arXiv subjects
Publications and source records attributed to Mohammed Bachir.
It is well known that in $R^n$ , G{â}teaux (hence Fr{é}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 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 structure. Then, we caracterize the probabilistic invariant complete Menger groups by the space of all probabilistic 1-Lipschitz maps in the sprit of the classical Banach-Stone theorem.
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 between the two classes of functions which preserves the problems of minimization.
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 idea is the introduction of new recursive assumptions and useful consequences of the Baire category theorem and of the Banach isomorphism theorem.
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 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 $Φ$-convexity. Under general conditions on the class of functions $Φ$, the Krein-Milman-Ky Fan theorem asserts then, that every compact $Φ$-convex subset of a Hausdorff space, is the $Φ$-convex hull of its $Φ$-extremal points. We prove in this paper that, in the metrizable case the situation is rather better. Indeed, we can replace the set of $Φ$-extremal points by the smaller subset of $Φ$-exposed points. We establish under general conditions on the class of functions $Φ$, that every $Φ$-convex compact metrizable subset of a Hausdorff space, is the $Φ$-convex hull of its $Φ$-exposed points. As a consequence we obtain that each convex weak compact metrizable (resp. convex weak$^*$ compact metrizable) subset of a Banach space (resp. of a dual Banach space), is the closed convex hull of its exposed points (resp. the weak$^*$ closed convex hull of its weak$^*$ exposed points). This result fails in general for compact $Φ$-convex subsets that are not metrizable.
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.
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 Fréchet differentiable at $y\in Y$ whenever $f$ is Gâteaux differentiable at $T(y)\in X$.
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 Principles for discrete-time problems.
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-convolution of functions on a group metric invariant is in reality an extension of the internal law of $X$ to spaces of functions on $X$. We give an example of monoid $(S(X),\oplus)$ for the inf-convolution structure, (which is dense in the set of all $1$-Lipschitz bounded from bellow functions) for which, the map $\arg\min : (S(X),\oplus) \rightarrow (X,.)$ is a (single valued) monoid morphism. It is also proved that, given a group complete metric invariant $(X,d)$, the complete metric space $(\mathcal{K}(X),d_{\infty})$ of all Katetov maps from $X$ to $\R$ equiped with the inf-convolution has a natural monoid structure which provides the following fact: the group of all isometric automorphisms $Aut_{Iso}(\mathcal{K}(X))$ of the monoid $\mathcal{K}(X)$, is isomorphic to the group of all isometric automorphisms $Aut_{Iso}(X)$ of the group $X$. On the other hand, we prove that the subset $\mathcal{K}_C(X)$ of $\mathcal{K}(X)$ of convex functions on a Banach space $X$, can be endowed with a convex cone structure in which $X$ embeds isometrically as Banach space.