# Algorithms for Solving System of Extended General Variational Inclusions and Fixed Points Problems

Authors: Narin Petrot; Javad Balooee
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

We introduce a new system of extended general nonlinear variational inclusions with different nonlinear operators and establish the equivalence between the aforesaid system and the fixed point problem. By using this equivalent formulation, we prove the existence and uniqueness theorem for solution of the system of extended general nonlinear variational inclusions. We suggest and analyze a new resolvent iterative algorithm to approximate the unique solution of the system of extended general nonlinear variational inclusions which is a fixed point of a nearly uniformly Lipschitzian mapping. Subsequently, the convergence analysis of the proposed iterative algorithm under some suitable conditions is considered. Furthermore, some related works to our main problem are pointed out and discussed.


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## Body

## 1. Introduction

Variational inequalities theory, as a very effective and powerful tool of the current mathematical technology, has been widely applied to mechanics, physics, optimization and control, economics and transportation equilibrium, engineering sciences, and so forth. Up until now variational inequalities have been very effective and powerful tools of the current mathematical technology; see for example [1–4] and references therein. In 1968, Brézis [5] initiated the study of the existence theory of a class of variational inequalities, later known as variational inclusions, using proximal-point mappings due to Moreau [6]. It is well known that variational inclusions include variational inequalities, quasivariational inequalities, and variational-like inequalities as special cases. For application of variational inclusions, see, for example, [7–21]. A number of problems leading to the system of variational inclusions/inequalities arise in applications to variational problems and engineering see, for example, [22–30]. Variational inclusions can be viewed as innovative and novel extension of the variational principles. It is well known that the system of variational inclusions/inequalities can provide new insight regarding problems being studied and can stimulate new and innovative ideas for problem solving.One of the most important and interesting problems in the theory of variational inclusions is the development of numerical methods which provide an efficient and implementable algorithm for solving variational inclusion and its generalizations. The method based on proximal-point mapping is a generalization of projection method and has been widely used to study the existence of solutions and to develop iterative algorithms for variational inclusions see, for example, [31–39]. In recent past, the methods based on different classes of proximal-point mappings have been developed to study the existence of solutions and to discuss convergence and stability analysis of proposed iterative algorithms; for various classes of variational/variational-like inclusions, see for example [34, 36, 37, 39–43].Recently, Noor [29] introduced and studied a new system of general mixed variational inequalities involving three different operators (SGMVID). By using the resolvent operator technique, he established the equivalence between the SGMVID and the fixed point problem. He used this equivalent formulation to suggest and analyze some new iterative methods for solving the SGMVID. He also studied the convergence analysis of the proposed iterative methods under some certain conditions.Very recently, M. A. Noor and K. I. Noor [44] introduced and considered the system of general variational inclusions involving seven different operators (SGVID). They suggested and analyzed two resolvent iterative algorithms for solving the SGVID and studied the convergence analysis of the proposed iterative schemes under some certain conditions.On the other hand, related to the variational inequalities/inclusions, we have the problem of finding the fixed points of the nonexpansive mappings, which is the subject of current interest in functional analysis. It is natural to consider a unified approach to these two different problems. Motivated and inspired by the research going in this direction, Noor and Huang [45] considered the problem of finding the common element of the set of the solutions of variational inequalities and the set of the fixed points of the nonexpansive mappings. It is well known that every nonexpansive mapping is a Lipschitzian mapping. Lipschitzian mappings have been generalized by various authors. Sahu [46] introduced and investigated nearly uniformly Lipschitzian mappings as generalization of Lipschitzian mappings.In this paper, we introduce and consider a new system of extended general nonlinear variational inclusions involving eight different nonlinear operators (SEGNVID). We first establish the equivalence between the SEGNVID and the fixed point problem, and then, by this equivalent formulation, we discuss the existence and uniqueness of solution of the SEGNVID. By using two nearly uniformly Lipschitzian mappingsS1 and S2 and the aforementioned equivalent formulation, we suggest and analyze a new resolvent iterative algorithm for finding an element of the set of the fixed points of the nearly uniformly Lipschitzian mapping 𝒬=(S1,S2) which is the unique solution of the SEGNVID. Finally, we consider the convergence analysis of the suggested iterative algorithms under some suitable conditions. Further, some related works, as appeared in [29, 44], are also discussed and improved.


## 2. Formulations and Basic Facts

Throughout this paper, we will letℋ be a real Hilbert space which is equipped with an inner product 〈·,·〉 and corresponding norm ∥·∥. To begin with, let us recall that a set-valued operator A:ℋ⊸ℋ is said to be monotone if and only if, for any x,y∈ℋ(2.1)〈u-v,x-y〉≥0,∀u∈A(x),v∈A(y).
A monotone set-valued operator A is called maximal if and only if its graph, Gph(A):={(x,y)∈ℋ×ℋ:y∈A(x)}, is not properly contained in the graph of any other monotone operator. It is well known that A is a maximal monotone operator if and only if (I+λA)(ℋ)=ℋ, for all λ>0, where I denotes the identity operator on ℋ.Definition 2.1 (see [47]).
For any maximal monotone operatorA, the resolvent operator associated with A of parameter λ is defined as
(2.2)JAλ(u)=(I+λA)-1(u),∀u∈H.
It is single valued and nonexpansive, that is,(2.3)‖JAλ(u)-JAλ(v)‖≤‖u-v‖,∀u,v∈H.
LetTi:ℋ×ℋ→ℋ and gi,hi:ℋ→ℋ(i=1,2) be six nonlinear single-valued operators and Ai:ℋ×ℋ⊸ℋ(i=1,2) be two set-valued operators such that, for all z,t∈ℋ, A1(·,z):ℋ⊸ℋ and A2(·,t):ℋ⊸ℋ are two maximal monotone operators with g2(u)∈dom(A1(·,z)) and h2(v)∈dom(A2(·,t)) for all u,v∈ℋ. For any given constants ρ,η>0, we consider the problem of finding (x*,y*)∈ℋ×ℋ such that
(2.4)0∈g2(x*)-g1(y*)+ρ(T1(y*,x*)+A1(g2(x*),x*)),0∈h2(y*)-h1(x*)+η(T2(x*,y*)+A2(h2(y*),y*)).
The problem (2.4) is called a system of extended general nonlinear variational inclusions involving eight different nonlinear operators (SEGNVID).
Some special cases of the SEGNVID (2.4) are as below.
IfTi:ℋ→ℋ and Ai=A:ℋ⊸ℋ(i=1,2) are univariate nonlinear operators, then taking g1=g, g2=g1, h1=h, and h2=h, the system (2.4) reduces to the system of finding (x*,y*)∈ℋ×ℋ such that
(2.5)0∈g1(x*)-g(y*)+ρ(T1(y*)+A(g1(x*))),0∈h1(y*)-h(x*)+η(T2(x*)+A(h1(y*))),
which is called the system of general nonlinear variational inclusions involving seven different nonlinear operators.Remark 2.2.
M. A. Noor and K. I. Noor [44] considered the system (2.5) where A:ℋ→ℋ is a single-valued operator. In view of the presented definitions and results in [44], we infer that the operator A in the system (2.5) (the system (1) in [44]) should be set valued not single valued and also be satisfied in the conditions Rangeg1⋂domA≠∅ and Rangeh1⋂domA≠∅. Therefore, if the mentioned corrections applied on the system (1) in [44], then the system (1) in [44] reduces to the system (2.5) which is a special case of the system (2.4).
Takingg1=g and h1=h in the system (2.5), the mentioned system collapses to the system of finding (x*,y*)∈ℋ×ℋ such that
(2.6)0∈T1(y*)+A(g(x*)),0∈T2(x*)+A(h(y*)).
The problem (2.6) is called the nonlinear variational inclusions system involving five different nonlinear operators.
If, for eachi=1,2, Ti=T, g1=h1=h=g, ρ=η, and x*=y*=x, then the system (2.5) reduces to the variational inclusion problem or finding the zero of the sum of two (more) monotone operators considered in [48–51].
If, for eachi=1,2, Ai:ℋ⊸ℋ is an univariate set-valued operator, A1(x)=∂φ(x) and A2(x)=∂ϕ(x) for all x∈ℋ, where φ,ϕ:ℋ→ℝ∪{+∞} are two proper, convex, and lower semi-continuous functionals, ∂φ and ∂ϕ denote subdifferential operators of φ and ϕ, respectively, then the system (2.4) reduces to the following system.
Find(x*,y*)∈ℋ×ℋ such that
(2.7)〈ρT1(y*,x*)+g2(x*)-g1(y*),g1(x)-g2(x*)〉≥ρφ(g2(x*))-ρφ(g1(x)),∀x∈H,〈ηT2(x*,y*)+h2(y*)-h1(x*),h1(x)-h2(y*)〉≥ηϕ(h2(y*))-ηϕ(h1(x)),∀x∈H,
which appears to be a new system of extended general mixed nonlinear variational inequalities involving eight different operators.
Ifg1=h1=g, g2=h2≡I (the identity operator), and φ=ϕ, then the system (2.7) is equivalent to that of finding (x*,y*)∈ℋ×ℋ such that
(2.8)〈ρT1(y*,x*)+x*-g(y*),g(x)-x*〉≥ρφ(x*)-ρφ(g(x)),∀x∈H,〈ηT2(x*,y*)+y*-g(x*),g(x)-y*〉≥ηφ(y*)-ηφ(g(x)),∀x∈H,
which was considered and studied by Noor [29]. Also, if T1=T2=T, then the system (2.8) is considered and studied in [29].
IfTi:ℋ→ℋ(i=1,2) are univariate nonlinear operaotrs and φ=ϕ, then the system (2.7) changes into that of finding (x*,y*)∈ℋ×ℋ such that
(2.9)〈ρT1(y*)+g2(x*)-g1(y*),g1(x)-g2(x*)〉≥ρφ(g2(x*))-ρφ(g1(x)),∀x∈H,〈ηT2(x*)+h2(y*)-h1(x*),h1(x)-h2(y*)〉≥ηφ(h2(y*))-ηφ(h1(x)),∀x∈H,
which was considered and investigated by M. A. Noor and K. I. Noor [44]. Also, if T1=T2=T, then the system (2.9) is considered and studied in [44]. When g1=h1=g and g2=h2≡I, the system (2.9) is introduced and studied in [29].
If, in the system (2.7), φ=ϕ=δK is the indicator function of a nonempty closed convex set K in ℋ defined by
(2.10)δK(y)={0y∈K,∞y∉K,
then the system (2.7) reduces to the system of finding (x*,y*)∈K×K such that
(2.11)〈ρT1(y*,x*)+g2(x*)-g1(y*),g1(x)-g2(x*)〉≥0,∀x∈H:g1(x)∈K,〈ηT2(x*,y*)+h2(y*)-h1(x*),h1(x)-h2(y*)〉≥0,∀x∈H:h1(x)∈K,
which has been introduced and considered by M. A. Noor and K. I. Noor [30].Remark 2.3.
Wheng1=h1=g and g2=h2≡I, the system (2.11) is considered and studied by Noor [52]. Also, if, for each i=1,2, Ti:ℋ→ℋ is an univariate nonlinear operator and gi=hi=g, then the system (2.11) is considered and studied by Yang et al. [53]. If, for each i=1,2, gi=hi≡I, then the system (2.11) is considered and studied by Huang and Noor [28]. If for each i=1,2, Ti=T and gi=hi≡I, then the system (2.11) introduced and studied by Chang et al. [26] and Verma [54]. If for each i=1,2, Ti=T, g1=h1=g, and g2=h2≡I, then the system (2.11) is considered and studied by Noor [52]. If for each i=1,2, Ti=T:ℋ→ℋ is an univariate nonlinear operator and gi=hi≡I, then the system (2.11) introduced and studied by Verma [55, 56]. If for each i=1,2, Ti=T:ℋ→ℋ is an univariate nonlinear operator, gi=hi≡I and x*=y*, then the system (2.11) reduces to the classical variational inequality introduced and studied by Stampacchia [57] in 1964. Other special cases of the above systems can be found in [29, 44] and the references therein. In brief, for suitable and appropriate choice of the operators Ti, Ai, gi, hi(i=1,2), and the constants ρ and η, one can obtain the various classes of variational inclusions and variational inequalities. This shows that the system of extended general nonlinear variational inclusions involving eight different operators (2.4) is more general and includes several classes of variational inclusions/inequalities and related optimization problems as special cases. For the recent applications, numerical methods and formulations of variational inequalities and variational inclusions, see [1–45, 47–62], and the references therein.


## 3. Existence of Solution and Uniqueness

In this section, we prove the existence and uniqueness theorem for a solution of the system of extended general nonlinear variational inclusions (2.4). For this end, we need the following lemma in which, by using resolvent operator technique, the equivalence between the system of extended general nonlinear variational inclusions (2.4) and a fixed point problem is proved.Lemma 3.1.
LetTi, Ai, gi, hi(i=1,2), ρ, and η be the same as in the system (2.4). Then (x*,y*)∈ℋ×ℋ is a solution of the system (2.4) if and only if
(3.1)g2(x*)=JA1(⋅,x*)ρ(g1(y*)-ρT1(y*,x*)),h2(y*)=JA2(⋅,y*)η(h1(x*)-ηT2(x*,y*)),
where for all z,t∈ℋ, JA1(.,z)ρ is the resolvent operator associated with A1(.,z) of parameter ρ and JA2(.,t)η is the resolvent operator associated with A2(.,t) of parameter η.Proof.
Let(x*,y*)∈ℋ×ℋ be a solution of the system (2.4). Then
(3.2)g1(y*)-ρT1(y*,x*)∈(I+ρA1(⋅,x*))(g2(x*)),h1(x*)-ηT2(x*,y*)∈(I+ηA2(⋅,y*))(h2(y*)),⟺g2(x*)=JA1(⋅,x*)ρ(g1(y*)-ρT1(y*,x*)),h2(y*)=JA2(⋅,y*)η(h1(x*)-ηT2(x*,y*)),
where I is identity operator.Definition 3.2.
A nonlinear single-valued operatorg:ℋ→ℋ is said to be(a)
monotone if
(3.3)〈g(x)-g(y),x-y〉≥0,∀x,y∈H;(b)
κ-strongly monotone if there exists a constant κ>0 such that
(3.4)〈g(x)-g(y),x-y〉≥κ‖x-y‖2,∀x,y∈H;(c)
(ϱ,ν)-relaxed cocoercive if there exist constants ϱ,ν>0 such that
(3.5)〈g(x)-g(y),x-y〉≥-ϱ‖g(x)-g(y)‖2+ν‖x-y‖2,∀x,y∈H;(d)
γ-Lipschitz continuous if there exists a constant γ>0 such that
(3.6)‖g(x)-g(y)‖≤γ‖x-y‖,∀x,y∈H.Definition 3.3.
A nonlinear single-valued operatorT:ℋ×ℋ→ℋ is called(a)
monotone in the first variable if for all x,y∈ℋ(3.7)〈T(x,u)-T(y,v),x-y〉≥0,∀u,v∈H;(b)
r-strongly monotone in the first variable if there exists a constant r>0 such that for all x,y∈ℋ(3.8)〈T(x,u)-T(y,v),x-y〉≥r‖x-y‖2,∀u,v∈H;(c)
(ξ,ς)-relaxed cocoercive in the first variable if there exist constants ξ,ς>0 such that for all x,y∈ℋ(3.9)〈T(x,u)-T(y,v),x-y〉≥-ξ‖T(x,u)-T(y,v)‖2+ς‖x-y‖2,∀u,v∈H;(d)
μ-Lipschitz continuous in the first variable if there exists a constant μ>0 such that for all x,y∈ℋ(3.10)‖T(x,u)-T(y,v)‖≤μ‖x-y‖,∀u,v∈H.
Now, we present the sufficient conditions for the existence solutions of our main considered problem (2.4).Theorem 3.4.
LetTi, Ai, gi, hi(i=1,2), ρ, and η be the same as in the system (2.4) and suppose further that, for i=1,2,(a)
Ti is ξi-strongly monotone and μi-Lipschitz continuous in the first variable;(b)
gi is ςi-strongly monotone and σi-Lipschitz continuous;(c)
hi is ϱi-strongly monotone and δi-Lipschitz continuous;(d)
there exists constantsτi such that
(3.11)‖JA1(⋅,u)ρ(z)-JA1(⋅,v)ρ(z)‖≤τ1‖u-v‖,‖JA2(⋅,u)η(z)-JA2(⋅,v)η(z)‖≤τ2‖u-v‖,∀u,v,z∈H.If two constants ρandη satisfy the following conditions
(3.12)|ρ-ξ1μ12|<ξ12-μ12μ(2-μ)μ12,|η-ξ2μ22|<ξ22-μ22ν(2-ν)μ22,ξ1>μ1μ(2-μ),ξ2>μ2ν(2-ν),κi=1-2ςi+σi2<1,2ςi<1+σi2,ωi=1-2ϱi+δi2<1,2ϱi<1+δi2,μ=τ2+ω2+κ1<1,ν=τ1+ω1+κ2<1,i=1,2,
then the system (2.4) admits a unique solution.Proof .
Firstly, let us defineψ,ϕ:ℋ×ℋ→ℋ by
(3.13)ψ(x,y)=x-g2(x)+JA1(⋅,x)ρ(g1(y)-ρT1(y,x)),ϕ(x,y)=y-h2(y)+JA2(⋅,y)η(h1(x)-ηT2(x,y)),
for all (x,y)∈ℋ×ℋ.
Also, defineF:ℋ×ℋ→ℋ×ℋ as follows:
(3.14)F(x,y)=(ψ(x,y),ϕ(x,y)),∀(x,y)∈H×H.
Consider a function ∥·∥* on ℋ×ℋ which is defined by
(3.15)‖(x,y)‖*=‖x‖+‖y‖,∀(x,y)∈H×H.
It is obvious that (ℋ×ℋ,∥·∥*) is a Hilbert space. Now, we establish that F is a contraction mapping on (ℋ×ℋ,∥·∥*). Let (x,y),(x^,y^)∈ℋ×ℋ be given. Since for all z∈ℋ, the resolvent operator JA1(·,z)ρ is nonexpansive, we have
(3.16)‖ψ(x,y)-ψ(x^,y^)‖≤‖x-x^-(g2(x)-g2(x^))‖+‖JA1(⋅,x)ρ(g1(y)-ρT1(y,x))-JA1(⋅,x^)ρ(g1(y^)-ρT1(y^,x^))‖≤‖x-x^-(g2(x)-g2(x^))‖+‖JA1(⋅,x)ρ(g1(y)-ρT1(y,x))-JA1(⋅,x^)ρ(g1(y)-ρT1(y,x))‖+‖JA1(⋅,x^)ρ(g1(y)-ρT1(y,x))-JA1(⋅,x^)ρ(g1(y^)-ρT1(y^,x^))‖≤‖x-x^-(g2(x)-g2(x^))‖+τ1‖x-x^‖+‖g1(y)-g1(y^)-ρ(T1(y,x)-T1(y^,x^))‖≤‖x-x^-(g2(x)-g2(x^))‖+τ1‖x-x^‖+‖y-y^-(g1(y)-g1(y^))‖+‖y-y^-ρ(T1(y,x)-T1(y^,x^))‖.
From ς2-strongly monotonicity and σ2-Lipschitz continuity of g2, it follows that
(3.17)‖x-x^-(g2(x)-g2(x^))‖2=‖x-x^‖2-2〈g2(x)-g2(x^),x-x^〉+‖g2(x)-g2(x^)‖2≤(1-2ς2+σ22)‖x-x^‖2,
which leads to
(3.18)‖x-x^-(g2(x)-g2(x^))‖≤1-2ς2+σ22‖x-x^‖.
In similar way, by using ς1-strongly monotonicity and σ1-Lipschitz continuity of g1, we deduce that
(3.19)‖y-y^-(g1(y)-g1(y^))‖≤1-2ς1+σ12‖y-y^‖.
Since T1 is ξ1-strongly monotone and μ1-Lipschitz continuous in the first variable, we conclude that
(3.20)‖y-y^-ρ(T1(y,x)-T1(y^,x^))‖2=‖y-y^‖2-2ρ〈T1(y,x)-T1(y^,x^),y-y^〉+ρ2‖T1(y,x)-T1(y^,x^)‖2≤(1-2ρξ1+ρ2μ12)‖y-y^‖2.
Substituting (3.18)–(3.20) in (3.16), we obtain
(3.21)‖ψ(x,y)-ψ(x^,y^)‖≤(τ1+κ2)‖x-x^‖+(κ1+θ1)‖y-y^‖,
where
(3.22)κi=1-2ςi+σi2,(i=1,2),θ1=1-2ρξ1+ρ2μ12.
On the other hand, since fori=1,2, hi is ϱi-strongly monotone and δi-Lipschitz continuous in the first variable, T2 is ξ2-strongly monotone and μ2-Lipschitz continuous in the first variable, in similar way to the proofs of (3.16)–(3.21), we can prove that
(3.23)‖ϕ(x,y)-ϕ(x^,y^)‖≤(ω1+θ2)‖x-x^‖+(τ2+ω2)‖y-y^‖,
where
(3.24)ωi=1-2ϱi+δi2,(i=1,2),θ2=1-2ηξ2+η2μ22.
It follows from (3.14), (3.21), and (3.23) that
(3.25)‖F(x,y)-F(x^,y^)‖*=‖ψ(x,y)-ψ(x^,y^)‖+‖ϕ(x,y)-ϕ(x^,y^)‖≤(τ1+ω1+κ2+θ2)‖x-x^‖+(τ2+ω2+κ1+θ1)‖y-y^‖≤ϑ‖(x,y)-(x^,y^)‖*,
where ϑ=max{τ1+ω1+κ2+θ2,τ2+ω2+κ1+θ1}. By condition (3.12), we note that 0≤ϑ<1, and so (3.25) guarantees that F is a contraction mapping. According to Banach fixed point theorem, there exists a unique point (x*,y*)∈ℋ×ℋ such that F(x*,y*)=(x*,y*). From (3.13) and (3.14), we conclude that g2(x*)=JA1(·,x*)ρ(g1(y*)-ρT1(y*,x*)) and h2(y*)=JA2(·,y*)η(h1(x*)-ηT2(x*,y*)). Now, Lemma 3.1 guarantees that (x*,y*)∈ℋ×ℋ is the unique solution of the system (2.4). This completes the proof.


## 4. Some New Resolvent Iterative Algorithms

In recent years, the nonexpansive mappings have been generalized and investigated by various authors. One of these generalizations is class of the nearly uniformly Lipschitzian mappings. In this section, we first recall some generalizations of the nonexpansive mappings which have been introduced in recent years, then we use two nearly uniformly Lipschitzian mappingsS1 and S2 and by using the equivalent alternative formulation (3.1), we suggest and analyze a new resolvent iterative algorithm for finding an element of the set of the fixed points of 𝒬=(S1,S2) which is the unique solution of the SEGNVID (2.4).In two next definitions, several generalizations of the nonexpansive mappings are stated.Definition 4.1.
A nonlinear mappingT:ℋ→ℋ is called as follows:(a)
nonexpansive if ∥Tx-Ty∥≤∥x-y∥, for all x,y∈ℋ;(b)
L-Lipschitzian if there exists a constant L>0 such that
(4.1)‖Tx-Ty‖≤L‖x-y‖,∀x,y∈H;(c)
generalized Lipschitzian if there exists a constant L>0 such that
(4.2)‖Tx-Ty‖≤L(‖x-y‖+1),∀x,y∈H;(d)
generalized(L,M)-Lipschitzian [46] if there exist two constants L,M>0 such that
(4.3)‖Tx-Ty‖≤L(‖x-y‖+M),∀x,y∈H;(e)
asymptotically nonexpansive [63] if there exists a sequence {kn}⊆[1,∞) with limn→∞kn=1 such that for each n∈ℕ,
(4.4)‖Tnx-Tny‖≤kn‖x-y‖,∀x,y∈H;(f)
pointwise asymptotically nonexpansive [64] if, for each integer n≥1,
(4.5)‖Tnx-Tny‖≤αn(x)‖x-y‖,x,y∈H,
where αn→1 pointwise on X;(g)
uniformlyL-Lipschitzian if there exists a constant L>0 such that for each n∈ℕ,
(4.6)‖Tnx-Tny‖≤L‖x-y‖,∀x,y∈H.Definition 4.2 (see [46]).
A nonlinear mappingT:ℋ→ℋ is said to be(a)
nearly Lipschitzian with respect to the sequence {an} if, for each n∈ℕ, there exists a constant kn>0 such that
(4.7)‖Tnx-Tny‖≤kn(‖x-y‖+an),∀x,y∈H,
where {an} is a fix sequence in [0,∞) with an→0, as n→∞.For an arbitrary, but fixedn∈ℕ, the infimum of constants kn in (4.7) is callednearly Lipschitz constant and is denoted by η(Tn). Notice that
(4.8)η(Tn)=sup⁡{‖Tnx-Tny‖‖x-y‖+an:x,y∈H,x≠y}.A nearly Lipschitzian mappingT with the sequence {(an,η(Tn))} is said to be(b)
nearly nonexpansive if η(Tn)=1 for all n∈ℕ, that is,
(4.9)‖Tnx-Tny‖≤‖x-y‖+an,∀x,y∈H;(c)
nearly asymptotically nonexpansive if η(Tn)≥1 for all n∈ℕ and limn→∞η(Tn)=1, in other words, kn≥1 for all n∈ℕ with limn→∞kn=1;(d)
nearly uniformlyL-Lipschitzian if η(Tn)≤L for all n∈ℕ, in other words, kn=L for all n∈ℕ.For some interesting examples to investigate relations between these mappings, introduced in Definitions4.1 and 4.2, ones may consult [58].LetS1:ℋ→ℋ be a nearly uniformly L1-Lipschitzian mapping with the sequence {an}n=1∞ and S2:ℋ→ℋ be a nearly uniformly L2-Lipschitzian mapping with the sequence {bn}n=1∞. We define the self-mapping 𝒬 of ℋ×ℋ as follows:
(4.10)Q(x,y)=(S1x,S2y),∀x,y∈H.
Then 𝒬=(S1,S2):ℋ×ℋ→ℋ×ℋ is a nearly uniformly max{L1,L2}-Lipschitzian mapping with the sequence {an+bn}n=1∞ with respect to the norm ∥·∥* in ℋ×ℋ, where ∥·∥* is defined as (3.15). Because, for any (x,y),(x′,y′)∈ℋ×ℋ and n∈ℕ, we have
(4.11)‖Qn(x,y)-Qn(x′,y′)‖*=‖(S1nx,S2ny)-(S1nx′,S2ny′)‖*=‖(S1nx-S1nx′,S2ny-S2ny′)‖*=‖S1nx-S1nx′‖+‖S2ny-S2ny′‖≤L1(‖x-x′‖+an)+L2(‖y-y′‖+bn)≤max⁡{L1,L2}(‖x-x′‖+‖y-y′‖+an+bn)=max⁡{L1,L2}(‖(x,y)-(x′,y′)‖*+an+bn).
We denote the sets of all the fixed points of Si(i=1,2) and 𝒬 by Fix(Si) and Fix(𝒬), respectively, and the set of all the solutions of the system (2.4) by SEGNVID(ℋ,Ti,Ai,gi,hi,i=1,2). In view of (4.10), for any (x,y)∈ℋ×ℋ, we see that (x,y)∈Fix(𝒬) if and only if x∈Fix(S1) and y∈Fix(S2). That is, Fix(𝒬)=Fix(S1,S2)=Fix(S1)×Fix(S2). We now characterize the following problem: if (x*,y*)∈Fix(𝒬)∩SEGNVID(ℋ,Ti,Ai,gi,hi,i=1,2), then by using Lemma 3.1, it is easy to see that for each n∈ℕ,
(4.12)x*=S1nx*=x*-g2(x*)+JA1(⋅,x*)ρ(g1(y*)-ρT1(y*,x*))=S1n[x*-g2(x*)+JA1(⋅,x*)ρ(g1(y*)-ρT1(y*,x*))],y*=S2ny*=y*-h2(y*)+JA2(⋅,y*)η(h1(x*)-ηT2(x*,y*))=S2n[y*-h2(y*)+JA2(⋅,y*)η(h1(x*)-ηT2(x*,y*))].The fixed point formulation (4.12) enables us to suggest the following iterative algorithm with mixed errors for finding an element of the set of the fixed points of the nearly uniformly Lipschitzian mapping 𝒬=(S1,S2) which is the unique solution of the system of extended general nonlinear variational inclusions (2.4).Algorithm 4.3.
LetTi, Ai, gi, hi(i=1,2), ρ, and η be the same as in the system (2.4). For an arbitrary chosen initial point (x1,y1)∈ℋ×ℋ, compute the iterative sequence {(xn,yn)}n=1∞ in the following way:
(4.13)xn+1=(1-αn-βn)xn+αnS1n[xn-g2(xn)+JA1(⋅,xn)ρ(g1(yn)-ρT1(yn,xn))]+αnen+βnjn+rn,yn+1=(1-αn-βn)yn+αnS2n[yn-h2(yn)+JA2(⋅,yn)η(h1(xn)-ηT2(xn,yn))]+αnpn+βnsn+ln,
where S1,S2:ℋ→ℋ are two nearly uniformly Lipschitzian mappings, {αn}n=1∞, {βn}n=1∞ are two sequences in interval [0,1] such that ∑n=1∞αn=∞, αn+βn≤1, ∑n=1∞βn<∞ and {en}n=1∞, {pn}n=1∞, {jn}n=1∞, {sn}n=1∞, {rn}n=1∞, {ln}n=1∞ are six sequences in ℋ to take into account a possible inexact computation of the resolvent operator point satisfying the following conditions: {jn}n=1∞, {sn}n=1∞ are two bounded sequences in ℋ, and {en}n=1∞, {pn}n=1∞, {rn}n=1∞, {ln}n=1∞ are four sequences in ℋ such that
(4.14)en=en′+en′′,pn=pn′+pn′′,n∈N,lim⁡n→∞‖(en′,pn′)‖*=0,∑n=1∞‍‖(en′′,pn′′)‖*<∞,∑n=1∞‍‖rn,ln‖*<∞.Remark 4.4.
If, for eachi=1,2, Si≡I, then Algorithm 4.3 reduces to the following iterative algorithm for solving the system (2.4).Algorithm 4.5.
Suppose thatTi, Ai, gi, hi(i=1,2), ρ, and η are the same as in the system (2.4). For an arbitrary chosen initial point (x1,y1)∈ℋ×ℋ, compute the iterative sequence {(xn,yn)}n=1∞ in the following way:
(4.15)xn+1=(1-αn-βn)xn+αn[xn-g2(xn)+JA1(⋅,xn)ρ(g1(yn)-ρT1(yn,xn))]+αnen+βnjn+rn,yn+1=(1-αn-βn)yn+αn[yn-h2(yn)+JA2(⋅,yn)η(h1(xn)-ηT2(xn,yn))]+αnpn+βnsn+ln,
where the sequences {αn}n=1∞, {βn}n=1∞, {en}n=1∞, {pn}n=1∞, {jn}n=1∞, {sn}n=1∞, {rn}n=1∞, and {ln}n=1∞ are the same as in Algorithm 4.3.


## 5. Convergence Analysis

In this section, under some suitable conditions, we establish the strong convergence of the sequence generated by iterative Algorithm4.3. We need the following lemma to prove our main result.Lemma 5.1.
Let{an}, {bn}, and {cn} be three nonnegative real sequences satisfying the following condition. There exists a natural number n0 such that
(5.1)an+1≤(1-tn)an+bntn+cn,∀n≥n0,
where tn∈[0,1], ∑n=0∞tn=∞, limn→∞bn=0, ∑n=0∞cn<∞. Then limn→∞an=0.Proof.
The proof directly follows from Lemma 2 in Liu [59].Theorem 5.2.
LetTi, Ai, gi, hi(i=1,2), ρ and η be the same as in Theorem 3.4 and let all the conditions Theorem 3.4 hold. Assume that S1:ℋ→ℋ is a nearly uniformly L1-Lipschitzian mapping with the sequence {bn}n=1∞, S2:ℋ→ℋ is a nearly uniformly L2-Lipschitzian mapping with the sequence {cn}n=1∞, and the self-mapping 𝒬 of ℋ×ℋ is defined by (4.10) such that Fix(𝒬)∩SEGNVID(ℋ,Ti,Ai,gi,hi,i=1,2)≠∅. Furthermore, let, for each i=1,2, Liϑ<1, where ϑ is the same as in (3.25). Then the iterative sequence {(xn,yn)}n=1∞, generated by Algorithm 4.3, converges strongly to the only element of Fix(𝒬)∩SEGNVID(ℋ,Ti,Ai,gi,hi,i=1,2).Proof.
According to Theorem3.4, the system (2.4) has a unique solution (x*,y*) in ℋ×ℋ, and so Lemma 3.1 implies that (x*,y*) satisfies (3.1). Since SEGNVID(ℋ,Ti,Ai,gi,hi,i=1,2) is a singleton set and Fix(𝒬)⋂SEGNVID(ℋ,Ti,Ai,gi,hi,i=1,2)≠∅, we conclude that x*∈Fix(S1) and y*∈Fix(S2). Therefore, for each n∈ℕ, we can write
(5.2)x*=(1-αn-βn)x*+αnS1n[x*-g2(x*)+JA1(⋅,x*)ρ(g1(y*)-ρT1(y*,x*))]+βnx*,y*=(1-αn-βn)y*+αnS2n[y*-h2(y*)+JA2(⋅,y*)η(h1(x*)-ηT2(x*,y*))]+βny*,
where the sequences {αn}n=1∞ and {βn}n=1∞ are the same as in Algorithm 4.3. Let K=supn≥1{∥jn-x*∥,∥sn-y*∥}. Then, by using (4.13), (5.2), and the assumptions, we have
(5.3)‖xn+1-x*‖≤(1-αn-βn)‖xn-x*‖+αn‖S1n[xn-g2(xn)+JA1(⋅,xn)ρ(g1(yn)-ρT1(yn,xn))]-S1n[x*-g2(x*)+JA1(.,x*)ρ(g1(y*)-ρT1(y*,x*))]‖+αn‖en‖+βn‖jn-x*‖+‖rn‖≤(1-αn-βn)‖xn-x*‖+αnL1({+‖JA1(.,xn)ρ(g1(yn)-ρT1(yn,xn))-JA1(.,x*)ρ(g1(y*)-ρT1(y*,x*))‖+bn}‖xn-x*-(g2(xn)-g2(x*))‖+‖JA1(⋅,xn)ρ(g1(yn)-ρT1(yn,xn))-JA1(⋅,x*)ρ(g1(y*)-ρT1(y*,x*))‖+bn)+αn(‖en′‖+‖en′′‖)+‖rn‖+βnK≤(1-αn-βn)‖xn-x*‖+αnL1({+‖JA1(.,x*)ρ(g1(yn)-ρT1(yn,xn))-JA1(.,x*)ρ(g1(y*)-ρT1(y*,x*))‖+bn}‖xn-x*-(g2(xn)-g2(x*))‖+‖JA1(⋅,xn)ρ(g1(yn)-ρT1(yn,xn))-JA1(⋅,x*)ρ(g1(yn)-ρT1(yn,xn))‖+‖JA1(⋅,x*)ρ(g1(yn)-ρT1(yn,xn))-JA1(⋅,x*)ρ(g1(y*)-ρT1(y*,x*))‖+bn)+αn‖en′‖+‖en′′‖+‖rn‖+βnK≤(1-αn-βn)‖xn-x*‖+αnL1(‖xn-x*-(g2(xn)-g2(x*))‖+τ1‖xn-x*‖+‖g1(yn)-g1(y*)-ρ(T1(yn,xn)-T1(y*,x*))‖+bn)+αn‖en′‖+‖en′′‖+‖rn‖+βnK≤(1-αn-βn)‖xn-x*‖+αnL1((τ1+κ2)‖xn-x*‖+(κ1+θ1)‖yn-y*‖+bn)+αn‖en′‖+‖en′′‖+‖rn‖+βnK,
where κi(i=1,2) and θ1 are the same as in (3.21).
In similar way to the proof of (5.3), one can establish that
(5.4)‖yn+1-y*‖≤(1-αn-βn)‖yn-y*‖+αnL2((ω1+θ2)‖xn-x*‖+(τ2+ω2)‖yn-y*‖+cn)+αn‖pn′‖+‖pn′′‖+‖ln‖+βnK,
where ωi(i=1,2) and θ2 are the same as in (3.23). Letting L=max{L1,L2} and applying (5.3) and (5.4), we obtain that
(5.5)‖(xn+1,yn+1)-(x*,y*)‖*≤(1-αn-βn)‖(xn,yn)-(x*,y*)‖*+αnLϑ‖(xn,yn)-(x*,y*)‖*+αnL(bn+cn)+αn‖(en′,pn′)‖*+‖(en′′,pn′′)‖*+‖(rn,ln)‖*+2βnK≤(1-(1-Lϑ)αn)‖(xn,yn)-(x*,y*)‖*+αnϑ‖(en′,pn′)‖*+L(bn+cn)ϑ+‖(en′′,pn′′)‖*+‖(rn,ln)‖*+2βnK,
where ϑ is the same as in (3.25). Since limn→∞bn=limn→∞cn=0 and ∑n=1∞βn<∞, in view of (4.14), it is obvious that the conditions of Lemma 5.1 are satisfied. Now, Lemma 5.1 and (5.5) guarantee that (xn,yn)→(x*,y*), as n→∞. Therefore, the sequence {(xn,yn)}n=1∞, generated by Algorithm 4.3, converges strongly to the only element (x*,y*) of Fix(𝒬)∩SEGNVID(ℋ,Ti,Ai,gi,hi,i=1,2). This completes the proof.Like in the proof of Theorem5.2, one can prove the strong convergence of the iterative sequence generated by Algorithm 4.5, and we omit its proof.Theorem 5.3.
Suppose thatTi, Ai, gi, hi(i=1,2), ρ, and η are the same as in Theorem 3.4 and let all the conditions Theorem 3.4 hold. Then the iterative sequence {(xn,yn)}n=1∞, generated by Algorithm 4.5, converges strongly to the unique solution (x*,y*) of the system (2.4).


## 6. Some Comments on Related Works

This section is devoting to investigate and analyze the results in [29, 44]. We state some remarks on main results in [29] and also the explicit iterative forms, which are related to Algorithms 3.1 and 3.2 from [44], are constructed. The incorrectness of Theorem 4.1 from [44] is discussed. Furthermore, the correct versions of the aforesaid algorithms and theorem are presented.Noor [29] proposed the following two-step iterative algorithm for solving the system of general mixed variational inequalities (2.8) and studied convergence analysis of the proposed iterative algorithm under some certain conditions.Algorithm 6.1 (see [29, Algorithm 3.1]).
For arbitrary chosen initial pointsx0,y0∈ℋ, compute the sequences {xn} and {yn} by
(6.1)xn+1=(1-an)xn+anJφ[g(yn)-ρT1(yn,xn)],yn+1=Jφ[g(xn+1)-ηT2(xn+1,yn)],
where an∈[0,1] for all n≥0.Theorem 6.2 (see [29, Theorem 3.1]).
Letx*, y* be the solution of SGMVID  (2.8). Suppose that T1:ℋ×ℋ→ℋ is relaxed (γ1,r1)-cocoercive and μ1-Lipschitzian in the first variable, and T2:ℋ×ℋ→ℋ is relaxed (γ2,r2)-cocoercive and μ2-Lipschitzian in the first variable. Let g be relaxed (γ3,r3)-cocoercive and μ3-Lipschitzian. If
(6.2)|ρ-r1-γ1μ12μ12|<(r1-γ1μ12)2-μ12k(2-k)μ12,r1>γ1μ12+μ1k(2-k),k<1,|η-r2-γ2μ22μ22|<(r2-γ2μ22)2-μ22k(2-k)μ22,r2>γ2μ22+μ2k(2-k),k<1,
where
(6.3)k=1-2(r3-γ3μ32)+μ32,
and an∈[0,1], ∑n=0∞‍an=∞, then for arbitrarily chosen initial points x0,y0∈ℋ, xn and yn obtained from Algorithm 6.1 converge strongly to x* and y*, respectively.By using Definition2.1, we note that the condition relaxed cocoercivity of the operator T is weaker than the condition of strong monotonicity of T. In other words, the class of relaxed cocoercive operators is more general than the class of strongly monotone operators. Now, we show that, unlike claim of Noor [29], he studied the convergence analysis of the proposed iterative algorithm under the condition of strong monotonicity, not the mild condition relaxed cocoercivity.Remark 6.3.
In view of the conditions (6.2) (the conditions (4.1) and (4.2) in [29]), we have k∈(0,1). The condition (6.3) (the condition (4.3) in [29]) and k>0 imply that 2(r3-γ3μ32)<1+μ32. Therefore, the condition 2(r3-γ3μ32)<1+μ32 should be added to the conditions (6.2)-(6.3). On the other hand, since k<1 from the condition (6.3), it follows that r3>γ3μ32. The conditions ri>γiμi2+μik(2-k) (i=1,2), and k<1 imply that ri>γiμi2 for each i=1,2. Since, for each i=1,2, the operator Ti is (γi,ri)-relaxed cocoercive and μi-Lipschitz continuous, the conditions ri>γiμi2(i=1,2) guarantee that, for each i=1,2, the operator Ti is (ri-γiμi2) strongly monotone. Similarly, since g is (γ3,r3) relaxed cocoercive and μ3-Lipschitz continuous, the condition r3>γ3μ32 implies that the operator g is (r3-γ3μ32)-strongly monotone.In view of the above remark, one can rewrite Theorem6.2 as follows.Theorem 6.4.
Letx* and y* be the solution of the SGMVID (2.8) and suppose that T1:ℋ×ℋ→ℋ is ξ1-strongly monotone and μ1-Lipschitz continuous in the first variable, and T2:ℋ×ℋ→ℋ is relaxed ξ2-strongly monotone and μ2-Lipschitz continuous in the first variable. Moreover, let g be ξ3-strongly monotone and μ3-Lipschitz continuous. If two constants ρ and η satisfy the following conditions:
(6.4)|ρ-ξ1μ12|<ξ12-μ12k(2-k)μ12,|η-ξ2μ22|<ξ22-μ22k(2-k)μ22,ξi>μik(2-k),(i=1,2),k=1-2ξ3+μ32<1,2ξ3<1+μ32,
and ∑n=0∞‍an=∞, then the iterative sequences {xn} and {yn} generated by Algorithm 6.1 converge strongly to x* and y*, respectively.M. A. Noor and K. I. Noor in [44] proposed the following iterative scheme for solving the system of general variational inclusions (1) from [44].Algorithm 6.5 (see [44, Algorithm 3.1]).
For arbitrary chosen initial pointsx0,y0∈K compute the sequences {xn} and {yn} by
(6.5)xn+1=(1-an)xn+an(xn+1-g1(xn+1))+anJA[g(yn)-ρT1(yn)],yn+1=yn+1-h1(yn+1)+JA[h(xn+1)-ηT2(xn+1)],
where an∈[0,1] for all n≥0 satisfies some suitable conditions.Takingg1=g and h1=h, Algorithm 6.5 reduces to the following iterative algorithm.Algorithm 6.6 (see [44, Algorithm 3.2]).
For arbitrary chosen initial pointsx0,y0∈K compute the sequences {xn} and {yn} by
(6.6)xn+1=(1-an)xn+an(xn+1-g(xn+1))+anJA[g(yn)-ρT1(yn)],yn+1=yn+1-h(yn+1)+JA[h(xn+1)-ηT2(xn+1)],
where an∈[0,1] for all n≥0 satisfies some suitable conditions.Remark 6.7.
By analyzing two Algorithms6.5 and 6.6, we note that the mentioned algorithms are in implicit forms. Further, in view of Remark 2.2, we should apply the system (2.5) instead of the system (1) from [44].
Next, we derive two explicit algorithms to solve the systems (2.5) and (2.6), respectively, as follows.Algorithm 6.8.
LetT1, T2, A, g, h, g1, and h1 be the same as in the system (2.5), and let h be an onto operator. For arbitrary chosen initial points x0,y0∈K compute the sequences {xn} and {yn} in the following way:
(6.7)xn+1=(1-an)xn+an(xn-g1(xn)+JA(g(yn+1)-ρT1(yn+1))),h1(yn+1)=JA(h(xn)-ηT2(xn))
where an∈[0,1] for all n≥0 satisfies some suitable conditions.Algorithm 6.9.
LetT1, T2, A, g, h, g1, and h1 be the same as in the system (2.5). For arbitrary chosen initial points x0,y0∈K compute the sequences {xn} and {yn} in the following way:
(6.8)xn+1=(1-an)xn+an(xn-g1(xn)+JA(g(yn)-ρT1(yn))),yn+1=(1-an)yn+an(yn-h1(yn)+JA(h(xn)-ηT2(xn))),
where an∈[0,1] for all n≥0 satisfies some suitable conditions.Takingg1=g and h1=h in two Algorithms 6.8 and 6.9, we can obtain the following two algorithms.Algorithm 6.10.
For arbitrary chosen initial pointsx0,y0∈K compute the sequences {xn} and {yn} in the following way:
(6.9)xn+1=(1-an)xn+an(xn-g(xn)+JA(g(yn+1)-ρT1(yn+1))),h(yn+1)=JA(h(xn)-ηT2(xn)),
where an∈[0,1] for all n≥0 satisfies some suitable conditions.Algorithm 6.11.
For arbitrary chosen initial pointsx0,y0∈K compute the sequences {xn} and {yn} in the following way:
(6.10)xn+1=(1-an)xn+an(xn-g(xn)+JA(g(yn)-ρT1(yn))),yn+1=(1-an)yn+an(yn-h(yn)+JA(h(xn)-ηT2(xn))),
where an∈[0,1] for all n≥0 satisfies some suitable conditions.We now recall some facts, which has presented in [44].Lemma 6.12 (see [44, Lemma 3.1]).
If the operatorA is maximal monotone, then (x*,y*)∈ℋ×ℋ is a solution of (2.5) (the correct version of the system (1) in [44]) if and only if (x*,y*)∈ℋ×ℋ satisfies
(6.11)g1(x*)=JAρ[g(y*)-ρT1(y*)],h1(y*)=JAη[h(x*)-ηT2(x*)].Remark 6.13.
In view of Lemma6.12, (x*,y*)∈ℋ×ℋ is a solution of the system (2.6) if and only if
(6.12)g(x*)=JAρ[g(y*)-ρT1(y*)],h(y*)=JAη[h(x*)-ηT2(x*)],
where ρ,η>0 are two constants.Theorem 6.14.
LetTi(i=1,2), g, and h be the same as in Algorithm 6.10, and let x*, y* be the solution of the system (2.6). Assume that for i=1,2, the operator Ti:ℋ→ℋ is ξi-strongly monotone and μi-Lipschitz continuous. Furthermore, let g be ξ3-strongly monotone and μ3-Lipschitz continuous and h be ξ4-strongly monotone and μ4-Lipschitz continuous. If there exist two constants ρ and η such that
(6.13)(k+θ1)(k1+θ2)<(1-k)(1-k1),
where
(6.14)k=1-2ξ3+μ32<1,2ξ3<1+μ32,k1=1-2ξ4+μ42<1,2ξ4<1+μ42,θ1=1-2ρξ1+ρ2μ12<1,2ρξ1<1+ρ2μ12,θ2=1-2ηξ2+η2μ22<1,2ηξ2<1+η2μ22
and ∑n=0∞‍an=∞, and then the sequences {xn} and {yn} generated by Algorithm 6.10 converge strongly to x* and y*, respectively.Proof.
Since(x*,y*)∈ℋ×ℋ is a solution of the system (2.6), in view of Remark 6.13, we have
(6.15)g(x*)=JAρ(g(y*)-ρT1(y*)),h(y*)=JAη(h(x*)-ηT2(x*)),
where ρandη are two constants. For each n≥0, one can rewrite the above equations as below:
(6.16)x*=(1-an)x*+an(x*-g(x*)+JAρ(g(y*)-ρT1(y*))),h(y*)=JAη(h(x*)-ηT2(x*)),
where the sequence {an} is the same as in Algorithm 6.10. It follows from (6.9), (6.16), and the nonexpansivity property of the resolvent operator JAρ, that
(6.17)‖xn+1-x*‖≤(1-an)‖xn-x*‖+an(‖xn-x*-(g(xn)-g(x*))‖+‖JAρ(g(yn+1)-ρT1(yn+1))-JAρ(g(y*)-ρT1(y*))‖)≤(1-an)‖xn-x*‖+an(‖xn-x*-(g(xn)-g(x*))‖+‖g(yn+1)-g(y*)-ρ(T1(yn+1)-T1(y*))‖)≤(1-an)‖xn-x*‖+an(‖xn-x*-(g(xn)-g(x*))‖+‖yn+1-y*-(g(yn+1)-g(y*))‖+‖yn+1-y*-ρ(T1(yn+1)-T1(y*))‖).
Since T1 is ξ1-strongly monotone and μ1-Lipschitz continuous, we have
(6.18)‖yn+1-y*-ρ(T1(yn+1)-T1(y*))‖2=‖yn+1-y*‖2-2ρ〈T1(yn+1)-T1(y*),yn+1-y*〉+ρ2‖T1(yn+1)-T1(y*)‖2≤(1-2ρξ1+ρ2μ12)‖yn+1-y*‖2
which leads to
(6.19)‖yn+1-y*-ρ(T1(yn+1)-T1(y*))‖≤1-2ρξ1+ρ2μ12‖yn+1-y*‖.
Since g is ξ3-strongly monotone and μ3-Lipschitz continuous, we get
(6.20)‖xn-x*-(g(xn)-g(x*))‖≤1-2ξ3+μ32‖xn-x*‖,(6.21)‖yn+1-y*-(g(yn+1)-g(y*))‖≤1-2ξ3+μ32‖yn+1-y*‖.
Combining (6.17)–(6.21), we get the following:
(6.22)‖xn+1-x*‖≤(1-an)‖xn-x*‖+ank‖xn-x*‖+an(k+θ1)‖yn+1-y*‖,
where k and θ1 are the same as in (6.14). Now, we find an estimation for ∥yn+1-y*∥. Using (6.9), (6.16), and the nonexpansivity property of the resolvent operator JAη, we obtain that
(6.23)‖yn+1-y*‖≤‖yn+1-y*-(h(yn+1)-h(y*))‖+‖h(yn+1)-h(y*)‖≤‖yn+1-y*-(h(yn+1)-h(y*))‖+‖JAη(h(xn)-ηT2(xn))-JAη(h(x*)-ηT2(x*))‖≤‖yn+1-y*-(h(yn+1)-h(y*))‖+‖h(xn)-h(x*)-η(T2(xn)-T2(x*))‖≤‖yn+1-y*-(h(yn+1)-h(y*))‖+‖xn-x*-(h(xn)-h(x*))‖+‖xn-x*-η(T2(xn)-T2(x*))‖.
Since T2 is ξ2-strongly monotone and μ2-Lipschitz continuous, and h is ξ4-strongly monotone and μ4-Lipschitz continuous, in similar way to the proofs of (6.19)–(6.21), we can establish that
(6.24)‖xn-x*-η(T2(xn)-T2(x*))‖≤1-2ηξ2+η2μ22‖xn-x*‖,‖xn-x*-(h(xn)-h(x*))‖≤1-2ξ4+μ42‖xn-x*‖,‖yn+1-y*-(h(yn+1)-h(y*))‖≤1-2ξ4+μ42‖yn+1-y*‖.
Substituting (6.24) in (6.23), we obtain that
(6.25)‖yn+1-y*‖≤k1‖yn+1-y*‖+(k1+θ2)‖xn-x*‖,
where k1 and θ2 are the same as in (6.14). From (6.25), we conclude that
(6.26)‖yn+1-y*‖≤k1+θ21-k1‖xn-x*‖.
It follows from (6.22) and (6.26) that
(6.27)‖xn+1-x*‖≤(1-an(1-k-(k+θ1)(k1+θ2)1-k1))‖xn-x*‖.
Letting ι=1-k-((k+θ1)(k1+θ2))/(1-k1), the condition (6.13) implies that ι∈(0,1). Since ∑n=0∞‍an=∞, setting bn=cn=0, for all n≥0, we note that all the conditions of Lemma 5.1 are satisfied. Now, Lemma 5.1 and (6.27) guarantee that ∥xn-x*∥*→0, as n→∞. The inequality (6.26) implies that ∥yn-y*∥*→0, as n→∞, and so the sequences {xn} and {yn} generated by Algorithm 6.10 converge strongly to x* and y*, respectively. This completes the proof.Theorem 6.15.
LetTi(i=1,2), g and h be the same as in Algorithm 6.11, and let x*and y* be the solution of the system (2.6). Suppose that, for i=1,2, the operator Ti:ℋ→ℋ is ξi-strongly monotone and μi-Lipschitz continuous. Moreover, let g be ξ3-strongly monotone and μ3-Lipschitz continuous and h be ξ4-strongly monotone and μ4-Lipschitz continuous. If there exist two constant ρ and η such that
(6.28)θ1+θ2<1-2(k+k1),
where k, k1, θ1, and θ2 are the same as in (6.14) and ∑n=0∞‍an=∞, and then the iterative sequences {xn} and {yn} generated by Algorithm 6.11 converge strongly to x* and y*, respectively.Proof.
Since(x*,y*)∈ℋ×ℋ is a solution of the system (2.6), in view of Remark 6.13, we have
(6.29)g(x*)=JAρ(g(y*)-ρT1(y*)),h(y*)=JAη(h(x*)-ηT2(x*)),
where ρ,η>0 are two constants. For each n≥0, one can rewrite the above equations as follows:
(6.30)x*=(1-an)x*+an(x*-g(x*)+JAρ(g(y*)-ρT1(y*))),y*=(1-an)y*+an(y*-h(y*)+JAη(h(x*)-ηT2(x*))),
where the sequence {an} is the same as in Algorithm 6.11. From (6.10), (6.30), and the nonexpansivity property of the resolvent operator JAρ, it follows that
(6.31)‖xn+1-x*‖≤(1-an)‖xn-x*‖+an(‖xn-x*-(g(xn)-g(x*))‖+‖JAρ(g(yn)-ρT1(yn))-JAρ(g(y*)-ρT1(y*))‖)≤(1-an)‖xn-x*‖+an(‖xn-x*-(g(xn)-g(x*))‖+‖g(yn)-g(y*)-ρ(T1(yn)-T1(y*))‖)≤(1-an)‖xn-x*‖+an(‖xn-x*-(g(xn)-g(x*))‖+‖yn-y*-(g(yn)-g(y*))‖+an+‖yn-y*-ρ(T1(yn)-T1(y*))‖).
Since T1 is ξ1-strongly monotone and μ1-Lipschitz continuous, and g is ξ3-strongly monotone and μ3-Lipschitz continuous, one can prove that
(6.32)‖yn-y*-ρ(T1(yn)-T1(y*))‖≤1-2ρξ1+ρ2μ12‖yn-y*‖,‖xn-x*-(g(xn)-g(x*))‖≤1-2ξ3+μ32‖xn-x*‖,∥yn-y*-(g(yn)-g(y*))∥≤1-2ξ3+μ32∥yn-y*∥.
Combining (6.31) and (6.32), we obtain that
(6.33)‖xn+1-x*‖≤(1-an)‖xn-x*‖+ank‖xn-x*‖+an(k+θ1)‖yn-y*‖,
where k and θ1 are the same as in (6.14). Because T2 is ξ2-strongly monotone and μ2-Lipschitz continuous, and h is ξ4-strongly monotone and μ4-Lipschitz continuous, in similar way to the proofs of (6.31)–(6.33), we can verify that
(6.34)‖yn+1-y*‖≤(1-an)‖yn-y*‖+ank1‖yn-y*‖+an(k1+θ2)‖xn-x*‖,
where k1 and θ2 are the same as in (6.14). From (6.33) and (6.34), it follows that
(6.35)‖(xn+1,yn+1)-(x*,y*)‖*≤(1-an)‖(xn,yn)-(x*,y*)‖*+an(k+k1)‖(xn,yn)-(x*,y*)‖*+an(k+k1+θ1+θ2)‖(xn,yn)-(x*,y*)‖*=(1-an(1-(2(k+k1)+θ1+θ2)))‖(xn,yn)-(x*,y*)‖*.
Letting ϖ=2(k+k1)+θ1+θ2, the condition (6.28) guarantees that ϖ∈(0,1). Since ∑n=0∞‍an=∞, setting bn=cn=0, for all n≥0, we infer that all the conditions of Lemma 5.1 are satisfied. Now, Lemma 5.1 and (6.35) guarantee that ∥(xn,yn)-(x*,y*)∥*→0, as n→∞, and so the sequences {xn} and {yn} generated by Algorithm 6.11 converge strongly to x* and y*, respectively. This completes the proof.Remark 6.16.
M. A. Noor and K. I. Noor [44] established the strong convergence of the sequences generated by iterative Algorithm 6.6. We would like to notice that, as we have made an observation in Remark 6.3, some assumptions should be added to ([44], Theorem 4.1).


## 7. Conclusions

In this paper, we have introduced and considered a new system of extended general nonlinear variational inclusions involving eight different nonlinear operators (SEGNVID). We have verified the equivalence between the SEGNVID and the fixed point problem and then by this equivalent formulation, and we have discussed the existence and uniqueness theorem for a solution of the SEGNVID. This equivalence and two nearly uniformly Lipschitzian mappingsSi(i=1,2) are used to suggest and analyze a new resolvent iterative algorithm with mixed errors for finding an element of the set of the fixed points of the nearly uniformly Lipschitzian mapping 𝒬=(S1,S2) which is the unique solution of the SEGNVID. In the final section, comments on some related works are presented. It is expected that the results proved in this paper may simulate further research regarding the numerical methods and their applications in various fields of pure and applied sciences.



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*Source: 569592-2012-08-30.xml*