and Applied Analysis 3 monotone and Lipschitz continuous mappings are cocoercive, and it follows that cocoercivity is an intermediate concept that lies between simple and strong monotonicity. Definition 2.3. A multivalued mapping M : X → 2 is said to be cocoercive if there exists a constant μ′′ > 0 such that 〈 u − v, x − y〉 ≥ μ′′‖u − v‖, ∀x, y ∈ X, u ∈ M x , v ∈ M(y). 2.7 Definition 2.4. Amappi...

We first show that a bounded linear operator $ T $ on a real Banach space $ E $ is quasicompact (Riesz, respectively) if and only if $T': E_{mathbb{C}}longrightarrow E_{mathbb{C}}$ is quasicompact  (Riesz, respectively), where the complex Banach space $E_{mathbb{C}}$ is a suitable complexification of $E$ and $T'$ is the complex linear operator on $E_{mathbb{C}}$ associated with $T$. Next, we pr...

An abstract Lipschitz stability estimate is proved for a class of inverse problems. It is then applied to the inverse Robin problem for the Laplace equation and to the inverse medium problem for the Helmholtz equation. Key-words: Derivative in the sense of Fréchet, Lipschitz stability, inverse Robin problem, inverse medium problem ha l-0 07 41 89 2, v er si on 1 15 O ct 2 01 2 Une remarque sur ...

The abscissa mapping on the affine variety Mn of monic polynomials of degree n is the mapping that takes a monic polynomial to the maximum of the real parts of its roots. This mapping plays a central role in the stability theory of matrices and dynamical systems. It is well known that the abscissa mapping is continuous on Mn, but not Lipschitz continuous. Furthermore, its natural extension to t...

Traditional finite-time convergence theory for numerical methods applied to stochastic differential equations (SDEs) requires a global Lipschitz assumption on the drift and diffusion coefficients. In practice, many important SDE models satisfy only a local Lipschitz property and, since Brownian paths can make arbitrarily large excursions, the global Lipschitz-based theory is not directly releva...

Remark 1. The Lipschitz condition (4) is called global because it holds for all x, y ∈ R with the same constant L. Klebaner Theorem 5.4 assumes only a local Lipschitz condition, where the Lipschitz constant may depend on the size of x, y. We use a global condition in order to make the presentation simpler. Later on we will also assume a Lipschitz condition with respect to t, see (21). (There is...

We present and analyse two implicit methods for Ito stochastic differential equations (SDEs) with Poisson-driven jumps. The first method, SSBE, is a split-step extension of the backward Euler method. The second method, CSSBE, arises from the introduction of a compensated, martingale, form of the Poisson process. We show that both methods are amenable to rigorous analysis when a one-sided Lipsch...

Note that the above does not require independence. To prove Theorem 1, it therefore suffices to bound ∑n i=1 α 2 i . Kontorovich & Ramanan (2008, Remark 2.1) showed that, if f is c-Lipschitz with respect to the Hamming metric, then ∑n i=1 α 2 i ≤ nc ‖Θ π n‖ 2 ∞. (Though the published results only prove this for countable spaces, Kontorovich later extended this analysis to continuous spaces in h...

Lipschitz domains. Our presentations here will almost exclusively be for bounded Lipschitz domains. Roughly speaking, a domain (a connected open set) Ω ⊂ R is called a Lipschitz domain if its boundary ∂Ω can be locally represented by Lipschitz continuous function; namely for any x ∈ ∂Ω, there exists a neighborhood of x, G ⊂ R, such that G ∩ ∂Ω is the graph of a Lipschitz continuous function und...