Linearization and Lemma of Newton for operator functions

Introduction. In [1] and [5] differential calculus is developed for (real or complex) Banach spaces in a dimension-free manner with scarcely more ado than in the 1-dimensional case. Our purpose is to demonstrate that a similar treatment, without reference to dimension, is available for analytic functions (real or complex Banach spaces) with equal simplicity. In case the domain or range is Rn or Cn the theory comprehends the classic one; but we need no polycylinders. We shall not write a treatise, but merely set up the basic definitions, derive a few classic theorems, and mention one or two points of caution. The reader will find further similar generalizations and details easy (not the fundamental theorem of algebra, to be sure) using some of the standard techniques of our references. We suppose more or less familiarity with those references and we shall use the notation of [I] and [5]. Cl A will denote the closure of A; Int A the interior of A. CnorCn(D: Y), n = 1, * . ., oo, a, denotes the class of maps with domain D and image in Y of continuous differentiability class n=1,***, oo, and n =a means analytic. X and Y denote Banach spaces, D an open set in X. Nr(x) stands for the open ball, center radius r. A diffeomorphism f: D-*f(D) C Y is a homeomorphism of class Cn, with f'(x) a topological isomorphism for each x (in D). We shall obtain, among other results, the inverse and implicit function theorems for analytic functions without reference to dimension or scalar field. Let xi, , xE,-X and an a continuous, symmetric, n-linear map of Xn into Y, i.e. an(Ln(X: Y). In an(xl, * * ., xn), we can restrict the xi to the diagonal, i.e. to be equal; an(x, * * *, x) will be abbreviated anX A power series in x with values in Y is a series of the form n-o a x, where ao is a point of Y. A power series with only a finite number of terms is a polynomial, O akx If an $0, then the polynomial has degree n. (The function an is 0 iff it vanishes identically on the diagonal, since the nth derivative Ofan Xn is n!a..)

Let f : U(x0) ⊂ E → F be a C1 map and f′(x0) be the Frechet derivative of f at x0. In local analysis of nonlinear functional analysis, implicit function theorem, inverse function theorem, local surjectivity theorem, local injectivity theorem, and the local conjugacy theorem are well known. Those theorems are established by using the properties: f′(x0) is double splitting and $R(f^{\prime}(x))\cap N(T_{0}^{+})=\{0\}$ near x0. However, in infinite dimensional Banach spaces, f′(x0) is not always double splitting (i.e., the generalized inverse of f′(x0) does not always exist), but its bounded outer inverse of f′(x0) always exists. Only using the C1 map f and the outer inverse ${T_{0}^{\#}}$ of f′(x0), the authors obtain two quasi-local conjugacy theorems, which imply the local conjugacy theorem if x0 is a locally fine point of f. Hence the quasi-local conjugacy theorems generalize the local conjugacy theorem in Banach spaces.

An Inverse Function Theorem in Sobolev Spaces and Applications to Quasi-Linear Schrödinger Equations

This chapter deals with the local inversion theorem and the implicit function theorem in Banach spaces. The Lyapunov—Schmidt reduction is discussed in Section 3.3. In Section 3.4 we prove the global inversion theorem, which goes back to Hadamard and Caccioppoli. Section 3.5 deals with a global inversion theorem in the presence of fold singularities.KeywordsBanach SpaceUnique SolutionNeighborhood VersusImplicit Function TheoremAuxiliary EquationThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

There are numerous cases of discrepancies between results obtained in the setting of real Banach spaces and those obtained in the complex context. This article is a modern exposition of the subtle differences between key results and theories for complex and real Banach spaces and the corresponding linear operators between them. We deeply discuss some aspects of the complexification of real Banach spaces and give several examples showing how drastically different can be the behavior of real Banach spaces versus their complex counterparts.

Summary In this article we formalize in Mizar [1], [2] the inverse function theorem for the class of C 1 functions between Banach spaces. In the first section, we prove several theorems about open sets in real norm space, which are needed in the proof of the inverse function theorem. In the next section, we define a function to exchange the order of a product of two normed spaces, namely 𝔼 ↶ ≂ (x, y) ∈ X × Y ↦ (y, x) ∈ Y × X, and formalized its bijective isometric property and several differentiation properties. This map is necessary to change the order of the arguments of a function when deriving the inverse function theorem from the implicit function theorem proved in [6]. In the third section, using the implicit function theorem, we prove a theorem that is a necessary component of the proof of the inverse function theorem. In the last section, we finally formalized an inverse function theorem for class of C 1 functions between Banach spaces. We referred to [9], [10], and [3] in the formalization.

Implicit function theorems for m-accretive and locally accretive set-valued mappings

In this chapter we will prove the inverse and implicit mapping theorems, which have far-reaching applications. We will begin with the inverse mapping theorem and then derive the implicit mapping theorem from it.

Introduction: One of the main difficulties-by no means the only onein the development of a theory of analytic functions in normed linear spaces stems from the fact that the modulus of a homogeneous polynomial and that of its polar are not always the same. We give here the theorenm that asserts that the modulus of a homogeneous polynomial and its polar are always equal in complex' Banach spaces. This has many important imi lications including the great simplifications brought about in the proofs of several furndamental theorems. The situation in real2 Banach spaces is quite different as we shall show. We base our proofs on a few fundamental lemmas in normed linear spaces: one generalizes Bernstein's theorem3 on the bounds and first derivative of S, a polynomial in the complex plane and two others generalize theorems of the brothers, A. Markoff4 and V. Markoff5 on the first and higher derivatives of polynomials of a real variable. These lemmas have an interest in themselves and have many applications not discussed in this paper. To present our results as briefly as possible, we assume familiarity with the Frechet differential calculus and analytic function theory in both real and complex Banach spaces. Theorems in Real Banach Spaces. The result of A. Markoff states that if pn(x) is a polynomial of degree n in a real variable x and if lPn(x)I < 1 in the interval xI < 1, then the derivative pn(x) satisfies the inequality lpA(x)l < nZ in |x| < 1. The following lelmma generalizes Markoff's result. Lemma 1. Let n

There is a subtle difference as far as the invariant subspace problem is concerned for operators acting on real Banach spaces and operators acting on complex Banach spaces. For instance, the classical hyperinvariant subspace theorem of Lomonosov [Funktsional. Anal. nal. i Prilozhen 7(3)(1973), 55–56. (Russian)], while true for complex Banach spaces is false for real Banach spaces. When one starts with a bounded operator on a real Banach space and then considers some “complexification technique” to extend the operator to a complex Banach space, there seems to be no pattern that indicates any connection between the invariant subspaces of the “real” operator and those of its “complexifications.” The purpose of this note is to examine two complexification methods of an operator T acting on a real Banach space and present some questions regarding the invariant subspaces of T and those of its complexifications

The implicit function theorem and the global methods of cohomology

Selfadjoint operators on real or complex Banach spaces

The phase-field system is a nonlinear model that has significant applications in material sciences. In this paper, we are concerned with the uniqueness of determining the nonlinear energy potential in a phase-field system consisting of Cahn–Hilliard and Allen–Cahn equations. This system finds widespread applications in the development of alloys engineered to withstand extreme temperatures and pressures. The goal is to reconstruct the nonlinear energy potential through the measurements of concentration fields. We establish the local well-posedness of the phase-field system based on the implicit function theorem in Banach spaces. Both of the uniqueness results for recovering time-independent and time-dependent energy potential functions are provided through the higher order linearization technique.

A local theorem of existence and uniqueness of solutions of the equations of stationary axially symmetric vacuum gravitational fields in the general theory of relativity close to the flat space solution is proved using the implicit function theorem in Banach spaces.

Assuming as starting point the validity of the Einstein-Rosen metric we study the hyperbolic system of P.D.E. to which the Einstein’s field equations can be reduced. We prove using the implicit function theorem in Banach spaces, the existence and uniqueness of gravitational waves of small amplitude. A class of solutions, not necessarily small, is also constructed. In the last Section a theorem of existence and uniqueness is given for the corresponding stationary problem.