On a~class of integral equations of convolution type and its applications

N. K. Karapetyants ; S. G. Samko

Izvestiya. Mathematics , Tome 5 (1971) no. 3, pp. 731-744.

Résumé

Operators of convolution type with variable coefficients are investigated in spaces of series $E$ commonly employed for such operators and supplemented by the spaces $M_0^{\sup}$, $M_0^{\mathrm{mes}}$, $M^{\sup}$, $M^{\mathrm{mes}}$ of bounded measurable functions. The properties of these new spaces are investigated. Conditions insuring that operators are of the Noether type are investigated and the index of operators with coefficients from these spaces are calculated. A theorem asserting the coincidence of zeros in these spaces is stated. The results are applied to investigate conditions for integral equations with homogeneous kernels to be of Noether type. For the last class the structure of zeros is clarified in the case of constant coefficients.

Bibliographie

[1] Rakovschik L. S., “K teorii integralnykh uravnenii tipa svertki”, Uspekhi matem. nauk, XVIII:4 (1963), 171–178

[2] Rakovschik L. S., “Integralnye uravneniya s pochti raznostnymi yadrami”, Vestn. Mosk. un-ta. Ser. matem. i astronom., 3:3 (1961), 52–72 | Zbl

[3] Rakovschik L. S., “Ob odnom integralnom uravnenii tipa svertki”, Sib. matem. zh., VI:1 (1965), 186–198

[4] Simonenko I. B., “O nekotorykh integro-differentsialnykh uravneniyakh tipa svertki”, Izv. VUZov. Matematika, 1959, no. 2, 213–226 | MR | Zbl

[5] Simonenko I. B., “O nekotorykh kraevykh zadachakh analiticheskikh funktsii”, Issledovaniya po sovremennym problemam teorii funktsii kompleksnogo peremennogo, M., 1961, 392–399 | Zbl

[6] Simonenko I. B., “Operatory tipa svertki v konusakh”, Matem. sb., 74(116):2 (1967), 298–313 | MR | Zbl

[7] Feldman I. A., “Ob effektivnom reshenii nekotorykh integralnykh uravnenii na pryamoi i polupryamoi”, Izv. AN Moldavskoi SSR, 1961, no. 10(88), 16–26

[8] Berkovich F. D., “Ob odnom obobschenii parnogo integralnogo uravneniya”, Izv. VUZov. Matematika, 1967, no. 12, 15–19 | Zbl

[9] Komyak I. I., “Ob integralnom uravnenii tipa svertki s $N$ yadrami”, Dokl. AN SSSR, 179:2 (1968), 279–282 | Zbl

[10] Krein M. G., “Integralnye uravneniya na poluosi s yadrom, zavisyaschim ot raznosti argumentov”, Uspekhi matem. nauk, XIII:5(83) (1958), 3–120 | MR

[11] Gokhberg I. Ts., Krein M. G., “O parnom integralnom uravnenii i ego transponirovannom. I”, Teor. i prikl. matem., 1, Lvovskii un-t, Lvov, 1959, 58–81

[12] Gokhberg I. Ts., Krein M. G., “Sistemy integralnykh uravnenii na polupryamoi s yadrami, zavisyaschimi ot raznosti argumentov”, Uspekhi matem. nauk, XIII:2 (1958), 3–72

[13] Mikhailov L. G., Integralnye uravneniya s yadrom, odnorodnym stepeni $-1$, AN TadzhSSR, Dushanbe, 1966

[14] Mikhailov L. G., “Integralnye uravneniya s lokalno odnorodnymi yadrami stepeni $-1$”, Differentsialnye i integralnye uravneniya s singulyarnymi koeffitsientami, AN TadzhSSR, Dushanbe, 1969, 54–72

[15] Titmarsh E., Vvedenie v teoriyu integralov Fure, Gostekhizdat, M., L., 1948

[16] Danford N., Shvarts Dzh. T., Lineinye operatory, t. I, IL, M., 1962

[17] Zabreiko P. P. i dr.(SMB), Integralnye uravneniya, Nauka, M., 1968

[18] Khermander L., Otsenki dlya operatorov, invariantnykh otnositelno sdviga, IL, M., 1962

[19] Karapetyants N. K., Samko S. G., “Ob odnom klasse integralnykh uravnenii tipa svertki i ego prilozhenii”, Dokl. AN SSSR, 193:5 (1970), 981–984 | MR | Zbl