+7 (3412) 91 60 92

## Archive of Issues

Russia Izhevsk
Year
2014
Issue
1
Pages
3-18
 Section Mathematics Title The spaces of regulated functions and differential equations with generalized functions in coefficients Author(-s) Derr V.Ya.a, Kim I.G.a Affiliations Udmurt State Universitya Abstract A function defined on an open (finite, semi-finite, infinite) interval is called regulated if it has finite one-sided limits at each point of its domain. In the present paper we study spaces of regulated functions, in particular, their dense subsets. Our motivation is applications to differential equations. Namely, we consider the Cauchy problem for a scalar linear differential equation with coefficients, which are derivatives of regulated functions. We immerse the Cauchy problem into the space of the Colombeau generalized functions. If the coefficients are derivatives of step functions, we find explicit solution $R(\varphi_\mu,t)$ of the Cauchy problem (in terms of representatives); its limit as $\mu \rightarrow +0$ is defined to be the solution of the original problem. In this way, we obtain a densely defined (on the space of regulated functions) operator $\mathbf T$, which associates the solution to a Cauchy problem with this problem. Next, using a well-known topological result on a continuous extension, we extend the operator $\mathbf T$ to the operator $\widehat{\mathbf T}$ defined on the entire space of regulated functions. We have given the explicit representation of solution of the Cauchy problem for the inhomogeneous differential equation. Illustrative examples are also offered. Keywords regulated functions, distributions, generalized functions of Colombeau, differential equations UDC 517.911 MSC 34A30 DOI 10.20537/vm140101 Received 17 February 2014 Language Russian Citation Derr V.Ya., Kim I.G. The spaces of regulated functions and differential equations with generalized functions in coefficients, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2014, issue 1, pp. 3-18. References Derr V.Ya., Dizendorf K.I. On the differential equations in $C$-generalized functions, Izv. Vyssh. Uchebn. Zaved. Mat., 1996, no. 11 (414), pp. 39-49 (in Russian). Kurzweil J. Generalized ordinary differential equations, Czechoslovak Mathematical Journal, 1958, vol.8, no. 3, pp. 360-388. Atkinson F.V. Discrete and continuous boundary problems, Mathematics in Science and Engineering, vol. 8, New York-London: Academic Press, 1964, 570 p. Translated under the title Diskretnye i nepreryvnye granichnye zadachi, Moscow: Mir, 1968, 749 p. Levin A.Yu. On the theory of ordinary differential equations. II, Vestnik Yaroslavskogo Universiteta, 1974, no. 8, pp. 122-144 (in Russian). Filippov A.F. Differential equations with discontinuous right hand sides, New York: Springer-Verlag, 1988, 304 p. Original Russian text published in Differentsial'nye uravneniya s razryvnoy pravoy chast'yu, Moscow: Nauka, 1985, 224 p. Derr V.Ya. To the definition of solution of a differential equation with generalized functions in coefficients, Dokl. Akad. Nauk SSSR, 1988, vol. 298, no. 2, pp. 269-272 (in Russian). Zavalischin S.T., Sesekin A.N. Impul'snye protsessy: modeli i prilozheniya (Impulse processes: models and applications), Moscow: Nauka, 1991, 256 p. Derr V.Ya., Kinzebulatov D.M. Differential equations with distributions admitting multiplication on discontinuous functions, Vestn. Udmurt. Univ. Mat. Mekh. Komp'yut. Nauki, 2005, no. 1, pp. 35-58 (in Russian). Colombeau J.F. Elementary introduction to new generalized functions, Amsterdam: North Holland Math. Studies, 1985, 300 p. Biagioni H.A., Colombeau J.F. New generalized functions and $C$${\infty}$ functions with values in generalized complex numbers, J. London Math. Soc., 1986, vol. 2, no. 33, pp. 169-179. Biagioni H.A. A nonlinear theory of generalized functions, Lecture notes in Mathematics, vol. 1421, New York: Springer-Verlag, 1990, 214 p. Grosser M., Kunziger M., Oberguggenberger M., Steinbauer R. Geometric theory of generalized functions with applications to general relativity, Dordrecht: Kluwer academic publishers, 2001, vol. 537, 505 p. Bourbaki N. General Topology: Chapters 1-4, New York: Springer-Verlag, 1998, 437 p. Dieudonne J. Foundations of Modern Analysis, New York: Academic Press, 2006, 408 p. Honig Ch.S. Volterra-Stieltjes integral equations, Amsterdam: North-Holland Math. Studies, 1975, vol. 16, 152 p. Rodionov V.I. On the space of the regular differentiable functions, Izv. Inst. Mat. Inform. Udmurt. Gos. Univ., 2004, no. 1 (29), pp. 3-32 (in Russian). Derr V.Ya., Kinzebulatov D.M. The Alpha-integral of Stieltjes type, Vestn. Udmurt. Univ. Mat. Mekh. Komp'yut. Nauki, 2006, no. 1, pp. 41-65 (in Russian). Derr V.Ya. Teoriya funktsii deistvitel'noi peremennoi. Lektsii i uprazhneniya (Theory of functions of real argument. Lectures and exercises), Moscow: Vyssh. Shkola, 2008, 384 p. Fedorov V.M. Teoriya funktsii i funktsional'nyi analiz. Chast' 2 (Theory of functions and functional analysis, part II), Moscow: Moscow State University, 2000, 191 p. Derr V.Ya. Funktsional'nyi analiz. Lektsii i uprazhneniya (Functional analysis. Lectures and exercises), Moscow: Knorus, 2013, 462 p. Full text