Convergence of a Time Discretization for a Nonlinear Second-Order Inclusion

We study an abstract second order inclusion involving two nonlinear single-valued operators and a nonlinear multi-valued term. Our goal is to establish the existence of solutions to the problem by applying numerical scheme based on time discretization. We show that the sequence of approximate solution converges weakly to a solution of the exact problem. We apply our abstract result to a dynamic, second-order-in-time differential inclusion involving a Clarke subdifferential of a locally Lipschitz, possibly non-convex and non-smooth potential. In the two presented examples the Clarke subdifferential appears either in a source term or in a boundary term.

Existence of solutions for a second order boundary value problem with the Clarke subdifferential

In this paper, we prove a theorem on the existence of solutions for a second order differential inclusion governed by the Clarke subdifferential of a Lipschitzian function and by a mixed semicontinuous perturbation.

Łojasiewicz-type inequalities with explicit exponents for the largest eigenvalue function of real symmetric polynomial matrices

Let [Formula: see text] be a real symmetric polynomial matrix of order [Formula: see text] and let [Formula: see text] be the largest eigenvalue function of the matrix [Formula: see text] We denote by [Formula: see text] the Clarke subdifferential of [Formula: see text] at [Formula: see text] In this paper, we first give the following nonsmooth version of Łojasiewicz gradient inequality for the function [Formula: see text] with an explicit exponent: For any [Formula: see text] there exist [Formula: see text] and [Formula: see text] such that we have for all [Formula: see text] [Formula: see text] where [Formula: see text] and [Formula: see text] is a function introduced by D’Acunto and Kurdyka: [Formula: see text] if [Formula: see text] and [Formula: see text] if [Formula: see text] Then we establish some local and global versions of Łojasiewicz inequalities which bound the distance function to the set [Formula: see text] by some exponents of the function [Formula: see text].

Implicit Multifunction Theorems in Banach Spaces

This paper is mainly devoted to the study of implicit multifunction theorems in terms of Clarke coderivative in general Banach spaces. We present new sufficient conditions for the local metric regularity, metric regularity, Lipschitz-like property, nonemptiness, and lower semicontinuity of implicit multifunctions in general Banach spaces. The basic tools of our analysis involve the Ekeland variational principle, the Clarke subdifferential, and the Clarke coderivative.

ON FIRST-ORDER NECESSARY CONDITIONS FOR OPTIMALITY OF REGULAR GENERALIZED SEMI-INFINITE PROGRAMMING

This paper is concerned with the optimality for generalized semi-infinite programming (GSIP) with nondifferentiable and nonconvex (but being regular in Clarke sense) constraint functions. The objective function is only locally Lipschitz. We consider a lower level constraint qualification which is based on the Clarke subdifferential. This constraint qualification is a natural extension of Mangasarian–Fromovitz one to the differentiable GSIP. The main results are Fritz-John type necessary conditions for optimal solutions.

NECESSARY OPTIMALITY CONDITIONS FOR NONSMOOTH VECTOR OPTIMIZATION PROBLEMS

In this paper we introduce a notion of generalized derivative for nonsmooth vector functions in order to obtain necessary optimality conditions for vector optimization problems. This definition generalizes to the vector case the notion introduced by Michel and Penot and extended by Yang and Jeyakumar. This generalized derivative is contained in the Clarke subdifferential and then the corresponding optimality conditions are sharper than the Clarke's ones.

Relaxed submonotone mappings

The notions ofrelaxed submonotoneandrelaxed monotonemappings in Banach spaces are introduced and many of their properties are investigated. For example, the Clarke subdifferential of a locally Lipschitz function in a separable Banach space is relaxed submonotone on a residual subset. For example, it is shown that this property need not be valid on the whole space. We prove, under certain hypotheses, the surjectivity of the relaxed monotone mappings.