Sobolev space estimates for solutions of equations with delay, and the basis of divided differences

Vlasov, V.; Ivanov, S.

doi:10.1090/S1061-0022-04-00821-0

Sobolev space estimates for solutions of equations with delay, and the basis of divided differences
HTML articles powered by AMS MathViewer

by V. V. Vlasov and S. A. Ivanov
Translated by: B. M. Bekker

St. Petersburg Math. J. 15 (2004), 545-561

DOI: https://doi.org/10.1090/S1061-0022-04-00821-0

Published electronically: July 6, 2004

PDF | Request permission

Abstract:

Sharp Sobolev space estimates for solutions of neutral difference-differential equations with arbitrary index are obtained without the assumption that the roots of the characteristic quasipolynomial are separated. The proof is based on the fact that the system of divided differences of the exponential solutions forms a Riesz basis. Moreover, it is proved that, under more general conditions, the system of exponential solutions is minimal and complete.

References

S. A. Avdonin and S. A. Ivanov, Riesz bases of exponentials and divided differences, Algebra i Analiz 13 (2001), no. 3, 1–17 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 13 (2002), no. 3, 339–351. MR 1850184
V. I. Vasjunin, Unconditionally convergent spectral decompositions and interpolation problems, Trudy Mat. Inst. Steklov. 130 (1978), 5–49, 223 (Russian). Spectral theory of functions and operators. MR 505683
N. K. Nikol′skiĭ, Treatise on the shift operator, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 273, Springer-Verlag, Berlin, 1986. Spectral function theory; With an appendix by S. V. Hruščev [S. V. Khrushchëv] and V. V. Peller; Translated from the Russian by Jaak Peetre. MR 827223, DOI 10.1007/978-3-642-70151-1
S. V. Hruščëv, N. K. Nikol′skiĭ, and B. S. Pavlov, Unconditional bases of exponentials and of reproducing kernels, Complex analysis and spectral theory (Leningrad, 1979/1980) Lecture Notes in Math., vol. 864, Springer, Berlin-New York, 1981, pp. 214–335. MR 643384
J.-L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications. Vol. 1, Travaux et Recherches Mathématiques, No. 17, Dunod, Paris, 1968 (French). MR 0247243
S. Ivanov and N. Kalton, Interpolation of subspaces and applications to exponential bases, Algebra i Analiz 13 (2001), no. 2, 93–115; English transl., St. Petersburg Math. J. 13 (2002), no. 2, 221–239. MR 1834861
V. V. Vlasov, On a class of differential-difference equations of neutral type, Izv. Vyssh. Uchebn. Zaved. Mat. 2 (1999), 20–29 (Russian); English transl., Russian Math. (Iz. VUZ) 43 (1999), no. 2, 17–26. MR 1685756
V. V. Vlasov, On the properties of a system of exponential solutions of differential-difference equations in Sobolev spaces, Izv. Vyssh. Uchebn. Zaved. Mat. 6 (2001), 23–29 (Russian); English transl., Russian Math. (Iz. VUZ) 45 (2001), no. 6, 21–27. MR 1858506
V. V. Vlasov and S. A. Ivanov, The basis property and estimates for solutions of equations with aftereffect in a scale of Sobolev spaces, Uspekhi Mat. Nauk 56 (2001), no. 3(339), 151–152 (Russian); English transl., Russian Math. Surveys 56 (2001), no. 3, 595–596. MR 1859727, DOI 10.1070/RM2001v056n03ABEH000397
Richard Bellman and Kenneth L. Cooke, Differential-difference equations, Academic Press, New York-London, 1963. MR 0147745
B. S. Pavlov, The basis property of a system of exponentials and the condition of Muckenhoupt, Dokl. Akad. Nauk SSSR 247 (1979), no. 1, 37–40 (Russian). MR 545940
G. E. Shilov, Mathematical analysis. Second special course, “Naukar7r7, Moscow, 1965. (Russian)
Norman Levinson, Gap and Density Theorems, American Mathematical Society Colloquium Publications, Vol. 26, American Mathematical Society, New York, 1940. MR 0003208
Daniel Henry, Linear autonomous neutral functional differential equations, J. Differential Equations 15 (1974), 106–128. MR 338520, DOI 10.1016/0022-0396(74)90089-8
Jack Hale, Theory of functional differential equations, 2nd ed., Applied Mathematical Sciences, Vol. 3, Springer-Verlag, New York-Heidelberg, 1977. MR 0508721
V. V. Vlasov, Estimates for solutions of differential-difference equations of neutral type, Izv. Vyssh. Uchebn. Zaved. Mat. 4 (2000), 14–22 (Russian); English transl., Russian Math. (Iz. VUZ) 44 (2000), no. 4, 12–20. MR 1782521
V. V. Vlasov, On the basis property of exponential solutions of functional-differential equations in Sobolev spaces, Dokl. Akad. Nauk 381 (2001), no. 3, 302–304 (Russian). MR 1892528
S. M. Verduyn Lunel, Series expansions and small solutions for Volterra equations of convolution type, J. Differential Equations 85 (1990), no. 1, 17–53. MR 1052326, DOI 10.1016/0022-0396(90)90087-6
M. C. Delfour and A. Manitius, The structural operator $F$ and its role in the theory of retarded systems. II, J. Math. Anal. Appl. 74 (1980), no. 2, 359–381. MR 572658, DOI 10.1016/0022-247X(80)90134-1
Sjoerd M. Verduyn Lunel and Dmitry V. Yakubovich, A functional model approach to linear neutral functional-differential equations, Integral Equations Operator Theory 27 (1997), no. 3, 347–378. MR 1433007, DOI 10.1007/BF01324734
Norman Levinson and Clement McCalla, Completeness and independence of the exponential solutions of some functional differential equations, Studies in Appl. Math. 53 (1974), 1–15. MR 342813, DOI 10.1002/sapm19745311
David L. Russell, On exponential bases for the Sobolev spaces over an interval, J. Math. Anal. Appl. 87 (1982), no. 2, 528–550. MR 658032, DOI 10.1016/0022-247X(82)90142-1
B. Ja. Levin, Distribution of zeros of entire functions, American Mathematical Society, Providence, R.I., 1964. MR 0156975
—, On bases for exponential functions in ${L}^2$, Khar′kov. Gos. Univ. Uchen. Zap. 115 (1961) (Zap. Mat. Otdel. Fiz.-Mat. Fak. Khar′kov. Gos. Univ. i Khar′kov. Mat. Obshch. (4) 27 (1961)), 39–48. (Russian).
A. A. Shkalikov, Boundary value problems for ordinary differential equations with a parameter in the boundary conditions, Trudy Sem. Petrovsk. 9 (1983), 190–229 (Russian, with English summary). MR 731903
A. M. Sedletskiĭ, Biorthogonal expansions of functions in exponential series on intervals of the real axis, Uspekhi Mat. Nauk 37 (1982), no. 5(227), 51–95, 248 (Russian). MR 676613
V. V. Vlasov and S. A. Ivanov, Estimates for solutions of equations with aftereffect in Sobolev spaces and a basis of divided differences, Mat. Zametki 72 (2002), no. 2, 303–306 (Russian); English transl., Math. Notes 72 (2002), no. 1-2, 271–274. MR 1942555, DOI 10.1023/A:1019862331288
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, Differencial′nye Uravnenija 4 (1968), 1774–1784 (Russian). MR 0241778