Estimates on the eigenvalues for some nonlinear ordinary differential operators

The nonlinear eigenvalue problem Lu+f(x,u)=λu in (a,b) , with u(a)=u(b)=0 , where Lu=−(p(x)u ′ ) ′ +q(x)u is the usual differential operator for Sturm-Liouville problems, f is continuous and such that f(x,0)=0 for all x , and λ is a real parameter, is considered in this paper. Here λ is called an eigenvalue if there is a nontrivial (nonzero) solution for the above problem. It is proved that if f is odd, such that 0≤f(x,s)≤αs 2 for some α>0 and all x and s , and satisfies additional assumptions, then all eigenvalues are contained in the intervals [λ 0 n ,λ 0 n +α] , where λ 0 n are the eigenvalues for the linear problem Lu=λu . The proof uses spectral properties of the associated linear operator. Some complementary results concerning families of eigenvalues are also included.