On alternative theorems and necessary conditions for efficiency

Trace Theorems for Sobolev Spaces on Lipschitz Domains. Necessary Conditions

A famous theorem of E. Gagliardo gives the characterization of traces for Sobolev spaces W 1, p (Ω) for 1 ≤ p < ∞ when Ω ⊂ R is a Lipschitz domain. The extension of this result to W m, p (Ω) for m ≥ 2 and 1 < p < ∞ is now well-known when Ω is a smooth domain. The situation is more complicated for polygonal and polyhedral domains since the characterization is given only in terms of local compati...

Donsker Theorems for Diffusions: Necessary and Sufficient Conditions

We consider the empirical process Gt of a one-dimensional diffusion with finite speed measure, indexed by a collection of functions F . By the central limit theorem for diffusions, the finite-dimensional distributions of Gt converge weakly to those of a zero-mean Gaussian random process G. We prove that the weak convergence Gt ⇒ G takes place in l(F) if and only if the limit G exists as a tight...

Necessary and Sufficient Conditions for Fixed-Point Theorems, Minimax Inequalities and Related Theorems

This paper provides necessary and sufficient conditions for fixed-point theorems, minimax inequalities and some related theorems defined on arbitrary topological spaces that may be discrete, continuum, non-compact or non-convex. We establish a single condition, γ-recursive transfer lower semicontinuity, which fully characterizes the existence of equilibrium of minimax inequality without imposin...

Some Results on Necessary Conditions for Tow Quasidifferentiable Optimization Problems

Alternative Theorems for Nonlinear

This paper introduces two concepts of exceptional families for a class of nonlinear projection equations which provide a uniied formulation of several special cases such as nite-dimensional variational inequalities and complementarity problems. Based on these concepts, two alternative theorems, and then two suucient existence conditions, are established for this class of nonlinear projection eq...