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A note on continuous dependence of solutions of Volterra integral equations


Author: Mats Gyllenberg
Journal: Proc. Amer. Math. Soc. 81 (1981), 546-548
MSC: Primary 45D05; Secondary 45G10
MathSciNet review: 601726
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Abstract: The topologies induced by two families of seminorms on a vector space of functions $ g:{R^ + } \times {R^ + } \times {E^n} \to {E^n}$ are compared. It is found that the continuous dependence of solutions of the Volterra equation $ x(t) = f(t) + \int {_0^tg(t,s,x(s))ds} $ does not hold for the weaker topology. This result corrects an error in the book of Miller, Benjamin, Menlo Park, Calif., 1971.


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

  • [1] Zvi Artstein, Continuous dependence of solutions of Volterra integral equations, SIAM J. Math. Anal. 6 (1975), 446–456. MR 0361656
  • [2] Richard K. Miller, Nonlinear Volterra integral equations, W. A. Benjamin, Inc., Menlo Park, Calif., 1971. Mathematics Lecture Note Series. MR 0511193

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1981-0601726-5
Keywords: Volterra integral equation, continuous dependence of solutions
Article copyright: © Copyright 1981 American Mathematical Society