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This book makes a significant inroad into the unexpectedly difficult question of existence of Fréchet derivatives of Lipschitz maps of Banach spaces into higher dimensional spaces. Because the question turns out to be closely related to porous sets in Banach spaces, it provides a bridge between descriptive set theory and the classical topic of existence of derivatives of vector-valued Lipschitz functions. The topic is relevant to classical analysis and descriptive set theory on Banach spaces. The book opens several new research directions in this area of geometric nonlinear functional analysis. The new methods developed here include a game approach to perturbational variational principles that is of independent interest. Detailed explanation of the underlying ideas and motivation behind the proofs of the new results on Fréchet differentiability of vector-valued functions should make these arguments accessible to a wider audience. The most important special case of the differentiability results, that Lipschitz mappings from a Hilbert space into the plane have points of Fréchet differentiability, is given its own chapter with a proof that is independent of much of the work done to prove more general results. The book raises several open questions concerning its two main topics.

This book is the first of a set dedicated to the mathematical tools used in partial differential equations derived from physics.Its focus is on normed or semi-normed vector spaces, including the spaces of Banach, Fréchet and Hilbert, with new developments on Neumann spaces, but also on extractable spaces.

This paper is mainly concerned with distributional chaos and the principal measure of C 0 -semigroups on a Frechet space. New definitions of strong irregular (semi-irregular) vectors are given. It is proved that if C 0 -semigroup T has strong irregular vectors, then T is distributional chaos in a sequence, and the principal measure μ p ( T ) is 1. Moreover, T is distributional chaos equivalent to that operator T t is distributional chaos for every ∀ t > 0 .

Extending a result of Mashreghi and Ransford, we prove that every complex separable infinite-dimensional Fréchet space with a continuous norm is isomorphic to a space continuously included in a space of holomorphic functions on the unit disc or the complex plane, which contains the polynomials as a dense subspace. As a consequence, we deduce the existence of nuclear Fréchet spaces of holomorphic functions without the bounded approximation.

In this article, it is shown that if 𝐺 is a strongly topological gyrogroup, 𝐻 is a closed strong subgyrogroup of 𝐺 and 𝐻 is inner neutral, then the quotient space 𝐺/𝐻 is a sequential 𝛼4-space if and only if it is a strongly Fréchet-Urysohn space, which deduces that the quotient space 𝐺/𝐻 is a weakly first-countable space if and only if it is metrizable. Then, it is shown that every Fréchet-Urysohn Hausdorff paratopological gyrogroup having the property (🞰🞰) is a strong 𝛼4-space, which deduces that every Fréchet- Urysohn Hausdorff paratopological gyrogroup having the property (🞰🞰) is a strongly Fréchet-Urysohn space. Moreover, it is shown that if a Hausdorff paratopological gyrogroup having the property (🞰🞰) is a sequential 𝛼4-space, then it is a strongly Fréchet-Urysohn space. Finally, we investigate the Fréchet-Urysohn Hausdorff paratopological gyrogroup with an 𝜔𝜔-base and show that every Fréchet-Urysohn Hausdorff paratopological gyrogroup having the property (🞰🞰) with an 𝜔𝜔-base is first-countable.

Topological gyrogroups, with a weaker algebraic structure than groups, were investigated lately. In this paper, to begin with, it is shown that every first-countable Hausdorff topological gyrogroup is metrizable, which gives an affirmative answer to an open question. Next, paratopological loops are introduced. Some convergence phenomena and the property of being an ℵ-space in paratopological loops are discussed, which improves some known conclusions. Finally, a few questions about paratopological loops are posed.

In this paper we will present abstract versions of fundamental theorem of calculus (FTC) in the setting of Kurzweil - Henstock integral for functions taking values in an infinite dimensional locally convex space. The result will also be dealt with weaker forms of primitives in a widespread setting of integration theories generalising Riemann integral.

We introduce the concepts of growth and spectral bound for strongly continuous semigroups acting on Fréchet spaces and show that the Banach space inequality s(A) ⩽ ω 0(T) extends to the new setting. Via a concrete example of an even uniformly continuous semigroup, we illustrate that for Fréchet spaces effects with respect to these bounds may happen that cannot occur on a Banach space.

In this paper, sufficient conditions are given to investigate the existence of mild solutions on a semi-infinite interval for first order semi linear impulsive neutral functional differential evolution inclusions with infinite delay using a recently developed nonlinear alternative for contractive multivalued maps in Frechet spaces due to Frigon combined with semigroup theory. The existence result has been proved without assumption of compactness of the semigroup. We introduced a new phase space for impulsive system with infinite delay and claim that the phase space considered by different authors are not correct.

This volume contains the proceedings of the Special Session on Several Complex Variables, which was held during the first USA-Uzbekistan Conference on Analysis and Mathematical Physics from May 20-23, 2014, at California State University, Fullerton.This volume covers a wide variety of topics in pluripotential theory, symplectic geometry and almost complex structures, integral formulas, holomorphic extension, and complex dynamics. In particular, the reader will find articles on Lagrangian submanifolds and rational convexity, multidimensional residues, S-parabolic Stein manifolds, Segre varieties, and the theory of quasianalytic functions.