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Nonlinear volterra equations with infinite delay

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Abstract

This paper is concerned with the existence and stability of nonlinear Volterra equations which have infinite delay and are of the form:

$$x (\varphi ) (t) = W (t, \tau ) \varphi (0) + \int\limits_\tau ^t {W (t, s)} F(s,x_s (\varphi )) ds, x_\tau (\varphi ) = \varphi \in C_u .$$

Here,X denotes a Banach space;W(t, s) is a linear evolution operator mappingX toX; C u is the space of uniformly continuous functions endowed with the supremum norm; andF(·,·) is a continuous mapping ofR×C u toX. The autonomous version of the preceding equation is also considered. A nonlinear semigroup is associated with its solutions and the infinitesimal generator of the semigroup is characterized. The generator is then used to represent and approximate solutions to the autonomous equation.

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  1. Department of Mathematics, University of Houston, Cullen Boulevard, 77004, Houston, TX, USA

    W. E. Fitzgibbon

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  1. W. E. Fitzgibbon
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Fitzgibbon, W.E. Nonlinear volterra equations with infinite delay. Monatshefte für Mathematik 84, 275–288 (1977). https://doi.org/10.1007/BF01366497

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  • DOI: https://doi.org/10.1007/BF01366497

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