Geodesic
On the solvability of one class of two-dimensional Urysohn integral equations
Matematičeskie trudy, Tome 20 (2017) no. 2, pp. 193-205.

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

We study one class of nonlinear Urysohn integral equations in a quadrant of the plane. It is assumed that, for the corresponding two-dimensional Urysohn operator, some Hammerstein operator with power nonlinearity serves as a minorant in the sense of M. A. Krasnosel'ski­ĭ. We prove the existence of a nonnegative (nontrivial) and bounded solution for such equations. 
@article{MT_2017_20_2_a7,
     author = {Kh. A. Khachatryan},
     title = {On the solvability of one class of two-dimensional {Urysohn} integral equations},
     journal = {Matemati\v{c}eskie trudy},
     pages = {193--205},
     publisher = {mathdoc},
     volume = {20},
     number = {2},
     year = {2017},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MT_2017_20_2_a7/}
}
TY  - JOUR
AU  - Kh. A. Khachatryan
TI  - On the solvability of one class of two-dimensional Urysohn integral equations
JO  - Matematičeskie trudy
PY  - 2017
SP  - 193
EP  - 205
VL  - 20
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MT_2017_20_2_a7/
LA  - ru
ID  - MT_2017_20_2_a7
ER  - 
%0 Journal Article
%A Kh. A. Khachatryan
%T On the solvability of one class of two-dimensional Urysohn integral equations
%J Matematičeskie trudy
%D 2017
%P 193-205
%V 20
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MT_2017_20_2_a7/
%G ru
%F MT_2017_20_2_a7
Kh. A. Khachatryan. On the solvability of one class of two-dimensional Urysohn integral equations. Matematičeskie trudy, Tome 20 (2017) no. 2, pp. 193-205. http://geodesic.mathdoc.fr/item/MT_2017_20_2_a7/

[1] Arabadzhyan L. G., “Resheniya odnogo integralnogo uravneniya tipa Gammershteina”, Izv. NAN Armenii. Matem., 32:1 (1997), 21–28 | Zbl

[2] Arabadzhyan L. G., Engibaryan N. B., “Uravneniya v svertkakh i nelineinye funktsionalnye uravneniya”, Itogi nauki i tekhn. Ser. matem. anal., 22, VINITI, M., 1984, 175–244 | Zbl

[3] Engibaryan N. B., “O nepodvizhnoi tochke monotonnogo operatora v kriticheskom sluchae”, Izv. RAN, Ser. matem., 70:5 (2006), 79–96 | DOI | Zbl

[4] Engibaryan N. B., Khachatryan A. Kh., “O tochnoi linearizatsii zadachi skolzheniya razrezhennogo gaza v BGK modeli”, TMF, 125:2 (2000), 339–342 | DOI | Zbl

[5] Zabreiko P. P., “O nepreryvnosti i polnoi nepreryvnosti operatorov P. S. Urysona”, Dokl. AN SSSR, 161:5 (1965), 1007–1010 | Zbl

[6] Zabreiko P. P., Pustylnik E. I., “O nepreryvnosti i polnoi nepreryvnosti nelineinykh integralnykh operatorov v prostranstvakh $L_p$”, UMN, 19:2 (1964), 204–205

[7] Kogan M. N., Dinamika razrezhennogo gaza, Nauka, M., 1967

[8] Kolmogorov A. N., Fomin S. V., Elementy teorii funktsii i funktsionalnogo analiza, Nauka, M., 1981

[9] Krasnoselskii M. A., “Nekotorye zadachi nelineinogo analiza”, UMN, 9:3 (1954), 57–114 | Zbl

[10] Krasnoselskii M. A., Polozhitelnye resheniya operatornykh uravnenii: glavy nelineinogo analiza, Fizmatgiz, M., 1962

[11] Krasnoselskii M. A., Zabreiko P. P., “O razreshimosti nelineinykh operatornykh uravnenii”, Funkts. analiz i ego pril., 5:3 (1971), 42–44 | Zbl

[12] Uryson P. S., “Ob odnom tipe nelineinykh integralnykh uravnenii”, Matem. sb., 31:2 (1923), 236–255 | Zbl

[13] Khachatryan Kh. A., “Dostatochnye usloviya razreshimosti integralnogo uravneniya Urysona na poluosi”, Dokl. RAN, 425:4 (2009), 462–465 | Zbl

[14] Khachatryan Kh. A., “Ob odnom klasse integralnykh uravnenii tipa Urysona s silnoi nelineinostyu”, Izv. RAN, Ser. matem., 76:1 (2012), 173–200 | DOI | Zbl

[15] Khachatryan A. Kh., Khachatryan Kh. A., “Kachestvennoe razlichie reshenii dlya statsionarnykh modelnykh uravnenii Boltsmana v lineinom i nelineinom sluchayakh”, TMF, 180:2 (2014), 272–288 | DOI | Zbl

[16] Banas J., “Integrable solutions of Hammerstein and Urysohn integral equations”, J. Austral. Math. Soc. Ser. A, 46:1 (1989), 61–68 | DOI | MR | Zbl

[17] Brezis H., Browder F. E., “Existence theorems for nonlinear integral equations of Hammerstein type”, Bull. Amer. Math. Soc., 81:1 (1975), 73–78 | DOI | MR | Zbl

[18] Golomb M., “Zur Theorie der nichtlinearen Integralgleichungen”, Math. Z., 39 (1934), 45–75 | DOI | MR

[19] Hammerstein A., “Nichtlineare Integralgleichungen nebst Anwendungen”, Acta. Math., 54:1 (1930), 117–176 | DOI | MR | Zbl

[20] Khachatryan Kh. A., “On solvability of some classes of Urysohn nonlinear integral equations with noncompact operator”, Ufim. Mat. Zh., 2:2 (2010), 102–117 | MR | Zbl

[21] Vladimirov V. S., “The equation of the p-adic closed strings for the scalar tachyon field”, Sci. China, Ser. A, 51:4 (2008), 754–764 | DOI | MR | Zbl

[22] Vladimirov V. S., Volovich Y. I., “Nonlinear dynamics equation in p-adic string theory”, Theor. Math. Phys., 138:3 (2004), 297–309 | DOI | MR | Zbl