zbMATH — the first resource for mathematics

On linear relations generated by an integro-differential equation with Nevanlinna measure in the infinite-dimensional case. (English. Russian original) Zbl 1316.45008
Math. Notes 96, No. 1, 10-25 (2014); translation from Mat. Zametki 96, No. 1, 5-21 (2014).
The author considers the following systems of integral equations with unknown functions \(y\) and \(y_{1}\) on \([a,b]\): \[ y(t)=c_{1}+\int_{t_{0}}^{t}y(s)ds \tag{1} \] and \[ y_{1}(t)=c_{2}-\int_{t_{0}}^{t}(d\widetilde{M}_{\lambda})y(s)-\int_{t_{0}}^{t}(d\widetilde{V})f(s), \tag{2} \] where \(\widetilde{M}_{\lambda}\) is the Nevanlinna measure whose values are bounded operators in a separable Hilbert space \(H\), \( \lambda\in\mathcal{C}_{0}\supset \mathcal{C}-\mathbb{R}\), \( \widetilde{V}=(Im\lambda_{0})^{-1}Im\widetilde{M}_{\lambda_{0}}\) and \(c_{1}\) and \(c_{2}\in H \). If \(H\) is one-dimensional and the generating function \( t\rightarrow M_{\lambda}(t)\) of the measure \( \widetilde{M}_{\lambda}\)is of the form \( M_{\lambda}(t)=-q(t)+\lambda m(t)\), where \(q\) is a function of bounded variation and the function \(m\) is nondecreasing, then the system (1) and (2) takes the form of an integro-differential equation \[ -y'(t)+y'(t_{0})+\int_{t_{0}}^{t}y(s)dq(s)-\lambda\int_{t_{0}}^{t}y(s) dm(s)=\int_{t_{0}}^{t}f(s)dm(s). \] The author determines the families of maximal and minimal relations generated by the system(1) and (2). The holomorphy of these families is established. Moreover, the operators inverse to continuously invertible restrictions of the maximal relations are described. This description is used to derive a formula for the generalized resolvents of the symmetric relation corresponding to (1) and (2).
45J05 Integro-ordinary differential equations
45F05 Systems of nonsingular linear integral equations
Full Text: DOI
[1] Bruk, V M, Linear relations generated by an integral equation with Nevanlinna operator measure, 3-19, (2012) · Zbl 1266.45014
[2] Bruk, V M, Invertible linear relations generated by an integral equation with Nevanlinna measure, 16-29, (2013) · Zbl 1266.45013
[3] Savchuk, A M; Shkalikov, A A, Sturm-Liouville operators with singular potentials, Mat. Zametki, 66, 897-912, (1999) · Zbl 0968.34072
[4] Rofe-Beketov, F S, Square-integrable solutions, self-adjoint extensions and spectrum of differential systems, Differential Equations, Sympos. Univ. Upsaliensis Ann. Quingentesimum Celebrantis, 7, 169-178, (1977) · Zbl 0405.34024
[5] V. I. Gorbachuk and M. L. Gorbachuk, Boundary-Value Problems for Differential-Operator Equations (Naukova Dumka, Kiev, 1984) [in Russian]. · Zbl 0567.47041
[6] Rofe-Beketov, F S; Khol’kin, A M, Spectral analysis of differential operators. interplay between spectral and oscillatory properties, (2005), Hackensack, NJ · Zbl 0544.34023
[7] Baskakov, A G; Chernyshev, K I, Spectral analysis of linear relations, and degenerate semigroups of operators, Mat. Sb., 193, 3-42, (2002)
[8] Baskakov, A G, Theory of representations of Banach algebras, and abelian groups and semigroups in the spectral analysis of linear operators, 3-151, (2004), Moscow · Zbl 1098.47005
[9] Baskakov, A G, Linear relations as generators of semigroups of operators, Mat. Zametki, 84, 175-192, (2008) · Zbl 1221.47076
[10] Baskakov, A G, Spectral analysis of differential operators with unbounded operator-valued coefficients, difference relations and semigroups of difference relations, Izv. Ross. Akad. Nauk Ser. Mat., 73, 3-68, (2009) · Zbl 1167.47006
[11] Baskakov, A G, Analysis of linear differential equations by methods of the spectral theory of difference operators and linear relations, Uspekhi Mat. Nauk, 68, 77-128, (2013)
[12] Bichegkuev, M S, On conditions for invertibility of difference and differential operators in weight spaces, Izv. Ross. Akad. Nauk Ser. Mat., 75, 3-20, (2011) · Zbl 1228.47034
[13] Didenko, V B, On the continuous invertibility and the Fredholm property of differential operators with multivalued impulse effects, Izv. Ross. Akad. Nauk Ser. Mat., 77, 5-22, (2013)
[14] Khrabustovsky, V I, On the characteristic operators and projections and on the solutions of Weyl type of dissipative and accumulative operator systems. I. general case, Zh. Mat. Fiz. Anal. Geom., 2, 149-175, (2006) · Zbl 1149.34036
[15] Bruk, VM, On linear relations generated by a differential expression and by a Nevanlinna operator function, Zh. Mat. Fiz. Anal.Geom., 7, 115-140, (2011) · Zbl 1226.47005
[16] Pokornyi, Yu V; Zvereva, M B; Shabrov, S A, Sturm-Liouville oscillation theory for impulsive problems, Uspekhi Mat. Nauk, 63, 111-154, (2008) · Zbl 1170.34313
[17] F. Atkinson, Discrete and Continuous Boundary Problems [New York, 1964; Mir, Moscow (1968); I. S. Katz and M. G. Krein, “On the spectral properties of a string,” [Suppl. 2 to the Russian translation of Atkinson’s book]. · Zbl 0117.05806
[18] Klotz, L P; Langer, H, Generalized resolvents and spectral functions of amatrix generalization of the Krein-Feller second-order derivative, Math. Nachr., 100, 163-186, (1981) · Zbl 0499.47002
[19] Yu. M. Berezanskii, Expansions in Eigenfunctions of Self-Adjoint Operators (Naukova Dumka, Kiev, 1965) [in Russian]. · Zbl 0142.37203
[20] T. Kato, Perturbation Theory for Linear Operators (Springer-Verlag, Heidelberg, 1966; Mir, Moscow, 1972). · Zbl 0148.12601
[21] Bruk, V M, On linear relations generated by a nonnegative operator function and a degenerate elliptic differential-operator expression, Zh. Mat. Fiz. Anal.Geom., 5, 123-144, (2009) · Zbl 1203.47024
[22] Shtraus, A V, Generalized resolvents of symmetric operators, Izv. Ross. Akad. Nauk Ser. Mat., 18, 51-86, (1954) · Zbl 0055.10903
[23] Dijksma, A; Snoo, H S V, Self-adjoint extensions of symmetric subspaces, Pacific J. Math., 54, 71-100, (1974) · Zbl 0304.47006
[24] Vernik, A N; Il’yazova, D Z, Generalized resolvents and spectral functions of an infinite-dimensional analog of a Krein-Feller differentiation operator, 20-26, (1986) · Zbl 0623.47028
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.