Two-parametric nonlinear eigenvalue problems

A. Gritsans, Daugavpils University, Daugavpils, Latvia
F. Sadyrbaev, Daugavpils University, Daugavpils, Latvia

E. J. Qualitative Theory of Diff. Equ., Proc. 8'th Coll. Qualitative Theory of Diff. Equ., No. 10. (2008), pp. 1-14.

Abstract: Eigenvalue problems of the form $x'' = -\lambda f(x^+) + \mu g(x^-),$ $\quad (i),$ $x(0) = 0, \; x(1) = 0,$ $\quad (ii)$ are considered, where $x^+$ and $x^-$ are the positive and negative parts of $x$ respectively. We are looking for $(\lambda, \mu)$ such that the problem $(i), (ii)$ has a nontrivial solution. This problem generalizes the famous Fu\v{c}\'{i}k problem for piece-wise linear equations. In our considerations functions $f$ and $g$ may be nonlinear functions of super-, sub- and quasi-linear growth in various combinations. The spectra obtained under the normalization condition $|x'(0)|=1$ are sometimes similar to usual Fu\v{c}\'{i}k spectrum for the Dirichlet problem and sometimes they are quite different. This depends on monotonicity properties of the functions $\xi t_1 (\xi)$ and $\eta \tau_1 (\eta),$ where $t_1 (\xi)$ and $\tau_1 (\eta)$ are the first zero functions of the Cauchy problems $x''= -f(x),$ $\: x(0)=0,$ $\: x'(0)=\xi> 0,$ $y''= g(y),$ $\: y(0)=0,$ $\: y'(0)=-\eta,$ $(\eta > 0)$ respectively.

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