Differential equations in metric spaces
Mathematica Bohemica, Tome 127 (2002) no. 2, pp. 353-360
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 $u^{\prime }(t)={\mathcal F}(u(t))$.
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 $u^{\prime }(t)={\mathcal F}(u(t))$.
DOI :
10.21136/MB.2002.134163
Classification :
34-02, 34A12, 34A25, 34A99, 34G99, 57R25
Keywords: differential equation; tangent space
Keywords: differential equation; tangent space
@article{10_21136_MB_2002_134163,
author = {Tabor, Jacek},
title = {Differential equations in metric spaces},
journal = {Mathematica Bohemica},
pages = {353--360},
year = {2002},
volume = {127},
number = {2},
doi = {10.21136/MB.2002.134163},
mrnumber = {1981539},
zbl = {1015.34003},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.21136/MB.2002.134163/}
}
Tabor, Jacek. Differential equations in metric spaces. Mathematica Bohemica, Tome 127 (2002) no. 2, pp. 353-360. doi: 10.21136/MB.2002.134163