A New System of Generalized Mixed Quasivariational Inclusions with Relaxed Cocoercive Operators and Applications

Abstract

A new system of generalized mixed quasivariational inclusions (for short, SGMQVI) with relaxed cocoercive operators, which develop some preexisting variational inequalities, is introduced and investigated in Banach spaces. Next, the existence and uniqueness of solutions to the problem (SGMQVI) are established in real Banach spaces. From fixed point perspective, we propose some new iterative algorithms for solving the system of generalized mixed quasivariational inclusion problem (SGMQVI). Moreover, strong convergence theorems of these iterative sequences generated by the corresponding algorithms are proved under suitable conditions. As an application, the strong convergence theorem for a class of bilevel variational inequalities is derived in Hilbert space. The main results in this paper develop, improve, and unify some well-known results in the literature.

1. Introduction

Generalized mixed quasivariational inclusion problems, which are extensions of variational inequalities introduced by Stampacchia [1] in the early sixties, are among the most interesting and extensively investigated classes of mathematics problems and have many applications in the fields of optimization and control, abstract economics, electrical networks, game theory, auction, engineering science, and transportation equilibria (see, e.g., [2–5] and the references therein). For the past few decades, existence results and iterative algorithms for variational inequality and variational inclusion problems have been obtained (see, e.g., [6–14] and the references cited therein). Recently, some new problems, which are called to be the system of variational inequality and equilibrium problems, received many attentions. Ansari et al. [2] considered a system of vector variational inequalities and obtained its existence results. In [3], Pang stated that the traffic equilibrium problem, the spatial equilibrium problem, the Nash equilibrium, and the general equilibrium programming problem can be modeled as a system of variational inequalities. Verma [15] and J. K. Kim and D. S. Kim [16] investigated a system of nonlinear variational inequalities. Cho et al. [17] introduced and studied a new system of nonlinear variational inequalities in Hilbert spaces and obtained the existence and uniqueness properties of solutions for the system of nonlinear variational inequalities. In [18], Peng and Zhu introduced a new system of generalized mixed quasivariational inclusions involving (𝐻,𝜂)-monotone operators. Very recently, Qin et al. [19] studied the approximation of solutions to a system of variational inclusions in Banach spaces and established a strong convergence theorem in uniformly convex and 2 uniformly smooth Banach spaces. In [20], Kamraksa and Wangkeeree gave a general iterative method for a general system of variational inclusions and proved a strong convergence theorem in strictly convex and 2 uniformly smooth Banach spaces. Further, Wangkeeree and Kamraksa [21] introduced an iterative algorithm for finding a common element of the set of solutions of a mixed equilibrium problem, the set of fixed points of an infinite family of nonexpansive mappings and the set of solutions of a general system of variational inequalities and then obtained the strong convergence of the iterative in Hilbert spaces. Petrot [22] applied the resolvent operator technique to find the common solutions for a generalized system of relaxed cocoercive mixed variational inequality problems and fixed point problems for Lipschitz mappings in Hilbert spaces. Zhao et al. [23] obtained some existence results for a system of variational inequalities by Brouwer’s fixed point theory and proved the convergence of an iterative algorithm in finite Euclidean spaces. Chen and Wan [24] also proved the existence of solutions and convergence analysis for a system of quasivariational inclusions in Banach spaces, proposed some iterative methods for finding the common element of the solutions set for the system of quasivariational inclusions and the fixed point set for Lipschitz mapping, and obtained the convergent rates of corresponding iterative sequences. On the other hand, various bilevel programming problems, bilevel decision problems, and mathematical program problems with equilibrium (variational inequalities) constraints have been wildly investigated (see, e.g., [25, 26]). To the best of our knowledge, there is few results concerning the algorithms and convergence analysis of solutions to bilevel variational inequalities in Hilbert spaces.

The aim of this paper is to introduce and study a new system of generalized mixed quasivariational inclusion problem (SGMQVI) in uniformly smooth Banach spaces which includes some previous variational inequalities as special cases. Furthermore, the existence and uniqueness theorems of solutions for the problem (SGMQVI) are established by using resolvent techniques. Thirdly, we also propose some new iterative algorithms for solving the problem (SGMQVI). Strong convergence of the iterative sequences generated by the corresponding iterative algorithms are proved under suitable conditions. As an application, we study the properties for the lower-level variational inequalities of a class of bilevel variational inequalities (for short, (BVI)) in Hilbert spaces and then suggest a reasonable iterative algorithm for (BVI). Finally, the strong convergence theorem for (BVI) are derived under appropriate assumptions. The results presented in this paper improve, develop, and extend the results of [8, 23, 24, 27].

2. Preliminaries

Throughout this paper, let 𝐸 be a real 𝑞-uniformly Banach space with its dual 𝐸^∗,𝑞>1, denote the duality between 𝐸 and 𝐸^∗ by ⟨⋅,⋅⟩ and the norm of 𝐸 by ‖⋅‖, and let 𝑇∶𝐸→𝐸 be a nonlinear mapping. If {𝑥_𝑛} is a sequence in 𝐸, we denote strong convergence of {𝑥_𝑛} to 𝑥∈𝐸 by 𝑥_𝑛→𝑥. A Banach space 𝐸 is called smooth if lim_𝑡→0(‖𝑥+𝑡𝑦‖−‖𝑥‖)/𝑡 exists for all 𝑥,𝑦∈𝐸 with ‖𝑥‖=‖𝑦‖=1. It is called uniformly smooth if the limit is attained uniformly for ‖𝑥‖=‖𝑦‖=1. The function 𝜌_𝐸(𝑡)+=sup𝑥+𝑦𝑥−𝑦2−1∶‖𝑥‖=1,𝑦≤𝑡 (2.1) is called the modulus of smoothness of 𝐸. 𝐸 is called 𝑞-uniformly smooth if there exists a constant 𝑐_𝑞>0 such that 𝜌_𝐸(𝑡)≤𝑐_𝑞𝑡^𝑞.

Example 2.1 (see [4]). All Hilbert spaces, 𝐿^𝑝(or𝑙^𝑝), and the Sobolev spaces 𝑊^𝑝_𝑚, (𝑝≥2) are 2-uniformly smooth, while 𝐿^𝑝(or𝑙^𝑝) and 𝑊^𝑝_𝑚 spaces (1<𝑝≤2) are 𝑝-uniformly smooth.
The generalized duality mapping 𝐽_𝑞∶𝐸→2^𝐸∗ defined as follows: 𝐽_𝑞(𝑥)=𝑓^∗∈𝐸^∗∶𝑓^∗,𝑥=𝑓^∗‖𝑥‖=‖𝑥‖^𝑞,𝑓^∗=‖𝑥‖^𝑞−1, (2.2) for all 𝑥∈𝐸. As we know that 𝐽=𝐽₂ is the usual normalized duality mapping, and 𝐽_𝑞(𝑥)=‖𝑥‖^𝑞−2𝐽(𝑥) for 𝑥≠0, 𝐽_𝑞(𝑡𝑥)=𝑡^𝑞−1𝐽_𝑞(𝑥), and 𝐽_𝑞(−𝑥)=−𝐽_𝑞(𝑥) for all 𝑥∈𝐸 and 𝑡∈[0,+∞), and 𝐽_𝑞 is single-valued if 𝐸 is smooth (see, e.g., [28]). If 𝐸 is a Hilbert space, then 𝐽=𝐼, where 𝐼 is the identity operator. Let 𝑔_𝑗∶𝐸→𝐸, let 𝐴_𝑗∶𝐸×𝐸→𝐸 be single-valued mappings, and let 𝑀_𝑗∶𝐸→2^𝐸 be set-valued mappings for all 𝑗∈{1,2,…,𝑛}. We consider the system of generalized mixed quasivariational inclusions problem (for short, (SGMQVI)) as follows: find (𝑥^∗₁,𝑥^∗₂,…,𝑥^∗_𝑛−1,𝑥^∗_𝑛)∈𝐸^𝑛 such that 0∈𝑥^∗₁−𝑔₁𝑥^∗₂+𝜌₁𝐴₁𝑥^∗₂,𝑥^∗₁+𝑀₁𝑥^∗₁,0∈𝑥^∗₂−𝑔₂𝑥^∗₃+𝜌₂𝐴₂𝑥^∗₃,𝑥^∗₂+𝑀₂𝑥^∗₂,⋮0∈𝑥^∗_𝑛−1−𝑔_𝑛−1𝑥^∗_𝑛+𝜌_𝑛−1𝐴_𝑛−1𝑥^∗_𝑛,𝑥^∗_𝑛−1+𝑀_𝑛−1𝑥^∗_𝑛−1,0∈𝑥^∗_𝑛−𝑔_𝑛𝑥^∗₁+𝜌_𝑛𝐴_𝑛𝑥^∗₁,𝑥^∗_𝑛+𝑀_𝑛𝑥^∗_𝑛, (2.3) where 𝜌_𝑖(𝑖=1,2,…,𝑛) are positive constants. Denote the set of solutions to (SGMQVI) by Ξ.
Special cases are as follows:
(I) If 𝑔_𝑗(𝑥)=𝑥 and 𝐴_𝑗(𝑥,𝑦)=𝑇_𝑗(𝑥)+𝑆_𝑗(𝑥) for all 𝑥,𝑦∈𝐸 and 𝑗=1,2,…,𝑛, where 𝑇_𝑗,𝑆_𝑗∶𝐸→𝐸 are single-valued mappings, then the problem (SGMQVI) is equivalent to find (𝑥^∗₁,𝑥^∗₂,…,𝑥^∗_𝑛−1,𝑥^∗_𝑛)∈𝐸^𝑛 such that 0∈𝑥^∗₁−𝑥^∗₂+𝜌₁𝑇₁𝑥^∗₂+𝑆₁𝑥^∗₂+𝑀₁𝑥^∗₁,0∈𝑥^∗₂−𝑥^∗₃+𝜌₂𝑇₂𝑥^∗₃+𝑆₂𝑥^∗₃+𝑀₂𝑥^∗₂,⋮0∈𝑥^∗_𝑛−1−𝑥^∗_𝑛+𝜌_𝑛−1𝑇_𝑛−1𝑥^∗_𝑛+𝑆_𝑛−1𝑥^∗_𝑛+𝑀_𝑛−1𝑥^∗_𝑛−1,0∈𝑥^∗_𝑛−𝑥^∗₁+𝜌_𝑛𝑇_𝑛𝑥^∗₁+𝑆_𝑛𝑥^∗₁+𝑀_𝑛𝑥^∗_𝑛, (2.4) where 𝜌_𝑖(𝑖=1,2,…,𝑛) are positive constants, which is called the system of generalized nonlinear mixed variational inclusions problem [8].
(II) If 𝑛=2,𝐴₁=𝐴₂=𝐴,𝐸=𝐻 is a Hilbert space, 𝑔₁(𝑥)=𝑔₂(𝑥)=𝑥 and 𝑀₁(𝑥)=𝑀₂(𝑥)=𝜕𝜙(𝑥) for all 𝑥∈𝐸, where 𝜙∶𝐸→𝑅∪{+∞} is a proper, convex, and lower semicontinuous functional, and 𝜕𝜙 denotes the subdifferential operator of 𝜙, then the problem (SGMQVI) is equivalent to find (𝑥^∗,𝑦^∗)∈𝐸×𝐸 such that 𝜌₁𝐴𝑦^∗,𝑥^∗+𝑥^∗−𝑦^∗,𝑥−𝑥^∗+𝜙(𝑥)−𝜙𝑥^∗𝜌≥0,∀𝑥∈𝐸,₂𝐴𝑥^∗,𝑦^∗+𝑦^∗−𝑥^∗,𝑥−𝑦^∗+𝜙(𝑥)−𝜙𝑦^∗≥0,∀𝑥∈𝐸, (2.5) where 𝜌_𝑖(𝑖=1,2) are positive constants, which is called the generalized system of relaxed cocoercive mixed variational inequality problem [29].
(III) If 𝑛=2,𝐸=𝐻 is a Hilbert space, and 𝐾 is a closed convex subset of 𝐸, and 𝜙(𝑥)=𝛿_𝐾(𝑥) for all 𝑥∈𝐾, where 𝛿_𝐾 is the indicator function of 𝐾 defined by 𝜙(𝑥)=𝛿_𝐾(𝑥)=0,if𝑥∈𝐾,+∞,otherwise, (2.6) then the problem (SGMQVI) is equivalent to find (𝑥^∗,𝑦^∗)∈𝐾×𝐾 such that 𝜌₁𝐴₁𝑦^∗,𝑥^∗+𝑥^∗−𝑔₁𝑦^∗,𝑔₁(𝑥)−𝑥^∗𝜌≥0,∀𝑥∈𝐾,₂𝐴₂𝑥^∗,𝑦^∗+𝑦^∗−𝑔₂𝑥^∗,𝑔₂(𝑥)−𝑦^∗≥0,∀𝑥∈𝐾, (2.7) where 𝜌_𝑖(𝑖=1,2) are positive constants, which is called the system of general variational inequalities problem [27].
(IV) If 𝑛=2,𝐴₁=𝐴₂=𝐴,𝐸=𝐻 is a Hilbert space, and 𝐾 is a closed convex subset of 𝐸, 𝑔₁(𝑦)=𝑔₂(𝑦)=𝑦, and 𝜙(𝑥)=𝛿_𝐾(𝑥) for all 𝑥∈𝐾,𝑦∈𝐸, where 𝛿_𝐾 is the indicator function of 𝐾 defined by 𝜙(𝑥)=𝛿_𝐾(𝑥)=0,if𝑥∈𝐾,+∞,otherwise, (2.8) then the problem (SGMQVI) is equivalent to find (𝑥^∗,𝑦^∗)∈𝐾×𝐾 such that 𝜌₁𝐴𝑦^∗,𝑥^∗+𝑥^∗−𝑦^∗,𝑥−𝑥^∗𝜌≥0,∀𝑥∈𝐾,₂𝐴𝑥^∗,𝑦^∗+𝑦^∗−𝑥^∗,𝑥−𝑦^∗≥0,∀𝑥∈𝐾, (2.9) where 𝜌_𝑖(𝑖=1,2) are positive constants, which is called the generalized system of relaxed cocoercive variational inequality problem [30].
(V) If for each 𝑖∈{1,2},𝑧∈𝐸,𝐴_𝑖(𝑥,𝑧)=Ψ_𝑖(𝑥), and 𝑔_𝑖(𝑥)=𝑥 for all 𝑥∈𝐸, where Ψ_𝑖∶𝐸→𝐸, then the problem (SGMQVI) is equivalent to find (𝑥^∗,𝑦^∗)∈𝐸×𝐸 such that 0∈𝑥^∗−𝑦^∗+𝜌₁Ψ₁𝑦^∗+𝑀₁𝑥^∗,0∈𝑦^∗−𝑥^∗+𝜌₂Ψ₂𝑥^∗+𝑀₂𝑦^∗, (2.10) where 𝜌_𝑖  (𝑖=1,2) are positive constants, which is called the system of quasivariational inclusion [19, 20].
(VI) If 𝑛=2, for each 𝑖∈{1,2},𝑧∈𝐸,𝐴_𝑖(𝑥,𝑧)=Ψ(𝑥), and 𝑔_𝑖(𝑥)=𝑥 for all 𝑥∈𝐸, where Ψ∶𝐸→𝐸 and 𝑀₁(𝑥)=𝑀₂(𝑥)=𝑀, then the problem (SGMQVI) is equivalent to find (𝑥^∗,𝑦^∗)∈𝐸×𝐸 such that 0∈𝑥^∗−𝑦^∗+𝜌₁Ψ𝑦^∗+𝑀𝑥^∗,0∈𝑦^∗−𝑥^∗+𝜌₂Ψ𝑥^∗+𝑀𝑦^∗, (2.11) where 𝜌_𝑖(𝑖=1,2) are positive constants, which is called the system of quasivariational inclusion [20].
We first recall some definitions and lemmas which are needed in our main results.

Definition 2.2. Let 𝑀∶dom(𝑀)⊂𝐸→2^𝐸 be a set-valued mapping, where dom(𝑀) is the effective domain of the mapping 𝑀. 𝑀 is said to be (i)accretive if, for any 𝑥,𝑦∈dom(𝑀),𝑢∈𝑀(𝑥) and 𝑣∈𝑀(𝑦), there exists 𝑗_𝑞(𝑥−𝑦)∈𝐽_𝑞(𝑥−𝑦) such that 𝑢−𝑣,𝑗_𝑞𝑥−𝑦≥0, (2.12)(ii)𝑚-accretive (maximal-accretive) if 𝑀 is accretive and (𝐼+𝜌𝑀)dom(𝑀)=𝐸 holds for every 𝜌>0, where 𝐼 is the identity operator on 𝐸.

Remark 2.3. If 𝐸 is a Hilbert space, then accretive operator and 𝑚-accretive operator are reduced to monotone operator and maximal monotone operator, respectively.

Definition 2.4 (see [24, 31]). Let 𝑇∶𝐸→𝐸 be a single-valued mapping. 𝑇 is said to be(i)𝛾-Lipschitz continuous mapping if there exists a constant 𝛾>0 such that 𝑇𝑥−𝑇𝑦≤𝛾𝑥−𝑦,∀𝑥,𝑦∈𝐸, (2.13)(ii)(𝑎,𝑏)-relaxed cocoercive if there exist two constants 𝑎≥0 and 𝑏>0 such that 𝑇(𝑥)−𝑇(𝑥),𝐽_𝑞(𝑥−𝑥)≥(−𝑎)‖𝑇(𝑥)−𝑇(𝑥)‖^𝑞+𝑏‖𝑥−𝑥‖^𝑞,∀𝑥,𝑥∈𝐸. (2.14)

Remark 2.5 (see [24]). (1) If 𝛾=1, then a 𝛾-Lipschitz continuous mapping reduces to a nonexpansive mapping.
(2) If 𝛾∈(0,1), then a 𝛾-Lipschitz continuous mapping reduces to a contractive mapping.
(3) It is easy to see that the identity operator 𝐼∶𝐸→𝐸 is (0,1) relaxed cocoercive, where 𝐼(𝑥)=𝑥 for all 𝑥∈𝐸.

Definition 2.6 (see [24]). Let 𝐴∶𝐸×𝐸→𝐸 be a mapping. 𝐴 is said to be (i)𝜏-Lipschitz continuous in the first variable if there exists a constant 𝜏>0 such that, for 𝑥,𝑥∈𝐸, 𝐴𝑥,𝑦−𝐴𝑥,𝑦≤𝜏‖𝑥−𝑥‖,∀𝑦,𝑦∈𝐸 (2.15)(ii)𝛼-strongly accretive if there exists a constant 𝛼>0 such that 𝐴𝑥,𝑦−𝐴𝑥,𝑦,𝐽_𝑞(𝑥−𝑥)≥𝛼‖𝑥−𝑥‖^𝑞,∀𝑥,𝑦,𝑥,𝑦∈𝐸×𝐸, (2.16) or equivalently, 𝐴𝑥,𝑦−𝐴𝑥,𝑦,𝐽(𝑥−𝑥)≥𝛼‖𝑥−𝑥‖,∀𝑥,𝑦,𝑥,𝑦∈𝐸×𝐸, (2.17)(iii)(𝜇,𝜈) relaxed cocoercive if there exist two constants 𝜇≥0 and 𝜈>0 such that 𝐴𝑥,𝑦−𝐴𝑥,𝑦,𝐽_𝑞(𝑥−𝑥)≥−𝜇𝐴𝑥,𝑦−𝐴𝑥,𝑦^𝑞+𝜈‖𝑥−𝑥‖^𝑞,∀𝑥,𝑦,𝑥,𝑦∈𝐸×𝐸. (2.18)

Remark 2.7. (1) Every 𝛼-strongly accretive mapping is a (𝜇,𝛼) relaxed cocoercive for any positive constant 𝜇. But the converse is not true in general.
(2) The conception of the cocoercivity is applied in several directions, especially for solving variational inequality problems by using the auxiliary problem principle and projection methods [14]. Several classes of relaxed cocoercive variational inequalities have been investigated in [5, 22, 28, 30].

Definition 2.8 (see [9, 32]). Let the set-valued mapping 𝑀∶dom(𝑀)⊂𝐸→2^𝐸 be 𝑚-accretive. For any positive number 𝜌>0, the mapping 𝑅^𝑀_𝜌∶𝐸→dom(𝑀) defined by 𝑅^𝑀_𝜌(𝑥)=𝐼+𝜌𝑀⁻¹(𝑥),𝑥∈𝐸, (2.19) is called the resolvent operator associated with 𝑀 and 𝜌, where 𝐼 is the identity operator on 𝐸.

Remark 2.9. Let 𝐶⊂𝐸 be a nonempty closed convex set. If 𝐸 is a Hilbert space and 𝑀=𝜕𝜙, the subdifferential of the indicator function 𝜙, that is, 𝜙(𝑥)=𝛿_𝐶(𝑥)=0,if𝑥∈𝐶,+∞,otherwise, (2.20) then 𝑅_(𝜌,𝑀)=𝑃_𝐶, the metric projection operator from 𝐸 onto 𝐶.

Lemma 2.10 (see [9, 32]). Let the set-valued mapping 𝑀∶dom(𝑀)⊂𝐸→2^𝐸 be 𝑚-accretive. Then the resolvent operator 𝑅^𝑀_𝜌 is single-valued and nonexpansive for all 𝜌>0.

Lemma 2.11 (see [33]). Let {𝐵_𝑛},{𝐶_𝑛}, and {𝐷_𝑛} be three nonnegative real sequences satisfying the following conditions: 𝐵_𝑛+1≤1−𝜆_𝑛𝐵_𝑛+𝐶_𝑛+𝐷_𝑛,∀𝑛≥𝑛₀, (2.21) for some 𝑛₀∈𝑁,{𝜆_𝑛}⊂(0,1) with ∑^∞_𝑛=0𝜆_𝑛=∞, 𝐶_𝑛=0(𝜆_𝑛) and ∑^∞_𝑛=0𝐷_𝑛<+∞. Then lim_𝑛→∞𝐵_𝑛=0.

Lemma 2.12 (see [34]). Let 𝐸 be a real 𝑞-uniformly Banach space. Then there exists a constant 𝑐_𝑞>0 such that 𝑥+𝑦^𝑞≤‖𝑥‖^𝑞+𝑞𝑦,𝐽_𝑞(𝑥)+𝑐_𝑞𝑦^𝑞,∀𝑥,𝑦∈𝐸. (2.22)

3. Existence Theorems

In this section, we will investigate the existence and uniqueness of solutions for the problem (SGMQVI) in 𝑞-uniformly smooth Banach space under some suitable conditions.

Theorem 3.1. Let (𝑥^∗₁,𝑥^∗₂,…,𝑥^∗_𝑛−1,𝑥^∗_𝑛)∈𝐸^𝑛, 𝑀_𝑖∶𝐸→2^𝐸(𝑖=1,2,…,𝑛) be maximal accretive. Then (𝑥^∗₁,𝑥^∗₂,…,𝑥^∗_𝑛−1,𝑥^∗_𝑛) is a solution of the problem (SGMQVI) if and only if 𝑥^∗₁=𝑅^𝑀1_𝜌1𝑔₁𝑥^∗₂−𝜌₁𝐴₁𝑥^∗₂,𝑥^∗₁,𝑥^∗₂=𝑅^𝑀2_𝜌2𝑔₂𝑥^∗₃−𝜌₂𝐴₂𝑥^∗₃,𝑥^∗₂,⋮𝑥^∗_𝑛−1=𝑅^𝑀𝑛−1_𝜌𝑛−1𝑔_𝑛−1𝑥^∗_𝑛−𝜌_𝑛−1𝐴_𝑛−1𝑥^∗_𝑛,𝑥^∗_𝑛−1,𝑥^∗_𝑛=𝑅^𝑀𝑛_𝜌𝑛𝑔_𝑛𝑥^∗₁−𝜌_𝑛𝐴_𝑛𝑥^∗₁,𝑥^∗_𝑛, (3.1) where 𝜌_𝑖(𝑖=1,2,…,𝑛) are positive constants.

Proof. Let (𝑥^∗₁,𝑥^∗₂,…,𝑥^∗_𝑛−1,𝑥^∗_𝑛)∈𝐸^𝑛 be a solution of the problem (SGMQVI). Then, for any given positive constants 𝜌_𝑖(𝑖=1,2,…,𝑛), the problem (SGMQVI) is equivalent to 𝑔₁𝑥^∗₂−𝜌₁𝐴₁𝑥^∗₂,𝑥^∗₁∈𝑥^∗₁+𝜌₁𝑀₁𝑥^∗₁,𝑔₂𝑥^∗₃−𝜌₂𝐴₂𝑥^∗₃,𝑥^∗₂∈𝑥^∗₂+𝜌₂𝑀₂𝑥^∗₂,⋮𝑔_𝑛−1𝑥^∗_𝑛−𝜌_𝑛−1𝐴_𝑛−1𝑥^∗_𝑛,𝑥^∗_𝑛−1∈𝑥^∗_𝑛−1+𝜌_𝑛−1𝑀_𝑛−1𝑥^∗_𝑛−1,𝑔_𝑛𝑥^∗₁−𝜌_𝑛𝐴_𝑛𝑥^∗₁,𝑥^∗_𝑛∈𝑥^∗_𝑛+𝜌_𝑛𝑀_𝑛𝑥^∗_𝑛. (3.2) From Definition 2.8 and Lemma 2.10, it yields that (3.2) is equivalent to (3.1). This completes the proof.

Theorem 3.2. Let 𝐸 be a real 𝑞-uniformly smooth Banach space. Let 𝑗∈{1,2,…,𝑛}, 𝑀_𝑗∶𝐸→2^𝐸 be 𝑚-accretive mapping, Let 𝐴_𝑗∶𝐸×𝐸→𝐸 be (𝜇_𝑗,𝜈_𝑗)-relaxed cocoercive and Lipschitz continuous in the first variable with constant 𝜏_𝑗, and Let 𝑔_𝑗∶𝐸→𝐸 be (𝑎_𝑗,𝑏_𝑗)-relaxed cocoercive and Lipschitz continuous with constant 𝜄_𝑗. Then, for each 𝑗∈{1,2,…,𝑛},𝑥∈𝐸, the mapping 𝑅^𝑀𝑗_𝜌𝑗(𝑔_𝑗(𝑥)−𝜌_𝑗𝐴_𝑗(𝑥,⋅))∶𝐸→𝐸 has at most one fixed point. If min1+𝑞𝑎_𝑗𝜄^𝑞_𝑗+𝑐_𝑞𝜄^𝑞_𝑗−𝑞𝑏_𝑗,1+𝑞𝜌_𝑗𝜇_𝑗𝜏^𝑞_𝑗+𝑐_𝑞𝜌^𝑞_𝑗𝜏^𝑞_𝑗−𝑞𝜌_𝑗𝜈_𝑗≥0, (3.3) then the implicit function 𝑦_𝑗(𝑥) determined by 𝑦_𝑗(𝑥)=𝑅^𝑀𝑗_𝜌𝑗𝑔_j(𝑥)−𝜌_𝑗𝐴_𝑗𝑥,𝑦_𝑗(𝑥) (3.4) is continuous on 𝐸.

Proof. Let 𝑥∈𝐸. We show by contradiction that 𝑅^𝑀𝑗_𝜌𝑗(𝑔_𝑗(𝑥)−𝜌_𝑗𝐴_𝑗(𝑥,⋅))∶𝐸→𝐸 has at most one fixed point. Suppose to the contrary that 𝑦,𝑦∈𝐸 and 𝑦≠𝑦 such that 𝑦=𝑅^𝑀𝑗_𝜌𝑗𝑔_𝑗(𝑥)−𝜌_𝑗𝐴_𝑗𝑥,𝑦,𝑦=𝑅^𝑀𝑗_𝜌𝑗𝑔_𝑗(𝑥)−𝜌_𝑗𝐴_𝑗𝑥,𝑦. (3.5) Since 𝐴_𝑗 is Lipschitz continuous in the first variable with constant 𝜏_𝑗, then 𝑦−𝑦=𝑅^𝑀𝑗_𝜌𝑗𝑔_𝑗(𝑥)−𝜌_𝑗𝐴_𝑗𝑥,𝑦−𝑅^𝑀𝑗_𝜌𝑗𝑔_𝑗(𝑥)−𝜌_𝑗𝐴_𝑗𝑦≤𝑔𝑥,_𝑗(𝑥)−𝜌_𝑗𝐴_𝑗𝑥,𝑦−𝑔_𝑗(𝑥)−𝜌_𝑗𝐴_𝑗𝑥,𝑦=𝜌_𝑗𝐴_𝑗𝑥,𝑦−𝐴_𝑗𝑥,𝑦≤𝜌_𝑗𝜏_𝑗‖𝑥−𝑥‖=0, (3.6) which is a contradiction. Therefore, the mapping 𝑅^𝑀𝑗_𝜌𝑗(𝑔_𝑗(𝑥)−𝜌_𝑗𝐴_𝑗(𝑥,⋅))∶𝐸→𝐸 has at most one fixed point.
Next, we show that the implicit function 𝑦_𝑗(𝑥) is continuous on 𝐸. For any sequence, {𝑥_𝑛}⊂𝐸,𝑥₀∈𝐸,𝑥_𝑛→𝑥₀ as 𝑛→∞. Since 𝐴_𝑗∶𝐸×𝐸→𝐸 is (𝜇_𝑗,𝜈_𝑗)-relaxed cocoercive and Lipschitz continuous in the first variable with constant 𝜏_𝑗 and 𝑔_𝑗∶𝐸→𝐸 is (𝑎_𝑗,𝑏_𝑗)-relaxed cocoercive and Lipschitz continuous with constant 𝜄_𝑗, one has 𝐿_𝐴𝑗=𝑥_𝑛−𝑥₀−𝜌_𝑗𝐴_𝑗𝑥_𝑛,𝑦𝑥_𝑛−𝐴_𝑗𝑥₀,𝑦𝑥₀^𝑞≤𝑥_𝑛−𝑥₀^𝑞−𝑞𝜌_𝑗𝐴_𝑗𝑥_𝑛,𝑦𝑥_𝑛−𝐴_𝑗𝑥₀,𝑦𝑥₀,𝐽_𝑞𝑥_𝑛−𝑥₀+𝑐_𝑞𝜌^𝑞_𝑗𝐴_𝑗𝑥_𝑛,𝑦𝑥_𝑛−𝐴_𝑗𝑥₀,𝑦𝑥₀^𝑞≤𝑞𝜌_𝑗𝜇_𝑗𝐴_𝑗𝑥_𝑛,𝑦𝑥_𝑛−𝐴_𝑗𝑥₀,𝑦𝑥₀^𝑞−𝜈_𝑗𝑥_𝑛−𝑥₀^𝑞+1+𝑐_𝑞𝜌^𝑞_𝑗𝜏^𝑞_𝑗𝑥_𝑛−𝑥₀^𝑞≤𝑞𝜌_𝑗𝜇_𝑗𝜏^𝑞_𝑗−𝜈_𝑗𝑥_𝑛−𝑥₀^𝑞+1+𝑐_𝑞𝜌^𝑞_𝑗𝜏^𝑞_𝑗𝑥_𝑛−𝑥₀^𝑞=1+𝑞𝜌_𝑗𝜇_𝑗𝜏^𝑞_𝑗+𝑐_𝑞𝜌^𝑞_𝑗𝜏^𝑞_𝑗−𝑞𝜌_𝑗𝜈_𝑗𝑥_𝑛−𝑥₀^𝑞,𝐿_𝑔𝑗=𝑥_𝑛−𝑥₀−𝑔_𝑗𝑥_𝑛−𝑔_j𝑥₀^𝑞≤𝑥_𝑛−𝑥₀^𝑞−𝑞𝑔_𝑗𝑥_𝑛−𝑔_𝑗𝑥₀,𝐽_𝑞𝑥_𝑛−𝑥₀+𝑐_𝑞𝑔_𝑗𝑥_𝑛−𝑔_𝑗𝑥₀^𝑞≤1+𝑐_𝑞𝜄^𝑞_𝑗𝑥_𝑛−𝑥₀^𝑞+𝑞𝑎_𝑗𝑔_𝑗𝑥_𝑛−𝑔_𝑗𝑥₀^𝑞−𝑏_𝑗𝑥_𝑛−𝑥₀^𝑞≤1+𝑐_𝑞𝜄^𝑞_𝑗𝑥_𝑛−𝑥₀^𝑞+𝑞𝑎_𝑗𝜄^𝑞_𝑗−𝑏_𝑗𝑥_𝑛−𝑥₀^𝑞=1+𝑞𝑎_𝑗𝜄^𝑞_𝑗+𝑐_𝑞𝜄^𝑞_j−𝑞𝑏_𝑗𝑥_𝑛−𝑥₀^𝑞. (3.7) Therefore, from Lemma 2.10, we get 𝑦_𝑗𝑥_𝑛−𝑦_𝑗𝑥₀=𝑅^𝑀𝑗_𝜌𝑗𝑔_𝑗𝑥_𝑛−𝜌_𝑗𝐴_𝑗𝑥_𝑛,𝑦_𝑗𝑥_𝑛−𝑅^𝑀𝑗_𝜌𝑗𝑔_𝑗𝑥₀−𝜌_𝑗𝐴_𝑗𝑥₀,𝑦_𝑗𝑥₀≤𝑔_𝑗𝑥_𝑛−𝜌_𝑗𝐴_𝑗𝑥_𝑛,𝑦_𝑗𝑥_𝑛−𝑔_𝑗𝑥₀−𝜌_𝑗𝐴_𝑗𝑥₀,𝑦_𝑗𝑥₀=𝑔_𝑗𝑥_𝑛−𝑔_𝑗𝑥₀−𝜌_𝑗𝐴_𝑗𝑥_𝑛,𝑦_𝑗𝑥_𝑛−𝐴_𝑗𝑥₀,𝑦_𝑗𝑥₀≤𝑥_𝑛−𝑥₀−𝑔_𝑗𝑥_𝑛−𝑔_𝑗𝑥₀+𝑥_𝑛−𝑥₀−𝜌_𝑗𝐴_𝑗𝑥_𝑛,𝑦_𝑗𝑥_𝑛−𝐴_𝑗𝑥₀,𝑦_𝑗𝑥₀≤^𝑞𝐿_𝐴𝑗+^𝑞𝐿_𝑔𝑗≤^𝑞1+𝑞𝜌_𝑗𝜇_𝑗𝜏^𝑞_𝑗+𝑐_𝑞𝜌^𝑞_𝑗𝜏^𝑞_𝑗−𝑞𝜌_𝑗𝜈_𝑗+^𝑞1+𝑞𝑎_𝑗𝜄^𝑞_𝑗+𝑐_𝑞𝜄^𝑞_𝑗−𝑞𝑏_𝑗𝑥_𝑛−𝑥₀. (3.8) From (3.3), it follows that the implicit function 𝑦_𝑗(𝑥) is continuous on 𝐸. This completes the proof.

If 𝑗=2 and 𝑔(𝑥)=𝑥 for all 𝑥∈𝐸, then Theorem 3.2 is reduced to the following result.

Corollary 3.3 (see [24]). Let 𝐸 be a real 𝑞-uniformly smooth Banach space. Let 𝑀₂∶𝐸→2^𝐸 be 𝑚-accretive mapping; Let 𝐴₂∶𝐸×𝐸→𝐸 be (𝜇₂,𝜈₂)-relaxed cocoercive and Lipschitz continuous in the first variable with constant 𝜏₂. Then, for each 𝑥∈𝐸, the mapping 𝑅^𝑀2_𝜌2(𝑥−𝜌₂𝐴₂(𝑥,⋅))∶𝐸→𝐸 has at most one fixed point. If 1+𝑞𝜌₂𝜇₂𝜏^𝑞₂+𝑐_𝑞𝜌^𝑞₂𝜏^𝑞₂−𝑞𝜌₂𝜈₂≥0, (3.9) then the implicit function 𝑦(𝑥) determined by 𝑦(𝑥)=𝑅^𝑀2_𝜌2𝑥−𝜌₂𝐴₂𝑥,𝑦₂(𝑥) (3.10) is continuous on 𝐸.

Theorem 3.4. Let 𝐸 be a real 𝑞-uniformly smooth Banach space. Let 𝑀_𝑗∶𝐸→2^𝐸 be 𝑚-accretive mapping, Let 𝐴_𝑗∶𝐸×𝐸→𝐸 be (𝜇_𝑗,𝜈_𝑗)-relaxed cocoercive and Lipschitz continuous in the first variable with constant 𝜏_𝑗, and let 𝑔_𝑗∶𝐸→𝐸 be (𝑎_𝑗,𝑏_𝑗)-relaxed cocoercive and Lipschitz continuous with constant 𝜄_𝑗 for 𝑗∈{1,2,…,𝑛}. Assume that min1+𝑞𝑎_𝑗𝜄^𝑞_𝑗+𝑐_𝑞𝜄^𝑞_𝑗−𝑞𝑏_𝑗,1+𝑞𝜌_𝑗𝜇_𝑗𝜏^𝑞_𝑗+𝑐_𝑞𝜌^𝑞_𝑗𝜏^𝑞_𝑗−𝑞𝜌_𝑗𝜈_𝑗≥0,𝑗=1,2,…,𝑛,0≤^𝑛_𝑗=1^𝑞1+𝑞𝑎_𝑗𝜄^𝑞_𝑗+𝑐_𝑞𝜄^𝑞_𝑗−𝑞𝑏_𝑗+^𝑞1+𝑞𝜌_𝑗𝜇_𝑗𝜏^𝑞_𝑗+𝑐_𝑞𝜌^𝑞_𝑗𝜏^𝑞_𝑗−𝑞𝜌_𝑗𝜈_𝑗<1. (3.11)(3.12) Then the problem (SGMQVI) has a solution. Moreover, the solutions set Ξ of (SGMQVI) is a singleton.

Proof. By Theorem 3.1, the problem (SGMQVI) has a solution if and only if (3.1) holds. For the convenience, we define a mapping 𝐹∶𝐸→𝐸 by 𝐹(𝑥)=𝑅^𝑀1_𝜌1𝑔₁𝑦₂(𝑥)−𝜌₁𝐴₁𝑦₂(𝑥),𝑦,𝑥₂(𝑥)=𝑅^𝑀2_𝜌2𝑔₂𝑦₃(𝑥)−𝜌₂𝐴₂𝑦₃(𝑥),𝑦₂(𝑥),⋮𝑦_𝑛−1(𝑥)=𝑅^𝑀𝑛−1_𝜌𝑛−1𝑔_𝑛−1𝑦_𝑛(𝑥)−𝜌_𝑛−1𝐴_𝑛−1𝑦_𝑛(𝑥),𝑦_𝑛−1(𝑥),𝑦_𝑛(𝑥)=𝑅^𝑀𝑛_𝜌𝑛𝑔_𝑛(𝑥)−𝜌_𝑛𝐴_𝑛𝑥,𝑦_𝑛(𝑥),𝑥∈𝐸. (3.13) Since 𝐴_𝑗∶𝐸×𝐸→𝐸 are (𝜇_𝑗,𝜈_𝑗)-relaxed cocoercive and Lipschitz continuous in the first variable with constant 𝜏_𝑗 and 𝑔_𝑗∶𝐸→𝐸 are (𝑎_𝑗,𝑏_𝑗)-relaxed cocoercive and Lipschitz continuous with constant 𝜄_𝑗 for 𝑗∈{1,2,…,𝑛}, by Theorem 3.2, we know that 𝐹(𝑥) and 𝑦_𝑖(𝑥)(𝑖=2,3,…,𝑛) are continuous on 𝐸. For any 𝑥,𝑧∈𝐸, 𝐿_𝐴𝑗=𝑦_𝑗+1(𝑥)−𝑦_𝑗+1(𝑧)−𝜌_𝑗𝐴_𝑗𝑦_𝑗+1(𝑥),𝑦_𝑗(𝑥)−𝐴_𝑗𝑦_𝑗+1(𝑧),𝑦_𝑗(𝑧)^𝑞≤−𝑞𝜌_𝑗𝐴_𝑗𝑦_𝑗+1(𝑥),𝑦_𝑗(𝑥)−𝐴_𝑗𝑦_𝑗+1(𝑧),𝑦_𝑗(𝑧),𝐽_𝑞𝑦_𝑗+1(𝑥)−𝑦_𝑗+1(𝑧+𝑦_𝑗+1)(𝑥)−𝑦_𝑗+1(𝑧)^𝑞+𝑐_𝑞𝜌^𝑞_𝑗𝐴_𝑗𝑦_𝑗+1(𝑥),𝑦_𝑗(𝑥)−𝐴_𝑗𝑦_𝑗+1(𝑧),𝑦_𝑗(𝑧)^𝑞≤𝑞𝜌_𝑗𝜇_𝑗𝐴_𝑗𝑦_𝑗+1(𝑥),𝑦_𝑗(𝑥)−𝐴_𝑗𝑦_𝑗+1(𝑧),𝑦_𝑗(𝑧)^𝑞−𝜈_𝑗𝑦_𝑗+1(𝑥)−𝑦_𝑗+1(𝑧)^𝑞+1+𝑐_𝑞𝜌^𝑞_𝑗𝜏^𝑞_𝑗𝑦_𝑗+1(𝑥)−𝑦_𝑗+1(𝑧)^𝑞≤1+𝑞𝜌_𝑗𝜇_𝑗𝜏^𝑞_𝑗+𝑐_𝑞𝜌^𝑞_𝑗𝜏^𝑞_𝑗−𝑞𝜌_𝑗𝜈_𝑗𝑦_𝑗+1(𝑥)−𝑦_𝑗+1(𝑧)^𝑞𝐿,𝑗=2,3,…,𝑛−1,_𝑔𝑗=𝑦_𝑗+1(𝑥)−𝑦_𝑗+1(𝑧)−𝑔_𝑗𝑦_𝑗+1(𝑥)−𝑔_𝑗𝑦_𝑗+1(𝑧)^𝑞≤𝑦_𝑗+1(𝑥)−𝑦_𝑗+1(𝑧)^𝑞−𝑞𝑔_𝑗𝑦_𝑗+1(𝑥)−𝑔_𝑗𝑦_𝑗+1(𝑧),𝐽_𝑞𝑦_𝑗+1(𝑥)−𝑦_𝑗+1(𝑧)+𝑐_𝑞𝑔_𝑗𝑦_𝑗+1(𝑥)−𝑔_𝑗𝑦_𝑗+1(𝑧)^𝑞≤1+𝑞𝑎_𝑗𝜄^𝑞_𝑗+𝑐_𝑞𝜄^𝑞_𝑗−𝑞𝑏_𝑗𝑦_𝑗+1(𝑥)−𝑦_𝑗+1(𝑧)^𝑞𝐿,𝑗=2,3,…,𝑛−1,_𝐴𝑛=𝑥−𝑧−𝜌_𝑛𝐴_𝑛𝑥,𝑦_𝑛(𝑥)−𝐴_𝑛𝑧,𝑦_𝑛(𝑧)^𝑞≤‖𝑥−𝑧‖^𝑞−𝑞𝜌_𝑛𝐴_𝑛𝑥,𝑦_𝑛(𝑥)−𝐴_𝑛𝑧,𝑦_𝑛(𝑧),𝐽_𝑞(𝑥−𝑧)+𝑐_𝑞𝜌^𝑞_𝑛𝐴_𝑛𝑥,𝑦_𝑛(𝑥)−𝐴_𝑛𝑧,𝑦_𝑛(𝑧)^𝑞≤1+𝑞𝜌_𝑛𝜇_𝑛𝜏^𝑞_𝑛+𝑐_𝑞𝜌^𝑞_𝑛𝜏^𝑞_𝑛−𝑞𝜌_𝑛𝜈_𝑛‖𝑥−𝑧‖^𝑞,𝐿_𝑔𝑛=𝑥−𝑧−𝑔_𝑛(𝑥)−𝑔_𝑛(𝑧)^𝑞≤‖𝑥−𝑧‖^𝑞−𝑞𝑔_𝑛(𝑥)−𝑔_𝑛(𝑧),𝐽_𝑞(𝑥−𝑧)+𝑐_𝑞𝑔_𝑛(𝑥)−𝑔_𝑛(𝑧)^𝑞≤1+𝑞𝑎_𝑛𝜄^𝑞_𝑛+𝑐_𝑞𝜄^𝑞_𝑛−𝑞𝑏_𝑛‖𝑥−𝑧‖^𝑞,𝐿_𝐴1=𝑦₂(𝑥)−𝑦₂(𝑧)−𝜌₁𝐴₁𝑦₂(𝑥),𝑥−𝐴₁𝑦₂(𝑧),𝑧^𝑞≤1+𝑞𝜌₁𝜇₁𝜏^𝑞₁+𝑐_𝑞𝜌^𝑞₁𝜏^𝑞₁−𝑞𝜌₁𝜈₁𝑦₂(𝑥)−𝑦₂(𝑧)^𝑞,𝐿_𝑔1=𝑦₂(𝑥)−𝑦₂(𝑧)−𝑔₁𝑦₂(𝑥)−𝑔₁𝑦₂(𝑧)^𝑞≤1+𝑞𝑎₁𝜄^𝑞₁+𝑐_𝑞𝜄^𝑞₁−𝑞𝑏₁𝑦₂(𝑥)−𝑦₂(𝑧)^𝑞. (3.14) From Lemma 2.10, it yields that ‖𝐹(𝑥)(𝑧=𝑅−𝐹)‖_𝑀1𝜌1𝑔₁𝑦₂(𝑥)−𝜌₁𝐴₁𝑦₂(𝑥),𝑥−𝑅^𝑀1_𝜌1𝑔₁𝑦₂(𝑧)−𝜌₁𝐴₁𝑦₂(𝑧)≤𝑔,𝑧₁𝑦₂(𝑥)−𝜌₁𝐴₁𝑦₂(𝑥),𝑥−𝑔₁𝑦₂(𝑧)−𝜌₁𝐴₁𝑦₂(𝑧)=𝑔,𝑧₁𝑦₂(𝑥)−𝑔₁𝑦₂(𝑧)−𝜌₁𝐴₁𝑦₂(𝑥),𝑥−𝐴₁𝑦₂(𝑧)≤𝑦,𝑧₂(𝑥)−𝑦₂(𝑧)−𝜌₁𝐴₁𝑦₂(𝑥),𝑥−𝐴₁𝑦₂(𝑧)+𝑦,𝑧₂(𝑥)−𝑦₂(𝑧)−𝑔₁𝑦₂(𝑥)−𝑔₁𝑦₂(𝑧)≤^𝑞1+𝑞𝑎₁𝜄^𝑞₁+𝑐_𝑞𝜄^𝑞₁−𝑞𝑏₁+^𝑞1+𝑞𝜌₁𝜇₁𝜏^𝑞₁+𝑐_𝑞𝜌^𝑞₁𝜏^𝑞₁−𝑞𝜌₁𝜈₁𝑦₂(𝑥)−𝑦₂(𝑧). (3.15) Note that, for each 𝑗∈{2,3,…,𝑛−1}, 𝑦_𝑗(𝑥)−𝑦_𝑗(𝑧=𝑅_{𝑀𝑗𝜌𝑗})𝑔_𝑗𝑦_𝑗+1(𝑥)−𝜌_𝑗𝐴_𝑗𝑦_𝑗+1(𝑥),𝑦_𝑗(𝑥)−𝑅^𝑀𝑗_𝜌𝑗𝑔_𝑗𝑦_𝑗+1(𝑧)−𝜌_𝑗𝐴_𝑗𝑦_𝑗+1(𝑧),𝑦_𝑗(𝑧≤𝑔_𝑗)𝑦_𝑗+1(𝑥)−𝜌_𝑗𝐴_𝑗𝑦_𝑗+1(𝑥),𝑦_𝑗(𝑥)−𝑔_𝑗𝑦_𝑗+1(𝑧)−𝜌_𝑗𝐴_𝑗𝑦_𝑗+1(𝑧),𝑦_𝑗(𝑧)=𝑔_𝑗𝑦_𝑗+1(𝑥)−𝑔_𝑗𝑦_𝑗+1(𝑧)−𝜌_𝑗𝐴_𝑗𝑦_𝑗+1(𝑥),𝑦_𝑗(𝑥)−𝐴_𝑗𝑦_𝑗+1(𝑧),𝑦_𝑗(𝑧)≤𝑦_𝑗+1(𝑥)−𝑦_𝑗+1(𝑧)−𝜌_𝑗𝐴_𝑗𝑦_𝑗+1(𝑥),𝑦_𝑗(𝑥)−𝐴_𝑗𝑦_𝑗+1(𝑧),𝑦_𝑗(𝑧)+𝑦_𝑗+1(𝑥)−𝑦_𝑗+1(𝑧)−𝑔_𝑗𝑦_𝑗+1(𝑥)−𝑔_𝑗𝑦_𝑗+1(𝑧)≤^𝑞1+𝑞𝜌_𝑗𝜇_𝑗𝜏^𝑞_𝑗+𝑐_𝑞𝜌^𝑞_𝑗𝜏^𝑞_𝑗−𝑞𝜌_𝑗𝜈_𝑗+^𝑞1+𝑞𝑎_𝑗𝜄^𝑞_𝑗+𝑐_𝑞𝜄^𝑞_𝑗−𝑞𝑏_𝑗𝑦_𝑗+1(𝑥)−𝑦_𝑗+1(𝑧,𝑦_𝑛)(𝑥)−𝑦_𝑛(𝑧=𝑅_{𝑀𝑛𝜌𝑛})𝑔_𝑛(𝑥)−𝜌_𝑛𝐴_𝑛𝑥,𝑦_𝑛(𝑥)−𝑅^𝑀𝑛_𝜌𝑛𝑔_𝑛(𝑧)−𝜌_𝑛𝐴_𝑛𝑧,𝑦_𝑛(𝑧)≤𝑔_𝑛(𝑥)−𝑔_𝑛(𝑧)−𝜌_𝑛𝐴_𝑛𝑥,𝑦_𝑛(𝑥)−𝐴_𝑛𝑧,𝑦_𝑛(𝑧)≤𝑥−𝑧−𝜌_𝑛𝐴_𝑛𝑥,𝑦_𝑛(𝑥)−𝐴_𝑛𝑧,𝑦_𝑛(𝑧)+𝑥−𝑧−𝑔_𝑛(𝑥)−𝑔_𝑛(𝑧≤_𝑞)1+𝑞𝜌_𝑛𝜇_𝑛𝜏^𝑞_𝑛+𝑐_𝑞𝜌^𝑞_𝑛𝜏^𝑞_𝑛−𝑞𝜌_𝑛𝜈_𝑛+^𝑞1+𝑞𝑎_𝑛𝜄^𝑞_𝑛+𝑐_𝑞𝜄^𝑞_𝑛−𝑞𝑏_𝑛‖𝑥−𝑧‖. (3.16) Therefore, we obtain ‖𝐹(𝑥)−𝐹(𝑧)‖≤^𝑛_𝑗=1^𝑞1+𝑞𝜌_𝑗𝜇_𝑗𝜏^𝑞_𝑗+𝑐_𝑞𝜌^𝑞_j𝜏^𝑞_𝑗−𝑞𝜌_𝑗𝜈_𝑗+^𝑞1+𝑞𝑎_𝑗𝜄^𝑞_𝑗+𝑐_𝑞𝜄^𝑞_𝑗−𝑞𝑏_𝑗‖𝑥−𝑧‖. (3.17) It follows from (3.12) that the mapping 𝐹 is contractive. By banach’s contraction principle, there exists a unique 𝑥^∗₁∈𝐸 such that 𝐹(𝑥^∗₁)=𝑥^∗₁. Therefore, by Theorem 3.2, there exists an unique (𝑥^∗₁,𝑥^∗₂,…,𝑥^∗_𝑛)∈𝐸^𝑛 such that (𝑥^∗₁,𝑥^∗₂,…,𝑥^∗_𝑛) is a solution of the problem (SGMQVI), where 𝑥^∗_𝑖=𝑦_𝑖(𝑥^∗₁) for 𝑖=2,3,…,𝑛, that is, Ξ={(𝑥^∗₁,𝑥^∗₂,…,𝑥^∗_𝑛)}. This completes the proof.

4. Convergence Analysis

In this section, we introduce several implicit algorithms with errors and explicit algorithms without errors for solving the system of generalized mixed quasivariational inclusions problem (SGMQVI) and then explore the convergence analysis of the iterative sequences generated by the corresponding algorithms.

From Section 3, we know that the system of generalized mixed quasivariational inclusions problem (SGMQVI) is equivalent to the fixed point problem (3.1). This equivalent formulation is crucial from the numerical analysis point of view. As we know, this fixed point formulation has been used to suggest and analyze some iterative methods for solving variational inequalities and related optimization problems. By using the relations between the problem (SGMQVI) and the fixed point problem (3.1), we construct the following iterative algorithms for solving the system of generalized mixed quasivariational inclusions problem (2.3).

Algorithm 4.1. Let 𝜌_𝑗 be positive constants for all 𝑗=1,2,…,𝑛. For any given points 𝑥_1,0∈𝐸, define sequences {𝑥_𝑗,𝑘}𝑗=1,2,…,𝑛 in 𝐸 by the following implicit algorithm: 𝑥_𝑛,𝑘=𝑅^𝑀𝑛_𝜌𝑛𝑔_𝑛𝑥_1,𝑘−𝜌_𝑛𝐴_𝑛𝑥_1,𝑘,𝑥_𝑛,𝑘,𝑥_{𝑛−1,𝑘}=𝑅^𝑀𝑛−1_𝜌𝑛−1𝑔_𝑛−1𝑥_𝑛,𝑘−𝜌_𝑛−1𝐴_𝑛−1𝑥_𝑛,𝑘,𝑥_{𝑛−1,𝑘},⋮𝑥_2,𝑘=𝑅^𝑀2_𝜌2𝑔₂𝑥_3,𝑘−𝜌₂𝐴₂𝑥_3,𝑘,𝑥_2,𝑘,𝑥_1,𝑘+1=1−𝛼_𝑘𝑥_1,𝑘+𝛼_𝑘𝑅^𝑀1_𝜌1𝑔₁𝑥_2,𝑘−𝜌₁𝐴₁𝑥_2,𝑘,𝑥_1,𝑘+𝑒_𝑘,𝑘=0,1,2,…, (4.1) where {𝑒_𝑘}⊂𝐸 and {𝛼_𝑘} is a real sequence in [0,1].

If 𝑒_𝑘≡0 for all 𝑘≥0, then Algorithm 4.1 is reduced to the following result.

Algorithm 4.2. Let 𝜌_𝑗 be positive constants for all 𝑗=1,2,…,𝑛. For any given points 𝑥_1,0∈𝐸, define sequences {𝑥_𝑗,𝑘}𝑗=1,2,…,𝑛 in 𝐸 by the following implicit algorithm 𝑥_𝑛,𝑘=𝑅^𝑀𝑛_𝜌𝑛𝑔_𝑛𝑥_1,𝑘−𝜌_𝑛𝐴_𝑛𝑥_1,𝑘,𝑥_𝑛,𝑘,𝑥_{𝑛−1,𝑘}=𝑅^𝑀𝑛−1_𝜌𝑛−1𝑔_𝑛−1𝑥_𝑛,𝑘−𝜌_𝑛−1𝐴_𝑛−1𝑥_𝑛,𝑘,𝑥_{𝑛−1,𝑘},⋮𝑥_2,𝑘=𝑅^𝑀2_𝜌2𝑔₂𝑥_3,𝑘−𝜌₂𝐴₂𝑥_3,𝑘,𝑥_2,𝑘,𝑥_1,𝑘+1=1−𝛼_𝑘𝑥_1,𝑘+𝛼_𝑘𝑅^𝑀1_𝜌1𝑔₁𝑥_2,𝑘−𝜌₁𝐴₁𝑥_2,𝑘,𝑥_1,𝑘,𝑘=0,1,2,..., (4.2) where {𝛼_𝑘} is a real sequence in [0,1].

Now we construct an explicit algorithms for solving the system of generalized mixed quasivariational inclusions problem (SGMQVI).

Algorithm 4.3. Let 𝜌_𝑗 be positive constants for all 𝑗=1,2,…,𝑛. For any given points (𝑥_1,0,𝑥_2,0,…,𝑥_𝑛−1,0,𝑥_𝑛,0)∈𝐸^𝑛, define sequences {𝑥_𝑗,𝑘}(𝑗=1,2,…,𝑛) in 𝐸 by the following explicit algorithm 𝑥_𝑛,𝑘+1=𝑅^𝑀𝑛_𝜌𝑛𝑔_𝑛𝑥_1,𝑘−𝜌_𝑛𝐴_𝑛𝑥_1,𝑘,𝑥_𝑛,𝑘,𝑥_{𝑛−1,𝑘+1}=𝑅^𝑀𝑛−1_𝜌𝑛−1𝑔_𝑛−1𝑥_𝑛,𝑘+1−𝜌_𝑛−1𝐴_𝑛−1𝑥_𝑛,𝑘+1,𝑥_{𝑛−1,𝑘},⋮𝑥_2,𝑘+1=𝑅^𝑀2_𝜌2𝑔₂𝑥_3,𝑘+1−𝜌₂𝐴₂𝑥_3,𝑘+1,𝑥_2,𝑘,𝑥_1,𝑘+1=1−𝛼_𝑘𝑥_1,𝑘+𝛼_𝑘𝑅^𝑀1_𝜌1𝑔₁𝑥_2,𝑘+1−𝜌₁𝐴₁𝑥_2,𝑘+1,𝑥_1,𝑘,𝑘=0,1,2,…, (4.3) where {𝛼_𝑘} is a real sequence in [0,1].

Remark 4.4. If 𝑛=2,𝐸=𝐻 is a Hilbert space, and 𝐾 is a closed convex subset of 𝐸, 𝜙(𝑥)=𝛿_𝐾(𝑥) for all 𝑥∈𝐾, and 𝑀₁(𝑥)=𝑀₂(𝑥)=𝜕𝜙(𝑥) for all 𝑥∈𝐸, where 𝛿_𝐾 is the indicator function of 𝐾, and 𝜕𝜙 denotes the subdifferential operator of 𝜙, then, from Remark 2.9, Algorithms 4.1 and 4.3 are reduced to the Algorithms 4.5 and 4.6 for solving the system of general variational inequalities problem (2.7).

Algorithm 4.5. Let 𝜌_𝑗 be positive constants for all 𝑗=1,2. For any given points 𝑥_1,0∈𝐸, define sequences {𝑥_𝑗,𝑘}𝑗=1,2 in 𝐸 by the following implicit algorithm: 𝑥_2,𝑘=𝑃_𝐾𝑔₂𝑥_1,𝑘−𝜌₂𝐴₂𝑥_1,𝑘,𝑥_2,𝑘,𝑥_1,𝑘+1=1−𝛼_𝑘𝑥_1,𝑘+𝛼_𝑘𝑃_𝐾𝑔₁𝑥_2,𝑘−𝜌₁𝐴₁𝑥_2,𝑘,𝑥_1,𝑘+𝑒_𝑘,𝑘=0,1,2,…, (4.4) where {𝑒_𝑘}⊂𝐸 and {𝛼_𝑘} is a real sequence in [0,1].

Algorithm 4.6. Let 𝜌_𝑗 be positive constants for all 𝑗=1,2. For any given points (𝑥_1,0,𝑥_2,0)∈𝐸², define sequences {𝑥_𝑗,𝑘}(𝑗=1,2) in 𝐸 by the following explicit algorithm: 𝑥_2,𝑘+1=𝑃_𝐾𝑔₂𝑥_1,𝑘−𝜌₂𝐴₂𝑥_1,𝑘,𝑥_2,𝑘,𝑥_1,𝑘+1=1−𝛼_𝑘𝑥_1,𝑘+𝛼_𝑘𝑃_𝐾𝑔₁𝑥_2,𝑘+1−𝜌₁𝐴₁𝑥_2,𝑘+1,𝑥_1,𝑘,𝑘=0,1,2,…, (4.5) where {𝛼_𝑘} is a real sequence in [0,1].

Theorem 4.7. Let 𝐸 be a real 𝑞-uniformly smooth Banach space, and let 𝑀_𝑗,𝐴_𝑗, and 𝑔_𝑗(𝑗=1,2,…,𝑛) be the same as in Theorem 3.4. Assume that {𝛼_𝑛} is a real sequence in (0,1] and satisfy the following conditions:(i)∑^∞_𝑘=0𝛼_𝑘=∞;(ii)∑^∞_𝑘=0‖𝑒_𝑘‖<+∞;(iii)min{1+𝑞𝑎_𝑗𝜄^𝑞_𝑗+𝑐_𝑞𝜄^𝑞_𝑗−𝑞𝑏_𝑗,1+𝑞𝜌_𝑗𝜇_𝑗𝜏^𝑞_𝑗+𝑐_𝑞𝜌^𝑞_𝑗𝜏^𝑞_𝑗−𝑞𝜌_𝑗𝜈_𝑗}≥0,𝑗=1,2,…,𝑛;(iv)0<∏^𝑛_𝑗=1(^𝑞1+𝑞𝑎_𝑗𝜄^𝑞_𝑗+𝑐_𝑞𝜄^𝑞_𝑗−𝑞𝑏_𝑗+^𝑞1+𝑞𝜌_𝑗𝜇_𝑗𝜏^𝑞_𝑗+𝑐_𝑞𝜌^𝑞_𝑗𝜏^𝑞_𝑗−𝑞𝜌_𝑗𝜈_𝑗)<1.Then the sequences {𝑥_𝑗,𝑘}(𝑗=1,2,…,𝑛) generated by Algorithm 4.1 converge strongly to 𝑥^∗_𝑗(𝑗=1,2,…,𝑛), respectively, such that (𝑥^∗₁,𝑥^∗₂,…,𝑥^∗_𝑛)∈Ξ.

Proof. By Theorem 3.4, we know that there exist an unique point (𝑥^∗₁,𝑥^∗₂,…,𝑥^∗_𝑛)∈𝐸^𝑛 such that (𝑥^∗₁,𝑥^∗₂,…,𝑥^∗_𝑛)∈Ξ. Then, from Theorem 3.1, one has 𝑥^∗₁=𝑅^𝑀1_𝜌1𝑔₁𝑥^∗₂−𝜌₁𝐴₁𝑥^∗₂,𝑥^∗₁,𝑥^∗₂=𝑅^𝑀2_𝜌2𝑔₂𝑥^∗₃−𝜌₂𝐴₂𝑥^∗₃,𝑥^∗₂,⋮𝑥^∗_𝑛−1=𝑅^𝑀𝑛−1_𝜌𝑛−1𝑔_𝑛−1𝑥^∗_𝑛−𝜌_𝑛−1𝐴_𝑛−1𝑥^∗_𝑛,𝑥^∗_𝑛−1,𝑥^∗_𝑛=𝑅^𝑀𝑛_𝜌𝑛𝑔_𝑛𝑥^∗₁−𝜌_𝑛𝐴_𝑛𝑥^∗₁,𝑥^∗_𝑛. (4.6) Therefore, from both (4.1) and (4.6), we have 𝑥_1,𝑘+1−𝑥^∗₁=1−𝛼_𝑘𝑥_1,𝑘+𝛼_𝑘𝑅^𝑀1_𝜌1𝑔₁𝑥_2,𝑘−𝜌₁𝐴₁𝑥_2,𝑘,𝑥_1,𝑘+𝑒_𝑘−𝑥^∗₁=1−𝛼_𝑘𝑥_1,𝑘−𝑥^∗₁+𝛼_𝑘𝑅^𝑀1_𝜌1𝑔₁𝑥_2,𝑘−𝜌₁𝐴₁𝑥_2,𝑘,𝑥_1,𝑘−𝑥^∗₁+𝑒_𝑘=𝛼_𝑘𝑅^𝑀1_𝜌1𝑔₁𝑥_2,𝑘−𝜌₁𝐴₁𝑥_2,𝑘,𝑥_1,𝑘−𝑅^𝑀1_𝜌1𝑔₁𝑥^∗₂−𝜌₁𝐴₁𝑥^∗₂,𝑥^∗₁+1−𝛼_𝑘𝑥_1,𝑘−𝑥^∗₁+𝑒_𝑘≤𝛼_𝑘𝑅^𝑀1_𝜌1𝑔₁𝑥_2,𝑘−𝜌₁𝐴₁𝑥_2,𝑘,𝑥_1,𝑘−𝑅^𝑀1_𝜌1𝑔₁𝑥^∗₂−𝜌₁𝐴₁𝑥^∗₂,𝑥^∗₁+1−𝛼_𝑘𝑥_1,𝑘−𝑥^∗₁+𝑒_𝑘≤𝛼_𝑘𝑔₁𝑥_2,𝑘−𝜌₁𝐴₁𝑥_2,𝑘,𝑥_1,𝑘−𝑔₁𝑥^∗₂−𝜌₁𝐴₁𝑥^∗₂,𝑥^∗₁+1−𝛼_𝑘𝑥_1,𝑘−𝑥^∗₁+𝑒_𝑘=𝛼_𝑘𝑔₁𝑥_2,𝑘−𝑔₁𝑥^∗₂−𝜌₁𝐴₁𝑥_2,𝑘,𝑥_1,𝑘−𝜌₁𝐴₁𝑥^∗₂,𝑥^∗₁+1−𝛼_𝑘𝑥_1,𝑘−𝑥^∗₁+𝑒_𝑘≤1−𝛼_𝑘𝑥_1,𝑘−𝑥^∗₁+𝑒_𝑘+𝛼_𝑘𝑥_2,𝑘−𝑥^∗₂−𝑔₁𝑥_2,𝑘−𝑔₁𝑥^∗₂+𝛼_𝑘𝑥_2,𝑘−𝑥^∗₂−𝜌₁𝐴₁𝑥_2,𝑘,𝑥_1,𝑘−𝐴₁𝑥^∗₂,𝑥^∗₁. (4.7) Since 𝐴_𝑗∶𝐸×𝐸→𝐸 are (𝜇_𝑗,𝜈_𝑗)-relaxed cocoercive and Lipschitz continuous in the first variable with constant 𝜏_𝑗 and 𝑔_𝑗∶𝐸→𝐸 are (𝑎_𝑗,𝑏_𝑗)-relaxed cocoercive and Lipschitz continuous with constant 𝜄_𝑗 for 𝑗∈{1,2,…,𝑛}, we can conclude 𝐿^∗_𝐴𝑖=𝑥_𝑖+1,𝑘−𝑥^∗_𝑖+1−𝜌_i𝐴_𝑖𝑥_𝑖+1,𝑘,𝑥_𝑖,𝑘−𝐴_𝑖𝑥^∗_𝑖+1,𝑥^∗_𝑖^𝑞≤−𝑞𝜌_𝑖𝐴_𝑖𝑥_𝑖+1,𝑘,𝑥_𝑖,𝑘−𝐴_𝑖𝑥^∗_𝑖+1,𝑥^∗_𝑖,𝐽_𝑞𝑥_𝑖+1,𝑘−𝑥^∗_𝑖+1+𝑥_𝑖+1,𝑘−𝑥^∗_𝑖+1^𝑞+𝑐_𝑞𝜌^𝑞_𝑖𝐴_𝑖𝑥_𝑖+1,𝑘,𝑥_𝑖,𝑘−𝐴_𝑖𝑥^∗_𝑖+1,𝑥^∗_𝑖^𝑞≤𝑞𝜌_𝑖𝜇_𝑖𝐴_𝑖𝑥_𝑖+1,𝑘,𝑥_𝑖,𝑘−𝐴_𝑖𝑥^∗_𝑖+1,𝑥^∗_𝑖^𝑞−𝜈_𝑖𝑥_𝑖+1,𝑘−𝑥^∗_𝑖+1^𝑞+1+𝑐_𝑞𝜌^𝑞_𝑖𝜏^𝑞_𝑖𝑥_𝑖+1,𝑘−𝑥^∗_𝑖+1^𝑞≤1+𝑞𝜌_𝑖𝜇_𝑖𝜏^𝑞_𝑖+𝑐_𝑞𝜌^𝑞_𝑖𝜏^𝑞_𝑖−𝑞𝜌_𝑖𝜈_𝑖𝑥_𝑖+1,𝑘−𝑥^∗_𝑖+1^𝑞𝐿,𝑖=1,2,…,𝑛−1,_∗𝑔𝑖=𝑥_𝑖+1,𝑘−𝑥^∗_𝑖+1−𝑔_𝑖𝑥_𝑖+1,𝑘−𝑔_𝑖𝑥^∗_𝑖+1^𝑞≤−𝑞𝑔_𝑖𝑥_𝑖+1,𝑘−𝑔_𝑖𝑥^∗_𝑖+1,𝐽_𝑞𝑥_𝑖+1,𝑘−𝑥^∗_𝑖+1+𝑥_𝑖+1,𝑘−𝑥^∗_𝑖+1^𝑞+𝑐_𝑞𝑔_𝑖𝑥_𝑖+1,𝑘−𝑔_𝑖𝑥^∗_𝑖+1^𝑞≤1+𝑞𝑎_𝑖𝜄^𝑞_𝑖+𝑐_𝑞𝜄^𝑞_𝑖−𝑞𝑏_𝑖𝑥_𝑖+1,𝑘−𝑥^∗_𝑖+1^𝑞𝐿,𝑖=1,2,…,𝑛−1,_∗𝐴𝑛=𝑥_1,𝑘−𝑥^∗₁−𝜌_𝑛𝐴_𝑛𝑥_1,𝑘,𝑥_𝑛,𝑘−𝐴_𝑛𝑥^∗₁,𝑥^∗_𝑛^𝑞≤𝑥_1,𝑘−𝑥^∗₁^𝑞−𝑞𝜌_𝑛𝐴_𝑛𝑥_1,𝑘,𝑥_𝑛,𝑘−𝐴_𝑛𝑥^∗₁,𝑥^∗_𝑛,𝐽_𝑞𝑥_1,𝑘−𝑥^∗₁+𝑐_𝑞𝜌^𝑞_𝑛𝐴_𝑛𝑥_1,𝑘,𝑥_𝑛,𝑘−𝐴_𝑛𝑥^∗₁,𝑥^∗_𝑛^𝑞≤1+𝑞𝜌_𝑛𝜇_𝑛𝜏^𝑞_𝑛+𝑐_𝑞𝜌^𝑞_𝑛𝜏^𝑞_𝑛−𝑞𝜌_𝑛𝜈_𝑛𝑥_1,𝑘−𝑥^∗₁^𝑞,𝐿^∗_𝑔𝑛=𝑥_1,𝑘−𝑥^∗₁−𝑔_𝑛𝑥_1,𝑘−𝑔_𝑛𝑥^∗₁^𝑞≤𝑥_1,𝑘−𝑥^∗₁^𝑞−𝑞𝑔_𝑛𝑥_1,𝑘−𝑔_𝑛𝑥^∗₁,𝐽_𝑞𝑥_1,𝑘−𝑥^∗₁+𝑐_𝑞𝑔_𝑛𝑥_1,𝑘−𝑔_𝑛𝑥^∗₁^𝑞≤1+𝑞𝑎_𝑛𝜄^𝑞_𝑛+𝑐_𝑞𝜄^𝑞_𝑛−𝑞𝑏_𝑛𝑥_1,𝑘−𝑥^∗₁^𝑞. (4.8) Noticing that, for each 𝑖∈{1,2,…,𝑛−2}, 𝑥_𝑖+1,𝑘−𝑥^∗_𝑖+1=𝑅^𝑀𝑖+1_𝜌𝑖+1𝑔_𝑖+1𝑥_𝑖+2,𝑘−𝜌_𝑖+1𝐴_𝑖+1x_𝑖+2,𝑘,𝑥_𝑖+1,𝑘−𝑅^𝑀𝑖+1_𝜌𝑖+1𝑔_𝑖+1𝑥^∗_𝑖+2−𝜌_𝑖+1𝐴_𝑖+1𝑥^∗_𝑖+2,𝑥^∗_𝑖+1≤𝑔_𝑖+1𝑥_𝑖+2,𝑘−𝜌_𝑖+1𝐴_𝑖+1𝑥_𝑖+2,𝑘,𝑥_𝑖+1,𝑘−𝑔_𝑖+1𝑥^∗_𝑖+2−𝜌_𝑖+1𝐴_𝑖+1𝑥^∗_𝑖+2,𝑥^∗_𝑖+1=𝑔_𝑖+1𝑥_𝑖+2,𝑘−𝑔_𝑖+1𝑥^∗_𝑖+2−𝜌_𝑖+1𝐴_𝑖+1𝑥_𝑖+2,𝑘,𝑥_𝑖+1,𝑘−𝐴_𝑖+1𝑥^∗_𝑖+2,𝑥^∗_𝑖+1≤𝑥_𝑖+2,𝑘−𝑥^∗_𝑖+2−𝜌_𝑖+1𝐴_𝑖+1𝑥_𝑖+2,𝑘,𝑥_𝑖+1,𝑘−𝐴_𝑖+1𝑥^∗_𝑖+2,𝑥^∗_𝑖+1+𝑥_𝑖+2,𝑘−𝑥^∗_𝑖+2−𝑔_𝑖+1𝑥_𝑖+2,𝑘−𝑔_𝑖+1𝑥^∗_𝑖+2≤^𝑞1+𝑞𝜌_𝑖+1𝜇_𝑖+1𝜏^𝑞_𝑖+1+𝑐_𝑞𝜌^𝑞_𝑖+1𝜏^𝑞_𝑖+1−𝑞𝜌_𝑖+1𝜈_𝑖+1+^𝑞1+𝑞𝑎_𝑖+1𝜄^𝑞_𝑖+1+𝑐_𝑞𝜄^𝑞_𝑖+1−𝑞𝑏_𝑖+1𝑥_𝑖+2,𝑘−𝑥^∗_𝑖+2,𝑥_𝑛,𝑘−𝑥^∗_𝑛=𝑅^𝑀𝑛_𝜌𝑛𝑔_𝑛𝑥_1,𝑘−𝜌_𝑛𝐴_𝑛𝑥_1,𝑘,𝑥_𝑛,𝑘−𝑅^𝑀𝑛_𝜌𝑛𝑔_𝑛𝑥^∗₁−𝜌_𝑛𝐴_𝑛𝑥^∗₁,𝑥^∗_𝑛≤𝑔_𝑛𝑥_1,𝑘−𝜌_𝑛𝐴_𝑛𝑥_1,𝑘,𝑥_𝑛,𝑘−𝑔_𝑛𝑥^∗₁−𝜌_𝑛𝐴_𝑛𝑥^∗₁,𝑥^∗_𝑛=𝑔_𝑛𝑥_1,𝑘−𝑔_𝑛𝑥^∗₁−𝜌_𝑛𝐴_𝑛𝑥_1,𝑘,𝑥_𝑛,𝑘−𝐴_𝑛𝑥^∗₁,𝑥^∗_𝑛≤𝑥_1,𝑘−𝑥^∗₁−𝜌_𝑛𝐴_𝑛𝑥_1,𝑘,𝑥_𝑛,𝑘−𝜌_𝑛𝐴_𝑛𝑥^∗₁,𝑥^∗_𝑛+𝑥_1,𝑘−𝑥^∗₁−𝑔_𝑛𝑥_1,𝑘−𝑔_𝑛𝑥^∗₁≤^𝑞1+𝑞𝜌_𝑛𝜇_𝑛𝜏^𝑞_𝑛+𝑐_𝑞𝜌^𝑞_𝑛𝜏^𝑞_𝑛−𝑞𝜌_𝑛𝜈_𝑛+^𝑞1+𝑞𝑎_𝑛𝜄^𝑞_𝑛+𝑐_𝑞𝜄^𝑞_𝑛−𝑞𝑏_𝑛×𝑥_1,𝑘−𝑥^∗₁. (4.9) As a consequence, we have 𝑥_1,𝑘+1−𝑥^∗₁≤𝛼_𝑘^𝑛_𝑗=1^𝑞1+𝑞𝜌_𝑗𝜇_𝑗𝜏^𝑞_𝑗+𝑐_𝑞𝜌^𝑞_𝑗𝜏^𝑞_𝑗−𝑞𝜌_𝑗𝜈_𝑗+^𝑞1+𝑞𝑎_𝑗𝜄^𝑞_𝑗+𝑐_𝑞𝜄^𝑞_𝑗−𝑞𝑏_𝑗𝑥_1,𝑘−𝑥^∗₁+1−𝛼_𝑘𝑥_1,𝑘−𝑥^∗₁+𝑒_𝑘=1−𝛼_𝑘1−^𝑛_𝑗=1^𝑞1+𝑞𝜌_𝑗𝜇_𝑗𝜏^𝑞_𝑗+𝑐_𝑞𝜌^𝑞_𝑗𝜏^𝑞_𝑗−𝑞𝜌_𝑗𝜈_𝑗+^𝑞1+𝑞𝑎_𝑗𝜄^𝑞_𝑗+𝑐_𝑞𝜄^𝑞_𝑗−𝑞𝑏_𝑗×𝑥_1,𝑘−𝑥^∗₁+𝑒_𝑘. (4.10) Putting 𝜆_𝑘=𝛼_𝑘(1−∏^𝑛_𝑗=1(^𝑞1+𝑞𝜌_𝑗𝜇_𝑗𝜏^𝑞_𝑗+𝑐_𝑞𝜌^𝑞_𝑗𝜏^𝑞_𝑗−𝑞𝜌_𝑗𝜈_𝑗+^𝑞1+𝑞𝑎_𝑗𝜄^𝑞_𝑗+𝑐_𝑞𝜄^𝑞_𝑗−𝑞𝑏_𝑗)), 𝐶_𝑘=0,𝐵_𝑘=‖𝑥_1,𝑘−𝑥^∗₁‖, and 𝐷_𝑘=‖𝑒_𝑘‖. Then 𝐵_𝑘+1≤(1−𝜆_𝑘)𝐵_𝑘+𝐶_𝑘+𝐷_𝑘. From the conditions (i)–(iv), it follows that ^∞_𝑘=0𝜆_𝑘=∞,𝐶_𝑘=0𝜆_𝑘,^∞_𝑘=0𝐷_𝑘<∞,0<𝜆_𝑘<1,∀𝑘∈𝑁. (4.11) Therefore, by Lemma 2.11 and (4.11), one has lim_𝑘→∞𝐵_𝑘=0, (4.12) that is, lim_𝑘→∞𝑥_1,𝑘=𝑥^∗₁. Again from (iii), this shows that ^𝑞1+𝑞𝜌_𝑗𝜇_𝑗𝜏^𝑞_𝑗+𝑐_𝑞𝜌^𝑞_𝑗𝜏^𝑞_𝑗−𝑞𝜌_𝑗𝜈_𝑗+^𝑞1+𝑞𝑎_𝑗𝜄^𝑞_𝑗+𝑐_𝑞𝜄^𝑞_𝑗−𝑞𝑏_𝑗≥0,𝑗=2,3,…,𝑛, (4.13) and so, lim_𝑘→∞𝑥_𝑗,𝑘−𝑥^∗_𝑗=0, (4.14) That is, 𝑥_𝑗,𝑘→𝑥^∗_𝑗 as 𝑘→∞ for 𝑗=2,3,…,𝑛. Thus, (𝑥_1,𝑘,𝑥_2,𝑘,…,𝑥_𝑛,𝑘) converges strongly to (𝑥^∗₁,𝑥^∗₂,…,𝑥^∗_𝑛). This completes the proof.

Theorem 4.8. Let 𝐸 be a real 𝑞-uniformly smooth Banach space, and let 𝑀_𝑗,𝐴_𝑗 and 𝑔_𝑗(𝑗=1,2,…,𝑛) be the same as in Theorem 3.4. Assume that {𝛼_𝑛} is a real sequence in (0,1] and satisfies the following conditions:(i)∑^∞_𝑘=0𝛼_𝑘=∞;(ii)min{1+𝑞𝑎_𝑗𝜄^𝑞_𝑗+𝑐_𝑞𝜄^𝑞_𝑗−𝑞𝑏_𝑗,1+𝑞𝜌_𝑗𝜇_𝑗𝜏^𝑞_𝑗+𝑐_𝑞𝜌^𝑞_𝑗𝜏^𝑞_𝑗−𝑞𝜌_𝑗𝜈_𝑗}≥0, 𝑗=1,2,…,𝑛;(iii)0<∏^𝑛_𝑗=1(^q1+𝑞𝑎_𝑗𝜄^𝑞_𝑗+𝑐_𝑞𝜄^𝑞_𝑗−𝑞𝑏_𝑗+^𝑞1+𝑞𝜌_𝑗𝜇_𝑗𝜏^𝑞_𝑗+𝑐_𝑞𝜌^𝑞_𝑗𝜏^𝑞_𝑗−𝑞𝜌_𝑗𝜈_𝑗)<1.Then the sequences {𝑥_𝑗,𝑘}(𝑗=1,2,…,𝑛) generated by Algorithm 4.2 converge strongly to 𝑥^∗_𝑗(𝑗=1,2,…,𝑛), respectively, such that (𝑥^∗₁,𝑥^∗₂,…,𝑥^∗_𝑛)∈Ξ.

Proof. It directly follows from Theorem 4.7, and so the proof is omitted. This completes the proof.

Theorem 4.9. Let 𝐸 be a real 𝑞-uniformly smooth Banach space, and let 𝑀_𝑗,𝐴_𝑗 and 𝑔_𝑗(𝑗=1,2,…,𝑛) be the same as in Theorem 3.4. Assume that {𝛼_𝑛} is a real sequence in (0,1] and satisfy the following conditions:(i)∑^∞_𝑘=0𝛼_𝑘=∞;(ii)min{1+𝑞𝑎_𝑗𝜄^𝑞_𝑗+𝑐_𝑞𝜄^𝑞_𝑗−𝑞𝑏_𝑗,1+𝑞𝜌_𝑗𝜇_𝑗𝜏^𝑞_𝑗+𝑐_𝑞𝜌^𝑞_𝑗𝜏^𝑞_𝑗−𝑞𝜌_𝑗𝜈_𝑗}≥0,𝑗=1,2,…,𝑛;(iii)0<∏^𝑛_𝑗=1(^𝑞1+𝑞𝑎_𝑗𝜄^𝑞_𝑗+𝑐_𝑞𝜄^𝑞_𝑗−𝑞𝑏_𝑗+^𝑞1+𝑞𝜌_𝑗𝜇_𝑗𝜏^𝑞_𝑗+𝑐_𝑞𝜌^𝑞_j𝜏^𝑞_𝑗−𝑞𝜌_𝑗𝜈_𝑗)<1.Then the sequences {𝑥_𝑗,𝑘}(𝑗=1,2,…,𝑛) generated by Algorithm 4.3 converge strongly to 𝑥^∗_𝑗(𝑗=1,2,…,𝑛), respectively, such that (𝑥^∗₁,𝑥^∗₂,…,𝑥^∗_𝑛)∈Ξ.

Proof. As in the proof of Theorem 4.7, we know that there exists an unique point (𝑥^∗₁,𝑥^∗₂,…,𝑥^∗_𝑛)∈𝐸^𝑛 such that (𝑥^∗₁,𝑥^∗₂,…,𝑥^∗_𝑛)∈Ξ, and so 𝑥^∗₁=𝑅^𝑀1_𝜌1𝑔₁𝑥^∗₂−𝜌₁𝐴₁𝑥^∗₂,𝑥^∗₁,𝑥^∗₂=𝑅^𝑀2_𝜌2𝑔₂𝑥^∗₃−𝜌₂𝐴₂𝑥^∗₃,𝑥^∗₂,⋮𝑥^∗_𝑛−1=𝑅^𝑀𝑛−1_𝜌𝑛−1𝑔_𝑛−1𝑥^∗_𝑛−𝜌_𝑛−1𝐴_𝑛−1𝑥^∗_𝑛,𝑥^∗_𝑛−1,𝑥^∗_𝑛=𝑅^𝑀𝑛_𝜌𝑛𝑔_𝑛𝑥^∗₁−𝜌_𝑛𝐴_𝑛𝑥^∗₁,𝑥^∗_𝑛, (4.15) Note that 𝑥_1,𝑘+1−𝑥^∗₁=1−𝛼_𝑘𝑥_1,𝑘+𝛼_𝑘𝑅^𝑀1_𝜌1𝑔₁𝑥_2,𝑘+1−𝜌₁𝐴₁𝑥_2,𝑘+1,𝑥_1,𝑘−𝑥^∗₁=1−𝛼_𝑘𝑥_1,𝑘−𝑥^∗₁+𝛼_𝑘𝑅^𝑀1_𝜌1𝑔₁𝑥_2,𝑘+1−𝜌₁𝐴₁𝑥_2,𝑘+1,𝑥_1,𝑘−𝑥^∗₁=𝛼_𝑘𝑅^𝑀1_𝜌1𝑔₁𝑥_2,𝑘+1−𝜌₁𝐴₁𝑥_2,𝑘+1,𝑥_1,𝑘−𝑅^𝑀1_𝜌1𝑔₁𝑥^∗₂−𝜌₁𝐴₁𝑥^∗₂,𝑥^∗₁+1−𝛼_𝑘𝑥_1,𝑘−𝑥^∗₁≤𝛼_𝑘𝑅^𝑀1_𝜌1𝑔₁𝑥_2,𝑘+1−𝜌₁𝐴₁𝑥_2,𝑘+1,𝑥_1,𝑘−𝑅^𝑀1_𝜌1𝑔₁𝑥^∗₂−𝜌₁𝐴₁𝑥^∗₂,𝑥^∗₁+1−𝛼_𝑘𝑥_1,𝑘−𝑥^∗₁≤𝛼_𝑘𝑔₁𝑥_2,𝑘+1−𝜌₁𝐴₁𝑥_2,𝑘+1,𝑥_1,𝑘−𝑔₁𝑥^∗₂−𝜌₁𝐴₁𝑥^∗₂,𝑥^∗₁+1−𝛼_𝑘𝑥_1,𝑘−𝑥^∗₁=𝛼_𝑘𝑔₁𝑥_2,𝑘+1−𝑔₁𝑥^∗₂−𝜌₁𝐴₁𝑥_2,𝑘+1,𝑥_1,𝑘−𝐴₁𝑥^∗₂,𝑥^∗₁+1−𝛼_𝑘𝑥_1,𝑘−𝑥^∗₁≤1−𝛼_𝑘𝑥_1,𝑘−𝑥^∗₁+𝛼_𝑘𝑥_2,𝑘+1−𝑥^∗₂−𝑔₁𝑥_2,𝑘−𝑔₁𝑥^∗₂+𝛼_𝑘𝑥_2,𝑘+1−𝑥^∗₂−𝜌₁𝐴₁𝑥_2,𝑘+1,𝑥_1,𝑘−𝐴₁𝑥^∗₂,𝑥^∗₁. (4.16) Since 𝐴_𝑗∶𝐸×𝐸→𝐸 are (𝜇_𝑗,𝜈_𝑗)-relaxed cocoercive and Lipschitz continuous in the first variable with constant 𝜏_𝑗 and 𝑔_𝑗∶𝐸→𝐸 are (𝑎_𝑗,𝑏_𝑗)-relaxed cocoercive and Lipschitz continuous with constant 𝜄_𝑗 for 𝑗∈{1,2,…,𝑛}, we can conclude that 𝐿^∗_𝐴𝑖=𝑥_{𝑖+1,𝑘+1}−𝑥^∗_𝑖+1−𝜌_𝑖𝐴_𝑖𝑥_{𝑖+1,𝑘+1},𝑥_𝑖,𝑘−𝐴_𝑖𝑥^∗_𝑖+1,𝑥^∗_𝑖^𝑞≤−𝑞𝜌_𝑖𝐴_𝑖𝑥_{𝑖+1,𝑘+1},𝑥_𝑖,𝑘−𝐴_𝑖𝑥^∗_𝑖+1,𝑥^∗_𝑖,𝐽_𝑞𝑥_{𝑖+1,𝑘+1}−𝑥^∗_𝑖+1+𝑥_{𝑖+1,𝑘+1}−𝑥^∗_𝑖+1^𝑞+𝑐_𝑞𝜌^𝑞_𝑖𝐴_𝑖𝑥_{𝑖+1,𝑘+1},𝑥_𝑖,𝑘−𝐴_𝑖𝑥^∗_𝑖+1,𝑥^∗_𝑖^𝑞≤𝑞𝜌_𝑖𝜇_𝑖𝐴_𝑖𝑥_{𝑖+1,𝑘+1},𝑥_𝑖,𝑘−𝐴_𝑖𝑥^∗_𝑖+1,𝑥^∗_𝑖^𝑞−𝜈_𝑖𝑥_{𝑖+1,𝑘+1}−𝑥^∗_𝑖+1^𝑞+1+𝑐_𝑞𝜌^𝑞_𝑖𝜏^𝑞_𝑖𝑥_{𝑖+1,𝑘+1}−𝑥^∗_𝑖+1^𝑞≤1+𝑞𝜌_𝑖𝜇_𝑖𝜏^𝑞_𝑖+𝑐_𝑞𝜌^𝑞_𝑖𝜏^𝑞_𝑖−𝑞𝜌_𝑖𝜈_𝑖𝑥_{𝑖+1,𝑘+1}−𝑥^∗_𝑖+1^𝑞𝐿,𝑖=1,2,…,𝑛−1,_∗𝑔𝑖=𝑥_{𝑖+1,𝑘+1}−𝑥^∗_𝑖+1−𝑔_𝑖𝑥_{𝑖+1,𝑘+1}−𝑔_𝑖𝑥^∗_𝑖+1^𝑞≤−𝑞𝑔_𝑖𝑥_{𝑖+1,𝑘+1}−𝑔_𝑖𝑥^∗_𝑖+1,𝐽_𝑞𝑥_{𝑖+1,𝑘+1}−𝑥^∗_𝑖+1+𝑥_{𝑖+1,𝑘+1}−𝑥^∗_𝑖+1^𝑞+𝑐_𝑞𝑔_𝑖𝑥_{𝑖+1,𝑘+1}−𝑔_𝑖𝑥^∗_𝑖+1^𝑞≤1+𝑞𝑎_𝑖𝜄^𝑞_𝑖+𝑐_𝑞𝜄^𝑞_𝑖−𝑞𝑏_𝑖𝑥_{𝑖+1,𝑘+1}−𝑥^∗_𝑖+1^𝑞𝐿,𝑖=1,2,…,𝑛−1,_∗𝐴𝑛=𝑥_1,𝑘−𝑥^∗₁−𝜌_𝑛𝐴_𝑛𝑥_1,𝑘,𝑥_𝑛,𝑘−𝐴_𝑛𝑥^∗₁,𝑥^∗_𝑛^𝑞≤𝑥_1,𝑘−𝑥^∗₁^𝑞−𝑞𝜌_𝑛𝐴_𝑛𝑥_1,𝑘,𝑥_𝑛,𝑘−𝐴_𝑛𝑥^∗₁,𝑥^∗_𝑛,𝐽_𝑞𝑥_1,𝑘−𝑥^∗₁+𝑐_𝑞𝜌^𝑞_𝑛𝐴_𝑛𝑥_1,𝑘,𝑥_𝑛,𝑘−𝐴_𝑛𝑥^∗₁,𝑥^∗_𝑛^𝑞≤1+𝑞𝜌_𝑛𝜇_𝑛𝜏^𝑞_𝑛+𝑐_𝑞𝜌^𝑞_𝑛𝜏^𝑞_𝑛−𝑞𝜌_𝑛𝜈_𝑛𝑥_1,𝑘−𝑥^∗₁^𝑞,𝐿^∗_𝑔𝑛=𝑥_1,𝑘−𝑥^∗₁−𝑔_𝑛𝑥_1,𝑘−𝑔_𝑛𝑥^∗₁^𝑞≤𝑥_1,𝑘−𝑥^∗₁^𝑞−𝑞𝑔_𝑛𝑥_1,𝑘−𝑔_𝑛𝑥^∗₁,𝐽_𝑞𝑥_1,𝑘−𝑥^∗₁+𝑐_𝑞𝑔_𝑛(𝑥_1,𝑘)−𝑔_𝑛(𝑥^∗₁)^𝑞≤1+𝑞𝑎_𝑛𝜄^𝑞_𝑛+𝑐_𝑞𝜄^𝑞_𝑛−𝑞𝑏_𝑛𝑥_1,𝑘−𝑥^∗₁^𝑞. (4.17) Noticing that, for each 𝑖∈{1,2,…,𝑛−2}, 𝑥_{𝑖+1,𝑘+1}−𝑥^∗_𝑖+1=𝑅^𝑀𝑖+1_𝜌𝑖+1𝑔_𝑖+1𝑥_{𝑖+2,𝑘+1}−𝜌_𝑖+1𝐴_𝑖+1𝑥_{𝑖+2,𝑘+1},𝑥_𝑖+1,𝑘−𝑅^𝑀𝑖+1_𝜌𝑖+1𝑔_𝑖+1𝑥^∗_𝑖+2−𝜌_𝑖+1𝐴_𝑖+1𝑥^∗_𝑖+2,𝑥^∗_𝑖+1≤𝑔_𝑖+1𝑥_{𝑖+2,𝑘+1}−𝜌_𝑖+1𝐴_𝑖+1𝑥_{𝑖+2,𝑘+1},𝑥_𝑖+1,𝑘−𝑔_𝑖+1𝑥^∗_𝑖+2−𝜌_𝑖+1𝐴_𝑖+1𝑥^∗_𝑖+2,𝑥^∗_𝑖+1=𝑔_𝑖+1𝑥_{𝑖+2,𝑘+1}−𝑔_𝑖+1𝑥^∗_𝑖+2−𝜌_𝑖+1×𝐴_𝑖+1𝑥_{𝑖+2,𝑘+1},𝑥_𝑖+1,𝑘−𝐴_𝑖+1𝑥^∗_𝑖+2,𝑥^∗_𝑖+1≤𝑥_{𝑖+2,𝑘+1}−𝑥^∗_𝑖+2−𝜌_𝑖+1𝐴_𝑖+1𝑥_{𝑖+2,𝑘+1},𝑥_𝑖+1,𝑘−𝐴_𝑖+1𝑥^∗_𝑖+2,𝑥^∗_𝑖+1+𝑥_{𝑖+2,𝑘+1}−𝑥^∗_𝑖+2−𝑔_𝑖+1𝑥_{𝑖+2,𝑘+1}−𝑔_𝑖+1𝑥^∗_𝑖+2≤^𝑞1+𝑞𝜌_𝑖+1𝜇_𝑖+1𝜏^𝑞_𝑖+1+𝑐_𝑞𝜌^𝑞_𝑖+1𝜏^𝑞_𝑖+1−𝑞𝜌_𝑖+1𝜈_𝑖+1+^𝑞1+𝑞𝑎_𝑖+1𝜄^𝑞_𝑖+1+𝑐_𝑞𝜄^𝑞_𝑖+1−𝑞𝑏_𝑖+1𝑥_{𝑖+2,𝑘+1}−𝑥^∗_𝑖+2,𝑥_𝑛,𝑘+1−𝑥^∗_𝑛=𝑅^𝑀𝑛_𝜌𝑛𝑔_𝑛𝑥_1,𝑘−𝜌_𝑛𝐴_𝑛𝑥_1,𝑘,𝑥_𝑛,𝑘−𝑅^𝑀𝑛_𝜌𝑛𝑔_𝑛𝑥^∗₁−𝜌_𝑛𝐴_𝑛𝑥^∗₁,𝑥^∗_𝑛≤𝑔_𝑛𝑥_1,𝑘−𝜌_𝑛𝐴_𝑛𝑥_1,𝑘,𝑥_𝑛,𝑘−𝑔_𝑛𝑥^∗₁−𝜌_𝑛𝐴_𝑛𝑥^∗₁,𝑥^∗_𝑛=𝑔_𝑛𝑥_1,𝑘−𝑔_𝑛𝑥^∗₁−𝜌_𝑛𝐴_𝑛𝑥_1,𝑘,𝑥_𝑛,𝑘−𝐴_𝑛𝑥^∗₁,𝑥^∗_𝑛≤𝑥_1,𝑘−𝑥^∗₁−𝜌_𝑛𝐴_𝑛𝑥_1,𝑘,𝑥_𝑛,𝑘−𝜌_𝑛𝐴_𝑛𝑥^∗₁,𝑥^∗_𝑛+𝑥_1,𝑘−𝑥^∗₁−𝑔_𝑛𝑥_1,𝑘−𝑔_𝑛𝑥^∗₁≤^𝑞1+𝑞𝜌_𝑛𝜇_𝑛𝜏^𝑞_𝑛+𝑐_𝑞𝜌^𝑞_𝑛𝜏^𝑞_𝑛−𝑞𝜌_𝑛𝜈_𝑛+^𝑞1+𝑞𝑎_𝑛𝜄^𝑞_𝑛+𝑐_𝑞𝜄^𝑞_𝑛−𝑞𝑏_𝑛×𝑥_1,𝑘−𝑥^∗₁. (4.18) Consequently, we have 𝑥_1,𝑘+1−𝑥^∗≤𝛼_𝑘^𝑛_𝑗=1^𝑞1+𝑞𝜌_𝑗𝜇_𝑗𝜏^𝑞_𝑗+𝑐_𝑞𝜌^𝑞_𝑗𝜏^𝑞_𝑗−𝑞𝜌_𝑗𝜈_𝑗+^𝑞1+𝑞𝑎_𝑗𝜄^𝑞_𝑗+𝑐_𝑞𝜄^𝑞_𝑗−𝑞𝑏_𝑗𝑥_1,𝑘−𝑥^∗₁+1−𝛼_𝑘𝑥_1,𝑘−𝑥^∗=1−𝛼_𝑘1−^𝑛_𝑗=1^𝑞1+𝑞𝜌_𝑗𝜇_𝑗𝜏^𝑞_𝑗+𝑐_𝑞𝜌^𝑞_𝑗𝜏^𝑞_𝑗−𝑞𝜌_𝑗𝜈_𝑗+^𝑞1+𝑞𝑎_𝑗𝜄^𝑞_𝑗+𝑐_𝑞𝜄^𝑞_𝑗−𝑞𝑏_𝑗𝑥_1,𝑘−𝑥^∗₁. (4.19) Taking 𝜆_𝑘=𝛼_𝑘(1−∏^𝑛_𝑗=1(^𝑞1+𝑞𝜌_𝑗𝜇_𝑗𝜏^𝑞_𝑗+𝑐_𝑞𝜌^𝑞_𝑗𝜏^𝑞_𝑗−𝑞𝜌_𝑗𝜈_𝑗+^𝑞1+𝑞𝑎_𝑗𝜄^𝑞_𝑗+𝑐_𝑞𝜄^𝑞_𝑗−𝑞𝑏_𝑗)), 𝐶_𝑘=𝐷_𝑘=0, and 𝐵_𝑘=‖𝑥_1,𝑘−𝑥^∗₁‖, then 𝐵_𝑘+1≤(1−𝜆_𝑘)𝐵_𝑘+𝐶_𝑘+𝐷_𝑘. It follows from the conditions (i)–(iii) that ^∞_𝑘=0𝜆_𝑘=∞,𝐶_𝑘=0𝜆_𝑘,^∞_𝑘=0𝐷_𝑘<∞,0<𝜆_𝑘<1,∀𝑘∈𝑁. (4.20) Therefore, by Lemma 2.11 and (4.20), one has lim_𝑘→∞𝐵_𝑘=0. By the same argument of Theorem 4.7, we get lim_𝑘→∞𝑥_𝑗,𝑘−𝑥^∗_𝑗=0,𝑗=1,2,…,𝑛, (4.21) that is, 𝑥_𝑗,𝑘→𝑥^∗_𝑗 as 𝑘→∞ for 𝑗=1,2,…,𝑛. Thus, (𝑥_1,𝑘,𝑥_2,𝑘,…,𝑥_𝑛,𝑘) converges strongly to (𝑥^∗₁,𝑥^∗₂,…,𝑥^∗_𝑛). This completes the proof.

By Remark 4.4, we have the following strong convergence theorems for the system of general variational inequalities problem (2.7).

Corollary 4.10. Let 𝐾 be a closed convex subset of a real Hilbert space 𝐸, and let 𝐴_𝑗 and 𝑔_𝑗  (𝑗=1,2) be the same as in Theorem 3.4. Assume that {𝛼_𝑛} is a real sequence in (0,1] and satisfies the following conditions:(i)∑^∞_𝑘=0𝛼_𝑘=∞;(ii)∑^∞_𝑘=0‖𝑒_𝑘‖<+∞;(iii)min{1+2𝑎_𝑗𝜄²_𝑗+𝜄²_𝑗−2𝑏_𝑗,1+2𝜌_𝑗𝜇_𝑗𝜏²_𝑗+𝜌²_𝑗𝜏²_𝑗−2𝜌_𝑗𝜈_𝑗}≥0,𝑗=1,2;(iv)0<∏^𝑛_𝑗=1(1+2𝑎_𝑗𝜄²_𝑗+𝜄²_𝑗−2𝑏_𝑗+1+2𝜌_𝑗𝜇_𝑗𝜏²_𝑗+𝜌²_𝑗𝜏²_𝑗−2𝜌_𝑗𝜈_𝑗)<1.Then the sequences {𝑥_𝑗,𝑘}(𝑗=1,2) generated by Algorithm 4.5 converge strongly to 𝑥^∗_𝑗(𝑗=1,2), respectively, such that (𝑥^∗₁,𝑥^∗₂) is the unique solution of the system of general variational inequalities problem (2.7).

Corollary 4.11. Let 𝐾 be a closed convex subset of a real Hilbert space 𝐸, and let 𝐴_𝑗 and 𝑔_𝑗(𝑗=1,2) be the same as in Theorem 3.4. Assume that {𝛼_𝑛} is a real sequence in (0,1] and satisfies the following conditions:(i)∑^∞_𝑘=0𝛼_𝑘=∞;(ii)min{1+2𝑎_𝑗𝜄²_𝑗+𝜄²_𝑗−2𝑏_𝑗,1+2𝜌_𝑗𝜇_𝑗𝜏²_𝑗+𝜌²_𝑗𝜏²_𝑗−2𝜌_𝑗𝜈_𝑗}≥0, 𝑗=1,2;(iii)0<∏^𝑛_𝑗=1(1+2𝑎_𝑗𝜄²_𝑗+𝜄²_𝑗−2𝑏_𝑗+1+2𝜌_𝑗𝜇_𝑗𝜏²_𝑗+𝜌²_𝑗𝜏²_𝑗−2𝜌_𝑗𝜈_𝑗)<1.Then the sequences {𝑥_𝑗,𝑘}(𝑗=1,2) generated by Algorithm 4.6 converge strongly to 𝑥^∗_𝑗(𝑗=1,2), respectively, such that (𝑥^∗₁,𝑥^∗₂) is the unique solution of the system of general variational inequalities problem (2.7).

5. An Application

In this section, we applied the obtained results to study a class of bilevel variational inequalities in Hilbert space, which includes some bilevel programming as special cases and widely used in many practical problems. Moreover, an iterative algorithm and convergence theorem for solutions to the bilevel variational inequalities are given in Hilbert space.

Let 𝐾₁ and 𝐾₂ be nonempty closed convex subsets of a Hilbert space 𝐸, and let 𝑔,ℎ∶𝐸→𝐸 and 𝐴∶𝐸×𝐸→𝐸 be single-valued mappings. We consider the following bilevel variational inequalities (for short, (BVI)): find (𝑥^∗,𝑦^∗)∈𝐾₁×𝐾₂ such that 𝑥^∗+ℎ𝑦^∗,ℎ(𝑥)−𝑥^∗≥0,∀𝑥∈𝐾₁,𝑦^∗∈Ψ𝑥^∗, (5.1) where Ψ(𝑥^∗) is the solutions set of the following parametric variational inequalities with respect to the parametric variable 𝑥^∗: 𝜌𝐴𝑥^∗,𝑦^∗+𝑦^∗−𝑔𝑥^∗,𝑔𝑦−𝑦^∗≥0,∀𝑦∈𝐾₂, (5.2) where 𝜌 is a positive constant. Equations (5.1) and (5.2) are called the upper-level variation inequality (for short, (UVI)) and the lower-level variation inequality (for short, LVI), respectively. Denote the set of solutions to the (BVI) by Θ. An important question for the (BVI) is how to solve the bilevel variational inequalities.

From Remark 2.9 and Theorem 3.1, one can easily conclude the following result.

Lemma 5.1. Let (𝑥^∗,𝑦^∗)∈𝐾₁×𝐾₂. Then (𝑥^∗,𝑦^∗) is a solution of the problem (BVI) if and only if 𝑥^∗=𝑃_𝐾1(−ℎ(𝑦^∗)), where 𝑦^∗=𝑃_𝐾2(𝑔(𝑥^∗)−𝜌𝐴(𝑥^∗,𝑦^∗)) and 𝜌 is a positive constant.

Lemma 5.2. Let 𝐾₁ and 𝐾₂ be nonempty closed convex subsets of a Hilbert space 𝐸. Let 𝐴∶𝐸×𝐸→𝐸 be (𝜇,𝜈)-relaxed cocoercive and Lipschitz continuous in the first variable with constant 𝜏, and let 𝑔∶𝐸→𝐸 be (𝑎₂,𝑏₂)-relaxed cocoercive and Lipschitz continuous with constant 𝑐₂. Assume that {1+2𝑎₂𝑐²₂+𝑐²₂−2𝑏₂,1+2𝜌𝜇𝜏²+𝜌²𝜏²−2𝜌𝜈}≥0 and 0≤1+2𝑎₂𝑐²₂+𝑐²₂−2𝑏₂+1+2𝜌𝜇𝜏²+𝜌²𝜏²−2𝜌𝜈<1. (5.3) Then, for each 𝑥∈𝐾₁, the parametric variational inequalities (5.2) have a uniquely solution. Further, the solution mapping 𝑦(𝑥) of the parametric variational inequalities (5.2) is continuous on 𝐾₁.

Proof. It directly follows from Theorems 3.2 and 3.4. This completes the proof.

Theorem 5.3. Let 𝐾₁ and 𝐾₂ be nonempty closed convex subsets of a Hilbert space 𝐸. Let 𝐴∶𝐸×𝐸→𝐸 be (𝜇,𝜈)-relaxed cocoercive and Lipschitz continuous in the first variable with constant 𝜏, ℎ∶𝐸→𝐸 a (𝑎₁,𝑏₁)-relaxed cocoercive and Lipschitz continuous with constant 𝑐₁, and let 𝑔∶𝐸→𝐸 be a (𝑎₂,𝑏₂)-relaxed cocoercive and Lipschitz continuous with constant 𝑐₂. Assume that {𝛼_𝑘} is a real sequence in (0,1] and satisfies the following conditions:(i)∑^∞_𝑘=0𝛼_𝑘=∞;(ii)min{1+2𝑎_𝑗𝑐²_𝑗+𝑐²_𝑗−2𝑏_𝑗,1+2𝜌𝜇𝜏²+𝜌²𝜏²−2𝜌𝜈,𝑗=1,2}≥0;(iii)0<∏^𝑛_𝑗=1(^𝑞1+𝑞𝑎_𝑗𝜄^𝑞_𝑗+𝑐_𝑞𝜄^𝑞_𝑗−𝑞𝑏_𝑗+^𝑞1+𝑞𝜌_𝑗𝜇_𝑗𝜏^𝑞_𝑗+𝑐_𝑞𝜌^𝑞_𝑗𝜏^𝑞_𝑗−𝑞𝜌_𝑗𝜈_𝑗)<1.The sequences {𝑥_𝑘} and {𝑦_𝑘} generated by the following algorithm: 𝑥₀∈𝐾₁,𝑦_𝑘=𝑃_𝐾2𝑔𝑥_𝑘−𝜌𝐴𝑥_𝑘,𝑦_𝑘,𝑥_𝑘+1=1−𝛼_𝑘𝑥_𝑘+𝛼_𝑘𝑃_𝐾1−ℎ𝑦_𝑘,𝑘=0,1,2,…, (5.4) where 𝜌 is a positive constant. Then the sequences {𝑥_𝑘} and {𝑦_𝑘} converge strongly to 𝑥^∗ and 𝑦^∗, respectively, such that (𝑥^∗,𝑦^∗) is a solution of the (BVI).

Proof. The proof is similar to Remark 2.9 and Theorem 4.7, and so the proof is omitted. This completes the proof.

Acknowledgments

The authors would like to thank two anonymous referees for their valuable comments and suggestions, which led to an improved presentation of the results, and are grateful to Professor Yongkun Li as the editor of their paper. This work was supported by the Natural Science Foundation of China (nos. 71171150, 60804065), the academic award for excellent Ph.D. Candidates funded by Wuhan University and the Fundamental Research Fund for the Central Universities.