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2014 | 12 | 12 | 1811-1828
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Some global results for nonlinear fourth order eigenvalue problems

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EN
In this paper, we consider the nonlinear fourth order eigenvalue problem. We show the existence of family of unbounded continua of nontrivial solutions bifurcating from the line of trivial solutions. These global continua have properties similar to those found in Rabinowitz and Berestycki well-known global bifurcation theorems.
Bibliografia
  • [1] Aliev Z.S., Basis properties in L p of systems of root functions of a spectral problem with spectral parameter in a boundary condition, Differential Equations, 2011, 47(6), 766–777 http://dx.doi.org/10.1134/S0012266111060024
  • [2] Allakhverdiev T.I., The study of some linear and nonlinear Sturm-Liouville problem with spectral parameter in boundary conditions, Dissertation, Baku, 1991 (in Russian)
  • [3] Ben Amara J., Vladimirov A.A., On oscillation of eigenfunctions of a fourth-order problem with spectral parameter in boundary condition, J. Math. Sciences, 2008, 150(5), 2317–2325 http://dx.doi.org/10.1007/s10958-008-0131-z
  • [4] Banks D.O., Kurowski G.J., A Prüfer transformation for the equation of the vibrating beam, Trans. Amer. Math. Soc., 1974, 199, 203–222
  • [5] Banks D.O., Kurowski G.J., A Prüfer transformation for the equation of a vibrating beam subject to axial forces, J. Differential Equations, 1977, 24, 57–74 http://dx.doi.org/10.1016/0022-0396(77)90170-X
  • [6] Berestycki H., On some nonlinear Sturm-Liouville problems, J. Differential Equations, 1977, 26, 375–390 http://dx.doi.org/10.1016/0022-0396(77)90086-9
  • [7] Binding P.A., Browne P.J., Watson B.A., Spectral problem for nonlinear Sturm-Liouville equations with eigenparameter dependent boundary conditions, Canad. J. Math., 2000, 52, 248–264 http://dx.doi.org/10.4153/CJM-2000-011-1
  • [8] Chiappinelli R., On eigenvalues and bifurcation for nonlinear Sturm-Liouville operators, Boll. Un. Math. Ital., 1985, 4-A, 77–83
  • [9] Chu J., O’Regan D., Positive solutions for regular and singular fourth-order boundary value problems, Comm. Appl. Anal., 2006, 10, 185–199
  • [10] Courant R., Zur Theorie der linear Integralgleichungen, Mathematishe Annalen, 1923, 89(1–2), 161–178
  • [11] Courant R., Hilbert D., Methods of mathematical phusics, I, Interscience, New York, 1953
  • [12] Crandall M.G., Rabinowitz P.H., Nonlinear Sturm-Liouville eigenvalue problems and topological degree, J. Math. Mech., 1970, 19, 1083–1102
  • [13] Dancer E.N., On the structure of solutions of nonlinear eigenvalue problems, Indiana Univ. Math. J., 1974, 23, 1069–1076 http://dx.doi.org/10.1512/iumj.1974.23.23087
  • [14] Janczewsky S.N., Oscillation theorems for the differential boundary value problems of the fourth order, Annals of Mathematics, 1928, 29(1–4), 521–542
  • [15] Kerimov N.B., Aliyev Z.S., On oscillation properties of the eigenfunctions of a fourth-order differential operator, Trans. Natl. Acad. Sci. Azerb. Ser. Phys.-Techn. Math. Sci., 2005, 25(4), 63–76
  • [16] Kerimov N.B., Aliyev Z.S., On the basis property of the system of eigenfunctions of a spectral problem with spectral parameter in the boundary condition, Differential Equations, 2007, 43(7), 905–915 http://dx.doi.org/10.1134/S0012266107070038
  • [17] Kranoselskii M.A., On a topologial method in the problem of eigenfunctions of nonlinear operators, Dokl.Akad. Nauk SSSR, 1950, 74, 5–7 (in Russian)
  • [18] Lazer A.C., McKenna P.J., Global bifurcation and a theorem of Tarantella, J. Math. Anal. Appl., 1994, 181, 648–655 http://dx.doi.org/10.1006/jmaa.1994.1049
  • [19] Leighton W., Nehari Z., On the oscillation of solutions of self-adjoint linear differential equations of the fourth order, Tras. Amer. Math. Soc., 1958, 98, 325–377 http://dx.doi.org/10.1090/S0002-9947-1958-0102639-X
  • [20] Li Y., Positive solutions of fourth-order boundary value problems with two parameters, J. Math. Anal. Appl., 2003, 281, 477–484 http://dx.doi.org/10.1016/S0022-247X(03)00131-8
  • [21] Ma R., Xu J., Bifurcation from interval and positive solutions of nonlinear fourth-order boundary value problem, Nonlinear Anal., 2010, 72, 113–122 http://dx.doi.org/10.1016/j.na.2009.06.061
  • [22] Ma R., Nodal solutions of boundary value problems of fourth-order ordinary differential equations, J. Math. Anal. Appl., 2006, 319, 424–434 http://dx.doi.org/10.1016/j.jmaa.2005.06.045
  • [23] Ma R., Tompson B., Nodal solutions for a nonlinear fourth-order eigenvalue problem, Acta Math. Sinica Eng. Ser., 2008, 24, 27–34 http://dx.doi.org/10.1007/s10114-007-1009-6
  • [24] Makhmudov A.P., Aliev Z.S., Global bifurcation of solutions of certain non-linearizable eigenvalue problems, Differential Equations, 1989, 25, 71–76
  • [25] Makhmudov A.P., Aliev Z.S., Some global results for linearizable and nonli-nearizable Sturm-Liouville problems of fourth order, Soviet Math. Dokl., 1990, 40, 472–476
  • [26] Makhmudov A.P., Aliev Z.S., Nondifferentiable perburbations of spectral problems for a pair of selfadjoint operators and global bifurgation, Soviet Math., 1990, 34(1), 51–60
  • [27] Rabinowitz P.H., Some global results for nonlinear eigenvalue problems, J. Funct. Anal., 1971, 7, 487–513 http://dx.doi.org/10.1016/0022-1236(71)90030-9
  • [28] Rynne B.P., Bifurcation from zero or infinity in Sturm-Liouville problems which are not linearizable, J. Math. Anal. Appl., 1998, 228(1), 141–156 http://dx.doi.org/10.1006/jmaa.1998.6122
  • [29] Rynne B.P., Infinitely many solutions of superlinear fourth-order boundary value problems, Topological Meth. in Nonlinear Anal. Journal of the Juliusz Schauder Center, 2002, 19, 303–312
  • [30] Rynne B.P., Global bifurcation for 2mth-order boundary value problems and infinitely many solutions superlinear problems, J. Differential Equations, 2003, 188(2), 461–472 http://dx.doi.org/10.1016/S0022-0396(02)00146-8
  • [31] Schmitt K., Smith H.L., On eigenvalue problems for nondifferentiable mappings, J. Differential Equations, 1979, 33, 294–319 http://dx.doi.org/10.1016/0022-0396(79)90067-6
  • [32] Walter J., Regular eigenvalue problems with eigenvalue parameter in the boundary conditions, Mathematische Zeitschrift, 1973, 133(4), 301–312 http://dx.doi.org/10.1007/BF01177870
  • [33] Webb J.R.L., Infante G., Franco D., Positive solutions of nonlinear fourth-order boundary value problems with local and nonlocal boundary conditions, Proc. Roy. Soc. Edinburgh Sect. A, 2008, 138, 427–446 http://dx.doi.org/10.1017/S0308210506001041
  • [34] Weyl H., Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Diferentialgleichungen, Mathematishe Annalen, 1912, 71(4), 441–479 http://dx.doi.org/10.1007/BF01456804
  • [35] Yao Q., Positive solutions for eigenvalue problems of fourth-order elastic beam equations, Appl. Math. Lett., 2004, 17, 237–243 http://dx.doi.org/10.1016/S0893-9659(04)90037-7
Typ dokumentu
Bibliografia
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bwmeta1.element.doi-10_2478_s11533-014-0416-z
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