Behavior of maximally defined solutions of a nonlinear Volterra equation
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- by Terry L. Herdman PDF
- Proc. Amer. Math. Soc. 67 (1977), 297-302 Request permission
Abstract:
This paper is concerned with the behavior of solutions of an n-dimensional nonlinear Volterra integral equation \[ x(t) = f(t) + \int _0^t {g(t,s,x(s))ds,\quad t \geqslant 0.} \] In particular, sufficient conditions for a solution $x(t)$ on its maximal interval of existence $[0,T)$ to possess the property that $|x(t)| \to + \infty$ 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
- Zvi Artstein, Continuous dependence of solutions of Volterra integral equations, SIAM J. Math. Anal. 6 (1975), 446–456. MR 361656, DOI 10.1137/0506039
- 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
- Richard K. Miller, Nonlinear Volterra integral equations, Mathematics Lecture Note Series, W. A. Benjamin, Inc., Menlo Park, Calif., 1971. MR 0511193
Additional Information
- © Copyright 1977 American Mathematical Society
- 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