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
M. A. Krasnosel'skii,Topological Methods in the Theory of Nonlinear Integral Equations [in Russian], Gostekhizdat, Moscow (1956).
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.
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).
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).
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.
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.
J. Keller and S. Antman (editors),Bifurcation Theory and Nonlinear Eigenvalue Problems, (Benjamin, New York-Amsterdam 1969).
Ch. Cosner, “Bifurcations from higher eigenvalues in nonlinear elliptic equations: continua that meet infinity,”Nonlin. Analysis,12, No. 3, 271–277 (1988).
A. A. Voronov,Foundtions of the Theory of Automatic Regulations and Control [in Russian], Vysshaya Shkola, Moscow (1977).
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.
R. Courant and D. Hilbert,Methoden der Mathematischen Physik, Vol. 1, Springer, Berlin (1931).
M. W. Hirsch,Differential Topology, Springer-Verlag, New York (1976).
Ya. M. Dymarskii, “On typical bifurcations in a certain class of operator equations,”Dokl. Akad. Nauk Rossii,338, No. 4, 446–449 (1994).
E. N. Dancer, “On the structure of solutions of nonlinear eigenvalue problems,”Indiana Univ. Math. J.,23, No. 11, 1069–1076 (1974).
J. Ize, “Bifurcation theory for Fredholm operators,”Memory AMS,7, No. 174, 1023–1039 (1976).
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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