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

and Applied Analysis 3 2. Formulations and Basic Facts Throughout this paper, we will let H 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 〈 u − v, x − y ≥ 0, ∀u ∈ A x , v ∈ Ay. 2.1 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 operator A, the resolvent operator associated with A of parameter λ is defined as J A u I λA −1 u , ∀u ∈ H. 2.2 It is single valued and nonexpansive, that is, ∥∥∥Jλ A u − J A v ∥∥ ≤ ‖u − v‖, ∀u, v ∈ H. 2.3 Let Ti : 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 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∗ )) . 2.4 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. If Ti : 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 0 ∈ g1 x∗ − g ( y∗ ) ρ ( T1 ( y∗ ) A ( g1 x∗ )) , 0 ∈ h1 ( y∗ ) − h x∗ ηT2 x∗ A ( h1 ( y∗ ))) , 2.5 which is called the system of general nonlinear variational inclusions involving seven different nonlinear operators. 4 Abstract and Applied Analysis 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 Range g1 ⋂ domA/ ∅ and Range h1 ⋂ 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 . Taking g1 g and h1 h in the system 2.5 , the mentioned system collapses to the system of finding x∗, y∗ ∈ H ×H such that 0 ∈ T1 ( y∗ ) A ( g x∗ ) , 0 ∈ T2 x∗ A ( h ( y∗ )) . 2.6 The problem 2.6 is called the nonlinear variational inclusions system involving five different nonlinear operators. If, for each i 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 each i 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 〈 ρT1 ( y∗, x∗ ) g2 x∗ − g1 ( y∗ ) , g1 x − g2 x∗ 〉 ≥ ρφg2 x∗ ) − ρφg1 x ) , ∀x ∈ H, 〈 ηT2 ( x∗, y∗ ) h2 ( y∗ ) − h1 x∗ , h1 x − h2 ( y∗ )〉 ≥ ηφh2 ( y∗ )) − ηφ h1 x , ∀x ∈ H, 2.7 which appears to be a new system of extended general mixed nonlinear variational inequalities involving eight different operators. If g1 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 〈 ρT1 ( y∗, x∗ ) x∗ − gy∗, g x − x∗ ≥ ρφ x∗ − ρφg x , ∀x ∈ H, 〈 ηT2 ( x∗, y∗ ) y∗ − g x∗ , g x − y∗ ≥ ηφy∗ − ηφg x , ∀x ∈ H, 2.8 which was considered and studied by Noor 29 . Also, if T1 T2 T , then the system 2.8 is considered and studied in 29 . If Ti : 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 〈 ρT1 ( y∗ ) g2 x∗ − g1 ( y∗ ) , g1 x − g2 x∗ 〉 ≥ ρφg2 x∗ ) − ρφg1 x ) , ∀x ∈ H, 〈 ηT2 x∗ h2 ( y∗ ) − h1 x∗ , h1 x − h2 ( y∗ )〉 ≥ ηφh2 ( y∗ )) − ηφ h1 x , ∀x ∈ H, 2.9 Abstract and Applied Analysis 5and Applied Analysis 5 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 inH defined by δK ( y ) { 0 y ∈ K, ∞ y / ∈ K, 2.10 then the system 2.7 reduces to the system of finding x∗, y∗ ∈ K ×K such that 〈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, 2.11 which has been introduced and considered by M. A. Noor and K. I. Noor 30 . Remark 2.3. When g1 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 byNoor 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. 6 Abstract and Applied Analysis Lemma 3.1. Let Ti, 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 g2 x∗ J ρ A1 ·,x∗ ( g1 ( y∗ ) − ρT1 ( y∗, x∗ )) , h2 ( y∗ ) J A2 ·,y∗ ( h1 x∗ − ηT2 ( x∗, y∗ )) , 3.1 where for all z, t ∈ H, J A1 .,z is the resolvent operator associated with A1 ., z of parameter ρ and J η A2 .,t 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 g1 ( y∗ ) − ρT1 ( y∗, x∗ ) ∈ I ρA1 ·, x∗ )( g2 x∗ ) , h1 x∗ − ηT2 ( x∗, y∗ ) ∈ I ηA2 ·, y∗h2 ( y∗ )) , ⇐⇒ g2 x∗ J ρ A1 ·,x∗ ( g1 ( y∗ ) − ρT1 ( y∗, x∗ )) , h2 ( y∗ ) J A2 ·,y∗ ( h1 x∗ − ηT2 ( x∗, y∗ )) , 3.2 where I is identity operator. Definition 3.2. A nonlinear single-valued operator g : H → H is said to be


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 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 mappings S 1 and S 2 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 S 1 , S 2 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.

Formulations and Basic Facts
Throughout this paper, we will let H 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 u − v, x − y ≥ 0, ∀u ∈ A x , v ∈ A y .

2.1
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 operator A, the resolvent operator associated with A of parameter λ is defined as It is single valued and nonexpansive, that is,

2.4
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.
If T i : H → H and A i A : H H i 1, 2 are univariate nonlinear operators, then taking g 1 g, g 2 g 1 , h 1 h, and h 2 h, the system 2.4 reduces to the system of finding 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 Range g 1 dom A / ∅ and Range h 1 dom A / ∅.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 .
Taking g 1 g and h 1 h in the system 2.5 , the mentioned system collapses to the system of finding x * , y * ∈ H × H such that

2.6
The problem 2.6 is called the nonlinear variational inclusions system involving five different nonlinear operators.
If, for each i 1, 2, T i T , g 1 h 1 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 each i 1, 2, A i : H H is an univariate set-valued operator, A 1 x ∂ϕ x and A 2 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 which appears to be a new system of extended general mixed nonlinear variational inequalities involving eight different operators.
If g 1 h 1 g, g 2 h 2 ≡ I the identity operator , and ϕ φ, then the system 2.7 is equivalent to that of finding x * , y * ∈ H × H such that which was considered and studied by Noor 29 .Also, if T 1 T 2 T , then the system 2.8 is considered and studied in 29 .
If T i : 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
Abstract and Applied Analysis 5 which was considered and investigated by M. A. Noor and K. I. Noor 44 .Also, if T 1 T 2 T , then the system 2.9 is considered and studied in 44 .When g 1 h 1 g and g 2 h 2 ≡ I, the system 2.9 is introduced and studied in 29 .
If, in the system 2.7 , ϕ φ δ K is the indicator function of a nonempty closed convex set K in H defined by then the system 2.7 reduces to the system of finding x * , y * ∈ K × K such that

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.Let T i , A i , g i , h i 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 where for all z, t ∈ H, J ρ A 1 .,z is the resolvent operator associated with A 1 ., z of parameter ρ and J η A 2 .,t is the resolvent operator associated with A 2 ., t of parameter η.
Proof.Let x * , y * ∈ H × H be a solution of the system 2.4 .Then where I is identity operator.

Definition 3.2.
A nonlinear single-valued operator g : H → H is said to be a monotone if b r-strongly monotone in the first variable if there exists a constant r > 0 such that for all x, y ∈ H

3.10
Now, we present the sufficient conditions for the existence solutions of our main considered problem 2.4 .Theorem 3.4.Let T i , A i , g i , h i i 1, 2 , ρ, and η be the same as in the system 2.4 and suppose further that, for i 1, 2, a T i is ξ i -strongly monotone and μ i -Lipschitz continuous in the first variable; b g i is ς i -strongly monotone and σ i -Lipschitz continuous; c h i is i -strongly monotone and δ i -Lipschitz continuous; d there exists constants τ i such that

3.11
If two constants ρ and η satisfy the following conditions 3.12 then the system 2.4 admits a unique solution.

3.15
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 From ς 2 -strongly monotonicity and σ 2 -Lipschitz continuity of g 2 , it follows that 3.17 which leads to

3.18
In similar way, by using ς 1 -strongly monotonicity and σ 1 -Lipschitz continuity of g 1 , we deduce that Since T 1 is ξ 1 -strongly monotone and μ 1 -Lipschitz continuous in the first variable, we conclude that

3.22
On the other hand, since for i 1, 2, h i is i -strongly monotone and δ i -Lipschitz continuous in the first variable, T 2 is ξ 2 -strongly monotone and μ 2 -Lipschitz continuous in the first variable, in similar way to the proofs of 3.16 -3.21 , we can prove that where

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 mappings S 1 and S 2 and by using the equivalent alternative formulation 3.
where {a n } is a fix sequence in 0, ∞ with a n → 0, as n → ∞.
For an arbitrary, but fixed n ∈ N, the infimum of constants k n in 4.

4.11
We denote the sets of all the fixed points of S i i 1, 2 and Q by Fix S i and Fix Q , respectively, and the set of all the solutions of the system 2.4 by SEGNVID H, T i , A i , g i , h i , i 1, 2 .
In view of 4.10 , for any x, y ∈ H × H, we see that T i , A i , g i , h i , i 1, 2 , then by using Lemma 3.1, it is easy to see that for each n ∈ N,

4.12
Abstract and Applied Analysis 13 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 S 1 , S 2 which is the unique solution of the system of extended general nonlinear variational inclusions 2.4 .Algorithm 4.3.Let T i , A i , g i , h i i 1, 2 , ρ, and η be the same as in the system 2.4 .For an arbitrary chosen initial point x 1 , y 1 ∈ H × H, compute the iterative sequence { x n , y n } ∞ n 1 in the following way: where S 1 , S 2 : H → H are two nearly uniformly Lipschitzian mappings, are six sequences in H to take into account a possible inexact computation of the resolvent operator point satisfying the following conditions: r n , l n * < ∞.

4.14
Remark 4.4.If, for each i 1, 2, S i ≡ I, then Algorithm 4.3 reduces to the following iterative algorithm for solving the system 2.4 .
Algorithm 4.5.Suppose that T i , A i , g i , h i i 1, 2 , ρ, and η are the same as in the system 2.4 .For an arbitrary chosen initial point x 1 , y 1 ∈ H × H, compute the iterative sequence { x n , y n } ∞ n 1 in the following way:

4.15
where the sequences are the same as in Algorithm 4.3.

Convergence Analysis
In this section, under some suitable conditions, we establish the strong convergence of the sequence generated by iterative Algorithm 4.3.We need the following lemma to prove our main result.
Lemma 5.1.Let {a n }, {b n }, and {c n } be three nonnegative real sequences satisfying the following condition.There exists a natural number n 0 such that where Proof.
Proof.According to Theorem 3.4, 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, T i , A i , g i , h i , i 1, 2 is a singleton set and Fix Q SEGNVID H, T i , A i , g i , h i , i 1, 2 / ∅, we conclude that x * ∈ Fix S 1 and y * ∈ Fix S 2 .Therefore, for each n ∈ N, we can write where the sequences {α n } ∞ n 1 and {β n } ∞ n 1 are the same as in Algorithm 4.3.Let K sup n≥1 { j n − x * , s n − y * }.Then, by using 4.13 , 5.2 , and the assumptions, we have Abstract and Applied Analysis 15 α n e n e n r n β n K,

5.3
where κ i i 1, 2 and θ 1 are the same as in 3.21 .
In similar way to the proof of 5.3 , one can establish that where ω i i 1, 2 and θ 2 are the same as in 3.23 .Letting L max{L 1 , L 2 } and applying 5.3 and 5.4 , we obtain that where ϑ is the same as in 3.25 .Since lim n → ∞ b n lim n → ∞ c n 0 and ∞ n 1 β n < ∞, 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 x n , y n → x * , y * , as n → ∞.Therefore, the sequence { x n , y n } ∞ n 1 , generated by Algorithm 4.3, converges strongly to the only element x * , y * of Fix Q ∩ SEGNVID H, T i , A i , g i , h i , i 1, 2 .This completes the proof.
Like in the proof of Theorem 5.2, one can prove the strong convergence of the iterative sequence generated by Algorithm 4.5, and we omit its proof.
Theorem 5.3.Suppose that T i , A i , g i , h i 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 { x n , y n } ∞ n 1 , generated by Algorithm 4.5, converges strongly to the unique solution x * , y * of the system 2.4 .

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 points x 0 , y 0 ∈ H, compute the sequences {x n } and {y n } by where a n ∈ 0, 1 for all n ≥ 0. where and a n ∈ 0, 1 , ∞ n 0 a n ∞, then for arbitrarily chosen initial points x 0 , y 0 ∈ H, x n and y n obtained from Algorithm 6.1 converge strongly to x * and y * , respectively.By using Definition 2.1, 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 r 3 − γ 3 μ 2 3 < 1 μ 2 3 .Therefore, the condition 2 r 3 − γ 3 μ 2 3 < 1 μ 2 3 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 1, 2 , and k < 1 imply that r i > γ i μ 2 i for each i 1, 2. Since, for each i 1, 2, the operator T i is γ i , r i -relaxed cocoercive and μ i -Lipschitz continuous, the conditions r i > γ i μ 2 i i 1, 2 guarantee that, for each i 1, 2, the operator T i is r i − γ i μ 2 i strongly monotone.Similarly, since g is γ 3 , r 3 relaxed cocoercive and μ 3 -Lipschitz continuous, the condition r 3 > γ 3 μ 2 3 implies that the operator g is r 3 −γ 3 μ 2 3 -strongly monotone.
In view of the above remark, one can rewrite Theorem 6.2 as follows.Theorem 6.4.Let x * and y * be the solution of the SGMVID 2.8 and suppose that T 1 : H×H → H is ξ 1 -strongly monotone and μ 1 -Lipschitz continuous in the first variable, and T 2 : H × H → H is relaxed ξ 2 -strongly monotone and μ 2 -Lipschitz continuous in the first variable.Moreover, let g be ξ 3 -strongly monotone and μ 3 -Lipschitz continuous.If two constants ρ and η satisfy the following conditions: and ∞ n 0 a n ∞, then the iterative sequences {x n } and {y n } 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 points x 0 , y 0 ∈ K compute the sequences {x n } and {y n } by where a n ∈ 0, 1 for all n ≥ 0 satisfies some suitable conditions.
Taking g 1 g and h 1 h, Algorithm 6.5 reduces to the following iterative algorithm.
Algorithm 6.6 see 44, Algorithm 3.2 .For arbitrary chosen initial points x 0 , y 0 ∈ K compute the sequences {x n } and {y n } by where a n ∈ 0, 1 for all n ≥ 0 satisfies some suitable conditions.Remark 6.7.By analyzing two Algorithms 6.5 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.Let T 1 , T 2 , A, g, h, g 1 , and h 1 be the same as in the system 2.5 , and let h be an onto operator.For arbitrary chosen initial points x 0 , y 0 ∈ K compute the sequences {x n } and {y n } in the following way: where a n ∈ 0, 1 for all n ≥ 0 satisfies some suitable conditions.Algorithm 6.9.Let T 1 , T 2 , A, g, h, g 1 , and h 1 be the same as in the system 2.5 .For arbitrary chosen initial points x 0 , y 0 ∈ K compute the sequences {x n } and {y n } in the following way: where a n ∈ 0, 1 for all n ≥ 0 satisfies some suitable conditions.
Taking g 1 g and h 1 h in two Algorithms 6.8 and 6.9, we can obtain the following two algorithms.Algorithm 6.10.For arbitrary chosen initial points x 0 , y 0 ∈ K compute the sequences {x n } and {y n } in the following way: 6.9 where a n ∈ 0, 1 for all n ≥ 0 satisfies some suitable conditions.Algorithm 6.11.For arbitrary chosen initial points x 0 , y 0 ∈ K compute the sequences {x n } and {y n } in the following way: where a n ∈ 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 operator A 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
Remark 6.13.In view of Lemma 6.12, x * , y * ∈ H × H is a solution of the system 2.6 if and only if where ρ, η > 0 are two constants.
Theorem 6.14.Let T i 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 T i : H → H is ξ i -strongly monotone and μ i -Lipschitz continuous.Furthermore, let g be ξ 3 -strongly monotone and μ 3 -Lipschitz continuous and h be ξ 4 -strongly monotone and μ 4 -Lipschitz continuous.If there exist two constants ρ and η such that where and ∞ n 0 a n ∞, and then the sequences {x n } and {y n } 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 where ρ and η are two constants.For each n ≥ 0, one can rewrite the above equations as below: where the sequence {a n } is the same as in Algorithm 6.10.It follows from 6.9 , 6.16 , and the nonexpansivity property of the resolvent operator J ρ A , that Since T 1 is ξ 1 -strongly monotone and μ 1 -Lipschitz continuous, we have which leads to Since g is ξ 3 -strongly monotone and μ 3 -Lipschitz continuous, we get Combining 6.17 -6.21 , we get the following: where k and θ 1 are the same as in 6.14 .Now, we find an estimation for y n 1 − y * .Using 6.9 , 6.16 , and the nonexpansivity property of the resolvent operator J η A , we obtain that

6.23
Since T 2 is ξ 2 -strongly monotone and μ 2 -Lipschitz continuous, and h is ξ 4 -strongly monotone and μ 4 -Lipschitz continuous, in similar way to the proofs of 6.19 -6.21 , we can establish that 6.24 Substituting 6.24 in 6.23 , we obtain that where k 1 and θ 2 are the same as in 6.14 .From 6.25 , we conclude that It follows from 6.22 and 6.26 that the condition 6.13 implies that ι ∈ 0, 1 .Since ∞ n 0 a n ∞, setting b n c n 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 x n − x * * → 0, as n → ∞.The inequality 6.26 implies that y n − y * * → 0, as n → ∞, and so the sequences {x n } and {y n } generated by Algorithm 6.10 converge strongly to x * and y * , respectively.This completes the proof.Theorem 6.15.Let T i 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 T i : H → H is ξ i -strongly monotone and μ i -Lipschitz continuous.Moreover, let g be ξ 3 -strongly monotone and μ 3 -Lipschitz continuous and h be ξ 4 -strongly monotone and μ 4 -Lipschitz continuous.If there exist two constant ρ and η such that where k, k 1 , θ 1 , and θ 2 are the same as in 6.14 and ∞ n 0 a n ∞, and then the iterative sequences {x n } and {y n } 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 where k and θ 1 are the same as in 6.14 .Because T 2 is ξ 2 -strongly monotone and μ 2 -Lipschitz continuous, and h is ξ 4 -strongly monotone and μ 4 -Lipschitz continuous, in similar way to the proofs of 6.31 -6.x n , y n − x * , y * * .

6.35
Letting 2 k k 1 θ 1 θ 2 , the condition 6.28 guarantees that ∈ 0, 1 .Since ∞ n 0 a n ∞, setting b n c n 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 x n , y n − x * , y * * → 0, as n → ∞, and so the sequences {x n } and {y n } 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 .

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 mappings S i 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 S 1 , S 2 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.

. 31 Since T 1 is ξ 1 - 1 − 2ξ 3 μ 2 3 y n − y * . 6 . 32 Combining 6 .
g x * J ρ A g y * − ρT 1 y * , h y * J η A h x * − ηT 2 x * , 6.29where ρ, η > 0 are two constants.For each n ≥ 0, one can rewrite the above equations as follows:x* 1 − a n x * a n x * − g x * J ρ A g y * − ρT 1 y * , y * 1 − a n y * a n y * − h y * J η A h x * − ηT 2 x * , 6.30where the sequence {a n } is the same as in Algorithm 6.11.From 6.10 , 6.30 , and the nonexpansivity property of the resolvent operator J ρ A , it follows thatx n 1 − x * ≤ 1 − a n x n − x * a n x n − x * − g x n − g x * J ρ A g y n − ρT 1 y n − J ρ A g y * − ρT 1 y * ≤ 1 − a n x n − x * a n x n − x * − g x n − g x * g y n − g y * − ρ T 1 y n − T 1 y * ≤ 1 − a n x n − x * a n x n − x * − g x n − g x * y n − y * − g y n − g y * y n − y * − ρ T 1 y n − T 1 y * .6stronglymonotone and μ 1 -Lipschitz continuous, and g is ξ 3 -strongly monotone and μ 3 -Lipschitz continuous, one can prove thaty n − y * − ρ T 1 y n − T 1 y * ≤ 1 − 2ρξ 1 ρ 2 μ 2 1 y n − y * , x n − x * − g x n − g x * ≤ 1 − 2ξ 3 μ 2 3 x n − x * , y n − y * − g y n − g y * ≤31and 6.32 , we obtain that x n 1 − x * ≤ 1 − a n x n − x * a n k x n − x * a n k θ 1 y n − y * , 6.33 When g 1 h 1 g and g 2 h 2 ≡ I, the system 2.11 is considered and studied byNoor 52.Also, if, for each i 1, 2, T i : H → H is an univariate nonlinear operator and g i h i g, then the system 2.11 is considered and studied byYang etal.53 .If, for each i 1, 2, g i h i ≡ I, then the system 2.11 is considered and studied by Huang and Noor 28 .If for each i 1, 2, T i T and g i h i ≡ I, then the system 2.11 introduced and studied by Chang et al. 26 and Verma 54 .If for each i 1, 2, T i T , g 1 h 1 g, and g 2 h 1 , we suggest and analyze a new resolvent iterative algorithm for finding an element of the set of the fixed points of Q S 1 , S 2 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 mapping T : H → H is called as follows: a nonexpansive if Tx − Ty ≤ x − y , for all x, y ∈ H; Tx − Ty ≤ L x − y M , ∀x, y ∈ H; 4.3 e asymptotically nonexpansive 63 if there exists a sequence {k n } ⊆ 1, ∞ with lim n → ∞ k n 1 such that for each n ∈ N, T n x − T n y ≤ k n x − y , ∀x, y ∈ H; 4.4 f pointwise asymptotically nonexpansive 64 if, for each integer n ≥ 1, T n x − T n y ≤ α n x x − y , x,y ∈ H, 4.5 where α n → 1 pointwise on X; g uniformly L-Lipschitzian if there exists a constant L > 0 such that for each n ∈ N, T n x − T n y ≤ L x − y , ∀x, y ∈ H. 4.6 Definition 4.2 see 46 .A nonlinear mapping T : H → H is said to be a nearly Lipschitzian with respect to the sequence {a n } if, for each n ∈ N, there exists a constant k n > 0 such that 7 is called nearly Lipschitz constant and is denoted by η T n .Notice that A nearly Lipschitzian mapping T with the sequence { a n , η T n } is said to be b nearly nonexpansive if η T n 1 for all n ∈ N, that is, nearly asymptotically nonexpansive if η T n ≥ 1 for all n ∈ N and lim n → ∞ η T n 1, in other words, k n ≥ 1 for all n ∈ N with lim n → ∞ k n 1; d nearly uniformly L-Lipschitzian if η T n ≤ L for all n ∈ N, in other words, k n L for all n ∈ N.For some interesting examples to investigate relations between these mappings, introduced in Definitions 4.1 and 4.2, ones may consult 58 .Let S 1 : H → H be a nearly uniformly L 1 -Lipschitzian mapping with the sequence {a n } ∞ n 1 and S 2 : H → H be a nearly uniformly L 2 -Lipschitzian mapping with the sequence {b n : x, y ∈ H, x / y .4.8 n } ∞ n 1 .We define the self-mapping Q of H × H as follows: Q x, y S 1 x, S 2 y , ∀x, y ∈ H. 4.10 Then Q S 1 , S 2 : H×H → H×H is a nearly uniformly max{L 1 , L 2 }-Lipschitzian mapping with the sequence {a n b n } ∞ 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 The proof directly follows from Lemma 2 in Liu 59 .
Theorem 5.2.Let T i , A i , g i , h i 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 S 1 : H → H is a nearly uniformly L 1 -Lipschitzian mapping with the sequence {b n } ∞ n 1 , S 2 : H → H is a nearly uniformly L 2 -Lipschitzian mapping with the sequence {c n } ∞ n 1 , and the self-mapping Q of H × H is defined by 4.10 such that Fix Q ∩ SEGNVID H, T i , A i , g i , h i , i 1, 2 / ∅.Furthermore, let, for each i 1, 2, L i ϑ < 1, where ϑ is the same as in 3.25 .Then the iterative sequence { x n , y n } ∞ n 1 , generated by Algorithm 4.3, converges strongly to the only element of Fix 33, we can verify thaty n 1 − y * ≤ 1 − a n y n − y * a n k 1 y n − y * a n k 1 θ 2 x n − x * , 6.34where k 1 and θ 2 are the same as in 6.14 .From 6.33 and 6.34 , it follows thatx n 1 , y n 1 − x * , y * * ≤ 1 − a n x n , y n − x * , y * * a n k k 1 x n , y n − x * , y * * a n k k 1 θ 1 θ 2 x n , y n − x * , y * * 1 − a n 1 − 2 k k 1 θ 1 θ 2