Geodesic
Semi-regular solutions of integral equations with discontinuous nonlinearities
Matematičeskie zametki, Tome 116 (2024) no. 1, pp. 109-121.

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We study integral equations with discontinuous nonlinearities in Lebesgue spaces. Using the variational method, based on the concept of a quasipotential operator, we establish a theorem on the existence of semi-regular solutions. For equations with a parameter, a theorem on the existence of nontrivial semi-regular solutions for sufficiently large parameter values is obtained. An example of an applied problem for which the conditions of these theorems are satisfied is given.
Keywords: integral equation, discontinuous nonlinearity, parameter, semi-regular solution, variational method. 
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V. N. Pavlenko; D. K. Potapov. Semi-regular solutions of integral equations with discontinuous nonlinearities. Matematičeskie zametki, Tome 116 (2024) no. 1, pp. 109-121. http://geodesic.mathdoc.fr/item/MZM_2024_116_1_a7/

[1] M. A. Goldshtik, “Matematicheskaya model otryvnykh techenii neszhimaemoi zhidkosti”, Dokl. AN SSSR, 147:6 (1962), 1310–1313

[2] M. A. Krasnoselskii, A. V. Pokrovskii, “Pravilnye resheniya uravnenii s razryvnymi nelineinostyami”, Dokl. AN SSSR, 226:3 (1976), 506–509 | MR | Zbl

[3] M. A. Krasnoselskii, A. V. Pokrovskii, “Ob ellipticheskikh uravneniyakh s razryvnymi nelineinostyami”, Dokl. RAN, 342:6 (1995), 731–734 | MR | Zbl

[4] V. N. Pavlenko, D. K. Potapov, “Cuschestvovanie polupravilnykh reshenii ellipticheskikh spektralnykh zadach s razryvnymi nelineinostyami”, Matem. sb., 206:9 (2015), 121–138 | DOI | MR | Zbl

[5] V. N. Pavlenko, D. K. Potapov, “Ob odnom klasse ellipticheskikh kraevykh zadach s parametrom i razryvnoi nelineinostyu”, Izv. RAN. Ser. matem., 84:3 (2020), 168–184 | DOI | MR

[6] V. N. Pavlenko, “O razreshimosti nekotorykh nelineinykh uravnenii s razryvnymi operatorami”, Dokl. AN SSSR, 204:6 (1972), 1320–1323 | MR | Zbl

[7] V. N. Pavlenko, “Variatsionnyi metod dlya uravnenii s razryvnymi operatorami”, Vestnik ChelGU, 1994, no. 2, 87–95

[8] V. N. Pavlenko, D. K. Potapov, “O suschestvovanii lucha sobstvennykh znachenii dlya uravnenii s razryvnymi operatorami”, Sib. matem. zhurn., 42:4 (2001), 911–919 | MR | Zbl

[9] M. M. Vainberg, I. M. Lavrentev, “Nelineinye kvazipotentsialnye operatory”, Dokl. AN SSSR, 205:5 (1972), 1022–1024 | MR | Zbl

[10] V. N. Pavlenko, D. K. Potapov, “Variatsionnyi metod dlya ellipticheskikh sistem s razryvnymi nelineinostyami”, Matem. sb., 212:5 (2021), 133–152 | DOI | MR

[11] V. N. Pavlenko, D. K. Potapov, “Polupravilnye resheniya ellipticheskikh kraevykh zadach s razryvnymi nelineinostyami eksponentsialnogo rosta”, Matem. sb., 213:7 (2022), 121–138 | DOI | MR

[12] F. Cianciaruso, G. Infante, P. Pietramala, “Solutions of perturbed Hammerstein integral equations with applications”, Nonlinear Anal. Real World Appl., 33 (2017), 317–347 | DOI | MR

[13] R. de Sousa, F. Minhós, “On coupled systems of Hammerstein integral equations”, Bound. Value Probl., 7 (2019), 1–14 | MR

[14] F. Minhós, “Solvability of generalized Hammerstein integral equations on unbounded domains, with sign-changing kernels”, Appl. Math. Lett., 65 (2017), 113–117 | DOI | MR

[15] F. Minhós, R. de Sousa, “Solvability of coupled systems of generalized Hammerstein-type integral equations in the real line”, Mathematics, 8 (2020), 1–9 | MR

[16] M. M. Vainberg, Variatsionnye metody issledovaniya nelineinykh operatorov, Gostekhizdat, M., 1956 | MR

[17] M. M. Vainberg, Variatsionnyi metod i metod monotonnykh operatorov v teorii nelineinykh uravnenii, Nauka, M., 1972 | MR

[18] I. V. Shragin, “Usloviya izmerimosti superpozitsii”, Dokl. AN SSSR, 197:2 (1971), 295–298 | MR | Zbl

[19] V. N. Pavlenko, “Teoremy suschestvovaniya dlya ellipticheskikh variatsionnykh neravenstv s kvazipotentsialnymi operatorami”, Differents. uravneniya, 24:8 (1988), 1397–1402 | MR

[20] M. A. Krasnoselskii, Topologicheskie metody v teorii nelineinykh integralnykh uravnenii, Gostekhizdat, M., 1956 | MR

[21] L. A. Lyusternik, V. I. Sobolev, Kratkii kurs funktsionalnogo analiza: Ucheb. posobie, Vyssh. shkola, M., 1982 | MR

[22] A. N. Kolmogorov, S. V. Fomin, Elementy teorii funktsii i funktsionalnogo analiza, Nauka, M., 1972 | MR