Subgradient Extragradient Method for Finite Lipschitzian Demicontractions and Variational Inequality Problems in a Hilbert Space

In this research, the modifed subgradient extragradient method and K -mapping generated by a fnite family of fnite Lipschitzian demicontractions are introduced. Ten, a strong convergence theorem for fnding a common element of the common fxed point set of fnite Lipschitzian demicontraction mappings and the common solution set of variational inequality problems is established. Furthermore, numerical examples are given to support the main theorem.


Introduction
Let H be a real Hilbert space and C be a nonempty closed convex subset of H with the inner product 〈•, •〉 and norm • ‖ ‖.
Te fxed point problem for the mapping S: H ⟶ H is to fnd u ∈ H such that u � Su. ( Te term F(S) is denoted by the set of fxed points of S, that is, F(S) � x ∈ H: Sx � x { }.Fixed point problem has been widely studied and developed in the various literature studies, see [1].Defnition 1.Let H be a real Hilbert space.
(i) A mapping S: H ⟶ H is said to be nonexpansive if (ii) A mapping S: H ⟶ H is said to be quasinonexpansive if Fix(S) ≠ ∅ and (iii) A mapping S: H ⟶ H is called κ-strictly pseudocontractive if there exists a constant κ ∈ [0, 1) such that If F(S) ≠ ∅, then a nonexpansive mapping is a quasinonexpansive mapping.Also, if κ � 0, then a strictly pseudocontractive reduces to a nonexpansive mapping.
In a real Hilbert space, the inequality (4) is equivalent to Defnition 2 (see [2]).A mapping S is called demicontractive if Fix(S) ≠ ∅ and there exists a constant κ ∈ [0, 1) such that Te class of demicontractive mappings covers a variety of nonlinear mappings, including strictly pseudocontractive mappings, quasinonexpansive mappings, and nonexpansive mappings.
By using the same technique as in the proof of (5), we see that (6) is equivalent to the inequality shown below if S: H ⟶ H is a demicontractive mapping.
Several mathematicians have taken an interest in studying the common fxed point of the fnite family of nonlinear mappings and their characteristics during the past few decades; see [3][4][5][6].
Te solution set of the problem ( 9) is denoted by VI(C, D).
Stampacchia [7] introduced and investigated variational inequalities in 1964.In addition to ofering a comprehensive, unifying framework for the study of optimization, equilibrium problems, and related problems, it also serves as a helpful computational framework for the resolution of various problems in a wide range of applications.For additional information, see [8][9][10][11][12][13]. Various approaches are investigated to solve variational inequality problems and the related optimization problems through iterative methods.
Several researchers have presented a variety of iterative algorithms designed for solving the variational inequality problem (VIP).Te projected gradient method (GM), which can be defned as follows, is the most fundamental projection technique for solving the VIP.
where P C denotes the metric projection mapping, D is the α-strongly monotone, and L− is Lipschitz continuous with ρ ∈ (0, 2α/L 2 ).
In 1976, Korpelevich [14] and Antipin [15] proposed the extragradient method (EGM) in a fnite-dimensional Euclidean space as follows: where ρ ∈ (0, 1/L) and D: R n ⟶ R 2 are monotones and L is Lipschitz continuous.If VI(C, D) is nonempty, the sequence generated by (11) converges to a solution of VIP.
According to [16][17][18] and related references, the EGM has undergone modifcations and enhancements in the past few years.
Later, in 2012, Censor et al. [19] defned the subgradient extragradient method (SEGM) by modifying the EGM and replacing the second projection with a projection onto a half-space which is presented as follows: Weak convergence theorem is obtained for SEGM (4) under some control conditions.
Recently, in 2021, Kheawborisut and Kangtunyakarn [20] introduced the modifed subgradient extragradient method (MSEGM) as follows: where and G is a nonexpansive mapping.Afterwards, under certain control settings, a strong convergence theorem is obtained.
Motivated by the recent research, the S-subgradient extragradient method (SSEGM) is introduced as follows: where S is a nonexpansive mapping.If σ n � 1, then the Ssubgradient extragradient method (SSEGM) reduces to the modifed subgradient extragradient method (MSEGM).
In this paper, inspired by [6,20], the S-subgradient extragradient method and K-mapping generated by a fnite family of fnite Lipschitzian demicontraction mappings are proposed.Under some control conditions, strong convergence theorems are proved.Moreover, numerical examples are given to support the main theorem.

Preliminaries
Te notations ″ ⇀ ″ and ″ ⟶ ″ are denoted weak convergence and strong convergence, respectively.For each u ∈ H, there exists a unique nearest point Te mapping P C is called the metric projection of H onto C. Also, P C is a frmly nonexpansive mapping from H onto C, that is, Moreover, for any u ∈ H and q ∈ C, q � P C u if and only Defnition 3. Let S: H ⟶ H be a mapping.Ten, (i) S is said to be μ-Lipschitz continuous if there is a positive real number μ > 0 such that (ii) S is called ξ-inverse strongly monotone if there is a positive real number ξ such that Lemma 4 (see [21]).Let p n   be a sequence of nonnegative real numbers satisfying where ε n   is a sequence in (0,1) and ρ n   is a sequence such that Ten, lim n⟶∞ p n � 0.
Te following lemmas are needed to prove the main result.
To show this, suppose u ∈ F(K) and v ∈ ∩ N i�1 F(S i ).By the defnition of K-mapping, we obtain By (23), it follows that Ten, we have Hence, u � S 1 u, that is,

Journal of Mathematics
By the defnition of V 1 and (26), we get that is, From ( 23) and (28), we have By the defnition of V 2 , ( 23) and ( 28), this implies that which yields that Using the same method, we get Next, we claim that u ∈ F(S N ).Since and ρ N ∈ (0, 1], we have which implies that Hence, Terefore, Finally, applying the same proof as in (23), K is a quasinonexpansive mapping.□ Lemma 9. Let C be a nonempty closed convex subset of a real Hilbert space H.For i � 1, 2, . . ., N, let S i : H ⟶ H be a fnite family of κ i -demicontractive mappings of C into itself and L i -Lipschitzian mappings with κ i ≤ c 1 and For each n ∈ N, let K and K n be the K-mappings generated by S 1 , S 1 , . . ., S N and ρ 1 , ρ 2 , . . ., ρ N and S 1 , S 2 , . . ., S N and ρ n 1 , ρ n 2 , . . ., ρ n N , respectively.Terefore, for each bounded sequence u n   in C, we have Proof.Let u n   be a bounded sequence in C and let V k and V n,k be generated by S 1 , S 1 , . . ., S N and ρ 1 , ρ 2 , . . ., ρ N and S 1 , S 1 , . . ., S N and ρ n 1 , ρ n 2 , . . ., ρ n N , respectively.For each n ∈ N, we have (41) From ( 40) and (41), we obtain Journal of Mathematics By (42) and the condition □ Lemma 10 (see [23]).Let Γ n   be a sequence of real numbers that do not decrease at infnity in the sense that there exists a subsequence Γ n j   of Γ n   such that Γ n j < Γ n j +1 for all j ≥ 0.
Also, we consider the sequence of integers τ n   n≥n 0 defned by Ten, τ n   n≥n 0 is a nondecreasing sequence verifying Lemma 11 (see [20]).Let H be a real Hilbert space, for every i � 1, 2, . . ., N, let D i : H ⟶ H be α i -inverse strongly monotone mappings with β � min α i  .Let u n   and v n   be sequences generated by Ten, the following inequality holds: for every i � 1, 2, . . ., N.

Journal of Mathematics
From Lemma 11 and ρ ∈ (0, η), we have From ( 50) and (51), we get By induction, we obtain Tis implies that Q n   is a bounded sequence.Next, from (50), observe that 10 Journal of Mathematics Tis follows that (55) Tus, we get (56) Next, the following two possible cases are considered.

Journal of Mathematics
From (71), the conditions (i), (ii), and Lemma 4, we can conclude that Q n   converges strongly to v � P Ω u.From (59), we also obtain that Z n   converges strongly to v � P Ω u.Case 14. Assume that there exists a subsequence Γ n i   ⊂ Γ n   such that Γn i ≤ Γ n i +1 for all i ∈ N. In this case, we can defne τ: N ⟶ N by τ(n) � max k ≤ n: Γ k < Γ k+1  .Ten, we obtain τ(n) ⟶ ∞ as n ⟶ ∞ and Γ τ(n) < Γ τ n +1 .Tis implies by (72) that Using the same method as in Case 13, it yields that Applying the same proof of Case 13 for x τ(n k )  , we get By Lemma 4, we have Hence, by Lemma 10, we obtain Terefore, we can conclude that Q n   converges strongly to v � P Ω u.From (59), we also have Z n   converging strongly to v � P Ω u.Te proof is complete.Remark 15.Since the S-subgradient extragradient method covers various type of iterations such as the modifed subgradient extragradient method (MSEGM), the 14 Journal of Mathematics subgradient extragradient method (SEGM), and the extragradient method (EGM), Teorem 12 can be seen as a modifcation and extension of several research papers, see, for example, [14,15,19,20].

Numerical Examples
Numerical examples are provided in this section to back up the main result.
Example 1.Let C � [0, 100] and R be the set of real numbers.Defne the mappings For i � 1, 2, . . ., N, let S i : R ⟶ R be defned by Hence, we obtain  Applying Teorem 12, the sequences Q n   and Z n   converge strongly to (0,0).Table 2 and Figure 2 show the values of sequences Q n   and Z n   where u � Q 1 � (2, 2) and n � N � 50.

Conclusion
Tis study proposes a new subgradient extragradient method for approximating a common fxed point of a fnite family of demicontractive mappings and Lipschitzian mappings and a common solution of variational inequality problems.It can also be considered as an extension and modifcation of several currently used techniques for solving variational inequality problems as well as a fxed point problem with some associated mappings.As special cases of Teorem 12, previous publications such as [14,15,19,20] can be considered.Also, numerical illustrations of the main theorem are given [24,25].

Figure 1 :Figure 2 :
Figure 1: Te convergence comparison of Z n   and Q n   with u � Q 1 � 5.

Table 1 :
Te values of Z n   and Q n   with n � N � 30.

Table 2 :
Te values of Z n   and Q n   with u � Q 1 � (2, 2) and n � N � 50.