Global bifurcation in nonlinear Dirac problems with spectral parameter in the boundary condition

Ziyatkhan S. Aliyev, Parvana R. Manafova

DOI: http://dx.doi.org/10.12775/TMNA.2019.072

Abstract


In this paper we consider nonlinear eigenvalue problems for a one-dimensional Dirac equation with spectral parameter in the boundary condition. We investigate local and global bifurcations of nontrivial solutions to these problems. The existence of unbounded continua of nontrivial solutions bifurcating from points and intervals of the line of trivial solutions is shown.

Keywords


Nonlinear Dirac problem; spectral parameter; bifurcation point; eigenvector-function; global continua

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References


Z.S. Aliyev, Some global results for nonlinear fourth order eigenvalue problems, Cent. Eur. J. Math. 12 (2014), no. 12, 1811–1828.

Z.S. Aliyev, Global bifurcation of solutions of certain nonlinear eigenvalue problems for ordinary differential equations of fourth order, Sb. Math. 207 (2016), no. 12, 1625–1649.

Z.S. Aliyev and G.M. Mamedova, Some global results for nonlinear Sturm–Liouville problems with spectral parameter in the boundary condition, Ann. Polon. Math. 115 (2015), no. 1, 75–87.

Z.S. Aliyev and P.R. Manafova, Oscillation theorems for the Dirac operator with spectral parameter in the boundary condition, Electron. J. Qual. Theory Differ. Equ. 115 (2016), 1–10.

Z.S. Aliyev and P.R. Manafova, Oscillation properties for the Dirac equation with spectral parameter in the boundary condition, Bull. Malays. Math. Sci. Soc., DOI: 10.1007/s40840-019-00749-1.

Z.S. Aliyev and H.Sh. Rzayeva, Oscillation properties for the equation of the relativistic quantum theory, Appl. Math. Comput. 271 (2015), 308–316.

Z.S. Aliyev and H.Sh. Rzayeva, On the oscillation of eigenvector functions of the onedimensional Dirac operator, Doklady Math. 94) (2016), no. 1, 401–405.

Z.S. Aliyev and H.Sh. Rzayeva, Global bifurcation for nonlinear Dirac problems, Electron. J. Qual. Theory Differ. Equ. 46 (2016), 1–14.

F.V. Atkinson, Discrete and Continuous Boundary Problems, Academic Press, New York-London, 1964.

H. Berestycki, On some nonlinear Sturm–Liouville problems, J. Differential Equations 26 (1977), 375–390.

P.A. Binding, P.J. Browne and B.A. Watson, Spectral problem for nonlinear Sturm–Liouville equations with eigenparameter dependent boundary conditions, Canad. J. Math. 52 (2000), no. 2, 248–264.

R. Chiappinelli, On eigenvalues and bifurcation for nonlinear Sturm–Liouville operators, Boll. Unione Mat. Ital. A4 (1985), 77–83.

G. Dai, Global bifurcation from intervals for Sturm–Liouville problems which are not linearizable, Electron. J. Qual. Theory Differ. Equ. 65 (2013), 1–7.

E.N. Dancer, On the structure of solutions of nonlinear eigenvalue problems, Indiana Univ. Math. J. 23 (1974), 1069–1076.

N.B. Kerimov, A boundary value problem for the Dirac system with a spectral parameter in the boundary conditions, Differ. Equ. 38 (2002), no. 2, 164–174.

M.A. Krasnosel’skiı̆, Topological Methods in the Theory of Nonlinear Integral Equations, Macmillan, New York, 1965.

A.C. Lazer and P.J. McKenna, Global bifurcation and a theorem of Tarantello, J. Math. Anal. Appl. 181 (1994), no. 3, 648–655.

B.M. Levitan and I.S. Sargsjan, Introduction to Spectral Theory; Selfadjoint Ordinary Differential Operators, Transl. Math. Monogr., vol. 39, Amer. Math. Soc., Providence, R.I., 1975.

R. Ma and B. Thompson, Nodal solutions for a nonlinear fourth-order eigenvalue problem, Acta Math. Sinica (Engl. Ser.) 24 (2008), no. 1, 27–34.

J. Przybycin, Some applications of bifurcation theory to ordinary differential equations of the fourth order, Ann. Polon. Math. 53 (1991), no. 2, 153–160.

P.H. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Funct. Anal. 7 (1971), 487–513.

B.P. Rynne, Bifurcation from zero or infinity in Sturm–Liouville problems which are not linearizable, J. Math. Anal. Appl. 228 (1998), 141–156.

K. Schmitt and H.L. Smith, On eigenvalue problems for nondifferentiable mappings, J. Differential Equations 33 (1979), no. 3, 294–319.


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