Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Invariance results for delay and Volterra equations in fractional order Sobolev spaces

Authors: F. Kappel and K. Kunisch
Journal: Trans. Amer. Math. Soc. 304 (1987), 1-51
MSC: Primary 45D05; Secondary 34K15, 45J05, 46E35
MathSciNet review: 906804
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Invariance of the trajectories of infinite delay- and Volterra-type equations in fractional order Sobolev spaces are derived under minimal assumptions on the problem data. Properties of fractional order Sobolev spaces defined over intervals are summarized.

References [Enhancements On Off] (What's this?)

  • [1] A. V. Balakrishnan, Applied functional analysis, Springer-Verlag, New York-Heidelberg, 1976. Applications of Mathematics, No. 3. MR 0470699
  • [2] H. T. Banks, J. A. Burns, and E. M. Cliff, Parameter estimation and identification for systems with delays, SIAM J. Control Optim. 19 (1981), no. 6, 791–828. MR 634954, 10.1137/0319051
  • [3] H. T. Banks and P. K. Daniel Lamm, Estimation of delays and other parameters in nonlinear functional-differential equations, SIAM J. Control Optim. 21 (1983), no. 6, 895–915. MR 719519, 10.1137/0321054
  • [4] John A. Burns, Terry L. Herdman, and Harlan W. Stech, Linear functional-differential equations as semigroups on product spaces, SIAM J. Math. Anal. 14 (1983), no. 1, 98–116. MR 686237, 10.1137/0514007
  • [5] Paul L. Butzer and Hubert Berens, Semi-groups of operators and approximation, Die Grundlehren der mathematischen Wissenschaften, Band 145, Springer-Verlag New York Inc., New York, 1967. MR 0230022
  • [6] Jack K. Hale and Junji Kato, Phase space for retarded equations with infinite delay, Funkcial. Ekvac. 21 (1978), no. 1, 11–41. MR 0492721
  • [7] Edwin Hewitt and Karl Stromberg, Real and abstract analysis, Springer-Verlag, New York-Heidelberg, 1975. A modern treatment of the theory of functions of a real variable; Third printing; Graduate Texts in Mathematics, No. 25. MR 0367121
  • [8] F. Kappel, Approximation of neutral functional-differential equations in the state space 𝑅ⁿ×𝐿², Qualitative theory of differential equations, Vol. I, II (Szeged, 1979), Colloq. Math. Soc. János Bolyai, vol. 30, North-Holland, Amsterdam-New York, 1981, pp. 463–506. MR 680607
  • [9 F] Kappel and K. Kunisch, An approximation scheme for parameter estimation in infinite delay equations of Volterra type. Numerical results, Technical Report, Institutes for Mathematics, University and Technical University of Graz, No. 51, 1984.
  • [10] F. Kappel and W. Schappacher, Some considerations to the fundamental theory of infinite delay equations, J. Differential Equations 37 (1980), no. 2, 141–183. MR 587220, 10.1016/0022-0396(80)90093-5
  • [11] J. L. Lions and E. Magenes, Nonhomogeneous boundary value problems and applications, Vol. I, Springer, Heidelberg, 1972.
  • [12] I. Lasiecka and A. Manitius, Differentiability and convergence rates of approximating semigroups for retarded functional-differential equations, SIAM J. Numer. Anal. 25 (1988), no. 4, 883–907. MR 954790, 10.1137/0725050
  • [13] Cora Sadosky, Interpolation of operators and singular integrals, Monographs and Textbooks in Pure and Applied Math., vol. 53, Marcel Dekker, Inc., New York, 1979. An introduction to harmonic analysis. MR 551747
  • [14] Konrad Schumacher, Remarks on semilinear partial functional-differential equations with infinite delay, J. Math. Anal. Appl. 80 (1981), no. 1, 261–290. MR 614255, 10.1016/0022-247X(81)90104-9
  • [15] Olof J. Staffans, Extended initial and forcing function semigroups generated by a functional equation, SIAM J. Math. Anal. 16 (1985), no. 5, 1034–1048. MR 800795, 10.1137/0516077
  • [16] Olof J. Staffans, Semigroups generated by a neutral functional-differential equation, SIAM J. Math. Anal. 17 (1986), no. 1, 46–57. MR 819211, 10.1137/0517006
  • [17] Hans Triebel, Interpolation theory, function spaces, differential operators, North-Holland Mathematical Library, vol. 18, North-Holland Publishing Co., Amsterdam-New York, 1978. MR 503903

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 45D05, 34K15, 45J05, 46E35

Retrieve articles in all journals with MSC: 45D05, 34K15, 45J05, 46E35

Additional Information

Article copyright: © Copyright 1987 American Mathematical Society