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Unbounded branches of solutions of some boundary-value problems

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Abstract

For nonlinear equations of a special type, in the case of double degeneration of a linearized problem, we prove the existence of unbounded branches of solutions originating at a bifurcation point.

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References

  1. M. A. Krasnosel'skii,Topological Methods in the Theory of Nonlinear Integral Equations [in Russian], Gostekhizdat, Moscow (1956).

    Google Scholar 

  2. P. H. Rabinowitz, “A global theorem for nonlinear eigenvalue problems and applications,” in:Contributions to Nonlinear Functional Analysis, Academic Press, New York (1971), pp. 11–36.

    Google Scholar 

  3. L. A. Lyusternik, “On one boundary-value problem in the theory of nonlinear differential equations,”Dokl. Akad. Nauk Ukr. SSSR, Ser. A,33, 5–8 (1941).

    Google Scholar 

  4. Ya. M. Dymarskii, “On normalized eigenfunctions of a two-point nonlinear boundary-value problems,”Dokl. Akad. Nauk Ukr. SSR, Ser. A,4, 4–8 (1984).

    MathSciNet  Google Scholar 

  5. Ya. M. Dymarskii, “On the Lysternik theorem for a two-point problem of the fourth order,” in:Qualitative and Approximate Methods for the Investigation of Operator Equations [in Russian], Yaroslavl' (1984), pp. 16–24.

  6. Ya. M. Dymarskii, “Existence, oscillation properties, and asymptotics of normalized eigenfunctions of nonlinear boundary-value problems,” in:Qualitative and Approximate Methods for the Investigation of Operator Equations [in Russian], Yaroslavl' (1984). pp. 133–139.

  7. J. Keller and S. Antman (editors),Bifurcation Theory and Nonlinear Eigenvalue Problems, (Benjamin, New York-Amsterdam 1969).

    MATH  Google Scholar 

  8. Ch. Cosner, “Bifurcations from higher eigenvalues in nonlinear elliptic equations: continua that meet infinity,”Nonlin. Analysis,12, No. 3, 271–277 (1988).

    Article  MATH  MathSciNet  Google Scholar 

  9. A. A. Voronov,Foundtions of the Theory of Automatic Regulations and Control [in Russian], Vysshaya Shkola, Moscow (1977).

    Google Scholar 

  10. Ya. M. Dymarskii, “On noncompact branches of eigenfunctions of nonlinear boundary-value problems,” in:Proceedings of the 26th Voronezh Winter School [in Russian], Voronezh University, Voronezh (1994), p. 45.

    Google Scholar 

  11. R. Courant and D. Hilbert,Methoden der Mathematischen Physik, Vol. 1, Springer, Berlin (1931).

    Google Scholar 

  12. M. W. Hirsch,Differential Topology, Springer-Verlag, New York (1976).

    MATH  Google Scholar 

  13. Ya. M. Dymarskii, “On typical bifurcations in a certain class of operator equations,”Dokl. Akad. Nauk Rossii,338, No. 4, 446–449 (1994).

    Google Scholar 

  14. E. N. Dancer, “On the structure of solutions of nonlinear eigenvalue problems,”Indiana Univ. Math. J.,23, No. 11, 1069–1076 (1974).

    Article  MATH  MathSciNet  Google Scholar 

  15. J. Ize, “Bifurcation theory for Fredholm operators,”Memory AMS,7, No. 174, 1023–1039 (1976).

    MathSciNet  Google Scholar 

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Lugansk Pedagogical Institute, Lugansk. Translated from Ukrainskii Matematischeskii Zhurnal, Vol. 48, No. 9, pp. 1194–1199, September, 1996.

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Dymarskii, Y.M. Unbounded branches of solutions of some boundary-value problems. Ukr Math J 48, 1353–1360 (1996). https://doi.org/10.1007/BF02595357

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  • DOI: https://doi.org/10.1007/BF02595357

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