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Stability of bounded solutions of linear functional equations


Author: Joel N. Franklin
Journal: Math. Comp. 25 (1971), 413-424
MSC: Primary 47A50; Secondary 35R25
DOI: https://doi.org/10.1090/S0025-5718-1971-0380461-7
MathSciNet review: 0380461
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Abstract: The weak sequential compactness of reflexive Banach spaces is used to explain the fact that certain ill-posed, linear problems become well-posed if the solutions are required to satisfy a prescribed bound. Applications are made to the computability of solutions of ill-posed problems associated with elliptic and parabolic partial differential equations.


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

  • [1] J. Hadamard, Lectures on Cauchy's Problem in Linear Partial Differential Equations, Yale Univ. Press, New Haven, Conn., 1923.
  • [2] F. John, ``Numerical solution of the equation of heat conduction for preceding times,'' Ann. Mat. Pura Appl. (4), v. 40, 1955, pp. 129-142. MR 19, 323. MR 0087224 (19:323e)
  • [3] F. John, Numerical Solution of Problems which are not Well Posed in the Sense of Hadamard, Proc. Sympos. Numerical Treatment of Partial Differential Equations with Real Characteristics (Rome, 1959), Libreria Eredi Virgilio Veschi, Rome, 1959, pp. 103-116. MR 21 #6704. MR 0107983 (21:6704)
  • [4] F. John, ``Continuous dependence on data for solutions of partial differential equations with a prescribed bound,'' Comm. Pure Appl. Math., v. 13, 1960, pp. 551-585. MR 24 #A317. MR 0130456 (24:A317)
  • [5] L. E. Payne, On Some Nonwell Posed Problems for Partial Differential Equations, Proc. Adv. Sympos. Numerical Solutions of Nonlinear Differential Equations (Madison, Wis., 1966), Wiley, New York, 1966, pp. 239-263. MR 35 #4606. MR 0213749 (35:4606)
  • [6] R. D. Richtmyer, Difference Methods for Initial-Value Problems, Interscience Tracts in Pure and Appl. Math., Tract 4, Interscience, New York, 1957. MR 20 #438. MR 0093918 (20:438)
  • [7] F. Riesz & B. Sz.-Nagy, Lecons d'Analyse Fonctionnelle, Akad. Kiadó, Budapest, 1953; English transl., Ungar, New York, 1955. MR 15, 132; MR 17, 175. MR 0056821 (15:132d)
  • [8] P. R. Garabedian, Partial Differential Equations, Wiley, New York, 1964. MR 28 #5247. MR 0162045 (28:5247)
  • [9] J. R. Gannon, Some Numerical Results for the Solution of the Heat Equation Backwards in Time, Proc. Adv. Sympos. Numerical Solutions of Nonlinear Differential Equations (Madison, Wis., 1966), Wiley, New York, 1966, pp. 21-54. MR 34 #7037. MR 0207221 (34:7037)
  • [10] J. Douglas, Jr., Approximate Continuation of Harmonic and Parabolic Functions, Proc. Sympos. Numerical Solution of Partial Differential Equations (Univ. of Maryland, 1965), Academic Press, New York, 1966, pp. 353-364. MR 34 #2206. MR 0202333 (34:2206)
  • [11] R. Saylor, ``Numerical elliptic continuation,'' SIAM J. Numer. Anal., v. 4, 1967, pp. 575-581. MR 36 #5513. MR 0222461 (36:5513)

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Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1971-0380461-7
Keywords: Well-posed, ill-posed, continuous dependence on data, stability, backwards heat-equation, final-value problem, elliptic continuation
Article copyright: © Copyright 1971 American Mathematical Society

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