Geodesic
Solution to the Volterra equations of the 1st kind with discontinuous kernels in the class of generalized functions
The Bulletin of Irkutsk State University. Series Mathematics, Tome 5 (2012) no. 1, pp. 80-95 Cet article a éte moissonné depuis la source Math-Net.Ru

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Sufficient conditions for existence and uniqueness of solutions of the equations of Volterra equation of the 1st kind with a piecewise continuous kernels in the class of generalized functions with point support are derived. An asymptotic approximation of a parametric family of generalized solutions is constructed. A method of the regular part of the solution’s improvement employs the method of successive approximations.
Keywords: Volterra integral equations; discontinuous kernel; generalized solution; successive approximations. 
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D. N. Sidorov. Solution to the Volterra equations of the 1st kind with discontinuous kernels in the class of generalized functions. The Bulletin of Irkutsk State University. Series Mathematics, Tome 5 (2012) no. 1, pp. 80-95. http://geodesic.mathdoc.fr/item/IIGUM_2012_5_1_a7/

[1] Apartsin A. S., Neklassicheskie uravneniya Volterra I roda: teoriya i chislennye metody, Nauka, Novosibirsk, 1999

[2] M. M. Vainberg, V. A. Trenogin, Teoriya vetvleniya reshenii nelineinykh uravnenii, Nauka, M., 1969 | MR

[3] V. S. Vladimirov, Obobschennye funktsii v matematicheskoi fizike, Nauka. Fizmatlit, M., 1976 | MR | Zbl

[4] A. O. Gelfond, Ischislenie konechnykh raznostei, Fizmatlit, M., 1959 | MR

[5] N. A. Magnitskii, “Asimptotika reshenii integralnogo uravneniya Volterry pervogo roda”, DAN SSSR, 269:1 (1983), 29–32 | MR

[6] E. V. Markova, I. V. Sidler, V. V. Trufanov, “O modelyakh razvivayuschikhsya sistem tipa Glushkova i ikh prilozheniyakh v elektroenergetike”, Avtomatika i telemekhanika, 2011, no. 7, 20–28 | MR | Zbl

[7] N. A. Sidorov, A. V. Trufanov, “Nelineinye operatornye uravneniya s funktsionalno vozmuschennym argumentom neitralnogo tipa”, Differents. uravneniya, 45:12 (2009), 1804–1808 | MR | Zbl

[8] N. A. Sidorov, D. N. Sidorov, “O malykh resheniyakh nelineinykh differentsialnykh uravnenii v okrestnosti tochek vetvleniya”, Izv. vuzov. Matematika, 2011, no. 5, 53–61 | MR | Zbl

[9] N. A. Sidorov, D. N. Sidorov, “Suschestvovanie i postroenie obobschennykh reshenii integralnykh uravnenii Volterry pervogo roda”, Differents. uravneniya, 42:9 (2006), 1243–1242 | MR

[10] N. A. Sidorov, D. N. Sidorov, A. V. Krasnik, “O reshenii operatorno-integralnykh uravnenii Volterry v neregulyarnom sluchae metodom posledovatelnykh priblizhenii”, Differents. uravneniya, 40:6 (2010), 874–882 | MR

[11] D. H. Sidorov, N. A. Sidorov, “Obobschennye resheniya v zadache modelirovaniya nelineinykh dinamicheskikh sistem polinomami Volterra”, Avtomatika i telemekhanika, 2011, no. 6, 127–132 | MR | Zbl

[12] V. A. Trenogin, Funktsionalnyi analiz, Nauka, M., 1993, 439 pp. | MR | Zbl

[13] L. E. Elsgolts, Kachestvennye metody v matematicheskom analize, GITTL, M., 1955

[14] Yu. P. Yatsenko, Integralnye modeli sistem s upravlyaemoi pamyatyu, Naukova dumka, Kiev, 1991 | MR | Zbl

[15] A. M. Denisov, A. Lorenzi, “On a special Volterra integral equation of the first kind”, Boll. Un. Mat. Ital. B. Vol., 7:9 (1995), 443–457 | MR

[16] D. Sidorov, “Volterra Equations of the First kind with Discontinuous Kernels in the Theory of Evolving Systems Control”, Studia Informatica Universalis, 9:3 (2011), 135–146 | MR

[17] D. N. Sidorov, N. A. Sidorov, “Convex majorants method in the theory of nonlinear Volterra equations”, Banach J. Math. Anal., 6:1 (2012), 1–10 | DOI | MR | Zbl

[18] N. Sidorov, B. Loginov, A. Sinitsyn, M. Falaleev, Lyapunov–Schmidt Methods in Nonlinear Analysis and Applications, Kluwer Academic Publisher, Dordrecht–Boston–London, 2002, 568 pp. | MR | Zbl

[19] D. Sidorov, “On impulsive control of nonlinear dynamical systems based on the Volterra series”, 10th IEEE International Conference on Environment and Electrical Engineering, EEEIC (8–11 May 2011, Rome, Italy), 2011, 1–6