×

Multiplicity in parameter-dependent problems for ordinary differential equations. (English) Zbl 1193.34030

The author presents some properties of the spectrum in connection with the number of solutions to the boundary value problems \[ x''+\lambda f(x)=0,\,\,\,x\in (0,1),\,\,\,x(0)=0,\,\,x(1)=0, \] where \(f\) is a continuously differentiable function and \(\lambda\) is a parameter, and \[ x''=-\lambda f(x^+)+\mu g(x^-),\,\,\,x\in (0,1),\,\,\,x(0)=0,\,\,x(1)=0, \] where \(x^+=\max\{x,0\}\), \(x^-=\max\{-x,0\}\), \(\lambda,\,\mu\) are nonnegative parameters and \(f,\,g\) are positive continuously differentiable functions defined on \(\mathbb{R}^+=[0,\infty)\).

MSC:

34B09 Boundary eigenvalue problems for ordinary differential equations
34B08 Parameter dependent boundary value problems for ordinary differential equations
34L15 Eigenvalues, estimation of eigenvalues, upper and lower bounds of ordinary differential operators
PDFBibTeX XMLCite
Full Text: DOI