Paper Title: A CRITICAL ANALYSIS ON VARIAIONAL INEQUALITIES
Author:
Abstract:
It is well known that the minimum of a linear functional on a convex set in a Hilbert space can be characterized by a variational inequality. The same is proved for a class of differentiable and non-differentiable nonlinear functionals. Representation theorems are proved for nonlinear problems in a Hilbert space. The Riesz-Frechet theorem and the Lax-Milgram lemma can be deduced from these theorems. Techniques based on the contraction mapping theorem are used to prove the existence of a unique solution to a new class of nonlinear variational inequalities. It has been shown that the linearization of the variational inequalities is useful for the regularization approximation. The equivalence of variational and weak formulations of nonlinear boundary value problems is proved. A finite element approximation to the solution of the weak problem in a finite dimensional subspace of the original Hilbert space is defined. Using the concept of pseudo projection, an inequality bounding the error in this approximation over all functions of the subspace is derived. In the study of variational inequalities, it is necessary that in addition to the usual finite dimensional subspace of the Hilbert space, we construct finite dimensional convex subset of the Hilbert space. We note that this finite dimensional convex subset is not necessarily contained in the original convex subset of the original Hilbert space. New inequalities bounding the error in these approximations over the functions of the finite dimensional convex subset are derived for the nonlinear variational inequalities
Keywords:Variaional , Inequalities
DOI Link – https://doi.org/10.63431/AIJITR/3.II.2026.119-128
Review By – Dr. Chandan Adhikary and Prof. Dr. Shishir Kumar Bej