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Journal of Applied Mathematics
Volume 2012 (2012), Article ID 487394, 24 pages
http://dx.doi.org/10.1155/2012/487394
Research Article

Algorithms for General System of Generalized Resolvent Equations with Corresponding System of Variational Inclusions

1Scientific Computing Key Laboratory of Shanghai Universities and Department of Mathematics, Shanghai Normal University, Shanghai 200234, China
2Center for General Education, Kaohsiung Medical University, Kaohsiung 807, Taiwan

Received 8 January 2012; Accepted 14 January 2012

Academic Editor: Yonghong Yao

Copyright © 2012 Lu-Chuan Ceng and Ching-Feng Wen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Very recently, Ahmad and Yao (2009) introduced and considered a system of generalized resolvent equations with corresponding system of variational inclusions in uniformly smooth Banach spaces. In this paper we introduce and study a general system of generalized resolvent equations with corresponding general system of variational inclusions in uniformly smooth Banach spaces. We establish an equivalence relation between general system of generalized resolvent equations and general system of variational inclusions. The iterative algorithms for finding the approximate solutions of general system of generalized resolvent equations are proposed. The convergence criteria of approximate solutions of general system of generalized resolvent equations obtained by the proposed iterative algorithm are also presented. Our results represent the generalization, improvement, supplement, and development of Ahmad and Yao corresponding ones.

1. Introduction and Preliminaries

It is well known that the theory of variational inequalities has played an important role in the investigation of a wide class of problems arising in mechanics, physics, optimization and control, nonlinear programming, elasticity, and applied sciences and so on; see, for example, [17] and the references therein. In recent years variational inequalities have been extended and generalized in different directions. A useful and significant generalization of variational inequalities is called mixed variational inequalities involving the nonlinear term [8], which enables us to study free, moving, obstacle, equilibrium problems arising in pure and applied sciences in a unified and general framework. Due to the presence of the nonlinear term, the projection method and its variant forms including the technique of the Wiener-Hopf equations cannot be extended to suggest the iterative methods for solving mixed variational inequalities. To overcome these drawbacks, Hassouni and Moudafi [9] introduced variational inclusions which contain mixed variational inequalities as special cases. They studied the perturbed method for solving variational inclusions. Subsequently, M. A. Noor and K. I. Noor [10] introduced and considered the resolvent equations by virtue of the resolvent operator concept and established the equivalence between the mixed variational inequalities and the resolvent equations. The technique of resolvent equations is being used to develop powerful and efficient numerical techniques for solving mixed (quasi)variational inequalities and related optimization problems. At the same time, some iterative algorithms for approximating a solution of some system of variational inequalities are also introduced and studied in Verma [11]. Pang [12], Cohen and Chaplais [13], Binachi [14], Ansari and Yao [15] considered a system of scalar variational inequalities and Pang showed that the traffic equilibrium problem, the Nash equilibrium, and the general equilibrium programming problem can be modeled as a system of variational inequalities. As generalizations of system of variational inequalities, Agarwal et al. [16] introduced a system of generalized nonlinear mixed quasi-variational inclusions and investigated the sensitivity analysis of solutions for the system of generalized mixed quasi-variational inclusions in Hilbert spaces. In 2007, Peng and Zhu [17] considered and studied a new system of generalized mixed quasi-variational inclusions with (𝐻,𝜂)-monotone operators and Lan et al. [18] studied a new system of nonlinear 𝐴-monotone multivalued variational inclusions. Furthermore, for more details in the related research work of this field, we invoke the readers to see, for instance, [1930]. Very recently, Ahmad and Yao [31] introduced and considered a new system of variational inclusions in uniformly smooth Banach spaces, which covers the system of variational inclusions in Hilbert spaces considered by [18]. They established an equivalence relation between this system of variational inclusions and a system of generalized resolvent equations, proposed a number of iterative algorithms for this system of variational inclusions, and also gave the convergence criteria.

Let 𝐸 be a real Banach space with its norm ,𝐸 the topological dual of 𝐸, and 𝑑 the metric induced by the norm . Let 𝐶𝐵(𝐸) (resp., 2𝐸) be the family of all nonempty closed and bounded subsets (resp., all nonempty subsets) of 𝐸 and 𝐷(,) the Hausdorff metric on 𝐶𝐵(𝐸) defined by𝐷(𝐴,𝐵)=maxsup𝑥𝐴𝑑(𝑥,𝐵),sup𝑦𝐵𝑑(𝐴,𝑦),(1.1) where 𝑑(𝑥,𝐵)=inf𝑦𝐵𝑑(𝑥,𝑦) and 𝑑(𝐴,𝑦)=inf𝑥𝐴𝑑(𝑥,𝑦). We write by 𝐽𝐸2𝐸 the normalized duality mapping defined as𝐽(𝑥)=𝑓𝐸𝑥,𝑓=𝑥2=𝑓2,𝑥𝐸,(1.2) where , denotes the duality pairing between 𝐸 and 𝐸.

The uniform convexity of a Banach space 𝐸 means that for any 𝜖>0, there exists 𝛿>0, such that for any 𝑥,𝑦𝐸,𝑥1,𝑦1,𝑥𝑦=𝜖 ensure the following inequality:𝑥+𝑦2(1𝛿).(1.3) The function𝛿𝐸(𝜖)=inf1𝑥+𝑦2𝑥=1,𝑦=1,𝑥𝑦=𝜖(1.4) is called the modulus of convexity of 𝐸.

The uniform smoothness of a Banach space 𝐸 means that for any given 𝜖>0, there exists 𝛿>0 such that𝑥+𝑦+𝑥𝑦21𝜖𝑦(1.5) holds. The function𝜏𝐸(𝑡)=sup𝑥+𝑦+𝑥𝑦21𝑥=1,𝑦=𝑡(1.6) is called the modulus of smoothness of 𝐸.

It is well known that the Banach space 𝐸 is uniformly convex if and only if 𝛿𝐸(𝜖)>0 for all 𝜖>0, and it is uniformly smooth if and only if lim𝑡0𝜏𝐸(𝑡)/𝑡=0. All Hilbert spaces, 𝐿𝑝 (or 𝑙𝑝) spaces (𝑝2), and the Sobolov spaces 𝑊𝑝𝑚(𝑝2) are 2-uniformly smooth, while, for 1<𝑝2,𝐿𝑝 (or 𝑙𝑝) and 𝑊𝑝𝑚 spaces are 𝑝-uniformly smooth.

Proposition 1.1 (see [15]). Let 𝐸 be a uniformly smooth Banach space. Then the normalized duality mapping  𝐽𝐸2𝐸 is single-valued, and for any 𝑥,𝑦𝐸 there holds the following:(i)𝑥+𝑦2𝑥2+2𝑦,𝐽(𝑥+𝑦),(ii)𝑥𝑦,𝐽(𝑥)𝐽(𝑦)2𝐶2𝜏𝐸(4𝑥𝑦/𝐶), where 𝐶=𝑥2+𝑦2/2.

Definition 1.2 (see [32]). A mapping 𝑔𝐸𝐸 is said to be
(i)𝑘-strongly accretive, 𝑘(0,1), if for any 𝑥,𝑦𝐸, there exists 𝑗(𝑥𝑦)𝐽(𝑥𝑦) such that𝑔(𝑥)𝑔(𝑦),𝑗(𝑥𝑦)𝑘𝑥𝑦2;(1.7)(ii)Lipschitz continuous if for any 𝑥,𝑦𝐸, there exists a constant 𝜆𝑔>0, such that𝑔(𝑥)𝑔(𝑦)𝜆𝑔𝑥𝑦.(1.8)

Definition 1.3 (see [13]). A set-valued mapping 𝐴𝐸2𝐸 is said to be
(i)accretive, if for any 𝑥,𝑦𝐸, there exists 𝑗(𝑥𝑦)𝐽(𝑥𝑦) such that for all 𝑢𝐴(𝑥) and 𝑣𝐴(𝑦), 𝑢𝑣,𝑗(𝑥𝑦)0;(1.9)(ii)𝑘-strongly accretive, 𝑘(0,1), if for any 𝑥,𝑦𝐸, there exists 𝑗(𝑥𝑦)𝐽(𝑥𝑦), such that for all 𝑢𝐴(𝑥) and 𝑣𝐴(𝑦), 𝑢𝑣,𝑗(𝑥𝑦)𝑘𝑥𝑦2;(1.10)(iii)𝑚-accretive if 𝐴 is accretive and (𝐼+𝜌𝐴)(𝐸)=𝐸, for every (equivalently, for some) 𝜌>0, where 𝐼 is the identity mapping (equivalently, if 𝐴 is accretive and (𝐼+𝐴)(𝐸)=𝐸). In particular, it is clear from [9] that if 𝐸=𝐻 is a Hilbert space, than 𝐴𝐸2𝐸 is an 𝑚-accretive mapping if and only if it is a maximal monotone mapping.

Definition 1.4 (see [31]). Let 𝑀𝐸2𝐸 be an 𝑚-accretive mapping. For any 𝜌>0, the mapping 𝐽𝜌𝑀𝐸𝐸 associated with 𝑀 defined by 𝐽𝜌𝑀(𝑥)=(𝐼+𝜌𝑀)1(𝑥),𝑥𝐸(1.11) is called the resolvent operator.

Definition 1.5 (see [33]). The resolvent operator 𝐽𝜌𝑀𝐸𝐸 is said to be a retraction if (𝐼+𝜌𝑀)1(𝐼+𝜌𝑀)1(𝑥)=(𝐼+𝜌𝑀)1(𝑥),𝑥𝐸.(1.12)
It is well known that 𝐽𝜌𝑀 is a single-valued and nonexpansive mapping.

Definition 1.6 (see [10]). A set-valued mapping 𝐻𝐸𝐶𝐵(𝐸) is said to be 𝐷-Lipschitz continuous if for any 𝑥,𝑦𝐸, there exists a constant 𝜆𝐷𝐻>0 such that 𝐷(𝐻(𝑥),𝐻(𝑦))𝜆𝐷𝐻𝑥𝑦.(1.13)

Let 𝐸1 and 𝐸2 be two real Banach spaces, 𝑆𝐸1×𝐸2𝐸1 and 𝑇𝐸1×𝐸2𝐸2 single-valued mappings, and 𝐺𝐸12𝐸1,𝐹𝐸22𝐸2,𝐻𝐸12𝐸1 and 𝑉𝐸22𝐸2 any four multivalued mappings. Let 𝑀𝐸12𝐸1 and 𝑁𝐸22𝐸2 be any nonlinear mappings, 𝑚𝐸2𝐸1,𝑛𝐸1𝐸2,𝑓𝐸1𝐸1 and 𝑔𝐸2𝐸2 nonlinear mappings with 𝑓(𝐸1)𝐷(𝑀) and 𝑔(𝐸2)𝐷(𝑁), respectively. Then we consider the problem of finding (𝑥,𝑦)𝐸1×𝐸2,(𝑠,𝑣)𝐺(𝑥)×𝐹(𝑦),(𝑢,𝑡)𝐻(𝑥)×𝑉(𝑦) such that𝑛𝑚(𝑦)𝑆(𝑠,𝑣)+𝑀(𝑓(𝑥)),(𝑥)𝑇(𝑢,𝑡)+𝑁(𝑔(𝑦)),(1.14) which is called a general system of variational inclusions. In particular, if 𝑚(𝑦)=0𝐸1,𝑛(𝑥)=0𝐸2,𝐺(𝑥)=𝑝(𝑥) and 𝑉(𝑦)=𝑞(𝑦), where 𝑝𝐸1𝐸1 and 𝑞𝐸2𝐸2 are single-valued mappings, then the general system of variational inclusions (1.14) reduces to the following system of variational inclusions0𝑆(𝑝(𝑥),𝑣)+𝑀(𝑓(𝑥)),0𝑇(𝑢,𝑞(𝑦))+𝑁(𝑔(𝑦)),(1.15) which was considered by Lan et al. [18] in Hilbert spaces and studied by Ahmad and Yao [31] in Banach spaces, respectively.

Proposition 1.7 (see [31, Lemma  2.1]). (𝑥,𝑦)𝐸1×𝐸2,𝑢𝐻(𝑥),𝑣𝐹(𝑦) is a solution of the system of variational inclusions (1.15) if and only if (𝑥,𝑦,𝑢,𝑣) satisfies 𝑓(𝑥)=𝐽𝜌𝑀(𝑓(𝑥)𝜌𝑆(𝑝(𝑥),𝑣)),𝜌>0,𝑔(𝑦)=𝐽𝛾𝑁(𝑔(𝑦)𝛾𝑇(𝑢,𝑞(𝑦))),𝛾>0.(1.16)

Proposition 1.8 (see [31, Proposition  3.1]). The system of variational inclusions (1.15) has a solution (𝑥,𝑦,𝑢,𝑣) with (𝑥,𝑦)𝐸1×𝐸2,𝑢𝐻(𝑥) and 𝑣𝐹(𝑦) if and only if the following system of generalized resolvent equations 𝑆(𝑝(𝑥),𝑣)+𝜌1𝑅𝜌𝑀𝑧=0,𝑅𝜌𝑀=𝐼𝐽𝜌𝑀𝑇,𝜌>0,(𝑢,𝑞(𝑦))+𝛾1𝑅𝛾𝑁𝑧=0,𝑅𝛾𝑁=𝐼𝐽𝛾𝑁,𝛾>0,(1.17) has a solution (𝑧,𝑧,𝑥,𝑦,𝑢,𝑣) with (𝑥,𝑦)𝐸1×𝐸2,𝑢𝐻(𝑥),𝑣𝐹(𝑦),𝑧𝐸1 and 𝑧𝐸2, where 𝑓(𝑥)=𝐽𝜌𝑀(𝑧),𝑔(𝑦)=𝐽𝛾𝑁(𝑧) and 𝑧=𝑓(𝑥)𝜌𝑆(𝑝(𝑥),𝑣),𝑧=𝑔(𝑦)𝛾𝑇(𝑢,𝑞(𝑦)).

Based on the above Propositions 1.7 and 1.8, Ahmad and Yao [31] presented the following algorithm and established the following strong convergence result for the sequences generated by the algorithm.

Algorithm 1.9 (see [31, Algorithm  3.1]). For given (𝑥0,𝑦0)𝐸1×𝐸2,𝑢0𝐻(𝑥0),𝑣0𝐹(𝑦0),𝑧0𝐸1 and 𝑧0𝐸2, compute {𝑧𝑘},{𝑧𝑘},{𝑥𝑘},{𝑦𝑘},{𝑢𝑘}, and {𝑣𝑘} by the iterative scheme: 𝑓𝑥𝑘=𝐽𝜌𝑀𝑧𝑘,𝑔𝑦𝑘=𝐽𝛾𝑁𝑧𝑘,𝑢𝑘𝑥𝐻𝑘𝑢𝑘+1𝑢𝑘𝐻𝑥𝐷𝑘+1𝑥,𝐻𝑘,𝑣𝑘𝑦𝐹𝑘𝑣𝑘+1𝑣𝑘𝐹𝑦𝐷𝑘+1𝑦,𝐹𝑘,𝑧𝑘+1𝑥=𝑓𝑘𝑝𝑥𝜌𝑆𝑘,𝑣𝑘,𝑧𝑘+1𝑦=𝑔𝑘𝑢𝛾𝑇𝑘𝑦,𝑞𝑘,𝑘=0,1,2,.(1.18)

Theorem 1.10 (see [31, Theorem  3.1]). Let 𝐸1 and 𝐸2 be two real uniformly smooth Banach spaces with modulus of smoothness 𝜏𝐸1(𝑡)𝐶1𝑡2 and 𝜏𝐸2(𝑡)𝐶2𝑡2 for 𝐶1,𝐶2>0, respectively. Let 𝐻𝐸1𝐶𝐵(𝐸1),𝐹𝐸2𝐶𝐵(𝐸2) be 𝐷-Lipschitz continuous mappings with constants 𝜆𝐷𝐻 and 𝜆𝐷𝐹, respectively, and let 𝑀𝐸12𝐸1,𝑁𝐸22𝐸2 be 𝑚-accretive mappings such that the resolvent operators associated with 𝑀 and 𝑁 are retractions. Let 𝑓𝐸1𝐸1,𝑔𝐸2𝐸2 be both strong accretive with constants 𝛼 and 𝛽, respectively, and Lipschitz continuous with constants 𝛿1 and 𝛿2, respectively. Let 𝑝𝐸1𝐸1,𝑞𝐸2𝐸2 be Lipschitz continuous with constants 𝜆𝑝 and 𝜆𝑞, respectively, and let 𝑆𝐸1×𝐸2𝐸1,𝑇𝐸1×𝐸2𝐸2 be Lipschitz continuous in the first and second arguments with constants 𝜆𝑆1,𝜆𝑆2 and 𝜆𝑇1,𝜆𝑇2, respectively.
If there exist constants 𝜌>0 and 𝛾>0, such that 0<𝐵+𝜃1+𝜃4𝐵1𝐵<1,0<+𝜃2+𝜃31𝐵<1,(1.19) where 𝐵=12𝛼+64𝐶1𝛿21,𝐵=12𝛽+64𝐶2𝛿22 and 𝜃1=(1+𝜌𝜆𝑆1𝜆𝑝)/(1𝜌(𝜆𝑆1𝜆𝑝+𝜆𝑆2𝜆𝐷𝐹)),𝜃2=(1+𝜌𝜆𝑆2𝜆𝐷𝐹)/(1𝜌(𝜆𝑆1𝜆𝑝+𝜆𝑆2𝜆𝐷𝐹)),𝜃3=(1+𝛾𝜆𝑇2𝜆𝑞)/(1𝛾(𝜆𝑇1𝜆𝐷𝐻+𝜆𝑇2𝜆𝑞)),𝜃4=𝛾𝜆𝑇1𝜆𝐷𝐻/(1𝛾(𝜆𝑇1𝜆𝐷𝐻+𝜆𝑇2𝜆𝑞)), then there exist (𝑥,𝑦)𝐸1×𝐸2,𝑢𝐻(𝑥),𝑣𝐹(𝑦) and (𝑧,𝑧)𝐸1×𝐸2 satisfying the system of generalized resolvent equations (1.17) (in this case, (𝑥,𝑦,𝑢,𝑣) is a solution of system of variational inclusions (1.15)), and the iterative sequences {𝑧𝑘},{𝑧𝑘},{𝑥𝑘},{𝑦𝑘},{𝑢𝑘}, and {𝑣𝑘} generated by Algorithm 1.9 converge strongly to 𝑧,𝑧,𝑥,𝑦,𝑢, and 𝑣, respectively.

In this paper we introduce and study a general system of generalized resolvent equations with corresponding general system of variational inclusions in uniformly smooth Banach spaces. Motivated and inspired by the above Proposition 1.8, we establish an equivalence relation between general system of generalized resolvent equations and general system of variational inclusions. By using Nadler [34] we propose some new iterative algorithms for finding the approximate solutions of general system of generalized resolvent equations, which include Ahmad and Yao’s corresponding algorithms as special cases to a great extent. Furthermore, the convergence criteria of approximate solutions of general system of generalized resolvent equations obtained by the proposed iterative algorithm are also presented. There is no doubt that our results represent the generalization, improvement, supplement, and development of Ahmad and Yao corresponding ones [31].

2. Main Results

Let 𝐸1 and 𝐸2 be two real Banach spaces, let 𝑆𝐸1×𝐸2𝐸1 and 𝑇𝐸1×𝐸2𝐸2 be single-valued mappings, and let 𝐺𝐸12𝐸1,𝐹𝐸22𝐸2,𝐻𝐸12𝐸1 and 𝑉𝐸22𝐸2 be any four multivalued mappings. Let 𝑀𝐸12𝐸1 and 𝑁𝐸22𝐸2 be any nonlinear mappings, 𝑚𝐸2𝐸1,𝑛𝐸1𝐸2,𝑓𝐸1𝐸1 and 𝑔𝐸2𝐸2 nonlinear mappings with 𝑓(𝐸1)𝐷(𝑀) and 𝑔(𝐸2)𝐷(𝑁), respectively. Then we consider the problem of finding (𝑥,𝑦)𝐸1×𝐸2,(𝑠,𝑣)𝐺(𝑥)×𝐹(𝑦),(𝑢,𝑡)𝐻(𝑥)×𝑉(𝑦),𝑧𝐸1,𝑧𝐸2 such that𝑆(𝑠,𝑣)+𝜌1𝑅𝜌𝑀𝑧𝑇=𝑚(𝑦),𝜌>0,(𝑢,𝑡)+𝛾1𝑅𝛾𝑁𝑧=𝑛(𝑥),𝛾>0,(2.1) where 𝑅𝜌𝑀=𝐼𝐽𝜌𝑀,𝑅𝛾𝑁=𝐼𝐽𝛾𝑁 and 𝐽𝜌𝑀,𝐽𝛾𝑁 are the resolvent operators associated with 𝑀 and 𝑁, respectively.

The corresponding general system of variational inclusions of (2.1) is the problem (1.14), that is, find (𝑥,𝑦)𝐸1×𝐸2,(𝑠,𝑣)𝐺(𝑥)×𝐹(𝑦),(𝑢,𝑡)𝐻(𝑥)×𝑉(𝑦) such that𝑛𝑚(𝑦)𝑆(𝑠,𝑣)+𝑀(𝑓(𝑥)),(𝑥)𝑇(𝑢,𝑡)+𝑁(𝑔(𝑦)).(2.2)

Proposition 2.1. (𝑥,𝑦)𝐸1×𝐸2,(𝑠,𝑣)𝐺(𝑥)×𝐹(𝑦),(𝑢,𝑡)𝐻(𝑥)×𝑉(𝑦) are solutions of general system of variational inclusions (1.14) if and only if (𝑥,𝑦,𝑢,𝑣,𝑠,𝑡) satisfies 𝑓(𝑥)=𝐽𝜌𝑀[]𝑓(𝑥)𝜌(𝑆(𝑠,𝑣)𝑚(𝑦)),𝜌>0,𝑔(𝑦)=𝐽𝛾𝑁[]𝑔(𝑦)𝛾(𝑇(𝑢,𝑡)𝑛(𝑥)),𝛾>0.(2.3)

Proof. The proof of Proposition 2.1 is a direct consequence of the definition of resolvent operator, and hence, is omitted.

Next we first establish an equivalence relation between general system of generalized resolvent equations (2.1) and general system of variational inclusions (1.14) and then prove the existence of a solution of (2.1) and convergence of sequences generated by the proposed algorithms.

Proposition 2.2. The general system of variational inclusions (1.14) has a solution (𝑥,𝑦,𝑢,𝑣,𝑠,𝑡) with (𝑥,𝑦)𝐸1×𝐸2,(𝑠,𝑣)𝐺(𝑥)×𝐹(𝑦) and (𝑢,𝑡)𝐻(𝑥)×𝑉(𝑦) if and only if general system of generalized resolvent equations (2.1) has a solution (𝑧,𝑧,𝑥,𝑦,𝑢,𝑣,𝑠,𝑡) with (𝑥,𝑦)𝐸1×𝐸2,(𝑠,𝑣)𝐺(𝑥)×𝐹(𝑦),(𝑢,𝑡)𝐻(𝑥)×𝑉(𝑦),(𝑧,𝑧)𝐸1×𝐸2, where 𝑓(𝑥)=𝐽𝜌𝑀𝑧,𝑔(𝑦)=𝐽𝛾𝑁𝑧,(2.4) and 𝑧=𝑓(𝑥)𝜌(𝑆(𝑠,𝑣)𝑚(𝑦)) and 𝑧=𝑔(𝑦)𝛾(𝑇(𝑢,𝑡)𝑛(𝑥)).

Proof. Let (𝑥,𝑦)𝐸1×𝐸2,(𝑠,𝑣)𝐺(𝑥)×𝐹(𝑦),(𝑢,𝑡)𝐻(𝑥)×𝑉(𝑦) be a solution of general system of variational inclusions (1.14). Then, by Proposition 2.1, it satisfies the following system of equations 𝑓(𝑥)=𝐽𝜌𝑀[],𝑓(𝑥)𝜌(𝑆(𝑠,𝑣)𝑚(𝑦))𝑔(𝑦)=𝐽𝛾𝑁[].𝑔(𝑦)𝛾(𝑇(𝑢,𝑡)𝑛(𝑥))(2.5)
Let 𝑧=𝑓(𝑥)𝜌(𝑆(𝑠,𝑣)𝑚(𝑦)) and 𝑧=𝑔(𝑦)𝛾(𝑇(𝑢,𝑡)𝑛(𝑥)). Then we have 𝑓(𝑥)=𝐽𝜌𝑀𝑧,𝑔(𝑦)=𝐽𝛾𝑁𝑧,(2.6) and hence 𝑧=𝐽𝜌𝑀(𝑧)𝜌(𝑆(𝑠,𝑣)𝑚(𝑦)) and 𝑧=𝐽𝛾𝑁(𝑧)𝛾(𝑇(𝑢,𝑡)𝑛(𝑥)). Thus it follows that 𝐼𝐽𝜌𝑀𝑧=𝜌(𝑆(𝑠,𝑣)𝑚(𝑦)),𝐼𝐽𝛾𝑁𝑧=𝛾(𝑇(𝑢,𝑡)𝑛(𝑥)),(2.7) that is, 𝑆(𝑠,𝑣)+𝜌1𝑅𝜌𝑀𝑧𝑇=𝑚(𝑦),(𝑢,𝑡)+𝛾1𝑅𝛾𝑁𝑧=𝑛(𝑥).(2.8) Therefore, (𝑧,𝑧,𝑥,𝑦,𝑢,𝑣,𝑠,𝑡) is a solution of general system of generalized resolvent equations (2.1).
Conversely, let (𝑧,𝑧,𝑥,𝑦,𝑢,𝑣,𝑠,𝑡) be a solution of general system of generalized resolvent equations (2.1). Then 𝜌(𝑆(𝑠,𝑣)𝑚(𝑦))=𝑅𝜌𝑀𝑧,𝛾(𝑇(𝑢,𝑡)𝑛(𝑥))=𝑅𝛾𝑁𝑧.(2.9)
Now observe that 𝜌(𝑆(𝑠,𝑣)𝑚(𝑦))=𝑅𝜌𝑀𝑧=𝐼𝐽𝜌𝑀𝑧=𝐽𝜌𝑀𝑧𝑧=𝐽𝜌𝑀[][],𝑓(𝑥)𝜌(𝑆(𝑠,𝑣)𝑚(𝑦))𝑓(𝑥)𝜌(𝑆(𝑠,𝑣)𝑚(𝑦))(2.10) which leads to 𝑓(𝑥)=𝐽𝜌𝑀[],𝑓(𝑥)𝜌(𝑆(𝑠,𝑣)𝑚(𝑦))(2.11) and also that 𝛾(𝑇(𝑢,𝑡)𝑛(𝑥))=𝑅𝛾𝑁𝑧=𝐼𝐽𝛾𝑁𝑧=𝐽𝛾𝑁𝑧𝑧=𝐽𝛾𝑁[][],𝑔(𝑦)𝛾(𝑇(𝑢,𝑡)𝑛(𝑥))𝑔(𝑦)𝛾(𝑇(𝑢,𝑡)𝑛(𝑥))(2.12) which leads to 𝑔(𝑦)=𝐽𝛾𝑁[].𝑔(𝑦)𝛾(𝑇(𝑢,𝑡)𝑛(𝑥))(2.13) Consequently, we have 𝑓(𝑥)=𝐽𝜌𝑀[],𝑓(𝑥)𝜌(𝑆(𝑠,𝑣)𝑚(𝑦))𝑔(𝑦)=𝐽𝛾𝑁[].𝑔(𝑦)𝛾(𝑇(𝑢,𝑡)𝑛(𝑥))(2.14) Therefore, by Proposition 2.1, (𝑥,𝑦,𝑢,𝑣,𝑠,𝑡) is a solution of general system of variational inclusions (1.14).

Proof (Alternative). Let 𝑧=𝑓(𝑥)𝜌(𝑆(𝑠,𝑣)𝑚(𝑦)),𝑧=𝑔(𝑦)𝛾(𝑇(𝑢,𝑡)𝑛(𝑥)).(2.15) Then, utilizing (2.4), we can write 𝑧=𝐽𝜌𝑀𝑧𝜌(𝑆(𝑠,𝑣)𝑚(𝑦)),𝑧=𝐽𝛾𝑁𝑧𝛾(𝑇(𝑢,𝑡)𝑛(𝑥))(2.16) which yield that 𝑆(𝑠,𝑣)+𝜌1𝑅𝜌𝑀𝑧𝑇=𝑚(𝑦),(𝑢,𝑡)+𝛾1𝑅𝛾𝑁𝑧=𝑛(𝑥),(2.17) the required general system of generalized resolvent equations.

Algorithm 2.3. For given (𝑥0,𝑦0)𝐸1×𝐸2,(𝑠0,𝑣0)𝐺(𝑥0)×𝐹(𝑦0),(𝑢0,𝑡0)𝐻(𝑥0)×𝑉(𝑦0),(𝑧0,𝑧0)𝐸1×𝐸2, compute 𝑧1𝑥=𝑓0𝑆𝑠𝜌0,𝑣0𝑦𝑚0,𝑧1𝑦=𝑔0𝑇𝑢𝛾0,𝑡0𝑥𝑛0.(2.18) For (𝑧1,𝑧1)𝐸1×𝐸2, we take (𝑥1,𝑦1)𝐸1×𝐸2 such that 𝑓(𝑥1)=𝐽𝜌𝑀(𝑧1) and 𝑔(𝑦1)=𝐽𝛾𝑁(𝑧1). Then, by Nadler [34], there exist (𝑠1,𝑣1)𝐺(𝑥1)×𝐹(𝑦1),(𝑢1,𝑡1)𝐻(𝑥1)×𝑉(𝑦1) such that 𝑢1𝑢0𝐻𝑥(1+1)𝐷1𝑥,𝐻0,𝑣1𝑣0𝐹𝑦(1+1)𝐷1𝑦,𝐹0,𝑠1𝑠0𝐺𝑥(1+1)𝐷1𝑥,𝐺0,𝑡1𝑡0𝑉𝑦(1+1)𝐷1𝑦,𝑉0,(2.19) where 𝐷(,) is the Hausdorff metric on 𝐶𝐵(𝐸1) (for the sake of convenience, we also denote by 𝐷(,) the Hausdorff metric on 𝐶𝐵(𝐸2)). Compute 𝑧2𝑥=𝑓1𝑆𝑠𝜌1,𝑣1𝑦𝑚1,𝑧2𝑦=𝑔1𝑇𝑢𝛾1,𝑡1𝑥𝑛1.(2.20) By induction, we can obtain sequences (𝑥𝑘,𝑦𝑘)𝐸1×𝐸2,(𝑠𝑘,𝑣𝑘)𝐺(𝑥𝑘)×𝐹(𝑦𝑘),(𝑢𝑘,𝑡𝑘)𝐻(𝑥𝑘)×𝑉(𝑦𝑘),(𝑧𝑘,𝑧𝑘)𝐸1×𝐸2 by the iterative scheme: 𝑓𝑥𝑘=𝐽𝜌𝑀𝑧𝑘,𝑔𝑦𝑘=𝐽𝛾𝑁𝑧𝑘,𝑢(2.21)𝑘𝑥𝐻𝑘𝑢𝑘+1𝑢𝑘11+𝐷𝐻𝑥𝑘+1𝑘+1𝑥,𝐻𝑘,𝑣𝑘𝑦𝐹𝑘𝑣𝑘+1𝑣𝑘11+𝐷𝐹𝑦𝑘+1𝑘+1𝑦,𝐹𝑘,𝑠𝑘𝑥𝐺𝑘𝑠𝑘+1𝑠𝑘11+𝐷𝐺𝑥𝑘+1𝑘+1𝑥,𝐺𝑘,𝑡𝑘𝑦𝑉𝑘𝑡𝑘+1𝑡𝑘11+𝐷𝑉𝑦𝑘+1𝑘+1𝑦,𝑉𝑘,𝑧(2.22)𝑘+1𝑥=𝑓𝑘𝑆𝑠𝜌𝑘,𝑣𝑘𝑦𝑚𝑘,𝑧𝑘+1𝑦=𝑔𝑘𝑇𝑢𝛾𝑘,𝑡𝑘𝑥𝑛𝑘,(2.23) for 𝑘=0,1,2,.
The general system of generalized resolvent equations (2.1) can also be rewritten as 𝑧=𝑓(𝑥)𝑆(𝑠,𝑣)+𝑚(𝑦)+𝐼𝜌1𝑅𝜌𝑀𝑧,𝑧=𝑔(𝑦)𝑇(𝑢,𝑡)+𝑛(𝑥)+𝐼𝛾1𝑅𝛾𝑁𝑧.(2.24)
Utilizing this fixed-point formulation, we suggest the following iterative algorithm.

Algorithm 2.4. For given (𝑥0,𝑦0)𝐸1×𝐸2,(𝑠0,𝑣0)𝐺(𝑥0)×𝐹(𝑦0),(𝑢0,𝑡0)𝐻(𝑥0)×𝑉(𝑦0),(𝑧0,𝑧0)𝐸1×𝐸2, compute 𝑧1𝑥=𝑓0𝑠𝑆0,𝑣0𝑦+𝑚0+𝐼𝜌1𝑅𝜌𝑀𝑧0,𝑧1𝑦=𝑔0𝑢𝑇0,𝑡0𝑥+𝑛0+𝐼𝛾1𝑅𝛾𝑁𝑧0.(2.25) For (𝑧1,𝑧1)𝐸1×𝐸2, we take (𝑥1,𝑦1)𝐸1×𝐸2 such that 𝑓(𝑥1)=𝐽𝜌𝑀(𝑧1) and 𝑔(𝑦1)=𝐽𝛾𝑁(𝑧1). Then, by Nadler [34], there exist (𝑠1,𝑣1)𝐺(𝑥1)×𝐹(𝑦1),(𝑢1,𝑡1)𝐻(𝑥1)×𝑉(𝑦1) such that 𝑢1𝑢0𝐻𝑥(1+1)𝐷1𝑥,𝐻0,𝑣1𝑣0𝐹𝑦(1+1)𝐷1𝑦,𝐹0,𝑠1𝑠0𝐺𝑥(1+1)𝐷1𝑥,𝐺0,𝑡1𝑡0𝑉𝑦(1+1)𝐷1𝑦,𝑉0,(2.26) where 𝐷(,) is the Hausdorff metric on 𝐶𝐵(𝐸1) (for the sake of convenience, we also denote by 𝐷(,) the Hausdorff metric on 𝐶𝐵(𝐸2)). Compute 𝑧2𝑥=𝑓1𝑠𝑆1,𝑣1𝑦+𝑚1+𝐼𝜌1𝑅𝜌𝑀𝑧1,𝑧2𝑦=𝑔1𝑢𝑇1,𝑡1𝑥+𝑛1+𝐼𝛾1𝑅𝛾𝑁𝑧1.(2.27) By induction, we can obtain sequences (𝑥𝑘,𝑦𝑘)𝐸1×𝐸2,(𝑠𝑘,𝑣𝑘)𝐺(𝑥𝑘)×𝐹(𝑦𝑘),(𝑢𝑘,𝑡𝑘)𝐻(𝑥𝑘)×𝑉(𝑦𝑘),(𝑧𝑘,𝑧𝑘)𝐸1×𝐸2 by the iterative scheme: 𝑓𝑥𝑘=𝐽𝜌𝑀𝑧𝑘,𝑔𝑦𝑘=𝐽𝛾𝑁𝑧𝑘,𝑢𝑘𝑥𝐻𝑘𝑢𝑘+1𝑢𝑘11+𝐷𝐻𝑥𝑘+1𝑘+1𝑥,𝐻𝑘,𝑣𝑘𝑦𝐹𝑘𝑣𝑘+1𝑣𝑘11+𝐷𝐹𝑦𝑘+1𝑘+1𝑦,𝐹𝑘,𝑠𝑘𝑥𝐺𝑘𝑠𝑘+1𝑠𝑘11+𝐷𝐺𝑥𝑘+1𝑘+1𝑥,𝐺𝑘,𝑡𝑘𝑦𝑉𝑘𝑡𝑘+1𝑡𝑘11+𝐷𝑉𝑦𝑘+1𝑘+1𝑦,𝑉𝑘,𝑧𝑘+1𝑥=𝑓𝑘𝑠𝑆𝑘,𝑣𝑘𝑦+𝑚𝑘+𝐼𝜌1𝑅𝜌𝑀𝑧𝑘,𝑧𝑘+1𝑦=𝑔𝑘𝑢𝑇𝑘,𝑡𝑘𝑥+𝑛𝑘+𝐼𝛾1𝑅𝛾𝑁𝑧𝑘,(2.28) for 𝑘=0,1,2,.
For positive stepsize 𝛿,𝛿, the general system of generalized resolvent equations (2.1) can also be rewritten as 𝑓𝑥,𝑧=𝑓𝑥,𝑧𝛿𝑧𝐽𝜌𝑀𝑧+𝜌(𝑆(𝑠,𝑣)𝑚(𝑦))=𝑓𝑥,𝑧𝛿𝑓(𝑥)𝐽𝜌𝑀(,𝑔𝑓(𝑥))+𝜌(𝑆(𝑠,𝑣)𝑚(𝑦))𝑦,𝑧=𝑔𝑦,𝑧𝛿𝑧𝐽𝛾𝑁𝑧+𝛾(𝑇(𝑢,𝑡)𝑛(𝑥))=𝑔𝑦,𝑧𝛿𝑔(𝑦)𝐽𝛾𝑁(.𝑔(𝑦))+𝛾(𝑇(𝑢,𝑡)𝑛(𝑥))(2.29)
This fixed point formulation enables us to propose the following iterative algorithm.

Algorithm 2.5. For given (𝑥0,𝑦0)𝐸1×𝐸2,(𝑠0,𝑣0)𝐺(𝑥0)×𝐹(𝑦0),(𝑢0,𝑡0)𝐻(𝑥0)×𝑉(𝑦0),(𝑧0,𝑧0)𝐸1×𝐸2, compute (𝑥1,𝑦1)𝐸1×𝐸2 and (𝑧1,𝑧1)𝐸1×𝐸2 such that 𝑓𝑥1,𝑧1𝑥=𝑓0,𝑧0𝛿𝑓𝑥0𝐽𝜌𝑀𝑓𝑥0𝑆𝑠+𝜌0,𝑣0𝑦𝑚0,𝑔𝑦1,𝑧1𝑦=𝑔0,𝑧0𝛿𝑔𝑦0𝐽𝛾𝑁𝑔𝑦0𝑇𝑢+𝛾0,𝑡0𝑥𝑛0.(2.30) Then, by Nadler [34], there exist (𝑠1,𝑣1)𝐺(𝑥1)×𝐹(𝑦1),(𝑢1,𝑡1)𝐻(𝑥1)×𝑉(𝑦1) such that 𝑢1𝑢0𝐻𝑥(1+1)𝐷1𝑥,𝐻0,𝑣1𝑣0𝐹𝑦(1+1)𝐷1𝑦,𝐹0,𝑠1𝑠0𝐺𝑥(1+1)𝐷1𝑥,𝐺0,𝑡1𝑡0𝑉𝑦(1+1)𝐷1𝑦,𝑉0,(2.31) where 𝐷(,) is the Hausdorff metric on 𝐶𝐵(𝐸1) (for the sake of convenience, we also denote by 𝐷(,) the Hausdorff metric on 𝐶𝐵(𝐸2)). Compute (𝑥2,𝑦2)𝐸1×𝐸2 and (𝑧2,𝑧2)𝐸1×𝐸2 such that 𝑓𝑥2,𝑧2𝑥=𝑓1,𝑧1𝛿𝑓𝑥1𝐽𝜌𝑀𝑓𝑥1𝑆𝑠+𝜌1,𝑣1𝑦𝑚1,𝑔𝑦2,𝑧2𝑦=𝑔1,𝑧1𝛿𝑔𝑦1𝐽𝛾𝑁𝑔𝑦1𝑇𝑢+𝛾1,𝑡1𝑥𝑛1.(2.32) By induction, we can obtain sequences (𝑥𝑘,𝑦𝑘)𝐸1×𝐸2,(𝑠𝑘,𝑣𝑘)𝐺(𝑥𝑘)×𝐹(𝑦𝑘),(𝑢𝑘,𝑡𝑘)𝐻(𝑥𝑘)×𝑉(𝑦𝑘),(𝑧𝑘,𝑧𝑘)𝐸1×𝐸2 by the iterative scheme: 𝑢𝑘𝑥𝐻𝑘𝑢𝑘+1𝑢𝑘11+𝐷𝐻𝑥𝑘+1𝑘+1𝑥,𝐻𝑘,𝑣𝑘𝑦𝐹𝑘𝑣𝑘+1𝑣𝑘11+𝐷𝐹𝑦𝑘+1𝑘+1𝑦,𝐹𝑘,𝑠𝑘𝑥𝐺𝑘𝑠𝑘+1𝑠𝑘11+𝐷𝐺𝑥𝑘+1𝑘+1𝑥,𝐺𝑘,𝑡𝑘𝑦𝑉𝑘𝑡𝑘+1𝑡𝑘11+𝐷𝑉𝑦𝑘+1𝑘+1𝑦,𝑉𝑘,𝑓𝑥𝑘+1,𝑧𝑘+1𝑥=𝑓𝑘,𝑧𝑘𝑓𝑥𝛿𝑘𝐽𝜌𝑀𝑓𝑥𝑘𝑆𝑠+𝜌𝑘,𝑣𝑘𝑦𝑚𝑘,𝑔𝑦𝑘+1,𝑧𝑘+1𝑦=𝑔𝑘,𝑧𝑘𝛿𝑔𝑦𝑘𝐽𝛾𝑁𝑔𝑦𝑘𝑇𝑢+𝛾𝑘,𝑡𝑘𝑥𝑛𝑘,(2.33) for 𝑘=0,1,2,.
Note that for 𝛿=𝛿=1,𝑓(𝑥𝑘,𝑧𝑘)=𝑓(𝑥𝑘),𝑔(𝑦𝑘,𝑧𝑘)=𝑔(𝑦𝑘), Algorithm 2.5 reduces to the following algorithm which solves the general system of variational inclusions (1.14).

Algorithm 2.6. For given (𝑥0,𝑦0)𝐸1×𝐸2,(𝑠0,𝑣0)𝐺(𝑥0)×𝐹(𝑦0),(𝑢0,𝑡0)𝐻(𝑥0)×𝑉(𝑦0), compute (𝑥1,𝑦1)𝐸1×𝐸2 such that 𝑓𝑥1=𝐽𝜌𝑀𝑓𝑥0𝑆𝑠𝜌0,𝑣0𝑦𝑚0,𝑔𝑦1=𝐽𝛾𝑁𝑔𝑦0𝑇𝑢𝛾0,𝑡0𝑥𝑛0.(2.34) Then, by Nadler [34], there exist (𝑠1,𝑣1)𝐺(𝑥1)×𝐹(𝑦1),(𝑢1,𝑡1)𝐻(𝑥1)×𝑉(𝑦1) such that 𝑢1𝑢0𝐻𝑥(1+1)𝐷1𝑥,𝐻0,𝑣1𝑣0𝐹𝑦(1+1)𝐷1𝑦,𝐹0,𝑠1𝑠0𝐺𝑥(1+1)𝐷1𝑥,𝐺0,𝑡1𝑡0𝑉𝑦(1+1)𝐷1𝑦,𝑉0,(2.35) where 𝐷(,) is the Hausdorff metric on 𝐶𝐵(𝐸1) (for the sake of convenience, we also denote by 𝐷(,) the Hausdorff metric on 𝐶𝐵(𝐸2)). Compute (𝑥2,𝑦2)𝐸1×𝐸2 such that 𝑓𝑥2=𝐽𝜌𝑀𝑓𝑥1𝑆𝑠𝜌1,𝑣1𝑦𝑚1,𝑔𝑦2=𝐽𝛾𝑁𝑔𝑦1𝑇𝑢𝛾1,𝑡1𝑥𝑛1.(2.36) By induction, we can obtain sequences (𝑥𝑘,𝑦𝑘)𝐸1×𝐸2,(𝑠𝑘,𝑣𝑘)𝐺(𝑥𝑘)×𝐹(𝑦𝑘),(𝑢𝑘,𝑡𝑘)𝐻(𝑥𝑘)×𝑉(𝑦𝑘) by the iterative scheme: 𝑓𝑥𝑘+1=𝐽𝜌𝑀𝑓𝑥𝑘𝑆𝑠𝜌𝑘,𝑣𝑘𝑦𝑚𝑘,𝑔𝑦𝑘+1=𝐽𝛾𝑁𝑔𝑦𝑘𝑇𝑢𝛾𝑘,𝑡𝑘𝑥𝑛𝑘,𝑢𝑘𝑥𝐻𝑘𝑢𝑘+1𝑢𝑘11+𝐷𝐻𝑥𝑘+1𝑘+1𝑥,𝐻𝑘,𝑣𝑘𝑦𝐹𝑘𝑣𝑘+1𝑣𝑘11+𝐷𝐹𝑦𝑘+1𝑘+1𝑦,𝐹𝑘,𝑠𝑘𝑥𝐺𝑘𝑠𝑘+1𝑠𝑘11+𝐷𝐺𝑥𝑘+1𝑘+1𝑥,𝐺𝑘,𝑡𝑘𝑦𝑉𝑘𝑡𝑘+1𝑡𝑘11+𝐷𝑉𝑦𝑘+1𝑘+1𝑦,𝑉𝑘,(2.37) for 𝑘=0,1,2,.

We now study the convergence analysis of Algorithm 2.3. In a similar way, one can study the convergence of other algorithms.

Theorem 2.7. Let 𝐸1 and 𝐸2 be two real uniformly smooth Banach spaces with modulus of smoothness 𝜏𝐸1(𝑡)𝐶1𝑡2 and 𝜏𝐸2(𝑡)𝐶2𝑡2 for 𝐶1,𝐶2>0, respectively. Let 𝐺𝐸1𝐶𝐵(𝐸1), 𝐹𝐸2𝐶𝐵(𝐸2),𝐻𝐸1𝐶𝐵(𝐸1),𝑉𝐸2𝐶𝐵(𝐸2) be 𝐷-Lipschitz continuous mappings with constants 𝜆𝐷𝐺,𝜆𝐷𝐹,𝜆𝐷𝐻, and 𝜆𝐷𝑉, respectively, and let 𝑀𝐸12𝐸1,𝑁𝐸22𝐸2 be 𝑚-accretive mappings such that the resolvent operators associated with 𝑀 and 𝑁 are retractions. Let 𝑓𝐸1𝐸1,𝑔𝐸2𝐸2 be both strong accretive with constants 𝛼 and 𝛽, respectively, and Lipschitz continuous with constants 𝛿1 and 𝛿2, respectively. Let 𝑚𝐸2𝐸1,𝑛𝐸1𝐸2 be Lipschitz continuous with constants 𝜆𝑚 and 𝜆𝑛, respectively, and 𝑆𝐸1×𝐸2𝐸1,𝑇𝐸1×𝐸2𝐸2 Lipschitz continuous in the first and second arguments with constants 𝜆𝑆1,𝜆𝑆2 and 𝜆𝑇1,𝜆𝑇2, respectively.
If there exist constants 𝜌>0 and 𝛾>0, such that 0<𝐵+𝜃1+𝜃4𝐵1𝐵<1,0<+𝜃2+𝜃31𝐵<1,(2.38) where 𝐵=12𝛼+64𝐶1𝛿21,𝐵=12𝛽+64𝐶2𝛿22, and 𝜃1=(1+𝜌𝜆𝑆1𝜆𝐷𝐺)/(1𝜌(𝜆𝑆1𝜆𝐷𝐺+𝜆𝑆2𝜆𝐷𝐹+𝜆𝑚)),𝜃2=𝜌(𝜆𝑆2𝜆𝐷𝐹+𝜆𝑚)/(1𝜌(𝜆𝑆1𝜆𝐷𝐺+𝜆𝑆2𝜆𝐷𝐹+𝜆𝑚)),𝜃3=(1+𝛾𝜆𝑇2𝜆𝐷𝑉)/(1𝛾(𝜆𝑇1𝜆𝐷𝐻+𝜆𝑇2𝜆𝐷𝑉+𝜆𝑛)),𝜃4=𝛾(𝜆𝑇1𝜆𝐷𝐻+𝜆𝑛)/(1𝛾(𝜆𝑇1𝜆𝐷𝐻+𝜆𝑇2𝜆𝐷𝑉+𝜆𝑛)), then there exist (𝑥,𝑦)𝐸1×𝐸2,(𝑠,𝑣)𝐺(𝑥)×𝐹(𝑦),(𝑢,𝑡)𝐻(𝑥)×𝑉(𝑦) and (𝑧,𝑧)𝐸1×𝐸2 satisfying the general system of generalized resolvent equations (2.1) (in this case, (𝑥,𝑦,𝑢,𝑣,𝑠,𝑡) is a solution of general system of variational inclusions (1.14)), and the iterative sequences {𝑧𝑘},{𝑧𝑘},{𝑥𝑘},{𝑦𝑘}, {𝑢𝑘},{𝑣𝑘},{𝑠𝑘}, and {𝑡𝑘} generated by Algorithm 2.3 converge strongly to 𝑧,𝑧,𝑥,𝑦,𝑢,𝑣,𝑠, and 𝑡, respectively.

Proof. From Algorithm 2.3 we have 𝑧𝑘+1𝑧𝑘=𝑓𝑥𝑘𝑆𝑠𝜌𝑘,𝑣𝑘𝑦𝑚𝑘𝑓𝑥𝑘1𝑆𝑠𝜌𝑘1,𝑣𝑘1𝑦𝑚𝑘1𝑥𝑘𝑥𝑘1𝑓𝑥𝑘𝑥𝑓𝑘1+𝑥𝑘𝑥𝑘1𝑆𝑠𝜌𝑘,𝑣𝑘𝑦𝑚𝑘𝑆𝑠𝑘1,𝑣𝑘1𝑦𝑚𝑘1.(2.39) By Proposition 1.1, we have (see, e.g., the proof of [32, Theorem  3]) 𝑥𝑘𝑥𝑘1𝑓𝑥𝑘𝑥𝑓𝑘1212𝛼+64𝐶𝛿21𝑥𝑘𝑥𝑘12.(2.40) Since 𝑆 is Lipschitz continuous in both arguments, 𝐺,𝐹 are 𝐷-Lipschitz continuous, and 𝑚 is Lipschitz continuous, we have 𝑆𝑠𝑘,𝑣𝑘𝑦𝑚𝑘𝑆𝑠𝑘1,𝑣𝑘1𝑦𝑚𝑘1=𝑆𝑠𝑘,𝑣𝑘𝑠𝑆𝑘1,𝑣𝑘𝑠+𝑆𝑘1,𝑣𝑘𝑠𝑆𝑘1,𝑣𝑘1𝑚𝑦𝑘𝑦𝑚𝑘1𝑆𝑠𝑘,𝑣𝑘𝑠𝑆𝑘1,𝑣𝑘+𝑆𝑠𝑘1,𝑣𝑘𝑠𝑆𝑘1,𝑣𝑘1+𝑚𝑦𝑘𝑦𝑚𝑘1𝜆𝑆1𝑠𝑘𝑠𝑘1+𝜆𝑆2𝑣𝑘𝑣𝑘1+𝜆𝑚𝑦𝑘𝑦𝑘1𝜆𝑆111+𝑘𝐷𝐺𝑥𝑘𝑥,𝐺𝑘1+𝜆𝑆211+𝑘𝐷𝐹𝑦𝑘𝑦,𝐹𝑘1+𝜆𝑚𝑦𝑘𝑦𝑘111+𝑘𝜆𝑆1𝜆𝐷𝐺𝑥𝑘𝑥𝑘1+11+𝑘𝜆𝑆2𝜆𝐷𝐹𝑦𝑘𝑦𝑘1+𝜆𝑚𝑦𝑘𝑦𝑘1=11+𝑘𝜆𝑆1𝜆𝐷𝐺𝑥𝑘𝑥𝑘1+11+𝑘𝜆𝑆2𝜆𝐷𝐹+𝜆𝑚𝑦𝑘𝑦𝑘1.(2.41) Utilizing (2.41) and Proposition 1.1, we have 𝑥𝑘𝑥𝑘1𝑆𝑠𝜌𝑘,𝑣𝑘𝑦𝑚𝑘𝑆𝑠𝑘1,𝑣𝑘1𝑦𝑚𝑘12𝑥𝑘𝑥𝑘12𝑆𝑠2𝜌𝑘,𝑣𝑘𝑦𝑚𝑘𝑆𝑠𝑘1,𝑣𝑘1𝑦𝑚𝑘1,𝐽𝑥𝑘𝑥𝑘1𝑆𝑠𝜌𝑘,𝑣𝑘𝑦𝑚𝑘𝑆𝑠𝑘1,𝑣𝑘1𝑦𝑚𝑘1𝑥𝑘𝑥𝑘12𝑆𝑠+2𝜌𝑘,𝑣𝑘𝑦𝑚𝑘𝑆𝑠𝑘1,𝑣𝑘1𝑦𝑚𝑘1×𝑥𝑘𝑥𝑘1𝑆𝑠𝜌𝑘,𝑣𝑘𝑦𝑚𝑘𝑆𝑠𝑘1,𝑣𝑘1𝑦𝑚𝑘1𝑥𝑘𝑥𝑘121+2𝜌1+𝑘𝜆𝑆1𝜆𝐷𝐺𝑥𝑘𝑥𝑘1+11+𝑘𝜆𝑆2𝜆𝐷𝐹+𝜆𝑚𝑦𝑘𝑦𝑘1×𝑥𝑘𝑥𝑘1𝑆𝑠𝜌𝑘,𝑣𝑘𝑦𝑚𝑘𝑆𝑠𝑘1,𝑣𝑘1𝑦𝑚𝑘1𝑥𝑘𝑥𝑘121+𝜌1+𝑘𝜆𝑆1𝜆𝐷𝐺×𝑥𝑘𝑥𝑘12+𝑥𝑘𝑥𝑘1𝑆𝑠𝜌𝑘,𝑣𝑘𝑦𝑚𝑘𝑆𝑠𝑘1,𝑣𝑘1𝑦𝑚𝑘121+𝜌1+𝑘𝜆𝑆2𝜆𝐷𝐹+𝜆𝑚×𝑦𝑘𝑦𝑘12+𝑥𝑘𝑥𝑘1𝑆𝑠𝜌𝑘,𝑣𝑘𝑦𝑚𝑘𝑆𝑠𝑘1,𝑣𝑘1𝑦𝑚𝑘12=11+𝜌1+𝑘𝜆𝑆1𝜆𝐷𝐺𝑥𝑘𝑥𝑘121+𝜌1+𝑘𝜆𝑆1𝜆𝐷𝐺+𝜆𝑆2𝜆𝐷𝐹+𝜆𝑚×𝑥𝑘𝑥𝑘1𝑆𝑠𝜌𝑘,𝑣𝑘𝑦𝑚𝑘𝑆𝑠𝑘1,𝑣𝑘1𝑦𝑚𝑘121+𝜌1+𝑘𝜆𝑆2𝜆𝐷𝐹+𝜆𝑚𝑦𝑘𝑦𝑘12,(2.42) which implies that 𝑥𝑘𝑥𝑘1𝑆𝑠𝜌𝑘,𝑣𝑘𝑦𝑚𝑘𝑆𝑠𝑘1,𝑣𝑘1𝑦𝑚𝑘121+𝜌(1+1/𝑘)𝜆𝑆1𝜆𝐷𝐺𝜆1𝜌(1+1/𝑘)𝑆1𝜆𝐷𝐺+𝜆𝑆2𝜆𝐷𝐹+𝜆𝑚𝑥𝑘𝑥𝑘12+𝜌(1+1/𝑘)𝜆𝑆2𝜆𝐷𝐹+𝜆𝑚𝜆1𝜌(1+1/𝑘)𝑆1𝜆𝐷𝐺+𝜆𝑆2𝜆𝐷𝐹+𝜆𝑚𝑦𝑘𝑦𝑘121+𝜌(1+1/𝑘)𝜆𝑆1𝜆𝐷𝐺𝜆1𝜌(1+1/𝑘)𝑆1𝜆𝐷𝐺+𝜆𝑆2𝜆𝐷𝐹+𝜆𝑚𝑥𝑘𝑥𝑘12+𝜌(1+1/𝑘)𝜆𝑆2𝜆𝐷𝐹+𝜆𝑚𝜆1𝜌(1+1/𝑘)𝑆1𝜆𝐷𝐺+𝜆𝑆2𝜆𝐷𝐹+𝜆𝑚𝑦𝑘𝑦𝑘12+21+𝜌(1+1/𝑘)𝜆𝑆1𝜆𝐷𝐺𝜆1𝜌(1+1/𝑘)𝑆1𝜆𝐷𝐺+𝜆𝑆2𝜆𝐷𝐹+𝜆𝑚𝜌(1+1/𝑘)𝜆𝑆2𝜆𝐷𝐹+𝜆𝑚𝜆1𝜌(1+1/𝑘)𝑆1𝜆𝐷𝐺+𝜆𝑆2𝜆𝐷𝐹+𝜆𝑚×𝑥𝑘𝑥𝑘1𝑦𝑘𝑦𝑘1=1+𝜌(1+1/𝑘)𝜆𝑆1𝜆𝐷𝐺𝜆1𝜌(1+1/𝑘)𝑆1𝜆𝐷𝐺+𝜆𝑆2𝜆𝐷𝐹+𝜆𝑚𝑥𝑘𝑥𝑘1+𝜌(1+1/𝑘)𝜆𝑆2𝜆𝐷𝐹+𝜆𝑚𝜆1𝜌(1+1/𝑘)𝑆1𝜆𝐷𝐺+𝜆𝑆2𝜆𝐷𝐹+𝜆𝑚𝑦𝑘𝑦𝑘12.(2.43) Thus, we have 𝑥𝑘𝑥𝑘1𝑆𝑠𝜌𝑘,𝑣𝑘𝑦𝑚𝑘𝑆𝑠𝑘1,𝑣𝑘1𝑦𝑚𝑘11+𝜌(1+1/𝑘)𝜆𝑆1𝜆𝐷𝐺𝜆1𝜌(1+1/𝑘)𝑆1𝜆𝐷𝐺+𝜆𝑆2𝜆𝐷𝐹+𝜆𝑚𝑥𝑘𝑥𝑘1+𝜌(1+1/𝑘)𝜆𝑆2𝜆𝐷𝐹+𝜆𝑚𝜆1𝜌(1+1/𝑘)𝑆1𝜆𝐷𝐺+𝜆𝑆2𝜆𝐷𝐹+𝜆𝑚𝑦𝑘𝑦𝑘1.(2.44) Note that lim𝑘1+𝜌(1+1/𝑘)𝜆𝑆1𝜆𝐷𝐺𝜆1𝜌(1+1/𝑘)𝑆1𝜆𝐷𝐺+𝜆𝑆2𝜆𝐷𝐹+𝜆𝑚=𝜃1,lim𝑘𝜌(1+1/𝑘)𝜆𝑆2𝜆𝐷𝐹+𝜆𝑚𝜆1𝜌(1+1/𝑘)𝑆1𝜆𝐷𝐺+𝜆𝑆2𝜆𝐷𝐹+𝜆𝑚=𝜃2,(2.45) where 𝜃1=(1+𝜌𝜆𝑆1𝜆𝐷𝐺)/(1𝜌(𝜆𝑆1𝜆𝐷𝐺+𝜆𝑆2𝜆𝐷𝐹+𝜆𝑚)) and 𝜃2=𝜌(𝜆𝑆2𝜆𝐷𝐹+𝜆𝑚)/(1𝜌(𝜆𝑆1𝜆𝐷𝐺+𝜆𝑆2𝜆𝐷𝐹+𝜆𝑚)).
Utilizing (2.40) and (2.44), we deduce from (2.39) that 𝑧𝑘+1𝑧𝑘12𝛼+64𝐶𝛿21+1+𝜌(1+1/𝑘)𝜆𝑆1𝜆𝐷𝐺𝜆1𝜌(1+1/𝑘)𝑆1𝜆𝐷𝐺+𝜆𝑆2𝜆𝐷𝐹+𝜆𝑚𝑥𝑘𝑥𝑘1+𝜌(1+1/𝑘)𝜆𝑆2𝜆𝐷𝐹+𝜆𝑚𝜆1𝜌(1+1/𝑘)𝑆1𝜆𝐷𝐺+𝜆𝑆2𝜆𝐷𝐹+𝜆𝑚𝑦𝑘𝑦𝑘1=𝐵+