Iterative Algorithms for a Generalized System of Mixed Variational-Like Inclusion Problems and Altering Points Problem

In this article, we introduce and study a generalized system of mixed variational-like inclusion problems involving αβ-symmetric η-monotone mappings. We use the resolvent operator technique to calculate the approximate common solution of the generalized system of variational-like inclusion problems involving αβ-symmetric η-monotone mappings and a fixed point problem for nonlinear Lipchitz mappings. We study strong convergence analysis of the sequences generated by proposed Mann type iterative algorithms. Moreover, we consider an altering points problem associated with a generalized system of variational-like inclusion problems. To calculate the approximate solution of our system, we proposed a parallel S-iterative algorithm and study the convergence analysis of the sequences generated by proposed parallel S-iterative algorithms by using the technique of altering points problem. The results presented in this paper may be viewed as generalizations and refinements of the results existing in the literature.


Introduction
The theory of variational inequality was introduced by Hartmann and Stampacchia [17] in 1966 as a tool to study partial differential equations with applications. It has emerged as a powerful tool for wide class of unrelated problems arises in various branches of physical, engineering, pure and applied sciences in a unified and general framework, see; for example, [3,4,5,8,9,10,21]. Variational inequalities have been extended and generalized in different directions by using novel and innovative techniques for their own sake as well as for their applications. In 1989, Parida et al. [26] studied a generalized form of variational inequalities. They called it variational-like inequality problem and established its relationship with mathematical programming problem. Variational-like inequality problem has many important and novel applications in economics and optimizations, see; for example, [7,32,35].
There are a substantial number of numerical methods including projection methods, Wiener-Hopf equations techniques, auxiliary principle technique, descent, and Newton methods, see; for example [11,12,13,14,24,25] to solve variational and variational-like inequalities. Projection method and its variant forms represent important 551 (iv) M (·, ·) is said to be ξ-mixed Lipschitz continuous with respect to A and B, if there exists a constant ξ > 0 satisfying ∥M (Ax, Bx) − M (Ay, By)∥ ≤ ξ∥x − y∥, ∀x, y ∈ H; (v) η is said to be τ -Lipschitz continuous, if there exists a constant τ > 0 such that ∥η(x, y)∥ ≤ τ ∥x − y∥, ∀x, y ∈ H. (iii) P is said to be strongly monotone with respect to N in the first argument, if there exists a constant c > 0 such that 2 , ∀x, y ∈ H and for some u ∈ P (x), v ∈ P (y); (iv) N is said to be γ-Lipschitz continuous in the first argument, if there exists a constant γ > 0 such that Similarly, one can define strong monotonicity of P in the second argument with respect to N and Lipschitz continuity of N in the second argument.
Let H be a real Hilbert space with norm ∥ · ∥. Define the norm ∥ · ∥ * on H × H by Note that (H × H, ∥ · ∥ * ) is a Banach space.
Definition 1.4. Let A and B be two nonempty subsets of a metric space X and let P : A → B and Q : B → A be mappings. Then x ∈ A and y ∈ B are called altering points of the mappings P and Q, if We denote the set of altering points of the mappings P : It is easy to see that each point of C is a fixed point of QP and each point of D is a fixed point of P Q. Thus, altering points x ∈ C and y ∈ D are given by 2], and D = [2,4]. Define P : C → D by P (x) = x + 2, ∀x ∈ C and Q : D → C by Q(x) = x 2 8 , ∀x ∈ D. Note that, P Q(x) = x 2 8 + 2, ∀x ∈ D and QP (x) = (x+2) 2

ITERATIVE ALGORITHMS FOR A GENERALIZED SYSTEM OF INCLUSIONS
Definition 1.5. [38]. A functional f : H × H → R ∪ {+∞} is said to be 0-diagonally quasi-concave (in short, 0-DQCV ) in x i , if for any finite set {x 1 , x 2 , · · · , x n } ⊂ H and for any y = then the mapping z → x, denoted by R ∂ηϕ ρ,M (·,·) (z) is called resolvent operator of ϕ. Then, we have z − M (Ax, Bx) ∈ ρ∂ η ϕ(x) and it follows that R    for any x ∈ R and k > 0. Then i.e., M is 3k c -strongly η-monotone with respect to A.
is αβ-symmetric η-monotone mapping with respect to A and B.
Next, for any x, z ∈ R, the mapping is 0-DQCV in y. If, it is false, then there exists a finite set {y 1 , y 2 , · · · , y m } and which is not possible. Thus, for any x, z ∈ R, the mapping h(y, x) is is 0-DQCV in y. Lemma 1.3. Let {a n } and {b n } be two nonnegative real sequences satisfying the following conditions: Then lim n→∞ a n = 0. Lemma 1.4. Let {a n } and {b n } be two nonnegative real sequences satisfying the following inequality: Then lim n→∞ a n = 0.
Problem (5) was considered and studied by Li and Li [22]. (iii) If E = F ≡ I, the identity mapping, then Problem (5) reduces to the problem of finding x, y ∈ K such that System (7) was studied by Petrot [27].
then the system (7) reduces to the following problem of finding x, y ∈ K such that A System of type (8) was studied by Chang et al. [6].
In the following theorem, we establish the fixed point formulation of GSM V LIP (3) then we prove the existence of common solution of GSM V LIP (3) for αβ-symmetric η-monotone mapping and a fixed point problem of nonlinear Lipschitz mappings. (3), if and only if (x, y, u, v) satisfies the following relation:
Based on Lemma 2.1, we suggest the following Mann-type iterative algorithm for finding a common element of the solution set of GSM V LIP (3) involving αβ-symmetric η-monotone mapping and the set of fixed points F (T ) of a nonlinear Lipschitz mapping T . Algorithm 2.1. For any x 0 , y 0 ∈ H, v 0 ∈ P (x 0 ), u 0 ∈ Q(y 0 ); compute the sequences {x n }, {y n }, {u n } and {v n } by the following iterative scheme: where n = 1, 2, 3, · · · , ρ i > 0 are constants and {α n }, {β n } are sequences in [

Proof
By applying Lemma 1.2, Algorithm 2.1, Lipschitz continuity of T and condition (12), we have Now, Since M 1 is mixed Lipschitz continuous with constant t 1 and mixed strongly monotone with constant k 1 with respect to A 1 and B 1 , therefore Stat., Optim. Inf. Comput. Vol. 8, June 2020 Since Q is strongly monotone with respect to N 1 with constant c 1 and N 1 is γ 1 -Lipschitz continuous in the first argument and Q is D-Lipschitz continuous with constant ξ 1 , we get Since N 1 is γ ′ 1 -Lipschitz continuous in the second argument and P is D-Lipschitz continuous with constant ξ 2 , we have It follows from (14), (15), (16) and (17) that Again employing Lemma 1.2, Algorithm 2.1, Lipschitz continuity of T and condition (13), we have Now, Since M 2 is mixed Lipschitz continuous with constant t 2 and mixed strongly monotone with constant k 2 with respect to A 2 and B 2 , we have Since P is strongly monotone with respect to N 2 with constant c 2 and N 2 is γ 2 -Lipschitz continuous in the first argument and P is D-Lipschitz continuous with constant ξ 2 , we get

ITERATIVE ALGORITHMS FOR A GENERALIZED SYSTEM OF INCLUSIONS
Since N 2 is γ ′ 2 -Lipschitz continuous in the second argument and Q is D-Lipschitz continuous with constant ξ 1 , we have It follows from (19), (20), (21) and (22) that which implies that Thus, from (18) and (23), we have Setting; a n = ∥x n − x∥, λ n = α n ∥x − y∥. It follows from condition (iii) that λ n ∈ (0, 1), ∀n ∈ N and condition (ii) implies that b n = o(λ n ). By using condition (iii), we Thus all the conditions of Lemma 1.3 are satisfied and so ∥x n − x∥ → 0 as n → ∞. Consequently, condition (ii) and (23) implies that y n → y as n → ∞. Since the mappings P and Q are D-Lipschitz continuous and using Algorithm 2.1, it follows that {u n } and {v n } are Cauchy sequences in H such that u n → u and v n → v as n → ∞. Now, we show that u ∈ P (x) and v ∈ Q(y); which implies that d(u, P (x)) = 0. Since P (x) ∈ CB(H), it follows that u ∈ P (x). Similarly, we can verify that v ∈ Q(y). Therefore, in view of Theorem 2.1 and Algorithm 2.1, we conclude that (x, y, u, v) such that x, y ∈ H, u ∈ P (x) and v ∈ Q(y) is a common solution of GSM V LIP (3) and F (T ).

Convergence Theorem via Altering Points Problem
In this section, we approximate the solution of GSM V LIP (3) by using the altering points problem. First, we propose a parallel S-iterative algorithm for altering points problem associated to generalized system of mixed variational-like inclusion problems involving αβ-symmetric η-monotone mappings and then we study strong convergence analysis of GSM V LIP (3) by using proposed iterative algorithm.
Let C and D be nonempty closed convex subsets of a real Hilbert space H. Let S 1 : C → D and S 2 : D → C be contraction mappings with Lipschitz constants τ 1 and τ 2 , respectively. Since S 2 S 1 : C → C is a contraction, there exists a unique element (x, y) ∈ C × D of the following altering points problem for operators S 1 and S 2 . Find It is well known that the Picard iterative algorithm converges faster than the Mann iterative algorithm [23] for contraction mappings, see; [2]. Sahu [28] has introduced Normal S-iteration Process, whose rate of convergence similar to the Picard iteration process and faster than other fixed point iteration processes (see; [28], Theorem 3.6) which is defined as follows: where T is a self mapping on a convex subset of a normed space X and α n ⊆ [0, 1] is a real sequence. Normal S-iterative algorithm is independent of Mann algorithm and it has attracted the attention of many researchers, see; for example, [34,15,30] due to fast convergence rate and its simplicity.
Very recently, Gursoy et al. [16] studied the following normal S-iterative algorithm: where {ξ n } ∞ n=0 ⊂ [0, 1]. They have approximated the solution of a generalized nonlinear variational inequalities and studied convergence analysis.
For approximate calculation of altering points of contraction mappings S 1 : C → D and S 2 : D → C, motivated by normal S-iteration process, Sahu [29] has introduced the following parallel S-iteration process: where α ∈ (0, 1). Further, Zhao et al. [37] generalized the parallel S-iteration process (29) and studied the following parallel S-iteration process: where {α n } and {β n } are sequences in (0, 1). The parallel S-iteration process (30) is a natural generalization of the parallel S-iteration process (29).
Based on Theorem 2.1 and Definition 1.4, we pose the following altering points problem associated to GSM V LIP (3).

ITERATIVE ALGORITHMS FOR A GENERALIZED SYSTEM OF INCLUSIONS
where S 1 : C → D and S 2 : D → C be contraction mappings with Lipschitz constants κ 1 and κ 2 , respectively. Now, we propose following parallel S-iterative algorithm to compute the approximate solution of GSM V LIP (3).
Algorithm 3.1. For any (x 0 , y 0 ) ∈ C × D, compute the sequences {x n , y n } ∈ C × D generated by the following parallel S-iteration scheme.

(35)
Then the iterative sequences {x n }, {y n }, {u n } and {v n } generated by parallel S-iterative algorithm 3.1 converges strongly to x, y, u and v, respectively.