Skip to main content
Log in

Some Properties of Functional-Differential Operators with Involution \(\nu(x)=1-x\) and Their Applications

  • Published:
Russian Mathematics Aims and scope Submit manuscript

Abstract

Functional-differential operators with involution \(\nu(x)=1-x\), related to integral operators whose kernels can have points of discontinuity on the lines \(t=x\) and \(t=1-x\) and to Dirac and Sturm–Liouville operators, are used in the study of these operators and various applications. This paper provides a survey on the spectral properties of such operators with involution and their applications in problems on geometric graphs, in the study of Dirac systems, and in the justification of the Fourier method in mixed problems for partial differential equations.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

REFERENCES

  1. Babbage, Ch. "An essay towards the calculus of functions", Philosophical Transactions of the Royal Soc. of London 11, 179-226 (1816).

    Google Scholar 

  2. Litvinchuk, G.S. Solvability Theory of Boundary Value Problems and Singular Integral Equations with Shift, Mathematics and Its Applications 523 (Springer, Netherlands, 2000 ).

    Book  MATH  Google Scholar 

  3. Karapetyzntz, N.K., Samko, S.G. Equations with Involutive Operators and Applications (Rostov State Univ, Rostov-on-Don, 1988 ) [in Russian].

    Google Scholar 

  4. Viner, I.Ya. "Differential equations with involutions", Differ. Equ. 5 (6), 1131-1137 (1969).

    MathSciNet  Google Scholar 

  5. Wiener, J. Generalized Solutions of Functional-Differential Equations (World Sci., 1993).

  6. Andreev, A.A. "Analogs of Classical Boundary Value Problems for a Second-Order Differential Equation with Deviating Argument", Differ. Equ. 40 (8), 1192-1194 (2004).

    Article  MathSciNet  MATH  Google Scholar 

  7. Andreev, A.A., Saushkin, I.N. "On analog of Tricomi problem for a model equation with involutive deviation in an unbounded domain", Vestn. Samarsk. Gos. Techn. Univ. Ser.: Phys.-Mat. nauki 34, 10-16 (2005).

    Google Scholar 

  8. Platonov, S.S. "Eigenfunction decomposition for some functional-differential operators", Trudi Petrozavodsk. Gos. Univ. Ser. Matem. 11, 15-35 (2004).

    Google Scholar 

  9. Watkins, W.T. "Asymptotic properties of differential equations with involutions", Int. J. Math. Math. Sci. 44 (4), 485 (2008).

    MathSciNet  MATH  Google Scholar 

  10. Kopzhassarova, A., Sarsenbi, A. "Basis properties of eigenfunctions of second-order differential operators with involution", Abstract and Appl. Anal. 2012, article 576843 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  11. Ashyralyev, A., Sarsenbi, A. "Well-posedness of an elliptic equation with involution", Electronic J. Diff. Equat. 284, 1-8 (2015).

    MathSciNet  MATH  Google Scholar 

  12. Kritskov, L.V., Sarsenbi, A.M. "Spectral properties of a nonlocal problem for a second-order differential equation with an involution", Diff. Equat. 51 (8), 984-990 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  13. Kritskov, L.V., Sarsenbi, A.M. "Basicity in Lp of root functions for differential equations with involution", Electronic J. Diff. Equat. 273, 1-9 (2015).

    MATH  Google Scholar 

  14. Kritskov, L.V., Sarsenbi, A.M. "Riesz basis property of system of root functions of second-order differential operator with involution", Diff. Equat. 53 (1), 33-46 (2017).

    Article  MathSciNet  MATH  Google Scholar 

  15. Tojo, F.A.F., Cabada, A. "Existence results for a linear equation with reflection, non-constant coefficient and periodic boundary conditions", J. Math. Anal. Appl. 412, 529-546 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  16. Cabada, A., Tojo, F.A.F. Differential Equations with Involutions (Atlantis Press, New York, 2015 ).

    Book  MATH  Google Scholar 

  17. Baskakov, A.G., Krishtal, I.A., Romanova, E.Yu. "Spectral analysis of a differential operator with an involution", J. Evolut. Equat. 17, 669-684 (2017).

    Article  MathSciNet  MATH  Google Scholar 

  18. Baskakov, A.G., Uskova, N.B. "Fourier method for first order differential equations with involution and groups of operators", Ufa Math. J. 10 (3), 11-34 (2018).

    Article  MathSciNet  MATH  Google Scholar 

  19. Vladykina, V.E., Shkalikov, A.A. "Regular Ordinary Differential Operators with Involution", Math. Notes 106 (5), 674-687 (2019).

    Article  MathSciNet  MATH  Google Scholar 

  20. Burlutskaya, M.Sh. "On some properties of differential equations and mixed problems with involution", Vestn. Voronezh. Gos. Univ. Ser.: Phys.-Mat. (1), 91-100 (2019).

  21. Burlutskaya, M.Sh., Kurdyumov, V.P., Lukonina, A.S., Khromov, A.P. "A functional-differential operator with involution", Dokl. Math. 75 (3), 399-402 (2007).

    Article  MathSciNet  MATH  Google Scholar 

  22. Khromov, A.P. "Inversion of integral operators with kernels discontinuous on the diagonal", Math. Notes 64 (6), 804-813 (1998).

    Article  MathSciNet  MATH  Google Scholar 

  23. Kornev, V.V., Khromov, A.P. "Equiconvergence of expansions in eigenfunctions of integral operators with kernels that can have discontinuities on the diagonals", Sb. Math. 192 (10), 1451-1469 (2001).

    Article  MathSciNet  MATH  Google Scholar 

  24. Khromov, A.P. "Equiconvergence theorem for an integral operator with variable upper limit of integration", in: Metr. teor. funktsii i smezhn. voprosi, 255-266 (AFTs, Moscow, 1999 ).

    Google Scholar 

  25. Kurdyumov, V.P., Khromov, A.P. "Riesz Bases of Eigenfunctions of an Integral Operator with a Variable Limit of Integration", Math. Notes 76 (1), 90-102 (2004).

    Article  MathSciNet  MATH  Google Scholar 

  26. Kornev, V.V., Khromov, A.P. "Absolute convergence of expansions in eigen- and adjoint functions of an integral operator with a variable limit of integration", Izv. Math. 69 (4), 703-717 (2005).

    Article  MathSciNet  MATH  Google Scholar 

  27. Kornev, V.V., Khromov, A.P. "Operator integration with an involution in the upper limit of integration", Doklady Mathematics 78 (2), 733-736 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  28. Khromov, A.P. "Integral operators with kernels that are discontinuous on broken lines", Sb. Math. 197 (11), 1669-1696 (2006).

    Article  MathSciNet  MATH  Google Scholar 

  29. Khromov, A.P., Kuvardina, L.P. "On the equiconvergence of expansions in eigen- and associated functions of an integral operator with involution", Russian Math. (Iz. VUZ) 52 (5), 58-66 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  30. Khalova, V.A., Khromov, A.P. "Integral operators with non-smooth involution", Izv. Saratov Univ. (N.S.), Ser. Math. Mech. Inform. 13 (1), 40-45 (2013).

    Article  MATH  Google Scholar 

  31. Koroleva, O.A. "An analogue of the Jordan–Dirichlet theorem for the integral operator with kernel having jumps on broken lines", Izv. Saratov Univ. (N.S.), Ser. Math. Mech. Inform. 13 (1), 14-23 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  32. Burlutskaya, M.Sh. Eigenfunction decomposition for differential and functional-differential operators on geometric graphs, Diss. kand. phys.-mat. nauk (Voronezh State Univ., Voronezh, 2007 ) [in Russian].

    Google Scholar 

  33. Kurdyumov, V.P., Khromov, A.P. "Riesz bases formed by root functions of a functional-differential equation with a reflection operator", Diff. Equ. 44 (2), 203-212 (2008).

    Article  MATH  Google Scholar 

  34. Kurdyumov, V.P., Khromov, A.P. "The Riesz bases consisting of eigen- and associated functions for a functional differential operator with variable structure", Russian Math. (Iz. VUZ) 54 (2), 33-45 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  35. Burlutskaya, M.Sh., Khromov, A.P. "The Steinhaus theorem on equiconvergence for functional-differential operators", Math. Notes 90 (1), 20-31 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  36. Burlutskaya, M.Sh. "Asympptotic formulas for eigenvalues and eigenfunctions of a functional-differential equation with involution", Vestn. Voronezh Gos. Univ. Ser. Phys. Matem. (2), 64-72 (2011).

  37. Pokornyi, Yu.V. Differential Equations on Geometric Graphs (Fizmatlit, Moscow, 2004 ) [in Russian].

    MATH  Google Scholar 

  38. Zavgorodnij, M.G. "On evolution problems on graphs", Russian Math. Surveys 46 (6), 199-200 (1991).

    MathSciNet  Google Scholar 

  39. Zavgorodnij, M.G. "Adjoint and self-adjoint boundary value problems on a geometric graph", Diff. Equat. 50 (4), 441-452 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  40. Zavgorodnij, M.G. "Quadratic functionals and nondegeneracy of boundary value problems on a geometric graph", Diff. Equat. 52 (1), 8-27 (2016).

    Article  MathSciNet  MATH  Google Scholar 

  41. Kulaev, R.Ch. "The Green function of the boundary value problem on a star-shaped graph", Russian Math. (Iz. VUZ) 57 (2), 48-57 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  42. Diab, A.T., Kuleshov, P.A., Penkin, O.M. "Estimate of the first eigenvalue of the laplacian on a graph", Math. Notes 96 (5–6), 948-956 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  43. Diab, A.T., Kaldybekova, B.K., Penkin, O.M. "On the multiplicity of eigenvalues of the Sturm–Liouville problem on graphs", Math. Notes 99 (3–4), 492-502 (2016).

    Article  MathSciNet  MATH  Google Scholar 

  44. Provotorov, V.V. "Eigenfunctions of the Sturm–Liouville problem on a star graph", Sb. Math. 199 (10), 1523-1545 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  45. Provotorov, V.V. "Expansion in eigenfunctions of the Sturm–Liouville problem on a graph bundle", Russian Math. (Iz. VUZ) 52 (3), 45-57 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  46. Volkova, A.S., Provotorov, V.V. "Generalized solutions and generalized eigenfunctions of boundary-value problems on a geometric graph", Russian Math. (Iz. VUZ) 58 (3), 1-13 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  47. Gomilko, A.M., Pivovarchik, V.N. "Inverse Sturm–Liouville problem on a figure-eight graph", Ukrainian Math. J. 60 (9), 1360-1385 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  48. Pryadiev, V.L. "An approach to description of finite form of solution to wave equation on spatial net", Spectral and Evolution Problems: Proc. of the 15-th Crim. Autumn Math. School-Symposium, Simferopol 15, 132-139 (2005).

    Google Scholar 

  49. Korovina, O.V., Pryadiev, V.L. "Structure of mixed problem solution for wave equation on compact geometrical graph in nonzero initial velocity case", Izv. Saratov Univ. (N.S.), Ser. Math. Mech. Inform. 9 (3), 37-46 (2009).

    Article  Google Scholar 

  50. Kulaev, R.Ch "Existence theorem for a parabolic mixed problem on a graph with boundary conditions containing time derivatives", Trudy Inst. Mat. i Mekh. UrO RAN 16 (2), 139-148 (2010).

    MathSciNet  Google Scholar 

  51. Volkova, A.S., Gnilitzkaya, Yu.A., Provotorov, V.V. "On solvability of boundary value problems for equations of parabolic and hyperbolic types on a geometric graph", Sistemi upravl. i inform. tekhnologii 51 (1), 11-15 (2013).

    Google Scholar 

  52. Golovko, N.I., Avereva, M.B., Shabrov, S.A. "On possibility of application of the Fourier method to a multi-order mathematical model", Vestn. Voronezh Univ. Ser. Phys. Mat. (1), 91-98 (2017).

  53. Burlutskaya, M.S. "On Riesz bases of root functions for a class of functional-differential operators on a graph", Diff. Equat. 45 (6), 779-788 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  54. Burlutskaya, M.S. Problems of Spectral Theory for Operators with Involuton and Applications (Nauchn. Kniga, Voronezh, 2020 ) [in Russian].

    Google Scholar 

  55. Djakov, P.V., Mityagin, B.S. "Instability zones of periodic 1-dimensional Schrödinger and Dirac operators", Russian Math. Surveys 61 (4), 663-766 (2006).

    Article  MathSciNet  MATH  Google Scholar 

  56. Djakov, P., Mityagin, B. "Bari–Markus property for Riesz projections of 1D periodic Dirac operators", Math. Nachr. 283 (3), 443-462 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  57. Trooshin, I., Yamamoto, M. "Riesz basis of root vectors of a nonsymmetric system of first-order ordinary differential operators and application to inverse eigenvalue problems", Appl. Anal. 80, 19-51 (2001).

    Article  MathSciNet  MATH  Google Scholar 

  58. Baskakov, A.G., Derbushev, A.V., Shcherbakov, A.O. "The method of similar operators in the spectral analysis of non-self-adjoint Dirac operators with non-smooth potentials", Izv. Math. 75 (3), 445-469 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  59. Savchuk, A.M., Sadovnichaya, I.V. "Asymptotic formulas for fundamental solutions of the Dirac system with complex-valued integrable potential", Diff. Equat. 49 (5), 273-284 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  60. Savchuk, A.M., Shkalikov, A.A. "Dirac operator with complex-valued summable potential", Math. Notes 96 (5), 777-810 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  61. Savchuk, A.M., Sadovnichaya, I.V. "The Riesz basis property of generalized eigenspaces for a Dirac system with integrable potential", Dokl. Math. 91 (3), 309-312 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  62. Malamud, M.M., Oridoroga, L.L. "On the completeness of root subspaces of boundary value problems for first order systems of ordinary differential equations", J. Func. Anal. 263 (7), 1939-1980 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  63. Burlutskaya, M.Sh., Kornev, V.V., Khromov, A.P. "Dirac system with non-differentiable potential and periodic boundary conditions", J. Vychisl. Mat. Mat. Fiz. 52 (9), 1621-1632 (2012).

    MATH  Google Scholar 

  64. Burlutskaya, M.Sh., Kurdyumov, V.P., Khromov, A.P. "Refined asymptotic formulas for eigenvalues and eigenfunctions of the Dirac system", Dokl. Math. 85 (2), 240-242 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  65. Burlutskaya, M.Sh., Khromov, A.P. "Dirac operator with a potential of special form and with the periodic boundary conditions", Diff. Equat. 54 (5), 586-595 (2018).

    Article  MathSciNet  MATH  Google Scholar 

  66. Djakov, P., Mityagin, B. "1D Dirac operators with special periodic potentials", arXiv:1007.3234v1 (2010).

  67. Il'in, V.A. "The solvability of mixed problems for hyperbolic and parabolic equations", Russian Math. Surveys 15 (1), 85-182 (1960).

    Article  MathSciNet  MATH  Google Scholar 

  68. Krylov, A.N. On some differential equations of mathematical physics having applications in technical problems (GITTL, Leningrad, 1950 ) [in Russian].

    Google Scholar 

  69. Chernyatin, V.A. Justification of the Fourier method in mixed problem for PDE (MGU, Moscow, 1950 ) [in Russian].

    Google Scholar 

  70. Burlutskaya, M.Sh., Khromov, A.P. "Fourier method in an initial-boundary value problem for a first-order partial differential equation with involution", Comput. Math. Math. Phys. 51 (12), 2102-2114 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  71. Burlutskaya, M.Sh. "Mixed problem for a first-order partial differential equation with involution and periodic boundary conditions", Comput. Math. Math. Phys. 54 (1), 1-10 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  72. Burlutskaya, M.Sh. "Mixed problem with involution on a two-edge graph containing a cycle", Doklady Mathematics 86 (3), 820-823 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  73. Burlutskaya, M.Sh. "Classical and generalized solutions of a mixed problem for a system of first-order equations with a continuous potential", Comput. Math. Math. Phys. 59 (3), 355-365 (2019).

    Article  MathSciNet  MATH  Google Scholar 

  74. Burlutskaya, M.Sh., Khromov, A.P. "The resolvent approach for the wave equation", Comput. Math. Math. Phys. 55 (2), 227-239 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  75. Burlutskaya, M.Sh. "Fourier method in a mixed problem for the wave equation on a graph", Dokl. Math. 92 (3), 735-738 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  76. Khromov, A.P. "Mixed problem for the wave equation with arbitrary two-point boundary conditions", Doklady Mathematics 91 (3), 294-296 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  77. Kornev, V.V., Khromov, A.P. "A resolvent approach in the Fourier method for the wave equation: The non-selfadjoint case", Comput. Math. Math. Phys. 55 (7), 1138-1149 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  78. Burlutskaya,, M.S., Khromov, A.P. "Mixed problem for the wave equation with integrable potential in the case of two-point boundary conditions of distinct orders", Diff. Equat. 53 (4), 497-508 (2017).

    Article  MathSciNet  MATH  Google Scholar 

  79. Khromov, A.P. "On the convergence of the formal Fourier solution of the wave equation with a summable potential", Comput. Math. Math. Phys. 56 (10), 1778-1792 (2016).

    Article  MathSciNet  MATH  Google Scholar 

Download references

Funding

The research is fulfilled under financial support of the Russian Science Foundation (project no. 19-11-00197, carried out at the Voronezh State University).

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to M. Sh. Burlutskaya.

Additional information

Communicated by V. G. Zvyagin.

Russian Text © The Author(s), 2021, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2021, No. 5, pp. 89–97.

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Burlutskaya, M.S. Some Properties of Functional-Differential Operators with Involution \(\nu(x)=1-x\) and Their Applications. Russ Math. 65, 69–76 (2021). https://doi.org/10.3103/S1066369X21050108

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.3103/S1066369X21050108

Keywords

Navigation