Some global results for nonlinear fourth order eigenvalue problems

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1. [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 Zbl1235.34222
2. [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. [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 Zbl1151.34338
4. [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 Zbl0291.34024
5. [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 Zbl0313.73051
6. [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. [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 Zbl0952.34018
8. [8] Chiappinelli R., On eigenvalues and bifurcation for nonlinear Sturm-Liouville operators, Boll. Un. Math. Ital., 1985, 4-A, 77–83 Zbl0565.34016
9. [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. [10] Courant R., Zur Theorie der linear Integralgleichungen, Mathematishe Annalen, 1923, 89(1–2), 161–178 Zbl49.0274.01
11. [11] Courant R., Hilbert D., Methods of mathematical phusics, I, Interscience, New York, 1953 Zbl0051.28802
12. [12] Crandall M.G., Rabinowitz P.H., Nonlinear Sturm-Liouville eigenvalue problems and topological degree, J. Math. Mech., 1970, 19, 1083–1102 Zbl0206.09705
13. [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 Zbl0276.47051
14. [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 Zbl54.0476.02
15. [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. [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 Zbl1189.34161
17. [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. [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 Zbl0797.34021
19. [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 Zbl0084.08104
20. [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 Zbl1030.34016
21. [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 Zbl1200.34023
22. [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 Zbl1098.34012
23. [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 Zbl1149.34312
24. [24] Makhmudov A.P., Aliev Z.S., Global bifurcation of solutions of certain non-linearizable eigenvalue problems, Differential Equations, 1989, 25, 71–76 Zbl0684.35014
25. [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. [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. [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. [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 Zbl0918.34028
29. [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 Zbl1017.34015
30. [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 Zbl1029.34015
31. [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. [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 Zbl0246.47058
33. [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 Zbl1167.34004
34. [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 Zbl43.0436.01
35. [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 Zbl1072.34022