Abstract
In this article we present a review of results on asymptotic behavior and stability of strong solutions for functional differential equations (FDE). We also formulate several results about spectral properties (completeness and basisness) of exponential solutions of the above-mentioned equations. It is relevant to emphasize that our approach for the research of FDE is based on the spectral analysis of operator pencils that are symbols (characteristic quasi-polynomials) with operator coefficients. The article is divided into two parts. The first part is devoted to the research on FDE in a Hilbert space; the second part is devoted to the research on FDE in a finite-dimensional space.
Similar content being viewed by others
REFERENCES
N.Azbelev, V.Maksimov,and L.Rakhmatulina,Introduction to the Theory of Functional Differential Equations [in Russian ],Nauka, Moscow (1991).
W.E. Brumley, “On the asymptotic behavior of solutions of Differential-difference equations of neutral type,” J.Differ.Equations,7,175-188 (1970).
R.Datko, “Representation of solutions and stability of linear Differential-difference equations in a Banach space,” J.Differ.Equations,29, No.1,105-166 (1978).
M.C. Delfour and A.Manitius, "The structural operator F and its role in the theory of retarded systems.Part I, "J.Math.Anal.Appl.,73,466-490 (1980).
M.C. Delfour and A.Manitius, "The structural operator F and its role in the theory of retarded systems.Part II, "J.Math.Anal.Appl.,74,359-381 (1980).
O.Diekman, S.A. van Gils, S.M. Verduyn Lunel,and H.O. Walther,Delay Equations:Functional, Complex and Nonlinear Analysis, Springer-Verlag, New York (1995).
E.L. El'sgol'ts and S.B. Norkin,Introduction to the Theory and Application of Differential Equations with Deviating Argument [in Russian ],Nauka, Moscow (1971);English transl.:Academic Press, New York (1973).
P.S. Gromova and A.M. Zverkin, "Trigonometric series whose sum is a continuous unbounded function on the real axis and is a solution of an equation with deviating argument, "Differ.Uravn., 4, 1774-1784 (1968);English transl.:Differ.Equations,4 (1968).
W.Hahn,“Über Differential-Differenzegleichungen mit anomalen Zosungun,”Math.Ann.,133, No.3,251-255 (1957).
J.Hale, Theory of Functional Differential Equations, Springer-Verlag, New York (1984).
D.Henry, "Small solutions of linear autonomous functional Differential equations, "J.Differ.Equations,8,494-501 (1970).
S.A. Ivanov,and V.V. Vlasov, "Basisness and estimates of solutions for the equations with delay in the scale of Sobolev spaces, "Usp.Mat.Nauk,56, No.3,151-152 (2001).
S.A. Ivanov and V.V. Vlasov, "The estimates of the solutions of the equations with delay and basis of the divided differences, "Math.Notes,72, No.2,303-306 (2002).
F.Kappel, "Laplace-transform methods and linear autonomous functional-Differential equations, "Ber.Math.Stat.Forshungsszentrum Graz.,64 (1976).
F.Kappel and K.Kunisch, "Invariance results for delay and Volterra equations in fractional order Sobolev spaces," Trans.Am.Math.Soc.,304, No.1,1-57 (1987).
T.Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin (1966).
V.B. Kolmanovskii and V.R. Nosov,Stability and Periodic Regimes of Regulated Systems with Aftereffect [in Russian ], Nauka, Moscow (1981).
K.Kunisch and W.Shappacher, "Necessary conditions for partial Differential equations with delay to generate a C 0-semigroup, "J.Differ.Equations,50, 49-79 (1983).
N.Levinson and C.McCalla "Completeness and independence of the exponential solutions of some functional Differential equations, "Stud.Appl.Math.,53, 1-15 (1974).
J.L. Lions and E. Magenes,Problemes aux Limites non Homogenes et Applications, Dunod, Paris (1968).
S.M. Verduyn Lunel, "Series expansions and small solutions for Volterra equations of convolutions type, "J.Differ.Equations,85, 17-53 (1990).
S.M. Verduyn Lunel, "The closure of the generalized eigenspace of a class of in nitesimal generators, "Proc.R.Soc.,Edinburgh,Sect.A,Math.,117A,171-192 (1991).
S.M. Verduyn Lunel, "Small solutions and completeness for linear functional Differential equations, "Contemp.Math.,129,127-152 (1992).
S.M. Verduyn Lunel and D.Yakubovich, "A functional model approach to linear neutral functional Differential equations, "Integral Equations Oper.Theory,27,347-378 (1997).
A.D. Myshkis,Linear Differential Equations with Retarded Argument, 2nd ed.[in Russian ],Nauka, Moscow (1972).
S.Nakagiri, "Structural properties of functional Differential equations in Banach space, "Osaka J. Math.,25,353-398 (1988).
A.A. Shkalikov, "Boundary-value problems for ordinary Differential equations with a parameter in the boundary conditions,J.Sov.Math.,33, No.6,1311-1342 (1986).
A.L. Skubachevskii,Elliptic Functional Differential Equations and Applications, Birkhauser, Basel-Boston-Berlin (1997).
O.J. Staffans, "Some well-posed functional equations which generate semigroups, "J.Differ.Equations,58, No.2,157-191 (1985).
V.V. Vlasov, "Solubility and properties of solutions of functional-Differential equations in Hilbert space, "Mat.Sb.,186, No.8,67-92 (1995);English transl.:Sb.Math.,186, No.8,1147-1172.
V.V. Vlasov, "Correct solubility of a certain class of functional-Differential equations in Hilbert pace, "Dokl.Akad.Nauk,Ross.Akad.Nauk,345, No.6,733-736 (1995);English transl.in Russian Acad.Sci.Dokl.Math.(dy1995).
V.V. Vlasov, "Correct solubility of a class of Differential equations with deviating argument in Hilbert space, "Izv.Vyssh.Uchebn.Zaved.,Mat.,No.1,22-35,(1996);English transl.:Russ.Math.,No.1 (1996).
V.V. Vlasov, "Certain properties of the system of elementary solutions of Differential-difference equations, "Usp.Mat.Nauk,51, No.1,143-144 (1996);English transl.:Russ.Math.Surv.,51, No.1(1996).
V.V. Vlasov, "On the behavior of solutions of a certain functional Differential equations in Hilbert space, "Izv.Vyssh.Uchebn.Zaved.,Mat.,1, 3-11 (1993);English transl.:Russ.Math.,1, 3-11 (1993).
V.V. Vlasov, "On properties of the solutions of a class Differential-difference equations and some spectral questions, "Usp.Mat.Nauk,47, No.5,173-174 (1992);English ransl.:Russ.Math.Surv., 47, No.sn5,209-211 (1992).
V.V. Vlasov, "On a class of Differential-difference equations in a Hilbert space and some spectral questions, "Dokl.Akad.Nauk,Ross.Akad.Nauk,327, No.4-6,428-432 (1992);English transl.: Russian Acad.Sci.Dokl.Math.,46, No.3,458-462 (1993).
V.V. Vlasov, "On some properties of solutions of linear Differential equations with delay in Hilbert space, "Izv.Vyssh.Uchebn.Zaved.,Mat.,5 (372),24-35 (1993);English transl.:Russ.Math.,5 (1993).
V.V. Vlasov, "Functional-Differential equations in a Hilbert space and related topics of spectral theory of operators, "Phystech J.,3, No.4,18-39 (1997).
V.V. Vlasov, "Certain properties of strong and exponential solutions difference-Differential equations of neutral type, "Dokl.Akad.Nauk,Ross.Akad.Nauk,364, No.5,583-585 (1999).
V.V. Vlasov, "On a certain class of difference-Differential equations of neutral type, "Izv.Vyssh. Uchebn.Zaved.,Mat.,441, No.2,22-35 (1999).
V.V. Vlasov, "On a certain class of functional Differential equations with retarded argument in a Hilbert space, "Mat.Zametki,62, No.5,782-786 (1997);English transl.:Math.Notes,62, No.5, 655-659 (1997).
V.V. Vlasov, "On certain spectral questions arising in the theory of Differential-difference equations, "Usp.Mat.Nauk,53, No.4,217-218 (1998);English transl.:Russ.Math.Surv.,53, No.4 (1998).
V.V. Vlasov, "On behavior of solutions of certain Differential-difference equations with operator coefficients, "Izv.Vyssh.Uchebn.Zaved.,Mat.,No.8,80-83 (1992);English transl.:Russ.Math., 36, No.8 (1992).
V.V. Vlasov, "On solvability of a certain class of difference-delay equations in Hilbert space, "Izv. Vyssh.Uchebn.Zaved.,Mat.,No.6,28-38 (1994);English transl.:Russ.Math.,38, No.6,26-36 (1994).
V.V. Vlasov, "On certain properties of solutions of functional-Differential equations in a Hilbert space, "Usp.Mat.Nauk,49, No.3,175-176 (1994);English transl.:Russ.Math.Surv.,49, No.3, 205-206 (1994).
V.V. Vlasov, "Solvability of boundary-value problems for a class of integroDifferential equations in the plane, "Differ.Uravn.,25, No.9,1589-1599 (1989);English transl.:Differ.Equations,25, 1124-1132 (1989).
V.V. Vlasov, "Boundary-value problems on a half-line for a class of integroDifferential equations, "Differ.Uravn.,25, No.10,1784-1787 (1989);English transl.:Differ.Equations,25, 1262-1265 (1989).
V.V. Vlasov, "On multiple minimality of a part of the system of root vectors of certain operator-valued functions, "Sov.Math.Dokl.,41, No.1,45-49 (1990).
V.V. Vlasov, "Solvability of a class of functional-Differential equations and some spectral questions, "Dokl.Akad.Nauk SSSR,319, No.1,22-26 (1991);English transl.:Soviet Math.Dokl.,44 (1992).
V.V. Vlasov, "On the behavior of solutions of a class of functional-Differential equations on the half-axis and some spectral questions, "Izv.Vyssh.Uchebn.Zaved.,Mat.,No.12,11-20 (1992);English transl.:Russ.Math.,36, No.12 (1992).
V.V. Vlasov, "On spectral problems arising in the theory of functional Differential equations, "Funct. Differ.Equ.,8, No.3-4,435-446 (2001).
V.V. Vlasov, "On estimates of the solutions for Differential-difference equations of neutral type, "Izv.Vyssh.Uchebn.Zaved.,Mat.,No.4,14-22 (2000).
V.V. Vlasov, "On basisness of the exponential solutions for functional-Differential equations in Sobolev spaces, "Dokl.Akad.Nauk,Ross.Akad.Nauk,381, No.5,302-304 (2001).
V.V. Vlasov, "On the solvability and estimates of solutions to functional Differential equations in Sobolev spaces, "Proc.Steklov Inst.Math.,227,104-115 (1999).
V.V. Vlasov, "On a class of delay functional-Differential equations in Hilbert space, "Math.Notes,62, No.5,655-659 (1997).
V.V. Vlasov and V.V. Malygina, "Correct solvability of linear Differential equations with after-effect in Hilbert spaces, "Differ.Uravn.,28, No.5,901-903 (1992).
V.V. Vlasov and V.Zh.Sakbaev, "Well-de ned solvability of some Differential-difference equations in Sobolev space, "Math.Notes,68, No.6,794-797 (2000).
V.V. Vlasov and V.Zh.Sakbaev, "The correct solvability of some Differential-difference equations in the scale of Sobolev spaces, "Differ.Equations,37, No.9,1252-1260 (2001).
J.Wu, "Semigroup and integral form on a class of partial Differential equations with in nite delay, "Differ.Integral Equ.,4, 1325-1351 (1991).
J.Wu, "Theory and applications of partial functional Differential equations, "Appl.Math.Sci.,119 (1996).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Vlasov, V.V. Spectral Problems Arising in the Theory of Differential Equations with Delay. Journal of Mathematical Sciences 124, 5176–5192 (2004). https://doi.org/10.1023/B:JOTH.0000047252.81054.77
Issue Date:
DOI: https://doi.org/10.1023/B:JOTH.0000047252.81054.77