Geodesic
Solvability of systems of integral Volterra equations of the first kind with piecewise continuous kernels
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 1 (2013), pp. 62-72.

Voir la notice de l'article provenant de la source Math-Net.Ru

We construct an asymptotic approximation for solutions of systems of integral Volterra equations of the first kind with piecewise continuous kernel. We employ the asymptotics as an initial approximation in the proposed method of successive approximations to the desired solutions. We prove the existence of a continuous solution depending on free parameters and establish sufficient conditions for the existence of a unique continuous solution. We illustrate the proved existence theorems with examples.
Keywords: systems of integral Volterra equations of the first kind, asymptotics, continuous kernel, successive approximations. 
@article{IVM_2013_1_a5,
     author = {D. N. Sidorov},
     title = {Solvability of systems of integral {Volterra} equations of the first kind with piecewise continuous kernels},
     journal = {Izvesti\^a vys\v{s}ih u\v{c}ebnyh zavedenij. Matematika},
     pages = {62--72},
     publisher = {mathdoc},
     number = {1},
     year = {2013},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IVM_2013_1_a5/}
}
TY  - JOUR
AU  - D. N. Sidorov
TI  - Solvability of systems of integral Volterra equations of the first kind with piecewise continuous kernels
JO  - Izvestiâ vysših učebnyh zavedenij. Matematika
PY  - 2013
SP  - 62
EP  - 72
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IVM_2013_1_a5/
LA  - ru
ID  - IVM_2013_1_a5
ER  - 
%0 Journal Article
%A D. N. Sidorov
%T Solvability of systems of integral Volterra equations of the first kind with piecewise continuous kernels
%J Izvestiâ vysših učebnyh zavedenij. Matematika
%D 2013
%P 62-72
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IVM_2013_1_a5/
%G ru
%F IVM_2013_1_a5
D. N. Sidorov. Solvability of systems of integral Volterra equations of the first kind with piecewise continuous kernels. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 1 (2013), pp. 62-72. http://geodesic.mathdoc.fr/item/IVM_2013_1_a5/

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

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

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

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

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

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

[7] Sidorov N. A., Trufanov A. V., Sidorov D. N., “Suschestvovanie i struktura reshenii integro-funktsionalnykh uravnenii Volterry pervogo roda”, Izv. IGU. Ser. matematika, 2007, no. 1, 267–274

[8] Sidorov D., Volterra equations of the first kind with discontinuous kernels in the theory of evolving systems control, 2011, arXiv: 1111.5903v1

[9] Bryuno A. D., “Asimptotiki i razlozheniya reshenii obyknovennogo differentsialnogo uravneniya”, UMN, 59:3 (2004), 31–80 | DOI | MR | Zbl

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

[11] Sidorov N. A., Falaleev M. V., Sidorov D. N., “Generalized solutions of Volterra integral equations of the first kind”, Bull. Malays. Math. Soc., 29:2 (2006), 1–5 | MR

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

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

[14] Elsgolts L. A., Kachestvennye metody v matematicheskom analize, GITTL, M., 1955 | MR

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

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

[17] Trenogin V. A., Funktsionalnyi analiz, izd. 4-e, Fizmatlit, M., 2007

[18] Sidorov N. A., Loginov B. V., Sinitsyn A. V., Falaleev M. V., Lyapunov–Schmidt methods in nonlinear analysis and applications, Series on mathematics and its applications, Kluwer Academic Publishers, Dordrecht, 2002 | MR | Zbl

[19] Sviridyuk G. A., Fedorov V. E., Linear Sobolev type equations and degenerate semigroups of operators, Inverse and ill-posed problems series, 42, VSP, 2003 | MR

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

[21] Sidorov N. A., Sidorov D. N., “Suschestvovanie i postroenie obobschennykh reshenii integralnykh uravnenii Volterra pervogo roda”, Differents. uravneniya, 42:9 (2006), 1243–1247 | MR | Zbl

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

[23] Messina E., Russo E., Vecchio A., “A stable numerical method for Volterra integral equations with discontinuous kernel”, J. Math. Anal. Appl., 337:2 (2008), 1383–1393 | DOI | MR | Zbl