Asymptotic of eigenvalues of periodic and antiperiodic boundary value problem for second order differential equations

The article considers asymptotic distribution of characteristic constants in periodic and antiperiodic boundary-value problems for the second-order linear equation with periodic coefficients. It allows getting asymptotics of stability and instability zones of solutions. It was shown that in the absence of turning points ($r(t) > 0$) the instability zones lengths converge to zero with their number increasing, and the stability zones lengths converge to a positive quantity. The situation, when ($r(t) \geqslant 0$) and there are zeroes $r(t),$ results in the fact that the lengths of stability and instability zones have a finite nonzero bound at an unbounded increase of the number of the corresponding zone. But if the function $r(t)$ is alternating, the lengths of all stability zones converge to zero, and the lengths of instability zones converge to some finite quantities. These conclusions allowed to formulate a series of interesting criteria of stability and instability of solutions of the second-order equation with periodic coefficients. The results given are illustrated by a substantial example. The methods of investigation are based on a detailed study of the so-called special standard equations and the consequent reduction of original equations to any particular type of standard equations. Here, asymptotic methods of the theory of singular perturbance, as well as certain properties of a series of special functions are used.