Asymptotic Expansions of Eigenvalues of Periodic and Antiperiodic Boundary Value Problems for Singularly Perturbed Second-Order Differential Equation with Turning Points

Abstract

For a second-order equation with a small factor at the highest derivative, the asymptotic behavior of all eigenvalues of periodic and antiperiodic boundary value problems is studied. The main assumption is that the coefficient at the first derivative in the equation is the sign of the variable, i.e., turning points exist. An algorithm to compute all coefficients of asymptotic series for every considered eigenvalue is developed. It turns out that the values of these coefficients are determined only by the values of the coefficients of the original equation in a neighborhood of turning points. The asymptotic behavior of the length of Lyapunov stability and instability zones is obtained. In particular, the stability problem is solved for solutions of second-order equations with periodic coefficients and small parameters at the highest derivative.