Related papers: On a global implicit function theorem for locally …
We present a fixed point theorem for a class of (potentially) non-monotonic functions over specially structured complete lattices. The theorem has as a special case the Knaster-Tarski fixed point theorem when restricted to the case of…
The paper proposes and justifies a new algorithm of the proximal Newton type to solve a broad class of nonsmooth composite convex optimization problems without strong convexity assumptions. Based on advanced notions and techniques of…
In this article we utilise abstract convexity theory in order to unify and generalize many different concepts from nonsmooth analysis. We introduce the concepts of abstract codifferentiability, abstract quasidifferentiability and abstract…
In this paper, we prove new existence and multiplicity results for critical points of lower semicontinuous functionals in Banach spaces, complementing the nonsmooth critical point theory set forth by Szulkin and avoiding the need of the…
We give an alternative proof of the Benedicks-Carleson theorem on the existence of strange attractors in H\'enon-like families in the plane. To bypass a huge inductive argument, we introduce an induction-free explicit definition of…
We study the fundamental properties of pointwise semi-Lipschitz functions between asymmetric spaces, which are the natural asymmetric counterpart of pointwise Lipschitz functions. We also study the influence that partial symmetries of a…
We consider homogenization of Dirichlet problems for semilinear elliptic systems with non-smooth data. We suppose that the diffusion tensors H-converge if the homogenization parameter tends to zero. Our result is of implicit function…
Majorization-minimization algorithms consist of successively minimizing a sequence of upper bounds of the objective function so that along the iterations the objective function decreases. Such a simple principle allows to solve a large…
Smooth Equations of the form G[z]=0 are investigated in Banach spaces with the aim of continuing the basic solution G[0]=0 to a solution curve of G[z]=0 with the implicit function theorem. If the linearization is surjective, then the…
We extend a recent result of Tim Austin (see arXiv:0905.0515) to the L^1 setting, thus providing a general version of the Birkhoff ergodic theorem for functions taking values in nonpositively curved spaces. In this setting, the notion of a…
We examine how implicit functions on ILB-Fr\'echet spaces can be obtained without metric or norm estimates which are classically assumed. We obtain implicit functions defined on a domain $D$ which is not necessarily open, but which contains…
We introduce a generalized version of the local Lipschitz number $\textrm{lip}\,u$, and show that it can be used to characterize Sobolev functions $u\in W_{\textrm{loc}}^{1,p}(\mathbb R^n)$, $1\le p\le \infty$, as well as functions of…
By means of non-smooth critical point theory we obtain existence of infinitely many weak solutions of the fractional Schr\"odinger equation with logarithmic nonlinearity. We also investigate the H\"older regularity of the weak solutions.
In this paper we study the nonlinear elliptic problem involving p(x)-Laplacian with nonsmooth potential, where the weighted function may change sign. By using critical point theory for locally Lipschitz functionals due to Chang, we obtain…
As a higher dimensional version of the theory of Morse functions, there have been various studies of smooth manifolds using generic smooth maps. As fundamental results, in these studies, they have found that inverse images of such maps…
The (global) Lipschitz smoothness condition is crucial in establishing the convergence theory for most optimization methods. Unfortunately, most machine learning and signal processing problems are not Lipschitz smooth. This motivates us to…
In this paper we present an inexact zeroth-order method suitable for the solution nonsmooth and nonconvex stochastic composite optimization problems, in which the objective is split into a real-valued Lipschitz continuous stochastic…
This is an exposition of some basic ideas in the realm of Global Inverse Function theorems. We address ourselves mainly to readers who are interested in the applications to Differential Equations. But we do not deal with those applications…
In this paper, we extend the method of invariant sets of descending flow that proposed by Sun Jingxian for smooth functionals to the locally Lipschitz functionals. By this way, we obtain the existence results for the positive, negative and…
Symmetry analysis can provide a suitable change of variables, i.e., in geometric terms, a suitable diffeomorphism that simplifies the given direction field, which can help significantly in solving or studying differential equations. Roughly…