A Smooth Variational Principle with Applications to Hamilton-Jacobi Equations in Infinite Dimensions

Deville, Robert; Godefroy, Gilles; Zizler, Václav

doi:10.1006/jfan.1993.1009

Cited by 112 publications

(103 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…We point out that Borwein-Preiss' result has been (partially) improved by Deville et al [ 14] who showed that, in the case where * denotes the Gâteaux, Hadamard or Fréchet notion of differentiability, the existence of a Lipschitz #-differentiable bump function on a Banach space X suffices to assure the dense #-subdifferentiability of lower semicontinuous functions on X. However, we do not develop this connection here since our main objective is to cover at once a large amount of cases in a systematic and simple way, for which a key role is played by the definition of a d -smooth norm.…”

Section: Introductionmentioning

confidence: 55%

Mean Value Property and Subdifferential Criteria for Lower Semicontinuous Functions

Aussel¹,

Corvellec²,

Lassonde³

1995

Transactions of the American Mathematical Society

View full text Add to dashboard Cite

Abstract.We define an abstract notion of subdifferential operator and an associated notion of smoothness of a norm covering all the standard situations. In particular, a norm is smooth for the Gâteaux (Fréchet, Hadamard, Lipschitzsmooth) subdifferential if it is Gâteaux (Fréchet, Hadamard, Lipschitz) smooth in the classical sense, while on the other hand any norm is smooth for the Clarke-Rockafellar subdifferential. We then show that lower semicontinüous functions on a Banach space satisfy an Approximate Mean Value Inequality with respect to any subdifferential for which the norm is smooth, thus providing a new insight on the connection between the smoothness of norms and the subdifferentiability properties of functions. The proof relies on an adaptation of the "smooth" variational principle of Bonvein-Preiss. Along the same vein, we derive subdifferential criteria for coercivity, Lipschitz behavior, conemonotonicity, quasiconvexity, and convexity of lower semicontinüous functions which clarify, unify and extend many existing results for specific subdifferentials.

show abstract

Section: Introductionmentioning

confidence: 55%

Mean Value Property and Subdifferential Criteria for Lower Semicontinuous Functions

Aussel¹,

Corvellec²,

Lassonde³

1995

Transactions of the American Mathematical Society

View full text Add to dashboard Cite

show abstract

“…They proved that there exists an everywhere dense G& set A o C A such that for each f € A o the problem (P f ) has a unique solution. Deville, Godefroy and Zizler [4] obtained an analogous result for a space of bounded continuous functions on a Banach space with the topology which is not weaker then the topology of the uniform convergence. The second author [9] established the existence result for optimal control problems with a generic integrand without convexity assumptions.…”

Section: Introductionmentioning

confidence: 86%

“…Recently Ioffe [5] discovered the connection between variational principles and generic existence results. In a joint paper with the second author [5] they obtained a general variational principle (an extension of the variational principle of Deville, Godefroy and Zizler [4]) and showed that generic existence results in optimization theory and calculus of variations are obtained as a realization of this principle.…”

Section: Introductionmentioning

confidence: 99%

Existence and uniqueness of a solution for a minimization problem with a generic increasing function

Rubinov¹,

Zaslavski²

1999

J Aust Math Soc A

View full text Add to dashboard Cite

In this paper we study the existence and uniqueness of a solution for minimization problems with generic increasing functions in an ordered Banach space X. The standard approaches are not suitable in such a setting. We propose a new type of perturbation adjusted for the problem under consideration, prove the existence and point out sufficient conditions providing the uniqueness of a solution. These results are proved by assuming that the space X enjoys the following property: each decreasing norm-bounded sequence has a limit. We supply a counterexample, which shows that this property is essential and give a modification of obtained results for the space C(T), which does not possess this property.1991 Mathematics subject classification (Amer. Math. Soc): primary 49J27, 90C30.

show abstract

“…The above scheme has been the case in the Ekeland variational principle [Ek1,Ek2], Borwein-Preiss smooth variational principle [BPr], Stegall variational principle [St], the generic variational principle ofČoban, Kenderov and Revalski [ČK,CKR1], Deville-Godefroy-Zizler variational principle [DGZ1,DGZ2] (see also a strengthening of the latter in [DR]) and Ioffe-Zaslavski principle [IZa]. With the exception of the first two principles, in the remaining ones the set of good perturbations contains a dense G δ -subset of the space of all perturbations.…”

Section: Introductionmentioning

confidence: 99%

Fragmentability of sequences of set-valued mappings with applications to variational principles

Lassonde¹,

Revalski

2005

Proc. Amer. Math. Soc.

View full text Add to dashboard Cite

show abstract

A Smooth Variational Principle with Applications to Hamilton-Jacobi Equations in Infinite Dimensions

Cited by 112 publications

References 0 publications

Mean Value Property and Subdifferential Criteria for Lower Semicontinuous Functions

Mean Value Property and Subdifferential Criteria for Lower Semicontinuous Functions

Existence and uniqueness of a solution for a minimization problem with a generic increasing function

Fragmentability of sequences of set-valued mappings with applications to variational principles

Contact Info

Product

Resources

About