Displaying similar documents to “Abstract version of the Cauchy-Kowalewski problem”

An Application Of The Theory Of Scale Of Banach Spaces

Łukasz Dawidowski (2015)

Annales Mathematicae Silesianae

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The abstract Cauchy problem on scales of Banach space was considered by many authors. The goal of this paper is to show that the choice of the space on scale is significant. We prove a theorem that the selection of the spaces in which the Cauchy problem ut − Δu = u|u|s with initial–boundary conditions is considered has an influence on the selection of index s. For the Cauchy problem connected with the heat equation we will study how the change of the base space influents the regularity...

On local convexity of nonlinear mappings between Banach spaces

Iryna Banakh, Taras Banakh, Anatolij Plichko, Anatoliy Prykarpatsky (2012)

Open Mathematics

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We find conditions for a smooth nonlinear map f: U → V between open subsets of Hilbert or Banach spaces to be locally convex in the sense that for some c and each positive ɛ < c the image f(B ɛ(x)) of each ɛ-ball B ɛ(x) ⊂ U is convex. We give a lower bound on c via the second order Lipschitz constant Lip2(f), the Lipschitz-open constant Lipo(f) of f, and the 2-convexity number conv2(X) of the Banach space X.

On the Cauchy problem for linear PDEs with retarded arguments at derivatives

Krzysztof A. Topolski (2015)

Annales Polonici Mathematici

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We present an existence theorem for the Cauchy problem related to linear partial differential-functional equations of an arbitrary order. The equations considered include the cases of retarded and deviated arguments at the derivatives of the unknown function. In the proof we use Tonelli's constructive method. We also give uniqueness criteria valid in a wide class of admissible functions. We present a set of examples to illustrate the theory.