On uniform continuous dependence of solution of Cauchy problem on a parameter

Abstract

Suppose that an $n$-dimensional Cauchy problem \frac{dx}{dt}=f(t,x,μ) (t \in I, μ\in M), x(t_0)=x^0 satisfies the conditions that guarantee existence, uniqueness and continuous dependence of solution x(t,t_0,μ) on parameter μin an open set M. We show that if one additionally requires that family \{f(t,x,\cdot)\}_{(t,x)} is equicontinuous, then the dependence of solution x(t,t_0,μ) on parameter μ\in M is uniformly continuous. An analogous result for a linear n \times n-dimensional Cauchy problem \frac{dX}{dt}=A(t,μ)X+Φ(t,μ) (t \in I, μ\in M), X(t_0,μ)=X^0(μ) is valid under the assumption that the integrals \int_I\|A(t,μ_1)-A(t,μ_2)\|dt and \int_I \|Φ(t,μ_1)-Φ(t,μ_2)\|dt can be made smaller than any given constant (uniformly with respect to μ_1, μ_2 \in M) provided that \|μ_1-μ_2\| is sufficiently small.