Éditeur(s) : HAL CCSD
Résumé : International audience
This paper deals with variational inclusions of the form 0 2 f(x) + g(x) K where f is smooth function from a re exive Banach space X into a Banach space Y , g is a Lipschitz function from X into Y and K is a nonempty closed convex cone in the space Y . We show that the previous problem can be solved by an extension of the Zincenko's method which can be seen as a perturbed Newton's method. Numerical results are given at the end of the paper.
Revista Investigacion Operacional

hal-00924143
https://hal.archives-ouvertes.fr/hal-00924143

Éditeur(s) : HAL CCSD
Résumé : International audience
It is established that the multivalued quasi variational inequalities in uniformly smooth Banach spaces are equivalent to the fixed-point problem. We use this equivalence to suggest and analyze some iterative algorithms for quasi variational inequalities with noncompact sets in Banach spaces. Our results are new and represent a significant improvement of the previously known results.;
JIPAM

hal-00778168
https://hal.univ-antilles.fr/hal-00778168

Éditeur(s) : HAL CCSD American Mathematical Society
Résumé : International audience
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, Lipschitz-smooth) 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 semicontinuous 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 Borwein-Preiss. Along the same vein, we derive subdifferential criteria for coercivity, Lipschitz behavior, cone-monotonicity, quasiconvexity, and convexity of lower semicontinuous functions which clarify, unify and extend many existing results for specific subdifferentials.
ISSN: 0002-9947

hal-00699220
https://hal.archives-ouvertes.fr/hal-00699220

Éditeur(s) : HAL CCSD
Résumé : International audience
This paper deals with variational inclusions of the form 0 ∈ K − f(x) − g(x) where f is a smooth function from a reflexive Banach space X into a Banach space Y , g is a function from X into Y admitting divided differences and K is a nonempty closed convex cone in the space Y . We show that the previous problem can be solved by a combination of two methods: the Newton and the Secant methods. We show that the order of the semilocal method obtained is equal to (1 + √5)/2. Numerical results are also given to illustrate the convergence at the end of the paper.
Serdica Math. J.

hal-00924150
https://hal.archives-ouvertes.fr/hal-00924150

Éditeur(s) : HAL CCSD Springer Verlag
Résumé : International audience
We provide a first-order necessary and sufficient condition for optimality of lower semicontinuous functions on Banach spaces using the concept of subdifferential. From the sufficient condition we derive that any subdifferential operator is monotone absorbing, hence maximal monotone when the function is convex.
ISSN: 0925-5001

hal-00761658
https://hal.archives-ouvertes.fr/hal-00761658
ARXIV : 1212.0532
DOI : 10.1007/s10898-013-0078-6

Éditeur(s) : HAL CCSD
Résumé : International audience
In this paper, we prove a new composition theorem for asymptotically antiperiodic and weighted pseudo antiperiodic functions. We also give some sufficient conditions to ensure invertibility of convolution operators in the space of antiperiodic functions. Then we prove the existence and uniqueness of asymptotically antiperiodic mild solutions to some fractional functional integro-differential equations in a Banach space using the Banach's fixed point theorem.
African Diaspora Journal of Mathematics

hal-01678067
https://hal.archives-ouvertes.fr/hal-01678067

Éditeur(s) : HAL CCSD
Résumé : International audience
We start from a basic version of the Hahn-Banach theorem, of which we provide a proof based on Tychonoff's theorem on the product of compact intervals. Then, in the first section, we establish conditions ensuring the existence of affine functions lying between a convex function and a concave one in the setting of vector spaces -- this directly leads to the theorems of Hahn-Banach, Mazur-Orlicz and Fenchel. In the second section, we caracterize those topological vector spaces for which certain convex functions are continuous -- this is connected to the uniform boundedness theorem of Banach-Steinhaus and to the closed graph and open mapping theorems of Banach. Combining both types of results readily yields topological versions of the theorems of the first section.
Nonconvex Optim. Appl.
Erice, Italy

Droits : info:eu-repo/semantics/OpenAccess
hal-00699215
https://hal.archives-ouvertes.fr/hal-00699215
https://hal.archives-ouvertes.fr/hal-00699215/document
https://hal.archives-ouvertes.fr/hal-00699215/file/M_NonconvexOptimAppl98.pdf

Éditeur(s) : HAL CCSD Taylor & Francis
Résumé : International audience
We provide an inequality relating the radial directional derivative and the subdifferential of proper lower semicontinuous functions, which extends the known formula for convex functions. We show that this property is equivalent to other subdifferential properties of Banach spaces, such as controlled dense subdifferentiability, optimality criterion, mean value inequality and separation principles. As an application, we obtain a first-order sufficient condition for optimality, which extends the known condition for differentiable functions in finite-dimensional spaces and which amounts to the maximal monotonicity of the subdifferential for convex lower semicontinuous functions. Finally, we establish a formula describing the subdifferential of the sum of a convex lower semicontinuous function with a convex inf-compact function in terms of the sum of their approximate epsilon-subdifferentials. Such a formula directly leads to the known formula relating the directional derivative of a convex lower semicontinuous function to its approximate epsilon-subdifferential.
ISSN: 0233-1934

hal-00694601
https://hal.archives-ouvertes.fr/hal-00694601
DOI : 10.1080/02331934.2011.645034

Éditeur(s) : HAL CCSD Springer Verlag
Résumé : International audience
Given a pair of Banach spaces X and Y such that one is the dual of the other, we study the relationships between generic Fréchet differentiability of convex continuous functions on Y (Asplund property), generic existence of linear perturbations for lower semicontinuous functions on X to have a strong minimum (Stegall variational principle), and dentability of bounded subsets of X (Radon-Nikodým Property).
ISSN: 0927-6947

hal-00694604
https://hal.archives-ouvertes.fr/hal-00694604

Éditeur(s) : HAL CCSD Heldermann
Résumé : International audience
Let f : X → R ∪ {+∞} be a lower semicontinuous function on a Banach space X. We show that f is quasiconvex if and only if its Clarke subdifferential ∂f is quasimonotone. As an immediate consequence, we get that f is convex if and only if ∂f is monotone.
Journal of Convex Analysis

hal-00699221
https://hal.archives-ouvertes.fr/hal-00699221

Éditeur(s) : HAL CCSD Springer Verlag
Résumé : International audience
We show that the lower limit of a sequence of maximal monotone operators on a reflexive Banach space is a representable monotone operator. As a consequence, we obtain that the variational sum of maximal monotone operators and the variational composition of a maximal monotone operator with a linear continuous operator are both representable monotone operators.
ISSN: 0927-6947

hal-00698802
https://hal.archives-ouvertes.fr/hal-00698802

Éditeur(s) : HAL CCSD Elsevier
Résumé : International audience
This paper is concerned with weighted pseudo almost automorphic functions, which are more general and complicated than weighted pseudo almost periodic functions. Using these properties, we establish an existence and uniqueness theorem for pseudo almost automorphic mild solutions to semilinear differential equations in a Banach spaces which extends known results.
ISSN: 0362-546X

hal-00617634
https://hal.archives-ouvertes.fr/hal-00617634

Éditeur(s) : HAL CCSD Society for Industrial and Applied Mathematics
Résumé : International audience
We establish a general Fermat rule for the problem of minimizing a lower semicontinuous function on a convex subset of a Banach space. Our basic tool is a constrained variational principle derived from the "smooth" variational principle of Borwein and Preiss. Specializing the Fermat rule to the case when the convex set is a "drop," we obtain a multidirectional Rolle-type inequality from which, in turn, we deduce a multidirectional mean value inequality, in the line of Clarke and Ledyaev. We follow the abstract approach of our previous paper [Trans. Amer. Math. Soc., 347 (1995), pp. 4147-4161], thus covering all standard situations met in applications, while stressing the links between the results and the few key properties that are needed.
ISSN: 1052-6234

hal-00699204
https://hal.archives-ouvertes.fr/hal-00699204

Éditeur(s) : HAL CCSD Springer Verlag
Résumé : International audience
The aim of this study is the approximation of a solution x* of a ganeralized equation 0 in f(x)+F(x) in Banach spaces, where f is a single-valued function whose second order Frechet derivative satisfies an Hölder condition and F stands for a set-valed map with closed graph. Using a fixed point theorem and the Aubin property of F, we show the existence and the superquadratic convergence of a sequence derived from a midpoint method.
ISSN: 0167-8019

hal-00642118
https://hal.archives-ouvertes.fr/hal-00642118

Éditeur(s) : HAL CCSD
Résumé : 6 pages
We provide a new proof that the subdifferential of a proper lower semicontinuous convex function on a Banach space is maximal monotone by adapting the pattern commonly used in the Hilbert setting. We then extend the arguments to show more precisely that subdifferentials of proper lower semicontinuous prox-bounded functions possess the Brondsted-Rockafellar property.
https://hal.archives-ouvertes.fr/hal-00838205

hal-00838205
https://hal.archives-ouvertes.fr/hal-00838205
ARXIV : 1306.5466