Introduction. In the study of uniform approximation to a function of a complex variable by polynomials or by bounded analytic functions, Lipschitz conditions have proved extremely useful in relating degree of approximation to continuity properties of the functions approximated. Parts of this theory are analogous to the older study (S. Bernstein, D. Jackson, de la Vallee Poussin, Montel) of appr...

In this paper, we study games with continuous action spaces and non-linear payoff functions. Our key insight is that Lipschitz continuity of the payoff function allows us to provide algorithms for finding approximate equilibria in these games. We begin by studying Lipschitz games, which encompass, for example, all concave games with Lipschitz continuous payoff functions. We provide an efficient...

3. The Implicit Function Theorem and its special case, the Inverse Function Theorem. Relation to the Contraction Mapping Fixed Point Theorem. Computation of several iterates of the approximating sequence to the implicit function. Continuity and differentiability of the implicit function. Application to differentiable variation of the fixed point in the Contraction Mapping Fixed Point Theorem. 4...

Solutions to optimization problems of convex type are typically characterized by saddle point conditions in which the primal vector is paired with a dual ‘multiplier’ vector. This paper investigates the behavior of such a primal-dual pair with respect to perturbations in parameters on which the problem depends. A necessary and sufficient condition in terms of certain matrices is developed for t...

We prove that Alexandrov spaces X of nonnegative curvature have Markov type 2 in the sense of Ball. As a corollary, any Lipschitz continuous map from a subset of X into a 2-uniformly convex Banach space is extended as a Lipschitz continuous map on the entire space X.

It is known that there are several approaches to define a Sobolev class for Banach valued functions. We compare the usual definition via weak derivatives with Reshetnyak–Sobolev space and Newtonian space; in particular, we provide sufficient conditions when all three agree. Also, revise difference quotient criterion property of Lipschitz mapping preserve it acting as superposition operator.

Motivated by problems arising in decentralized control problems and non-cooperative Nash games, we consider a class of strongly monotone Cartesian variational inequality (VI) problems, where the mappings either contain expectations or their evaluations are corrupted by error. Such complications are captured under the umbrella of Cartesian stochastic variational inequality problems and we consid...

Abstract. We will prove an interior estimate for the mappings on the space, which is generalization of the Heinz-Bernstein theorem for the plane. In other worlds we prove the theorem. Let s : Bn → R (s : Bn → Bm) be a continuous function from the closed unit ball Bn ⊂ R into the real line (closed unit ball in R) satisfying the conditions: (i) s is C on U, (ii) s is C in ∂B = Sn−1 and (iii) |∆s|...

Abstract In this study, we investigate the boundedness of composition operators acting on Morrey spaces and weak spaces. The primary aim study is to a necessary sufficient condition operator induced by diffeomorphism particular, detailed information derived from boundedness, i.e., bi-Lipschitz continuity mapping that induces follows mapping. idea proof determine norm characteristic functions, e...

We investigate typical properties of nonexpansive mappings on unbounded complete hyperbolic metric spaces. For two families metrics uniform convergence bounded sets, we show that the mapping is a Rakotch contraction every subset and there set which mapped into itself by this mapping. In particular, obtain in setting has unique fixed point can be reached iterating Nevertheless, it turns out not ...