scholarly journals On the Clarke subdifferential of the distance function of a closed set

1992 ◽  
Vol 166 (1) ◽  
pp. 199-213 ◽  
Author(s):  
James V Burke ◽  
Michael C Ferris ◽  
Maijian Qian
Keyword(s):  
Distance Function ◽  
Clarke Subdifferential ◽  
Closed Set ◽  
The Clarke Subdifferential

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

2016 ◽  
Vol 27 (02) ◽  
pp. 1650012 ◽  
Author(s):  
Si Tiep Dinh ◽  
Tien Son Pham
Keyword(s):  
Distance Function ◽  
Polynomial Matrix ◽  
Symmetric Polynomial ◽  
Clarke Subdifferential ◽  
Polynomial Matrices ◽  
Largest Eigenvalue ◽  
The Matrix ◽  
Matrix Formula ◽  
The Clarke Subdifferential ◽  
The Largest Eigenvalue

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].

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

Filomat ◽  
2017 ◽  
Vol 31 (9) ◽  
pp. 2763-2771 ◽  
Author(s):  
Dalila Azzam-Laouir ◽  
Samira Melit
Keyword(s):  
Boundary Value Problem ◽  
Differential Inclusion ◽  
Existence Of Solutions ◽  
Boundary Value ◽  
Second Order ◽  
Clarke Subdifferential ◽  
Lipschitzian Function ◽  
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.

Extensions Of Subdifferential Calculus Rules in Banach Spaces

1996 ◽  
Vol 48 (4) ◽  
pp. 834-848 ◽  
Author(s):  
A. Jourani ◽  
L. Thibault
Keyword(s):  
Banach Spaces ◽  
Clarke Subdifferential ◽  
Subdifferential Calculus ◽  
Finite Dimensional ◽  
Calculus Rules ◽  
Approximate Subdifferential ◽  
The Clarke Subdifferential

AbstractThis paper is devoted to extending formulas for the geometric approximate subdifferential and the Clarke subdifferential of extended-real-valued functions on Banach spaces. The results are strong enough to include completely the finite dimensional setting.

scholarly journals On convergence of closed sets in a metric space and distance functions

1985 ◽  
Vol 31 (3) ◽  
pp. 421-432 ◽  
Author(s):  
Gerald Beer
Keyword(s):  
Distance Function ◽  
Metric Space ◽  
Pointwise Convergence ◽  
Distance Functions ◽  
Convergence Theorems ◽  
Closed Set ◽  
Closed Sets ◽  
The Family ◽  
Kuratowski Convergence ◽  
General Convergence

Let CL(X) denote the nonempty closed subsets of a metric space X. We answer the following question: in which spaces X is the Kuratowski convergence of a sequence {Cn} in CL(X) to a nonempty closed set C equivalent to the pointwise convergence of the distance functions for the sets in the sequence to the distance function for C ? We also obtain some related results from two general convergence theorems for equicontinuous families of real valued functions regarding the convergence of graphs and epigraphs of functions in the family.

scholarly journals Nearest points to closed sets and directional derivatives of distance functions

1989 ◽  
Vol 39 (2) ◽  
pp. 233-238 ◽  
Author(s):  
Simon Fitzpatrick
Keyword(s):  
Distance Function ◽  
Directional Derivative ◽  
Distance Functions ◽  
Directional Derivatives ◽  
Closed Set ◽  
Closed Sets ◽  
Set Of Points ◽  
Derivatives Of ◽  
Nearest Points ◽  
Dense Set

We investigate the circumstances under which the distance function to a closed set in a Banach space having a one-sided directional derivative equal to 1 or −1 implies the existence of nearest points. In reflexive spaces we show that at a dense set of points outside a closed set the distance function has a directional derivative equal to 1.

scholarly journals Relaxed submonotone mappings

2003 ◽  
Vol 2003 (1) ◽  
pp. 19-31 ◽  
Author(s):  
Tzanko Donchev ◽  
Pando Georgiev
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Lipschitz Function ◽  
Clarke Subdifferential ◽  
Monotone Mappings ◽  
Locally Lipschitz Function ◽  
Residual Subset ◽  
Locally Lipschitz ◽  
Whole Space ◽  
The Clarke Subdifferential

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.

On the Clarke Subdifferential of an Integral Functional on Lp, 1 ≤ p < ∞

1998 ◽  
Vol 41 (1) ◽  
pp. 41-48 ◽  
Author(s):  
E. Giner
Keyword(s):  
Growth Condition ◽  
Upper Bound ◽  
Directional Derivative ◽  
Integral Functional ◽  
Clarke Subdifferential ◽  
The Clarke Subdifferential

AbstractGiven an integral functional defined on Lp, 1 ≤ p < ∞, under a growth condition we give an upper bound of the Clarke directional derivative and we obtain a nice inclusion between the Clarke subdifferential of the integral functional and the set of selections of the subdifferential of the integrand.

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