥−𝑇𝑦≤𝑥−𝑦,∀𝑥,𝑦∈𝐶, (1.6)(ii)Lipschitzian if there exists a constant 𝐿>0 such that 𝑇𝑥−𝑇𝑦≤𝐿𝑥−𝑦,∀𝑥,𝑦∈𝐶, (1.7)(iii)contraction if there exists a constant 𝛼∈(0,1) such that 𝑇𝑥−𝑇𝑦≤𝛼𝑥−𝑦,∀𝑥,𝑦∈𝐶. (1.8)

Remark 1.2. We denote 𝐹(𝑇) as the set of fixed points of 𝑇. We know that 𝐹(𝑇) is nonempty if 𝐶 is bounded; for more detail see [2].

Definition 1.3. A one-parameter family 𝒮={𝑇(𝑡)∶𝑡∈ℝ⁺} from 𝐶 of 𝑋 into itself is said to be a nonexpansive semigroup on 𝐶 if it satisfies the following conditions: (i)𝑇(0)𝑥=𝑥 for all 𝑥∈𝐶,(ii)𝑇(𝑠+𝑡)=𝑇(𝑠)∘𝑇(𝑡) for all 𝑠,𝑡∈ℝ⁺,(iii)for each 𝑥∈𝐶 the mapping 𝑡↦𝑇(𝑡)𝑥 is continuous,(iv)‖𝑇(𝑡)𝑥−𝑇(𝑡)𝑦‖≤‖𝑥−𝑦‖ for all 𝑥,𝑦∈𝐶 and 𝑡∈ℝ⁺.

Remark 1.4. We denote by 𝐹(𝒮) the set of all common fixed points of 𝒮, that is 𝐹(𝒮)∶=⋂_𝑡∈ℝ⁺𝐹(𝑇(𝑡))={𝑥∈𝐶∶𝑇(𝑡)𝑥=𝑥}. We know that 𝐹(𝒮) is nonempty if 𝐶 is bounded, see [3].

Let 𝐶 be a nonempty closed convex subset of a smooth Banach space 𝑋. Recall the following definitions of a nonlinear mapping 𝐵∶𝐶→𝑋, the following are mentioned.

Definition 1.5. Given a mapping 𝐵∶𝐶→𝑋, (i)𝐵 is said to be accretive𝐵𝑥−𝐵𝑦,𝐽𝑥−𝑦≥0,∀𝑥,𝑦∈𝐶, (1.9)(ii)𝐵 is said to be 𝛼-strongly accretive if there exists a constant 𝛼>0 such that 𝐵𝑥−𝐵𝑦,𝐽𝑥−𝑦≥𝛼𝑥−𝑦²,∀𝑥,𝑦∈𝐶, (1.10)(iii)𝐵 is said to be 𝛼-inverse-strongly accretive or 𝛼-cocoercive if there exists a constant 𝛼>0 such that 𝐵𝑥−𝐵𝑦,𝐽𝑥−𝑦≥𝛼𝐵𝑥−𝐵𝑦²,∀𝑥,𝑦∈𝐶, (1.11)(iv)𝐵 is said to be 𝛼-relaxed cocoercive if there exists a constant 𝛼>0 such that 𝐵𝑥−𝐵𝑦,𝐽𝑥−𝑦≥−𝛼𝐵𝑥−𝐵𝑦²,∀𝑥,𝑦∈𝐶, (1.12)(v)𝐵 is said to be (𝛼,𝛽)-relaxed cocoercive if there exist positive constants 𝛼>0 and 𝛽>0 such that ⟨𝐵𝑥−𝐵𝑦,𝐽𝑥−𝑦⟩≥−𝛼𝐵𝑥−𝐵𝑦²+𝛽𝑥−𝑦²,∀𝑥,𝑦∈𝐶. (1.13)

Remark 1.6. (1) Every 𝛼-strongly accretive mapping is an accretive mapping.
(2) Every 𝛼-strongly accretive mapping is a (𝛽,𝛼)-relaxed cocoercive mapping for any positive constant 𝛽 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., [4]).
(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 [5]. Several classes of relaxed cocoercive variational inequalities have been studied in [6, 7].

The resolvent operator technique for solving variational inequalities and variational inclusions is interesting and important. The resolvent equation technique is used to develop powerful and efficient numerical techniques for solving various classes of variational inequalities, inclusions, and related optimization problems.

Definition 1.7. Let 𝑀∶𝑋→2^𝑋 be a multivalued maximal accretive mapping. The single-valued mapping 𝐽_(𝑀,𝜌)∶𝑋→𝑋, defined by 𝐽_(𝑀,𝜌)(𝑢)=𝐼+𝜌𝑀⁻¹(𝑢),∀𝑢∈𝑋, (1.14) is called resolvent operator associated with 𝑀, where 𝜌 is any positive number and 𝐼 is the identity mapping.

In 2010, Qin et al. [8] introduced a system of quasivariational inclusions as follows. Find (𝑥^∗,𝑦^∗)∈𝑋×𝑋 such that0∈𝑥^∗−𝑦^∗+𝜌₁𝐵₁𝑦^∗+𝑀₁𝑥^∗,0∈𝑦^∗−𝑥^∗+𝜌₂𝐵₂𝑥^∗+𝑀₂𝑦^∗, (1.15) where 𝐵_𝑖∶𝑋→𝑋 and 𝑀_𝑖∶𝑋→2^𝑋 are nonlinear mappings for all 𝑖=1,2. As special cases of problem (1.15), we have the following. (1)If 𝐵₁=𝐵₂=𝐵 and 𝑀₁=𝑀₂=𝑀, then problem (1.15) is reduced to the following. Find (𝑥^∗,𝑦^∗)∈𝑋×𝑋 such that 0∈𝑥^∗−𝑦^∗+𝜌₁𝐵𝑦^∗+𝑀𝑥^∗,0∈𝑦^∗−𝑥^∗+𝜌₂𝐵𝑥^∗+𝑀𝑦^∗. (1.16)(2)Further, if 𝑥^∗=𝑦^∗ in problem (1.16), then problem (1.16) is reduced to the following. Find 𝑥^∗∈𝑋 such that 0∈𝐵𝑥^∗+𝑀𝑥^∗. (1.17)

The problem (1.17) is called variational inclusion problem denoted by 𝑉(𝑋,𝐵,𝑀).

Here we have examples of the variational inclusion (1.17). If 𝑀=𝜕𝛿_𝐶, where 𝐶 is a nonempty closed convex subset of 𝑋, and 𝛿_𝐶∶𝑋→[0,∞)𝛿_𝐶(𝑥)=0,𝑥∈𝐶,+∞,𝑥∉𝐶, (1.18) then the variational inclusion problem (1.17) is equivalent (see [9]) to finding 𝑢∈𝐶 such that⟨𝐵𝑢,𝑣−𝑢⟩≥0,∀𝑥∈𝐶. (1.19) This problem is called Hartman-Stampacchia variational inequality problem denoted by VI(𝐶,𝐵).

Let 𝐷 be a subset of 𝐶, and let 𝑃 be a mapping of 𝐶 into 𝐷. Then, 𝑃 is said to be sunny if𝑃(𝑃𝑥+𝑡(𝑥−𝑃𝑥))=𝑃𝑥, (1.20) whenever 𝑃𝑥+𝑡(𝑥−𝑃𝑥)∈𝐶 for 𝑥∈𝐶 and 𝑡≥0. A mapping 𝑃 of 𝐶 into itself is called a retraction if 𝑃²=𝑃. If a mapping 𝑃 of 𝐶 into itself is a retraction, then 𝑃𝑧=𝑧 for all 𝑧∈𝑅(𝑃), where 𝑅(𝑃) is the range of 𝑃. A subset 𝐷 of 𝐶 is called a sunny nonexpansive retract of 𝐶 if there exists a sunny nonexpansive retraction from 𝐶 onto 𝐷.

In 2006, Aoyama et al. [10] considered the following problem: find 𝑢∈𝐶 such that⟨𝐴𝑢,𝐽(𝑣−𝑢)⟩≥0,∀𝑣∈𝐶. (1.21) They proved that the variational inequality (1.21) is equivalent to a fixed point problem. The element 𝑢∈𝐶 is a solution of the variational inequality (1.21) if and only if 𝑢∈𝐶 satisfies the following equation:𝑢=𝑃_𝐶(𝑢−𝜆𝐴𝑢), (1.22) where 𝜆>0 is a constant and 𝑃_𝐶 is a sunny nonexpansive retraction from 𝑋 onto 𝐶.

The following results describe a characterization of sunny nonexpansive retractions on a smooth Banach space.

Proposition 1.8 (see [11]). Let 𝑋 be a smooth Banach space and 𝐶 a nonempty subset of 𝑋. Let 𝑃∶𝑋→𝐶 be a retraction and 𝐽 the normalized duality mapping on 𝑋. Then the following are equivalent: (1)𝑃 is sunny and nonexpansive,(2)⟨𝑥−𝑃𝑥,𝐽(𝑦−𝑃𝑥)⟩≤0, for all 𝑥∈𝑋, 𝑦∈𝐶.

Proposition 1.9 (see [12]). Let 𝐶 be a nonempty closed convex subset of a uniformly convex and uniformly smooth Banach space 𝐸 and 𝑇 a nonexpansive mapping of 𝐶 into itself with 𝐹(𝑇)≠∅. Then the set 𝐹(𝑇) is a sunny nonexpansive retract of 𝐶.

For the class of nonexpansive mappings, one classical way to study nonexpansive mappings is to use contractions to approximate a nonexpansive mapping [13, 14]. More precisely, take 𝑡∈(0,1), and define a contraction 𝑇_𝑡∶𝐶→𝐶 by𝑇_𝑡𝑥=𝑡𝑢+(1−𝑡)𝑇_𝑡𝑥,∀𝑥∈𝐶, (1.23) where 𝑢∈𝐶 is a fixed point. Banachs contraction mapping principle guarantees that 𝑇_𝑡 has a unique fixed point 𝑥_𝑡∈𝐶, that is,𝑥_𝑡=𝑡𝑢+(1−𝑡)𝑇_𝑡𝑥_𝑡. (1.24) It is unclear, in general, what the behavior of 𝑥_𝑡 is as 𝑡→0, even if 𝑇 has a fixed point. However, in the case of 𝑇 having a fixed point, Ceng et al. [15] proved that, if 𝑋 is a Hilbert space, then 𝑥_𝑡 converges strongly to a fixed point of 𝑇. Reich [14] extended Browders result to the setting of Banach spaces and proved that, if 𝑋 is a uniformly smooth Banach space, then 𝑥_𝑡 converges strongly to a fixed point of 𝑇, and the limit defines the (unique) sunny nonexpansive retraction from 𝐶 onto 𝐹(𝑇).

Reich [14] showed that, if 𝑋 is uniformly smooth and 𝐷 is the fixed point set of a nonexpansive mapping from 𝐶 into itself, then there is a unique sunny nonexpansive retraction from 𝐶 onto 𝐷 and it can be constructed as follows.

Proposition 1.10 (see [14]). Let 𝑋 be a uniformly smooth Banach space and 𝑇∶𝐶→𝐶 a nonexpansive mapping such that 𝐹(𝑇)≠∅. For each fixed 𝑢∈𝐶 and every 𝑡∈(0,1), the unique fixed point 𝑥_𝑡∈𝐶 of the contraction 𝐶∋𝑥↦𝑡𝑢+(1−𝑡)𝑇𝑥 converges strongly as 𝑡→0 to a fixed point of 𝑇. Define 𝑃∶𝐶→𝐷 by 𝑃𝑢=𝑠−lim_𝑡→0𝑥_𝑡. Then 𝑃 is the unique sunny nonexpansive retract from 𝐶 onto 𝐷; that is, 𝑃 satisfies the property, 𝑥−𝑃𝑥,𝐽𝑦−𝑃𝑥≤0,∀𝑥∈𝐶,𝑦∈𝐷. (1.25)

Many authors have studied the problems of finding a common element of the set of fixed points of a nonexpansive mapping and one of the sets of solutions to the variational inclusion and variational inequalities (1.15)–(1.17) and (1.21) by using different iterative methods (see, e.g., [10, 16–27]).

Recently, Qin et al. [8] considered the problem of finding the solutions of a general system of variational inclusion (1.15) with 𝛼-inverse strongly accretive mappings. To be more precise, they obtained the following results.

Lemma 1.11. For any (𝑥^∗,𝑦^∗)∈𝑋×𝑋, where 𝑦^∗=𝐽_{(𝑀2,𝜌2)}(𝑥^∗−𝜌₂𝐵₂𝑥^∗), (𝑥^∗,𝑦^∗) is a solution of the problem (1.15) if and only if 𝑥^∗ is a fixed point of the mapping 𝑄 defined by 𝑄𝑥∶=𝐽_{(𝑀1,𝜌1)}𝐽_{(𝑀2,𝜌2)}𝑥−𝜌₂𝐵₂𝑥−𝜌₁𝐵₁𝐽_{(𝑀2,𝜌2)}𝑥−𝜌₂𝐵₂𝑥. (1.26)

Theorem QCCK (see [8]). Let 𝑋 be a uniformly convex and 2-uniformly smooth Banach space with the smoothness constant 𝐾. Let 𝑀_𝑖∶𝑋→2^𝑋 be a maximal monotone mapping and 𝐵_𝑖∶𝑋→𝑋 be a 𝜌_𝑖-inverse-strongly accretive mapping, respectively, for all 𝑖=1,2. Let 𝑇∶𝑋→𝑋 be a 𝜖-strict pseudocontraction such that 𝐹(𝑇)≠∅. Define a mapping 𝑆 by 𝑆𝑥=(1−𝜖/𝐾²)𝑥+(𝜖/𝐾²)𝑇𝑥, for all 𝑥∈𝑋. Assume that Ω=𝐹(𝑇)∩𝐹(𝑄), where 𝑄 is defined as in Lemma 1.11. Let 𝑥₁=𝑢∈𝐶 and let {𝑥_𝑛} be a sequence defined by 𝑧_𝑛=𝐽_{(𝑀2,𝜌2)}𝑥_𝑛−𝜌₂𝐵₂𝑥_𝑛,𝑦_𝑛=𝐽_{(𝑀1,𝜌1)}𝑧_𝑛−𝜌₁𝐵₁𝑧_𝑛,𝑥_𝑛+1=𝛼_𝑛𝑢+𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛−𝛼_𝑛𝜇𝑆𝑥_𝑛+1−𝜇𝑦_𝑛,∀𝑛≥1, (1.27) where 𝜇∈(0,1), 𝜌₁∈(0,𝛾₁/𝐾²], 𝜌₂∈(0,𝛾₂/𝐾²], and {𝛼_𝑛}, {𝛽_𝑛} are sequences in (0,1). If the control consequences {𝛼_𝑛} and {𝛽_𝑛} satisfy the following restrictions: (C1)0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1,(C2)lim_𝑛→∞𝛼_𝑛=0 and ∑^∞_𝑛=1𝛼_𝑛=∞, then {𝑥_𝑛} converges strongly to 𝑥^∗=𝑃_Ω𝑢, where 𝑃_Ω is the sunny nonexpansive retraction from 𝑋 onto Ω and (𝑥^∗,𝑦^∗) is a solution of the problem (1.15), where 𝑦^∗=𝐽_{(𝑀2,𝜌2)}(𝑥^∗−𝜌₂𝐵₂𝑥^∗).

Iterative methods for nonexpansive mappings have recently been applied to solve minimization problems; see, for example, [28–32]. Let 𝐻 be a real Hilbert space, whose inner product and norm are denoted by ‖⋅,⋅‖ and ⟨⋅⟩, respectively. Let 𝐴 be a strongly positive bounded linear operator on 𝐻: that is, there is a constant 𝛾>0 with property⟨𝐴𝑥,𝑥⟩≥𝛾‖𝑥‖²,∀𝑥∈𝐻. (1.28) A typical problem is to minimize a quadratic function over the set of the fixed points of a nonexpansive mapping on a real Hilbert space 𝐻min_𝑥∈𝐹12⟨𝐴𝑥,𝑥⟩−⟨𝑥,𝑢⟩, (1.29) where 𝐹 is the fixed point set of a nonexpansive mapping 𝑇 on 𝐻 and 𝑢 is a given point in 𝐻.

In [33], Moudafi introduced the viscosity approximation method and proved that if 𝐻 is a real Hilbert space, the sequence {𝑥_𝑛} generated by the following algorithm:𝑥₀𝑥=𝑢∈𝐻chosenarbitrarily,_𝑛+1=𝛼_𝑛𝑓𝑥_𝑛+1−𝛼_𝑛𝑇𝑥_𝑛,∀𝑛≥0, (1.30) where 𝑓∶𝐶→𝐶 is a contraction mapping with a constant 𝛼∈(0,1) and {𝛼_𝑛}⊂(0,1) satisfies certain conditions, converges strongly to a fixed point of 𝑇 in 𝐶 which is the unique solution of the following variational inequality:𝑓−𝐼𝑥^∗,𝑥−𝑥^∗≤0,∀𝑥∈𝐹(𝑇). (1.31) In 2006, Marino and Xu [34] introduced the following general iterative method:𝑥₀𝑥=𝑢∈𝐻chosenarbitrarily,_𝑛+1=𝛼_𝑛𝛾𝑓𝑥_𝑛+𝐼−𝛼_𝑛𝐴𝑇𝑥_𝑛,∀𝑛≥0, (1.32) where 𝐴 is a strongly positive bounded linear operator on a Hilbert space 𝐻. They proved that, if the sequence {𝛼_𝑛} of parameters satisfies appropriate conditions, then the sequence {𝑥_𝑛} generated by (1.32) converges strongly to the unique solution of the variational inequality:𝛾𝑓−𝐴𝑥^∗,𝑥−𝑥^∗≤0,∀𝑥∈𝐹(𝑇), (1.33) which is the optimality condition for the minimization problem:min_𝑥∈𝐶12⟨𝐴𝑥,𝑥⟩−ℎ(𝑥), (1.34) where 𝐶 is the fixed point set of a nonexpansive mapping 𝑇 and ℎ is a potential function for 𝛾𝑓 (i.e., ℎ^′(𝑥)=𝛾𝑓(𝑥) for all 𝑥∈𝐻).

In a smooth Banach space, we always assume that 𝐴 is strongly positive (see [35]), that is, a constant 𝛾>0 with the property⟨𝐴𝑥,𝐽(𝑥)⟩≥𝛾‖𝑥‖²,‖𝑎𝐼−𝑏𝐴‖=sup_{‖𝑥‖≤1}|⟨(𝑎𝐼−𝑏𝐴)𝑥,𝐽(𝑥)⟩|,𝑎∈[0,1],𝑏∈[−1,1], (1.35) where 𝐼 is the identity mapping and 𝐽 is the normalized duality mapping.

Recently, Sunthrayuth and Kumam [36] introduced the following iterative method for nonexpansive semigroup 𝒮={𝑇(𝑡)∶𝑡∈ℝ⁺} in Banach spaces,𝑥₀𝑥=𝑢∈𝐶chosenarbitrarily,_𝑛+1=𝛼_𝑛𝛾𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝐼−𝛼_𝑛𝐴1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑥_𝑛𝑑𝑠,∀𝑛≥0. (1.36) They proved strong convergence theorem of the iterative scheme {𝑥_𝑛} defined by (1.36) converges strongly to the common fixed point of 𝑥^∗∈𝐹(𝒮) solving the variational inequality𝛾𝑓−𝐴𝑥^∗,𝐽𝑥−𝑥^∗≤0,∀𝑥∈𝐹(𝒮). (1.37) In 2010, Kamraksa and Wangkeeree [37] introduced a general iterative approximation method for finding common elements of the set of solutions to a general system of variational inclusions with Lipschitzian and relaxed cocoercive mappings and the set common fixed points of a countable family of strict pseudocontractions. They proved the strong convergence theorems of such iterative scheme for finding a common element of such two sets which is a unique solution of some variational inequality and is also the optimality condition for some minimization problems in a strictly convex and 2-uniformly smooth Banach space.

In this paper, we are motivated and inspire by idea of Qin et al. [8] and Sunthrayuth and Kumam [36].

First, we introduce a new general system of variational inclusions in Banach spaces as follows.

Let 𝑋 be Banach spaces. We consider a system of quasivariational inclusions as follows. Finding (𝑥^∗,𝑦^∗,𝑧^∗)∈𝑋×𝑋×𝑋 such that0∈𝑥^∗−𝑦^∗+𝜌₁𝐵₁𝑦^∗+𝑀₁𝑥^∗,0∈𝑦^∗−𝑧^∗+𝜌₂𝐵₂𝑧^∗+𝑀₂𝑦^∗,0∈𝑧^∗−𝑥^∗+𝜌₃𝐵₃𝑥^∗+𝑀₃𝑧^∗, (1.38) which is called a new general system of variational inclusions in Banach spaces, 𝐵_𝑖∶𝑋→𝑋 and 𝑀_𝑖∶𝑋→2^𝑋 are nonlinear mappings for all 𝑖=1,2,3. As special cases of problem (1.38), we have the following. (1)If 𝐵₁=𝐵₂=𝐵₃=𝐵 and 𝑀₁=𝑀₂=𝑀₃=𝑀, then problem (1.38) is reduced to the following. Finding (𝑥^∗,𝑦^∗,𝑧^∗)∈𝑋×𝑋×𝑋 such that 0∈𝑥^∗−𝑦^∗+𝜌₁𝐵𝑦^∗+𝑀𝑥^∗,0∈𝑦^∗−𝑧^∗+𝜌₂𝐵𝑧^∗+𝑀𝑦^∗,0∈𝑧^∗−𝑥^∗+𝜌₃𝐵𝑥^∗+𝑀𝑧^∗. (1.39)(2)Further, if 𝐵₃=𝑀₃=0, 𝑧^∗=𝑥^∗, then problem (1.38) is reduced to problem (1.15).

Second, we study a general iterative approximation method (3.1) below, for finding common elements of the set of solutions of a new general system of variational inclusions (1.38) with set-valued maximal monotone mapping and Lipschitzian relaxed cocoercive mappings and the set common fixed points of nonexpansive semigroup in the framework of Banach spaces. Moreover, we prove the strong convergence of the proposed iterative method under some certain control conditions. The results presented in this paper extend and improve the results of Qin et al. [8] and Sunthrayuth and Kumam [36], and many authors.

2. Preliminaries

This section collects some results that will be used in the proofs of our main results.

Lemma 2.1 (see [38]). The resolvent operator 𝐽_(𝑀,𝜌) associated with 𝑀 is single valued and nonexpansive for all 𝜌>0.

Lemma 2.2 (see [39]). Let 𝑋 be a real 2-uniformly smooth Banach space with the best smoothness constant 𝐾. Then, the following inequality holds: 𝑥+𝑦²≤‖𝑥‖²+2⟨𝑦,𝐽𝑥⟩+2𝐾𝑦²,∀𝑥,𝑦∈𝑋. (2.1)

Lemma 2.3 (see [40]). In a real Banach space 𝑋, the following inequality holds: 𝑥+𝑦²≤‖𝑥‖²+2𝑦,𝐽𝑥+𝑦,∀𝑥,𝑦∈𝑋. (2.2)

Now, we present the concept of a uniformly asymptotically regular semigroup (see [41–43]).

Definition 2.4. Let 𝐶 be a nonempty closed convex subset of a Banach space 𝑋, 𝒮={𝑇(𝑡)∶𝑡∈ℝ⁺} be a continuous operator semigroup on 𝐶. Then 𝒮 is said to be uniformly asymptotically regular (in short, u.a.r.) on 𝐶 if for all ℎ≥0 and any bounded subset 𝐵 of 𝐶 such that lim_𝑡→∞sup_𝑥∈𝐵‖𝑇(ℎ)𝑇(𝑡)𝑥−𝑇(𝑡)𝑥‖=0. (2.3)

Lemma 2.5 (see [44]). Let 𝐶 be a nonempty closed convex subset of a uniformly Banach space 𝑋, 𝐵 be a bounded closed convex subset of 𝐶. If we denote 𝒮={𝑇(𝑡)∶𝑡∈ℝ⁺} a nonexpansive semigroup on 𝐶 such that 𝐹(𝒮)∶=⋂_𝑡∈ℝ⁺𝐹(𝑇(𝑡))≠∅. For all ℎ≥0, the set 𝜎_𝑡(𝑥)=(1/𝑡)∫^𝑡₀𝑇(𝑠)𝑥𝑑𝑠, then lim_𝑡→∞sup_𝑥∈𝐵𝜎_𝑡(𝑥)−𝑇(ℎ)𝜎_𝑡(𝑥)=0. (2.4)

Remark 2.6. It is easy to check that the set {𝜎_𝑡∶𝑡∈ℝ⁺} defined by Lemma 2.5 is a u.a.r. nonexpansive semigroup on 𝐶 (see [45] for more detail).

Lemma 2.7 (see [46]). Let 𝐶 be a nonempty closed convex subset of 𝑋 and let 𝒮={𝑇(𝑡)∶𝑡∈ℝ⁺} be a u.a.r. nonexpansive semigroup on 𝐶 such 𝐹(𝒮)∶=⋂_𝑡∈ℝ⁺𝐹(𝑇(𝑡))≠∅ and at least there exists a 𝑇(𝑡) which is demicompact. Then, for each 𝑥∈𝐶, there exists a sequence {𝑇(𝑡_𝑛)∶𝑡_𝑛∈ℝ⁺,𝑛∈ℕ}⊂𝑇(𝑡), such that {𝑇(𝑡_𝑛)𝑥} converges strongly to some point in 𝐹(𝒮), where lim_𝑛→∞𝑡_𝑛=∞.

Remark 2.8. By Lemma 2.7, we can see that, for each 𝑥∈𝐶, there is a corresponding unique point 𝑦∈𝐹(𝒮), thus we can define a mapping 𝑇 such that 𝑇𝑥=𝑦 and it is easy to see that 𝐹(𝑇)=𝐹(𝒮).

Remark 2.9. From the definition of 𝑇, we can see that 𝑇 is a nonexpansive mapping. Actually, by Lemma 2.7, let 𝑥₁∈𝐶; then there exists a sequence {𝑇(𝑡_𝑛)} of {𝑇(ℎ)} such that 𝑇𝑥₁=lim_𝑛→∞𝑇(𝑡_𝑛)𝑥₁=𝑦₁∈𝐹(𝒮). Further, for any other point 𝑥₂∈𝐶, by the definition of a u.a.r., we can get a subsequence {𝑇(𝑡_𝑛𝑘)} of {𝑇(𝑡_𝑛)} such that 𝑇(𝑡_𝑛𝑘)→𝑦₂∈𝐹(𝒮), then 𝑇𝑥₁−𝑇𝑥₂=lim_𝑘→∞𝑇𝑡_𝑛𝑘𝑥₁−𝑇𝑡_𝑛𝑘𝑥₂≤𝑥₁−𝑥₂. (2.5)

Lemma 2.10 (see [35]). Let 𝐶 be a nonempty closed convex subset of a reflexive, smooth Banach space 𝑋 which admits a weakly sequentially continuous duality mapping 𝐽 from 𝑋 into 𝑋^∗, 𝑇 be a nonexpansive mapping such that 𝐹(𝑇)≠∅, 𝑓∶𝐶→𝐶 be a contraction mapping with a coefficient 𝛼∈(0,1) and 𝐴 be a strongly positive bounded linear operator with a coefficient 𝛾>0. Let 𝑡∈(0,1) such that 𝑡≤‖𝐴‖⁻¹ and 0<𝛾<𝛾/𝛼 which satisfies 𝑡→0. Then the sequence {𝑥_𝑡} defined by 𝑥_𝑡=𝑡𝛾𝑓𝑥_𝑡+(𝐼−𝑡𝐴)𝑇𝑥_𝑡, (2.6) converges strongly to the common fixed point 𝑥^∗ as 𝑡→0, where 𝑥^∗ is a unique solution in 𝐹(𝒮) of the variational inequality 𝛾𝑓−𝐴𝑥^∗,𝐽𝑧−𝑥^∗≤0,∀𝑧∈𝐹(𝒮). (2.7)

Lemma 2.11 (see [47]). Let 𝐶 be a closed convex subset of a strictly convex Banach space 𝑋. Let {𝑇_𝑛∶𝑛∈ℕ} be a sequence of nonexpansive mappings on 𝐶. Suppose ⋂^∞_𝑛=1𝐹(𝑇_𝑛) is nonempty. Let {𝜇_𝑛} be a sequence of positive numbers with ∑^∞_𝑛=1𝜇_𝑛=1. Then a mapping 𝑆 on 𝐶 defined by 𝑆𝑥=∑^∞_𝑛=1𝜇_𝑛𝑇_𝑛𝑥 for all 𝑥∈𝐶 is well defined, nonexpansive and 𝐹(𝑆)=⋂^∞_𝑛=1𝐹(𝑇_𝑛) holds.

Lemma 2.12 (see [48]). Let {𝑥_𝑛} and {𝑣_𝑛} be bounded sequences in a Banach space 𝑋 and let {𝛽_𝑛} be a sequence in [0,1] with 0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1. Suppose 𝑥_𝑛+1=(1−𝛽_𝑛)𝑣_𝑛+𝛽_𝑛𝑥_𝑛 for all integers 𝑛≥0 and limsup_𝑛→∞(‖𝑣_𝑛+1−𝑣_𝑛‖−‖𝑥_𝑛+1−𝑥_𝑛‖)≤0. Then, lim_𝑛→∞‖𝑣_𝑛−𝑥_𝑛‖=0.

Lemma 2.13 (see [35]). Assume that 𝐴 is a strongly positive linear bounded operator on a smooth Banach space 𝑋 with coefficient 𝛾>0 and 0<𝜌≤‖𝐴‖⁻¹. Then ‖𝐼−𝜌𝐴‖≤1−𝜌𝛾.

If a Banach space 𝑋 admits a sequentially continuous duality mapping 𝐽 from weak topology to weak star topology, then by Lemma 1 of [49], we have that duality mapping 𝐽 is a single value. In this case, the duality mapping 𝐽 is said to be a weakly sequentially continuous duality mapping, that is, for each {𝑥_𝑛}⊂𝑋 with 𝑥_𝑛⇀𝑥, we have 𝐽(𝑥_𝑛)⇀^∗𝐽(𝑥) (see [49–51] for more details).

A Banach space 𝑋 is said to be satisfying Opial's condition if for any sequence 𝑥_𝑛⇀𝑥 for all 𝑥∈𝑋 implies limsup_𝑛→∞𝑥_𝑛−𝑥<limsup_𝑛→∞𝑥_𝑛−𝑦,∀𝑦∈𝑋,with𝑥≠𝑦. (2.8)

By Theorem 1 in [49], it is well known that if 𝑋 admits a weakly sequentially continuous duality mapping, then 𝑋 satisfies Opial's condition, and 𝑋 is smooth.

Lemma 2.14 (see [50], Demiclosed principle). Let 𝐶 be a nonempty closed convex subset of a reflexive Banach space 𝑋 which satisfies Opial's condition and suppose 𝑇∶𝐶→𝑋 is nonexpansive. Then the mapping 𝐼−𝑇 is demiclosed at zero, that is, 𝑥_𝑛⇀𝑥 and 𝑥_𝑛−𝑇𝑥_𝑛→0 implies 𝑥=𝑇𝑥.

Lemma 2.15 (see [30]). Assume that {𝑎_𝑛} is a sequence of nonnegative real numbers such that 𝑎_𝑛+1≤1−𝜏_𝑛𝑎_𝑛+𝜎_𝑛, (2.9) where {𝜏_𝑛} is a sequence in (0,1) and {𝜎_𝑛} is a sequence in ℝ such that (i)∑^∞_𝑛=0𝜏_𝑛=∞, (ii)limsup_𝑛→∞(𝜎_𝑛/𝜏_𝑛)≤0 or ∑^∞_𝑛=0|𝜎_𝑛|<∞. Then, lim_𝑛→∞𝑎_𝑛=0.

Let 𝑋 be a real 2-uniformly smooth Banach space 𝑋 with the smoothness constant 𝐾. Let 𝐵∶𝑋→𝑋 be an 𝐿_𝐵-Lipschitzian and relaxed (𝑐,𝑑)-cocoercive mapping, we defined a function 𝑝∶(0,+∞)→(−∞,+∞) by 𝑝𝜌∶=1+2𝜌𝑐𝐿²_𝐵−2𝜌𝑑+2𝜌²𝐾²𝐿²_𝐵,∀𝜌∈(0,+∞). (2.10) Consequence, we put 𝜃_𝜌=𝑝𝜌𝜌1,if𝑝>0,𝜌1+𝐾,if𝑝≤0. (2.11)

In order to prove our main result, the following lemmas are needed.

Lemma 2.16. Let 𝑋 be a real 2-uniformly smooth Banach space 𝑋 with the smoothness constant 𝐾. Let 𝐵∶𝑋→𝑋 be an 𝐿_𝐵-Lipschitzian and relaxed (𝑐,𝑑)-cocoercive mapping. Then (𝐼−𝜌𝐵)𝑥−(𝐼−𝜌𝐵)𝑦²≤𝜃²_𝜌𝑥−𝑦². (2.12) In particular, if 0<𝜌≤(𝑑−𝑐𝐿²_𝐵)/𝐾²𝐿²_𝐵, then 𝐼−𝜌𝐵 is nonexpansive.

Proof. For all 𝑥,𝑦∈𝑋, from Lemma 2.2 and by the cocoercivity of the mapping 𝐵, we have (𝐼−𝜌𝐵)𝑥−(𝐼−𝜌𝐵)𝑦²=(𝑥−𝑦)−(𝜌𝐵𝑥−𝜌𝐵𝑦)²≤𝑥−𝑦²−2𝜌⟨𝐵𝑥−𝐵𝑦,𝐽𝑥−𝑦⟩+2𝜌²𝐾²𝐵𝑥−𝐵𝑦²≤𝑥−𝑦²−2𝜌−𝑐𝐵𝑥−𝐵𝑦²+𝑑𝑥−𝑦²+2𝜌²𝐾²𝐵𝑥−𝐵𝑦²=𝑥−𝑦²−2𝜌𝑑𝑥−𝑦²+2𝜌𝑐𝐵𝑥−𝐵𝑦²+2𝜌²𝐾²𝐵𝑥−𝐵𝑦²≤1+2𝜌𝑐𝐿²_𝐵−2𝜌𝑑+2𝜌²𝐾²𝐿²_𝐵𝑥−𝑦²=𝜃²_𝜌𝑥−𝑦². (2.13) It follows that 𝐼−𝜌𝐵𝑥−𝐼−𝜌𝐵𝑦≤𝜃_𝜌𝑥−𝑦. (2.14) It is clear that, if 0<𝜌≤(𝑑−𝑐𝐿²_𝐵)/𝐾²𝐿²_𝐵. Thus, we have 𝐼−𝜌𝐵 is nonexpansive.

Lemma 2.17. Let 𝑋 be a 2-uniformly smooth Banach space 𝑋. Let 𝑀_𝑖∶𝑋→2^𝑋 be the a maximal monotone mapping and 𝐵_𝑖∶𝑋→𝑋 be an 𝐿_𝑖-Lipschitzian and relaxed (𝑐_𝑖,𝑑_𝑖)-cocoercive mapping for 𝑖=1,2,3. Let 𝑄∶𝑋→𝑋 be a mapping defined by 𝑄𝑥∶=𝐽_{(𝑀1,𝜌1)}𝐽_{(𝑀2,𝜌2)}𝐽_{(𝑀3,𝜌3)}𝑥−𝜌₃𝐵₃𝑥−𝜌₂𝐵₂𝐽_{(𝑀3,𝜌3)}𝑥−𝜌₃𝐵₃𝑥−𝜌₁𝐵₁𝐽_{(𝑀2,𝜌2)}𝐽_{(𝑀3,𝜌3)}𝑥−𝜌₃𝐵₃𝑥−𝜌₂𝐵₂𝐽_{(𝑀3,𝜌3)}𝑥−𝜌₃𝐵₃𝑥,∀𝑥∈𝑋. (2.15) If 0<𝜌_𝑖≤(𝑑_𝑖−𝑐_𝑖𝐿²_𝑖)/𝐾²𝐿²_𝑖 and 𝜃_𝜌=max{𝜃_𝜌𝑖} for all 𝑖=1,2,3, then the mapping 𝑄 is nonexpansive.

Proof. For all 𝑥,𝑦∈𝑋, we have 𝑄𝑥−𝑄𝑦≤𝐽_{(𝑀1,𝜌1)}𝐽_{(𝑀2,𝜌2)}𝐽_{(𝑀3,𝜌3)}𝑥−𝜌₃𝐵₃𝑥−𝜌₂𝐵₂𝐽_{(𝑀3,𝜌3)}𝑥−𝜌₃𝐵₃𝑥−𝜌₁𝐵₁𝐽_{(𝑀2,𝜌2)}𝐽_{(𝑀3,𝜌3)}𝑥−𝜌₃𝐵₃𝑥−𝜌₂𝐵₂𝐽_{(𝑀3,𝜌3)}𝑥−𝜌₃𝐵₃𝑥−𝐽_{(𝑀1,𝜌1)}𝐽_{(𝑀2,𝜌2)}𝐽_{(𝑀3,𝜌3)}𝑦−𝜌₃𝐵₃𝑦−𝜌₂𝐵₂𝐽_{(𝑀3,𝜌3)}𝑦−𝜌₃𝐵₃𝑦−𝜌₁𝐵₁𝐽_{(𝑀2,𝜌2)}𝐽_{(𝑀3,𝜌3)}𝑦−𝜌₃𝐵₃𝑦−𝜌₂𝐵₂𝐽_{(𝑀3,𝜌3)}𝑦−𝜌₃𝐵₃𝑦≤𝐽_{(𝑀2,𝜌2)}𝐽_{(𝑀3,𝜌3)}𝑥−𝜌₃𝐵₃𝑥−𝜌₂𝐵₂𝐽_{(𝑀3,𝜌3)}𝑥−𝜌₃𝐵₃𝑥−𝜌₁𝐵₁𝐽_{(𝑀2,𝜌2)}𝐽_{(𝑀3,𝜌3)}𝑥−𝜌₃𝐵₃𝑥−𝜌₂𝐵₂𝐽_{(𝑀3,𝜌3)}𝑥−𝜌₃𝐵₃𝑥−𝐽_{(𝑀2,𝜌2)}𝐽_{(𝑀3,𝜌3)}𝑦−𝜌₃𝐵₃𝑦−𝜌₂𝐵₂𝐽_{(𝑀3,𝜌3)}𝑦−𝜌₃𝐵₃𝑦−𝜌₁𝐵₁𝐽_{(𝑀2,𝜌2)}𝐽_{(𝑀3,𝜌3)}𝑦−𝜌₃𝐵₃𝑦−𝜌₂𝐵₂𝐽_{(𝑀3,𝜌3)}𝑦−𝜌₃𝐵₃𝑦=𝐽_{(𝑀2,𝜌2)}𝐼−𝜌₂𝐵₂𝐽_{(𝑀3,𝜌3)}𝐼−𝜌₃𝐵₃𝐼−𝜌₁𝐵₁𝑥−𝐽_{(𝑀2,𝜌2)}𝐼−𝜌₂𝐵₂𝐽_{(𝑀3,𝜌3)}𝐼−𝜌₃𝐵₃𝐼−𝜌₁𝐵₁𝑦≤𝐼−𝜌₂𝐵₂𝐽_{(𝑀3,𝜌3)}𝐼−𝜌₃𝐵₃𝐼−𝜌₁𝐵₁𝑥−𝐼−𝜌₂𝐵₂𝐽_{(𝑀3,𝜌3)}𝐼−𝜌₃𝐵₃𝐼−𝜌₁𝐵₁𝑦. (2.16) From Lemma 2.16 and by the nonexpansiveness of 𝐽_{(𝑀𝑖,𝜌𝑖)} for all 𝑖=1,2,3, we have (𝐼−𝜌₂𝐵₂)𝐽_{(𝑀3,𝜌3)}(𝐼−𝜌₃𝐵₃)(𝐼−𝜌₁𝐵₁) is a nonexpansive mapping, which implies that the mapping 𝑄 is nonexpansive.

Lemma 2.18. For all (𝑥^∗,𝑦^∗,𝑧^∗)∈𝑋×𝑋×𝑋, where 𝑦^∗=𝐽_{(𝑀2,𝜌2)}(𝑧^∗−𝜌₂𝐵₂𝑧^∗) and 𝑧^∗=𝐽_{(𝑀3,𝜌3)}(𝑥^∗−𝜌₃𝐵₃𝑥^∗), (𝑥^∗,𝑦^∗,𝑧^∗) is a solution of the problem (1.38) if and only if 𝑥^∗ is a fixed point of the mapping 𝑄 defined as in Lemma 2.17.

Proof. Let (𝑥^∗,𝑦^∗,𝑧^∗)∈𝑋×𝑋×𝑋 be a solution of the problem (1.38). Then, we have 𝑦^∗−𝜌₁𝐵₁𝑦^∗∈𝐼+𝜌₁𝑀₁𝑥^∗,𝑧^∗−𝜌₂𝐵₂𝑧^∗∈𝐼+𝜌₂𝑀₂𝑦^∗,𝑥^∗−𝜌₃𝐵₃𝑥^∗∈𝐼+𝜌₃𝑀₃𝑧^∗, (2.17) which implies that 𝑥^∗=𝐽_{(𝑀1,𝜌1)}𝑦^∗−𝜌₁𝐵₁𝑦^∗,𝑦^∗=𝐽_{(𝑀2,𝜌2)}𝑧^∗−𝜌₂𝐵₂𝑧^∗,𝑧^∗=𝐽_{(𝑀3,𝜌3)}𝑥^∗−𝜌₃𝐵₃𝑥^∗. (2.18) We can deduce that (2.18) is equivalent to 𝑥^∗=𝐽_{(𝑀1,𝜌1)}𝐽_{(𝑀2,𝜌2)}𝐽_{(𝑀3,𝜌3)}𝑥^∗−𝜌₃𝐵₃𝑥^∗−𝜌₂𝐵₂𝐽_{(𝑀3,𝜌3)}𝑥^∗−𝜌₃𝐵₃𝑥^∗−𝜌₁𝐵₁𝐽_{(𝑀2,𝜌2)}𝐽_{(𝑀3,𝜌3)}𝑥^∗−𝜌₃𝐵₃𝑥^∗−𝜌₂𝐵₂𝐽_{(𝑀3,𝜌3)}𝑥^∗−𝜌₃𝐵₃𝑥^∗. (2.19) This completes the proof.

3. Main Results

In this section, we prove that the iterative scheme (3.1) below converges strongly to common element of the set of solutions of the variational inclusion with set-valued maximal monotone mapping and Lipschitzian relaxed cocoercive mapping and the set of fixed point of a family of nonexpansive semigroup in a uniformly convex and 2-uniformly smooth Banach space under some certain control conditions.

Now, we prove the strong convergence theorem of the sequence (3.1) for solving the problem (1.38).

Theorem 3.1. Let 𝑋 be a uniformly convex and 2-uniformly smooth Banach space which admits a weakly continuous duality mapping and has the smoothness constant 𝐾. Let 𝑀_𝑖∶𝑋→2^𝑋 be a maximal monotone mapping and 𝐵_𝑖∶𝑋→𝑋 be a 𝐿_𝑖-Lipschitzian and relaxed (𝑐_𝑖,𝑑_𝑖)-cocoercive mapping with 𝜌_𝑖∈(0,(𝑑_𝑖−𝑐_𝑖𝐿²_𝑖)/𝐾²𝐿²_𝑖] and 𝜃_𝜌=max{𝜃_𝜌𝑖} for all 𝑖=1,2,3. Let 𝒮={𝑇(𝑡)∶𝑡∈ℝ⁺} be a nonexpansive semigroup from 𝑋 into itself and at least there exists a 𝑇(𝑡) which is demicompact. Assume that Ω∶=𝐹(𝒮)∩𝐹(𝑄)≠∅, where 𝑄 is defined as in Lemma 2.17. Let 𝑓∶𝑋→𝑋 be a contraction mapping with a coefficient 𝛼∈(0,1) and 𝐴 be a strongly positive linear bounded self adjoint operator with a coefficient 𝛾∈(0,1) such that ‖𝐴‖≤1 and 0<𝛾<𝛾/𝛼. Let {𝑥_𝑛} be a sequence defined by 𝑥₁𝑧∈𝑋chosenarbitrarily,_𝑛=𝐽_{(𝑀3,𝜌3)}𝑥_𝑛−𝜌₃𝐵₃𝑥_𝑛,𝑦_𝑛=𝐽_{(𝑀2,𝜌2)}𝑧_𝑛−𝜌₂𝐵₂𝑧_𝑛,𝑣_𝑛=𝐽_{(𝑀1,𝜌1)}𝑦_𝑛−𝜌₁𝐵₁𝑦_𝑛,𝑥_𝑛+1=𝛼_𝑛𝛾𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝐼−𝛼_𝑛𝐴𝜇1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑥_𝑛𝑑𝑠+1−𝜇𝑣_𝑛,∀𝑛≥1, (3.1) where 𝜇∈(0,1), {𝛼_𝑛}^∞_𝑛=1, {𝛽_𝑛}^∞_𝑛=1 are the sequences in (0,1) which satisfies 𝛼_𝑛+𝛽_𝑛≤1 and {𝑡_𝑛}^∞_𝑛=1 is a positive real divergent sequence satisfy the following restrictions: (C1)lim_𝑛→∞𝛼_𝑛=0 and ∑^∞_𝑛=1𝛼_𝑛=∞, (C2)0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1, (C3)lim_𝑛→∞|𝑡_𝑛+1−𝑡_𝑛|/𝑡_𝑛+1=0. Then the sequence {𝑥_𝑛} defined by (3.1) converges strongly to 𝑥^∗, which 𝑥^∗ solves the variational inequality 𝛾𝑓−𝐴𝑥^∗,𝐽𝑧−𝑥^∗≤0,∀𝑧∈Ω, (3.2) and (𝑥^∗,𝑦^∗,𝑧^∗) is a solution of general system of variational inequality problem (1.38), where 𝑦^∗=𝐽_{(𝑀2,𝜌2)}(𝑧^∗−𝜌₂𝐵₂𝑧^∗) and 𝑧^∗=𝐽_{(𝑀3,𝜌3)}(𝑥^∗−𝜌₃𝐵₃𝑥^∗).

Proof. First, we show {𝑥_𝑛} is bounded. By the condition (C1), we may assume, with no loss of generality, that 𝛼_𝑛≤(1−𝛽_𝑛)‖𝐴‖⁻¹. Since 𝐴 is a linear bounded operator on 𝑋, by (1.35), we have ‖𝐴‖=sup{|⟨𝐴𝑢,𝐽(𝑢)⟩|∶𝑢∈𝑋, ‖𝑢‖=1}. Observe that 1−𝛽_𝑛𝐼−𝛼_𝑛𝐴𝑢,𝐽(𝑢)=1−𝛽_𝑛−𝛼_𝑛⟨𝐴𝑢,𝐽(𝑢)⟩≥1−𝛽_𝑛−𝛼_𝑛‖𝐴‖≥0, (3.3) that is, (1−𝛽_𝑛)𝐼−𝛼_𝑛𝐴 is positive. It follows that 1−𝛽_𝑛𝐼−𝛼_𝑛𝐴=sup⟨1−𝛽_𝑛𝐼−𝛼_𝑛𝐴𝑢,𝐽(𝑢)⟩∶𝑢∈𝑋,‖𝑢‖=1=sup1−𝛽_𝑛−𝛼_𝑛⟨𝐴𝑢,𝐽(𝑢)⟩∶𝑢∈𝑋,‖𝑢‖=1≤1−𝛽_𝑛−𝛼_𝑛𝛾. (3.4) Taking 𝑥∈Ω, it follows from Lemma 2.17 that 𝑥=𝐽_{(𝑀1,𝜌1)}𝐽_{(𝑀2,𝜌2)}𝐽_{(𝑀3,𝜌3)}𝑥−𝜌₃𝐵₃𝑥−𝜌₂𝐵₂𝐽_{(𝑀3,𝜌3)}𝑥−𝜌₃𝐵₃𝑥−𝜌₁𝐵₁𝐽_{(𝑀2,𝜌2)}𝐽_{(𝑀3,𝜌3)}𝑥−𝜌₃𝐵₃𝑥−𝜌₂𝐵₂𝐽_{(𝑀3,𝜌3)}𝑥−𝜌₃𝐵₃𝑥. (3.5) Putting 𝑦=𝐽_{(𝑀2,𝜌2)}(𝑧−𝜌₂𝐵₂𝑧) and 𝑧=𝐽_{(𝑀1,𝜌1)}(𝑥−𝜌₃𝐵₃𝑥). Then 𝑥=𝐽_{(𝑀1,𝜌1)}(𝑦−𝜌₁𝐵₁𝑦). It follows from Lemmas 2.1 and 2.16 that 𝑣_𝑛−𝑥=𝐽_{(𝑀1,𝜌1)}𝑦_𝑛−𝜌₁𝐵₁𝑦_𝑛−𝐽_{(𝑀1,𝜌1)}𝑦−𝜌₁𝐵₁𝑦≤𝑦_𝑛−𝜌₁𝐵₁𝑦_𝑛−𝑦−𝜌₁𝐵₁𝑦≤𝑦_𝑛−𝑦=𝐽_{(𝑀2,𝜌2)}𝑧_𝑛−𝜌₂𝐵₂𝑧_𝑛−𝐽_{(𝑀2,𝜌2)}𝑧−𝜌₂𝐵₂𝑧≤𝑧_𝑛−𝜌₂𝐵₂𝑧_𝑛−𝑧−𝜌₂𝐵₂𝑧≤𝑧_𝑛−𝑧=𝐽_{(𝑀3,𝜌3)}𝑥_𝑛−𝜌₃𝐵₃𝑥_𝑛−𝐽_{(𝑀3,𝜌3)}𝑥−𝜌₃𝐵₃𝑥≤𝑥_𝑛−𝜌₃𝐵₃𝑥_𝑛−𝑥−𝜌₃𝐵₃𝑥≤𝑥_𝑛−𝑥 (3.6) and setting 𝜖_𝑛=𝜇(1/𝑡_𝑛)∫^𝑡𝑛₀𝑇(𝑠)𝑥_𝑛𝑑𝑠+(1−𝜇)𝑣_𝑛. From (3.6), we obtain 𝜖_𝑛−𝑥=𝜇1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑥_𝑛𝑑𝑠−𝑥+1−𝜇𝑣_𝑛−𝑥1≤𝜇𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑥_𝑛𝑑𝑠−𝑥+1−𝜇𝑣_𝑛−𝑥≤𝜇𝑥_𝑛−𝑥+1−𝜇𝑥_𝑛−𝑥=𝑥_𝑛−𝑥. (3.7) It follows from (3.7) that 𝑥_𝑛+1−𝑥=𝛼_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑥+𝛽_𝑛𝑥_𝑛−𝑥+1−𝛽_𝑛𝐼−𝛼_𝑛𝐴𝜖_𝑛−𝑥≤𝛼_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑥+𝛽_𝑛𝑥_𝑛−𝑥+1−𝛽_𝑛−𝛼_𝑛𝛾𝜖_𝑛−𝑥≤𝛼_𝑛𝛾𝑓𝑥_𝑛−𝑓(𝑥)+𝛼_𝑛𝛾𝑓(𝑥)−𝐴𝑥+𝛽_𝑛𝑥_𝑛−𝑥+1−𝛽_𝑛−𝛼_𝑛𝛾𝑥_𝑛−𝑥≤1−𝛼_𝑛𝛾𝑥_𝑛−𝑥+𝛼_𝑛𝛾𝛼𝑥_𝑛−𝑥+𝛼_𝑛𝛾𝑓(𝑥)−𝐴𝑥=1−𝛼_𝑛𝛾−𝛾𝛼𝑥_𝑛−𝑥+𝛼_𝑛𝛾𝑓(𝑥)−𝐴𝑥≤max𝑥_𝑛−𝑥,𝛾𝑓(𝑥)−𝐴𝑥.𝛾−𝛾𝛼 (3.8) By induction, we have 𝑥_𝑛−𝑥≤max𝑥₁−𝑥,𝛾𝑓(𝑥)−𝐴𝑥𝛾−𝛾𝛼,∀𝑛≥1. (3.9) Hence, {𝑥_𝑛} is bounded, so are {𝑧_𝑛}, {𝑦_𝑛}, and {𝑣_𝑛}. On the other hand, by nonexpansiveness of 𝐽_{(𝑀𝑖,𝜌𝑖)} and 𝐼−𝜌_𝑖𝐵_𝑖 for all 𝑖=1,2,3, we have 𝑣_𝑛+1−𝑣_𝑛=𝐽_{(𝑀1,𝜌1)}𝑦_𝑛+1−𝜌₁𝐵₁𝑦_𝑛+1−𝐽_{(𝑀1,𝜌1)}𝑦_𝑛−𝜌₁𝐵₁𝑦_𝑛≤𝑦_𝑛+1−𝜌₁𝐵₁𝑦_𝑛+1−𝑦_𝑛−𝜌₁𝐵₁𝑦_𝑛≤𝑦_𝑛+1−𝑦_𝑛=𝐽_{(𝑀2,𝜌2)}𝑧_𝑛+1−𝜌₂𝐵₂𝑧_𝑛+1−𝐽_{(𝑀2,𝜌2)}𝑧_𝑛−𝜌₂𝐵₂𝑧_𝑛≤𝑧_𝑛+1−𝑧_𝑛=𝐽_{(𝑀3,𝜌3)}𝑥_𝑛+1−𝜌₃𝐵₃𝑥_𝑛+1−𝐽_{(𝑀3,𝜌3)}𝑥_𝑛−𝜌₃𝐵₃𝑥_𝑛≤𝑥_𝑛+1−𝑥_𝑛. (3.10) Now, we estimate 1𝑡_𝑛+1^𝑡𝑛+1₀𝑇(𝑠)𝑥_𝑛+11𝑑𝑠−𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑥_𝑛=1𝑑𝑠𝑡_𝑛+1^𝑡𝑛+1₀𝑇(𝑠)𝑥_𝑛+1−𝑇(𝑠)𝑥_𝑛1𝑑𝑠+𝑡_𝑛+1^𝑡𝑛+1₀𝑇(𝑠)𝑥_𝑛1𝑑𝑠−𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑥_𝑛=1𝑑𝑠𝑡_𝑛+1^𝑡𝑛+1₀𝑇(𝑠)𝑥_𝑛+1−𝑇(𝑠)𝑥_𝑛+1𝑑𝑠𝑡_𝑛+1^𝑡𝑛₀𝑇(𝑠)𝑥_𝑛1𝑑𝑠+𝑡_𝑛+1^𝑡𝑛+1_𝑡𝑛𝑇(𝑠)𝑥_𝑛1𝑑𝑠−𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑥_𝑛=1𝑑𝑠𝑡_𝑛+1^𝑡𝑛+1₀𝑇(𝑠)𝑥_𝑛+1−𝑇(𝑠)𝑥_𝑛1𝑑𝑠+𝑡_𝑛+1−1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑥_𝑛1𝑑𝑠+𝑡_𝑛+1^𝑡𝑛+1_𝑡𝑛𝑇(𝑠)𝑥_𝑛𝑑𝑠, (3.11) for 𝑥∈Ω, it follows that 1𝑡_𝑛+1^𝑡𝑛+1₀𝑇(𝑠)𝑥_𝑛+11𝑑𝑠−𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑥_𝑛=1𝑑𝑠𝑡_𝑛+1^𝑡𝑛+1₀𝑇(𝑠)𝑥_𝑛+1−𝑇(𝑠)𝑥_𝑛+1𝑑𝑠𝑡_𝑛+1−1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑥_𝑛−𝑥1𝑑𝑠+𝑡_𝑛+1^𝑡𝑛+1_𝑡𝑛𝑇(𝑠)𝑥_𝑛−𝑥≤1𝑑𝑠𝑡_𝑛+1^𝑡𝑛+1₀𝑇(𝑠)𝑥_𝑛+1−𝑇(𝑠)𝑥_𝑛1𝑑𝑠+𝑡_𝑛+1−1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑥_𝑛−𝑥+1𝑑𝑠𝑡_𝑛+1^𝑡𝑛+1_𝑡𝑛𝑇(𝑠)𝑥_𝑛−𝑥≤𝑥𝑑𝑠_𝑛+1−𝑥_𝑛+1𝑡_𝑛+1−1𝑡_𝑛𝑡_𝑛+1𝑡_𝑛+1𝑡_𝑛+1−𝑡_𝑛𝑥_𝑛−𝑥≤𝑥_𝑛+1−𝑥_𝑛+2𝑡_𝑛+1−𝑡_𝑛𝑡_𝑛+1𝑥_𝑛−𝑥. (3.12) It follows from (3.10) and (3.12) that 𝜖_𝑛+1−𝜖_𝑛=𝜇1𝑡_𝑛+1^𝑡𝑛+1₀𝑇(𝑠)𝑥_𝑛+1𝑑𝑠+1−𝜇𝑣_𝑛+1−𝜇1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑥_𝑛𝑑𝑠+1−𝜇𝑣_𝑛1≤𝜇𝑡_𝑛+1^𝑡𝑛+1₀𝑇(𝑠)𝑥_𝑛+11𝑑𝑠−𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑥_𝑛𝑑𝑠+1−𝜇𝑣_𝑛+1−𝑣_𝑛≤𝑥_𝑛+1−𝑥_𝑛+2𝑡_𝑛+1−𝑡_𝑛𝑡_𝑛+1𝑥_𝑛−𝑥+1−𝜇𝑥_𝑛+1−𝑥_𝑛≤𝑥_𝑛+1−𝑥_𝑛+2𝑡_𝑛+1−𝑡_𝑛𝑡_𝑛+1𝑀₁, (3.13) where 𝑀₁>0 is an appropriate constant such that 𝑀₁=sup_𝑛≥1{‖𝑥_𝑛−𝑥‖}.
Setting 𝑙_𝑛=(𝑥_𝑛+1−𝛽_𝑛𝑥_𝑛)/(1−𝛽_𝑛), for all 𝑛∈ℕ. Then 𝑥_𝑛+1=𝛽_𝑛𝑥_𝑛+(1−𝛽_𝑛)𝑙_𝑛, for all 𝑛∈ℕ, we have 𝑙_𝑛+1−𝑙_𝑛=𝑥_𝑛+2−𝛽_𝑛+1𝑥_𝑛+11−𝛽_𝑛+1−𝑥_𝑛+1−𝛽_𝑛𝑥_𝑛1−𝛽_𝑛=𝛼_𝑛+1𝛾𝑓𝑥_𝑛+1+1−𝛽_𝑛+1𝐼−𝛼_𝑛+1𝐴𝜖_𝑛+11−𝛽_𝑛+1−𝛼_𝑛𝛾𝑓𝑥_𝑛+1−𝛽_𝑛𝐼−𝛼_𝑛𝐴𝜖_𝑛1−𝛽_𝑛=𝛼_𝑛+11−𝛽_𝑛+1𝛾𝑓𝑥_𝑛+1−𝐴𝜖_𝑛+1+𝛼_𝑛1−𝛽_𝑛𝐴𝜖_𝑛−𝛾𝑓𝑥_𝑛+𝜖_𝑛+1−𝜖_𝑛 (3.14) and, hence, 𝑙_𝑛+1−𝑙_𝑛≤𝛼_𝑛+11−𝛽_𝑛+1𝛾𝑓𝑥_𝑛+1−𝐴𝜖_𝑛+1+𝛼_𝑛1−𝛽_𝑛𝐴𝜖_𝑛−𝛾𝑓𝑥_𝑛+𝜖_𝑛+1−𝜖_𝑛, (3.15) which, combined with (3.13) yields that 𝑙_𝑛+1−𝑙_𝑛≤𝛼_𝑛+11−𝛽_𝑛+1𝛾𝑓𝑥_𝑛+1−𝐴𝜖_𝑛+1+𝛼_𝑛1−𝛽_𝑛𝐴𝜖_𝑛−𝛾𝑓𝑥_𝑛+𝑥_𝑛+1−𝑥_𝑛+2𝑡_𝑛+1−𝑡_𝑛𝑡_𝑛+1𝑀₁. (3.16) It follows that 𝑙_𝑛+1−𝑙_𝑛−𝑥_𝑛+1−𝑥_𝑛≤𝛼_𝑛+11−𝛽_𝑛+1𝛾𝑓𝑥_𝑛+1−𝐴𝜖_𝑛+1+𝛼_𝑛1−𝛽_𝑛𝐴𝜖_𝑛−𝛾𝑓𝑥_𝑛+2𝑡_𝑛+1−𝑡_𝑛𝑡_𝑛+1𝑀₁. (3.17) By conditions (C1)–(C3), we have limsup_𝑛→∞𝑙_𝑛+1−𝑙_𝑛−𝑥_𝑛+1−𝑥_𝑛≤0. (3.18) Then, from Lemma 2.12, we obtain lim_𝑛→∞𝑙_𝑛−𝑥_𝑛=0, (3.19) observing that 𝑥_𝑛+1−𝑥_𝑛=1−𝛽_𝑛𝑙_𝑛−𝑥_𝑛, (3.20) and, hence, lim_𝑛→∞𝑥_𝑛+1−𝑥_𝑛=0. (3.21) Since 𝑥_𝑛+1=𝛼_𝑛𝛾𝑓(𝑥_𝑛)+𝛽_𝑛𝑥_𝑛+((1−𝛽_𝑛)𝐼−𝛼_𝑛𝐴)𝜖_𝑛, then we have 𝑥_𝑛+1−𝑥_𝑛=𝛼_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑥_𝑛+1−𝛽_𝑛𝐼−𝛼_𝑛𝐴𝜖_𝑛−𝑥_𝑛. (3.22) It follows that 1−𝛽_𝑛−𝛼_𝑛𝛾𝜖_𝑛−𝑥_𝑛≤𝑥_𝑛−𝑥_𝑛+1+𝛼_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑥_𝑛. (3.23) From (3.21) and by the conditions (C1), (C2), we obtain that lim_𝑛→∞𝜖_𝑛−𝑥_𝑛=0. (3.24) Define the mapping 𝐺_𝑛 by 𝐺_𝑛1𝑥=𝜇𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑥𝑑𝑠+1−𝜇𝑄𝑥,∀𝑥∈𝑋, (3.25) where 𝑄 is defined as in Lemma 2.17. By Lemma 2.11, we see that 𝐺_𝑛 is nonexpansive such that 𝐹𝐺_𝑛=𝐹(𝒮)∩𝐹(𝑄). (3.26) From (3.24), it follows that lim_𝑛→∞𝐺_𝑛𝑥_𝑛−𝑥_𝑛=0. (3.27) Since {𝑇(𝑡)∶𝑡∈ℝ⁺} is a u.a.r. nonexpansive semigroup by Remark 2.6, and at least there exists a 𝑇(𝑡) which is demicompact, then by Lemma 2.7, we can define the mapping 𝑇∶𝑋→𝑋 such that 𝑇𝑦=lim_𝑛→∞(1/𝑡_𝑛)∫^𝑡𝑛₀𝑇(𝑠)𝑦𝑑𝑠 for all 𝑦∈𝑋. Furthermore, let 𝐺∶𝑋→𝑋 be the mapping define by 𝐺𝑦=lim_𝑛→∞𝐺_𝑛𝑦, for all 𝑦∈𝑋. It follows from the nonexpansivity of 𝐺 and 𝐺𝑦∶=𝜇𝑇𝑦+1−𝜇𝑄𝑦,∀𝑦∈𝑋, (3.28) such that 𝐹(𝐺)=𝐹(𝑇)∩𝐹(𝑄)=𝐹(𝒮)∩𝐹(𝑄)=𝐹𝐺_𝑛. (3.29) On the other hand, we observe that for any bounded subset 𝐶 of 𝑋 containing {𝑥_𝑛}, 𝑥_𝑛−𝐺𝑥_𝑛≤𝑥_𝑛−𝐺_𝑛𝑥_𝑛+𝐺_𝑛𝑥_𝑛−𝐺𝑥_𝑛≤𝑥_𝑛−𝐺_𝑛𝑥_𝑛+sup_𝜔∈𝐶𝐺_𝑛𝜔−𝐺𝜔. (3.30) By Lemma 2.7 and (3.27), we obtain that lim_𝑛→∞𝑥_𝑛−𝐺𝑥_𝑛=0. (3.31) Since 𝑋 is a uniformly convex, hence it is reflexive and {𝑥_𝑛} is bounded, then there exists a subsequence {𝑥_𝑛𝑗} of {𝑥_𝑛} which converges weakly to some 𝑧∈𝑋 as 𝑗→∞. Again, since Banach space 𝑋 has a weakly sequentially continuous duality mapping satisfying Opial's condition, then by (3.32), we have by Lemma 2.14, that 𝑧∈𝐹(𝐺)=Ω.
Next, we show that limsup_𝑛→∞⟨𝛾𝑓(𝑥^∗)−𝐴𝑥^∗,𝐽(𝑥_𝑛−𝑥^∗)⟩≤0, where 𝑥^∗=lim_𝑡→0𝑥_𝑡 with 𝑥_𝑡 be the fixed point of the contraction 𝑥↦𝑡𝛾𝑓(𝑥)+(𝐼−𝑡𝐴)𝐺𝑥. Since 𝐺 is a nonexpansive mapping, it follows from Lemma 2.10 that 𝑥^∗∈𝐹(𝐺)=Ω, which solves the variational inequality𝛾𝑓−𝐴𝑥^∗,𝐽𝑧−𝑥^∗≤0,∀𝑧∈𝐹(𝐺) (3.32) and (𝑥^∗,𝑦^∗,𝑧^∗) is a solution of general system of variational inequality problem (1.38) such that 𝑦^∗=𝐽_{(𝑀2,𝜌2)}(𝑧^∗−𝜌₂𝐵₂𝑧^∗) and 𝑧^∗=𝐽_{(𝑀3,𝜌3)}(𝑥^∗−𝜌₃𝐵₃𝑥^∗).
Let {𝑥_𝑛𝑗} be a subsequence of {𝑥_𝑛} such thatlimsup_𝑛→∞𝛾𝑓𝑥^∗−𝐴𝑥^∗,𝐽𝑥_𝑛−𝑥^∗=lim_𝑗→∞𝛾𝑓𝑥^∗−𝐴𝑥^∗,𝐽𝑥_𝑛𝑗−𝑥^∗. (3.33) Since the duality mapping 𝐽 is single-valued and weakly sequentially continuous from 𝑋 to 𝑋^∗, we obtain that limsup_𝑛→∞𝛾𝑓𝑥^∗−𝐴𝑥^∗,𝐽𝑥_𝑛−𝑥^∗=lim_𝑗→∞𝛾𝑓𝑥^∗−𝐴𝑥^∗,𝐽𝑥_𝑛𝑗−𝑥^∗=𝛾𝑓𝑥^∗−𝐴𝑥^∗,𝐽𝑧−𝑥^∗≤0, (3.34) as required.
Finally, we show that lim_𝑛→∞‖𝑥_𝑛−𝑥^∗‖=0. Now, from Lemma 2.3, we have 𝑥_𝑛+1−𝑥^∗2=𝛼_𝑛(𝛾𝑓(𝑥_𝑛)−𝐴𝑥^∗)+𝛽_𝑛(𝑥_𝑛−𝑥^∗)+((1−𝛽_𝑛)𝐼−𝛼_𝑛𝐴)(𝜖_𝑛−𝑥^∗)²≤1−𝛽_𝑛𝐼−𝛼_𝑛𝐴(𝜖_𝑛−𝑥^∗)+𝛽_𝑛(𝑥_𝑛−𝑥^∗)²+2𝛼_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑥^∗,𝐽𝑥_𝑛+1−𝑥^∗≤1−𝛽_𝑛−𝛼_𝑛𝛾𝜖_𝑛−𝑥^∗+𝛽_𝑛𝑥_𝑛−𝑥^∗2+2𝛼_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑥^∗,𝐽𝑥_𝑛+1−𝑥^∗≤1−𝛼_𝑛𝛾²𝑥_𝑛−𝑥^∗2+2𝛼_𝑛𝛾⟨𝑓𝑥_𝑛−𝑓𝑥^∗,𝐽𝑥_𝑛+1−𝑥^∗⟩+2𝛼_𝑛𝛾𝑓𝑥^∗−𝐴𝑥^∗,𝐽𝑥_𝑛+1−𝑥^∗≤1−𝛼_𝑛𝛾²𝑥_𝑛−𝑥^∗2+2𝛼_𝑛𝛾𝛼𝑥_𝑛−𝑥^∗𝑥_𝑛+1−𝑥^∗+2𝛼_𝑛𝛾𝑓𝑥^∗−𝐴𝑥^∗,𝐽𝑥_𝑛+1−𝑥^∗≤1−𝛼_𝑛𝛾²𝑥_𝑛−𝑥^∗2+𝛼_𝑛𝛾𝛼𝑥_𝑛−𝑥^∗2+𝑥_𝑛+1−𝑥^∗2+2𝛼_𝑛𝛾𝑓𝑥^∗−𝐴𝑥^∗,𝐽𝑥_𝑛+1−𝑥^∗, (3.35) which implies that 𝑥_𝑛+1−𝑥^∗2≤1−𝛼_𝑛𝛾²+𝛼_𝑛𝛾𝛼1−𝛼_𝑛𝑥𝛾𝛼^𝑛−𝑥^∗2+2𝛼_𝑛1−𝛼_𝑛𝑥𝛾𝛼𝛾𝑓^∗−𝐴𝑥^∗,𝐽𝑥_𝑛+1−𝑥^∗=1−2𝛼_𝑛𝛾+𝛼_𝑛𝛾𝛼1−𝛼_𝑛𝑥𝛾𝛼^𝑛−𝑥^∗2+𝛼²_𝑛𝛾²1−𝛼_𝑛𝑥𝛾𝛼^𝑛−𝑥^∗2+2𝛼_𝑛1−𝛼_𝑛𝑥𝛾𝛼𝛾𝑓^∗−𝐴𝑥^∗,𝐽𝑥_𝑛+1−𝑥^∗≤1−2𝛼_𝑛𝛾−𝛾𝛼1−𝛼_𝑛𝑥𝛾𝛼^𝑛−𝑥^∗2+2𝛼_𝑛𝛾−𝛾𝛼1−𝛼_𝑛1𝛾𝛼𝑥𝛾−𝛾𝛼𝛾𝑓^∗−𝐴𝑥^∗,𝐽𝑥_𝑛+1−𝑥^∗+𝛼_𝑛𝛾²2𝑀𝛾−𝛾𝛼², (3.36) where 𝑀₂>0 is an appropriate constant such that 𝑀₂=sup_𝑛≥1{‖𝑥_𝑛−𝑥^∗‖}. Put 𝜏_𝑛=2𝛼_𝑛(𝛾−𝛾𝛼)/(1−𝛼_𝑛𝛾𝛼) and 𝛿_𝑛=(1/(𝛾−𝛾𝛼))⟨𝛾𝑓(𝑥^∗)−𝐴𝑥^∗,𝐽(𝑥_𝑛+1−𝑥^∗)⟩+(𝛼_𝑛𝛾²/2(𝛾−𝛾𝛼))𝑀₂. Then (3.32) reduces to formula 𝑥_𝑛+1−𝑥^∗2≤1−𝜏_𝑛𝑥_𝑛−𝑥^∗2+𝜏_𝑛𝛿_𝑛. (3.37) By the condition (C1) and (3.31), it easy to see that lim_𝑛→∞𝜏_𝑛=0,^∞_𝑛=1𝜏_𝑛=∞,limsup_𝑛→∞𝛿_𝑛≤0. (3.38) Applying Lemma 2.15 to (3.37), we obtain 𝑥_𝑛→𝑥^∗ as 𝑛→∞, that is lim_𝑛→∞‖𝑥_𝑛−𝑥^∗‖=0. This completes the proof.

Remark 3.2. Theorem 3.1 mainly improves Theorem 2.1 of Qin et al. [8], in the following respects: (1)from a system variational inclusion to a general system of variational inclusions,(2)from the class of inverse-strongly accretive mappings to the class of Lipchitzian and relaxed cocoercive mappings.

Taking 𝐴≡𝐼 and 𝛾=1 in Theorem 3.1, we can obtain the following result.

Corollary 3.3. Let 𝑋 be a uniformly convex and 2-uniformly smooth Banach space which admits a weakly continuous duality mapping and has the smoothness constant 𝐾. Let 𝑀_𝑖∶𝑋→2^𝑋 be a maximal monotone mapping and 𝐵_𝑖∶𝑋→𝑋 be a 𝐿_𝑖-Lipschitzian and relaxed (𝑐_𝑖,𝑑_𝑖)-cocoercive mapping with 𝜌_𝑖∈(0,(𝑑_𝑖−𝑐_𝑖𝐿²_𝑖)/𝐾²𝐿²_𝑖] and 𝜃_𝜌=max{𝜃_𝜌𝑖} for all 𝑖=1,2,3. Let 𝒮={𝑇(𝑡)∶𝑡∈ℝ⁺} be a nonexpansive semigroup from 𝑋 into itself and at least there exists a 𝑇(𝑡) which is demicompact. Assume that Ω∶=𝐹(𝒮)∩𝐹(𝑄)≠∅, where 𝑄 is defined as in Lemma 2.17. Let 𝑓∶𝑋→𝑋 be a contraction mapping with a coefficient 𝛼∈(0,1). Let {𝑥_𝑛} be a sequence defined by 𝑥₁𝑧∈𝑋chosenarbitrarily,_𝑛=𝐽_{(𝑀3,𝜌3)}𝑥_𝑛−𝜌₃𝐵₃𝑥_𝑛,𝑦_𝑛=𝐽_{(𝑀2,𝜌2)}𝑧_𝑛−𝜌₂𝐵₂𝑧_𝑛,𝑣_𝑛=𝐽_{(𝑀1,𝜌1)}𝑦_𝑛−𝜌₁𝐵₁𝑦_𝑛,𝑥_𝑛+1=𝛼_𝑛𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛−𝛼_𝑛𝜇1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑥_𝑛𝑑𝑠+1−𝜇𝑣_𝑛,∀𝑛≥1, (3.39) where 𝜇∈(0,1), {𝛼_𝑛}^∞_𝑛=1,{𝛽_𝑛}^∞_𝑛=1 are the sequences in (0,1) which satisfies 𝛼_𝑛+𝛽_𝑛≤1 and {𝑡_𝑛}^∞_𝑛=1 is a positive real divergent sequence satisfy the following restrictions: (C1)lim_𝑛→∞𝛼_𝑛=0 and ∑^∞_𝑛=1𝛼_𝑛=∞, (C2)0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1, (C3)lim_𝑛→∞(|𝑡_𝑛+1−𝑡_𝑛|/𝑡_𝑛+1)=0. Then the sequence {𝑥_𝑛} defined by (3.39) converges strongly to 𝑥^∗, which 𝑥^∗ solves the variational inequality 𝑓−𝐼𝑥^∗,𝐽𝑧−𝑥^∗≤0,∀𝑧∈Ω, (3.40) and (𝑥^∗,𝑦^∗,𝑧^∗) is a solution of general system of variational inequality problem (1.38), where 𝑦^∗=𝐽_{(𝑀2,𝜌2)}(𝑧^∗−𝜌₂𝐵₂𝑧^∗) and 𝑧^∗=𝐽_{(𝑀3,𝜌3)}(𝑥^∗−𝜌₃𝐵₃𝑥^∗).

Taking 𝑓(𝑥)=𝑢 in Corollary 3.3, we can obtain the following result.

Corollary 3.4. Let 𝑋 be a uniformly convex and 2-uniformly smooth Banach space which admits a weakly continuous duality mapping and has the smoothness constant 𝐾. Let 𝑀_𝑖∶𝑋→2^𝑋 be a maximal monotone mapping and 𝐵_𝑖∶𝑋→𝑋 be a 𝐿_𝑖-Lipschitzian and relaxed (𝑐_𝑖,𝑑_𝑖)-cocoercive mapping with 𝜌_𝑖∈(0,(𝑑_𝑖−𝑐_𝑖𝐿²_𝑖)/𝐾²𝐿²_𝑖] and 𝜃_𝜌=max{𝜃_𝜌𝑖} for all 𝑖=1,2,3. Let 𝒮={𝑇(𝑡)∶𝑡∈ℝ⁺} be a nonexpansive semigroup from 𝑋 into itself and at least there exists a 𝑇(𝑡) which is demicompact. Assume that Ω∶=𝐹(𝒮)∩𝐹(𝑄)≠∅, where 𝑄 is defined as in Lemma 2.17. Let {𝑥_𝑛} be a sequence defined by 𝑥₁𝑧∈𝑋chosenarbitrarily,_𝑛=𝐽_(𝑀,𝜌3)𝑥_𝑛−𝜌₃𝐵₃𝑥_𝑛,𝑦_𝑛=𝐽_(𝑀,𝜌2)𝑧_𝑛−𝜌₂𝐵₂𝑧_𝑛,𝑣_𝑛=𝐽_(𝑀,𝜌1)𝑦_𝑛−𝜌₁𝐵₁𝑦_𝑛,𝑥_𝑛+1=𝛼_𝑛𝑢+𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛−𝛼_𝑛𝜇1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑥_𝑛𝑑𝑠+1−𝜇𝑣_𝑛,∀𝑛≥1, (3.41) where 𝜇∈(0,1), {𝛼_𝑛}^∞_𝑛=1, {𝛽_𝑛}^∞_𝑛=1 are the sequences in (0,1) which satisfies 𝛼_𝑛+𝛽_𝑛≤1 and {𝑡_𝑛}^∞_𝑛=1 is a positive real divergent sequence satisfy the following restrictions: (C1)lim_𝑛→∞𝛼_𝑛=0 and ∑^∞_𝑛=1𝛼_𝑛=∞, (C2)0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1, (C3)lim_𝑛→∞(|𝑡_𝑛+1−𝑡_𝑛|/𝑡_𝑛+1)=0. Then the sequence {𝑥_𝑛} defined by (3.41) converges strongly to 𝑥^∗, which 𝑥^∗ solves the variational inequality 𝑢−𝑥^∗,𝐽𝑧−𝑥^∗≤0,∀𝑧∈Ω, (3.42) and (𝑥^∗,𝑦^∗,𝑧^∗) is a solution of general system of variational inequality problem (1.38), where 𝑦^∗=𝐽_{(𝑀2,𝜌2)}(𝑧^∗−𝜌₂𝐵₂𝑧^∗) and 𝑧^∗=𝐽_{(𝑀3,𝜌3)}(𝑥^∗−𝜌₃𝐵₃𝑥^∗).

4. Some Applications

As some applications of Theorem 3.1, we obtain the following results.

Lemma 4.1. For all (𝑥^∗,𝑦^∗,𝑧^∗)∈𝑋×𝑋×𝑋, where 𝑦^∗=𝐽_(𝑀,𝜌2)(𝑧^∗−𝜌₂𝐵𝑧^∗) and 𝑧^∗=𝐽_(𝑀,𝜌3)(𝑥^∗−𝜌₃𝐵𝑥^∗), (𝑥^∗,𝑦^∗,𝑧^∗) is a solution of the problem (1.39) if and only if 𝑥^∗ is a fixed point of the mapping 𝑄^′ defined by 𝑄^′𝑥∶=𝐽_(𝑀,𝜌1)𝐽_(𝑀,𝜌2)𝐽_(𝑀,𝜌3)𝑥−𝜌₃𝐵𝑥−𝜌₂𝐵𝐽_(𝑀,𝜌3)𝑥−𝜌₃𝐵𝑥−𝜌₁𝐵𝐽_(𝑀,𝜌2)𝐽_(𝑀,𝜌3)𝑥−𝜌₃𝐵𝑥−𝜌₂𝐵𝐽_(𝑀,𝜌3)𝑥−𝜌₃𝐵𝑥,∀𝑥∈𝑋. (4.1)

Corollary 4.2. Let 𝑋 be a uniformly convex and 2-uniformly smooth Banach space which admits a weakly continuous duality mapping and has the smoothness constant 𝐾. Let 𝑀∶𝑋→2^𝑋 be a maximal monotone mapping and 𝐵∶𝑋→𝑋 be a 𝐿-Lipschitzian and relaxed (𝑐,𝑑)-cocoercive mapping with 𝜌_𝑖∈(0,(𝑑−𝑐𝐿²)/𝐾²𝐿²] and 𝜃_𝜌=max{𝜃_𝜌𝑖} for all 𝑖=1,2,3. Let 𝒮={𝑇(𝑡)∶𝑡∈ℝ⁺} be a nonexpansive semigroup from 𝑋 into itself and at least there exists a 𝑇(𝑡) which is demicompact. Assume that Ω∶=𝐹(𝒮)∩𝐹(𝑄^′)≠∅, where 𝑄^′ is defined as in Lemma 4.1. Let 𝑓∶𝑋→𝑋 be a contraction mapping with a coefficient 𝛼∈(0,1) and 𝐴 be a strongly positive linear bounded self adjoint operator with a coefficient 𝛾∈(0,1) such that ‖𝐴‖≤1 and 0<𝛾<𝛾/𝛼. Let {𝑥_𝑛} be a sequence defined by 𝑥₁𝑧∈𝑋chosenarbitrarily,_𝑛=𝐽_(𝑀,𝜌3)𝑥_𝑛−𝜌₃𝐵𝑥_𝑛,𝑦_𝑛=𝐽_(𝑀,𝜌2)𝑧_𝑛−𝜌₂𝐵𝑧_𝑛,𝑣_𝑛=𝐽_(𝑀,𝜌1)𝑦_𝑛−𝜌₁𝐵𝑦_𝑛,𝑥_𝑛+1=𝛼_𝑛𝛾𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝐼−𝛼_𝑛𝐴𝜇1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑥_𝑛𝑑𝑠+1−𝜇𝑣_𝑛,∀𝑛≥1, (4.2) where 𝜇∈(0,1), {𝛼_𝑛}^∞_𝑛=1,{𝛽_𝑛}^∞_𝑛=1 are the sequences in (0,1) which satisfies 𝛼_𝑛+𝛽_𝑛≤1 and {𝑡_𝑛}^∞_𝑛=1 is a positive real divergent sequence satisfy the following restrictions: (C1)lim_𝑛→∞𝛼_𝑛=0 and ∑^∞_𝑛=1𝛼_𝑛=∞, (C2)0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1, (C3)lim_𝑛→∞(|𝑡_𝑛+1−𝑡_𝑛|/𝑡_𝑛+1)=0. Then the sequence {𝑥_𝑛} defined by (4.2) converges strongly to 𝑥^∗, which 𝑥^∗ solves the variational inequality 𝛾𝑓−𝐴𝑥^∗,𝐽𝑧−𝑥^∗≤0,∀𝑧∈Ω, (4.3) and (𝑥^∗,𝑦^∗,𝑧^∗) is a solution of system of variational inequality problem (1.39), where 𝑦^∗=𝐽_(𝑀,𝜌2)(𝑧^∗−𝜌₂𝐵𝑧^∗) and 𝑧^∗=𝐽_(𝑀,𝜌3)(𝑥^∗−𝜌₃𝐵𝑥^∗).

Lemma 4.3 (see [8]). For all (𝑥^∗,𝑦^∗)∈𝑋×𝑋, where 𝑦^∗=𝐽_{(𝑀2,𝜌2)}(𝑥^∗−𝜌₂𝐵₂𝑥^∗), (𝑥^∗,𝑦^∗) is a solution of the problem (1.15) if and only if 𝑥^∗ is a fixed point of the mapping 𝑄^′ defined by 𝑄^′𝑥∶=𝐽_{(𝑀1,𝜌1)}𝐽_{(𝑀2,𝜌2)}𝑥−𝜌₂𝐵₂𝑥−𝜌₁𝐵₁𝐽_{(𝑀2,𝜌2)}𝑥−𝜌₂𝐵₂𝑥. (4.4)

Corollary 4.4. Let 𝑋 be a uniformly convex and 2-uniformly smooth Banach space which admits a weakly continuous duality mapping and has the smoothness constant 𝐾. Let 𝑀_𝑖∶𝑋→2^𝑋 be a maximal monotone mapping and 𝐵_𝑖∶𝑋→𝑋 be a 𝐿_𝑖-Lipschitzian and relaxed (𝑐_𝑖,𝑑_𝑖)-cocoercive mapping with 𝜌_𝑖∈(0,(𝑑_𝑖−𝑐_𝑖𝐿²_𝑖)/𝐾²𝐿²_𝑖] and 𝜃_𝜌=max{𝜃_𝜌𝑖} for all 𝑖=1,2. Let 𝒮={𝑇(𝑡)∶𝑡∈ℝ⁺} be a nonexpansive semigroup from 𝑋 into itself and at least there exists a 𝑇(𝑡) which is demicompact. Assume that Ω∶=𝐹(𝒮)∩𝐹(𝑄^′)≠∅, where 𝑄^′ is defined as in Lemma 4.3. Let 𝑓∶𝑋→𝑋 be a contraction mapping with a coefficient 𝛼∈(0,1) and 𝐴 be a strongly positive linear bounded self adjoint operator with a coefficient 𝛾∈(0,1) such that ‖𝐴‖≤1 and 0<𝛾<𝛾/𝛼. Let {𝑥_𝑛} be a sequence defined by 𝑥₁𝑦∈𝑋chosenarbitrarily,_𝑛=𝐽_(𝑀,𝜌2)𝑥_𝑛−𝜌₂𝐵₂𝑥_𝑛,𝑣_𝑛=𝐽_(𝑀,𝜌1)𝑦_𝑛−𝜌₁𝐵₁𝑦_𝑛,𝑥_𝑛+1=𝛼_𝑛𝛾𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝐼−𝛼_𝑛𝐴𝜇1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑥_𝑛𝑑𝑠+1−𝜇𝑣_𝑛,∀𝑛≥1, (4.5) where 𝜇∈(0,1), {𝛼_𝑛}^∞_𝑛=1,{𝛽_𝑛}^∞_𝑛=1 are the sequences in (0,1) which satisfies 𝛼_𝑛+𝛽_𝑛≤1 and {𝑡_𝑛}^∞_𝑛=1 is a positive real divergent sequence satisfy the following restrictions: (C1)lim_𝑛→∞𝛼_𝑛=0 and ∑^∞_𝑛=1𝛼_𝑛=∞, (C2)0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1, (C3)lim_𝑛→∞(|𝑡_𝑛+1−𝑡_𝑛|/𝑡_𝑛+1)=0. Then the sequence {𝑥_𝑛} defined by (4.5) converges strongly to 𝑥^∗, which 𝑥^∗ solves the variational inequality 𝛾𝑓−𝐴𝑥^∗,𝐽𝑧−𝑥^∗≤0,∀𝑧∈Ω, (4.6) and (𝑥^∗,𝑦^∗) is a solution of system of variational inequality problem (1.15), where 𝑦^∗=𝐽_{(𝑀2,𝜌2)}(𝑥^∗−𝜌₂𝐵₂𝑥^∗).

If we set 𝐵₁=𝐵₂=𝐵 and 𝑀₁=𝑀₂=𝑀, we obtain the following result.

Lemma 4.5 (see [8]). For all (𝑥^∗,𝑦^∗)∈𝑋×𝑋, where 𝑦^∗=𝐽_(𝑀,𝜌2)(𝑥^∗−𝜌₂𝐵𝑥^∗), (𝑥^∗,𝑦^∗) is a solution of the problem (1.16) if and only if 𝑥^∗ is a fixed point of the mapping 𝑄^′′ defined by 𝑄^′′𝑥∶=𝐽_(𝑀,𝜌1)𝐽_(𝑀,𝜌2)𝑥−𝜌₂𝐵𝑥−𝜌₁𝐵𝐽_(𝑀,𝜌2)𝑥−𝜌₂𝐵𝑥. (4.7)

Corollary 4.6. Let 𝑋 be a uniformly convex and 2-uniformly smooth Banach space which admits a weakly continuous duality mapping and has the smoothness constant 𝐾. Let 𝑀∶𝑋→2^𝑋 be a maximal monotone mapping and 𝐵∶𝑋→𝑋 be a 𝐿-Lipschitzian and relaxed (𝑐,𝑑)-cocoercive mapping with 𝜌_𝑖∈(0,(𝑑−𝑐𝐿²)/𝐾²𝐿²] and 𝜃_𝜌=max{𝜃_𝜌𝑖} for all 𝑖=1,2. Let 𝒮={𝑇(𝑡)∶𝑡∈ℝ⁺} be a nonexpansive semigroup from 𝑋 into itself and at least there exists a 𝑇(𝑡) which is demicompact. Assume that Ω∶=𝐹(𝒮)∩𝐹(𝑄^′)≠∅, where 𝑄^′′ is defined as in Lemma 4.5. Let 𝑓∶𝑋→𝑋 be a contraction mapping with a coefficient 𝛼∈(0,1) and 𝐴 be a strongly positive linear bounded self adjoint operator with a coefficient 𝛾∈(0,1) such that ‖𝐴‖≤1 and 0<𝛾<𝛾/𝛼. Let {𝑥_𝑛} be a sequence defined by 𝑥₁𝑦∈𝑋chosenarbitrarily,_𝑛=𝐽_(𝑀,𝜌2)𝑥_𝑛−𝜌₂𝐵𝑥_𝑛,𝑣_𝑛=𝐽_(𝑀,𝜌1)𝑦_𝑛−𝜌₁𝐵𝑦_𝑛,𝑥_𝑛+1=𝛼_𝑛𝛾𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝐼−𝛼_𝑛𝐴𝜇1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑥_𝑛𝑑𝑠+1−𝜇𝑣_𝑛,∀𝑛≥1, (4.8) where 𝜇∈(0,1), {𝛼_𝑛}^∞_𝑛=1, {𝛽_𝑛}^∞_𝑛=1 are the sequences in (0,1) which satisfies 𝛼_𝑛+𝛽_𝑛≤1 and {𝑡_𝑛}^∞_𝑛=1 is a positive real divergent sequence satisfy the following restrictions: (C1)lim_𝑛→∞𝛼_𝑛=0 and ∑^∞_𝑛=1𝛼_𝑛=∞, (C2)0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1, (C3)lim_𝑛→∞(|𝑡_𝑛+1−𝑡_𝑛|/𝑡_𝑛+1)=0. Then the sequence {𝑥_𝑛} defined by (4.8) converges strongly to 𝑥^∗, which 𝑥^∗ solves the variational inequality 𝛾𝑓−𝐴𝑥^∗,𝐽𝑧−𝑥^∗≤0,∀𝑧∈Ω, (4.9) and (𝑥^∗,𝑦^∗) is a solution of system of variational inequality problem (1.16), where 𝑦^∗=𝐽_(𝑀,𝜌2)(𝑥^∗−𝜌₂𝐵𝑥^∗).

Acknowledgments

The authors wish to express gratitude to the referees for their careful reading of the manuscript and helpful suggestions. This research was partially supported by the Centre of Excellence in Mathematics under the Commission on Higher Education, Thailand, and P. Sunthrayuth was partially supported by the Centre of Excellence in Mathematics for Ph.D. Program at KMUTT. Moreover, we also would like to thank the Higher Education Research Promotion and National Research University Project of Thailand, Office of the Higher Education Commission for financial support (under the CSEC project no. 54000267).