Abstract:

In this paper, by using the relation between left-definite and right-definite problems, the characterization of the periodic and semi-periodic eigenvalues of left-definite Sturm-Liouville problems with periodic coefficients on the interval[a, a+kh] are obtained. For each k >2 we identify explicitly which of the uncountable number of complex conditions generates these periodic eigenvalues. Based on this condition, we give the inequality relation of periodic eigenvalues on the interval[a, a+kh] and 1-1 correspondence between the periodic and semi-periodic eigenvalues on the interval[a, a+kh], k >1 and the corresponding eigenvalues from the interval[a, a+h].

Key words: Left-definite Sturm-liouville problems, Periodic coefficients, Eigenvalues inequalities, 1-1 correspondence