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

TL;DR

This paper establishes conditions under which the solutions of Cauchy problems depend uniformly continuously on parameters, emphasizing equicontinuity of the family of functions involved.

Contribution

It proves that equicontinuity of the family (t,x,ullet){} ensures uniform continuous dependence of solutions on parameters in nonlinear problems, and provides analogous results for linear systems.

Findings

Uniform continuity of solutions w.r.t. parameters under equicontinuity conditions.

Extension of results to linear systems with integral bounds on coefficient differences.

Conditions for uniform dependence in parameterized Cauchy problems.

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.