The parity as an invariant for detecting bifurcation of the zeroes of one parameter families of nonlinear Fredholm maps

1993 ◽  
pp. 1-31 ◽  
Author(s):  
Patrick Fitzpatrick
Keyword(s):  
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 Nonlinear Fredholm maps of index zero and their singularities

1978 ◽  
Vol 68 (3) ◽  
pp. 317-317 ◽  
Author(s):  
R. A. Plastock
Keyword(s):  
Index Zero ◽  
Fredholm Maps

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.

A Local Bifurcation Theorem for C 1 -Fredholm Maps

1990 ◽  
Vol 109 (4) ◽  
pp. 995
Author(s):  
P. M. Fitzpatrick ◽  
Jacobo Pejsachowicz
Keyword(s):  
Local Bifurcation ◽  
Bifurcation Theorem ◽  
Fredholm Maps

scholarly journals Critical values of Fredholm maps

1988 ◽  
Vol 104 (3) ◽  
pp. 993-993 ◽  
Author(s):  
P. T. Church ◽  
J. G. Timourian
Keyword(s):  
Critical Values ◽  
Fredholm Maps

scholarly journals On the concept of orientability for Fredholm maps between real Banach manifolds

2000 ◽  
Vol 16 (2) ◽  
pp. 279 ◽  
Author(s):  
Pierluigi Benevieri ◽  
Massimo Furi
Keyword(s):  
Banach Manifolds ◽  
Fredholm Maps

The degree of Fredholm maps equivariant with respect to the actions of the circle and the torus

1990 ◽  
Vol 45 (2) ◽  
pp. 229-230 ◽  
Author(s):  
V G Zvyagin
Keyword(s):  
Fredholm Maps

Counting Zeros of C 1 Fredholm Maps of Index 1

2005 ◽  
Vol 37 (5) ◽  
pp. 778-792 ◽  
Author(s):  
J. López-gómez ◽  
C. Mora-Corral
Keyword(s):  
Fredholm Maps

Semi-Fredholm maps ofFK spaces

1975 ◽  
Vol 144 (1) ◽  
pp. 9-12 ◽  
Author(s):  
Albert Wilansky
Keyword(s):  
Fredholm Maps
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