An analytical study of bifurcation problems for equations involving Fredholm mappings

1988 ◽  
Vol 110 (3-4) ◽  
pp. 199-225 ◽  
Author(s):  
N.X. Tan
Keyword(s):  
Banach Space ◽  
Open Subset ◽  
Bifurcation Point ◽  
Normed Space ◽  
Analytical Study ◽  
Fredholm Mapping ◽  
Characteristic Value ◽  
Bifurcation Problems ◽  
Image Position

Let us consider equations in the formwhere Λ is an open subset of a normed space. For any fixed λ ∊ Λ, T, L(λ,.) and M(λ,.) are mappings from the closure D0 of a neighbourhood D0 of the origin in a Banach space X into another Banach space Y with T(0) = L(λ, 0) = M(λ, 0) = 0. Let λ be a characteristic value of the pair (T, L) such that T − L(λ,.) is a Fredholm mapping with nullity p and index s, p> s≧ 0. Under sufficient hypotheses on T, L and M, (λ, 0) is a bifurcation point of the above equations. Some well-known results obtained by Crandall and Rabinowitz [2], McLeod and Sattinger [5] and others will be generalised. The results in this paper are extensions of the results obtained by the author in [7].

TWO NEW GENERALISED HYPERSTABILITY RESULTS FOR THE DRYGAS FUNCTIONAL EQUATION

2017 ◽  
Vol 95 (2) ◽  
pp. 269-280 ◽  
Author(s):  
LADDAWAN AIEMSOMBOON ◽  
WUTIPHOL SINTUNAVARAT
Keyword(s):  
Banach Space ◽  
Functional Equation ◽  
Normed Space ◽  
Nonempty Subset ◽  
Image Position

Let $X$ be a nonempty subset of a normed space such that $0\notin X$ and $X$ is symmetric with respect to $0$ and let $Y$ be a Banach space. We study the generalised hyperstability of the Drygas functional equation $$\begin{eqnarray}f(x+y)+f(x-y)=2f(x)+f(y)+f(-y),\end{eqnarray}$$ where $f$ maps $X$ into $Y$ and $x,y\in X$ with $x+y,x-y\in X$. Our first main result improves the results of Piszczek and Szczawińska [‘Hyperstability of the Drygas functional equation’, J. Funct. Space Appl.2013 (2013), Article ID 912718, 4 pages]. Hyperstability results for the inhomogeneous Drygas functional equation can be derived from our results.

scholarly journals The evaluation functionals associated with an algebra of bounded operators

1969 ◽  
Vol 10 (1) ◽  
pp. 73-76 ◽  
Author(s):  
J. Duncan
Keyword(s):  
Banach Space ◽  
Normed Space ◽  
Finite Rank ◽  
Bounded Operators ◽  
Image Position

In this note we shall employ the notation of [1] without further mention. Thus X denotes a normed space and P the subset of X × X′ given byGiven a subalgebra of B(X), the set {Φ(X,f):(x,f) ∈ P} of evaluation functional on is denoted by II. We shall prove that if X is a Banach space and if contains all the bounded operators of finite rank, then Π is norm closed in ′. We give an example to show that Π need not be weak* closed in ″. We show also that FT need not be norm closed in ″ if X is not complete.

The Generalized Rayleigh Quotient

1974 ◽  
Vol 17 (2) ◽  
pp. 251-256 ◽  
Author(s):  
M. V. Pattabhiraman
Keyword(s):  
Banach Space ◽  
Symmetric Matrix ◽  
Matrix Pencil ◽  
Complex Banach Space ◽  
Rayleigh Quotient ◽  
Positive Definite ◽  
Characteristic Vector ◽  
Characteristic Value ◽  
Image Position ◽  
Generalized Rayleigh Quotient

In this paper we generalize the concept of the Rayleigh quotient to a complex Banach space. Lord Rayleigh investigated the quotient(1)considered as a function of the components of q, in the case of a symmetric matrix pencil Aλ+C with A positive definite. It is known that R(q) has a stationary value when q is a characteristic vector of Aλ+C and that(2)where qi is a characteristic vector corresponding to the characteristic value λi

The wigner property for CL-spaces and finite-dimensional polyhedral banach spaces

2021 ◽  
pp. 1-17
Author(s):  
Dongni Tan ◽  
Xujian Huang
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Phase Function ◽  
Linear Isometry ◽  
Basic Properties ◽  
Finite Dimensional ◽  
Image Position

Abstract We say that a map $f$ from a Banach space $X$ to another Banach space $Y$ is a phase-isometry if the equality \[ \{\|f(x)+f(y)\|, \|f(x)-f(y)\|\}=\{\|x+y\|, \|x-y\|\} \] holds for all $x,\,y\in X$ . A Banach space $X$ is said to have the Wigner property if for any Banach space $Y$ and every surjective phase-isometry $f : X\rightarrow Y$ , there exists a phase function $\varepsilon : X \rightarrow \{-1,\,1\}$ such that $\varepsilon \cdot f$ is a linear isometry. We present some basic properties of phase-isometries between two real Banach spaces. These enable us to show that all finite-dimensional polyhedral Banach spaces and CL-spaces possess the Wigner property.

scholarly journals On uniformly distributed orbits of certain horocycle flows

1982 ◽  
Vol 2 (2) ◽  
pp. 139-158 ◽  
Author(s):  
S. G. Dani
Keyword(s):  
Probability Measure ◽  
Periodic Point ◽  
Open Subset ◽  
Invariant Probability Measure ◽  
Zero Measure ◽  
Lebesque Measure ◽  
Image Position

AbstractLet(where t ε ℝ) and let μ be the G-invariant probability measure on G/Γ. We show that if x is a non-periodic point of the flow given by the (ut)-action on G/Γ then the (ut)-orbit of x is uniformly distributed with respect to μ; that is, if Ω is an open subset whose boundary has zero measure, and l is the Lebesque measure on ℝ then, as T→∞, converges to μ(Ω).

On the centres of hereditary JBW-subalgebras of a JBW-algebra

1979 ◽  
Vol 85 (2) ◽  
pp. 317-324 ◽  
Author(s):  
C. M. Edwards
Keyword(s):  
Banach Space ◽  
General Theory ◽  
Jordan Algebra ◽  
Identity Element ◽  
Image Position

A JB-algebra A is a real Jordan algebra, which is also a Banach space, the norm in which satisfies the conditions thatandfor all elements a and b in A. It follows from (1.1) and (l.2) thatfor all elements a and b in A. When the JB-algebra A possesses an identity element then A is said to be a unital JB-algebra and (1.2) is equivalent to the condition thatfor all elements a and b in A. For the general theory of JB-algebras the reader is referred to (2), (3), (7) and (10).

scholarly journals TOPOLOGICAL CELL DECOMPOSITION AND DIMENSION THEORY IN P-MINIMAL FIELDS

2017 ◽  
Vol 82 (1) ◽  
pp. 347-358 ◽  
Author(s):  
PABLO CUBIDES KOVACSICS ◽  
LUCK DARNIÈRE ◽  
EVA LEENKNEGT
Keyword(s):  
Algebraic Geometry ◽  
Open Subset ◽  
Cell Decomposition ◽  
Real Algebraic Geometry ◽  
Dimension Theory ◽  
Definable Set ◽  
Image Position ◽  
Definable Function

AbstractThis paper addresses some questions about dimension theory for P-minimal structures. We show that, for any definable set A, the dimension of $\bar A\backslash A$ is strictly smaller than the dimension of A itself, and that A has a decomposition into definable, pure-dimensional components. This is then used to show that the intersection of finitely many definable dense subsets of A is still dense in A. As an application, we obtain that any definable function $f:D \subseteq {K^m} \to {K^n}$ is continuous on a dense, relatively open subset of its domain D, thereby answering a question that was originally posed by Haskell and Macpherson.In order to obtain these results, we show that P-minimal structures admit a type of cell decomposition, using a topological notion of cells inspired by real algebraic geometry.

scholarly journals On the essential spectrum of Banach-space operators

2000 ◽  
Vol 43 (3) ◽  
pp. 511-528 ◽  
Author(s):  
Jörg Eschmeier
Keyword(s):  
Banach Space ◽  
Essential Spectrum ◽  
Negative Index ◽  
Resolvent Set ◽  
Image Position ◽  
Nuclear Spaces ◽  
Simply Connected ◽  
Banach Space Operators

AbstractLet T and S be quasisimilar operators on a Banach space X. A well-known result of Herrero shows that each component of the essential spectrum of T meets the essential spectrum of S. Herrero used that, for an n-multicyclic operator, the components of the essential resolvent set with maximal negative index are simply connected. We give new and conceptually simpler proofs for both of Herrero's results based on the observation that on the essential resolvent set of T the section spaces of the sheavesare complete nuclear spaces that are topologically dual to each other. Other concrete applications of this result are given.

CLASSES OF SEQUENTIALLY LIMITED OPERATORS

2015 ◽  
Vol 58 (3) ◽  
pp. 573-586
Author(s):  
JAN H. FOURIE ◽  
ELROY D. ZEEKOEI
Keyword(s):  
Banach Space ◽  
Vector Space ◽  
Normed Space ◽  
Operator Ideal ◽  
Operator Ideals ◽  
Ideal Norm ◽  
Relevant Operator

AbstractThe purpose of this paper is to present a brief discussion of both the normed space of operator p-summable sequences in a Banach space and the normed space of sequentially p-limited operators. The focus is on proving that the vector space of all operator p-summable sequences in a Banach space is a Banach space itself and that the class of sequentially p-limited operators is a Banach operator ideal with respect to a suitable ideal norm- and to discuss some other properties and multiplication results of related classes of operators. These results are shown to fit into a general discussion of operator [Y,p]-summable sequences and relevant operator ideals.

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