Differential equations in metric spaces

Jacek Tabor

Mathematica Bohemica (2002)

  • Volume: 127, Issue: 2, page 353-360
  • ISSN: 0862-7959

Abstract

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We give a meaning to derivative of a function u X , where X is a complete metric space. This enables us to investigate differential equations in a metric space. One can prove in particular Gronwall’s Lemma, Peano and Picard Existence Theorems, Lyapunov Theorem or Nagumo Theorem in metric spaces. The main idea is to define the tangent space 𝒯 x X of x X . Let u , v [ 0 , 1 ) X , u ( 0 ) = v ( 0 ) be continuous at zero. Then by the definition u and v are in the same equivalence class if they are tangent at zero, that is if lim h 0 + d ( u ( h ) , v ( h ) ) h = 0 . By 𝒯 x X we denote the set of all equivalence classes of continuous at zero functions u [ 0 , 1 ) X , u ( 0 ) = x , and by 𝒯 X the disjoint sum of all 𝒯 x X over x X . By u ' ( t ) 𝒯 u ( t ) X , where u X , we understand the equivalence class of a function [ 0 , 1 ) h u ( t + h ) X . Given a function X 𝒯 X such that ( x ) 𝒯 x X we are now able to investigate solutions to the differential equation u ' ( t ) = ( u ( t ) ) .

How to cite

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Tabor, Jacek. "Differential equations in metric spaces." Mathematica Bohemica 127.2 (2002): 353-360. <http://eudml.org/doc/249048>.

@article{Tabor2002,
abstract = {We give a meaning to derivative of a function $u\:\mathbb \{R\}\rightarrow X$, where $X$ is a complete metric space. This enables us to investigate differential equations in a metric space. One can prove in particular Gronwall’s Lemma, Peano and Picard Existence Theorems, Lyapunov Theorem or Nagumo Theorem in metric spaces. The main idea is to define the tangent space $\{\mathcal \{T\}\}_xX$ of $x \in X$. Let $u,v\:[0,1) \rightarrow X$, $u(0)=v(0)$ be continuous at zero. Then by the definition $u$ and $v$ are in the same equivalence class if they are tangent at zero, that is if \[ \lim \_\{h \rightarrow 0^+\} \frac\{d(u(h),v(h))\}\{h\}=0. \] By $\{\mathcal \{T\}\}_xX$ we denote the set of all equivalence classes of continuous at zero functions $u\:[0,1) \rightarrow X$, $u(0)=x$, and by $\{\mathcal \{T\}\}X$ the disjoint sum of all $\{\mathcal \{T\}\}_xX$ over $x \in X$. By $u^\{\prime \}(t) \in \{\mathcal \{T\}\}_\{u(t)\}X$, where $u\:\mathbb \{R\}\rightarrow X$, we understand the equivalence class of a function $[0,1) \ni h \rightarrow u(t+h) \in X$. Given a function $\{\mathcal \{F\}\}\:X \rightarrow \{\mathcal \{T\}\}X$ such that $\{\mathcal \{F\}\}(x) \in \{\mathcal \{T\}\}_x X$ we are now able to investigate solutions to the differential equation $u^\{\prime \}(t)=\{\mathcal \{F\}\}(u(t))$.},
author = {Tabor, Jacek},
journal = {Mathematica Bohemica},
keywords = {differential equation; tangent space; differential equation; tangent space},
language = {eng},
number = {2},
pages = {353-360},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Differential equations in metric spaces},
url = {http://eudml.org/doc/249048},
volume = {127},
year = {2002},
}

TY - JOUR
AU - Tabor, Jacek
TI - Differential equations in metric spaces
JO - Mathematica Bohemica
PY - 2002
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 127
IS - 2
SP - 353
EP - 360
AB - We give a meaning to derivative of a function $u\:\mathbb {R}\rightarrow X$, where $X$ is a complete metric space. This enables us to investigate differential equations in a metric space. One can prove in particular Gronwall’s Lemma, Peano and Picard Existence Theorems, Lyapunov Theorem or Nagumo Theorem in metric spaces. The main idea is to define the tangent space ${\mathcal {T}}_xX$ of $x \in X$. Let $u,v\:[0,1) \rightarrow X$, $u(0)=v(0)$ be continuous at zero. Then by the definition $u$ and $v$ are in the same equivalence class if they are tangent at zero, that is if \[ \lim _{h \rightarrow 0^+} \frac{d(u(h),v(h))}{h}=0. \] By ${\mathcal {T}}_xX$ we denote the set of all equivalence classes of continuous at zero functions $u\:[0,1) \rightarrow X$, $u(0)=x$, and by ${\mathcal {T}}X$ the disjoint sum of all ${\mathcal {T}}_xX$ over $x \in X$. By $u^{\prime }(t) \in {\mathcal {T}}_{u(t)}X$, where $u\:\mathbb {R}\rightarrow X$, we understand the equivalence class of a function $[0,1) \ni h \rightarrow u(t+h) \in X$. Given a function ${\mathcal {F}}\:X \rightarrow {\mathcal {T}}X$ such that ${\mathcal {F}}(x) \in {\mathcal {T}}_x X$ we are now able to investigate solutions to the differential equation $u^{\prime }(t)={\mathcal {F}}(u(t))$.
LA - eng
KW - differential equation; tangent space; differential equation; tangent space
UR - http://eudml.org/doc/249048
ER -

References

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  1. Mutational equations in metric spaces, Set-Valued Analysis 1 (1993), 3–46. (1993) MR1230367
  2. Ordinary Differential Equations in Banach Spaces, Springer, Berlin, 1977. (1977) MR0463601

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