dissipativity of the backward euler method for nonlinear volterra functional differential equations in banach space

R E S E A R C H Open AccessDissipativity of the backward Euler method for nonlinear Volterra functional differential equations in Banach space Siqing Gan* * Correspondence: sqgan@csu.edu

R E S E A R C H Open Access

Dissipativity of the backward Euler

method for nonlinear Volterra functional

differential equations in Banach space

Siqing Gan*

* Correspondence:

sqgan@csu.edu.cn

School of Mathematics and

Statistics, Central South University,

Changsha, Hunan 410083, China

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

Keywords: dissipativity; Volterra functional differential equation; Banach space;

backward Euler method

1 Introduction

Many dynamical systems are characterized by the property of possessing a bounded ab-sorbing 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 so-lutions 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 s considerable process has been made in dissipativity analysis of numer-ical methods The papers [–] focus on the numernumer-ical methods for ordinary differential equations For the delay differential equations (DDEs) with constant delay, sufficient con-ditions for the dissipativity of analytical and numerical solutions are presented in [–] Since that, the analysis is extended to DDEs with variable lags [, ] and Volterra func-tional differential equations [–]

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 dissipa-tivity of nonlinear VFDEs in Banach space and their numerical discretization

The main contributions of this paper could be summarized as follows

© 2015 Gan; licensee Springer This article is distributed under the terms of the Creative Commons Attribution 4.0 Interna-tional 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

(a) Sufficient conditions for the dissipativity of nonlinear VFDEs in Banach space are de-rived 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 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 un-derlying system Theorem . and Theorem . show the close relationship between the

absorbing set of VFDEs and that of the numerically discrete system generated by the

back-ward 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 , some basic concepts for nonlinear VFDEs in Banach space are presented In Section , some sufficient conditions for the

dissipativity of nonlinear VFDEs are given In Section , 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 ·  For any given closed interval

I ⊂ R, let the symbol C X (I) denote a Banach space consisting of all continuous mappings

x : I → X, on which the norm is defined by x∞= maxt ∈I x(t).

Consider the following initial value problem (IVP) []:



y(t) = f (t, y(t), y), t ≥ a,

where a, τ are constants,  ≤ τ ≤ +∞, ϕ ∈ C X [a – τ , a] is a given initial function, f :

[a, +∞) × X × C X [a – τ , +∞) → X is a given continuous mapping satisfying the

condi-tions



 – λα(t)

D f (, t, u, ψ)

≤ D f (λ, t, u, ψ) + λ



γ (t) + β(t) max

t –μ(t)≤ξ≤t–μ(t)

ψ (ξ )

,

∀λ ≥ , t ∈ [a, +∞), u ∈ X, ψ ∈ C X [a – τ , +∞), (.) where

D f (λ, t, u, ψ) =u – λf (t, u, ψ)

Here α(t), β(t), γ (t) are continuous functions, μ(t) and μ(t) satisfy

≤ μ (t) ≤ μ (t) ≤ t – a + τ, ∀t ∈ [a, +∞). (.)

μ() = inf

a ≤t<∞ μ(t)≥ , μ() (ξ, ξ) = inf

ξ≤t≤ξ



t – μ(t)

≥ a – τ,

We always assume that problem (.) has a unique solution on the interval [a – τ , +∞).

Condition (.) implies that the mapping f (t, ψ(t), ψ) is independent of the values of the

function ψ(ξ ) with t < ξ ≤ b, i.e., f (t, ψ(t), ψ) is a Volterra functional.

For simplicity, we use the symbolA(α, β, γ , μ, μ) to denote the problem class consist-ing of all problems (.) satisfyconsist-ing condition (.)

For the special case where X is a Hilbert space with the inner product·, · and the cor-responding norm · , condition (.) is equivalent to

 Re

u , f (t, u, ψ)

≤ γ (t) + α(t)u+ β(t) max

t –μ(t)≤ξ≤t–μ(t)

ψ (ξ )

,

∀t ∈ [a, +∞), u ∈ X, ψ ∈ C X [a – τ , +∞) (.) The dissipativity analysis of (.) can be found in [, ]

3 Dissipativity of nonlinear Volterra functional differential equations

Definition .[] The evolutionary equation (.) is said to be dissipative in X if there

is a bounded setB ⊂ X such that for all bounded sets Ψ ⊂ X there is a time t= t(Ψ )

such that for all initial functions ϕ(t) contained in Ψ , the corresponding solution y(t) is

contained inB for all t ≥ t.B is called an absorbing set in X.

Lemma . Equation (.) implies that γ (t) ≥  and β(t) ≥ .

Proof Setting u≡  in (.) we obtain

≤ λf (t, , ψ)

+ γ (t) + β(t) max

t –μ(t)≤ξ≤t–μ(t)

ψ (ξ ) ,

∀λ > , t ∈ [a, +∞), ψ ∈ C X [a – τ , +∞),

which yields

≤ γ (t) + β(t) max

t –μ(t)≤ξ≤t–μ(t)

ψ (ξ )

, ∀t ∈ [a, +∞), ψ ∈ C X [a – τ , +∞) (.)

as λ →  Let ψ(t) ≡  It follows from (.) that γ (t) ≥  If there is ˜t ≥ a such that β(˜t) < ,

it is easy to find a function ψ ∈ C X [a – τ , +∞) which satisfies

γ (˜t) + β(˜t) max

˜t–μ(˜t)≤ξ≤˜t–μ(˜t)

ψ (ξ )

< ,

which contradicts (.) Therefore, β(t)≥ 

For a continuous real-valued function y(t) of a real variable, the Dini derivatives D+y (t) and D–y (t) are defined as

D+y (t) = lim sup



y (t + δ) – y(t)

δ and D–y (t) = lim inf

δ

y (t + δ) – y(t)

Lemma . If u (t) ≥ , t ∈ (–∞, +∞), and

u(t) ≤ γ (t) + α(t)u(t) + β(t) sup

t –τ (t)≤ξ≤t u (ξ ), t ≥ a,

u (t) = ψ (t), t ≤ a,

(.)

where ψ (t) is bounded and continuous for t ≤ a, continuous functions γ (t) ≥ , β(t) ≥ 

and α (t) <  for t ∈ [a, +∞), τ(t) ≥  and

lim

t→+∞



t – τ (t)

a ≤t<+∞ α (t) < , sup

a ≤t<+∞

β (t)

|α(t)|< . (.)

Then , for any given > , there exists ˆt = ˆt(G, ) > a such that

u (t)≤γ∗

where

γ∗= sup

a ≤t<+∞ γ (t), G= sup

–∞<ξ≤aψ (ξ )

Proof The last two inequalities of (.) imply that (.) and (.) of [] hold The

con-clusion follows from Theorem . of [] directly 

Theorem . Suppose problem(.)∈A(α, β, γ , μ, μ) and that

lim

t→+∞



t – μ(t)

a ≤t<+∞ α (t) < , sup

a ≤t<+∞

β (t)

|α(t)|< . (.)

Then , for any given > , there exists ˇt = ˇt( ¯ϕ, ) such that

y (t)

≤γ∗

where γ∗= supt ≥a γ (t), ¯ϕ = sup t ≤a ϕ(t) Hence the system is dissipative with an

absorb-ing set B = B(,√

γ∗/σ + ).

Proof By the definition of Dini derivative, we have

D–y (t)

= lim inf

δ→–

y(t + δ)–y (t)

δ

= lim inf

δ→–

y(t + δ) – y(t)

δ y (t + δ)+y (t),

D+y (t)

= lim inf

δ→+

y(t + δ)–y(t)

δ

= lim inf

δ→+

y(t + δ) – y(t)

δ y (t + δ)+y (t).

(.)

Applying Lemma .. of [], we see that the limits

lim

δ→–

y(t) + δy(t)  – y(t)

δ→+

y(t) + δy(t)  – y(t)

δ

exist Lemma .. of [] tells us that

lim inf

δ→–

y(t + δ) – y(t)

δ→–

y(t) + δy(t) – y(t)

lim inf

δ→+

y(t + δ) – y(t)

δ→+

y(t) + δy(t) – y(t)

(.)

Then (.) and (.) together imply that D–(y(t)) = D+(y(t)) Let u(t) = y(t)

Therefore, u(t) exists, and

u(t) = lim

δ→–

y(t + δ)–y(t)

δ

= lim

δ→–

y(t) + δy(t) + o(δ)–y(t) + δy(t)+y(t) + δy(t)–y(t)

δ

= lim

δ→–

y(t) + δy(t)–y(t)

δ

= lim

δ→–

y(t) + δf (t, y(t), y)–y(t)

Using condition (.), we have

u(t)≤ lim

δ→–

( + δα(t))y(t)+ δ(γ (t) + β(t) max t –μ(t)≤ξ≤t–μ(t) y(ξ)) –y(t)

δ

= γ (t) + α(t)y (t)

+ β(t) max

t –μ(t)≤ξ≤t–μ(t)

y (ξ ) , that is,

u(t) ≤ γ (t) + α(t)u(t) + β(t) max

t –μ(t)≤ξ≤t–μ(t) u (ξ ). (.) The desired result follows from Theorem . The proof is complete 

Remark . Specializing Theorem . to Hilbert spaces, we can obtain the corresponding

result which is in accordance with that obtained in []

Remark . Specializing Theorem . to DDEs in Hilbert spaces with constant delays

and α(t) ≡ α, β(t) ≡ β, γ (t) ≡ γ , we can obtain the corresponding result which is in

ac-cordance with that obtained in []

Remark . Specializing Theorem . to ODEs in Euclidean spaces with α(t) ≡ α,

γ (t) ≡ γ , we can obtain the corresponding result which is in accordance with that

ob-tained in []

Remark . Theorem . covers most of the existing dissipativity results of DDEs with

a wide variety of delay arguments such as constant delays [], bounded varying delays

[] and unbounded varying delays [], discrete and distributed delays [, ] and so on.

In brief, Theorem . provides 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

4 Dissipativity of the backward Euler method

For simplicity, from now on we assume that

γ (t) ≡ γ , α (t) ≡ α, β (t) ≡ β, t ∈ [a, +∞).

Theorem . can be rewritten as follows

Theorem . Suppose problem(.)∈A(α, β, γ , μ, μ) and that

lim

t→+∞



t – μ(t)

→ +∞, α + β < .

Then , for any given > , there exist ˇt = ˇt( ¯ϕ, ), ¯ϕ = sup t ≤a ϕ(t)such that

y (t) ≤ γ

–(α + β) + , t > ˇt.

The backward Euler method applied to (.) gives



y n+= y n + hf (t n+, y n+, y h(·)), n = , , , ,

y h (t) = π h (t, ϕ, y, y, , y n+), a – τ ≤ t ≤ t n+, (.)

where π his an appropriate interpolation operator which approximates to the exact

solu-tion y(t) on the interval [a – τ , b], h >  is the stepsize, y nis 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 fol-lowing piecewise linear interpolation:

y h (t) =





h [(t i+– t)y i + (t – t i )y i+], t i ≤ t ≤ t i+,

Theorem . Assume that problem(.)∈A(α, β, γ , μ, μ) and that

lim

t→+∞



t – μ(t)

→ +∞, α + β < .

Let {y n } be the sequence of numerical solutions obtained by (.)-(.) Then, for any given

> , there exists n= n(¯ϕ, ) such that

y n ≤

γ

–(α + β)·  – hα

 – h(α + β) + , n ≥ n

Proof It follows from (.) that

y – hf

t , y , y h(·)

Using (.), we have

y n+– hf

t n+, y n+, y h(·)

≥ ( – hα)y n+– h



t n+–μ(t n+ )≤ξ≤tn+–μ(t n+ )

y h (ξ )

It follows from (.) and (.) that

( – hα)y n+≤ y n+ h



t n+–μ(t n+ )≤ξ≤tn+–μ(t n+ )

y h (ξ )

In view of (.), we have

max

t n+–μ(t n+ )≤ξ≤tn+–μ(t n+ )

y h (ξ )

≤ max max

≤i≤n+y i, max

a –τ≤t≤aϕ (t)

where we used the following inequality:

( – δ)y i + δy i+≤ ( – δ)y i+ δy i+≤ maxy i,y i+

A combination of (.) and (.) leads to

( – hα)y n+≤ y n+ h



γ + β max max

≤i≤n+y i, max

a –τ≤t≤aϕ (t)

For simplicity, for any given nonnegative integer n, we write

Q n= max max

≤i≤ny i, max

a –τ≤t≤aϕ (t)

, n≥ ,

Q= max

a –τ≤t≤aϕ (t)

We now consider two cases:

(a) max max

≤i≤n+y i, max

a –τ≤t≤aϕ (t)

=y n+,

(b) max max

≤i≤n+y i, max

a –τ≤t≤aϕ (t)

= y n+

In the case of (a), it follows from (.) that

y n+≤ hγ

 – h(α + β)+



In the case of (b), it follows from (.) that

y n+≤ hγ

 – hα+



 – hα



y n+ hβQ n

To summarize both of the two cases, we have shown that

y n+≤ max



hγ

 – h(α + β),

hγ

 – hα

 + max





 – h(α + β),

 + hβ

 – hα



Q n,

which yields

y n+≤ hγ

 – h(α + β)+

 + hβ

 – hα Q n =: d+ dQ n, (.)

where

 – h(α + β), d=

 + hβ

 – hα< .

Considering Q n=y nor Q n = y nand inserting (.) repeatedly, we obtain

y n≤ d



 + d+· · · + d n–





+ d nQ

≤ d

 – d + d

n

Q= γ

–(α + β)

 – hα

 – h(α + β) + d

n

Therefore, for any given > , there exists n= n(¯ϕ, ) such that

y n ≤

γ

–(α + β)·  – hα

 – h(α + β) + , n ≥ n

Remark . Theorem . and Theorem . show the close relationship between the

ab-sorbing 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 –h(α+β) –hα ≥  Furthermore,

Theo-rem . implies the following facts

• In the case of ODEs, that is β = , for any given > , the ball B(,



γ

–α + )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 β > , h > , if |α| 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–h(α+β) –hα →  as h → , hence given any > , there exists h= h()such

that for h < hthe ball B(,

γ

–(α+β) + )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

Competing interests

The author declares that he has no competing interests.

Acknowledgements

This work is supported by the National Natural Science Foundation of China (No 11171352).

Received: 9 January 2015 Accepted: 13 April 2015

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of numerical methods for VFDEs are independent of the size of the absorbing set of the

underlying system... + )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 β > ,... )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