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Positive solutions for nonlinear eigenvalue problems. (English) Zbl 0876.34023

The authors are concerned with determining values of λ (eigenvalues), for which there exist positive solutions of the boundary value problem

(1 λ )u '' +λa(t)f(u)=0,0<t<1,(2)u(0)=u(1)=0,

where f:[0,)[0,) is continuous; a:[0,1][0,) is continuous and does not vanish identically on any subinterval, and

f 0 =lim x0 (f(x)/x),f =lim x (f(x)/x)

exist. To research the problem (1 λ ), (2) the Krasnosel’skij methods of solutions of nonlinear operator equations in a space with a cone are applicable.

34B15Nonlinear boundary value problems for ODE
34B27Green functions
34B24Sturm-Liouville theory
34L15Eigenvalues, estimation of eigenvalues, upper and lower bounds for OD operators