Geodesic
Approximate solutions of nonlinear convolution type equations on segment
Ufa mathematical journal, Tome 5 (2013) no. 2, pp. 3-11 Cet article a éte moissonné depuis la source Math-Net.Ru

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For various classes of integral convolution type equations with a monotone nonlinearity, we prove global solvability and uniqueness theorems as well as theorems on the ways for finding the solutions in real Lebesgue spaces. It is shown that the solutions can be found in space $L_2(0, 1)$ by a Picard's type successive approximations method and we prove the estimates for the rate of convergence. The obtained results cover, in particular, linear integral convolution type equations. In the case of a power nonlinearity, it is shown that the solutions can be found by the gradient method in the space $L_p(0, 1)$ and weighted spaces $L_p(\varrho)$.
Keywords: nonlinear integral equations, convolution type operator, potential operator, monotone operator. 
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S. N. Askhabov; A. L. Dzhabrailov. Approximate solutions of nonlinear convolution type equations on segment. Ufa mathematical journal, Tome 5 (2013) no. 2, pp. 3-11. http://geodesic.mathdoc.fr/item/UFA_2013_5_2_a0/

[1] Askhabov S. N., “Nelineinye integralnye uravneniya tipa svertki na otrezke”, Izvestiya vuzov. Sev-Kav. region. Estestv. nauki, 2007, no. 1, 3–5

[2] Askhabov S. N., “Priblizhennoe reshenii nelineinykh uravnenii s vesovymi operatorami tipa potentsiala”, Ufimskii matematicheskii zhurnal, 3:4 (2011), 8–13 | Zbl

[3] Vainberg M. M., Variatsionnyi metod i metod monotonnykh operatorov v teorii nelineinykh uravnenii, Nauka, M., 1972, 416 pp. | MR | Zbl

[4] Askhabov S. N., Nelineinye uravneniya tipa svertki, Fizmatlit, M., 2009, 304 pp. | MR

[5] Nakhushev A. M., Drobnoe ischislenie i ego primenenie, Fizmatlit, M., 2003, 272 pp. | Zbl

[6] Gaevskii Kh., Greger K., Zakharias K., Nelineinye operatornye uravneniya i operatornye differentsialnye uravneniya, Mir, M., 1978, 336 pp. | MR

[7] Khvedelidze B. V., “Lineinye razryvnye granichnye zadachi teorii funktsii, singulyarnye integralnye uravneniya i nekotorye ikh prilozheniya”, Trudy Tbilis. mat. in-ta AN GruzSSR, 23, 1956, 3–158

[8] Askhabov S. N., “Primenenie metoda monotonnykh operatorov k nekotorym nelineinym uravneniyam tipa svertki i singulyarnym integralnym uravneniyam”, Izvestiya vuzov. Matematika, 1981, no. 9, 64–66 | MR | Zbl

[9] Askhabov S. N., “Nelineinye singulyarnye integralnye uravneniya v prostranstvakh Lebega”, Sovremennaya matematika i ee prilozheniya, 67 (2010), 33–48