scholarly journals Orientation and Degree for Fredholm Maps of Index Zero Between Banach Spaces

2001 ◽  
pp. 201-213
Author(s):  
Pierluigi Benevieri
Keyword(s):  
Banach Spaces ◽  
Index Zero ◽  
Fredholm Maps

Linear Operators in Banach Spaces

2018 ◽  
Author(s):  
D. E. Edmunds ◽  
W. D. Evans
Keyword(s):  
Banach Spaces ◽  
Linear Operators ◽  
Fredholm Maps

Three main themes run through this chapter: compact linear operators, measures of non-compactness, and Fredholm and semi-Fredholm maps. Connections are established between these themes so as to derive important results later in the book.

scholarly journals Finite-dimensional Banach spaces with numerical index zero

2004 ◽  
Vol 53 (5) ◽  
pp. 1279-1289 ◽  
Author(s):  
Miguel Martin ◽  
Javier Meri ◽  
Angel Rodriguez-Palacios
Keyword(s):  
Banach Spaces ◽  
Index Zero ◽  
Finite Dimensional ◽  
Numerical Index

scholarly journals On a local degree for a class of multi-valued vector fields in infinite dimensional Banach spaces

1996 ◽  
Vol 1 (4) ◽  
pp. 381-396 ◽  
Author(s):  
N. M. Benkafadar ◽  
B. D. Gel'man
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Vector Fields ◽  
Existence Of Solutions ◽  
Index Zero ◽  
Compact Mapping ◽  
Infinite Dimensional ◽  
Upper Semicontinuous ◽  
Local Degree

This paper is devoted to the development of a local degree for multi-valued vector fields of the formf−F. Here,fis a single-valued, proper, nonlinear, Fredholm,C1-mapping of index zero andFis a multi-valued upper semicontinuous, admissible, compact mapping with compact images. The mappingsfandFare acting from a subset of a Banach spaceEinto another Banach spaceE1. This local degree is used to investigate the existence of solutions of a certain class of operator inclusions.

scholarly journals GLOBAL BRANCH OF SOLUTIONS FOR NON-LINEAR SCHRÖDINGER EQUATIONS WITH DEEPENING POTENTIAL WELL

2006 ◽  
Vol 92 (3) ◽  
pp. 655-681 ◽  
Author(s):  
C. A. STUART ◽  
HUAN-SONG ZHOU
Keyword(s):  
Potential Well ◽  
Index Zero ◽  
Topological Methods ◽  
Open Set ◽  
Connected Sets ◽  
Non Linear ◽  
Asymptotic Linearization ◽  
Positive Eigenfunction ◽  
Fredholm Maps ◽  
Global Branch

We consider the stationary non-linear Schrödinger equation\begin{equation*}\Delta u + \{1 + \lambda g(x)\} u = f(u)\mbox{with}u \in H^{1} (\mathbb{R}^{N}), u \not\equiv 0,\end{equation*} where $\lambda >0$ and the functions $f$ and $g$ are such that\begin{equation*} \lim_{s \rightarrow 0}\frac{f(s)}{s} = 0 \mbox{and} 1 < \alpha + 1 = \lim _{|s| \rightarrow \infty}\frac{f(s)}{s} < \infty\end{equation*} and \begin{equation*} g(x)\equiv 0 \mbox{on} \bar{\Omega}, g(x)\in (0, 1] \mbox{on} {\mathbb{R}^{N}} \setminus {\overline{\Omega}} \mbox{and} \lim_{|x| \rightarrow + \infty} g(x) = 1 \end{equation*} for some bounded open set $\Omega \in \mathbb{R}^{N}$. We use topological methods to establish the existence of two connected sets $\mathcal{D}^{\pm}$ of positive/negative solutions in $\mathbb{R} \times W^{2, p} (\mathbb{R}^{N})$ where $p \in [2, \infty) \cap (\frac{N}{2},\infty)$ that cover the interval $(\alpha,\Lambda(\alpha))$ in the sense that \begin{align*} P \mathcal{D}^{\pm} & = (\alpha, \Lambda(\alpha)) \text{where}P(\lambda, u) = \lambda \text{and furthermore,} \\ \lim_{\lambda \rightarrow \Lambda(\alpha)-}\left\Vert u_{\lambda} \right\Vert _{L^{\infty} (\mathbb{R}^{N})} & = \lim_{\lambda \rightarrow \Lambda (\alpha )-} \left\Vert u_{\lambda} \right\Vert _{W^{2, p}(\mathbb{R}^{N})} = \infty \text{ for }(\lambda, u_{\lambda}) \in \mathcal{D}^{\pm}. \end{align*} The number $\Lambda(\alpha)$ is characterized as the unique value of $\lambda$ in the interval $(\alpha, \infty)$ for which the asymptotic linearization has a positive eigenfunction. Our work uses a degree for Fredholm maps of index zero.

scholarly journals Two-dimensional Banach spaces with polynomial numerical index zero

2009 ◽  
Vol 430 (8-9) ◽  
pp. 2488-2500 ◽  
Author(s):  
Domingo García ◽  
Bogdan C. Grecu ◽  
Manuel Maestre ◽  
Miguel Martín ◽  
Javier Merí
Keyword(s):  
Banach Spaces ◽  
Two Dimensional ◽  
Index Zero ◽  
Numerical Index

scholarly journals A note on continuation problems

1986 ◽  
Vol 28 (1) ◽  
pp. 55-61 ◽  
Author(s):  
Rita Nugari
Keyword(s):  
Banach Spaces ◽  
Topological Structure ◽  
Vector Fields ◽  
Existence Of Solution ◽  
Index Zero ◽  
Bounded Linear ◽  
Solution Sets ◽  
Connected Sets ◽  
Continuation Principle ◽  
Parameter Dependent

Recently M. Martelli [6] and M. Furi and M. P. Pera [1] proved some interesting results about the existence and the global topological structure of connected sets of solutions to problems of the form:Lx = N(λ, x)with L:E → F a bounded linear Fredholm operator of index zero (where E, F are real Banach spaces), and N:ℝ × E → F a nonlinear map satisfying suitable conditions.While the existence of solution sets for this kind of problem follows from the Leray–Schauder continuation principle, it is our aim to show in this note that their global topological structure can be obtained as a consequence of the theory developed by J. Ize, I. Massabò, J. Pejsachowicz and A. Vignoli in [3, 4] about parameter dependent compact vector fields in Banach spaces.

The G-invariant implicit function theorem in infinite dimensions II

1986 ◽  
Vol 102 (3-4) ◽  
pp. 211-220 ◽  
Author(s):  
E. N. Dancer
Keyword(s):  
Banach Spaces ◽  
Group Action ◽  
Implicit Function Theorem ◽  
Implicit Function ◽  
Infinite Dimensions ◽  
Regularity Theorem ◽  
Fredholm Maps

SynopsisIn this paper, we study the perturbation of zeros of maps of Banach spaces where the maps are invariant under continuous groups of symmetries. In some cases, we allow the perturbed maps partially to break the symmetries. Our results improve earlier results of the author by removing smoothness conditions on the group action. The key new idea is a regularity theorem for the zeros of invariant Fredholm maps.

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