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differential equation; tangent space
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 $u^{\prime }(t)={\mathcal F}(u(t))$.
[1] Jean-Pierre Aubin: Mutational equations in metric spaces. Set-Valued Analysis 1 (1993), 3–46. DOI 10.1007/BF01039289 | MR 1230367
[2] Klaus Deimling: Ordinary Differential Equations in Banach Spaces. Springer, Berlin, 1977. MR 0463601
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