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## Archive of Issues

Russia Izhevsk
Year
2012
Issue
4
Pages
22-29
 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. References Coddington E.A., Levinson N. Theory of ordinary differential equations, New York-Toronto-London: McGraw-Hill, 1955, 415 p. Translated under the title Teoriya obyknovennykh differentsialnykh uravnenii, Moscow: Inostrannaya literatura, 1958, 475 p. Hartman P. Ordinary differential equations, New York-London-Sydney: John Wiley & Sons, 1964. Translated under the title Obyknovennye differentsialnye uravneniya, Moscow: Mir, 1970. Colombeau J.F. Elementary introduction to new generalized functions, Amsterdam: Elsevier Science Publishers B.V., 1985, 300 p. Full text