Asymptotical Distributions of Eigenvalues of Periodic and Antiperiodic Boundary Value Problems for Second-Order Differential Equations

Abstract

Abstract—We consider asymptotical distributions of characteristic constants in periodic and antiperiodic boundary value problems for a second-order linear equation with periodic coefficients. This allows one to obtain asymptotical properties of stability and instability zones of solutions. We show that if there are no turning points, i.e., if \(r(t) > 0\), then the lengths of instability zones converge to zero as their number increases, while the lengths of stability zones converge to a positive number. If \(r(t) \geqslant 0\) and function \(r(t)\) has zeroes, then the lengths of stability and instability zones have finite nonzero limits as the numbers of the corresponding zones infinitely increase. If function \(r(t)\) is alternating, then the lengths of all stability zones converge to zero and the lengths of all instability zones converge to finite numbers. This yields various stability and instability criteria for solutions of second-order equations with periodic coefficients. The presented results are illustrated by a substantial example. The investigation methods are based on a detailed study of so-called special standard equations and the reduction of original equations to standard equations. Here, asymptotical methods of the theory of singular perturbations and properties of series of special functions are used.