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Behavior of maximally defined solutions of a nonlinear Volterra equation


Author: Terry L. Herdman
Journal: Proc. Amer. Math. Soc. 67 (1977), 297-302
MSC: Primary 45D05
DOI: https://doi.org/10.1090/S0002-9939-1977-0474745-9
MathSciNet review: 474745
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Abstract: This paper is concerned with the behavior of solutions of an n-dimensional nonlinear Volterra integral equation

$\displaystyle x(t) = f(t) + \int_0^t {g(t,s,x(s))ds,\quad t \geqslant 0.} $

In particular, sufficient conditions for a solution $ x(t)$ on its maximal interval of existence $ [0,T)$ to possess the property that $ \vert x(t)\vert \to + \infty$ as $ t \to {T^ - }$ are obtained. Thus these additional conditions give a positive answer to the problem posed by Miller [3, p. 145]. One can construct examples, satisfying the hypotheses given in [3], which provide a negative answer to this problem, see Artstein [1, Appendix A].

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, https://doi.org/10.1137/0506039
  • [2] Terry L. Herdman, Existence and continuation properties of solutions of a nonlinear Volterra integral equation, Dynamical systems (Proc. Internat. Sympos., Brown Univ., Providence, R.I., 1974) Academic Press, New York, 1976, pp. 307–310. MR 0622528
  • [3] 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/S0002-9939-1977-0474745-9
Keywords: Nonlinear Volterra integral equation, maximal interval of existence, maximally defined solution
Article copyright: © Copyright 1977 American Mathematical Society