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On nonlinear scalar Volterra integral equations. I


Author: Hans Engler
Journal: Trans. Amer. Math. Soc. 291 (1985), 319-336
MSC: Primary 45D05
DOI: https://doi.org/10.1090/S0002-9947-1985-0797063-7
MathSciNet review: 797063
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Abstract | References | Similar Articles | Additional Information

Abstract: The scalar nonlinear Volterra integral equation

$\displaystyle u(t) + \int_0^t {g(t,s,u(s))\,ds = f(t)\qquad (0 \leqslant t)} $

is studied. Conditions are given under which the difference of two solutions can be estimated by the variation of the difference of the corresponding right-hand sides. Criteria for the existence of $ \lim u(t)$ (as $ t \to \infty $) are given, and existence and uniqueness questions are also studied.

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

  • [1] H. Engler, Bounds and asymptotics for a scalar Volterra integral equation, J. Integral Equations 7 (1984), 209-227. MR 770148 (86j:45005)
  • [2] G. Gripenberg, An abstract nonlinear Volterra equation, Israel J. Math. 34 (1979), 198-212. MR 570881 (81g:45011)
  • [3] -, Bounded solutions of a Volterra equation, J. Differential Equations 28 (1978), 18-22. MR 0463827 (57:3766)
  • [4] T. Kiffe and M. Stecher, An abstract Volterra integral equation in a reflexive Banach space, J. Differential Equations 34 (1979), 303-325. MR 550048 (81d:45020)
  • [5] -, $ {L^2}$-solutions of Volterra integral equations, SIAM J. Math. Anal. 10 (1979), 274-280. MR 523843 (80c:45005)
  • [6] V. Lakshmikantham and S. Leela, Differential and integral inequalities. I, II, Academic Press, New York, 1969.
  • [7] J. J. Levin, A bound on the solutions of a Volterra equation, Arch. Rational Mech. Anal. 52 (1973), 339-349. MR 0336269 (49:1045)
  • [8] -, Some a priori bounds for nonlinear Volterra equations, SIAM J. Math. Anal. 7 (1976), 872-896. MR 0420170 (54:8185)
  • [9] S. O. London, On the solutions of a nonlinear Volterra equation, J. Math. Anal. Appl. 39 (1972), 564-573. MR 0313743 (47:2297)
  • [10] R. K. Miller, Nonlinear Volterra integral equations, Benjamin, Menlo Park, 1971. MR 0511193 (58:23394)
  • [11] J. T. Schwartz, Nonlinear functional analysis, Gordon & Breach, New York, 1969. MR 0433481 (55:6457)
  • [12] O. J. Staffans, A bound on the solutions of a nonlinear Volterra equation, J. Math. Anal. Appl. 83 (1981), 127-134. MR 632331 (82j:45004)
  • [13] -, A priori bounds for a discontinuous Volterra equation, J. Integral Equations 3 (1981), 231-243. MR 628353 (82j:45003)
  • [14] J. H. Roberts and W. R. Mann, On a certain nonlinear integral equation of the Volterra type, Pacific J. Math. 1 (1951), 431-445. MR 0044009 (13:354a)
  • [15] J. J. Levin, Remarks on a Volterra equation, Delay and Functional Differential Equations and their Applications, Academic Press, London, New York, 1972, pp. 233-255. MR 0402437 (53:6257)
  • [16] O. J. Staffans, A note on a Volterra equation with several nonlinearities, J. Integral Equations 7 (1984), 249-252. MR 770151 (86e:45019)
  • [17] H. Engler, A note on scalar Volterra integral equations. II, J. Math. Anal. Appl. (to appear). MR 836233 (87g:45001)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1985-0797063-7
Keywords: Nonlinear scalar Volterra equation, uniqueness, existence, continuous dependence on data, a priori estimates, asymptotic behavior, log-convex kernels
Article copyright: © Copyright 1985 American Mathematical Society

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