In this paper, we study stochastic optimization of two-level composition functions without Lipschitz continuous gradient. The smoothness property is generalized by the notion relative which provokes Bregman gradient method. We propose three algorithms for possible relatively smooth compositional scenarios and provide their sample complexities to achieve an $$\epsilon $$ -approximate stationary ...

Uncertain delay differential equation (UDDE) is a type of differential equations driven by Liu process. It has been proved that uncertain delay differential equation has a unique solution in the finite domain, under the conditions that the coefficients are global Lipschitz continuous. This paper will extend this existence and uniqueness theorem from finite domain to infinite domain under the lo...

Let X, Y be complete metric spaces and E, F be Banach spaces. A bijective linear operator from a space of E-valued functions on X to a space of F -valued functions on Y is said to be biseparating if f and g are disjoint if and only if Tf and Tg are disjoint. We introduce the class of generalized Lipschitz spaces, which includes as special cases the classes of Lipschitz, little Lipschitz and uni...

In a paper of 1950 Graves proved that for a function f acting between Banach spaces and an interior point x̄ in its domain, if there exists a continuous linear mapping A which is surjective and the Lipschitz modulus of the difference f −A at x̄ is sufficiently small, then f is (linearly) open at x̄. This is an extension of the Banach open mapping principle from continuous linear mappings to Lipsch...

The paper is concerned with the synchronization problem of a general class of multi-input multi-output (MIMO) nonlinear continuous-time systems under sampled-data output feedback control. The main contributions of the present paper are twofold: (i) we provide a unified synthesis method and synchronization criteria for MIMO Lipschitz nonlinear continuous-time systems; (ii) we present a systemati...

This paper discusses several (sub)gradient methods attaining the optimal complexity for smooth problems with Lipschitz continuous gradients, nonsmooth problems with bounded variation of subgradients, weakly smooth problems with Hölder continuous gradients. The proposed schemes are optimal for smooth strongly convex problems with Lipschitz continuous gradients and optimal up to a logarithmic fac...

We consider inverse boundary value problems in Rn, n ≥ 3, for operators which may be written as first order perturbations of the Laplacian. The purpose is to obtain global uniqueness theorems for such problems when the coefficients are nonsmooth. We use complex geometrical optics solutions of Sylvester-Uhlmann type to achieve this. A main tool is an extension of the Nakamura-Uhlmann intertwinin...

The notion of gradient flows is generalized to a metric space setting without any linear structure. The metric spaces considered are a generalization of Hilbert spaces, and the properties of such metric spaces are used to set up a finite-difference scheme of variational form. The proof of the Crandall–Liggett generation theorem is adapted to show convergence. The resulting flow generates a stro...

We study existence and uniqueness of a mild solution in the space of continuous functions and existence of an invariant measure for a class of reaction-diffusion systems on bounded domains of R , perturbed by a multiplicative noise. The reaction term is assumed to have polynomial growth and to be locally Lipschitz-continuous and monotone. The noise is white in space and time if d = 1 and colour...

It is a well-known result of Kadec that every two separable infinite dimensional Banach spaces are homeomorphic. Also in large classes of nonseparable Banach spaces (perhaps all) the density character of a Banach space is its only topological invariant (see the book [2] for details). The situation changes considerably if we consider uniform homeomorphisms. Several results are known which prove ...