Dissipativity of the backward Euler method for nonlinear Volterra functional differential equations in Banach space
 Siqing Gan^{1}Email author
https://doi.org/10.1186/s1366201504698
© Gan; licensee Springer. 2015
Received: 9 January 2015
Accepted: 13 April 2015
Published: 25 April 2015
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), integrodifferential 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.
Keywords
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. Such systems are called dissipative. Dissipativity means that the eventual time evolution of solutions is confined to a bounded absorbing set. In the study of numerical methods, it is natural to ask whether those discrete systems preserve the dissipativity of the continuous system.
Since the 1990s considerable process has been made in dissipativity analysis of numerical methods. The papers [1–5] focus on the numerical methods for ordinary differential equations. For the delay differential equations (DDEs) with constant delay, sufficient conditions for the dissipativity of analytical and numerical solutions are presented in [6–8]. Since that, the analysis is extended to DDEs with variable lags [9, 10] and Volterra functional differential equations [11–16].
The dissipativity analysis of numerical methods for 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.
The main contributions of this paper could be summarized as follows.
(a) Sufficient conditions for the dissipativity of nonlinear VFDEs in Banach space are derived. The general results provide a unified theoretical treatment for dissipativity analysis to ordinary differential equations (ODEs), delay differential equations (DDEs), integrodifferential equations (IDEs) and VFDEs of other type appearing in practice. In particular, the theory covers the existing dissipativity results of DDEs with a wide variety of delay arguments such as constant delays, bounded and unbounded vary delays, discrete and distributed delays and so on.
(b) It is proved that the backward Euler method can inherit the dissipativity of the underlying system. Theorem 4.1 and Theorem 4.2 show the close relationship between the absorbing set of VFDEs and that of the numerically discrete system generated by the backward Euler method. It implies that the radius of the absorbing set of the discrete system approaches to that of the underlying system as the stepsize approaches to zero. On the contrary, most of the existing dissipativity results of numerical methods for VFDEs are independent of the size of the absorbing set of the underlying system.
This paper is organized as follows. In Section 2, some basic concepts for nonlinear VFDEs in Banach space are presented. In Section 3, some sufficient conditions for the dissipativity of nonlinear VFDEs are given. In Section 4, it is shown that the backward Euler method can inherit the dissipativity of the underlying system.
2 Some concepts
Let X be a real or complex Banach space with the norm \(\ \cdot \\). For any given closed interval \(I\subset\mathbb{R}\), let the symbol \(C_{X}(I)\) denote a Banach space consisting of all continuous mappings \(x: I\rightarrow X\), on which the norm is defined by \(\x\_{\infty}=\max_{t\in I}\x(t)\\).
For simplicity, we use the symbol \(\mathcal{A}(\alpha,\beta,\gamma,\mu_{1},\mu_{2})\) to denote the problem class consisting of all problems (2.1) satisfying condition (2.2).
3 Dissipativity of nonlinear Volterra functional differential equations
Definition 3.1
[6]
The evolutionary equation (2.1) is said to be dissipative in X if there is a bounded set \(\mathcal{B}\subset X\) such that for all bounded sets \(\varPsi \subset X\) there is a time \(t_{0}=t_{0}(\varPsi )\) such that for all initial functions \(\varphi(t)\) contained in Ψ, the corresponding solution \(y(t)\) is contained in \(\mathcal{B}\) for all \(t\geq t_{0}\). \(\mathcal{B}\) is called an absorbing set in X.
Lemma 3.2
Equation (2.2) implies that \(\gamma(t) \geq0\) and \(\beta(t) \geq0\).
Proof
Lemma 3.3
Proof
The last two inequalities of (3.3) imply that (2.10) and (2.12) of [16] hold. The conclusion follows from Theorem 2.3 of [16] directly. □
Theorem 3.4
Proof
Remark 3.5
Specializing Theorem 3.4 to Hilbert spaces, we can obtain the corresponding result which is in accordance with that obtained in [16].
Remark 3.6
Specializing Theorem 3.4 to DDEs in Hilbert spaces with constant delays and \(\alpha(t)\equiv\alpha\), \(\beta(t)\equiv\beta\), \(\gamma(t)\equiv\gamma\), we can obtain the corresponding result which is in accordance with that obtained in [6].
Remark 3.7
Specializing Theorem 3.4 to ODEs in Euclidean spaces with \(\alpha(t)\equiv\alpha\), \(\gamma(t)\equiv\gamma\), we can obtain the corresponding result which is in accordance with that obtained in [4].
Remark 3.8
Theorem 3.4 covers most of the existing dissipativity results of DDEs with a wide variety of delay arguments such as constant delays [6], bounded varying delays [10] and unbounded varying delays [9], discrete and distributed delays [11, 12] and so on. In brief, Theorem 3.4 provides a unified theoretical treatment for dissipativity analysis to ordinary differential equations (ODEs), delay differential equations (DDEs), integrodifferential equations (IDEs) and VFDEs of other type appearing in practice.
4 Dissipativity of the backward Euler method
Theorem 4.1
Theorem 4.2
Proof
Remark 4.3

In the case of ODEs, that is \(\beta=0\), for any given \(\epsilon >0\), the ball \(B (0,\sqrt{\frac{\gamma}{\alpha}+\epsilon} )\) is an absorbing set of the discrete system as well as the underlying system. The absorbing set is independent of the stepsize of the backward Euler method.

For fixed \(\beta>0\), \(h>0\), if \(\alpha\) is sufficiently large, then the difference between the radius of the absorbing set of the numerically discrete system and that of the underlying system is sufficiently small.

Notice that \(\frac{1h\alpha}{1h(\alpha+\beta)}\rightarrow1\) as \(h\rightarrow0\), hence given any \(\epsilon>0\), there exists \(h_{0}=h_{0}(\epsilon)\) such that for \(h< h_{0}\) the ball \(B (0,\sqrt{\frac {\gamma}{(\alpha+\beta)}+\epsilon} )\) is an absorbing set of the discrete system. In other words, the radius of the absorbing set of the discrete system approaches to that of the underlying system as the stepsize approaches to zero.
Declarations
Acknowledgements
This work is supported by the National Natural Science Foundation of China (No. 11171352).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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