An inverse spectral problem for second-order functional-differential pencils with two delays

Recently, there appeared a considerable interest in inverse Sturm–Liouville-type problems with constant delay. However, necessary and sufficient conditions for solvability of such were obtained only one very particular situation. Here we address this gap by obtaining the case functional-differential pencils possessing more general form along nonlinear dependence on spectral parameter. For purpose, develop so-called transformation operator approach, which allows reducing problem to vectorial integral equation. In Appendix A, obtain as corollary analogous result Sturm–Liouville operators Remarkably, present paper is first work dealing an any form. Besides generality under consideration, important advantage studying them possibility recovering both delayed terms, impossible two delays. The latter, turn, illustrated even different values these delays counterexample B. We also provide brief survey contemporary state theory delay observing recently answered long-term open questions.