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Transformation and canonical forms of functional-differential equations

Published online by Cambridge University Press:  14 November 2011

František Neuman
Affiliation:
Mathematical Institute of the Czechoslovak Academy of Sciences, branch Brno, Mendelovo nám. 1, 66282 Brno, Czechoslovakia.

Synopsis

Functional-differential equations, especially linear ones, are considered with respect to global pointwise transformations. Two types of canonical forms for certain classes of these equations are introduced. These transformations and the corresponding canonical forms preserve oscillatory or non-oscillatory behaviour of solutions. They are also suitable for studying both-side solutions of equivalent functional-differential equations.

Type
Research Article
Copyright
Copyright © Royal Society of Edinburgh 1990

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References

1Barvínek, E.. O rozloŶení nulových bodŭu řešení lineární diferenciální rovnice y N + Q(t)y a jejich derivací. Acta Math. Univ. Comenian 5 (1961), 465474.Google Scholar
2Blanton, G. and Baker, J. A.. Iteration groups generated by Cn-functions. Arch. Math. (Brno) 19 (1982), 121127.Google Scholar
3Borŭvka, O.. Lineare Differentialtransformationen 2. Ordnung. (Berlin: VEB, 1967); translated as Linear Differential Transformations of the Second Order (London: English University Press, 1971),Google Scholar
4čadek, M.. Form of general pointwise transformations of linear differential equations. Czechoslovak Math. J. 35 (1985), 617624.CrossRefGoogle Scholar
5Choczewski, B. On differentiable solutions of a functional equation. Ann. Polon. Math. 13 (1963), 133138.CrossRefGoogle Scholar
6Erbe, L. H. and Zhang, B. G.. Oscillation for first order linear differential equations with deviating arguments. Diff. Integral Equations 1 (1988), 305314.Google Scholar
7Everitt, W. N.. On the transformation theory of ordinary second-order linear symmetric differential expressions. Czechoslovak Math. J. 32 (1982), 275306.CrossRefGoogle Scholar
8Forsyth, A. R.. Invariants, covariants and quotient-derivatives associated with linear differential equations. Philos. Trans. Roy. Soc. London Ser. A 179 (1899), 377489.Google Scholar
9Friedman, H. I. and Gopalsamy, K.. Nonoccurence of stability switching in systems with discrete delays. Canad. Math. Bull. 31 (1988), 5258.CrossRefGoogle Scholar
10Györi, I. and Ladas, G.. Oscillation of systems of neutral differential equations. Diff. Integral Equations 1 (1988), 281286.Google Scholar
11Kitamura, Y. and Kusano, T.. An oscillation theorem for a superlinear functional differential equation with general deviating arguments. Bull. Austral. Math. Soc. 18 (1978), 395402.CrossRefGoogle Scholar
12Koplatadze, R. G. and Chanturija, T. A.. Ob oscilljacionnych svojstvach reshenij linijnogo differencial'nogo uraunĕnija s otklonjajushchimsja argumentom (Tbilisi, 1977).Google Scholar
13Kummer, E. E.. De generali quadam aequatione differentiali tertii ordinis. Progr. Evang. Königl. Stadtgymnasium Liegnitz (1834); reprinted in J. Reine Angew. Math. 100 (1887), 110.Google Scholar
14Laguerre, E.. Sur les ùquations diffùrentielles linùaires du troisiøme ordre. C R. Acad. Sci Paris 88 (1879), 116118.Google Scholar
15Myshkys, A. D.. Linějnyje differencial'nyje uravnenija s zapazdyvajushchim argumentom (Moscow: Nauka, 1972).Google Scholar
16Neuman, F.. Geometrical approach to linear differential equations of the n-th order. Rend. Mat. 5 (1972), 579602; (Abstract: some results on geometrical approach to linear differential equations of the n-th order. Comment. Math. Univ. Carolin. 12 (1971), 307–315).Google Scholar
17Neuman, F.. On transformations of differential equations and systems with deviating argument. Czechoslovak Math. J. 31 (1981), 8790.CrossRefGoogle Scholar
18Neuman, F.. Simultaneous solutions of a system of Abel equations and differential equations with several deviations. Czechoslovak Math. J. 32 (1982), 488494.CrossRefGoogle Scholar
19Neuman, F.. Global theory of ordinary linear homogeneous differential equations in the real domain I and II, Aequationes Math. 33 (1987), 123149; 34 (1987), 1–22.CrossRefGoogle Scholar
20Norkin, S. B.. Prostranstvo dvustoronnych reshenij s eksponencial'nym rostom pri t→∞ linějnoj sistěmy s asimptoticheski postojannymi matricej i zapazdyvanijem. In Issledovanija po teorii differencial'nych urauněnij, 2734 (Moscow, 1986).Google Scholar
21Rjabov, A. Ju.. Dvustoronnyje reshenija linějnych integro-differencial'nych uravněnij tipa Volterra s beskoněchnym nizhnim predelom. In Issledovanija po teorii defferencial'nych urauněnij, 316 (Moscow, 1986).Google Scholar
22Tryhuk, V.. The most general transformation of homogeneous linear differential retarded equations of the first order, Arch. Math. (Brno) 16 (1980), 225230.Google Scholar
23Tryhuk, V.. The most general transformation of homogeneous retarded linear differential equations of the n-th order. Math. Slovaka 33 (1983), 1521.Google Scholar
24Wilczynski, E. J.. Projective differential geometry of curves and ruled surfaces (Leipzig: Teubner, 1906).Google Scholar
25Zdun, C. M.. Note on commutable functions. Aequationes Math. 36 (1988), 153164.CrossRefGoogle Scholar