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Russia Izhevsk
Section Mathematics
Title On uniform continuous dependence of solution of Cauchy problem on parameter
Author(-s) Derr V.Ya.a
Affiliations Udmurt State Universitya
Abstract 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 $\dfrac{dx}{dt}=f(t,x,\mu)$ $(t\in \mathcal I, \mu\in \mathcal M),$ $x(t_0)=x^0$ 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 $\dfrac{dX}{dt}=A(t,\mu)X+\Phi(t,\mu)$ $(t\in \mathcal I, \mu\in \mathcal M),$ $X(t_0,\mu)=X^0(\mu)$ 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.
Keywords uniformly continuity, equipower continuity
UDC 517.91.4
MSC 34A12
DOI 10.20537/vm120402
Received 11 November 2011
Language Russian
Citation Derr V.Ya. On uniform continuous dependence of solution of Cauchy problem on parameter, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2012, issue 4, pp. 22-29.
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