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Proceedings of the American Mathematical Society

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The final value problem for Sobolev equations


Author: John Lagnese
Journal: Proc. Amer. Math. Soc. 56 (1976), 247-252
MSC: Primary 34G05; Secondary 35R20
DOI: https://doi.org/10.1090/S0002-9939-1976-0419971-9
Erratum: Proc. Amer. Math. Soc. 63 (1977), 375.
MathSciNet review: 0419971
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Abstract: Let $ A$ and $ B$ be $ m$-accretive linear operators in a complex Hilbert space $ H$ with $ D(A) \subset D(B)$. The method of quasi-reversibility is used to obtain a solution to the Sobolev equation $ (d/dt)[(I + B)u(t)] + Au(t) = 0,0 < t < 1$, which approximates a specified final value $ u(1) = f$. In general, when $ D(A) \subset D(B)$, it is not possible to find a solution which achieves exactly the final value $ u(1) = f$.


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

  • [1] N. Dunford and J. T. Schwartz, Linear operators. II: Spectral theory. Selfadjoint operators in Hilbert space, Wiley, New York, 1963. MR 32 #6181.
  • [2] Tosio Kato, Perturbation theory for linear operators, Die Grundlehren der mathematischen Wissenschaften, Band 132, Springer-Verlag New York, Inc., New York, 1966. MR 0203473
  • [3] John Lagnese, Existence, uniqueness and limiting behavior of solutions of a class of differential equations in Banach space, Pacific J. Math. 53 (1974), 473–485. MR 0361336
  • [4] R. Lattés and J.-L. Lions, The method of quasi-reversibility, Modern Analytic and Computational Methods in Sci. and Math., no. 18, Elsevier, New York, 1969. MR 39 #5067.
  • [5] R. E. Showalter, Equations with operators forming a right angle, Pacific J. Math. 45 (1973), 357–362. MR 0318971
  • [6] R. E. Showalter, The final value problem for evolution equations, J. Math. Anal. Appl. 47 (1974), 563–572. MR 0352644, https://doi.org/10.1016/0022-247X(74)90008-0
  • [7] Kôsaku Yosida, Functional analysis, Die Grundlehren der Mathematischen Wissenschaften, Band 123, Academic Press, Inc., New York; Springer-Verlag, Berlin, 1965. MR 0180824

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1976-0419971-9
Article copyright: © Copyright 1976 American Mathematical Society