On the study of the spectrum of a functional-differential operator with a summable potential

The paper deals with a functional-differential operator of the eighth order with a summable potential. The boundary conditions are separated. Functional-differential operators of this kind arise in the study of vibrations of beams and bridges made up of materials of different density. To solve the functional-differential equation that defines a differential operator, the method of variation of constants is applied. The solution of the initial functional-differential equation is reduced to the solution of the Volterra integral equation. The resulting Volterra integral equation is solved by Picard's method of successive approximations. As a result of the investigation of the integral equation, asymptotic formulas and estimates for the solutions of the functional-differential equation that defines the differential operator are obtained. For large values of the spectral parameter, the asymptotics of the solutions of the differential equation defining the differential operator is derived...