Diffeomorphisms with Banach space domains
Gaetano Zampieri1992, Nonlinear Analysis: Theory, Methods & Applications
Sign up for access to the world's latest research
checkGet notified about relevant papers
checkSave papers to use in your research
checkJoin the discussion with peers
checkTrack your impact
Abstract
AI
AI
This paper investigates the conditions under which a C1 mapping between Banach spaces can be a global diffeomorphism. The study focuses on sufficient conditions for injectivity and bijectivity, leveraging nonnegative auxiliary scalar coercive functions to establish global properties of the mappings. The results contribute to the understanding of diffeomorphisms in the context of infinite-dimensional spaces.
Key takeaways
- In this paper we use the hypothesis that the operator norm f ′ (x) −1 is bounded on bounded sets, i.e. sup x ≤r f ′ (x) −1 < +∞ , ∀r : 0 < r < +∞ , (1.2) Theorem 2.1 (in the particular case we mentioned above) says that the local diffeomorphism f is injective if (1.2) holds and there exist a point x 0 ∈ X, and a coercive function k ∈ C 1 (X; R + ), such that
- We are going to use nonnegative auxiliary scalar coercive functions, that is continuous mappings k : X → R + with k(x) → +∞ as x → +∞ .
- if conditions (i) and (ii) in Theorem 2.1 are satisfied, and there exist points x 0 , x 1 ∈ X, and nonnegative real numbers a, b, c, such that
- So (3.2) is satisfied, (3.1) holds by (3.6), and Theorem 3.1 gives Corollary 3.2.
- The 'if' in Theorem 3.1 and Corollary 3.3 can be substituted by 'if and only if'.
Related papers
A criterion of invertibility in the large for local diffeomorphisms between Banach spaces1
Nonlinear Analysis Theory Methods Applications, 1993
A criterion of invertibility in the large for local diffeomorphisms between Banach spaces
Nonlinear Analysis: Theory, Methods & Applications, 1993
An inverse function theorem
Contemporary Mathematics, 1999
For k = 1, 2 ,. .. ∞ and a Frölicher-Kriegl order k Lipschitz differentiable map f : E ⊇ U → E having derivative at x 0 ∈ U a linear homeomorphism E → E and satisfying a Colombeau type tameness condition, we prove that x 0 has a neighbourhood V ⊆ U with f | V a local order k Lipschitz diffeomorphism. As a corollary we obtain a similar result for Keller C ∞ c maps with E in a class including Fréchet and Silva spaces. We also indicate a procedure for verifying the tameness condition for maps of the type x → ϕ • [ id , x ] and spaces E = C ∞ (Q) when Q is compact by considering the case Q = [ 0 , 1 ]. Our considerations are motivated by the wish to try to retain something valuable in an interesting but defective treatment of integrability of Lie algebras by J. Leslie.
Equivalence of Families of Diffeomorphisms on Banach Spaces
2008
We consider two families of C ∞-diffeomorphisms (with hyperbolic linear part at 0) on a Banach space. Suppose that these two families formally conjugate at 0. We prove that they admit local conjugation, which is infinitely smooth in both, the space variable and the family parameter. In particular, subject to non-resonance condition, there exists a family of local C ∞ linearizations of the family of diffeomorphisms. The linearizing family has C ∞ smoothness in parameter. The results are proved under the assumption that the Banach space allows a special extension of the maps. We discuss corresponding properties of Banach spaces.
On local convexity of nonlinear mappings between Banach spaces
Central European Journal of Mathematics, 2012
We find conditions for a smooth nonlinear map : U → V between open subsets of Hilbert or Banach spaces to be locally convex in the sense that for some and each positive ε < the image (B ε ( )) of each ε-ball B ε ( ) ⊂ U is convex. We give a lower bound on via the second order Lipschitz constant Lip 2 ( ), the Lipschitz-open constant Lip o ( ) of , and the 2-convexity number conv 2 (X ) of the Banach space X . MSC: 46B20, 49J50, 52A41, 58C20 Keywords: Locally convex mapping • Hilbert and Banach spaces • Modulus of convexity • Modulus of smoothness • Lipschitzopen maps © Versita Sp. z o.o. *
Local mappings on spaces of differentiable functions
Manuscripta Mathematica, 1993
On the range of the derivate of a smooth function and applications
2006
We survey recent results on the structure of the range of the derivative of a smooth real valued function f defined on a real Banach space X and of a smooth mapping F between two real Banach spaces X and Y . We recall some necessary conditions and some sufficient conditions on a subset A of L(X; Y ) for the existence of a Fr?echet-differentiable mapping F from X into Y so that F0(X) = A. Whenever F is only assumed G?ateaux-differentiable, new phenomena appear : we discuss the existence of a mapping F from a Banach space X into a Banach space Y , which is bounded, Lipschitz-continuous, and so that for all x; y 2 X, if x 6= y, then kF0(x) i F0(y)kL(X;Y ) > 1. Applications are given to existence and uniqueness of solutions of Hamilton-Jacobi equations.
Local inversion theorems without assuming continuous differentiability
Journal of mathematical analysis and …, 1989
On properties of subdifferential mappings in Frechet spaces
2005
UDC 517.9 We present conditions under which the subdifferential of a proper convex lower-semicontinuous functional in a Fréchet space is a bounded upper-semicontinuous mapping. The theorem on the boundedness of a subdifferential is also new for Banach spaces. We prove a generalized Weierstrass theorem in Fréchet spaces and study a variational inequality with a set-valued mapping. In recent years, operator and differential operator inclusions, multivariational inequalities, and systems containing both evolutionary and operator inclusions have been extensively studied. In reflexive Banach spaces, these objects were investigated by many authors (in particular, in [1-4]). Variational inequalities with a convex proper lower-semicontinuous functional ϕ are among the sources that generate operator inclusions [5]. In Banach spaces, the subdifferential ∂ϕ (⋅) of a proper lower-semicontinuous functional possesses several important properties [6-8], which play the key role in the investigation of variational inequalities. However, in separated locally convex spaces, analogous properties have not been investigated.
Related topics
Related papers
On the range of the derivative of a smooth mapping between Banach spaces
Abstract and Applied Analysis, 2005
We survey recent results on the structure of the range of the derivative of a smooth mapping f between two Banach spaces X and Y. We recall some necessary conditions and some sufficient conditions on a subset A of ᏸ(X,Y) for the existence of a Fréchet differentiable mapping f from X into Y so that f (X) = A. Whenever f is only assumed Gâteaux differentiable, new phenomena appear: for instance, there exists a mapping f from 1 (N) into R 2 , which is bounded, Lipschitz-continuous, and so that for all x, y ∈ 1 (N), if x = y, then f (x) − f (y) > 1.
Serdica Math. J. 21 (1995), 59-66 A MEAN VALUE THEOREM FOR NON DIFFERENTIABLE MAPPINGS IN BANACH SPACES
2015
Abstract. We prove that if f is a real valued lower semicontinuous function on a Banach space X and if there exists a C1, real valued Lipschitz continuous function on X with bounded support and which is not identically equal to zero, then f is Lipschitz continuous of constant K provided all lower subgradients of f are bounded by K. As an application, we give a regularity result of viscosity supersolutions (or subsolutions) of Hamilton-Jacobi equations in infinite dimensions which satisfy a coercive condition. This last result slightly improves some earlier work by G. Barles and H. Ishii. Let X be a Banach space and f: X → R be an arbitrary function. It has been proved by D. Preiss [12] that if X is an Asplund space and if f is Lipschitz continuous, then the set D of all points of differentiability of f is dense in X and f satisfies the mean value theorem; that is, for every x, y ∈ X: ‖f(x) − f(y) ‖ ≤ L‖x − y‖ where L = sup{‖f ′(x)‖;x ∈ D} ( = Lipschitz constant of f).
A note on a global invertibility of locally lipschitz functions on $R^n$
arXiv (Cornell University), 2015
We provide sufficient conditions for a locally lipschitz mapping f : R n → R n to be invertible. We use classical local invertibility conditions together with the non-smooth critical point theory.
Estimates for domains of local invertibility of diffeomorphisms
Proceedings of the American Mathematical Society, 2006
Using a novel Wintner-type formulation of the classical Peano's existence theorem [Math. Ann. 37 (1890), 182-228], we enhance Ważewski's result on invertibility of maps defined on closed balls [Ann. Soc. Pol. Math. 20 (1947), 81-125] securing the size of the domain of invertibility that agrees with the bounds derived by John [Comm.
Differentiable and Lipschitzian mappings of Banach spaces
Mathematical Notes of the Academy of Sciences of the USSR, 1988
A Mean Value Theorem for Non Differentiable Mappings in Banach Spaces
1995
We prove that if f is a real valued lower semicontinuous function on a Banach space X and if there exists a C1, real valued Lipschitz continuous function on X with bounded support and which is not identically equal to zero, then f is Lipschitz continuous of constant K provided all lower subgradients of f are bounded by K. As
Local homeo- and diffeomorphisms: invertibility and convex image
Bulletin of The Australian Mathematical Society, 1994
We prove a necessary and sufficient condition for a local homeomorphism defined on an open, connected subset of a Euclidean space to be globally one-to-one and, at the same time, for the image to be convex. Among the applications we give a practical sufficiency test for invertibility for twice differentiable local diffeomorphisms defined on a ball.
An inverse function theorem in Fréchet spaces
Annales de l'Institut Henri Poincaré C, Analyse non linéaire, 2011
I present an inverse function theorem for differentiable maps between Fréchet spaces which contains the classical theorem of Nash and Moser as a particular case. In contrast to the latter, the proof does not rely on the Newton iteration procedure, but on Lebesgue's dominated convergence theorem and Ekeland's variational principle. As a consequence, the assumptions are substantially weakened: the map F to be inverted is not required to be C 2 , or even C 1 , or even Fréchet-differentiable. 2 k := p1+...+pn≤p Ω ∂ p1+...+pn x ∂ p1 ω 1 ...∂ pn ω n 2 dω Date: October 14, 2010; to appear, Annales de l'Institut Henri Poincaré, Analyse Non Linéaire. Key words and phrases. Inverse function theorem, implicit function theorem, Fréchet space, Nash-Moser theorem. The author thanks Eric Séré, Louis Nirenberg, Massimiliano Berti and Philippe Bolle, who were the first to see this proof. Particular thanks go to Philippe Bolle for spotting a significant mistake in an earlier version, and to Eric Séré, whose careful reading led to several improvements and clarifications. Their contribution and friendship is gratefully acknowledged.
Global univalence and global inversion theorems in Banach spaces
Nonlinear analysis, 1989
Perturbation of the mapping and solvability theorems in Banach space
Nonlinear Analysis: Theory, Methods & Applications, 2010
Here we consider perturbations of continuous mappings on Banach spaces, and investigate their images under various conditions. Consequently we study the solvability of some classes of equations and inclusions. For these we start by the investigation of local properties of the considered mapping and local comparisons of this mapping with certain smooth mappings. Moreover, we study different mixed problems.
Loading Preview
Sorry, preview is currently unavailable. You can download the paper by clicking the button above.