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A nonlinear singularly perturbed Volterra integrodifferential equation of nonconvolution type

  • G. S. Jordan (a1)

We consider the nonconvolution initial value problem

where μ is a small positive parameter, b(t, s) is a given real kernel, and F, g are given real functions. For the convolution case b(t,s) = a(t − s). Lodge, McLeod, and Nohel recently established many qualitative properties of the solution of (+); we extend their results to the general nonconvolution problem. In particular, conditions are given that ensure that the solution of (+) decreases to a limiting value α(μ) > 1 as t → ∞.

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1Kiffe T. R. On Nonlinear Volterra Equations of Nonconvolution Type. J. Differential Equations 22 (1976), 349367.
2Levin J. J. A Nonlinear Volterra Equation not of Convolution Type. J. Differential Equations 4 (1968), 176186.
3Lodge A. S., McLeod J. B. and Nohel J. A. A Nonlinear Singularly Perturbed Volterra Integrodifferential Equation Occurring in Polymer Rheology. Proc. Roy. Soc. Edinburgh Sect. A, 80 (1978), 99137.
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Proceedings of the Royal Society of Edinburgh Section A: Mathematics
  • ISSN: 0308-2105
  • EISSN: 1473-7124
  • URL: /core/journals/proceedings-of-the-royal-society-of-edinburgh-section-a-mathematics
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