Inequalities Among Eigenvalues Of Left-definite Sturm-liouville Problems With Periodic Coefficients

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].