Spectral problems for operators with deviating arguments

The topic of this paper are direct and inverse spectral boundary problems of the Sturm-Liouville type with two deviating arguments, one delay and one advance. This type of problem was firstly introduced by M. Pikula, E. C atrnja, I. Kal c o and A. S ari c at the 9th International Scientific Conference "Science and Higher Education in Function of Sustainable Development - SED 2016" and further developed M. Pikula, E. C atrnja , I. Kal c o at the International Conference "Contemporary Problems of Mathematical Physics and Computational Mathematics" dedicated to the 110th anniversary of A. N. Tikhonov. In this paper we take both delays to have the same value and in its first part solve the direct boundary problem, construct the corresponding characteristic function and find the asymptotic behavior of eigenvalues. In the second part of the paper, we give the necessary and sufficient conditions for the existence of the solution of the inverse problem and give its solution by the method of Fourier coefficients.