An implicit function theorem in Banach spaces

Banach Families and the Implicit Function Theorem

We generalise the classical implicit function theorem (IFT) for a family of Banach spaces, with the resulting implicit function having derivatives that are locally Lipschitz to very strong operator norms. Notation. For Banach spaces X and Y , L(X,Y ) is the Banach space of continuous linear maps from X to Y ; so L(X,X) is the Banach algebra of operators on X . X × Y and X ∩ Y have by default th...

Implicit Function Theorem in Carnot–carathéodory Spaces

In this paper, we study the notion of regular surface in the Carnot–Carathéodory spaces and we prove the implicit function theorem in this setting. We fix at every point of R a subset of the tangent space, called horizontal tangent plane, and we assume that it has a basis X1, . . . , Xm of C∞ vector fields satisfying the Hörmander condition of hypoellipticity, (see [16]). This choice induces a ...

Convergence of an Implicit Iteration for Affine Mappings in Normed and Banach Spaces

On an atomic decomposition in Banach spaces

An atomic decomposition is considered in Banach space.  A method for constructing an atomic decomposition of Banach  space, starting with atomic decomposition of  subspaces  is presented. Some relations between them are established. The proposed method is used in the  study  of the  frame  properties of systems of eigenfunctions and associated functions of discontinuous differential operators.

On Peano’s Theorem in Banach Spaces

We show that if X is a separable Banach space (or more generally a Banach with an infinite-dimensional separable quotient) then there is a continuous mapping f : X → X such that the autonomous differential equation x′ = f(x) has no solution at any point. In order to put our results into context, let us start by formulating the classical theorem of Peano. Theorem 1. (Peano) Let X = R, f : R×X → ...