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Section
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Mathematics
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Title
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On uniform continuous dependence of solution of Cauchy problem on parameter
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Author(-s)
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Derr V.Ya.a
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Affiliations
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Udmurt State Universitya
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Abstract
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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.
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Keywords
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uniformly continuity, equipower continuity
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UDC
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517.91.4
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MSC
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34A12
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DOI
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10.20537/vm120402
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Received
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11 November 2011
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Language
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Russian
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Citation
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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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References
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- 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.
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Full text
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