Abstract
The primary objective of this study is to introduce two novel extragradienttype iterative schemes for solving variational inequality problems in a real Hilbert space. The proposed iterative schemes extend the wellknown subgradient extragradient method and are used to solve variational inequalities involving the pseudomonotone operator in real Hilbert spaces. The proposed iterative methods have the primary advantage of using a simple mathematical formula for step size rule based on operator information rather than the Lipschitz constant or another line search method. Strong convergence results for the suggested iterative algorithms are wellestablished for mild conditions, such as Lipschitz continuity and mapping monotonicity. Finally, we present many numerical experiments that show the effectiveness and superiority of iterative methods.
1. Introduction
The primary objective of this research is to investigate the iterative methodologies used to estimate the solution of variational inequalities in a real Hilbert space. To establish the convergence analysis theorems, the following conditions need to be satisfied:
Condition 1. The solution set of the problem (VIP) denoted by and it is nonempty.
Condition 2. A mapping is said to be pseudomonotone if
Condition 3. A mapping is said to be Lipschitz continuous with constant if
Condition 4. A mapping is said to be weakly sequentially continuous if converges weakly to for each sequence converges weakly to an element
Let be any real Hilbert space and be any nonempty convex closed subset of a Hilbert space Assume that be an arbitrary mapping. The variational inequality problem for an operator on is defined in the following manner [1, 2]:
Let stand for the solution set for the problem (VIP). The mathematical model of variational inequalities covers many mathematical problems, such as partial differential equations, optimization, optimal control, mechanics, finance, and mathematical programming (see for details [3–9]) and others in [10–20]. Since it is a fundamental problem in the applied sciences and nonlinear functional analysis, many researchers are investigating not only the stability and existence of solutions to such problems but also iterative methods for solving them numerically. In order to solve variational inequalities numerically, projection iterative methods are a very important tool. Many researchers have provided various projection method extensions and modifications to solve the problem (VIP) (see [21–33]). The extragradient method described below was developed by Korpelevich [25] and Antipin [34]. Their method takes the form of where For each iteration of the above iterative scheme, two projections on the feasible set are required to be figured out. Of course, if the feasible set has a complicated framework, this can affect the method’s computational effectiveness. The first one is to follow the subgradient extragradient method designed by Censor et al. [22] to overcome this deficiency. This method is in the form of where and
It is a key point to note that the abovementioned wellestablished methods have two major drawbacks. The first is the fixed constant step size, which needs knowledge or approximation of the appropriate operator Lipschitz constant, and also is only weakly convergent in Hilbert spaces. Using a fixed step size can be difficult in terms of computation, affecting the method convergence rate and efficiency.
Hence, a natural question arises:
“Is it possible to propose two new strongly convergent subgradient extragradient algorithms with a nonmonotone selfadaptive step size rule to solve the problem (VIP)?”
The primary objective of this study is to introduce two new strongly convergent subgradient extragradient methods for enhancing the convergence rate of an iterative sequence. The answer to the above question is given in this study, which would be the subgradient extragradient algorithms, which set up a strong convergent iterative sequence by letting a variable nonmonotone step size rule. The suggested methods are employed to solve variational inequality problems involving pseudomonotone and Lipschitz regular operators in real Hilbert space. The proposed methods are based on the projection method [22] as well as the methods proposed in [26, 35]. The established method only needs to compute one projection onto the feasible set and one projection onto the halfspace for each iteration. The iterative sequences established by the proposed method strongly converge to some solution of the underlined problem in the framework of some appropriate conditions on control parameters. A number of numerical examples are also added to elaborate on the computational effectiveness of the new methods over some existing methods presented in [36, 37].
The paper is arranged in the following manner: In Section 2, we provide some basic identities and preliminary results that were used in this paper. Section 3 includes the proposed methods and proves their convergence analysis. Finally, Section 4 presents some numerical results to illustrate the convergence and the effectiveness of the proposed methods.
2. Preliminaries
This section contains a number of important identities, as well as useful lemmas and definitions. For all we have
A metric projection of an element is evaluated by
Next, we list some of the important identities that are used to prove the convergence analysis.
Lemma 1 (see [38]). Let be a metric projection on set For each and then the following inequalities are satisfied: (i) is true if and only if(ii) (iii) (iv) (v)
Lemma 2 (see [39]). Assuming that , a sequence meets the following criteria: Moreover, and are two sequences such that Then,
Lemma 3 (see [40]). Assume that a sequence of real numbers and there is subsequence of such that
(102)
Thus, there exists a natural nondecreasing sequence with as and satisfies the following criteria for :
Indeed,
Lemma 4 (see [41]). Assume that is a pseudomonotone and continuous mapping. Then, is a solution of the problem (VIP) if and only if is a solution of the following problem:
3. Main Results
In this part of the research article, we propose two new methods and the corresponding strong convergence theorems. Both methods are presented in the following manner. The first method is of the following form.

Assume that is a contraction having constant The second major contribution of this study work is as follows. The second main algorithm has the following form.

Lemma 5. A sequence generated by (3.1) is convergent to and satisfies the following inequality:
Proof. Let such that By using mathematical induction on the definition of , we have Let Due to expression of we can write Thus, is convergent. Next, we have to prove the convergence of the following series: Let . Thus, we have Thus, we have Letting in (24), we obtain as This is a contradiction. Due to the convergence of the series and taking in (24), we obtain This completes the proof.☐☐
Lemma 6. Let be an operator satisfies the criteria Condition 1–Condition 4. For a given we have
Proof. We have to evaluate It is given that such that Furthermore, it implies that By using expressions (26) and (28), we obtain Since is the solution of problem (VIP), we have We have condition on mapping with feasible set we get By substituting we get Therefore, we have From (29) and (33), we get Since and from we have Combining expressions (34) and (35), we obtain ☐☐
Lemma 7. Let be a mapping meet the items Condition 1–Condition 4. If there exists a subsequence convergent weakly to and then is the solution of (VIP).
Proof. We need to prove that Indeed, we have that is equivalent to The above inequality implies that Thus, we obtain Since and is a bounded sequence. By taking and in (18), we obtain Moreover, we have Since and is Lipschitz continuity on we have which together with (42) and (43), we obtain Next, we have Due to decreasing, this implies that is increasing.☐☐
Case I. Suppose that a subsequence of such as (). Let we get Then, which further implies that
Case II. Suppose that there exists such that for all , Consider that Due to the above definition, we obtain Moreover, expressions (45) and (48) for all we have Due to the pseudomonotonicity of for we have For all we have Let us consider that ; we obtain Thus, we obtain By letting in (51), we obtain Due to the Minty Lemma in [41], we infer that
Theorem 8. Let be a mapping that satisfies the conditions Condition 1–Condition 4. Then, sequence generated by Algorithm 1 strongly converges to an element
Proof. Since , there exists a fixed number such that Thus, expression (36) gives that It is given that ; we obtain Next, we have to evaluate the following: Substituting (56) into (59), we obtain Next, we have From expressions (57) and (61), we obtain Thus, from the above relation, we obtain that is bounded sequence. From sequence we can write By the use of expression (59), we have Combining expressions (63) and (64) (for some ), we obtain
The remainder of the proof is now split into two parts:
Case 1. Suppose that there exists a fixed number () such that Then, exists. From (65), we have Due to the existence of and we infer that It follows that It follows from expression (69) and that which implies that We have and by using Lemma 1 (i), we can write By using expression (72) and Lemma 7, we obtain Due to . It gives that Next, we assume that Thus, we have where