Abstract
This paper concerns the dissipativity of nonlinear Volterra functional differential equations (VFDEs) in Banach space and their numerical discretization. We derive sufficient conditions for the dissipativity of nonlinear VFDEs. The general results provide a unified theoretical treatment for dissipativity analysis to ordinary differential equations (ODEs), delay differential equations (DDEs), integro-differential equations (IDEs) and VFDEs of other type appearing in practice. Then the dissipativity property of the backward Euler method for VFDEs is investigated. It is shown that the method can inherit the dissipativity of the underlying system. The close relationship between the absorbing set of the numerically discrete system generated by the backward Euler method and that of the underlying system is revealed.
Highlights
1 Introduction Many dynamical systems are characterized by the property of possessing a bounded absorbing set where all trajectories enter in a finite time and thereafter remain inside
The dissipativity analysis of numerical methods for Volterra functional differential equations (VFDEs) in the literature was limited in Euclidean spaces or Hilbert spaces
The aim of this paper is to investigate the dissipativity of nonlinear VFDEs in Banach space and their numerical discretization
Summary
Many dynamical systems are characterized by the property of possessing a bounded absorbing set where all trajectories enter in a finite time and thereafter remain inside. In the study of numerical methods, it is natural to ask whether those discrete systems preserve the dissipativity of the continuous system. For the delay differential equations (DDEs) with constant delay, sufficient conditions for the dissipativity of analytical and numerical solutions are presented in [ – ].
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