Fredholm Mappings and Banach Manifolds

Acta Mathematica Scientia, 2010

Sufficient conditions are given to assert that two C 1-mappings share only one value in a connected compact Banach manifold modelled over R n. The proof of the result, which is based upon continuation methods, is constructive.

Communications in Applied Analysis

In some previous papers we presented a fairly simple construction of a topo-logical degree for C 1 Fredholm maps of index zero between Banach manifolds which verifies the three fundamental properties of the classical degree theory: normalization, additivity and homotopy invariance. We show here that this degree is unique. Precisely, by an axiomatic approach similar to the one due to Amann-Weiss, we prove that there exists at most one real function satisfying the above properties, and this function must be integer valued. 2000 Mathematics Subject Classification. 47H11, 47A53, 58Cxx.

Dedicated to Alfonso Vignoli on the occasion of his 60th anniversary Summary: We give a version of the classical Invariance of Domain Theorem for nonlinear Fredholm maps of index zero between Banach spaces (and Banach manifolds). The proof is based on a finite dimensional reduction technique combined with a mod 2 degree argument for continuous maps between (finite dimensional) differentiable manifolds.

Bulletin of the Brazilian Mathematical Society, New Series, 2007

Sufficient conditions are given to assert that a perturbed mapping has a zero in a Banach manifold modelled over R n. The zero is estimated by means of sequences of Newton's iterations. The proof of the result is constructive and is based upon continuation methods.

Ukrainian Mathematical Journal, 2011

In this article a new class of Banach manifolds and a new class of mappings between them are presented and also the theory of degree of such mappings is given. Представлено новий клас многовидів Банаха та новий клас відображень між ними, а також наведено теорію степеня таких відображень. 0. Introduction. As it is known, the degree theory for infinite-dimensional mappings (of the kind "identical+compact") for the first time was given by Leray and Schauder. Afterwards, this theory was expanded up to various classes of mappings (for example, up to class Fredholm proper mappings) 1. However, these theories were not appropriate for solution of non-linear Hilbert problem. For solution of this problem the class of Fredholm Quasi-Linear (FQL) mappings, determined on Banach space, was introduced by A. I. Shnirelman, and was determined the degree of such a mapping, which has all the main properties of classical (finite-dimensional) degree (see [8]). Later, M. A. Efendiyev expanded this theory up to FQL-mappings, determined on quasicylindrical domains (see [6]). In the given article, this theory is expanded up to FQLmappings, determined between FQL-manifolds. In more details: In first part of this article an example of FQL-manifold, given in [2], is extended up to example of Banach manifold from a wide class, namely up to space H s (M, N) , where M and N are compact smooth manifolds of dimensions m , respectively n and N doesn't have boundary. First such structure is given in H s (M, N) at m < n , and later, at m ! n. In the last case m ! n

Rendiconti del Circolo Matematico di Palermo, 1980

Geometric and Functional Analysis, 2009

Abstract and Applied Analysis, 2006

We present an integer valued degree theory for locally compact perturbations of Fredholm maps of index zero between (open sets in) Banach spaces (quasi-Fredholm maps, for short). The construction is based on the Brouwer degree theory and on the notion of orientation for nonlinear Fredholm maps given by the authors in some previous papers. The theory includes in a natural way the celebrated Leray-Schauder degree.

Fixed Point Theory and Applications, 2013

In this paper, the concepts of conditionally sequential absorbing and pseudo-reciprocal continuous maps are introduced in connection to giving a brief discussion on the role of various types of commutativity (e.g., weakly compatible, occasionally weakly compatible, subcompatible, pseudo-compatible, etc.) and continuity-type conditions (e.g., reciprocal, weak reciprocal, g-reciprocal, conditionally reciprocal, subsequential and sequential continuity of type (A g ) and (A f )) in the context of existence of common fixed points of a pair of maps. Here, the utility of newly introduced maps (i.e., conditionally sequential absorbing and pseudo-reciprocal continuous) in view of common fixed points for a pair of maps satisfying contractive as well as nonexpansive Lipschitz-type conditions is shown. MSC: 47H10; 54H25