Differential Equations in Metric Spaces

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 TxX 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 TxX 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 TxX over x ∈ X. By u′(t) ∈ Tu(t)X, where u : → X, we understand the equivalence class of a function [0, 1) h → u(t + h) ∈ X. Given a function F : X → T X such that F(x) ∈ TxX we are now able to investigate solutions to the differential equation u′(t) = F(u(t)).