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,\mu) (t \in I, \mu \in M), x(t_0)=x^0 satisfies the conditions that guarantee existence, uniqueness and continuous dependence of solution x(t,t_0,\mu) on parameter \mu in 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,\mu) on parameter \mu \in M is uniformly continuous. An analogous result for a linear n \times n-dimensional Cauchy problem \frac{dX}{dt}=A(t,\mu)X+\Phi(t,\mu) (t \in I, \mu \in M), X(t_0,\mu)=X^0(\mu) is valid under the assumption that the integrals \int_I\|A(t,\mu_1)-A(t,\mu_2)\|dt and \int_I \|\Phi(t,\mu_1)-\Phi(t,\mu_2)\|dt can be made smaller than any given constant (uniformly with respect to \mu_1, \mu_2 \in M) provided that \|\mu_1-\mu_2\| is sufficiently small.