1 Introduction
In 1966, in order to discuss the stability of the zero solution of the delay differential equation
Halanay introduced the following lemma (see [1] p. 378).
$$u'(t)=Au(t)+Bu(t-\tau), \quad \tau>0, $$
Lemma 1.1
(Basic Halanay inequality)
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Assume that
\(\tau\geq0\)
and
\(v(t)\)
is a positive function defined on
\([t_{0}-\tau,+\infty)\), with derivative
\(v'(t)\)
on
\([t_{0},+\infty)\). If
where
\(\alpha>\beta>0\), then there exist
\(\gamma>0\)
and
\(k>0\)
such that
$$v'(t)\leq-\alpha v(t)+\beta\sup_{t-\tau\leq s\leq t}v(s),\quad t\geq t_{0}, $$
$$v(t)\leq k \exp\bigl(-\gamma(t-t_{0})\bigr), \quad t\geq t_{0}. $$
The Halanay inequality became a powerful tool in the stability theory of delay differential equations, therefore many authors improved or generalized it to more general type and used it for investigating the stability and dissipativity of various functional differential equations. We refer the reader to the papers, for instance, of Baker and Tang [2], Agarwal, Kim and Sen [3, 4], Baker [5], Liz and Trofimchuk [6], Tian [7], Wen, Yu and Wang [8, 9], Liu et al. [10], Wang [11], Hien et al. [12], and Gan [13].
On the other hand, many interesting problems in physics and engineering are modeled by dissipative dynamical systems. These systems are characterized by the property of possessing a bounded absorbing set, which all trajectories enter in a finite time and thereafter remain inside of. In the study of dissipative systems, this asymptotic behavior of the system is of interest and important (see [14]). In 1994, Humphries and Stuart [15] first studied the analytical and numerical dissipativity of initial value problems (IVPs) in ODEs. Hereafter, a number of results on the analytical and numerical dissipativity with respect to various types of differential equations are presented (such as found in [16‐21]).
In this paper, we first present a more general Halanay-type inequality in Sect. 2. Then, in Sect. 3, we use this inequality to discuss the analytical dissipativity of two classes of nonlinear neutral delay integro-differential equations (NDIDEs) and some sufficient conditions which ensure the systems to be dissipative are given. Finally, the paper ends with a conclusion.
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2 The generalized Halanay inequality
For simplicity of presentation, we denote \(f^{[t_{1},t_{2}]}:=\sup_{t_{1} \leq\xi\leq t_{2}}f(\xi)\) and \(f^{[t_{1},+\infty)}:=\sup_{\xi\geq t_{1}}f(\xi)\) for a bounded function f.
Theorem 2.1
Assume that
\(\tau\geq0\)
and
\(u(t), w(t)\)
are non-negative functions defined on
\([t_{0}-\tau,+\infty)\), with derivative
\(u'(t)\)
on
\([t_{0},+\infty)\). If
for
\(t\geq t_{0}\)
and
and there exists a constant
\(\sigma>0\)
such that
Then, for
\(t\geq t_{0}\), we have
where
\(R_{1}(t), R_{2}(t), -A(t), B(t), C(t), D(t), F(t), G(t), H(t)\)
are non-negative, continuous and bounded functions defined on
\([t_{0}, +\infty)\);
and
\(0< H(t)\leq H_{0}<1\), \(\gamma^{*}=\gamma_{1}+\frac{C_{0}+D_{0}}{1-H_{0}} \gamma _{2}\). The constant
\(\mu^{\ast}> 0\)
is defined as
$$ \textstyle\begin{cases}u'(t)\leq R_{1}(t)+A(t) u(t)+B(t) u^{[t-\tau,t]}+C(t) w(t)+D(t)w^{[t-\tau,t]}, \\ w(t)\leq R_{2}(t)+F(t) u(t)+G(t) u^{[t-\tau,t]}+H(t)w^{[t-\tau,t]}, \end{cases} $$
(2.1)
$$ w^{[t_{0}-\tau,t_{0}]} \leq \frac{r_{2}}{1-H_{0}}+\frac{F_{0}+G_{0}}{1-H_{0}} u^{[t_{0}-\tau,t_{0}]}, $$
(2.2)
$$ A(t)+B(t)+\bigl(C(t)+D(t)\bigr)\frac{F_{0}+G_{0}}{1-H_{0}}\leq-\sigma,\quad \forall t\geq t_{0}. $$
(2.3)
$$ \textstyle\begin{cases}u(t)\leq\frac{\gamma^{*}}{\sigma}+\phi e^{-\mu^{\ast }(t-t_{0})},\\ w(t)\leq\frac{\gamma_{2}}{1-H_{0}} +\frac{F_{0}+G_{0}}{1-H_{0}} \frac{\gamma ^{*}}{\sigma}+\frac{F_{0}+G_{0}e^{\mu^{\ast}\tau}}{1-H_{0} e^{\mu^{\ast}\tau}} \phi e^{-\mu^{\ast}(t-t_{0})}, \end{cases} $$
(2.4)
$$\textstyle\begin{cases} F_{0}=F^{[t_{0},+\infty)}, \qquad G_{0}=G^{[t_{0},+\infty)},\qquad C_{0}=C^{[t_{0},+\infty)},\qquad D_{0}=D^{[t_{0},+\infty)},\\ H_{0}=H^{[t_{0},+\infty)},\qquad \gamma_{1}=R_{1}^{[t_{0},+\infty)}, \qquad \gamma _{2}=R_{2}^{[t_{0},+\infty)},\qquad \phi= u^{[t_{0}-\tau,t_{0}]}, \end{cases} $$
$$\mu^{\ast}=\inf_{t \geq t_{0}} \biggl\{ \mu(t): \mu(t)+A(t)+B(t)e^{\mu (t)\tau}+\bigl(C(t)+D(t)e^{\mu(t)\tau}\bigr) \frac{F_{0}+G_{0} e^{\mu(t) \tau}}{1-H_{0} e^{\mu(t)\tau}}=0 \biggr\} . $$
Specially, if
\(R_{1}(t)=R_{2}(t)\equiv0\), (2.4) degenerates into following form:
$$ \textstyle\begin{cases} u(t)\leq\phi e^{-\mu^{\ast}(t-t_{0})},\\ w(t)\leq\frac{F_{0}+G_{0} e^{\mu^{\ast}\tau}}{1-H_{0} e^{\mu^{\ast}\tau}} \phi e^{-\mu^{\ast}(t-t_{0})}, \end{cases}\displaystyle t\geq t_{0}. $$
(2.5)
Proof
First, if \(\tau=0\), from the second formula of (2.1), we have
Substituting (2.6) into the first formula of (2.1) shows that (2.1) degenerates into a differential inequality
where
$$ w(t)\leq\frac{R_{2}(t)}{1-H(t)}+\frac{F(t)+G(t)}{1-H(t)}u(t). $$
(2.6)
$$u'(t)\leq R(t)+\widetilde{A}(t) u(t),\quad t\geq t_{0}, $$
$$\textstyle\begin{cases}R(t)=R_{1}(t)+(C(t)+D(t))\frac{R_{2}(t)}{1-H(t)},\\ \widetilde{A}(t)=A(t)+B(t)+(C(t)+D(t))\frac{F(t)+G(t)}{1-H(t)}. \end{cases} $$
Noting the condition (2.3), it is can be proved that
The combination of this formula and (2.6) shows that (2.4) holds with
It is obvious that \(\mu^{\ast}>0\) under the assumption (2.3).
$$u(t)\leq\frac{\gamma^{*}}{\sigma} \biggl( 1-\exp \biggl( \int_{t_{0}}^{t} \widetilde{A} (s) \,ds \biggr) \biggr)+u(t_{0})\exp \biggl( \int_{t_{0}}^{t} \widetilde{A} (s) \,ds \biggr). $$
$$\mu^{\ast}=\inf_{t \geq t_{0}} \biggl\{ -A(t)-B(t)- \bigl(C(t)+D(t)\bigr)\frac {F(t)+G(t)}{1-H_{0}} \biggr\} . $$
In the following we assume that \(\tau>0\). For any given \(t\in[t_{0}, +\infty)\), we define function \(E(\mu)\) on \([0,\frac{1}{\tau}\ln\frac {1}{H_{0}})\) by
From (2.7) we can see that
Therefore, there exists a unique \(\mu\in(0,\frac{1}{\tau}\ln\frac {1}{H_{0}})\) such that
which defines an implicit function \(\mu(t)\) for \(t\geq t_{0}\). It is obvious that \(\mu^{\ast}\geq0\). Now we prove that \(\mu^{\ast}> 0\).
$$ E(\mu):=\mu+A(t)+B(t)e^{\mu\tau}+ \bigl(C(t)+D(t)e^{\mu\tau}\bigr)\frac{F_{0}+G_{0} e^{\mu \tau}}{1-H_{0} e^{\mu\tau}}. $$
(2.7)
$$E(0)< 0,\qquad \lim_{\mu\rightarrow\frac{1}{\tau}\ln\frac{1}{H_{0}}-0} E(\mu) =+\infty,\qquad E'(\mu)>0. $$
$$ \mu+A(t)+B(t)e^{\mu\tau}+\bigl(C(t)+D(t)e^{\mu\tau} \bigr)\frac{F_{0}+G_{0} e^{\mu \tau}}{1-H_{0} e^{\mu\tau}}=0, $$
(2.8)
In fact, if this is not true. Let \(\widetilde{H}_{0}\) satisfying \(0< H_{0}< \widetilde{H}_{0} <1\) and let \(0< \varepsilon_{1} <\min\{\frac{\sigma}{2}, -\frac{1}{\tau} \ln\widetilde {H}_{0}, \frac{1}{\tau}\ln (\frac{\sigma}{2Q}+1)\}\), where
and \(B_{0}=B^{[t_{0},+\infty)}\).
$$Q=B_{0} +\frac{C_{0}\widetilde{H}_{0} (G_{0} + F_{0} H_{0})+D_{0} \widetilde {H}_{0}(F_{0}+G_{0})+D_{0} G_{0}(1-H_{0})}{(\widetilde{H}_{0}-H_{0} )(1-H_{0})}, $$
Then there would exist \(t^{\ast}\geq t_{0}\) such that \(\hat{\mu}:=\mu (t^{*})<\varepsilon_{1}\) and
Substituting (2.3) into (2.9) gives
which is a contradiction.
$$ \hat{\mu}+A\bigl(t^{\ast}\bigr)+B \bigl(t^{\ast}\bigr)e^{\hat{\mu}\tau}+\bigl(C\bigl(t^{\ast }\bigr)+D \bigl(t^{\ast}\bigr)e^{\hat{\mu}\tau}\bigr)\frac{F_{0}+G_{0} e^{\hat{\mu} \tau}}{1-H_{0} e^{\hat{\mu}\tau}}=0. $$
(2.9)
$$\begin{aligned} 0&=\hat{\mu}+A\bigl(t^{\ast}\bigr)+B\bigl(t^{\ast} \bigr)e^{\hat{\mu}\tau }+\bigl(C\bigl(t^{\ast}\bigr)+D\bigl(t^{\ast} \bigr)e^{\hat{\mu}\tau}\bigr)\frac{F_{0}+G_{0} e^{\hat{\mu} \tau}}{1-H_{0} e^{\hat{\mu}\tau}} \\ &\leq\varepsilon_{1}-\sigma+ \biggl(B_{0}+ \frac{C_{0} (F_{0} H_{0}+G_{0})+D_{0}(F_{0}+G_{0})+D_{0} G_{0}e^{\varepsilon_{1}\tau}(1-H_{0}))}{(1-H_{0} e^{\varepsilon_{1}\tau})(1-H_{0})} \biggr) \bigl(e^{\varepsilon_{1}\tau}-1\bigr) \\ &\leq\varepsilon_{1}-\sigma+Q\bigl(e^{\varepsilon_{1}\tau}-1\bigr) < \varepsilon_{1}-\sigma+\frac{\sigma}{2}< 0, \end{aligned}$$
In order to verify (2.4), we first show that, for any \(\varepsilon>0\),
In fact, when \(t=t_{0}\), (2.10) is evident by using (2.2).
$$ \textstyle\begin{cases} u(t)< \frac{\gamma^{*}}{\sigma}+\varepsilon+\phi e^{-\mu ^{\ast}(t-t_{0})},\\ w(t)< \frac{\gamma_{2}}{1-H_{0}}+\frac{F_{0}+G_{0}}{1-H_{0}} (\frac{\gamma^{*}}{\sigma}+\varepsilon)+\frac{F_{0}+G_{0} e^{\mu \tau}}{1-H_{0} e^{\mu \tau }} \phi e^{-\mu^{\ast}(t-t_{0})}, \end{cases}\displaystyle t\geq t_{0}. $$
(2.10)
If we suppose (2.10) is not true for \(t>t_{0}\), then there would exist some \(\varepsilon_{0}>0\) and \(\varsigma>t_{0}\) such that when \(t< \varsigma\)
while when \(t=\varsigma\), at least one of the following two equalities is true:
and
$$ \textstyle\begin{cases} u(t)< \frac{\gamma^{*}}{\sigma}+\varepsilon_{0}+\phi e^{-\mu^{\ast}(t-t_{0})},\\ w(t)< \frac{\gamma_{2}}{1-H_{0}}+ \frac{G_{0}+F_{0}}{1-H_{0}} (\frac{\gamma^{*}}{\sigma}+\varepsilon_{0})+\frac{F_{0}+G_{0}e^{\mu^{\ast}\tau}}{1-H_{0} e^{\mu^{\ast}\tau}} \phi e^{-\mu^{\ast}(t-t_{0})}, \end{cases} $$
(2.11)
$$ u(\varsigma)=\frac{\gamma^{*}}{\sigma}+\varepsilon_{0}+\phi e^{-\mu^{\ast}(\varsigma-t_{0})} $$
(2.12)
$$ w(\varsigma)= \frac{\gamma_{2}}{1-H_{0}}+\frac{G_{0}+F_{0}}{1-H_{0}} \biggl( \frac{\gamma^{*}}{\sigma}+\varepsilon_{0}\biggr)+\frac{F_{0}+G_{0}e^{\mu^{\ast}\tau}}{1-H_{0} e^{\mu^{\ast}\tau}} \phi e^{-\mu^{\ast}(\varsigma-t_{0})}. $$
(2.13)
However, from the second formula of (2.1), when \(\varsigma-\tau \geq t_{0}\), we have
and, when \(\varsigma-\tau< t_{0}\), we have
Hence (2.14) and (2.15) show that (2.13) is not true. Therefore we need only consider the case that (2.12) holds and we shall obtain a contradiction. Set
Then \(z(t)>0\) for \(t<\varsigma\) and \(z(\varsigma)=0\) and \(z'(\varsigma)\leq0\). Hence from the first formula of (2.1) we have
If \(\varsigma-\tau\geq t_{0}\), it follows from (2.11), (2.12), (2.16) and the definition of \(\gamma^{*}\) that
From the definition of the function \(\mu(t)\), we have
Therefore, it is easy to see that
which substituting into (2.17) and noting the condition (2.3), gives
$$\begin{aligned} w(\varsigma)\leq{}& R_{2}(\varsigma)+F(\varsigma) u(\varsigma)+G( \varsigma)\sup_{\varsigma-\tau\leq\xi\leq \varsigma}u(\xi)+H(\varsigma)\sup_{\varsigma-\tau\leq\xi\leq \varsigma}w( \xi) \\ < {}& R_{2}(\varsigma)+F(\varsigma) \biggl( \frac{\gamma^{*}}{\sigma }+ \varepsilon_{0}+\phi e^{-\mu^{\ast}(\varsigma-t_{0})} \biggr)+G(\varsigma) \biggl( \frac{\gamma ^{*}}{\sigma}+\varepsilon_{0}+\phi e^{-\mu^{\ast}(\varsigma-\tau-t_{0})} \biggr) \\ &{}+H(\varsigma) \biggl(\frac{\gamma_{2}}{1-H_{0}}+\frac{F_{0}+G_{0}}{1-H_{0}} \biggl( \frac{\gamma^{*}}{\sigma}+\varepsilon_{0}\biggr)+\frac{F_{0}+G_{0}e^{\mu^{\ast}\tau}}{1-H_{0} e^{\mu^{\ast}\tau}} \phi e^{-\mu^{\ast}(\varsigma-\tau-t_{0})} \biggr) \\ ={}& R_{2}(\varsigma)+H(\varsigma)\frac{\gamma_{2}}{1-H_{0}}+ \biggl( F(\varsigma )+G(\varsigma)+H(\varsigma)\frac{F_{0}+G_{0}}{1-H_{0}} \biggr) \biggl( \frac{\gamma }{\sigma}+ \varepsilon_{0}\biggr) \\ &{}+ \biggl(F(\varsigma)+G(\varsigma)e^{\mu^{\ast}\tau} +\frac{F_{0}+G_{0}e^{\mu^{\ast}\tau}}{1-H_{0} e^{\mu^{\ast}\tau}}H_{0} e^{\mu^{\ast}\tau} \biggr)\phi e^{-\mu^{\ast }(\varsigma-t_{0})} \\ \leq{}&\frac{\gamma_{2}}{1-H_{0}}+\frac{F_{0}+G_{0}}{1-H_{0}} \biggl( \frac{\gamma ^{*}}{\sigma}+\varepsilon_{0}\biggr)+\frac{F_{0}+G_{0}e^{\mu^{\ast}\tau}}{1-H_{0} e^{\mu^{\ast}\tau}}\phi e^{-\mu^{\ast}(\varsigma-t_{0})}, \end{aligned}$$
(2.14)
$$\begin{aligned} w(\varsigma)\leq{}& R_{2}(\varsigma)+ F(\varsigma) u(\varsigma)+G( \varsigma)\max\Bigl\{ \sup_{t_{0}-\tau\leq\xi\leq t_{0}}u(\xi), \sup_{t_{0}\leq\xi\leq\varsigma}u( \xi)\Bigr\} \\ &{}+H(\varsigma)\max\Bigl\{ \sup_{t_{0}-\tau\leq\xi\leq t_{0}}w(\xi), \sup _{t_{0}\leq\xi\leq\varsigma}w(\xi)\Bigr\} \\ < {}& R_{2}(\varsigma)+F(\varsigma) \biggl(\frac{\gamma^{*}}{\sigma}+ \varepsilon_{0}+\phi e^{-\mu^{\ast}(\varsigma-t_{0})}\biggr)+G(\varsigma) \biggl( \frac{\gamma^{*}}{\sigma }+\varepsilon_{0}+\phi\biggr) \\ &{}+H(\varsigma) \biggl(\frac{\gamma_{2}}{1-H_{0}}+\frac{F_{0}+G_{0}}{1-H_{0}}\biggl( \frac{\gamma ^{*}}{\sigma}+\varepsilon_{0}\biggr)+\frac{F_{0}+G_{0}e^{\mu^{\ast}\tau}}{1-H_{0} e^{\mu^{\ast}\tau}} \phi \biggr) \\ \leq{}& \frac{\gamma_{2}}{1-H_{0}}+\frac{F_{0}+G_{0}}{1-H_{0}}\biggl( \frac{\gamma^{*}}{\sigma }+\varepsilon_{0}\biggr)+\frac{F_{0}+G_{0}e^{\mu^{\ast}\tau}}{1-H_{0} e^{\mu^{\ast}\tau}} \phi e^{-\mu^{\ast}(\varsigma-t_{0})}. \end{aligned}$$
(2.15)
$$v(t)=\frac{\gamma^{*}}{\sigma}+\varepsilon_{0}+\phi e^{-\mu^{\ast}(t-t_{0})},\qquad z(t)=v(t)-u(t). $$
$$\begin{aligned} z'( \varsigma)={}&v'(\varsigma)-u'(\varsigma) \\ \geq{}& {-}\phi \mu^{\ast} e^{-\mu^{\ast}(\varsigma-t_{0})} \\ &{}- \bigl( R_{1}(\varsigma)+A(\varsigma) u(\varsigma)+B(\varsigma )u^{[\varsigma-\tau,\varsigma]}+C(\varsigma) w(\varsigma)+D(\varsigma )w^{[\varsigma-\tau,\varsigma]} \bigr). \end{aligned}$$
(2.16)
$$\begin{aligned} z'(\varsigma)\geq{}& {-} \gamma^{*}-\biggl( \frac{\gamma^{*}}{\sigma}+\varepsilon_{0}\biggr) \biggl(A(\varsigma )+B( \varsigma)+\bigl(C(\varsigma)+D(\varsigma)\bigr)\frac{F_{0}+G_{0}}{1-H_{0}} \biggr) \\ &{}- \phi e^{-\mu^{\ast}(\varsigma-t_{0})} \\ &{}\times\biggl( \mu^{\ast}+A(\varsigma)+B( \varsigma)e^{\mu^{\ast}\tau}+\bigl(C(\varsigma )+D(\varsigma)e^{\mu^{\ast}\tau}\bigr) \frac{F_{0}+G_{0} e^{\mu^{\ast} \tau}}{1-H_{0} e^{\mu^{\ast}\tau}} \biggr). \end{aligned}$$
(2.17)
$$\mu(\varsigma)+A(\varsigma)+B(\varsigma)e^{\mu(\varsigma)\tau }+\bigl(C(\varsigma)+D( \varsigma)e^{\mu(\varsigma)\tau}\bigr)\frac{F_{0}+G_{0} e^{\mu (\varsigma) \tau}}{1-H_{0} e^{\mu(\varsigma)\tau}}=0. $$
$$\begin{aligned} &\mu^{\ast}+A(\varsigma)+B( \varsigma)e^{\mu^{\ast}\tau}+\bigl(C(\varsigma )+D(\varsigma)e^{\mu^{\ast}\tau}\bigr) \frac{F_{0}+G_{0} e^{\mu^{\ast} \tau}}{1-H_{0} e^{\mu^{\ast}\tau}} \\ &\quad= \mu^{\ast}-\mu(\varsigma)+B(\varsigma) \bigl(e^{\mu^{\ast}\tau}-e^{\mu (\varsigma)\tau} \bigr)+C(\varsigma) \biggl(\frac{F_{0}+G_{0} e^{\mu^{\ast} \tau}}{1-H_{0} e^{\mu^{\ast}\tau}}-\frac{F_{0}+G_{0} e^{\mu(\varsigma) \tau}}{1-H_{0} e^{\mu(\varsigma)\tau}} \biggr) \\ &\qquad{}+D(\varsigma) \biggl(\frac{F_{0}+G_{0} e^{\mu^{\ast} \tau}}{1-H_{0} e^{\mu^{\ast}\tau}}e^{\mu^{\ast}\tau}-\frac{F_{0}+G_{0} e^{\mu (\varsigma) \tau}}{1-H_{0} e^{\mu(\varsigma)\tau}}e^{\mu(\varsigma)\tau} \biggr) \\ &\quad\leq0, \end{aligned}$$
$$ w'(\varsigma)=v'(\varsigma)-u'( \varsigma)\geq\sigma\varepsilon_{0}>0. $$
(2.18)
If \(\varsigma-\tau< t_{0}\), it follows from (2.16) that
Thus we also can get (2.18) by simple derivation. This is in contradiction with our result \(w'(\varsigma)\leq0\). Therefore the inequality (2.10) must hold for any given \(\varepsilon>0\). Since \(\varepsilon>0\) is arbitrary, we let \(\varepsilon\rightarrow0\) and obtain (2.4), which completes the proof of Theorem 2.1. □
$$\begin{aligned} z'(\varsigma)\geq{}& {-}\phi\mu^{\ast} e^{-\mu^{\ast}(\varsigma-t_{0})}- R_{1}(\varsigma)-A(\varsigma) u(\varsigma )-B(\varsigma)\max\bigl\{ \phi, u^{[t_{0},\varsigma]}\bigr\} \\ &{}-C(\varsigma) w(\varsigma)-D(\varsigma)\max\bigl\{ w^{[t_{0}-\tau,t_{0}]}, w^{[t_{0},\varsigma]}\bigr\} . \end{aligned}$$
3 Dissipativity of two classes of nonlinear neutral functional differential equations
In this section, we consider several simple applications of Theorem 2.1 to the study of dissipativity for two classes of nonlinear neutral functional differential equations.
Let X be a real or complex, finite-dimensional or infinite-dimensional Hilbert space with the inner product \(\langle\cdot ,\cdot\rangle\) and the corresponding norm \(\Vert \cdot \Vert \).
3.1 Dissipativity of nonlinear neutral delay integro-differential equations (NNDIDEs)
Consider the IVPs in NNDIDEs as follows:
where τ are positive constant, the functions \(f:[t_{0},+\infty) \times X \times X \times X \times X \rightarrow X\), \(g:[t_{0},+\infty) \times[t_{0}-\tau,+\infty) \times X \rightarrow X\), \(\phi:[t_{0}-\tau,t_{0}] \rightarrow X\) are assumed to be continuous functions and for any \(t\geq t_{0}, y, u, v, w\in X\), f and g satisfy the conditions:
and
where \(-\alpha, \beta, \gamma_{1}, \gamma_{2}, \omega, \lambda, L_{y}, L_{u}, L_{v}, L_{w}\) are all non-negative real constants.
$$ \textstyle\begin{cases} y'(t)=f(t,y(t),y(t-{\tau}),y'(t-\tau),\int^{t}_{t-{\tau}}g(t,\xi,y(\xi ))\,d\xi), \quad t\geq t_{0},\\ y(t) =\phi(t),\qquad y'(t) =\phi'(t), \quad t_{0}-\tau \leq t\leq t_{0}, \end{cases} $$
(3.1)
$$ \textstyle\begin{cases} 2 \operatorname{Re}\langle f(t,y,u,v,w),y\rangle \leq \alpha \Vert y \Vert ^{2}+ \beta \Vert f(t,0,u,v,w) \Vert ^{2}, \\ \Vert f(t,y,u,v,w) \Vert ^{2} \leq \gamma_{1}+L_{y} \Vert y \Vert ^{2} +\omega \Vert f(t,0,u,v,w) \Vert ^{2}, \\ \Vert f(t,0,u,v,w) \Vert ^{2} \leq \gamma_{2}+L_{u} \Vert u \Vert ^{2}+L_{v} \Vert v \Vert ^{2}+L_{w} \Vert w \Vert ^{2}, \end{cases} $$
(3.2)
$$ \bigl\Vert g(t,\xi,u) \bigr\Vert \leq \lambda \Vert u \Vert ,\quad t - \tau \leq \xi \leq t, t\geq t_{0}, $$
(3.3)
Theorem 3.1
Let problem (3.1) satisfy (3.2) and (3.3) with
\(L_{v} \omega<1\), and initial value function
\(\phi(t)\)
satisfy
Let
\(y(t)\)
be the solution of (3.1). Assume that there exists a constant
\(\sigma>0\)
such that
Then
$$ \max_{t_{0}-\tau\leq t \leq t_{0}} \bigl\Vert \phi'(t) \bigr\Vert ^{2} \leq \frac{\gamma_{1}+\omega\gamma_{2}}{1-\omega L_{v}}+\frac{L_{y}+\omega (L_{u}+\lambda^{2} \tau^{2} L_{w} )}{1-\omega L_{v}} \max _{t_{0}-\tau\leq t \leq t_{0}} \bigl\Vert \phi(t) \bigr\Vert ^{2}. $$
$$ \alpha+ \frac{\beta( L_{u} + L_{v} L_{y} + \lambda^{2} \tau^{2} L_{w} )}{1- L_{v} \omega} \leq -\sigma. $$
(3.4)
(1) for any
\(t \geq t_{0}\)
we have
where
\(\phi_{0} =\max_{t_{0}-\tau\leq t \leq t_{0}}\|\phi(t)\|^{2}\), \({\mu }^{*}>0\)
is given as follows:
$$ \textstyle\begin{cases} \Vert y(t) \Vert ^{2} \leq \frac{\beta(\gamma_{2} + L_{v} \gamma_{1})}{(1-L_{v}\omega)\sigma} + \phi_{0} e^{-{\mu}^{*}(t-t_{0})},\\ \Vert y'(t) \Vert ^{2} \leq\frac{\gamma_{1}+\omega\gamma_{2}}{1-\omega L_{v}} +\frac{L_{y}+\omega(L_{u}+\lambda^{2} \tau^{2} L_{w} )}{1-\omega L_{v}} \frac{\beta(\gamma_{2} + L_{v} \gamma_{1})}{(1-L_{v}\omega)\sigma}\\ \phantom{ \Vert y'(t) \Vert ^{2} \leq}{}+ \frac{L_{y}+ \omega(L_{u}+\lambda^{2} \tau^{2} L_{w})e^{\mu^{*}\tau}}{1-\omega L_{v}e^{\mu^{*}\tau}} \phi_{0} e^{-\mu^{\ast}(t-t_{0})}, \end{cases} $$
(3.5)
$$\begin{aligned} \mu^{*} ={}&\inf_{t\geq t_{0}}\biggl\{ \mu(t): \mu(t)+\alpha+\beta \bigl(L_{u}+\lambda^{2} \tau^{2} L_{w}\bigr)e^{\mu(t)\tau} \\ &{} + \frac{L_{y}+ \omega(L_{u}+L_{w}\lambda^{2} \tau^{2})e^{\mu(t)\tau}}{1-\omega L_{v}e^{\mu(t)\tau}} \beta L_{v} e^{\mu(t)\tau} = 0 \biggr\} . \end{aligned}$$
(3.6)
(2) the system is dissipative, for any
\(\varepsilon>0\)
the open ball
is an absorbing set.
$$B=B \biggl(0,\sqrt{\frac{\beta(\gamma_{2} + L_{v} \gamma_{1})}{(1-L_{v}\omega )\sigma}+\varepsilon} \biggr) $$
Proof
Let
and
Then when \(t\geq t_{0}\), from (3.2) we have
Noting (3.3) one obtains
which gives
Substituting (3.9) into (3.8), we have
On the other hand, from the second formula of (3.2) and (3.9) we have
Therefore, combining of (3.10) and (3.11), for \(t\geq t_{0}\) we have
$$ \textstyle\begin{cases}u(t) = \Vert y(t) \Vert ^{2}, \\ w(t)= \Vert y'(t) \Vert ^{2}, \end{cases}\displaystyle t\geq t_{0} - \tau $$
(3.7)
$$z(t)= \int_{t-\tau}^{t}g\bigl(t,\xi,y(\xi)\bigr)\,d\xi,\quad t\geq t_{0}. $$
$$\begin{aligned} u^{\prime}(t)={}& \frac{d}{dt}\bigl\langle y(t), y(t)\bigr\rangle \\ ={}& 2 \operatorname{Re} \bigl\langle y(t),f\bigl(t,y(t),y(t-\tau),y'(t- \tau), z(t)\bigr)\bigr\rangle \\ \leq {}& \alpha u(t)+\beta \bigl\Vert f\bigl(t,0,y(t-\tau),y'(t- \tau), z(t)\bigr) \bigr\Vert ^{2} \\ \leq {}&\alpha u(t)+ \beta\bigl(\gamma_{2} + L_{u} u(t- \tau)+L_{v} w(t-\tau) + L_{w} \bigl\Vert z(t) \bigr\Vert ^{2}\bigr). \end{aligned}$$
(3.8)
$$\begin{aligned} \bigl\Vert z(t) \bigr\Vert & \leq\lambda \int_{t-\tau}^{t} \bigl\Vert y(\xi ) \bigr\Vert \,d \xi \\ &\leq\lambda\tau\max_{t-\tau \leq \xi \leq t} \bigl\Vert y(\xi) \bigr\Vert , \end{aligned}$$
$$ \bigl\Vert z(t) \bigr\Vert ^{2} \leq \lambda^{2}\tau^{2}\max_{t-\tau\leq \xi\leq t} u(\xi). $$
(3.9)
$$ u^{\prime}(t) \leq \beta\gamma_{2} + \alpha u(t) + \beta\bigl(L_{u} + L_{w} \lambda^{2}\tau^{2}\bigr) u^{[t-\tau,t]}+ \beta L_{v} w(t-\tau). $$
(3.10)
$$\begin{aligned} w(t)&= \bigl\Vert f\bigl(t,y(t),y(t- \tau),y'(t-\tau), z(t)\bigr) \bigr\Vert ^{2} \\ &\leq \gamma_{1} + L_{y} u(t) + \omega \bigl( \gamma_{2} + L_{u} u(t-\tau )+L_{v} w(t-\tau)+ L_{w} \bigl\Vert z(t) \bigr\Vert ^{2} \bigr) \\ & \leq \gamma_{1} + \omega\gamma_{2} + L_{y} u(t) + \omega L_{v} w(t-\tau) + \omega \bigl(L_{u} + L_{w} \lambda^{2} \tau^{2} \bigr)u^{[t-\tau,t]}. \end{aligned}$$
(3.11)
$$ \textstyle\begin{cases} u^{\prime}(t)\leq \beta\gamma_{2} + \alpha u(t) + \beta(L_{u} + L_{w} \lambda^{2}\tau^{2}) u^{[t-\tau,t]}+ \beta L_{v} w^{[t-\tau,t]},\\ w(t)\leq \gamma_{1} + \omega\gamma_{2} + L_{y} u(t) +\omega(L_{u} + L_{w} \lambda^{2} \tau^{2} )u^{[t-\tau,t]} + \omega L_{v} w^{[t-\tau,t]}. \end{cases} $$
(3.12)
Let
$$\textstyle\begin{cases} R_{1}=\beta\gamma_{2},\qquad A=\alpha,\qquad B=\beta(L_{u}+L_{w}\lambda^{2}\tau^{2}),\qquad C=0,\qquad D=\beta L_{v},\\ R_{2}=\gamma_{1} + \omega\gamma_{2}, \qquad F=L_{y}, \qquad G=\omega(L_{u}+L_{w}\lambda^{2}\tau ^{2}),\qquad H=\omega L_{v}. \end{cases} $$
3.2 Dissipativity of nonlinear neutral Volterra integro-differential equations (NNVIDEs)
Consider the IVPs in NNVIDEs as follows:
where \(\tau>0\) is constant, ϕ is a continuous function, and the functions \(f:[t_{0},+\infty)\times X \times X \times X \rightarrow X\) and \(K:[t_{0},+\infty)\times[t_{0}-\tau,+\infty)\times X \times X \rightarrow X\) satisfy the conditions for any \(t\geq t_{0}, y, u, v\in X\):
where \(D=\{(t,s):t\in[t_{0},+\infty),s\in[t-\tau,t]\}\), \(\gamma, \beta_{1}, \beta_{2}, \mu, L_{y}, L_{u}, L_{v}\) are non-negative real constants and \(\alpha \leq0\).
$$ \textstyle\begin{cases} y^{\prime}(t)=f (t,y(t),y(t-\tau),\int_{t-\tau }^{t}K(t,s,y(s),y^{\prime}(s))\,ds ),\quad t \geq t_{0},\\ y(t)=\phi(t), \qquad y^{\prime}(t)=\phi{'}(t), \quad t \in [t_{0}-\tau,t_{0}], \end{cases} $$
(3.13)
$$ \textstyle\begin{cases} 2 \operatorname{Re}\langle f(t,y,u,v),y\rangle \leq \gamma+\alpha \Vert y \Vert ^{2}+\beta_{1} \Vert f(t,0,u,v) \Vert ^{2},\\ \Vert f(t,y,u,v) \Vert ^{2}\leq L_{y} \Vert y \Vert ^{2}+\beta_{2} \Vert f(t,0,u,v) \Vert ^{2}, \\ \Vert f(t,0,u,v) \Vert ^{2}\leq L_{u} \Vert u \Vert ^{2}+L_{v} \Vert v \Vert ^{2}, \\ \Vert K(t,s,u,v) \Vert \leq \mu \Vert u \Vert +L_{k} \Vert v \Vert ,\quad (t,s)\in D, \end{cases} $$
(3.14)
Theorem 3.2
Assume that (3.13) satisfies (3.14) with
\(2\beta_{2} \tau^{2} L_{v}L_{k}^{2}<1\), and initial value function
\(\phi(t)\)
satisfies
Assume there exists a constant
\(\sigma>0\)
such that
Let
\(y(t)\)
be the solution of (3.13). Then
$$ \max_{t_{0}-\tau\leq t \leq t_{0}} \bigl\Vert \phi'(t) \bigr\Vert ^{2} \leq \frac{L_{y}+\beta_{2}(L_{u}+2\tau^{2}\mu^{2}L_{v})}{1-2\beta_{2} \tau^{2} L_{v} L_{k}^{2}} \max_{t_{0}-\tau\leq t \leq t_{0}} \bigl\Vert \phi(t) \bigr\Vert ^{2}. $$
$$ \alpha+\beta_{1} \frac{L_{u}+2 \tau^{2} L_{v} (L_{k}^{2} L_{y}+ \mu^{2})}{1-2\beta_{2} \tau^{2} L_{v} L_{k}^{2}} \leq -\sigma. $$
(3.15)
(1) for any
\(t \geq t_{0}\)
we have
where
\(\phi_{0} =\sup_{ t_{0}-\tau \leq \xi \leq t_{0}} \Vert \phi(\xi) \Vert ^{2}\), \({\mu}^{*}>0\)
is defined as
$$ \textstyle\begin{cases} \Vert y(t) \Vert ^{2} \leq \frac{\gamma}{\sigma}+\phi_{0} e^{-\mu^{\ast}(t-t_{0})},\\ \Vert y'(t) \Vert ^{2} \leq \frac{L_{y}+\beta_{2}(L_{u}+2\tau^{2}\mu^{2}L_{v})}{1-2\beta _{2} \tau^{2} L_{v} L_{k}^{2}} \frac{\gamma}{\sigma} +\frac{L_{y}+\beta _{2}(L_{u}+2\tau^{2}\mu^{2}L_{v})e^{{\mu}^{*}\tau}}{1-2\beta_{2}\tau^{2} L_{v} L_{k}^{2} e^{{\mu}^{*}\tau}}\phi_{0} e^{-{\mu}^{*}(t-t_{0})}, \end{cases} $$
$$\begin{aligned} {\mu}^{*}=\inf_{t\geq t_{0}} \biggl\{ \mu(t): \mu(t)+\alpha+ \frac {L_{u}+2 \tau^{2} L_{v} (L_{k}^{2} L_{y}+ \mu^{2})}{1-2\beta_{2} \tau^{2} L_{v} L_{k}^{2} e^{\mu(t)\tau}}\beta_{1} e^{\mu(t)\tau}=0 \biggr\} . \end{aligned}$$
(2) the system is dissipative, for any
\(\varepsilon>0\)
the open ball
is an absorbing set.
$$\mathbf{B}=\mathbf{B} \biggl(0,\sqrt{\frac{\gamma}{\sigma}+\varepsilon } \biggr) $$
Proof
Let
and
From (3.14) we can obtain
and
and
It can be summarized from (3.16) and (3.17) that
We denote
Then from Theorem 2.1 we can complete the proof of Theorem 3.2. □
$$ \textstyle\begin{cases}u(t) = \Vert y(t) \Vert ^{2}, \\ w(t)= \Vert y'(t) \Vert ^{2}, \end{cases}\displaystyle t\geq t_{0} - \tau $$
$$z(t)= \int_{t-\tau}^{t}K\bigl(t,s,y(s),y^{\prime}(s) \bigr)\,ds),\quad t\geq t_{0}. $$
$$\begin{aligned} \bigl\Vert z(t) \bigr\Vert ^{2}&\leq\tau^{2} \Bigl(\mu \max_{t-\tau \leq \xi \leq t} \bigl\Vert y(\xi ) \bigr\Vert +L_{k} \max_{t-\tau \leq \xi \leq t} \bigl\Vert y^{\prime}(\xi) \bigr\Vert \Bigr)^{2} \\ &\leq2 \tau^{2} \bigl(\mu^{2} u^{[t-\tau,t]}+L_{k}^{2} w^{[t-\tau,t]}\bigr) \end{aligned}$$
$$\begin{aligned} u^{\prime}(t)&=2\operatorname{Re}\bigl\langle f \bigl(t,y(t),y(t-\tau),z(t)\bigr),y(t)\bigr\rangle \\ &\leq \gamma+\alpha u(t)+\beta_{1} \bigl\Vert f\bigl(t,0,y(t- \tau),z(t)\bigr) \bigr\Vert ^{2} \\ &\leq \gamma+\alpha u(t)+\beta_{1} \bigl(L_{u} u(t- \tau)+L_{v} \bigl\Vert z(t) \bigr\Vert ^{2}\bigr) \\ &\leq \gamma+\alpha u(t)+\beta_{1}\bigl(L_{u}+2 \mu^{2}\tau^{2}L_{v}\bigr)u^{[t-\tau ,t]}+2 \beta_{1}\tau^{2} L_{v} L_{k}^{2} w^{[t-\tau,t]} \end{aligned}$$
(3.16)
$$\begin{aligned} w(t) &= \bigl\Vert f\bigl(t,y(t),y(t-\tau),z(t)\bigr) \bigr\Vert ^{2} \\ &\leq L_{y} \bigl\Vert y(t) \bigr\Vert ^{2}+ \beta_{2} \bigl\Vert f\bigl(t,0,y(t-\tau),z(t)\bigr) \bigr\Vert ^{2} \\ &\leq L_{y} \bigl\Vert y(t) \bigr\Vert ^{2}+ \beta_{2}\bigl[L_{u} u(t-\tau)+L_{v} \bigl\Vert z(t) \bigr\Vert ^{2}\bigr] \\ &\leq L_{y} u(t)+\beta_{2}\bigl(L_{u}+2L_{v} \tau^{2}\mu^{2}\bigr)u^{[t-\tau,t]}+2\beta_{2} \tau^{2} L_{v} L_{k}^{2} w^{[t-\tau,t]}. \end{aligned}$$
(3.17)
$$ \textstyle\begin{cases} u^{\prime}(t)\leq \gamma+\alpha u(t)+\beta_{1}(L_{u}+2 \tau^{2} \mu^{2} L_{v})u^{[t-\tau,t]}+2\beta_{1}\tau^{2} L_{v} L_{k}^{2} w^{[t-\tau,t]},\\ w(t)\leq L_{y} u(t)+\beta_{2}(L_{u}+2\tau^{2}\mu^{2} L_{v})u^{[t-\tau,t]}+2\beta _{2} \tau^{2} L_{v} L_{k}^{2} w^{[t-\tau,t]}. \end{cases} $$
(3.18)
$$\begin{aligned} &\gamma_{1}=\gamma, \qquad A=\alpha,\qquad B=\beta_{1} \bigl(L_{u}+2\tau^{2}\mu ^{2}L_{v} \bigr),\qquad C=0,\qquad D=2\beta_{1} \tau^{2} L_{v} L_{k}^{2}, \\ &\gamma_{2}=0,\qquad F=L_{y},\qquad G=\beta_{2} \bigl(L_{u}+2\tau^{2}\mu^{2}L_{v}\bigr),\qquad H=2\beta_{2} \tau^{2} L_{v} L_{k}^{2}. \end{aligned}$$
Remark 3.3
From a numerical point of view, it is important to study the potential of numerical methods in preserving the qualitative behavior of the analytical solutions. Therefore, the results of Theorem 3.1 and Theorem 3.2 presented in this paper, provide the theoretical foundation for analyzing the dissipativity of the numerical methods when they are applied to the underlying systems.
Acknowledgements
This work is supported by Hunan Key Laboratory for Computation and Simulation in Science and Engineering.
Competing interests
The authors declare that they have no competing interests.
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