PERIODIC SOLUTION OF GENERALIZED ABEL INTEGRAL EQUATION OF THE FIRST KIND 

Malyutina Mariya Vyacheslavovna, Master’s degree student, Irkutsk State University (1 Karla Marxa street, Irkutsk, Russia), fanofevanescence2008@yandex.ru
Orlov Sergey Sergeevich, Candidate of physical and mathematical sciences, associate professor, subdepartment of mathematical analysis and differential equations, Institute of Mathematics, Economics and Informatics, Irkutsk State University (1 Karla Marxa street, Irkutsk, Russia), orlov_sergey@inbox.ru

517.968.22

10.21685/2072-3040-2017-4-5

Background. The problem of existence of periodic solutions of Volterra integral equations has not been sufficiently studied even in the linear case. In the academic literature little attention is paid to this problem. Therefore, there is a need to create a methodology for investigating the existence of periodic solutions of exactly integral equations that considers the specificity of these mathematical objects. In the article this problem is solved for the class of generalized Abel integral equations of the first kind. At the present time these equations have remained relevant as objects of the research. Firstly, they have many important applications. Secondly, the research of Abel integral equations greatly contributed to the emergence and development of a whole mathematical direction such as fractional calculus, which is very popular in Russia and abroad.
Materials and methods. Methods of mathematical analysis, functional analysis, and the theory of differential and integral equations are used to solve the problems posed in the paper.
Results. The criterion of the existence and uniqueness of a continuous periodic solution of the generalized Abel integral equation of the first kind is proved. The cases of natural and positive real exponents of the kernel are considered. The formulas of periodic solutions are obtained, and their main periods are found.
Conclusions. The theorems formulated in the article characterize the image of the class of continuous periodic functions under a linear map, given by the Riemann – Liouville operator. These theorems can be useful for research in the field of fractional integro-differentiation. Abel integral equations with the natural and real exponents of the kernels are in obvious relation, namely, the first equation is a particular case of the second equation, but the proved criterions of the periodicity of their solutions are not in such relation. This is due to the fact that the nonlocal Riemann – Liouville operators of the natural and fractional orders of integration differ from each other by the properties. The first operator has a differential operator as an inverse, which is a local operator. The locality of differential operators and the nonlocality of Volterra integral operators also explain the differences in the research of the problem of the existence of periodic solutions of appropriate equations.

Abel integral equation, Riemann – Liouville operator, periodic function, basic period

1. Vol'terra V. Teoriya funktsionalov, integral'nykh i integrodifferentsial'nykh uravneniy [The theory of functionals, integral and integrodifferential equations]. Transl. from English. Moscow: Nauka, 1982, 304 p.
2. Bykov Ya. V. O nekotorykh zadachakh teorii integrodifferentsial'nykh uravneniy [On some problems of the theory of integrodifferential equations]. Frunze: Izd-vo Kirg. gos. un-ta, 1957, 328 p.
3. Botashev A. I., Talipova L. A. Izvestiya Akademii nauk Kirg. SSR [Proceedings of AS of Kyrg. SSR]. 1974, iss. 1, pp. 8–11.
4. Pulyaev V. F., Tsalyuk Z. B. Differentsial'nye uravneniya [Differential equations]. 1974, vol. 10, no. 6, pp. 1103–1110.
5. Tsalyuk Z. B. Itogi nauki i tekhniki. Ser.: Matematicheskiy analiz [Results of science and technology. Series: Mathematical analysis]. 1977, vol. 15, pp. 131–198.
6. Levin J. J. Proc. Amer. Math. Soc. 1963, vol. 14, pp. 534–541. DOI: 10.2307/2034270. 7. Burton T. A. Tohoku Math. J. 1994, vol. 46, no. 2, pp. 207–220. DOI: 10.2748/tmj/1178225758
8. Burton T. A., Furumochi T. Funkcial. Ekvac. 1996, vol. 39, pp. 87–107. MR1401654
9. Burton T. A., Zhang B. CUBO A Math. J. 2012, vol. 14, no. 1, pp. 55–79. DOI: 10.4067/S0719-06462012000100006
10. Islam M. N. Elect. J. Diff. Eqs. 2016, vol. 2016, no. 83, pp. 1–9.
11. Islam M. N. Internat. J. Math. & Math. Sci. 1988, vol. 11, no. 4, pp. 781–792. DOI: 10.1155/S016117128800095X
12. Cromer T. L. J. Math. Anal. Appl. 1985, vol. 110, pp. 483–494. DOI: 10.1016/0022- 247X(85)90310-5
13. Cushing J. M. Volterra Equations. Lecture Notes in Mathematics. Eds.: S.-O. Londen, O. J. Staffans. 1979, vol. 737, pp. 50–66. DOI: 10.1007/BFb0064495.
14. Nussbaum R. D. A SIAM J. Math. Anal. 1978, vol. 9, pp. 356–376. DOI: 10.1137/0509024.
15. Butris R. N., Fars B. S. Gen. Math. Notes. 2014, vol. 21, no. 1, pp. 137–156.
16. Gorenflo R., Vessella S. Abel Integral Equations: Analysis and Application. Berlin– Heidelberg: Springer–Verlag, 1991, 217 p. DOI: 10.1007/BFb0084665
17. Samko S. G., Kilbas A. A., Marichev O. I. Integraly i proizvodnye drobnogo poryadka i nekotorye ikh prilozheniya [Integrals and fractional derivatives and some applications thereof]. Minsk: Nauka i tekhnika, 1987, 688 p.
18. Tenreiro Machado J. A., Galhano A. M., Trujillo J. J. Fract. Calc. Appl. Anal. 2013, vol. 16, no. 2, pp. 479–500. DOI: 10.2478/s13540-013-0030-y.
19. Akhiezer N. I. Elementy teorii ellipticheskikh funktsiy [Elements of the theory of elliptic functions]. Moscow: Nauka, 1970, 304 p.
20. Budak B. M., Fomin S. V. Kurs vysshey matematiki i matematicheskoy fiziki. Kratnye integraly i ryady [A course of higher mathematics and mathematical physics. Multiple integrals and series]. Moscow: Nauka, 1965, 608 p.
21. Erugin N. P. Kniga dlya chteniya po obshchemu kursu differentsial'nykh uravneniy [A reader on the general course of differential equations]. Minsk: Nauka i tekhnika, 1979, 744 p.
22. Malyutina M. V. Yuzhno-ural'skaya molodezhnaya shkola po matematicheskomu modelirovaniyu: sb. tr. II Vseros. nauch.-prakt. konf. [The Southern Ural Youth School on Mathematical Modeling: proceedings of II All-Russian scientific and practical conference].
Chelyabinsk: Izd. tsentr YuUrGU, 2015, pp. 110–115.
23. Azbelev N. V., Maksimov V. P., Rakhmatullina L. F. Introduction to the Theory of Functional Differential Equations: Methods and Applications. New York – Nasr: Hindawi Publ. Corp., 2007, 318 p. DOI: 10.1155/9789775945495.