arXiv subjects
Publications and source records attributed to Mohammed Bachir.
We generalize the classical Monge-Kantorovich duality--typically established for tight (Radon) probability measures--to separable Baire probability measures, which are strictly more general than tight measures on completely regular Hausdorff spaces. Within this broader framework, we also demonstrate the existence of solutions.
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 inequality constraints. This extension is achieved using the Dini differentiability concept which is more general than Fréchet or Gâteaux differentiability. We will illustrate our results by giving examples of optimisation problems with a finite or countable number of inequality constraints where the functions are not Gateaux-differentiable but only upper Dini-differentiable.
The aim of this article is to establish new two-functions minimax inequalities extending classical results such as Simons' minimax theorem. Our results will be proved in a non-compact setting. We also prove, under general conditions, that the one-function minimax equality is in fact equivalent to the well known Simons inequality. 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 J. Jahn in \cite{Ja}, replacing Fréchet-differentiability assumptions on the functions by the Gateaux-differentiability. Moreover, the closed convex cone with a nonempty interior in the constraints is replaced by a strictly general class of closed subsets introduced in the paper and called {\it ``admissible sets"}. Examples illustrating our results are given.
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 consist in the extension of the results known for the Fréchet and Gateaux-differentiable functions as well as for the Clarke's subdifferential of Lipschitz functions, to the more general Dini-differentiable functions. As consequences, we extend the result of B.H. Pourciau in \cite[Theorem 6, p. 445]{Po} from the convexity to the {\it "Dini-pseudoconvexity"}.
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 version of the {\it ``Bishop-Phelps-Bollob\'as"} theorem will be also given. The density and the residuality of bounded linear operators attaining their maximum on $C$ (known in the literature) will be replaced, for this class of sets, by being the complement of a $\sigma$-porous set. The result of the paper is applicable for both linear operators and non-linear mappings. When we apply our result to subsets (from this class) whose closed convex hull is the closed unit ball, we obtain a new class of Banach spaces involving property $(A)$ introduced by Lindenstrauss. We also establish that this class of Banach spaces is stable under $\ell_1$-sum when the spaces have a same ``modulus". Applications to norm attaining bounded multilinear mappings and Lipschitz mappings will also be given.
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 but also the complement of a $σ$-porous subset. In fact, we generalize the notion of numerical radius to a large class $\mathcal{Z}$ of vector-valued operators defined from $X\times X^*$ into a Banach space $W$ and we prove that the set of all elements of $\mathcal{Z}$ strongly (up to a symmetry) attaining their {\it numerical radius} is the complement of a $σ$-porous subset of $\mathcal{Z}$ and moreover the {\it "numerical radius"} {\it Bishop-Phelps-Bollobás property} is also satisfied for this class. Our results extend (up to the assumption on $X$) some known results in several directions: $(1)$ the density is replaced by being the complement of a $σ$-porous subset, $(2)$ the operators attaining their {\it numerical radius} are replaced by operators strongly (up to a symmetry) attaining their {\it numerical radius} and $(3)$ the results are obtained in the vector-valued framework for general linear and non-linear vector-valued operators (including bilinear mappings and the classical space of bounded linear operators).
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 depend on the variables $(x,t)\in X\times \R$. Several examples will be given, in various function spaces, to illustrate our results. The vector-valued framework is also considered.
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, limited and completely continuous linear operators. Moreover, our results yield a characterization of Gelfand-Phillips spaces and recover some known result of Schur spaces and reflexive spaces concerning the differentiability of real-valued Lipschitz functions.
Let (X, d) be a bounded metric space with a base point 0 X , (Y, $\bullet$) be a Banach space and Lip $α$ 0 (X, Y) be the space of all $α$-H{ö}lderfunctions that vanish at 0 X , equipped with its natural norm (0 < $α$ $\le$ 1). Let 0 < $α$ < $β$ $\le$ 1. We prove that Lip $β$ 0 (X, Y) is a $σ$-porous subset of Lip $α$ 0 (X, Y), if (and only if) inf{d(x, x ') : x, x ' $\in$ X; x = x ' } = 0 (i.e. d is non-uniformly discrete). A more general result will be given.
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.
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 $(α)$ 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 $σ$-porous set. Moreover, the results of the paper applies also to an abstract class of (linear and nonlinear) operator spaces.
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-Milman theorem. We show that the class of upper semicontinuous convex-trace functions attaining their maximum at exactly one Choquet-boundary point is residual and we obtain several enhanced versions of the maximum principle which generalize both the classical Bauer's theorem as well as its abstract version in the Choquet theory. We illustrate our notions and results with concrete examples of three different types.
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 class of functions but still with Lipschitz flavour. As an application we obtain a Banach-Stone-like theorem. On the other hand, we give an extension of a result of Bourgain and Diestel related to limited operators and cosingularity.
Schweizer, Sklar and Thorp proved in 1960 that a Menger space $(G,D,T)$ under a continuous $t$-norm $T$, induce a natural topology $τ$ 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 $(G,τ)$ is uniformly homeomorphic to a (deterministic) metric space $(G,σ_D)$ for some canonical metric $σ_D$ on $G$. As applications, we extend the fixed point theorem of Hicks to probabilistic metric spaces which are not necessarily Menger spaces and we prove a probabilistic Arzela-Ascoli type theorem.
We prove that, in the space of all probabilistic continuous functions from a probabilistic metric space G to the set $Δ$ + 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 prove that the set of all convex upper semi-continuous functions attaining there maximum at exactly one extremal point of X is a G $δ$ dense subset of the space of all convex upper semi-continuous functions equipped with a metric compatible with the uniform convergence .
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.