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.
Citation:
A. G. Baskakov, “Analysis of linear differential equations by methods of the spectral theory of difference operators and linear relations”, Russian Math. Surveys, 68:1 (2013), 69–116
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\by A.~G.~Baskakov
\paper Analysis of linear differential equations by methods of the spectral theory of difference operators and linear relations
\jour Russian Math. Surveys
\yr 2013
\vol 68
\issue 1
\pages 69--116
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Mykhailo Horodnii, Oleksii Pecherytsia, “Bounded Solutions of a Second-Order Differential Equation with Piecewise-Constant Operator Coefficients”, J Math Sci, 282:6 (2024), 935
A. G. Baskakov, G. V. Garkavenko, N. B. Uskova, L. N. Kostina, “Ob ekvivalentnykh operatorakh”, Materialy Voronezhskoi mezhdunarodnoi vesennei matematicheskoi shkoly «Sovremennye metody kraevykh zadach. Pontryaginskie chteniya—XXXV», Voronezh, 26-30 aprelya 2024 g. Chast 1, Itogi nauki i tekhn. Sovrem. mat. i ee pril. Temat. obz., 235, VINITI RAN, M., 2024, 3–14
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Jian Wei-Gang, Ding Hui-Sheng, “Loomis type 定理 on the half-line and its application”, Sci. Sin.-Math., 53:9 (2023), 1241
M. F. Horodnii, O. A. Pecherytsia, “Bounded Solutions of a Differential Equation with Piecewise Constant Operator Coefficients”, J Math Sci, 270:2 (2023), 237
Hui-Sheng Ding, Wei-Gang Jian, Nguyen Van Minh, Gaston M. N'Guérékata, “Kadets type and Loomis type theorems for asymptotically almost periodic functions”, Journal of Differential Equations, 373 (2023), 389
Vladislav Bruk, “On characteristic functions of generalized resolvents generated by integral equations with operator measures”, Filomat, 37:23 (2023), 7699
M. F. Horodnii, “Obmezhenі ta sumovnі rozv'yazki odnogo rіznitsevogo rіvnyannya z kuskovostalimi operatornimi koefіtsієntami”, Ukr. Mat. Zhurn., 74:7 (2022), 930
I. I. Strukova, “On Some Properties of Functions Almost Periodic at Infinity from Homogeneous Spaces”, J Math Sci, 263:5 (2022), 643
V. E. Strukov, “On Distributions That Are Almost Periodic at Infinity”, J Math Sci, 263:4 (2022), 511
Vladislav Bruk, “Generalized resolvents of linear relations generated by integral equations with operator measures”, Filomat, 36:14 (2022), 4793
M. F. Horodnii, “Bounded and Summable Solutions of a Difference Equation with Piecewise-Constant Operator Coefficients”, Ukr Math J, 74:7 (2022), 1063
I. A. Vysotskaya, I. I. Strukova, “Issledovanie nekotorykh klassov pochti periodicheskikh na beskonechnosti funktsii”, Izv. Sarat. un-ta. Nov. ser. Ser.: Matematika. Mekhanika. Informatika, 21:1 (2021), 4–14
A. B. Antonevich, “Pravostoronnyaya obratimost dvuchlennykh funktsionalnykh operatorov i graduirovannaya dikhotomiya”, Posvyaschaetsya pamyati professora N.D. Kopachevskogo, SMFN, 67, no. 2, Rossiiskii universitet druzhby narodov, M., 2021, 208–236
I. I. Strukova, “Garmonicheskii analiz pochti periodicheskikh na beskonechnosti funktsii v banakhovykh modulyakh”, Izv. Sarat. un-ta. Nov. ser. Ser.: Matematika. Mekhanika. Informatika, 21:4 (2021), 448–457
Bruk V.M., “Invertible Linear Relations Generated By Integral Equations With Operator Measures”, Filomat, 35:5 (2021), 1589–1607
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