ON ASYMPTOTIC UNIFORM SMOOTHNESS AND NONLINEAR GEOMETRY OF BANACH SPACES

These notes concern the nonlinear geometry of Banach spaces, asymptotic uniform smoothness and several Banach–Saks-like properties. We study the existence of certain concentration inequalities in asymptotically uniformly smooth Banach spaces as well as weakly sequentially continuous coarse (Lipschitz) embeddings into those spaces. Some results concerning the descriptive set theoretical complexity of those properties are also obtained. We finish the paper with a list of open problem.

Nonlinear Weakly Sequentially Continuous Embeddings Between Banach Spaces

In these notes, we study nonlinear embeddings between Banach spaces that are also weakly sequentially continuous. In particular, our main result implies that if a Banach space $X$ coarsely (resp. uniformly) embeds into a Banach space $Y$ by a weakly sequentially continuous map, then every spreading model $(e_n)_n$ of a normalized weakly null sequence in $X$ satisfies $$ \|e_1+\ldots+e_k\|_{\overline{\delta}_Y}\lesssim\|e_1+\ldots+e_k\|_S,$$where $\overline{\delta }_Y$ is the modulus of asymptotic uniform convexity of $Y$. Among other results, we obtain Banach spaces $X$ and $Y$ so that $X$ coarsely (resp. uniformly) embeds into $Y$, but so that $X$ cannot be mapped into $Y$ by a weakly sequentially continuous coarse (resp. uniform) embedding.

ON BOUNDED WEAK AND PSEUDO-SOLUTIONS OF NONLINEAR DIFFERENTIAL EQUATIONS HAVING TRICHOTOMY WITH AND WITHOUT DELAY IN BANACH SPACES

In the present work we consider E is a Banach space, E* is its dual space and L(E) is the space of continuous linear operators from E to itself. A function x: ℝ → E is said to be a pseudo-solution of the equation [Formula: see text] where A:ℝ → L(E) is strongly measurable and Bochner integrable function on every finite subinterval of ℝ with f:ℝ × E → E is only assumed to be weakly weakly sequentially continuous or Pettis-integrable and the linear equation [Formula: see text] has a trichotomy with constants α ≥ 1 and σ > 0, if x is absolutely continuous function and for each x* ∈ E* there exists a negligible set ℵx* such that for each t ∉ ℵx*, then we have [Formula: see text] We give an existence theorem for bounded weak and pseudo-solutions of the nonlinear differential equations [Formula: see text] Let T, r, d > 0, Br = {x > E: ‖x‖ ≤ r} and CE([-d,0]) be the Banach space of continuous functions from [-d,0] into E. Finally we prove an existence result for the differential equation with delay [Formula: see text] where fd : [a,b] × CE([-d,0]) → E is weakly weakly sequentially continuous function, [Formula: see text] is strongly measurable and Bochner integrable operator on [a,b] and θtx(s) = x(t + s) for all s ∈ [-d,0].

Integrodifferential Equations on Time Scales with Henstock-Kurzweil-Pettis Delta Integrals

We prove existence theorems for integro-differential equations , , , , where denotes a time scale (nonempty closed subset of real numbers ), and is a time scale interval. The functions are weakly-weakly sequentially continuous with values in a Banach space , and the integral is taken in the sense of Henstock-Kurzweil-Pettis delta integral. This integral generalizes the Henstock-Kurzweil delta integral and the Pettis integral. Additionally, the functions and satisfy some boundary conditions and conditions expressed in terms of measures of weak noncompactness. Moreover, we prove Ambrosetti's lemma.