ISSN 1816-9791 (Print)
ISSN 2541-9005 (Online)

For citation:

Yurko V. A. On Determination of Functional-Differential Pencils on Closed Sets from the Weyl-Type Function. Izv. Saratov Univ. (N. S.), Ser. Math. Mech. Inform., 2020, vol. 20, iss. 3, pp. 343-350. DOI: 10.18500/1816-9791-2020-20-3-343-350

Published online: 
Full text:
(downloads: 239)
Article type: 

On Determination of Functional-Differential Pencils on Closed Sets from the Weyl-Type Function

Yurko Vyacheslav Anatol'evich, Saratov State University

Second-order functional-differential pencils on closed sets are considered with nonlinear dependence on the spectral parameter. Properties of their spectral characteristics are obtained and the inverse problem is studied, which consists in recovering coefficients of the pencil from the given Weyl-type function. The statement and the study of inverse spectral problems essentially depend on the structure of the closed set. We consider an important subclass of closed sets when the set is a unification of a finite number of closed intervals and isolated points. In order to solve the inverse spectral problem for this class of closed sets, we develop ideas of the method of spectral mappings. We also establish and use connections between the Weyl-type functions related to different subsets of the main closed set. Using these ideas and properties we obtain a global constructive procedure for the solution of the nonlinear inverse problem considered, and we establish the uniqueness of the solution of the inverse problem.

  1. Bohner M., Peterson A. Dynamic Equations on Time Scales. Boston, MA, Birkh¨auser, 2001. 358 p. DOI:
  2. Freiling G., Yurko V. A. Inverse Sturm – Liouville Problems and Their Applications. New York, NOVA Science Publ. Inc., 2001. 305 p.
  3. Yurko V. A. Method of Spectral Mappings in the Inverse Problem Theory. Utrecht, VSP, 2002. 316 p. Inverse and Ill-posed Problems Series.
  4. Gasymov M. G., Gusejnov G. S. Determination of diffusion operators from the spectral data. DAN Azer. SSR, 1981, vol. 37, no. 2, pp. 19–23.
  5. Yurko V. A. Boundary value problems with a parameter in the boundary conditions. Soviet J. Contemporary Math. Anal., 1984, vol. 19, no. 5, pp. 62–73.
  6. Yurko V. A. An inverse problem for pencils of differential operators. Sb. Math., 2000, vol. 191, iss. 10, pp. 1561–1586. DOI:
  7. Nabiev I. M. Inverse spectral problem for the diffusion operator on an interval. Mat. Fiz. Anal. Geom., 2004, vol. 11, no. 3, pp. 302–313.
  8. Guseinov I., Nabiev I. The inverse spectral problem for pencils of differential operators. Sb. Math., 2007, vol. 198, iss. 11, pp. 1579–1598. DOI:
  9. Buterin S. A., Yurko V. A. Inverse problems for second-order differential pencils with Dirichlet boundary conditions. J. Inverse Ill-Posed Probl., 2012, vol. 20, iss. 5–6, pp. 855–881. DOI:
  10. Yurko V. A. Inverse problems for non-selfadjoint quasi-periodic differential pencils. Anal. Math. Phys., 2012, vol. 2, no. 3, pp. 215–230. DOI:
  11. Yurko V. A. Inverse problems for Sturm– Liouville differential operators on closed sets. Tamkang Journal of Mathematics, 2019, vol. 50, no. 3, pp. 199–206. DOI: