On eigenvalues and bifurcation for nonlinear Sturm-Liouville operators. (English) Zbl 0565.34016

Let u satisfy (1) \(Lu+f(.,u)=\lambda u\) where all u in the domain of the regular Sturm-Liouville operator L satisfy \(\alpha u(a)+\beta u'(a)=0=\gamma u(b)+\delta u'(b)\) and \(Lu:=-(pu')'+qu\). It is shown that if, for some \(k\in R\), \(| f(x,u)| \leq k| u|\) for all (x,u)\(\in [a,b]\times R\) then either \(u=0\) or there exists an eigenvalue \(\lambda_ n\) of \(Lu=\lambda u\) such that \(| \lambda -\lambda_ n| \leq k\). An existence result for (1) is also proved.
Reviewer: A.L.Andrew


34B15 Nonlinear boundary value problems for ordinary differential equations
34L99 Ordinary differential operators
47J05 Equations involving nonlinear operators (general)