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...

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...

We show that the Scott topology induces a topology for real-valued Lipschitz maps on Banach spaces which we call the L-topology. It is the weakest topology with respect to which the L-derivative operator, as a second order functional which maps the space of Lipschitz functions into the function space of non-empty weak* compact and convex valued maps equipped with the Scott topology, is continuo...

We show that if Y is a separable subspace of a Banach space X such that both X and the quotient X/Y have Cp-smooth Lipschitz bump functions, and U is a bounded open subset of X, then, for every uniformly continuous function f : Y ∩U → R and every ε > 0, there exists a Cp-smooth Lipschitz function F : X → R such that |F (y)− f(y)| ≤ ε for every y ∈ Y ∩U . If we are given a separable subspace Y o...

where u : (0, T ]× Ω→ R is the unknown function, Ω ⊂ R is a bounded domain with a Lipschitz continuous boundary Γ, 0 < T <∞ and ∂νβ(u) denotes outside normal derivative. Function β : R → R is nondecreasing Lipschitz continuous function satisfying (2.1). Function f : (0, T ]×Ω×R→ R is Lipschitz continuous with Lipschitz constant Lf and satisfies (2.2). We denote g(t, x, s) : = γs+φ(t, x), where ...

We characterize the local single-valuedness and continuity of multifunctions (set-valued mappings) in terms of their submonotonicity and lower semicontinuity. This result completes the well-known condition that lower semicontinuous, monotone multifunctions are single-valued and continuous. We also show that a multifunction is actually a Lipschitz single-valued mapping if and only if it is submo...

In this paper, an implicit iterative process is considered for strongly continuous semigroup of Lipschitz pseudocontraction mappings. Weak and strong convergence theorems for common fixed points of strongly continuous semigroup of Lipschitz pseudocontraction mappings are established in a real Banach space. c ©2016 All rights reserved.

This paper studies the global output convergence of a class of recurrent neural networks with globally Lipschitz continuous and monotone nondecreasing activation functions and locally Lipschitz continuous time-varying inputs. We establish two sufficient conditions for global output convergence of this class of neural networks. Symmetry in the connection weight matrix is not required in the pres...

We show that the Scott topology induces a topology for real-valued Lipschitz maps on Banach spaces which we call the L-topology. It is the weakest topology with respect to which the L-derivative operator, as a second order functional which maps the space of Lipschitz functions into the function space of non-empty weak* compact and convex valued maps equipped with the Scott topology, is continuo...