On eigenvalues of third order composite type equations with regular boundary value conditions

Abstract

In the paper the question about distribution of eigenvalues of third - order composite type equations with regular, more precisely, with periodic boundary value conditions is studied. After, applying the Fourier method, the original problem splits into two problems on eigenvalues of third - order ordinary differential operators with periodic boundary value conditions in L₂(0,1). Characteristic determinants are calculated and zeros of entire analytic functions are found, and their location on the complex plane is determined. Existence of an infinite number of eigenvalues of a third order composite type operator is proved. Distance between the neighboring eigenvalues of the third order composite type operator of each series, which lie on rays, perpendicular to sides of a conjugate indicator diagram, that is, a regular hexagon on the complex plane, is determined. Moreover, it is determined that zero is not an eigenvalue of a third order composite type operator, in other words, zero is a regular point of the operator that belongs to resolvent set of the original operator. Adjoint operator with periodic boundary value conditions is constructed.