On uniform continuous dependence of solution of Cauchy problem on parameter

V. Ya. Derr

Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, no. 4 (2012), pp. 22-29

Résumé

We prove that if, in addition to the assumptions that guarantee existence, uniqueness and continuous dependence on parameter $\mu\in\mathcal M$ of solution $x(t,t_0,\mu)$ of a $n$-dimensional Cauchy problem $\frac{dx}{dt}=f(t,x,\mu)$ $(t\in\mathcal I,\mu\in\mathcal M)$, $x(t_0)=x^0$ one requires that the family $\{f(t,x,\cdot)\}_{(t,x)}$ is equicontinuous, then the dependence of $x(t,t_0,\mu)$ on parameter $\mu$ in an open $\mathcal M$ is uniformly continuous. Analogous result for a linear $n\times n$-dimensional Cauchy problem $\frac{dX}{dt}=A(t,\mu)X+\Phi(t,\mu)$ $(t\in\mathcal I,\mu\in\mathcal M)$, $X(t_0,\mu)=X^0(\mu)$ is valid under the assumption that the integrals $\int_\mathcal I\|A(t,\mu_1)-A(t,\mu_2)\|\,dt $ and $\int_\mathcal I\|\Phi(t,\mu_1)-\Phi(t,\mu_2)\|\,dt$ are uniformly arbitrarily small, provided that $\|\mu_1-\mu_2\|$, $\mu_1,\mu_2\in\mathcal M$, is sufficiently small.

Détail

Keywords: uniformly continuity, equipower continuity.