# Differential equations in metric spaces

Mathematica Bohemica (2002)

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

## Access Full Article

top## Abstract

top## How to cite

topTabor, 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

top## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.