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Reducible Volterra and Levin–Nohel Retarded Equations with Infinite Delay

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Abstract

We consider two cases of reducible Volterra and Levin–Nohel retarded equations with infinite delay. In these cases reducibility arises from the use of a special type of memory functions with an exponential behavior. We address global questions like the existence of Liapunov functions and, consequently, of attractors for the nonlinear systems generated by these equations as well as the attractors for the reduced systems. For the reducible Volterra equations we exhibit cases of nontrivial Hamiltonian behaviour and for the reducible Levin–Nohel equation we identify Hopf and saddle connection bifurcations.

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Authors and Affiliations

  1. Center for Mathematical Analysis, Geometry, and Dynamical Systems and Institute for Systems and Robotics, Instituto Superior Técnico, 1049-001, Lisbon, Portugal

    Waldyr M. Oliva

  2. Departamento de Matematica, Center for Mathematical Analysis, Geometry, and Dynamical Systems, Instituto Superior Técnico, 1049-001, Lisbon, Portugal

    Carlos Rocha

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  1. Waldyr M. Oliva
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  2. Carlos Rocha
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Correspondence to Waldyr M. Oliva.

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This paper is dedicated to Professor Jack Hale on the occasion of his 80th birthday.

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Oliva, W.M., Rocha, C. Reducible Volterra and Levin–Nohel Retarded Equations with Infinite Delay. J Dyn Diff Equat 22, 509–532 (2010). https://doi.org/10.1007/s10884-010-9177-y

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  • DOI: https://doi.org/10.1007/s10884-010-9177-y

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