 Research
 Open Access
 Published:
Dissipativity of the backward Euler method for nonlinear Volterra functional differential equations in Banach space
Advances in Difference Equations volume 2015, Article number: 128 (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.
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.
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)\\).
Consider the following initial value problem (IVP) [17]:
where a, τ are constants, \(0\leq\tau \leq+\infty\), \(\varphi\in C_{X}[a\tau,a]\) is a given initial function, \(f:[a,+\infty)\times X\times C_{X}[a\tau, +\infty)\rightarrow X\) is a given continuous mapping satisfying the conditions
where
Here \(\alpha(t)\), \(\beta(t)\), \(\gamma(t)\) are continuous functions, \(\mu_{1}(t)\) and \(\mu_{2}(t)\) satisfy
Define
We always assume that problem (2.1) has a unique solution on the interval \([a\tau,+\infty)\). Condition (2.2) implies that the mapping \(f(t,\psi(t),\psi)\) is independent of the values of the function \(\psi(\xi)\) with \(t<\xi\leq b\), i.e., \(f(t,\psi(t),\psi)\) is a Volterra functional.
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).
For the special case where X is a Hilbert space with the inner product \(\langle\cdot,\cdot\rangle\) and the corresponding norm \(\\cdot\\), condition (2.2) is equivalent to
The dissipativity analysis of (2.6) can be found in [14, 16].
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
Setting \(u \equiv0\) in (2.2) we obtain
which yields
as \(\lambda\rightarrow0\). Let \(\psi(t)\equiv0\). It follows from (3.1) that \(\gamma(t)\geq0\). If there is \(\tilde{t}\geq a\) such that \(\beta(\tilde {t})<0\), it is easy to find a function \(\psi\in C_{X}[a\tau,+\infty)\) which satisfies
which contradicts (3.1). Therefore, \(\beta(t)\geq0\).
For a continuous realvalued function \(y(t)\) of a real variable, the Dini derivatives \(D^{+}y(t)\) and \(D_{}y(t)\) are defined as
□
Lemma 3.3
If \(u(t)\geq0\), \(t\in(\infty,+\infty)\), and
where \(\psi(t)\) is bounded and continuous for \(t\leq a\), continuous functions \(\gamma(t)\geq0\), \(\beta(t)\geq0\) and \(\alpha(t)<0\) for \(t\in [a,+\infty)\), \(\tau(t)\geq0\) and
Then, for any given \(\epsilon>0\), there exists \(\hat{t}=\hat {t}(G,\epsilon)>a\) such that
where
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
Suppose problem \(\mbox{(2.1)} \in\mathcal{A}(\alpha,\beta,\gamma,\mu _{1},\mu_{2})\) and that
Then, for any given \(\epsilon>0\), there exists \(\check{t}=\check{t}(\bar {\varphi},\epsilon)\) such that
where \(\gamma^{*}=\sup_{t\geq a}\gamma(t)\), \(\bar{\varphi}=\sup_{t\leq a}\ \varphi(t)\^{2}\). Hence the system is dissipative with an absorbing set \(B=B(0,\sqrt{\gamma^{*}/\sigma+\epsilon})\).
Proof
By the definition of Dini derivative, we have
Applying Lemma 4.6.2 of [18], we see that the limits
exist. Lemma 4.6.3 of [18] tells us that
Then (3.7) and (3.8) together imply that \(D_{}(\ y(t)\^{2})=D^{+}(\y(t)\^{2})\). Let \(u(t)=\y(t)\^{2}\). Therefore, \(u'(t)\) exists, and
Using condition (2.2), we have
that is,
The desired result follows from Theorem 3.3. The proof is complete. □
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.
Dissipativity of the backward Euler method
For simplicity, from now on we assume that
Theorem 3.4 can be rewritten as follows.
Theorem 4.1
Suppose problem \(\mbox{(2.1)} \in\mathcal{A}(\alpha,\beta,\gamma,\mu _{1},\mu_{2})\) and that
Then, for any given \(\epsilon>0\), there exist \(\check{t}=\check{t}(\bar {\varphi},\epsilon)\), \(\bar{\varphi}=\sup_{t\leq a}\\varphi(t)\^{2}\) such that
The backward Euler method applied to (2.1) gives
where \(\pi^{h}\) is an appropriate interpolation operator which approximates to the exact solution \(y(t)\) on the interval \([a\tau, b]\), \(h>0\) is the stepsize, \(y_{n}\) is an approximation to the exact solution \(y(t_{n})\) with \(t_{n}=a+nh\).
Noting that the backward Euler method for ODEs is of order one, we can use the following piecewise linear interpolation:
Theorem 4.2
Assume that problem \(\mbox{(2.1)} \in\mathcal{A}(\alpha,\beta,\gamma ,\mu_{1},\mu_{2})\) and that
Let \(\{y_{n}\}\) be the sequence of numerical solutions obtained by (4.1)(4.2). Then, for any given \(\epsilon>0\), there exists \(n_{0}=n_{0}(\bar{\varphi},\epsilon)\) such that
Proof
It follows from (4.1) that
Using (2.2), we have
It follows from (4.3) and (4.4) that
In view of (4.2), we have
where we used the following inequality:
A combination of (4.5) and (4.6) leads to
For simplicity, for any given nonnegative integer n, we write
We now consider two cases:
In the case of (a), it follows from (4.7) that
In the case of (b), it follows from (4.7) that
To summarize both of the two cases, we have shown that
which yields
where
Considering \(Q_{n}=\y_{n}\^{2}\) or \(Q_{n}\ne\y_{n}\^{2}\) and inserting (4.10) repeatedly, we obtain
Therefore, for any given \(\epsilon>0\), there exists \(n_{0}=n_{0}(\bar {\varphi},\epsilon)\) such that
This completes the proof of the theorem. □
Remark 4.3
Theorem 4.1 and Theorem 4.2 show the close relationship between the absorbing set of the underlying system and that of the numerically discrete system generated by the backward Euler method. 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. It is obvious that the radius of the absorbing set of the discrete system is longer than that of the underlying system because of \(\frac{1h\alpha}{1h(\alpha+\beta)}\geq1\). Furthermore, Theorem 4.2 implies the following facts.

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.
References
 1.
Hill, AT: Global dissipativity for Astable methods. SIAM J. Numer. Anal. 34, 119142 (1997)
 2.
Hill, AT: Dissipativity of RungeKutta methods in Hilbert spaces. BIT Numer. Math. 37, 3742 (1997)
 3.
Humphries, AR, Stuart, AM: Model problems in numerical stability theory for initial value problems. SIAM Rev. 36, 226257 (1994)
 4.
Humphries, AR, Stuart, AM: RungeKutta methods for dissipative and gradient dynamical systems. SIAM J. Numer. Anal. 31, 14521485 (1994)
 5.
Xiao, A: Dissipativity of general linear methods for dissipative dynamical systems in Hilbert spaces. Math. Numer. Sin. 22, 429436 (2000) (in Chinese)
 6.
Huang, C: Dissipativity of RungeKutta methods for dissipative systems with delays. IMA J. Numer. Anal. 20, 153166 (2000)
 7.
Huang, C: Dissipativity of oneleg methods for dissipative systems with delays. Appl. Numer. Math. 35, 1122 (2000)
 8.
Huang, C, Chang, Q: Dissipativity of multistep RungeKutta methods for dynamical systems with delays. Math. Comput. Model. 40(1112), 12851296 (2004)
 9.
Gan, S: Exact and discretized dissipativity of the pantograph equation. J. Comput. Math. 25(1), 8188 (2007)
 10.
Tian, H: Numerical and analytic dissipativity of the θmethod for delay differential equations with a bounded lag. Int. J. Bifurc. Chaos 14, 18391845 (2004)
 11.
Gan, S: Dissipativity of linear θmethods for integrodifferential equations. Comput. Math. Appl. 52, 449458 (2006)
 12.
Gan, S: Dissipativity of θmethods for nonlinear Volterra delayintegrodifferential equations. J. Comput. Appl. Math. 206, 898907 (2007)
 13.
Tian, H, Guo, N: Asymptotic stability, contractivity and dissipativity of oneleg θmethod for nonautonomous delay functional differential equations. Appl. Math. Comput. 203, 333342 (2008)
 14.
Wen, L: Numerical stability analysis for nonlinear Volterra functional differential equations in abstract spaces. Ph.D. thesis, Xiangtan University (2005) (in Chinese)
 15.
Wen, L, Li, S: Dissipativity of Volterra functional differential equations. J. Math. Anal. Appl. 324(1), 696706 (2006)
 16.
Wen, L, Yu, Y, Wang, W: Generalized Halanay inequalities for dissipativity of Volterra functional differential equations. J. Math. Anal. Appl. 347, 169178 (2008)
 17.
Li, S: Stability analysis of solutions to nonlinear stiff Volterra functional differential equations in Banach space. Sci. China Ser. A 48, 372387 (2005)
 18.
Li, S: Theory of Computational Methods for Stiff Differential Equations. Hunan Science and Technology Publisher, Changsha (1997) (in Chinese)
Acknowledgements
This work is supported by the National Natural Science Foundation of China (No. 11171352).
Author information
Affiliations
Corresponding author
Additional information
Competing interests
The author declares that he has no competing interests.
Rights and permissions
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.
About this article
Cite this article
Gan, S. Dissipativity of the backward Euler method for nonlinear Volterra functional differential equations in Banach space. Adv Differ Equ 2015, 128 (2015). https://doi.org/10.1186/s1366201504698
Received:
Accepted:
Published:
Keywords
 dissipativity
 Volterra functional differential equation
 Banach space
 backward Euler method