Alessandro Trombetta

PROPER 1-BALL CONTRACTIVE RETRACTIONS IN BANACH SPACES OF MEASURABLE FUNCTIONS

In this paper we consider the Wosko problem of evaluating, in an infinite-dimensional Banach space X, the infimum of all k > 1 for which there exists a k-ball contractive retraction of the unit ball onto its boundary. We prove that in some classical Banach spaces the best possible value 1 is attained. Moreover we give estimates of the lower H-measure of noncompactness of the retractions we construct. 1. Introduction Let X be an infinite-dimensional Banach space with unit closed ball B(X) and unit sphere S(X). It is well known that, in this setting, there is a retraction of B(X) onto S(X), that is, a continuous mapping R : B(X) ! S(X) with Rx = x for all x 2 S(X). In (4) Benyamini and Sternfeld, following Nowak ((13)), proved that such a retraction can be chosen among Lipschitz mappings. The problem of evaluating the infimum k0(X) of the Lipschitz constants of such retractions is of considerable interest in the literature. A general result states that in any Banach space X, 3 6 k0(X) 6 k0 (see (8, 10)), where k0 is a universal constant. In special spaces more precise estimates have been obtained by means of constructions which depend on each space. We refer the reader to (9, 10) for a collection of results on this problem and related ones.

ON THE ADMISSIBILITY OF THE SPACE L 0 (

We prove the admissibility of the space L0(A,X) of vector- valued measurable functions determined by real-valued finitely additive set functions defined on algebras of sets. The notion of admissibility introduced by Klee (7) guarantees that a com- pact mapping into an admissible Hausdorff topological vector space E can be approximated by compact finite dimensional mappings. This notion is very important in degree theory and fixed point theory. It is known that locally convex spaces are admissible (see (10)). There are some classes of nonlocally convex spaces which are admissible. Riedrich in (13) proved the admissibility of the space S(0,1) of measurable functions and in (12) the admissibility of the space Lp(0,1) for 0 < p < 1. The admissibility of other function spaces has been proved by Mach (6) and Ishii (8). In (14) it is proved the admissibility of spaces of Besov-Triebel-Lizorkin type. Definition 1 ((7)). Let E be a Haudorff topological vector space. A subset Z of E is said to be admissible if for every compact subset K of Z and for every neighborhood V of zero in E there exists a continuous mapping H : K → Z such that dim(span (H(K)))< +∞ and x−Hx ∈ V for every x ∈ K. If Z = E we say that the space E is admissible. In this paper we deal with spaces of vector-valued measurable functions and, as a major fact, instead of σ-additive measures we consider finitely additive set functions defined on algebras of sets. Let X be a Banach space, a nonempty set, A a subalgebra of the power set P() of and µ : A → R a finitely additive set function. We prove

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We present a theorem about calculation of fixed point index for k-ψ-contractive operators with 0 ≤ k < 1 defined on a radial set of a wedge of an infinite-dimensional Banach space. Then, results on the existence of eigenvectors and nonzero fixed points are obtained.

,X) OF VECTOR-VALUED MEASURABLE FUNCTIONS

Abstract We present new boundary conditions under which the fixed point index of a strict- ψ -contractive wedge operator is zero. Then we investigate eigenvalues and corresponding eigenvectors of k – ψ -contractive wedge operators.

On Boundary Conditions for Wedge Operators on Radial Sets

Abstract Let Ω be a Lebesgue-measurable set in ℝn of finite positive Lebesgue measure. In this note we calculate the lack of equi-measurability of the set Kc(Ω), c > 0, of all Lebesgue-measurable functions f : Ω → ℝ such that 0 ≤ f ≤ c, a.e. on Ω. From our result we repair a gap in the Example 2.3 of the paper [APPELL, J.—DE PASCALE, E.: Su alcuni parametri connessi con la misura di non compattezza di Hausdorff in spazi di funzioni misurabili, Boll. Un. Mat. Ital. B (6) 3 (1984), 497–515].

Eigenvectors of k–ψ-contractive wedge operators

We estimate the measure of nonconvex total boundedness in terms of simpler quantitative characteristics in the space of measurable functions   𝐿 0 . A Frechet-Smulian type compactness criterion for convexly totally bounded subsets of 𝐿 0 is established.

On the lack of equi-measurability for certain sets of Lebesgue-measurable functions

An implicit function theorem in locally convex spaces is proved. As an application we study the stability, with respect to a parameter , of the solutions of the Hammerstein equation in a locally convex space.

On Convex Total Bounded Sets in the Space of Measurable Functions

We prove that the convergence of a sequence of functions in the space of measurable functions, with respect to the topology of convergence in measure, implies the convergence-almost everywhere ( denotes the Lebesgue measure) of the sequence of rearrangements. We obtain nonexpansivity of rearrangement on the space, and also on Orlicz spaces with respect to a finitely additive extended real-valued set function. In the space and in the space, of finite elements of an Orlicz space of a-additive set function, we introduce some parameters which estimate the Hausdorff measure of noncompactness. We obtain some relations involving these parameters when passing from a bounded set of, or, to the set of rearrangements.