• Unification of optimality and duality frameworks for variational problems in Banach spaces, consolidating generalized and asymptotic Kuhn–Tucker conditions and Banach-space dual representations to address constraint qualifications and sufficiency in non-convex settings [1] [9] [11].

• Regularity results and convergence analysis for variational inequality solutions, including elliptic-variational regularity, convergence of solution sets under convexity, and structural properties like the coincidence set [2] [3] [12].

• Existence, smoothness, and approximation for nonlinear and non-coercive variational inequalities, complemented by dual and numerical solution strategies that address existence/approximability in challenging VI contexts [19] [8] [7].

• Numerical methods and approximation theory for variational problems, highlighting dual solution methods, constrained Chebyshev-type approximations, and L2-approximation frameworks that bridge VIs and computational methods [7] [10] [5].

• Bounds and extremal constants for differential-inequality contexts, focusing on best constants in integral inequalities and upper/lower bounds for differential-inequality solutions (e.g., Poisson–Boltzmann) to bound VI-related quantities [13] [15] [20].