MATHEMATICA BOHEMICA, Vol. 127, No. 2, pp. 353-360 (2002)

Differential equations in metric spaces

Jacek Tabor

Jacek Tabor, Institute of Mathematics, Jagiellonian University, Reymonta 4, 30-059 Krakow, Poland, e-mail: tabor@im.uj.edu.pl

Abstract: We give a meaning to derivative of a function $u \R \to X$, 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. \endgraf The main idea is to define the tangent space $\T _xX$ of $x \in X$. Let $u,v [0,1) \to X$, $u(0)=v(0)$ 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 $$\lim _{h \to 0^+} \frac {d(u(h),v(h))}{h}=0.$$ By $\T _xX$ we denote the set of all equivalence classes of continuous at zero functions $u [0,1) \to X$, $u(0)=x$, and by $\T X$ the disjoint sum of all $\T _xX$ over $x \in X$. \endgraf By $u'(t) \in \T _{u(t)}X$, where $u \R \to X$, we understand the equivalence class of a function $[0,1) \ni h \to u(t+h) \in X$. Given a function $\F X \to \T X$ such that $\F (x) \in \T _x X$ we are now able to investigate solutions to the differential equation $u'(t)=\F (u(t))$.

Keywords: differential equation, tangent space

Classification (MSC2000): 34A99, 34G99, 57R25

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