On a~class of integral equations of Urysohn type with strong non-linearity

Kh. A. Khachatryan

Izvestiya. Mathematics , Tome 76 (2012) no. 1, pp. 163-189.

Résumé

We study a class of homogeneous and non-homogeneous integral equations of Urysohn type with strong non-linearity on the positive semi-axis. It is assumed that some non-linear integral operator of Wiener–Hopf–Hammerstein type is a local minorant of the corresponding Urysohn operator. Using special methods of the linear theory of convolution-type integral equations, we construct positive solutions for these classes of Urysohn equations. We also study the asymptotic behaviour of these solutions at infinity. As an auxiliary fact in the course of the proof of these assertions, we construct a one-parameter family of positive solutions for non-linear integral equations of Wiener–Hopf–Hammerstein type whose operator is a minorant for the original Urysohn operator. We give particular examples of non-linear integral equations for which all the hypotheses of the main theorems hold.

Détail

Keywords: Urysohn equation, one-parameter family of solutions, factorization.
Mots-clés : minorant

Bibliographie

[1] M. A. Krasnoselskii, Polozhitelnye resheniya operatornykh uravnenii, Fizmatlit, M., 1962 | MR | Zbl

[2] P. P. Zabrejko, “Continuity and complete continuity of Urysohn operators”, Soviet Math. Dokl., 6 (1965), 540–544 | MR | Zbl

[3] M. A. Krasnoselskii, “Priznaki nepreryvnosti nekotorykh nelineinykh operatorov”, Ukr. matem. zhurn., 2:3 (1950), 70–86 | MR | Zbl

[4] G. Emmanuele, “An existence theorem for Hammerstein integral equations”, Portugal. Math., 51:4 (1994), 607–611 | MR | Zbl

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

[6] P. P. Zabreiko, “O nepreryvnosti nelineinogo operatora”, Sib. matem. zhurn., 5:4 (1964), 958–960 | MR | Zbl

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

[8] 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

[9] A. Kh. Khachatryan, Kh. A. Khachatryan, “A nonlinear integral equation of Hammerstein type with a noncompact operator”, Sb. Math., 201:4 (2010), 595–606 | DOI | MR | Zbl

[10] Kh. A. Khachatryan, “On solvability of some classes of Urysohn nonlinear integral equations with noncompact operators”, Ufimsk. matem. zhurn., 2:2 (2010), 102–117

[11] Kh. A. Khachatryan, “Solubility of a class of second-order integro-differential equations with monotone non-linearity on a semi-axis”, Izv. Math., 74:5 (2010), 1069–1082 | DOI | MR | Zbl

[12] I. C. Gohberg, I. A. Fel'dman, Convolution equations and projection methods for their solution, Amer. Math. Soc., Providence, RI, 1974 | MR | MR | Zbl

[13] S. G. Krein, Lineinye uravneniya v banakhovom prostranstve, Nauka, M., 1971 | MR | Zbl

[14] N. B. Engibaryan, L. G. Arabadzhyan, “Some factorization problems for convolution integral operators”, Differential Equations, 26:8 (1990), 1069–1078 | MR | Zbl | Zbl

[15] A. N. Kolmogorov, S. V. Fomin, Introductory real analysis, Prentice-Hall, Englewood Cliffs, NJ, 1970 | MR | MR | Zbl | Zbl

[16] G. G. Gevorkyan, N. B. Engibaryan, “New theorems for the renewal integral equation”, J. Contemp. Math. Anal., 32:1 (1997), 2–16 | MR | Zbl

[17] L. G. Arabadzhyan, “Ob odnom integralnom uravnenii teorii perenosa v neodnorodnoi srede”, Differents. uravneniya, 23:9 (1987), 1618–1622 | MR | Zbl

[18] L. G. Arabadzhyan, “Solutions of certain integral equations of the Hammerstein type”, J. Contemp. Math. Anal., 32:1 (1997), 17–24 | MR | Zbl

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