In this paper, we introduce a new three-step iterative scheme for finding the common solutions of the variational inequality using the technique of updating the solution. We suggest, iterative algorithms involving three-steps for the predictor-corrector method of variational inequality in real Hilbert spaces H. Our results include the Takahashi and Toyoda, extra gradient, Mann and Noor iterations as special cases. We also investigate the convergence criteria of the three-step iterative scheme. As special cases, the earlier findings are included in our results, which can be seen as an advancement and improvement over the previous investigation. This is a new refinement in our existing literature and previously known algorithms. A numerical example is given to illustrate the efficiency and performance of the proposed self-adaptive scheme.

A literature review revealed that the general variational inequalities, fixed-point problems, and Winner–Hopf equations are equivalent. In this study, general variational inequality and fixed-point problem are considered. We introduced a new iterative method based on a self-adaptive predictor-corrector approach for finding a solution to the GVI. Adaptations in the fixed-point formulation and self-adaptive techniques have been used to predict a novel iterative approach. Convergence analyses of the suggested algorithm are demonstrated. Moreover, numerical analysis shows that we establish the new best method for solving general variational inequality which performs better than the previous one. Furthermore, it is known that GVI consisted of several classes including variational inequalities and related optimization problems, and results obtained in this study continue to hold for these problems.

In this paper, we examine aspects of general variational inequalities (GVIs) that are similar to fixed-point problems. Special cases of the proposed unique iterative methods are introduced, including the implicit, explicit, and extra-gradient methods of the proximal point approach. The convergence of the derived expression is analyzed under appropriate assumptions to demonstrate the effectiveness of the proposed method. Numerical results are also presented to evaluate the algorithms’ performance in practice. In these experiments, matrix-scaling tests were conducted, and parameter sensitivity analyses were performed; the convergence speed, computational efficiency, robustness, and stability of the experiments were evaluated. From the results, the implicit proximal method seems to give very good results, and convergence is achieved after fewer iterations than other algorithms. Additionally, we examine the non-convergent behavior of other algorithms under varying parameters. Overall, our study validates the theoretical findings and highlights the effectiveness of advanced proximal methods for large-scale GVI problems, while also suggesting directions for future research in this area.

In this study, we investigate an extended class of hierarchical variational inequalities (SEGHVIDs, i.e., generalized hierarchical variational inequalities with differentiable operators, enabling a unified framework that incorporates five distinct nonlinear mappings through fixed-point problems) in real Hilbert spaces, with special cases being found in many important engineering practical applications, such as image recognition, signal processing, and machine learning. To solve the (SEGHVID) of potential application value, inspired by the projection technique, we developed a novel explicit projection-based iterative algorithm that combines the projection method with a correction term and a self-adaptive step-size technique. Under appropriate assumptions, we establish the strong convergence of the iterative schemes. The results generalize and extend several existing hierarchical variational inequality (HVIPs) models in the literature and provide an explicit computational strategy for solving a broad class of multi-operator hierarchical problems. Finally, we apply our novel iterative approximation to solve multiple-set split feasibility problems and verify the effectiveness of the proposed iterative algorithm using numerical results.