A fixed point result in strictly convex Banach spaces

We define an alternate convexically nonexpansive map T on a bounded, closed, convex subset C of a Banach space X and prove that if X is a strictly convex Banach space and C is a nonempty weakly compact convex subset of X, then every alternate convexically nonexpansive map T : C → C has a fixed point. As its application, we give an existence result for the solution of an integral equation.     

On a fixed point result of Amini-Harandi in strictly convex Banach spaces

Summary Amini-Harandi proved that alternate convexically nonexpansive mappings on non-empty weakly compact convex subsets of strictly convex Banach spaces have fixed points. We prove that Amini-Harandi's result holds also in Banach spaces with the Kadec--Klee property and the result is true for a larger class of mappings. Moreover, we show that the Alspach mapping in L¹[0,1] is not a 2-alternate convexically nonexpansive mapping.     

On equivalent strictly G-convex renormings of Banach spaces

We study strictly G-convex renormings and extensions of strictly G-convex norms on Banach spaces. We prove that ℓ_ω(Γ) space cannot be strictly G-convex renormed given Γ is uncountable and G is bounded and separable.     

Bilinear maps of locally convex spaces

Translated from Matematicheskie Zametki, Vol. 47, No. 3, pp. 58–64, March, 1990.     

Porosity and unique completion in strictly convex spaces

We are concerned in this paper with topological and stochastic properties of the family ${\mathcal G}$ of all closed convex sets with a unique extension to a complete set. With the help of a strengthened version of a lemma by Groemer we show that, in Minkowski spaces with a strictly convex norm, ${\mathcal G}$ is lower porous. This improves a previous result from Groemer (Geom. Dedicata 20:319?C334, 1986) where, in the same context, ${\mathcal G}$ was proved to be nowhere dense. In contrast to this fact we show that, in these spaces, there is a stochastic construction procedure which provides a complete set with probability one. This generalizes an earlier result of Bavaud (Trans. Amer. Math. Soc. 333(1):315?C324, 1992) proved for the particular case of the Euclidean plane.     

Quasireflexive locally convex spaces without Banach subspaces

Translated from Teoriya Funktsii, Funktsional'nyi Analiz i Ikh Prilozheniya, No. 44, pp. 78–84, 1985.     

Non convex control problems in Banach spaces

We consider a general control problem which includes, as particular cases, Bolza, Lagrange and Mayer problems. We show that it can be reduced to a free problem and we give sufficient conditions for the existence of a minimum over all absolutely continuous arcs with values in a reflexive, separable Banach space. A regularization result is also proved and an application to explicit control problems is considered.This work was supported by the Laboratorio per la Matematica Applicata del C. N. R.-Istituto di Matematica della Università di Genova.     

k-super-strongly convex and k-super-strongly smooth Banach spaces

In this paper, we introduce two types of new Banach spaces: k-super-strongly convex spaces and k-super-strongly smooth spaces. It is proved that these two notions are dual. We also prove that the class of k-super-strongly convexifiable spaces is strictly between locally k-uniformly rotund spaces and k-strongly convex spaces, and obtain some necessary and sufficient conditions of k-super-strongly convex space (respectively k-super-strongly smooth space). Also, for each k?2, it is shown that there exists a k-super-strongly convex (respectively k-super-strongly smooth) space which is not (k−1)-super-strongly convex (respectively (k−1)-super-strongly smooth) space.     

Remarks on coarse embeddings of metric spaces into uniformly convex Banach spaces

We study the support and convergence conditions for a metric space to be coarsely embeddable into a uniformly convex Banach space. By using ultraproducts we also show that the coarse embeddability of a metric space into a uniformly convex Banach space is determined by its finite subspaces.     

Extensions of lipschitz maps into Banach spaces

It is proved that ifY ⊂X are metric spaces withY havingn≧2 points then any mapf fromY into a Banach spaceZ can be extended to a map fromX intoZ so that wherec is an absolute constant. A related result is obtained for the case whereX is assumed to be a finite-dimensional normed space andY is an arbitrary subset ofX. Supported in part by US-Israel Binational Science Foundation and by NSF MCS-7903042. Supported in part by NSF MCS-8102714.