Existence theorems on unbounded sets in Banach spaces

Fixed points for multivalued mappings defined on unbounded sets in Banach spaces

ABSTRACTFor a bounded closed convex subset C of a Banach space X with the origin as an interior point, we study the convexity issue of proximinal sets G in Banach spaces in the sense of generalized best approximation determined by the Minkowski functional generated by C. Some sufficient conditions such as the smoothness and compactly locally uniform convexity of C, or the weakly uniform convexity of are provided for a proximinal set G to be convex.

A "purely geometric" proof of the Lindenstrauss-Troyanski result ([2], [6]) on strongly exposed points of weakly compact sets in Banach spaces is given.I would like to thank the referee for his suggestions.We consider a Banach-space X, \\ ||.If G X and if e > 0, then B(,e) = {x G X; \\x -|| < e}.The closed convex hull of a set A G X, is denoted c(A).Let C be a convex subset of X, then Ce is the set of the extremal points of C. If F is a convex, weakly compact, F a convex, closed, bounded set of X and J a closed subinterval of [0,1 ], (K, B,J) denotes the closed, convex setWe have to introduce a few geometrical definitions, referring to [4].Suppose that C is a nonempty, bounded, closed and convex subset of X.Let"strongly exposed" if there exists / G X* (\\f\\ = 1) such that Ve > 0, 3a > 0 with G S(fa, C) G B(e).Let S be the set of all/ G X* such that 11/11 = 1 and/strongly exposes some point of C. Proposition I.If K is a nonempty, convex, weakly-compact subset of X and if B is convex, closed and bounded with K fl B = 0, then the set D = c(K U F) = (K,B, [0,1]) has the following property: Ve > 0, 3 G K such that c(D\B(,e)).First we observe that it is sufficient to prove the proposition for X separable.Indeed, suppose D does not have the required property.Then there exists e > 0 with Vx G K: x G c(D\B(x,e)) and hence, Vx G K, 3A(x) G D with the properties: x G c(A(x)), A(x) n B(x, e) = 0 and A(x) is countable.By induction we construct a sequence (Kn,Bn)n where Kn is countable in K and Bn is countable in F. Let (K0,B0) = ({x},0), where x is some element of K. Suppose we already found (Kn,Bn).Consider Vy G UxeKnA(x) an element ky in K and by in B, with y G c({ky,by)).Let

We establish a necessary and sufficient condition for the differentiability of the distance function generated by a nonempty closed set K in a real normed linear space X under a proximinality condition on K. We do not assume the uniform differentiability constraints on the norm of the space as in Giles (Proc Am Math Soc 104: 458–464, 1988). Hence, our result advances that of Giles (Proc Am Math Soc 104: 458–464, 1988). We prove that the proximinal condition of Giles (Proc Am Math Soc 104: 458–464, 1988) is true for almost suns. The proximinal condition ensures convexity of an almost sun in some class of strongly smooth spaces under a differentiability condition of the distance function. A necessary and sufficient condition is obtained for the convexity of Chebyshev sets in Banach spaces with rotund dual.

This paper is concerned with a class of problem of best simultaneous approximations from $RS$-sets in Banach spaces $X$. It is shown that the best simultaneous approximations from an $RS$-set is strongly unique in the case when $X$ is a real Banach space, and strongly unique of order $\alpha \geq 2$ in the case when $X$ is a complex Banach space.

In this paper, we first introduce a Bregman prox-convexity notion as the generalization of the convexity for Bregman prox-convexity in Banach spaces. We verify the single-valudeness and Bregman firm nonexpansiveness of the Bregman proximity operator of the functions and provide examples of the notion. Using this notion, we show that the Bregman proximal point scheme remains convergent under appropriate conditions on the functions to be minimized in the setting of Banach spaces.

In this paper, we define the prox-regularity for functions on Banach spaces by adapting the original definition in R n . In this context, we establish a subdifferential characterization and show that qualified convexly C 1,+-composite functions and primal lower nice functions belong to this class, as already known in the setting of Hilbert spaces. We also study, in a geometrical point of view, the epigraphs of prox-regular functions. The subdifferential characterization allows us to show that some Moreau-envelope-like regularizations of such functions are of class C 1 in the context of certain uniformly convex spaces.

We study the notion of (I)-generating introduced by V. Fonf and J. Lindendstrauss and a related notion of (I)-envelope. As a consequence of our results we get an easy proof of the James characterization of weak* compactness in Banach spaces with weak angelic dual unit ball and an easy proof of the James characterization of reflexivity within a large class of spaces. We also show by an example that the general James theorem cannot be proved by this method.

Weak convergence of random sets in Banach spaces

Introduction. S. Mazur [10] was the first to consider the following smoothness property in normed linear spaces, called the Mazur Intersection Property (MIP

An iterative procedure for constructing zeros of accretive sets in Banach spaces

Constructing constant width sets or, more generally, complete sets in Banach spaces seems to be a not so easy task. A new construction working in separable Banach spaces is presented, and Bavaud’s and Lachand-Robert and Oudet’s constructions of complete sets are extended to a more general context.

We prove that every boundedly compact m-connected (Menger-connected) set is monotone path-connected and is a sun in a broad class of Banach spaces (in particular, in separable spaces). We show that the intersection of a boundedly compact monotone path-connected (m-connected) set with a closed ball is cell-like (of trivial shape) and, in particular, acyclic (contractible in the finite-dimensional case) and is a sun. We also prove that every boundedly weakly compact m-connected set is monotone path-connected. In passing, we extend the Rainwater–Simons weak convergence theorem to the case of convergence with respect to the associated norm (in the sense of Brown).

Let E \mathbf {E} be a separable strictly convex Banach space of dimension at least 2. It is shown that there exists a nonempty compact connected set X ⊂ E X \subset \mathbf {E} such that the nearest point mapping p X : E → 2 E p_X: \mathbf {E}\to 2^{ \mathbf {E}} is not single valued on a set of points dense in E \mathbf {E} . Furthermore, it is proved that most (in the sense of the Baire category) nonempty compact connected sets X ⊂ E X\subset \mathbf {E} have the above property. Similar results hold for the furthest point mapping.

Relations between certain classes of sets in Banach spaces