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A nonlinear singularly perturbed Volterra integrodifferential equation of nonconvolution type

Published online by Cambridge University Press:  14 November 2011

G. S. Jordan
Affiliation:
University of Tennessee, Knoxville, Tenn., U.S.A.

Synopsis

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 → ∞.

Type
Research Article
Copyright
Copyright © Royal Society of Edinburgh 1978

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References

1Kiffe, T. R.On Nonlinear Volterra Equations of Nonconvolution Type. J. Differential Equations 22 (1976), 349367.CrossRefGoogle Scholar
2Levin, J. J.A Nonlinear Volterra Equation not of Convolution Type. J. Differential Equations 4 (1968), 176186.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar