Abstract
A review of the results on degenerate differential-operator equations mainly obtained by Khar’kov and Donetsk mathematicians starting from the 1970s is presented. Applications to evolutionary problems of electrodynamics and mathematical circuit theory, including the construction of models of the corresponding physical systems, are considered.
Similar content being viewed by others
References
M. Benadbdallakh, A. G. Rutkas, and A. A. Solov’ev, “Application of asymptotic expansions to studying the infinite system Ax n+1 + Bx n = f n in a Banach space,” Teor. Funkts. Funkts. Anal. Prilozh. (Khar’kov), 12, 20–35 (1970).
M. Benabdallakh and A. G. Rutkas, “Normal solutions of the system Ax n+1 + Bx n = f n in a Banach space,” Vestn. Khar’kov. Univ., 254, 62–65 (1984).
M. Benabdallakh, A. G. Rutkas, and A. A. Solov’ev, “On the stability of degenerate difference systems in Banach spaces,” Dinam. Sistemy, Kiev, Simferopol’, 6, 103–109 (1987).
M. F. Bondarenko, L. A. Vlasenko, and A. G. Rutkas, “Periodic solutions of a certain class of implicit difference equations,” Dokl. Nats. Akad. Nauk Ukrainy, 1, 9–14 (1999).
M. F. Bondarenko and A. G. Rutkas, “Determinancy tests of implicit discrete nonautonomous systems,” Dokl. Nats. Akad. Nauk Ukrainy, 2, 7–11 (2001).
S. L. Campbell, “Singular systems of differential equations,” Res. Notes Math., 40, Pitman Publishing, New York (1980)
S. L. Campbell, “Singular systems of differential equations, II,” Res. Notes Math., 61, Pitman Publishing, New York (1981).
A. Favini, “Laplace transform method for a class of degenerate evolution problems,” Rend. Math., 12, Nos. 3–4, 511–536 (1979).
A. Favini, “Abstract potential operators and spectral methods for a class of degenerate evolution problems,” J. Differ. Equations, 39, No. 2, 212–225 (1981).
A. Favini, “Degenerate and singular evolution equations in Banach spaces,” J. Math. Ann., 273, 17–44 (1985).
A. Favini and P. Plazzi, “Some results concerning the abstract nonlinear equation D t Mu(t)+Lu(t) = f(t, Ku(t)),” Circuits Systems Signal Processing, 5, 261–274 (1986).
A. Favini and P. Plazzi, “On some abstract degenerate problems of parabolic type. II. The nonlinear case,” Nonlin. Anal. Theory Meth. Appl., 13, 23–31 (1989).
A. Favini and A. Rutkas, “Existence and uniqueness of solution of some abstract degenerate nonlinear equations,” Differ. Integr. Equations, 12, No. 3, 373–394 (1999).
A. Favini and L. Vlasenko, “On solvability of degenerate nonstationary differential-difference equations in Banach spaces,” Differ. Integr. Equations, 14 (2001).
I. Ts. Gokhberg and M. G. Krein, Introduction to the Theory of Self-Adjoint Operators [in Russian], Nauka, Moscow (1965).
R. Kalman, M. Arbib, and P. Falb, Topics in Mathematical System Theory [Russian translation], Mir, Moscow (1971).
M. V. Keldysh, “On eigenvalues and eigenfunctions of certain classes of non-self-adjoint equations,” Dokl. Akad. Nauk SSSR, 77, No. 1, 11–14 (1951).
S. G. Krein, Linear Differential Equations in a Banach Space [in Russian], Nauka, Moscow (1967).
M. S. Lefshits, Operators, Oscillations, and Waves (Open Systems) [in Russian], Nauka, Moscow (1966).
A. L. Liven’, “Solvability of the Cauchy problem and estimators for the initial manifold for a certain implicit operator-differential equation,” Vestn. Khar’kov Univ, Ser. Mat., Prikl. Mat., Mekh., 458, 101–108 (1999).
Yu. I. Lyubich, “Classical and local Laplace transform in the abstract Cauchy problem,” Usp. Mat. Nauk, 21, No. 3, 3–51 (1966).
N. I. Radbel’, “On the initial manifold and dissipativity of the Cauchy problem for the equation Ax′(t) + Bx(t) = 0,” Differ. Uravn., 15, No. 6, 1142–1143 (1979).
A. G. Rutkas, “Non-self-adjoint operators in the theory of multipole networks.” Vestn. Khar’kov. Univ., 32, 154–180 (1966).
A. G. Rutkas, “Applications of non-self-adjoint operators to a certain class of electric circuits,” Radiotekhn., 4, 139–150 (1967).
A. G. Rutkas, “Pairs of non-self-adjoint operators and operator hypernodes,” Ukr. Mat. Zh., 1, 37–52 (1970).
A. G. Rutkas, “Cauchy problem for the equation Ax′(t) + Bx(t) = f(t),” Differ. Uravn., 11, No. 11, 1996–2010 (1975).
A. G. Rutkas, “Properties of the distance functions and propagation of waves in structures with a given geometry,” Dokl. Akad. Nauk SSSR, 230, No. 1, 38–40 (1976).
A. G. Rutkas, “On the theory of characteristic functions of linear operators,” Dokl. Akad. Nauk SSSR, 229, No. 3, 546–549 (1976).
A. G. Rutkas, “Characteristic function and model of a linear operator pencil,” Teor. Funkts. Funkts. Anal Prilozh. (Khar’kov), 45, 98–111 (1986).
A. G. Rutkas, “Modal fields in a wave guide with layered dispersive medium,” Preprint, No. 360, Akad. Nauk USSR, Khar’kov (1987).
A. G. Rutkas, “On classification and properties of solutions of the equation Ax′ + Bx = f(t),” Differ. Uravn., 25, No. 7, 1150–1155 (1989).
A. G. Rutkas, “Perturbations of skew-Hermitian pencils and the Cauchy problem,” Ukr. Mat. Zh., 41, No. 8, 1082–1088 (1989).
A. Rutkas, “The solvability of a nonlinear differential equation in a Banach space. Spectral and evolutional problems,” Proc. 6th Crimean Math. School-Symposium, Simferopol (1996), pp. 317–320.
A. G. Rutkas and D. M. Chausovskii, “Some applications of operator nodes and hypernodes,” Mat. Fizika (Kiev), 5, 173–178 (1968).
A. G. Rutkas and N. I. Khirgii, “Semigroups of monomorphisms of graphs in discrete structures,” Teor. Funkts. Funkts. Anal. Prilozh. (Khar’kov), 19, 111–125 (1974).
A. G. Rutkas and N. I. Radbel’, “On linear operator pencils and noncanonical systems,” Teor. Funkts. Funkts. Anal. Prilozh. (Khar’kov), 17, 3–14 (1973).
A. G. Rutkas and L. A. Vlasenko, “Solvability and completeness for an electrodynamical system of non-Kowalewski type,” Mat. Zametki, 53, 138–140 (1993).
A. G. Rutkas and L. A. Vlasenko, “Uniqueness and approximation theorems for a certain degenerate operator-differential equation,” Mat. Zametki, 60, 597–600 (1996).
A. Rutkas and L. Vlasenko, “Implicit differential equations and applications to electrodynamics,” Math. Meth. Appl. Sci., 23, 1–15 (2000).
A. G. Rutkas and L. A. Vlasenko, “Existence of solutions of degenerate nonlinear differential operator equations,” in: Nonlinear Oscilations, Kiev (2001).
A. G. Rutkas and L. A. Vlasenko, “The solvability of a nonlinear degenerate differential equation,” in: Proc. XXth Joint Session of the Petrovskii Seminar and the Moscow Mathematical Society, Moscow Univ. (2001), pp. 351–352.
S. L. Sobolev, “On a certain new problem of mathematical physics,” Izv. Akad. Nauk, Ser. Mat., 18, No. 1, 3–50 (1954).
L. A. Vlasenko, “Construction of solutions for certain classes of equations Au′(t) + Bu(t) = f(t),” Vestn. Khar’kov Univ., 286, 24–28 (1986).
L. A. Vlasenko, “On completeness of normal solutions of the equation Au′(t) + Bu(t) = f(t),” Teor. Funkts. Funkts. Anal. Prilozh., (Khar’kov), 48, 46–51 (1987).
L. A. Vlasenko, “Completeness of elementary solutions of a certain operator-differential equation with delay,” Dokl. Nats. Akad. Nauk Ukrainy, 11, 15–19 (1998).
L. A. Vlasenko, “Existence and uniqueness theorems for a certain implicit differential equation with delay,” Differ. Uravn., 36, No 5, 624–628 (2000).
L. Vlasenko, “Implicit linear time-dependent differential-difference equations and applications,” Math. Meth. Appl. Sci., 23, 937–948 (2000).
Additional information
__________
Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 35, Voronezh Conference-2, 2005.
Rights and permissions
About this article
Cite this article
Rutkas, A.G. Spectral methods for studying degenerate differential-operator equations. I. J Math Sci 144, 4246–4263 (2007). https://doi.org/10.1007/s10958-007-0267-2
Issue Date:
DOI: https://doi.org/10.1007/s10958-007-0267-2