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Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2012, Issue 4, Pages 22–29
(Mi vuu346)
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This article is cited in 1 scientific paper (total in 1 paper)
MATHEMATICS
On uniform continuous dependence of solution of Cauchy problem on parameter
V. Ya. Derr Department of Mathematical Analysis, Udmurt State University, Izhevsk, Russia
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 $\frac{dx}{dt}=f(t,x,\mu)$ $(t\in\mathcal I,\mu\in\mathcal M)$, $x(t_0)=x^0$ 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.
Keywords:
uniformly continuity, equipower continuity.
Received: 11.11.2011
Citation:
V. Ya. Derr, “On uniform continuous dependence of solution of Cauchy problem on parameter”, Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 2012, no. 4, 22–29
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https://www.mathnet.ru/eng/vuu346 https://www.mathnet.ru/eng/vuu/y2012/i4/p22
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