Search results for "Integrability"

Painlev\'{e} analysis for a generalized nonlinear Schr\"{o}dinger equation

2008

Measurable selectors and set-valued Pettis integral in non-separable Banach spaces

2009

AbstractKuratowski and Ryll-Nardzewski's theorem about the existence of measurable selectors for multi-functions is one of the keystones for the study of set-valued integration; one of the drawbacks of this result is that separability is always required for the range space. In this paper we study Pettis integrability for multi-functions and we obtain a Kuratowski and Ryll-Nardzewski's type selection theorem without the requirement of separability for the range space. Being more precise, we show that any Pettis integrable multi-function F:Ω→cwk(X) defined in a complete finite measure space (Ω,Σ,μ) with values in the family cwk(X) of all non-empty convex weakly compact subsets of a general (n…

Pettis integrability of fuzzy mappings with values in arbitrary Banach spaces

2017

Abstract In this paper we study the Pettis integral of fuzzy mappings in arbitrary Banach spaces. We present some properties of the Pettis integral of fuzzy mappings and we give conditions under which a scalarly integrable fuzzy mapping is Pettis integrable.

Integrability via Reversibility

2017

Abstract A class of left-invariant second order reversible systems with functional parameter is introduced which exhibits the phenomenon of robust integrability: an open and dense subset of the phase space is filled with invariant tori carrying quasi-periodic motions, and this behavior persists under perturbations within the class. Real-analytic volume preserving systems are found in this class which have positive Lyapunov exponents on an open subset, and the complement filled with invariant tori.

Equilibrium measures for uniformly quasiregular dynamics

2012

We establish the existence and fundamental properties of the equilibrium measure in uniformly quasiregular dynamics. We show that a uniformly quasiregular endomorphism $f$ of degree at least 2 on a closed Riemannian manifold admits an equilibrium measure $\mu_f$, which is balanced and invariant under $f$ and non-atomic, and whose support agrees with the Julia set of $f$. Furthermore we show that $f$ is strongly mixing with respect to the measure $\mu_f$. We also characterize the measure $\mu_f$ using an approximation property by iterated pullbacks of points under $f$ up to a set of exceptional initial points of Hausdorff dimension at most $n-1$. These dynamical mixing and approximation resu…

Kurzweil-Henstock type integration on Banach spaces

2004

In this paper properties of Kurzweil-Henstock and Kurzweil-Henstock-Pettis integrals for vector-valued functions are studied. In particular, the absolute integrability for Kurzweil-Henstock integrable functions is characterized and a Kurzweil-Henstock version of the Vitali Theorem for Pettis integrable functions is given.

Riesz and Wolff potentials and elliptic equations in variable exponent weak Lebesgue spaces

2015

Submitted by Alexandre Almeida (jaralmeida@ua.pt) on 2015-11-12T11:41:07Z No. of bitstreams: 1 RieszWolff_RIA.pdf: 159825 bytes, checksum: d99abdf3c874f47195619a31ff5c12c7 (MD5) Approved for entry into archive by Bella Nolasco(bellanolasco@ua.pt) on 2015-11-17T12:18:41Z (GMT) No. of bitstreams: 1 RieszWolff_RIA.pdf: 159825 bytes, checksum: d99abdf3c874f47195619a31ff5c12c7 (MD5) Made available in DSpace on 2015-11-17T12:18:41Z (GMT). No. of bitstreams: 1 RieszWolff_RIA.pdf: 159825 bytes, checksum: d99abdf3c874f47195619a31ff5c12c7 (MD5) Previous issue date: 2015-04

A New Analysis of the Three-Body Problem

2022

In the recent papers [5, 18], respectively, the existence of motions where the perihelions afford periodic oscillations about certain equilibria and the onset of a topological horseshoe have been proved. Such results have been obtained using, as neighbouring integrable system, the so-called two-centre (or Euler) problem and a suitable canonical setting proposed in [16, 17]. Here we review such results.

Godbillon–Vey sequence and Françoise algorithm

2019

Abstract We consider foliations given by deformations d F + ϵ ω of exact forms dF in C 2 in a neighborhood of a family of cycles γ ( t ) ⊂ F − 1 ( t ) . In 1996 Francoise gave an algorithm for calculating the first nonzero term of the displacement function Δ along γ of such deformations. This algorithm recalls the well-known Godbillon–Vey sequences discovered in 1971 for investigation of integrability of a form ω. In this paper, we establish the correspondence between the two approaches and translate some results by Casale relating types of integrability for finite Godbillon–Vey sequences to the Francoise algorithm settings.

MR3191427 Naralenkov, Kirill M., A Lusin type measurability property for vector- valued functions. J. Math. Anal. Appl. 417 (2014), no. 1, 293307. 28…

2014

In the paper under review the author introduces the notion of Riemann measurability for vector-valued functions, generalizing the classical Lusin condition, which is equivalent to the Lebesgue measurability for real valued functions. Let X be a Banach space, let f : [a; b] ! X and let E be a measurable subset of [a; b]. The function f is said to be Riemann measurable on E if for each " > 0 there exist a closed set F E with (E n F) < 0 (where is the Lebesgue measure) and a positive number such that k XK k=1 ff(tk) ?? f(t0 k)g (Ik)k < " whenever fIkgKk =1 is a nite collection of pairwise non-overlapping intervals with max1 k K (Ik) < and tk; t0 k 2 Ik T F. The Riemann measurabilit…