Sturm-Liouville eigenvalue problems for half-linear ordinary differential equations

In this paper we discuss the half-linear Sturm-Liouville eigenvalue problem [GRAPHICS] for the case where q(t) may change signs in the interval [a, b]. As a typical result we have the following theorem. If q(t) takes both a positive value and a negative value, then the totality of eigenvalues consists of two sequences {lambda(N)(+)}(n=0)(infinity) and {lambdaN-}(N=0)(infinity) such that...< λ(-)(n) < (...) < lambda(1)(-) < λ(-)(0) < 0 < lambda(0)(+) < λ(+)(1) < (...) < lambda(n)(+) < (...), lim(n-->infinity) lambda(n)(+) = +infinity and lim(n-->infinity) lambda(n)(-) = -infinity. The eigenfunctions associated with lambda = lambda(n)(+) and lambda(n)(-) have exactly n zeros in (a, b). This gives a complete generalization of the well-known results for the linear case (alpha = 1).