Approximate Rolle's theorems for the proximal subgradient and the generalized gradient

Azagra, Daniel; Ferrera, Juan; López-Mesas, Fernando

doi:10.1016/s0022-247x(03)00267-1

Cited by 5 publications

(2 citation statements)

References 7 publications

(9 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Namely, these conditions, combined with the Ekeland principle, make it possible to show that the

\mathrm{argmin}\kern0.1em

set of a studied functionals is nonempty, which leads to the existence of equilibria via Fermat's rule. A similar approach has been applied in case of infinite‐dimensional versions of another milestone result of analysis, that is, the Rolle theorem—see Azagra et al, 4 where the approximate Rolle theorem was established and de B e Silva and Teixeira, 5 where the Palais–Smale condition was employed.…”

Section: Introductionmentioning

confidence: 99%

Remarks on the infinite‐dimensional counterparts of the Darboux theorem

Juszczyk

Kryszewski

2022

Math Methods in App Sciences

View full text Add to dashboard Cite

The Darboux theorem, one of the fundamental results in analysis, states that the derivative of a real (not necessarily continuously) differentiable function defined on a compact interval has the intermediate value property, that is, attains each value between the derivatives at the endpoints. The Bolzano intermediate value theorem, which implies Darboux's theorem when the derivative is continuous, states that a continuous real‐valued function f$$ f $$ defined on false[−1,1false]$$ \left[-1,1\right] $$ satisfying ffalse(−1false)<0$$ f\left(-1\right)<0 $$ and ffalse(1false)>0$$ f(1)>0 $$, has a zero, that is, ffalse(xfalse)=0$$ f(x)&#x0003D;0 $$ for at least one number −1

show abstract

“…Namely, these conditions, combined with the Ekeland principle, make it possible to show that the

\mathrm{argmin}\kern0.1em

Section: Introductionmentioning

confidence: 99%

Remarks on the infinite‐dimensional counterparts of the Darboux theorem

Juszczyk

Kryszewski

2022

Math Methods in App Sciences

View full text Add to dashboard Cite

show abstract

“…For the first time this implication was mentioned in [32] with a short proof. The opposite implication based on separable reduction was proved by Fabian [24,25] 4 . A short proof based on Theorem 9.8 follows.…”

mentioning

confidence: 98%

On the theory of subdifferentials

Ioffe

2012

Advances in Nonlinear Analysis

View full text Add to dashboard Cite

The theory presented in the paper consists of two parts. The first is devoted to basic concepts and principles such as the very concept of a subdifferential, trustworthiness and its characterizations, geometric consistence, fuzzy principles and calculus rules, methods of creation of new subdifferentials etc. In the second part we study certain specific subdifferentials, namely, subdifferentials associated with bornologies, their limiting versions and metric modifications. For each subdifferential we verify that the basic properties discussed in the first part are satisfied and prove calculus rules for two main operations: summation and partial minimization. Separate sections are devoted to the Fréchet and limiting Fréchet subdifferentials and to the approximate G-subdifferential. For the last two new definitions are given which lead to a certain unification and simplification of analysis. The sum rule for these two subdifferentials is proved with the so-called "linear metric qualification condition", so far the most general. In the last section we briefly discuss how other operations reduce to the two mentioned basic operations and give the corresponding calculus rules (with suitable versions of the metric qualification conditions).

show abstract

Proximal Calculus on Riemannian Manifolds

Azagra

Ferrera

2005

MedJM

Self Cite

View full text Add to dashboard Cite

We introduce a proximal subdifferential and develop a calculus for nonsmooth functions defined on any Riemannian manifold M . We give some applications of this theory, concerning, for instance, a Borwein-Preiss type variational principle on a Riemannian manifold M , as well as differentiability and geometrical properties of the distance function to a closed subset C of M . (2000). 49J52, 58E30, 58C30, 47H10. Mathematics Subject Classification

show abstract

Approximate Rolle's theorems for the proximal subgradient and the generalized gradient

Cited by 5 publications

References 7 publications

Remarks on the infinite‐dimensional counterparts of the Darboux theorem

Remarks on the infinite‐dimensional counterparts of the Darboux theorem

On the theory of subdifferentials

Proximal Calculus on Riemannian Manifolds

Contact Info

Product

Resources

About