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Uspekhi Mat. Nauk, 2013, Volume 68, Issue 1(409), Pages 77–128 (Mi umn9505)  

This article is cited in 66 scientific papers (total in 66 papers)

Analysis of linear differential equations by methods of the spectral theory of difference operators and linear relations

A. G. Baskakov

Voronezh State University

Abstract: Many properties of solutions to linear differential equations with unbounded operator coefficients (their boundedness, almost periodicity, stability) are closely connected with the corresponding properties of the differential operator defining the equation and acting in an appropriate function space. The structure of the spectrum of this operator and whether it is invertible, correct, and Fredholm depend on the dimension of the kernel of the operator, the codimension of its range, and the existence of complemented subspaces. The notion of a state of a linear relation (multivalued linear operator) is introduced, and is associated with some properties of the kernel and range. A linear difference operator (difference relation) is assigned to the differential operator under consideration (or the corresponding equation), the sets of their states are proved to be the same, and necessary and sufficient conditions for them to have the Fredholm property are found. Criteria for the almost periodicity at infinity of solutions of differential equations are derived. In the proof of the main results, the property of exponential dichotomy of a family of evolution operators and the spectral theory of linear relations are heavily used.
Bibliography: 98 titles.

Keywords: linear differential operators, set of states of an operator, Fredholm operator, difference operators and difference relations, spectrum of an operator or linear relation, functions almost periodic at infinity.

Funding Agency Grant Number
Russian Foundation for Basic Research 10-01-00276
13-01-00378


DOI: https://doi.org/10.4213/rm9505

Full text: PDF file (975 kB)
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English version:
Russian Mathematical Surveys, 2013, 68:1, 69–116

Bibliographic databases:

UDC: 517.937+517.983
MSC: Primary 34D09, 34G10, 47A25; Secondary 46B45, 46E15, 47A06, 47A53, 47D06
Received: 31.10.2012

Citation: A. G. Baskakov, “Analysis of linear differential equations by methods of the spectral theory of difference operators and linear relations”, Uspekhi Mat. Nauk, 68:1(409) (2013), 77–128; Russian Math. Surveys, 68:1 (2013), 69–116

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    5. Baskakov A.G., Krishtal I.A., “On Completeness of Spectral Subspaces of Linear Relations and Ordered Pairs of Linear Operators”, J. Math. Anal. Appl., 407:1 (2013), 157–178  crossref  mathscinet  zmath  isi  elib  scopus
    6. Kuznetsova V.I., Kurbatov V.G., “Ob obratimosti raznostno-integralnogo operatora v prostranstve medlenno menyayuschikhsya funktsii”, Vestnik Voronezhskogo gosudarstvennogo universiteta. Seriya: Fizika. Matematika, 2013, no. 2, 219–229  mathscinet  elib
    7. M. S. Bichegkuev, “Spectral Analysis of Differential Operators with Unbounded Operator Coefficients in Weighted Spaces of Functions”, Math. Notes, 95:1 (2014), 15–21  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    8. V. M. Bruk, “On Linear Relations Generated by an Integro-Differential Equation with Nevanlinna Measure in the Infinite-Dimensional Case”, Math. Notes, 96:1 (2014), 10–25  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    9. V. B. Didenko, “O sostoyaniyakh obratimosti lineinykh differentsialnykh operatorov s neogranichennymi periodicheskimi koeffitsientami”, Izv. Sarat. un-ta. Nov. ser. Ser. Matematika. Mekhanika. Informatika, 14:2 (2014), 136–144  mathnet  crossref  elib
    10. V. M. Bruk, “Dissipative expansions of a symmetric relation generated by a system of integral equations with operator measures”, Russian Math. (Iz. VUZ), 58:12 (2014), 7–22  mathnet  crossref
    11. I. I. Strukova, “Harmonic Analysis of Periodic Vectors and Periodic at Infinity Functions”, J. Math. Sci., 211:6 (2015), 874–885  mathnet  crossref
    12. S. V. Besaeva, “On the spectra of difference relations and differential operators on weighted spaces of sequences and functions”, St. Petersburg Math. J., 26:4 (2015), 499–513  mathnet  crossref  mathscinet  isi  elib  elib
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    15. A. G. Chshiev, “Ob obraschenii polugruppy operatorov”, Vladikavk. matem. zhurn., 16:2 (2014), 79–89  mathnet
    16. A. G. Baskakov, “Harmonic and Spectral Analysis of Power Bounded Operators and Bounded Semigroups of Operators on Banach Spaces”, Math. Notes, 97:2 (2015), 164–178  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    17. A. Yu. Duplischeva, “O differentsialnykh operatorakh i matritsakh vtorogo poryadka”, Izv. Sarat. un-ta. Nov. ser. Ser. Matematika. Mekhanika. Informatika, 15:1 (2015), 31–37  mathnet  crossref  elib
    18. A. A. Ryzhkova, I. A. Trishina, “O pochti periodicheskikh na beskonechnosti resheniyakh raznostnykh uravnenii”, Izv. Sarat. un-ta. Nov. ser. Ser. Matematika. Mekhanika. Informatika, 15:1 (2015), 45–49  mathnet  crossref  elib
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    22. M. S. Bichegkuev, “Spectral analysis of differential operators with unbounded operator coefficients on the half-line”, Differ. Equ., 51:4 (2015), 431–439  crossref  crossref  mathscinet  zmath  isi  elib  elib  scopus
    23. R. K. Romanovskii, E. M. Nazaruk, “Dichotomy of solutions of functional-differential equations in the Sobolev space”, Differ. Equ., 51:4 (2015), 464–476  crossref  crossref  mathscinet  zmath  isi  elib  elib  scopus
    24. Meshkov V.Z., Polovinkin I.P., Polovinkina M.V., Ermakova Yu.D., Rabeeakh S.A., “Raznostnaya formula srednego znacheniya dlya dvumernogo lineinogo giperbolicheskogo uravneniya tretego poryadka”, Vestnik Voronezhskogo gosudarstvennogo universiteta. Seriya: Fizika. Matematika, 2015, 112–119  zmath  elib
    25. Strukov V.E., “Ob opredeleniyakh lokalnogo spektra i spektra Karlemana vektorov”, Vestnik Voronezhskogo gosudarstvennogo universiteta. Seriya: Fizika. Matematika, 2015, 153–160  zmath  elib
    26. Strukova I.I., “Spektry algebr medlenno menyayuschikhsya i periodicheskikh na beskonechnosti funktsii i banakhovy predely”, Vestnik Voronezhskogo gosudarstvennogo universiteta. Seriya: Fizika. Matematika, 2015, 161–165  zmath  elib
    27. Bruk V.M., “O skhodimosti reshenii granichnykh zadach dlya integralnykh uravnenii”, Tavricheskii vestnik informatiki i matematiki, 2015, 20–33  elib
    28. M. S. Bichegkuev, “Lyapunov Transformation of Differential Operators with Unbounded Operator Coefficients”, Math. Notes, 99:1 (2016), 24–36  mathnet  crossref  crossref  mathscinet  isi  elib
    29. A. B. Antonevich, E. V. Panteleeva, “Well-Posed Boundary-Value Problems, Right Hyperbolicity, and Exponential Dichotomy”, Math. Notes, 100:1 (2016), 11–23  mathnet  crossref  crossref  mathscinet  isi  elib
    30. I. I. Strukova, “On Wiener's Theorem for functions periodic at infinity”, Siberian Math. J., 57:1 (2016), 145–154  mathnet  crossref  crossref  mathscinet  isi  elib
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    34. T. Álvarez, F. Fakhfakh, M. Mnif, “Left-right fredholm and left-right browder linear relations”, Filomat, 31:2 (2017), 255–271  crossref  mathscinet  isi  scopus
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    36. I. I. Strukova, “Garmonicheskii analiz periodicheskikh na beskonechnosti funktsii v prostranstvakh Stepanova”, Izv. Sarat. un-ta. Nov. ser. Ser. Matematika. Mekhanika. Informatika, 17:2 (2017), 172–182  mathnet  crossref  elib
    37. A. A. Ryzhkova, “Garmonicheskii analiz periodicheskikh na beskonechnosti posledovatelnostei”, Vestn. Volgogr. gos. un-ta. Ser. 1, Mat. Fiz., 2017, no. 1(38), 22–32  mathnet  crossref
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    41. I. I. Strukova, “Garmonicheskii analiz periodicheskikh na beskonechnosti funktsii v odnorodnykh prostranstvakh”, Vestn. Volgogr. gos. un-ta. Ser. 1, Mat. Fiz., 2017, no. 2(39), 29–38  mathnet  crossref
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    57. V. E. Strukov, “O raspredeleniyakh, pochti periodicheskikh na beskonechnosti”, Materialy Voronezhskoi zimneimatematicheskoi shkolySovremennye metody teorii funktsiii smezhnye problemy.28 yanvarya–2 fevralya 2019 g.Chast 1, Itogi nauki i tekhn. Ser. Sovrem. mat. i ee pril. Temat. obz., 170, VINITI RAN, M., 2019, 51–61  mathnet  crossref  elib
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    60. A. G. Baskakov, D. B. Didenko, “Spectral Analysis of Operator Polynomials and Second-Order Differential Operators”, Math. Notes, 108:4 (2020), 477–491  mathnet  crossref  crossref  mathscinet  isi  elib
    61. V. M. Bruk, “On self-adjoint and invertible linear relations generated by integral equations”, Bul. Acad. Ştiinţe Repub. Mold. Mat., 2020, no. 1, 106–121  mathnet
    62. A. G. Baskakov, V. E. Strukov, I. I. Strukova, “Almost periodic at infinity functions from homogeneous spaces as solutions to differential equations with unbounded operator coefficients”, Eurasian Math. J., 11:4 (2020), 8–24  mathnet  crossref
    63. Baskakov A.G., Krishtal I.A., Uskova N.B., “Closed Operator Functional Calculus in Banach Modules and Applications”, J. Math. Anal. Appl., 492:2 (2020), 124473  crossref  mathscinet  zmath  isi
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