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Sibirsk. Mat. Zh., 2010, Volume 51, Number 2, Pages 404–409 (Mi smj2093)  

This article is cited in 12 scientific papers (total in 12 papers)

Solving the Hammerstein integral equation in the irregular case by successive approximations

N. A. Sidorova, D. N. Sidorovb

a Irkutsk State University, Irkutsk
b L. A. Melentiev Energy Systems Institute, Siberian Branch of the Russian Academy of Sciences, Irkutsk

Abstract: The branches of a solution of the nonlinear integral equation
$$ u(x)=\int_a^bK(x,s)q(s,u(s),\lambda) ds, $$
where $q(s,u,\lambda)=u(s)+\sum_{i=0}^\infty\sum_{k=1}^\infty q_{ik}(s)u^i\lambda^k$ and $\lambda$ is a parameter, are constructed by successive approximations. Under consideration is the case when unity is a characteristic number of the kernel $K(x,s)$ of rank $n\ge1$, and $\lambda=0$ is a bifurcation point. The principal term of the asymptotic expansion constructed is used as an initial approximation. The uniform convergence is established in some neighborhood about the bifurcation point on using the implicit function theorem and the Schmidt lemma.

Keywords: Hammerstein equation, successive approximation, bifurcation.

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English version:
Siberian Mathematical Journal, 2010, 51:2, 325–329

Bibliographic databases:

UDC: 517.988.67
Received: 14.02.2009

Citation: N. A. Sidorov, D. N. Sidorov, “Solving the Hammerstein integral equation in the irregular case by successive approximations”, Sibirsk. Mat. Zh., 51:2 (2010), 404–409; Siberian Math. J., 51:2 (2010), 325–329

Citation in format AMSBIB
\by N.~A.~Sidorov, D.~N.~Sidorov
\paper Solving the Hammerstein integral equation in the irregular case by successive approximations
\jour Sibirsk. Mat. Zh.
\yr 2010
\vol 51
\issue 2
\pages 404--409
\jour Siberian Math. J.
\yr 2010
\vol 51
\issue 2
\pages 325--329

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    This publication is cited in the following articles:
    1. Sidorov D.N., Sidorov N.A., Leontev R.Yu., “Asimptoticheskie priblizheniya reshenii nelineinykh kraevykh zadach s vektornym parametrom v okrestnosti tochki bifurkatsii”, Sovremennye tekhnologii. Sistemnyi analiz. Modelirovanie, 2011, no. 3, 16–22  mathscinet  elib
    2. R. Yu. Leontev, N. A. Sidorov, “Uniformizatsiya i posledovatelnye priblizheniya reshenii nelineinykh uravnenii s vektornym parametrom”, Izvestiya Irkutskogo gosudarstvennogo universiteta. Seriya Matematika, 4:3 (2011), 116–123  mathnet
    3. N. A. Sidorov, D. N. Sidorov, R. Yu. Leont'ev, “Successive approximations to solutions of nonlinear equations with vector parameter in the irregular case”, J. Appl. Industr. Math., 6:3 (2012), 387–392  mathnet  crossref  mathscinet
    4. N. A. Sidorov, D. N. Sidorov, “O posledovatelnykh priblizheniyakh reshenii vyrozhdennoi zadachi Koshi”, Tr. IMM UrO RAN, 18, no. 2, 2012, 238–244  mathnet  elib
    5. I. R. Muftahov, D. N. Sidorov, N. A. Sidorov, “On perturbation method for the first kind equations: regularization and application”, Vestn. YuUrGU. Ser. Matem. modelirovanie i programmirovanie, 8:2 (2015), 69–80  mathnet  crossref  elib
    6. N. A. Sidorov, D. N. Sidorov, I. R. Muftakhov, “O roli metoda vozmuschenii i teoremy Banakha–Shteingauza v voprosakh regulyarizatsii uravnenii pervogo roda”, Izvestiya Irkutskogo gosudarstvennogo universiteta. Seriya Matematika, 14 (2015), 82–99  mathnet
    7. N. A. Sidorov, “Classic solutions of boundary value problems for partial differential equations with operator of finite index in the main part of equation”, Izvestiya Irkutskogo gosudarstvennogo universiteta. Seriya Matematika, 27 (2019), 55–70  mathnet  crossref
    8. N. A. Sidorov, A. I. Dreglya, “Differentsialnye uravneniya v banakhovykh prostranstvakh s neobratimym operatorom v glavnoi chasti i neklassicheskimi nachalnymi usloviyami”, Differentsialnye uravneniya i optimalnoe upravlenie, Itogi nauki i tekhn. Ser. Sovrem. mat. i ee pril. Temat. obz., 183, VINITI RAN, M., 2020, 120–129  mathnet  crossref
    9. Ameer E., Aydi H., Hammad H.A., Shatanawi W., Mlaiki N., “on (Phi, Psi)-Metric Spaces With Applications”, Symmetry-Basel, 12:9 (2020), 1459  crossref  isi  scopus
    10. Sidorov N., Sidorov D., Dreglea A., “Solvability and Bifurcation of Solutions of Nonlinear Equations With Fredholm Operator”, Symmetry-Basel, 12:6 (2020), 912  crossref  isi  scopus
    11. Yuldashev T.K., “Nonlocal Inverse Problem For a Pseudohyperbolic-Pseudoelliptic Type Integro-Differential Equations”, Axioms, 9:2 (2020), 45  crossref  isi  scopus
    12. Minhos F., de Sousa R., “Solvability of Coupled Systems of Generalized Hammerstein-Type Integral Equations in the Real Line”, Mathematics, 8:1 (2020), 111  crossref  mathscinet  isi  scopus
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