SINGLE-STEP ALGORITHM FOR VARIATIONAL INEQUALITY PROBLEMS IN BANACH SPACES

ABSTRACT

In this study, we introduce a novel one-step algorithm designed to solve variational inequality problems within a 2-uniformly convex Banach space. We establish the weak convergence of this algorithm, demonstrating its ability to converge towards a solution of the variational inequality under reasonable assumptions. Specifically, we prove the following theorem:

Theorem: Let E denote a real 2-uniformly convex and uniformly smooth Banach space. Consider a nonempty closed convex subset C of E. Let A : E → E∗ be a monotone and Lipschitz function with Lipschitz constant L. Take x0 and x−1 as elements of E, and define the sequence {xn} iteratively as xn+1 = ΠC J−1 (Jxn − λnAxn − λn−1(Axn − Axn−1)), for n ≥ 0. Here, {λn} is a subset of h satisfying 1−2 2µL i for some > 0 and µ ≥ 1. Assume that Γ is nonempty and that the normalized duality mapping J is weakly sequentially continuous. In such a case, the sequence {xn} weakly converges to an element within Γ.

Furthermore, The proposed algorithm in this work offers a one-step solution to variational inequality problems within a 2-uniformly convex Banach space. Variational inequality problems are mathematical optimization problems that involve finding a solution that satisfies a certain inequality constraint.

The key aspect of this algorithm is its ability to achieve weak convergence, meaning that the sequence of iterates generated by the algorithm approaches a solution of the variational inequality in a weak sense. Weak convergence is a desirable property as it allows for flexibility and robustness in finding approximate solutions.

The algorithm operates within the framework of a real 2-uniformly convex and uniformly smooth Banach space. These properties of the Banach space ensure certain geometric and smoothness conditions that facilitate the convergence analysis of the algorithm.

The algorithm involves several components and parameters. The function A : E → E∗ represents a monotone and Lipschitz operator mapping the Banach space E to its dual space E∗. The Lipschitz constant L measures the rate of growth of this operator.

The sequence {xn} is iteratively computed using the update rule xn+1 = ΠC J−1 (Jxn − λnAxn − λn−1(Axn − Axn−1)), where ΠC is the projection operator onto the closed convex subset C of E, J is the normalized duality mapping, and {λn} is a sequence of parameters satisfying certain conditions.

The theorem established in the work proves that under reasonable assumptions, such as the non-emptiness of Γ (the set of solutions) and the weak sequential continuity of the normalized duality mapping J, the sequence {xn} generated by the algorithm converges weakly to an element within Γ, thereby providing a solution to the variational inequality problem.

Additionally, the study highlights practical applications of the algorithm, demonstrating its relevance and usefulness in addressing real-life problems. These applications serve as examples to illustrate how the algorithm can be applied in various contexts to solve specific problems that involve variational inequalities.

Overall, this work contributes to the field of variational inequality theory by introducing a one-step algorithm with proven weak convergence properties in the setting of a 2-uniformly convex Banach space, and by providing applications that showcase its practical utility.