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Mat. Sb. (N.S.), 1971, Volume 85(127), Number 3(7), Pages 440–454 (Mi msb3265)  

This article is cited in 12 scientific papers (total in 13 papers)

The use of group properties to determine milti-parameter families of solutions of nonlinear equations

B. V. Loginov, V. A. Trenogin


Abstract: We consider a nonlinear equation in a Banach space which is invariant relative to a continuous group. We give conditions which allow us to reduce Lyapunov–Schmidt branch equations in both the number of equations and the number of unknowns, which makes it possible to simplify significantly the search for multi-parameter families of solutions of the given problem.
Bibliography: 15 titles.

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English version:
Mathematics of the USSR-Sbornik, 1971, 14:3, 438–452

Bibliographic databases:

UDC: 517.93+513.881
MSC: Primary 34G05; Secondary 34B15
Received: 15.06.1970

Citation: B. V. Loginov, V. A. Trenogin, “The use of group properties to determine milti-parameter families of solutions of nonlinear equations”, Mat. Sb. (N.S.), 85(127):3(7) (1971), 440–454; Math. USSR-Sb., 14:3 (1971), 438–452

Citation in format AMSBIB
\Bibitem{LogTre71}
\by B.~V.~Loginov, V.~A.~Trenogin
\paper The use of group properties to determine milti-parameter families of solutions of nonlinear equations
\jour Mat. Sb. (N.S.)
\yr 1971
\vol 85(127)
\issue 3(7)
\pages 440--454
\mathnet{http://mi.mathnet.ru/msb3265}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=283642}
\zmath{https://zbmath.org/?q=an:0221.47040}
\transl
\jour Math. USSR-Sb.
\yr 1971
\vol 14
\issue 3
\pages 438--452
\crossref{https://doi.org/10.1070/SM1971v014n03ABEH002627}


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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. T. S. Èrgashbaev, “On the branching theory for a non-linear operator invariant under a group”, Russian Math. Surveys, 39:6 (1984), 207–208  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    2. B. V. Loginov, N. A. Sidorov, “Group symmetry of the Lyapunov–Schmidt branching equation and iterative methods in the problem of a bifurcation point”, Math. USSR-Sb., 73:1 (1992), 67–77  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    3. B. V. Loginov, V. A. Trenogin, “Branching equation of Andronov-Hopf bifurcation under group symmetry conditions”, Chaos, 7:2 (1997), 229  crossref  mathscinet  zmath  adsnasa  isi  elib
    4. B.V. Loginov, “Determination of the branching equation by its group symmetry—Andronov—Hopf bifurcation”, Nonlinear Analysis: Theory, Methods & Applications, 28:12 (1997), 2033  crossref  mathscinet  zmath
    5. V. S. Vladimirov, L. D. Kudryavtsev, S. M. Nikol'skii, D. M. Klimov, F. L. Chernous'ko, “Vladilen Aleksandrovich Trenogin (on his 70th birthday)”, Russian Math. Surveys, 56:6 (2001), 1199–1207  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    6. N. A. Sidorov, V. R. Abdullin, “Interlaced branching equations in the theory of non-linear equations”, Sb. Math., 192:7 (2001), 1035–1052  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    7. Abdullin V., Sidorov N., “Intertwined Equations in Bifurcation Theory”, Dokl. Math., 63:2 (2001), 179–181  zmath  isi
    8. Karasozen, B, “Invariant reduction of partially potential branching equations and iterative methods in the problem on a bifurcation point with a symmetry”, Differential Equations, 40:3 (2004), 410  mathnet  crossref  mathscinet  zmath  isi  elib
    9. L.R. Kim-Tyan, B.V. Loginov, Yu.B. Rousak, “On the stability of periodic solutions for differential equations with a Fredholm operator at the highest derivative”, Nonlinear Analysis: Theory, Methods & Applications, 67:5 (2007), 1570  crossref  mathscinet  zmath
    10. A. N. Andronov, “Ob ustoichivosti razvetvlyayuschikhsya semeistv reshenii zadachi o kapillyarno-gravitatsionnykh volnakh v glubokom prostranstvennom sloe flotiruyuschei zhidkosti”, Vestn. SamGU. Estestvennonauchn. ser., 2009, no. 2(68), 10–25  mathnet
    11. E. N. Dancer, “On the existence of bifurcating solutions in the presence of symmetries”, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 85:3-4 (2011), 321  crossref  mathscinet
    12. Andronov A.N., “Sistema uravnenii laplasa v poluprostranstve so svobodnoi granitsei razdela, kapillyarno-gravitatsionnye volny, bifurkatsiya i simmetriya”, Differentsialnye uravneniya, 47:5 (2011), 756–760  mathscinet  zmath  elib
    13. N. I. Makarenko, Z. V. Makridin, “Periodic oscillations and waves in nonlinear weakly coupled dispersive systems”, Proc. Steklov Inst. Math., 300 (2018), 149–158  mathnet  crossref  crossref  isi  elib
  • Математический сборник (новая серия) - 1964–1988 Sbornik: Mathematics (from 1967)
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