Positive solubility of some classes of non-linear integral equations

Kh. A. Khachatryan

Izvestiya. Mathematics , Tome 79 (2015) no. 2, pp. 411-430.

Résumé

We study certain classes of non-linear Hammerstein integral equations on the semi-axis and the whole line. These classes of equations arise in the theory of radiative transfer in nuclear reactors, in the kinetic theory of gases, and for travelling waves in non-linear Richer competition systems. By combining special iteration methods with the methods of construction of invariant cone segments for the appropriate non-linear operator, we are able to prove constructive existence theorems for positive solutions in various function spaces. We give illustrative examples of equations satisfying all the hypotheses of our theorems.

Détail

Keywords: Hammerstein equation, Carathéodory condition, monotonicity, induction, iterations
Mots-clés : convergence.

Bibliographie

[1] N. B. Engibaryan, “Ob odnoi zadache nelineinogo perenosa izlucheniya”, Astrofizika, 2:1 (1966), 31–36

[2] M. N. Kogan, Dinamika razrezhennogo gaza, Nauka, M., 1967, 440 pp. | Zbl

[3] N. B. Engibaryan, A. Kh. Khachatryan, “Exact linearization of the sliding problem for a dilute gas in the Bhatnagar–Gross–Krook model”, Theoret. and Math. Phys., 125:2 (2000), 1589–1592 | DOI | DOI | MR | Zbl

[4] A. Kh. Khachatryan, Kh. A. Khachatryan, “Qualitative difference between solutions for a model of the Boltzmann equation in the linear and nonlinear cases”, Theoret. and Math. Phys., 172:3 (2012), 1315–1320 | DOI | DOI | MR | Zbl

[5] Li Kun, Li Xiong, “Asymptotic behavior and uniquness of traveling wave solutions in Richer competition system”, J. Math. Anal. Appl., 389:1 (2012), 486–497 | DOI | MR | Zbl

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

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

[8] M. A. Krasnoselskii, L. A. Ladyzhenskii, “Usloviya polnoi nepreryvnosti operatora P. S. Urysona, deistvuyuschego v prostranstve $L^p$”, Tr. MMO, 3, GITTL, M., 1954, 307–320 | MR | Zbl

[9] V. Ya. Stetsenko, A. R. Esayan, “Teoremy o polozhitelnykh resheniyakh uravnenii vtorogo roda s nelineinymi operatorami”, Matem. sb., 68(110):4 (1965), 473–486 | MR | Zbl

[10] P. P. Zabreiko, M. A. Krasnosel'skii, “Solvability of nonlinear operator equations”, Funct. Anal. Appl., 5:3 (1971), 206–208 | DOI | MR | Zbl

[11] M. A. Krasnoselskii, Ya. B. Rutitskii, “Prostranstva Orlicha i nelineinye integralnye uravneniya”, Tr. MMO, 7, GIFML, M., 1958, 63–120 | MR | Zbl

[12] P. P. Zabreiko, E. I. Pustylnik, “O nepreryvnosti i polnoi nepreryvnosti nelineinykh integralnykh operatorov v prostranstvakh $L_p$”, UMN, 19:2(116) (1964), 204–205

[13] F. E. Browder, “Strongly nonlinear integral equations of Hammerstein type”, Proc. Nat. Acad. Sci. U.S.A., 72:5 (1975), 1937–1939 | DOI | MR | Zbl

[14] H. Brézis, F. E. Browder, “Some new results about Hammerstein equations”, Bull. Amer. Math. Soc., 80:3 (1974), 567–572 | DOI | MR | Zbl

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

[16] L. G. Arabadzhyan, “Existence of nontrivial solutions of certain linear and nonlinear convolution-type equations”, Ukrainian Math. J., 41:12 (1989), 1359–1367 | DOI | MR | Zbl

[17] Kh. A. Khachatryan, “One-parameter family of solutions for one class of Hammerstein nonlinear equations on a half-line”, Dokl. Math., 80:3 (2009), 872–876 | DOI | MR | Zbl

[18] Kh. A. Khachatryan, “On a class of nonlinear integral equations with a noncompact operator”, J. Contemp. Math. Anal., 46:2 (2011), 89–100 | DOI | MR | Zbl

[19] Kh. A. Khachatryan, “On a class of integral equations of Urysohn type with strong non-linearity”, Izv. Math., 76:1 (2012), 163–189 | DOI | DOI | MR | Zbl

[20] Kh. A. Khachatryan, “Sufficient conditions for the solvability of the Urysohn integral equation on a half-line”, Dokl. Math., 79:2 (2009), 246–249 | DOI | MR | Zbl

[21] A. Kh. Khachatryan, Kh. A. Khachatryan, “Existence and uniqueness theorem for a Hammerstein nonlinear integral equation”, Opuscula Math., 31:3 (2011), 393–398 | DOI | MR | Zbl

[22] Kh. A. Khachatryan, “Nekotorye klassy nelineinykh integralnykh uravnenii Urysona na poluosi”, Dokl. NAN Belarusi, 55:1 (2011), 5–9 | MR | Zbl

[23] V. S. Vladimirov, “The equation of the $p$-adic open string for the scalar tachyon field”, Izv. Math., 69:3 (2005), 487–512 | DOI | DOI | MR | Zbl

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

[25] I. Ya. Arefeva, I. V. Volovich, “O nelokalnykh kosmologicheskikh uravneniyakh na poluosi”, Vestn. Sam. gos. tekhn. un-ta. Ser. Fiz.-mat. nauki, 1(22) (2011), 16–27 | DOI

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

[27] L. G. Arabadzhyan, A. S. Khachatryan, “A class of integral equations of convolution type”, Sb. Math., 198:7 (2007), 949–966 | DOI | DOI | MR | Zbl

[28] L. G. Arabadzhyan, N. B. Engibaryan, “Convolution equations and nonlinear functional equations”, J. Soviet Math., 36:6 (1987), 745–791 | DOI | MR | Zbl

[29] W. Rudin, Functional analysis, McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., 1973, xiii+397 pp. | MR | MR | Zbl

[30] N. B. Yengibarian, “Renewal equation on the whole line”, Stochastic Process. Appl., 85:2 (2000), 237–247 | DOI | MR | Zbl