Continuous Dependence and Uniqueness

Abstract

Theorem 5.1. Suppose D is an open set in R × C, f_k D → Rⁿ, k = 0,1,2,… are continuous, f_k(s,Ψ) → f_o(s,ψ) as k → ∞,ψ →ϕ for all (s,ϕ) ∈ D, and for every compact set W in D, there is an open neighborhood V(W) of W and a constant M > O such that

$$\left| {{\rm{f}}_{\rm{k}} \left( {{\rm{t}},{\rm{\psi }}} \right)} \right| \le {\rm{M, }}\left( {{\rm{t}},{\rm{\psi }}} \right) \in {\rm{V}}\left( {\rm{W}} \right),{\rm{k = 0,1,2,}} \ldots $$

((5.1))

Finally, suppose σ_k ∈ R, ϕ_k ∈ C, (σ_k,ϕ_k) ∈ D, k = 0,1,2,…, σ_k → σ_o, ϕ_k →σ_o as k →∞ and suppose x^k = x^k (σ_k,ϕ_k), k = 0,1,2,… is a solution of

$${\rm{\dot x}}^{\rm{k}} \left( {\rm{t}} \right) = {\rm{f}}_{\rm{k}} \left( {{\rm{t}},{\rm{x}}_{\rm{t}}^{\rm{k}} } \right),{\rm{t}} \ge \sigma _{\rm{k}}$$

((5.2))

with initial value ϕ_k at σ_k. If x^o is defined on [σ_o−r,b] and is the only solution through (σ_o,ϕ_o),then there is an integer k_o such that each x^k, k ≥ k_o, can be defined on [σ_k−r,b] and x^k(t) →x^o (t) uniformly on [σ_o−r,b]. Since all x^k may not be defined on [σ_o−r,b], by x^k →x^o uniformly on [σ_o−r,b], we mean that for any ε > o, there is a k_o = k_o(ε) ≥ 0 such that x^k (t), k ≥ k_o, is defined on [σ−r−ε,b] and x^k (t) →x^o(t) uniformly for t ∈ [σ−r−ε,b].