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 on the space of Lipschitz maps into the domain of non-empty, convex and weak*-compact valued functions ordered by reverse inclusion, is continuous. For finite dime...

For a linear-quadratic state constrained optimal control problem, it is proved in [11] that under an independence condition for the active constraints, the optimal control is Lipschitz continuous. We now give a new proof of this result based on an analysis of the Euler discretization given in [9]. There we exploit the Lipschitz continuity of the control to estimate the error in the Euler discre...

Proof. Let L be a Lipschitz constant that applies to all utility functions. Suppose each message in the set V is Lipschitz continuous with Lipschitz constant L. Consider the message from an activity a to a resource r ∈ R(a). Define X a\r , ∏r∈R(a)\r Xr to be the space of consumption bundles for activity a, excluding resource r. Without loss of generality, assume that (FV )a→r(xar) ≥ (FV )a→r(xa...

is called the modulus of (uniform) continuity of f . The mapping f is said to be uniformly continuous if Ω f (d) → 0 as d ↓ 0. In this case the modulus of continuity is a subadditive monotone continuous function. The definition of Ω f implies that f (Br(x)) ⊂ BΩ f (r)( f (x)). (By Bρ(y) and Bρ(y) we denote, respectively, the open and the closed ball of radius ρ, centered at y.) One important cl...

We consider a class of Lipschitz vector fields S :Ω →Rn whose values lie in a suitable cone and we show that the trajectories of the system x′ = S(x) admit a parametrization that is invertible and Lipschitz with its inverse. As a consequence, every v in W1,1(Ω) admits a representative that is absolutely continuous on almost every trajectory of x′ = S(x). If S is an arbritrary Lipschitz field th...

The construction of a Lyapunov function characterizing the pullback attraction of a cocycle attractor of a nonautonomous discrete time dynamical system involving Lipschitz continuous mappings is presented.

Based on a formula of Tseng, we show that the squared norm of the matrix-valued Fischer-Burmeister function has a Lipschitz continuous gradient.

We prove existence of weak solutions to an eigenvalue Steklov problem defined in a bounded domain with a Lipschitz continuous boundary.