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Izv. RAN. Ser. Mat., 2012, Volume 76, Issue 1, Pages 173–200 (Mi izv5402)  

This article is cited in 10 scientific papers (total in 10 papers)

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

Kh. A. Khachatryan

Institute of Mathematics, National Academy of Sciences of Armenia

Abstract: 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.

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


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English version:
Izvestiya: Mathematics, 2012, 76:1, 163–189

Bibliographic databases:

UDC: 517.968
MSC: 45G05, 45M05, 45M20
Received: 29.09.2010
Revised: 09.03.2011

Citation: Kh. A. Khachatryan, “On a class of integral equations of Urysohn type with strong non-linearity”, Izv. RAN. Ser. Mat., 76:1 (2012), 173–200; Izv. Math., 76:1 (2012), 163–189

Citation in format AMSBIB
\by Kh.~A.~Khachatryan
\paper On a~class of integral equations of Urysohn type with strong non-linearity
\jour Izv. RAN. Ser. Mat.
\yr 2012
\vol 76
\issue 1
\pages 173--200
\jour Izv. Math.
\yr 2012
\vol 76
\issue 1
\pages 163--189

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    This publication is cited in the following articles:
    1. Kh. A. Khachatryan, “O nekotorykh klassakh nelineinykh integralnykh uravnenii s nekompaktnymi operatorami”, Vestn. Sam. gos. tekhn. un-ta. Ser. Fiz.-mat. nauki, 1(30) (2013), 181–188  mathnet  crossref
    2. M. F. Broyan, Kh. A. Khachatryan, “On some nonlinear integral and integro-differential equations with noncompact operators on positive half-line”, Ufa Math. J., 5:2 (2013), 31–42  mathnet  crossref  mathscinet  elib
    3. Khachatryan Kh.A., Broyan M.F., “One-Parameter Family of Positive Solutions for a Class of Non-Linear Infinite Algebraic Systems with Toeplitz-Hankel Type Matrices”, J. Contemp. Math. Anal.-Armen. Aca., 48:5 (2013), 209–220  crossref  mathscinet  zmath  isi  scopus
    4. Kh. A. Khachatryan, T. H. Sardaryan, “On solvability of a class of nonlinear integral equations with Hammerstein type noncompact operator in the space $L_1(R^+)$”, Uch. zapiski EGU, ser. Fizika i Matematika, 2014, no. 3, 16–23  mathnet
    5. Kh. A. Khachatryan, “Positive solubility of some classes of non-linear integral equations of Hammerstein type on the semi-axis and on the whole line”, Izv. Math., 79:2 (2015), 411–430  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    6. K. A. Khachatryan, T. E. Terdzhyan, “On the solvability of one class of nonlinear integral equations in $L_1(0,+\infty)$”, Siberian Adv. Math., 25:4 (2015), 268–275  mathnet  crossref  crossref  mathscinet  elib
    7. Kh. A. Khachatryan, A. S. Petrosyan, A. A. Sisakyan, “O netrivialnoi razreshimosti odnogo klassa nelineinykh integralnykh uravnenii tipa Urysona”, Tr. IMM UrO RAN, 23, no. 2, 2017, 266–273  mathnet  crossref  elib
    8. Kh. A. Khachatryan, “On the solvability of one class of two-dimensional Urysohn integral equations”, Siberian Adv. Math., 28:3 (2018), 166–174  mathnet  crossref  crossref  elib
    9. Kh. A. Khachatryan, H. S. Petrosyan, “One parameter families of positive solutions of some classes of convolution type nonlinear integral equations”, J. Math. Sci., 231:2 (2018), 153–167  mathnet  crossref  crossref
    10. A. S. Petrosyan, Ts. E. Terdzhyan, Kh. A. Khachatryan, “Edinstvennost resheniya odnoi sistemy integralnykh uravnenii na poluosi s vypukloi nelineinostyu”, Matem. tr., 23:2 (2020), 187–203  mathnet  crossref
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