Mohammed Bachir

• Paris 1 Panthéon-Sorbonne University

We introduce a new intermediate optimization problem situated between Kantorovich's primal and dual formulations. This new problem extends Kantorovich's duality to separable Baire measures, which are strictly more general than tight (or Radon) measures in completely regular Hausdorff spaces. In the special case where the measures are Radon, our int...

We present a new alternative theorems for sequences of functions. As applications, we extend recent results in the literature related to first-order necessary conditions for optimality problems. Our contributions involve extending well-known results, previously established for a finite number of inequality constraints to a countable number of inequ...

We establish an alternative theorem and deduce some new minimax theorems extending classical results such as Fan, K$\ddot{\textnormal{o}}$nig and Simons theorems. Some applications will be given.

We give a general Lagrange multiplier rule for mathematical programming problems in a Hausdorff locally convex space. We consider infinitely many inequality and equality constraints. Our results gives in particular a generalisation of the result of Jahn (Introduction to the theory of nonlinear optimization, Springer, Berlin, 2007), replacing Fréche...

We give a class of bounded closed sets C in a Banach space satisfying a generalized and stronger form of the Bishop-Phelps property studied by Bourgain in [8] for dentable sets. A version of the "Bishop-Phelps-Bollobás" theorem will be also given. The density and the residuality of bounded linear operators attaining their maximum on C (known in the...

We provide new results of first-order necessary conditions of optimality problem in the form of John's theorem and in the form of Karush-Kuhn-Tucker's theorem. We establish our result in a topological vector space for problems with inequality constraints and in a Banach space for problems with equality and inequality constraints. Our contributions...

We give a general Lagrange multiplier rule for mathematical programming problems in a Hausdorff locally convex space. We consider infinitely many inequality and equality constraints. Our results gives in particular a generalisation of the result of J. Jahn in \cite{Ja}, replacing Fr\'echet-differentiability assumptions on the functions by the Gatea...

We give a class of bounded closed sets $C$ in a Banach space satisfying a generalized and stronger form of the Bishop-Phelps property studied by Bourgain in \cite{Bj} for dentable sets. A {\it "quantitative version of the Bishop-Phelps-Bollob\'as"} will be also given. On the one hand, the density and the risiduality of bounded linear operators atta...

In this work we provide a characterization of distinct types of (linear and non-linear) maps between Banach spaces in terms of the differentiability of certain class of Lipschitz functions. Our results are stated in an abstract bornological and non-linear framework. Restricted to the linear case, we can apply our results to compact, weakly-compact,...

It is well known that under certain conditions on a Banach space $X$, the set of bounded linear operators attaining their numerical radius is a dense subset. We prove in this paper that if $X$ is assumed to be uniformly convex and uniformly smooth then the set of bounded linear operators attaining their numerical radius is not only a dense subset b...

We prove that an asymmetric normed space is never a Baire space if the topology induced by the asymmetric norm is not equivalent to the topology of a norm. More precisely, we show that a biBanach asymmetric normed space is a Baire space if and only if it is isomorphic to its associated normed space.

Let (X,d) be a bounded metric space with a base point 0X, (Y,‖⋅‖) be a Banach space and Lip0α(X,Y) (0<α≤1) be the space of all α-Hölder-functions that vanish at 0X, equipped with its natural norm. Let 0<α<β≤1. We prove that Lip0β(X,Y) is σ-porous in Lip0α(X,Y), if (and only if) inf⁡{d(x,x′):x,x′∈X;x≠x′}=0 (i.e. d is non-uniformly discrete). A more...

We prove results of existence of a solution (resp. existence and uniqness of a solution) for nonlinear differential equations of type $x'(t) +G(x,t) x(t) = F(x,t),$ in an abstract Banach subspace $X$ of the space of bounded real-valued continuous functions, satisfying some general and natural property. In our work, the functions $F$ and $G$ jointly...

In this work we provide a characterization of distinct type of (linear and non-linear) maps between Banach spaces in terms of the differentiability of certain class of Lipschitz functions. Our results are stated in an abstract bornological and non-linear framework. Restricted to the linear case, we can apply our results to compact, weakly-compact,...

We establish a linear variational principle extending the Deville-Godefroy-Zizler's one. We use this variational principle to prove that if $X$ is a Banach space having property $(\alpha)$ of Schachermayer and $Y$ is any banach space, then the set of all norm strongly attaining linear operators from $X$ into $Y$ is a complement of a $\sigma$-porous...

Let (X, d) be a bounded metric space with a base point 0 X , (Y, $\bullet$) be a Banach space and Lip $\alpha$ 0 (X, Y) be the space of all $\alpha$-H{\"o}lderfunctions that vanish at 0 X , equipped with its natural norm (0 < $\alpha$ $\le$ 1). Let 0 < $\alpha$ < $\beta$ $\le$ 1. We prove that Lip $\beta$ 0 (X, Y) is a $\sigma$-porous subset of Lip...

We prove that an asymmetric normed space is never a Baire space if the topology induced by the asymmetric norm is not equivalent to the topology of a norm. More precisely, we show that a biBanach asymmetric normed space is a Baire space if and only if it is isomorphic to its associated normed space.

Let T : Y → X be a bounded linear operator between two real normed spaces. We characterize compactness of T in terms of differentiability of the Lipschitz functions defined on X with values in another normed space Z. Furthermore, using a similar technique we can also characterize finite rank operators in terms of differentiability of a wider class...

This work is devoted to the metrization of probabilistic spaces. More precisely, given such a space (G,D,⋆) and provided that the triangle function ⋆ is continuous, we exhibit an explicit and canonical metric σD on G such that the associated topology is homeomorphic to the so-called strong topology. As applications, we make advantage of this explic...

We introduce the notion of trace convexity for functions and respectively, for subsets of a compact topological space. This notion generalizes both classical convexity of vector spaces, as well as Choquet convexity for compact metric spaces. We provide new notions of trace-convexification for sets and functions as well as a general version of Krein...

Let X, Y be asymmetric normed spaces and Lc(X, Y) the convex cone of all linear continuous operators from X to Y. It is known that in general, Lc(X, Y) is not a vector space. The aim of this note is to prove, using the Baire category theorem, that if Lc(X, Y) is a vector space for some asymmetric normed space Y , then X is isomorphic to its associa...

We study the notion of finitely determined functions defined on a topological vector space E equipped with a biorthogonal system. We prove that, for real-valued convex functions defined on a Banach space with a Schauder basis, the notion of finitely determined function coincides with the classical continuity but outside the convex case there are ma...

Let $T:Y\to X$ be a bounded linear operator between two normed spaces. We characterize compactness of $T$ in terms of differentiability of the Lipschitz functions defined on $X$ with values in another normed space $Z$. Furthermore, using a similar technique we can also characterize finite rank operators in terms of differentiability of a wider clas...

We introduce and study a natural notion of probabilistic 1-Lipschitz maps. We prove that the space of all probabilistic 1-Lipschitz maps defined on a probabilistic metric space G is also a probabilistic metric space. Moreover, when G is a group, then the space of all prob-abilistic 1-Lipschitz maps defined on G can be endowed with a monoid structur...

Schweizer, Sklar and Thorp proved in 1960 that a Menger space $(G,D,T)$ under a continuous $t$-norm $T$, induce a natural topology $\tau$ wich is metrizable. We extend this result to any probabilistic metric space $(G,D,\star)$ provided that the triangle function $\star$ is continuous. We prove in this case, that the topological space $(G,\tau)$ is...

We prove that, in the space of all probabilistic continuous functions from a probabilistic metric space G to the set $\Delta$ + of all cumulative distribution functions vanishing at 0, the space of all 1-Lipschitz functions is compact if and only if the space G is compact. This gives a probabilistic Arzela-Ascoli type Theorem.

Let X be a nonempty convex compact subset of some Haus-dorff locally convex topological vector space S. The well know Bauer's maximum principle stats that every convex upper semi-continuous function from X into R attains its maximum at some extremal point of X. We give some extensions of this result when X is assumed to be compact metrizable. We pr...

We propose a simple mathematical model based on two axioms and the set theory to approach the problem developed by the philosopher B. Spinoza in the Ethics. We then use the Knaster-Tarski Theorem to prove the existence and uniqueness of the Substance asserted by Spinoza.

We extend the extension by Ky Fan of the Krein–Milman theorem. The Φ-extreme points of a Φ-convex compact metrizable space are replaced by the Φ-exposed points in the statement of Ky Fan theorem. Our main results are based on new variational principles which are of independent interest. Several applications will be given.

We derive conditions for optimality and Gâteaux differentiability of convex functions in certain spaces of infinite dimension. The Karush-Kuhn-Tucker theorem naturally extends to the case of a convex functions with a countable number of variables (such as series) and a finite number of constraints. Thus, under minimal hypotheses, a solution of firs...

We prove that the notion of Tykhonov well-posed problems is stable under the operation of inf-convolution. We deal with lower semicontinuous functions (not necessarily convex) defined on a metric magma. Several applications are given, in particular to the study of the map argmin\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym}...

It is well known that in $R^n$ , G{\^a}teaux (hence Fr{\'e}chet) differ-entiability of a convex continuous function at some point is equivalent to the existence of the partial derivatives at this point. We prove that this result extends naturally to certain infinite dimensional vector spaces, in particular to Banach spaces having a Schauder basis.

We introduce and study the natural notion of probabilistic 1-Lipschitz maps. We use the space of all probabilistic 1-Lipschitz maps to give a new method for the construction of probabilistic metric completion (respec-tively of probabilistic invariant metric group completion). Our construction is of independent interest. We prove that the space of a...

We establish an extension of the Banach–Stone theorem to a class of isomorphisms more general than isometries in a noncompact framework. Some applications are given. In particular, we give a canonical representation of some (not necessarily linear) operators between products of function spaces. Our results are established for an abstract class of f...

We propose a simple mathematical model based on two axioms and the set theory to approach the problem developed by the philosopher B. Spinoza in "the Ethics". We then use the Knaster-Tarski Theorem to prove the existence and uniqueness of the Substance asserted by Spinoza.

The aim of this paper is to establish Pontryagin's principles in a dicrete-time infinite-horizon setting when the state variables and the control variables belong to infinite dimensional Banach spaces. In comparison with previous results on this question, we delete conditions of finiteness of codi-mension of subspaces. To realize this aim, the main...

We prove that, any problem of minimization of proper lower semicontinuous function defined on a normal Hausdorff space, is canonically equivalent to a problem of minimization of a proper weak * lower semicontinuous convex function defined on a weak * convex compact subset of some dual Banach space. We estalish the existence of an bijective operator...

We prove that each isometric isomorphism, between the monoids of all nonegative $1$-Lipschitz maps defined on invariant metric groups and equiped with the inf-convolution law, is given canonically from an isometric isomorphism between their groups of units.

The Krein-Milman theorem (1940) states that every convex compact subset of a Hausdorff locally convex topological space, is the closed convex hull of its extreme points. In 1963, Ky Fan extended the Krein-Milman theorem to the general framework of $\Phi$-convexity. Under general conditions on the class of functions $\Phi$, the Krein-Milman-Ky Fan t...

The aim of this paper is to provide improvments to Pontryagin principles in infinite-horizon discrete-time framework when the space of states and of space of controls are infinite-dimensional. We use the method of reduction to finite horizon and several functional-analytic lemmas to realize our aim.

Given an invariant metric group $(X,d)$, we prove that the set $Lip^1_+(X)$ of all nonnegative and $1$-Lipschitz maps on $(X,d)$ endowed with the inf-convolution structure is a monoid which completely determine the group completion of $(X,d)$. This gives a Banach-Stone type theorem for the inf-convolution structure in the group framework.

Given two normed spaces X, Y , the aim of this paper is establish that the existence of an isomorphism isometric between X × R and Y × R is equivalent to the existence of an isometric isomorphism between X and Y , provided the norms satisfy an appropriate condition. By means of a counterexample, it is shown that this result fails for arbitrary norm...

The Krein-Milman theorem (1940) states that every convex compact subset of a Hausdorfflocally convex topological space, is the closed convex hull of its extreme points. In 1963, Ky Fan extended the Krein-Milman theorem to the general framework of $\Phi$-convexity. Under general conditions on the class of functions $\Phi$, the Krein-Milman-Ky Fan th...

We characterize the limited operators by differentiability of convex continuous functions. Given Banach spaces $Y$ and $X$ and a linear continuous operator $T: Y \longrightarrow X$, we prove that $T$ is a limited operator if and only if, for every convex continuous function $f: X \longrightarrow \R$ and every point $y\in Y$, $f\circ T$ is Fr\'echet...

We give some reasonable and usable conditions on a sequence of norm one in a dual banach space under which the sequence does not converges to the origin in the $w^*$-topology. These requirements help to ensure that the Lagrange multipliers are nontrivial, when we are interested for example on the infinite dimensional infinite-horizon Pontryagin Pri...

In this article, we bring a new light on the concept of the inf-convolution operation $\oplus$ and provides additional informations to the work started in \cite{Ba1} and \cite{Ba2}. It is shown that any internal law of group metric invariant (even quasigroup) can be considered as an inf-convolution. Consequently, the operation of the inf-convolutio...

This work generalize and extend results obtained reently in [2] from the Banah spaces framework to the groups framework. We study abstract classe of functions monoids for the inf-onvolution struture and gives a complete description of the group of unit of such monoids. We then apply this results to obtain various versions of the Banah-Stone theorem...

The aim of this paper is the extension to setting of the infinite-dimensional state and control spaces of the method of the reduction at finite horizon. This method is a way to establish Pontryagin principles for infinite-horizon discrete-time optimal control problems. We establish several new Pontryagin principles using this approach in the infini...

In this article we study the operation of inf-convolution in a new direction. We prove that the inf-convolution gives a monoid structure to the space of convex k-Lipschitz and bounded from below real-valued functions on a Banach space X. Then we show that the structure of the space X is completely determined by the structure of this monoid by estab...

We establish in this article a formula which will allow to classify isometries as well as partial isometries between spaces of functions. Our result applies in a non compact framework and for abstract class of functions spaces included in the space of continuous and bounded functions on complete metric space. The result of this article is in partic...

We give a multidirectional mean value inequality with second order information. This result extends the classical Clarke-Ledyaev's inequality to the second order. As application, we give the uniqueness of viscosity solution of second order Hamilton-Jacobi equations in finite dimensions.

We prove that a Banach space X has the Schur property if and only if every X-valued weakly differentiable function is Fréchet differ-entiable. We give a general result on the Fréchet differentiability of f • T , where f is a Lipschitz function and T is a compact linear operator. Finally we study, using in particular a smooth variational principle,...

We consider the question of integration of a multivalued operator T, that is the question of finding a function f such that Tf. If is the Fenchel–Moreau subdifferential, the above problem has been completely solved by Rockafellar, who introduced cyclic monotonicity as a necessary and sufficient condition. In this article we consider the case where...

We introduce and study a new notion of conjugacy, similar to Fenchel conjugacy, in a non-convex setting. Dual versions of Smulyan's classical result are established in the framework of this conjugacy, which reveal a relation between well-posed problems and the differentiability. As an application we deduce the generic Frechet differentiability of t...

We prove that a Banach space X has the Radon-Nikodym property if, and only if, every weak*-lower semicontinuous convex continuous function f of X* is Gâteaux differentiable at some point of its domain with derivative in the predual space x.

We introduce a new notion of conjugacy, similar to Fenchel conjugacy, in a non-convex setting. We establish relation between well-posed problems and differentiability. As an application we deduce results of generic differentiability of the norm ∥ · ∥∞ in certain spaces of bounded continuous functions. On the other hand, we extend the Banach-Stone's...

We introduce a new notion of conjugacy, similar to Fenchel conjugacy, in a non-convex setting. We establish relation between well-posed problems and differentiability. As an application we deduce results of generic differentiability of the norm ‖·‖∞ in certain spaces of bounded continuous functions. On the other hand, we extend the Banach–Stone's t...