# On the existence and exponential attractivity of a unique positive almost
periodic solution to an impulsive hematopoiesis model with delays

Trinh Tuan Anh, Tran Van Nhung and Le Van Hien  Trinh Tuan Anh
Department of Mathematics, Hanoi National University of Education
136 Xuan Thuy Road, Hanoi, Vietnam<EMAIL_ADDRESS>Tran Van Nhung
Vietnam State Council for Professor Promotion, No. 1, Dai Co Viet Road, Hanoi,
Vietnam<EMAIL_ADDRESS>Le Van Hien
Department of Mathematics, Hanoi National University of Education
136 Xuan Thuy Road, Hanoi, Vietnam<EMAIL_ADDRESS>
###### Abstract.

In this paper, a generalized model of hematopoiesis with delays and impulses
is considered. By employing the contraction mapping principle and a novel type
of impulsive delay inequality, we prove the existence of a unique positive
almost periodic solution of the model. It is also proved that, under the
proposed conditions in this paper, the unique positive almost periodic
solution is globally exponentially attractive. A numerical example is given to
illustrate the effectiveness of the obtained results.

###### Key words and phrases:

Hematopoiesis model; almost periodic solution; impulsive systems.

Accepted for publication in AMV

##  1\. Introduction

The nonlinear delay differential equation

(1.1) $\dot{x}(t)=-ax(t)+\frac{b}{1+x^{n}(t-\tau)},\quad n>0,$

where $a,b,\tau$ are positive constants, proposed by Mackey and Glass [17],
has been used as an appropriate model for the dynamics of hematopoiesis (blood
cells production) [5, 7, 17]. In medical terms, $x(t)$ denotes the density of
mature cells in blood circulation at time $t$ and $\tau$ is the time delay
between the production of immature cells in the bone marrow and their
maturation for release in circulating bloodstream.

As we may know, the periodic or almost periodic phenomena are popular in
various natural problems of real world applications [2, 5, 6, 9, 10, 16, 19,
22, 23]. In comparing with periodicity, almost periodicity is more frequent in
nature and much more complicated in studying for such model [20, 21]. On the
other hand, many dynamical systems describe the real phenomena depend on the
history as well as undergo abrupt changes in their states. This kind of models
are best described by impulsive delay differential equations [3, 18, 20]. A
great deal of effort from researchers has been devoted to study the existence
and asymptotic behavior of almost periodic solutions of (1.1) and its
generalizations due to their extensively realistic significance. We refer the
reader to [8, 13, 15, 21, 24, 25] and the references therein. Particularly, in
[21], Wang and Zhang investigated the existence, nonexistence and uniqueness
of positive almost periodic solution of the following model

(1.2) $\dot{x}(t)=-a(t)x(t)+\frac{b(t)x(t-\tau(t))}{1+x^{n}(t-\tau(t))},\quad
n>1,$

by using a new fixed point theorem in cone. Very recently, using a fixed point
theorem for contraction mapping combining with the Lyapunov functional method,
Zhang et al. [25] obtained sufficient conditions for the existence and
exponential stability of a positive almost periodic solution to a generalized
model of (1.1)

(1.3)
$\dot{x}(t)=-a(t)x(t)+\sum_{i=1}^{m}\frac{b_{i}(t)}{1+x^{n}(t-\tau_{i}(t))},\quad
n>0.$

By employing a novel argument, a delay-independent criteria was established in
[13] ensuring the existence, uniqueness, and global exponential stability of
positive almost periodic solutions of a non-autonomous delayed model of
hematopoiesis with almost periodic coefficients and delays. In [1], Alzabut et
al. considered the following model of hematopoiesis with impulses

(1.4) $\displaystyle\dot{x}(t)=-a(t)x(t)+\frac{b(t)}{1+x^{n}(t-\tau)},\;t\geq
0,\;t\neq t_{k},$ $\displaystyle\Delta
x(t_{k})=\gamma_{k}x\left(t_{k}^{-}\right)+\delta_{k},\;k\in\mathbb{N},$

where $t_{k}$ represents the instant at which the density suffers an increment
of $\delta_{k}$ unit and $\Delta
x(t_{k})=x\left(t_{k}^{+}\right)-x\left(t^{-}_{k}\right)$. The density of
mature cells in blood circulation decreases at prescribed instant $t_{k}$ by
some medication and it is proportional to the density at that time
$t^{-}_{k}$. By employing the contraction mapping principle and applying
Gronwall-Bellman’s inequality, sufficient conditions which guarantee the
existence and exponential stability of a positive almost periodic solution of
system (1.4) were given in [1] as follows.

######  Theorem 1.1 ([1]).

Assume that

  * (C1)

The function $a\in C\left(\mathbb{R}^{+},\mathbb{R}^{+}\right)$ is almost
periodic in the sense of Bohr and there exists a positive constant $\mu$ such
that $a(t)\geq\mu$.

  * (C2)

The sequence $\\{\gamma_{k}\\}$ is almost periodic and $-1\leq\gamma_{k}\leq
0,\ k\in\mathbb{N}$.

  * (C3)

The sequences $\\{t_{k}^{p}\\}$ are uniformly almost periodic and there exists
a positive constant $\eta$ such that $\inf_{k\in\mathbb{N}}t^{1}_{k}=\eta$,
where $0<\sigma\leq t_{k}<t_{k+1},\forall k\in\mathbb{N}$,
$\lim_{k\to\infty}t_{k}=\infty$ and
$t_{k}^{p}=t_{k+p}-t_{k},k,p\in\mathbb{N}$.

  * (C4)

The function $b\in C\left(\mathbb{R}^{+},\mathbb{R}^{+}\right)$ is almost
periodic in the sense of Bohr, $b(0)=0$, and there exists a positive constant
$\nu$ such that $\sup_{t\in\mathbb{R}^{+}}|b(t)|<\nu$.

  * (C5)

The sequence $\\{\delta_{k}\\}$ is almost periodic and there exists a constant
$\delta>0$ such that $\sup_{k\in\mathbb{N}}|\delta_{k}|<\delta$.

If $\nu<\mu$, then equation (1.4) has a unique positive almost periodic
solution.

Unfortunately, the above theorem is incorrect. For this, let us consider the
following example.

######  Example 1.1.

Consider the following equation

(1.5) $\dot{x}(t)=-x(t),\quad t\geq 0,t\neq k\in\mathbb{N},\quad\Delta
x(k)=-1,\;k\in\mathbb{N}.$

Note that (1.5) is a special case of (1.4). Moreover, we can easily see that
equation (1.5) satisfies conditions (C1)-(C5), where $t_{k}=k,\gamma_{k}=0$
and $\delta_{k}=-1$.

Suppose that system (1.5) has a positive almost periodic solution $x^{*}(t)$.
It is obvious that

$x^{*}(t)=e^{-t}x^{*}(0)-\sum_{k\in\mathbb{N},\;k\leq t}e^{-(t-k)},\quad t>0.$

For any positive integer $n$, we have

$0<x^{*}(n)=e^{-n}x^{*}(0)-e^{-1}\left(\frac{1-e^{-n}}{1-e^{-1}}\right)\longrightarrow\frac{-e^{-1}}{1-e^{-1}}<0\text{
as }n\to\infty$

which yields a contradiction. This shows that (1.5) has no positive almost
periodic solution. Thus, Theorem 1.1 is incorrect, and Theorem 3.2 in [1] is
also incorrect.

Motivated by the aforementioned discussions, in this paper we consider a
generalized model of hematopoiesis with delays, harvesting terms [4, 12, 14]
and impulses of the form

(1.6) $\begin{split}&\begin{aligned}
\dot{x}(t)=-a(t)x(t)&+\sum_{i=1}^{m}\bigg{[}\frac{b_{i}(t)}{1+x^{\alpha_{i}}(t-\tau_{i}(t))}\\\
&+c_{i}(t)\int_{0}^{T}\frac{v_{i}(s)}{1+x^{\beta_{i}}(t-s)}ds-
H_{i}(t,x(t-\sigma_{i}(t)))\bigg{]},\quad t\neq t_{k},\end{aligned}\\\ &\Delta
x(t_{k})=x\left(t_{k}^{+}\right)-x\left(t^{-}_{k}\right)=\gamma_{k}x\left(t_{k}^{-}\right)+\delta_{k},\quad
k\in\mathbb{Z},\end{split}$

where $m$ is given positive integer,
$a,b_{i},c_{i}:\mathbb{R}\to\mathbb{R},i\in\underline{m}:=\\{1,2,\ldots,m\\}$,
are nonnegative functions;
$H_{i}:\mathbb{R}\times\mathbb{R}\to\mathbb{R}_{+},i\in\underline{m},$ are
nonnegative functions represent harvesting terms;
$\tau_{i}(t),\sigma_{i}(t)\geq 0,i\in\underline{m}$, are time delays;
$\alpha_{i},\beta_{i},i\in\underline{m},$ are positive numbers and $T>0$ is a
constant; $\gamma_{k},\delta_{k},k\in\mathbb{Z},$ are constants;
$v_{i}(t),i\in\underline{m},$ are nonnegative integrable functions on $[0,T]$
with $\int_{0}^{T}v_{i}(s)ds=1$; $\\{t_{k}\\},k\in\mathbb{Z}$, is an
increasing sequence involving the fixed impulsive points with
$\lim_{k\to\pm\infty}t_{k}=\pm\infty$.

The main goal of the present paper is to establish conditions for the
existence of a unique positive almost periodic solution of model (1.6). It is
also proved that, under the proposed conditions, the unique positive almost
periodic solution of (1.6) is globally exponentially attractive.

The rest of this paper is organized as follows. Section 2 introduces some
notations, basic definitions and technical lemmas. Main results on the
existence and exponential attractivity of a unique positive almost periodic
solution of (1.6) are presented in section 3. An illustrative example is given
in section 4. The paper ends with the conclusion and cited references.

##  2\. Preliminaries

Let $\\{t_{k}\\}_{k\in\mathbb{Z}}$ be a fixed sequence of real numbers
satisfying $t_{k}<t_{k+1}$ for all $k\in\mathbb{Z}$,
$\lim_{k\to\pm\infty}t_{k}=\pm\infty$. Let $X$ be an interval of $\mathbb{R}$,
denoted by $PLC(X,\mathbb{R})$ the space of all piecewise left continuous
functions $\phi:X\to\mathbb{R}$ with points of discontinuity of the first kind
at $t=t_{k}$, $k\in\mathbb{Z}$.

The following notations will be used in this paper. For bounded functions
$f:\mathbb{R}\to\mathbb{R}$, $F:\mathbb{R}\times\mathbb{R}_{+}\to\mathbb{R}$
and a bounded sequence $\\{z_{k}\\}$, we set

$\displaystyle
f_{L}=\inf_{t\in\mathbb{R}}f(t),\;f_{M}=\sup_{t\in\mathbb{R}}f(t),$
$\displaystyle
F_{L}=\inf_{(t,x)\in\mathbb{R}\times\mathbb{R}_{+}}F(t,x),\;F_{M}=\sup_{(t,x)\in\mathbb{R}\times\mathbb{R}_{+}}F(t,x),$
$\displaystyle
z_{L}=\inf_{k\in\mathbb{Z}}z_{k},\;z_{M}=\sup_{k\in\mathbb{Z}}z_{k}.$

The following definitions are borrowed from [18].

######  Definition 2.1 ([18, 20]).

The set of sequences $\\{t_{k}^{p}\\}$, where
$t_{k}^{p}=t_{k+p}-t_{k},p,k\in\mathbb{Z}$, is said to be uniformly almost
periodic if for any positive number $\epsilon$, there exists a relatively
dense set of $\epsilon$-almost periods common for all sequences.

######  Definition 2.2 ([10, 18]).

A function $\phi\in PLC(\mathbb{R},\mathbb{R})$ is said to be almost periodic
if the following conditions hold

  * (i)

The set of sequences $\\{t_{k}^{p}\\}$ is uniformly almost periodic.

  * (ii)

For any $\epsilon>0$, there exists $\delta=\delta(\epsilon)>0$ such that, if
$t,\bar{t}$ belong to the same interval of continuity of $\phi(t)$,
$|t-\bar{t}|<\delta$, then $|\phi(t)-\phi(\bar{t})|<\epsilon$.

  * (iii)

For any $\epsilon>0$, there exists a relatively dense set $\Omega$ of
$\epsilon$-almost periods such that, if $\omega\in\Omega$ then
$|\phi(t+\omega)-\phi(t)|<\epsilon$ for all $t\in\mathbb{R}$, $k\in\mathbb{Z}$
satisfying $|t-t_{k}|>\epsilon$.

For equation (1.6), we introduce the following assumptions

  * (A1)

The function $a(t)$ is almost periodic in the sense of Bohr and $a_{L}>0$.

  * (A2)

The functions $b_{i}(t),c_{i}(t),i\in\underline{m}$, are nonnegative and
almost periodic in the sense of Bohr.

  * (A3)

The function $H_{i}(t,x),i\in\underline{m}$, are bounded nonnegative and
almost periodic in the sense of Bohr in $t\in\mathbb{R}$ uniformly in
$x\in\mathbb{R}_{+}$ and there exist positive constants $L_{i}$ such that

$|H_{i}(t,x)-H_{i}(t,y)|\leq
L_{i}|x-y|,\;\forall(t,x),(t,y)\in\mathbb{R}\times\mathbb{R}_{+}.$

  * (A4)

The functions $\tau_{i}(t),\sigma_{i}(t),i\in\underline{m}$, are almost
periodic in the sense of Bohr, $\dot{\tau}_{i}(t),\dot{\sigma}_{i}(t)$ are
bounded, $\inf_{t\in\mathbb{R}}\left(1-\dot{\tau}_{i}(t)\right)>0$,
$\inf_{t\in\mathbb{R}}\left(1-\dot{\sigma}_{i}(t)\right)>0$.

  * (A5)

The sequence $\\{\delta_{k}\\}$ is almost periodic.

  * (A6)

The sequence $\\{\gamma_{k}\\}$ is almost periodic satisfying

$\gamma_{L}>-1,\quad\Gamma_{M}=\sup_{p,q\in\mathbb{Z},p\geq
q}\Gamma(q,p)<\infty,\quad\Gamma_{L}=\inf_{p,q\in\mathbb{Z},p\geq
q}\Gamma(q,p)>0,$

where $\Gamma(q,p)=\prod\limits_{i=q}^{p}(1+\gamma_{i}),p\geq q$.

  * (A7)

The set of sequences $\\{t_{k}^{p}\\}$ is uniformly almost periodic,
$\eta=\inf_{k\in\mathbb{Z}}t_{k}^{1}>0$.

######  Remark 2.1.

It should be noted that model (1.6) includes (1.4) as a special case. For that
model, assumptions (A3), (A4) obviously be removed. Furthermore, we make
assumption (A6) in order to correct condition (C2) in [1].

The following lemmas will be used in the proof of our main results.

######  Lemma 2.1 ([18]).

Let Assumption (A7) holds. Assume that functions $g_{i}(t),i\in\underline{m}$,
are almost periodic in the sense of Bohr, a function $\phi(t)$ and sequences
$\\{\delta_{k}\\}$, $\\{\gamma_{k}\\}$ are almost periodic. Then for any
$\epsilon>0$, there exist $\epsilon_{1}\in(0,\epsilon)$, relatively dense sets
$\Omega\subset\mathbb{R}$, $\mathcal{P}\subset\mathbb{Z}$ such that

  * (a1)

$|\phi(t+\omega)-\phi(t)|<\epsilon,t\in\mathbb{R},|t-t_{k}|>\epsilon,k\in\mathbb{Z}$;

  * (a2)

$|g_{i}(t+\omega)-g_{i}(t)|<\epsilon,t\in\mathbb{R},i\in\underline{m}$;

  * (a3)

$|\gamma_{k+p}-\gamma_{k}|<\epsilon,|\delta_{k+p}-\delta_{k}|<\epsilon,|t^{p}_{k}-\omega|<\epsilon_{1},\omega\in\Omega,p\in{\mathcal{P}},k\in\mathbb{Z}.$

######  Lemma 2.2.

Let Assumption (A7) holds. Assume that functions
$f_{i}(t,x),i\in\underline{m}$, are almost periodic in $t\in\mathbb{R}$
uniformly in $x\in\mathbb{R}$ in the sense of Bohr, a function $\phi(t)$ and
sequences $\\{\delta_{k}\\}$, $\\{\gamma_{k}\\}$ are almost periodic. Then for
any compact set ${\mathcal{M}}\subset\mathbb{R}$ and positive number
$\epsilon$, there exist $\epsilon_{1}\in(0,\epsilon)$, relatively dense sets
$\Omega\subset\mathbb{R}$, $\mathcal{P}\subset\mathbb{Z}$ such that

  * (b1)

$|\phi(t+\omega)-\phi(t)|<\epsilon,t\in\mathbb{R},|t-t_{k}|>\epsilon,\omega\in\Omega$;

  * (b2)

$|f_{i}(t+\omega,x)-f_{i}(t,x)|<\epsilon,t\in\mathbb{R},x\in{\mathcal{M}},\omega\in\Omega,i\in\underline{m}$;

  * (b3)

$|\gamma_{k+p}-\gamma_{k}|<\epsilon,|\delta_{k+p}-\delta_{k}|<\epsilon,|t^{p}_{k}-\omega|<\epsilon_{1},\omega\in\Omega,p\in{\mathcal{P}},k\in\mathbb{Z}.$

###### Proof.

The proof of this lemma is similar to the proof of Lemma 2.1 in [20] so let us
omit it here. ∎

######  Lemma 2.3.

For given $\epsilon>\epsilon_{1}>0$, real number $\omega$ and integers $k,p$
such that $|t^{p}_{k}-\omega|<\epsilon_{1}$, if $|t-t_{i}|>\epsilon$ for all
$i\in\mathbb{Z}$ and $t_{k-1}<t<t_{k}$ then $t_{k+p-1}<t+\omega<t_{k+p}$.

###### Proof.

The proof is straight forward, so let us omit it here. ∎

######  Lemma 2.4.

Let Assumptions (A6) and (A7) hold. If $p\in\mathbb{Z}$ satisfies
$|\gamma_{i+p}-\gamma_{i}|\leq\epsilon$ for all $i\in\mathbb{Z}$, then

$\left|\Gamma(n+p,k+p)-\Gamma(n,k)\right|\leq\frac{\Gamma_{M}}{1+\gamma_{L}}(k-n+1)\epsilon,\quad\forall
k,n\in\mathbb{Z},k\geq n.$

###### Proof.

Using the facts that
$\left|e^{u}-e^{v}\right|\leq|u-v|\max\\{e^{u},e^{v}\\},\forall
u,v\in\mathbb{R}$ and
$\left|\ln(1+u)-\ln(1+v)\right|\leq\dfrac{1}{1+\min\\{u,v\\}}|u-v|,\forall
u,v>-1$, from (A6), we have

$\displaystyle\left|\Gamma(n+p,k+p)-\Gamma(n,k)\right|$
$\displaystyle\leq\left|\exp\bigg{(}\sum_{i=n+p}^{k+p}\ln(1+\gamma_{i})\bigg{)}-\exp\bigg{(}\sum_{i=n}^{k}\ln(1+\gamma_{i})\bigg{)}\right|$
$\displaystyle\leq\frac{{\Gamma_{M}}}{1+\gamma_{L}}\sum\limits_{i=n}^{k}|\gamma_{i+p}-\gamma_{i}|\leq\frac{{\Gamma_{M}}}{1+\gamma_{L}}(k-n+1)\epsilon,\
k\geq n.$

The proof is completed. ∎

######  Lemma 2.5.

Let Assumption (A7) holds. For any $\alpha>0$, $0<\epsilon<\eta/2$, we have

$\sum_{t_{k}<t}e^{-\alpha(t-t_{k})}\leq\frac{1}{1-e^{-\alpha\eta}},\quad\sum_{t_{k}<t}\int_{t_{k}-\epsilon}^{t_{k}+\epsilon}e^{-\alpha(t-s)}ds\leq
2e^{\frac{1}{2}\alpha\eta}\frac{\epsilon}{1-e^{-\alpha\eta}}.$

###### Proof.

The proof follows by some direct estimates and, thus, is omitted here. ∎

Now, let $\sigma=\max_{i\in\underline{m}}\\{T,\tau_{iM},\sigma_{iM}\\}$, then
$0<\sigma<+\infty$. From biomedical significance, we only consider the initial
condition

(2.1) $x(s)=\xi(s)\geq 0,\ s\in[\alpha-\sigma,\alpha),\ \xi(\alpha)>0,\ \xi\in
PLC([\alpha-\sigma,\alpha],\mathbb{R}).$

It should be noted that problem (1.6) and (2.1) has a unique solution
$x(t)=x(t;\alpha,\xi)$ defined on $[\alpha-\sigma,\infty)$ which is piecewise
continuous with points of discontinuity of the first kind, namely
$t_{k},k\in\mathbb{Z}$, at which it is left continuous and the following
relations are satisfied [18]

$x\left(t^{-}_{k}\right)=x(t_{k}),\ \Delta
x(t_{k}):=x\left(t_{k}^{+}\right)-x\left(t^{-}_{k}\right)=\gamma_{k}x\left(t_{k}^{-}\right)+\delta_{k}.$

Related to (1.6), we consider the following linear equation

(2.2) $\dot{y}(t)=-a(t)y(t),\ t\neq t_{k},\quad\Delta
y(t_{k})=\gamma_{k}y(t^{-}_{k}),\ k\in\mathbb{Z}.$

######  Lemma 2.6.

Let Assumptions (A1), (A6) and (A7) hold. Then

$Be^{-a_{M}(t-s)}\leq H(t,s)\leq Ae^{-a_{L}(t-s)},\ s\leq t,$

where

(2.3) $H(t,s)=\begin{cases}\exp\left(-\int_{s}^{t}a(r)dr\right),\ \text{ if }\
t_{k-1}<s\leq t\leq t_{k},\\\
\Gamma(n,k)\exp\left(-\int_{s}^{t}a(r)dr\right),\ \text{ if }\ t_{n-1}<s\leq
t_{n}\leq t_{k}<t\leq t_{k+1}\end{cases}$

is the Cauchy matrix of (2.2), $A=\max\left\\{\Gamma_{M},1\right\\}$ and
$B=\min\left\\{\Gamma_{L},1\right\\}$.

###### Proof.

The proof is straight forward from (2.3), so let us omit it here. ∎

Similar to Lemma 36 in [18] and Lemma 2.6 in [20] we have the following lemma.

######  Lemma 2.7.

Let Assumptions (A1), (A6) and (A7) hold. Then, for given
$0<\epsilon_{1}<\epsilon$, relatively dense sets $\Omega\subset\mathbb{R}$,
$\mathcal{P}\subset\mathbb{Z}$, satisfying

  * (c1)

$|a(t+\omega)-a(t)|<\epsilon,\ t\in\mathbb{R},\omega\in\Omega$;

  * (c2)

$|\gamma_{k+p}-\gamma_{k}|<\epsilon,\
|t^{p}_{k}-\omega|<\epsilon_{1},\omega\in\Omega,\ p\in{\mathcal{P}},\
k\in\mathbb{Z}$,

the following estimate holds

$\left|H(t+\omega,s+\omega)-H(t,s)\right|\leq\epsilon
Me^{-\frac{1}{2}a_{L}(t-s)},$

for any $\omega\in\Omega$, $t,s\in\mathbb{R}$ satisfying $t\geq s$,
$|t-t_{k}|>\epsilon,\ |s-t_{k}|>\epsilon,k\in\mathbb{Z}$, where

(2.4)
$M=\max\left\\{\frac{2}{a_{L}},\Gamma_{M}\left[\frac{2}{a_{L}}+\frac{1}{1+\gamma_{L}}\left(1+\frac{2}{a_{L}\eta}\right)\right]\right\\}.$

###### Proof.

We divide the proof into two possible cases as follows.

Case 1: $t_{k-1}<s\leq t\leq t_{k}$. By Lemma 2.3, $t_{k+p-1}<s+\omega\leq
t+\omega<t_{k+p}$. Since $|a(t+\omega)-a(t)|\leq\epsilon$, $\forall
t\in\mathbb{R}$, $\epsilon<\eta/2$ and
$\frac{1}{2}a_{L}(t-s)e^{-\frac{1}{2}a_{L}(t-s)}<1$, it follows from (2.3),
(2.4) that

(2.5) $\displaystyle|H(t+\omega,s+\omega)-H(t,s)|$
$\displaystyle=\left|\exp\left(-\int_{s}^{t}a(r+\omega)dr\right)-\exp\left(-\int_{s}^{t}a(r)dr\right)\right|$
$\displaystyle\leq e^{-a_{L}(t-s)}\int_{s}^{t}|a(r+\omega)-a(r)|dr$
$\displaystyle\leq\frac{2}{a_{L}}\epsilon
e^{-\frac{1}{2}a_{L}(t-s)}\leq\epsilon Me^{-\frac{1}{2}a_{L}(t-s)}.$

Case 2: $t_{n-1}<s\leq t_{n}\leq t_{k}<t\leq t_{k+1}$. Similarly, we have

$t_{n+p-1}<s+\omega<t_{n+p}\leq t+\omega<t_{k+p+1}.$

By Lemma 2.4, from (2.3)-(2.5) we obtain

$\displaystyle|H(t+\omega,s+\omega)-H(t,s)|$
$\displaystyle=\Gamma(n+p,k+p)\left|\exp\left(-\int_{s+\omega}^{t+\omega}a(r)dr\right)-\exp\left(-\int_{s}^{t}a(r)dr\right)\right|$
$\displaystyle\quad+|\Gamma(n+p,k+p)-\Gamma(n,k)|\exp\left(-\int_{s}^{t}a(r)dr\right)$
$\displaystyle\leq\frac{2\Gamma_{M}\epsilon}{a_{L}}e^{-\frac{1}{2}a_{L}(t-s)}+\frac{\Gamma_{M}\epsilon}{1+\gamma_{L}}(k-n+1)e^{-a_{L}(t-s)}$
$\displaystyle\leq\frac{2\Gamma_{M}\epsilon}{a_{L}}e^{-\frac{1}{2}a_{L}(t-s)}+\frac{\Gamma_{M}\epsilon}{1+\gamma_{L}}\left(\frac{t-s}{\eta}+1\right)e^{-a_{L}(t-s)}$
$\displaystyle\leq\frac{2\Gamma_{M}\epsilon}{a_{L}}e^{-\frac{1}{2}a_{L}(t-s)}+\frac{\Gamma_{M}\epsilon}{1+\gamma_{L}}\left(1+\frac{2}{a_{L}\eta}\right)e^{-\frac{1}{2}a_{L}(t-s)}$
$\displaystyle\leq\epsilon Me^{-\frac{1}{2}a_{L}(t-s)}.$

The proof is completed. ∎

It is worth noting that, the proof of Lemma 2.7 is different from those in [1,
18, 20]. By employing Lemma 2.4, we obtain a new bound for constant $M$ given
in (1.4).

######  Lemma 2.8 ([11]).

Assume that there exist constants $R,S>0,\tau\geq 0$, $T_{0}\in\mathbb{R}$ and
a function $y\in PLC([T_{0}-\tau,\infty),\mathbb{R}^{+})$ satisfying

  * (d1)

$\Delta y(t_{k})\leq\gamma_{k}y(t^{-}_{k})$ for $t_{k}\geq T_{0}$, where
$\gamma_{k}>-1$ and $\max_{t_{k}\geq
T_{0}}\left\\{(\gamma_{k}+1)^{-1},1\right\\}<\dfrac{R}{S}$;

  * (d2)

$D^{+}y(t)\leq-Ry(t)+S\overline{y}(t)$ for $t\geq T_{0},t\neq t_{k}$, where
$\overline{y}(t)=\sup_{t-\tau\leq s\leq t}y(s)$ and $D^{+}$ denotes the upper-
right Dini derivative;

  * (d3)

$\tau\leq t_{k}-t_{k-1}$ for all $k\in\mathbb{Z}$ satisfies $t_{k}\geq T_{0}$.

Then

$y(t)\leq\overline{y}(T_{0})\bigg{(}\prod_{T_{0}<t_{k}\leq
t}(\gamma_{k}+1)\bigg{)}e^{-\lambda(t-T_{0})},\forall t\geq T_{0},$

where $0<\lambda\leq R-S\max_{t_{k}\geq
T_{0}}\left\\{(\gamma_{k}+1)^{-1},1\right\\}e^{\lambda\tau}.$

##  3\. Main results

Let us set $\mathcal{D}_{1}=\Big{\\{}\phi\in
PLC(\mathbb{R},\mathbb{R}):\phi\text{ is almost periodic},\phi(t)\geq 0\text{
for all }t\in\mathbb{R}\Big{\\}}$ and
$\|\phi\|=\sup_{t\in\mathbb{R}}|\phi(t)|$. We define an operator
$F:\mathcal{D}_{1}\to PLC(\mathbb{R},\mathbb{R})$ as follows

(3.1) $\displaystyle
F\phi(t)=\int_{-\infty}^{t}H(t,s)\sum_{i=1}^{m}\bigg{\\{}$
$\displaystyle\frac{b_{i}(s)}{1+\phi^{\alpha_{i}}(s-\tau_{i}(s))}+c_{i}(s)\int_{0}^{T}\frac{v_{i}(r)}{1+\phi^{\beta_{i}}(s-r)}dr$
$\displaystyle-
H_{i}(s,\phi(s-\sigma_{i}(s)))\bigg{\\}}ds+\sum_{t_{k}<t}H(t,t_{k})\delta_{k}.$

It can be verified that $x^{*}(t)=\phi(t)$ is an almost periodic solution on
$\mathcal{D}_{1}$ of (1.6) if and only if $F\phi=\phi$.

We define the following constants

(3.2) $\displaystyle\underline{\delta}$
$\displaystyle=\inf_{k\in\mathbb{Z}}|\delta_{k}|,\
\overline{\delta}=\sup_{k\in\mathbb{Z}}|\delta_{k}|,\
\overline{\eta}=\sup_{k\in\mathbb{Z}}t^{1}_{k},$ $\displaystyle M_{1}$
$\displaystyle=\frac{A}{a_{L}}\sum_{i=1}^{m}\left(b_{iM}+c_{iM}-H_{iL}\right)+\frac{A\delta_{M}}{1-e^{-a_{L}\eta}},$
$\displaystyle M_{2}$
$\displaystyle=\begin{cases}\displaystyle\frac{B}{a_{M}}\sum_{i=1}^{m}\bigg{(}\frac{b_{iL}}{1+M_{1}^{\alpha_{i}}}+\frac{c_{iL}}{1+M_{1}^{\beta_{i}}}-H_{iM}\bigg{)}+\frac{B\delta_{L}e^{-a_{M}{\overline{\eta}}}}{1-e^{-a_{M}{\bar{\eta}}}}\
\text{ if }\delta_{L}\geq 0,\\\
\frac{B}{a_{M}}\sum_{i=1}^{m}\bigg{(}\dfrac{b_{iL}}{1+M_{1}^{\alpha_{i}}}+\frac{c_{iL}}{1+M_{1}^{\beta_{i}}}-H_{iM}\bigg{)}+\frac{A\delta_{L}}{1-e^{-a_{L}\eta}}\
\text{ if }\delta_{L}<0.\end{cases}$

######  Lemma 3.1.

Let Assumptions (A1)-(A7) hold. If $\phi\in\mathcal{D}_{1}$ then

$M_{2}\leq F\phi(t)\leq M_{1},\ \forall t\in\mathbb{R}.$

###### Proof.

Let $\phi\in\mathcal{D}_{1}$. By Lemma 2.5 and Lemma 2.6, from (3.1) we have

(3.3) $\displaystyle F\phi(t)$
$\displaystyle\leq\int_{-\infty}^{t}Ae^{-a_{L}(t-s)}\sum_{i=1}^{m}\left(b_{iM}+c_{iM}-H_{iL}\right)ds+A\delta_{M}\sum_{t_{k}<t}e^{-a_{L}(t-t_{k})}$
$\displaystyle\leq\frac{A}{a_{L}}\sum_{i=1}^{m}\left(b_{iM}+c_{iM}-H_{iL}\right)+\frac{A\delta_{M}}{1-e^{-a_{L}\eta}}=M_{1},\
\forall t\in\mathbb{R}.$

For each $t\in\mathbb{R}$, let $n_{0}$ be an integer such that
$t_{n_{0}}<t\leq t_{n_{0}+1}$. If $\delta_{L}\geq 0$ then, by Lemma 2.6, it
follows from the fact $(n-k)\eta\leq t_{n}-t_{k}\leq(n-k)\bar{\eta},\ \forall
k\leq n$ that

(3.4) $\displaystyle\sum_{t_{k}<t}H(t,t_{k})\delta_{k}$
$\displaystyle\geq\sum_{t_{k}<t}B\delta_{L}e^{-a_{M}(t-t_{k})}\geq\sum_{t_{k}\leq
t_{n_{0}}}B\delta_{L}e^{-a_{M}(t_{n_{0}+1}-t_{k})}$ $\displaystyle=\sum_{k\leq
n_{0}}B\delta_{L}e^{-a_{M}(t_{n_{0}+1}-t_{k})}\geq\sum_{q=1}^{\infty}B\delta_{L}e^{-a_{M}\bar{\eta}q}=\frac{B\delta_{L}e^{-a_{M}\bar{\eta}}}{1-e^{-a_{M}\bar{\eta}}}.$

If $\delta_{L}<0$ then from (A7) and Lemma 2.6, we have

(3.5) $\displaystyle\sum_{t_{k}<t}H(t,t_{k})\delta_{k}$
$\displaystyle\geq\sum_{t_{k}<t}A\delta_{L}e^{-a_{L}(t-t_{k})}\geq\sum_{t_{k}\leq
t_{n_{0}}}A\delta_{L}e^{-a_{L}(t_{n_{0}}-t_{k})}$
$\displaystyle\geq\sum_{q=0}^{\infty}A\delta_{L}e^{-a_{L}q\eta}=\frac{A\delta}{1-e^{-a_{L}\eta}}.$

From (3.4) and (3.5) we obtain

(3.6) $\displaystyle
F\phi(t)\geq\frac{B}{a_{M}}\sum_{i=1}^{m}\bigg{(}\frac{b_{iL}}{1+M_{1}^{\alpha_{i}}}+\frac{c_{iL}}{1+M_{1}^{\beta_{i}}}-H_{iM}\bigg{)}+\sum_{t_{k}<t}H(t,t_{k})\delta_{k}\geq
M_{2}.$

The proof is completed. ∎

Now we are in position to introduce our main results as follows.

######  Theorem 3.1.

Under the Assumptions (A1)-(A7), if $\phi\in\mathcal{D}_{1}$ then $F\phi(t)$
is almost periodic.

###### Proof.

Let $\phi\in\mathcal{D}_{1}$. For given $\epsilon\in(0,\eta/2)$, there exists
$0<\delta<\epsilon/2$ such that, if $t,\bar{t}$ belong to the same interval of
continuity of $\phi(t)$ then

(3.7)
$\left|\phi(t)-\phi\left(\bar{t}\right)\right|<\epsilon,\;|t-\bar{t}|<\delta.$

By Lemma 2.2 and Lemma 2.7, there exist $0<\epsilon_{1}<\delta$, relatively
dense sets $\Omega\subset\mathbb{R}$, $\mathcal{P}\subset\mathbb{Z}$ such
that, for all $\omega\in\Omega$, we have

(3.8) $\displaystyle|H(t+\omega,s+\omega)-H(t,s)|\leq\delta
Me^{-\frac{1}{2}a_{L}(t-s)},\ t\geq s,|t-t_{k}|>\delta,|s-t_{k}|>\delta;$
$\displaystyle|\phi(t+\omega)-\phi(t)|<\delta,\
t\in\mathbb{R},|t-t_{k}|>\delta,k\in\mathbb{Z};$
$\displaystyle|H_{i}(t+\omega,x)-H_{i}(t,x)|<\delta,\
t\in\mathbb{R},x\in[\phi_{L},\|\phi\|],i\in\underline{m};$
$\displaystyle|a(t+\omega)-a(t)|<\delta,\ t\in\mathbb{R};$
$\displaystyle|b_{i}(t+\omega)-b_{i}(t)|<\delta,\;|c_{i}(t+\omega)-c_{i}(t)|<\delta,\
t\in\mathbb{R},i\in\underline{m};$
$\displaystyle|\tau_{i}(t+\omega)-\tau_{i}(t)|<\delta,\;|\sigma_{i}(t+\omega)-\sigma_{i}(t)|<\delta,\
t\in\mathbb{R},i\in\underline{m};$
$\displaystyle|\gamma_{k+p}-\gamma_{k}|<\delta,\;|\delta_{k+p}-\delta_{k}|<\delta,\;|t^{p}_{k}-\omega|<\epsilon_{1},p\in\mathcal{P},k\in\mathbb{Z}.$

Let $\omega\in\Omega$, $p\in\mathcal{P}$. One can easily see that

(3.9) $\displaystyle F\phi(t+\omega)=$
$\displaystyle\int_{-\infty}^{t}H(t+\omega,s+\omega)\sum_{i=1}^{m}\bigg{\\{}\frac{b_{i}(s+\omega)}{1+\phi^{\alpha_{i}}(s+\omega-\tau_{i}(s+\omega))}$
$\displaystyle+\int_{0}^{T}\frac{c_{i}(s+\omega)v_{i}(r)}{1+\phi^{\beta_{i}}(s+\omega-r)}dr-
H_{i}(s+\omega,\phi(s+\omega-\sigma_{i}(s+\omega)))\bigg{\\}}ds$
$\displaystyle+\sum_{t_{k}<t}H(t+\omega,t_{k+p})\delta_{k+p}.$

We define
$E_{\epsilon}(\\{t_{k}\\})=\\{t\in\mathbb{R}:\;|t-t_{k}|>\epsilon,\forall
k\in\mathbb{Z}\\}$. For $t\in E_{\epsilon}(\\{t_{k}\\})$, $i\in\underline{m}$,
let us set

(3.10) $\displaystyle C_{i}$
$\displaystyle=\int_{-\infty}^{t}\left|\frac{H(t+\omega,s+\omega)b_{i}(s+\omega)}{1+\phi^{\alpha_{i}}(s+\omega-\tau_{i}(s+\omega))}-\frac{H(t,s)b_{i}(s)}{1+\phi^{\alpha_{i}}(s-\tau_{i}(s))}\right|ds,$
$\displaystyle D_{i}$
$\displaystyle=\int_{-\infty}^{t}\left|H(t+\omega,s+\omega)\int_{0}^{T}\frac{c_{i}(s+\omega)v_{i}(r)}{1+\phi^{\beta_{i}}(s+\omega-r)}dr-H(t,s)\int_{0}^{T}\frac{c_{i}(s)v_{i}(r)}{1+\phi^{\beta_{i}}(s-r)}dr\right|ds,$
$\displaystyle E_{i}$
$\displaystyle=\int_{-\infty}^{t}\left|H(t+\omega,s+\omega)H_{i}(s+\omega,\phi(s+\omega-\sigma_{i}(s+\omega)))-H(t,s)H_{i}(s,\phi(s-\sigma_{i}(s)))\right|ds,$
$\displaystyle G$
$\displaystyle=\sum_{t_{k}<t}\left|H(t+\omega,t_{k+p})\delta_{k+p}-H(t,t_{k})\delta_{k}\right|,$

then we have

(3.11)
$|F\phi(t+\omega)-F\phi(t)|\leq\sum_{i=1}^{m}(C_{i}+D_{i}+E_{i})+G,\;t\in
E_{\epsilon}(\\{t_{k}\\}).$

We also define

(3.12) $\displaystyle C_{i1}$
$\displaystyle=\int_{-\infty}^{t}|H(t+\omega,s+\omega)-H(t,s)|\frac{|b_{i}(s+\omega)|}{1+\phi^{\alpha_{i}}(s+\omega-\tau_{i}(s+\omega))}ds,$
$\displaystyle C_{i2}$
$\displaystyle=\int_{-\infty}^{t}H(t,s)\frac{|b_{i}(s+\omega)-b_{i}(s)|}{1+\phi^{\alpha_{i}}(s+\omega-\tau_{i}(s+\omega))}ds,$
$\displaystyle C_{i3}$
$\displaystyle=\int_{-\infty}^{t}H(t,s)|\phi(s+\omega-\tau_{i}(s+\omega))-\phi(s-\tau_{i}(s+\omega))|ds,$
$\displaystyle C_{i4}$
$\displaystyle=\int_{-\infty}^{t}H(t,s)|\phi(s-\tau_{i}(s+\omega))-\phi(s-\tau_{i}(s))|ds$

and $K_{i}=\sup_{\phi_{L}\leq x\leq\|\phi\|}\alpha_{i}x^{\alpha_{i}-1}$. It
can be seen from (3.10) and (3.12) that

(3.13) $C_{i}\leq
C_{i1}+C_{i2}+b_{iM}K_{i}(C_{i3}+C_{i4}),\;i\in\underline{m}.$

By Lemma 2.5 and Lemma 2.6, from (3.8), (3.12) and the fact that
$\int_{-\infty}^{t}e^{-\frac{1}{2}a_{L}(t-s)}ds=2/a_{L}$, we have

(3.14)
$C_{i1}\leq\frac{2b_{iM}M}{a_{L}}+\sum_{t_{k}<t}2Ab_{iM}\int_{t_{k}-\epsilon}^{t_{k}+\epsilon}e^{-a_{L}(t-s)}ds\leq
b_{iM}\bigg{(}\frac{2M}{a_{L}}+\frac{4Ae^{\frac{1}{2}a_{L}\eta}}{1-e^{-a_{L}\eta}}\bigg{)}\epsilon,$

and

(3.15) $\displaystyle C_{i2}\leq\int_{-\infty}^{t}A\epsilon
e^{-a_{L}(t-s)}ds=\frac{A}{a_{L}}\epsilon,$ $\displaystyle
C_{i3}\leq\int_{-\infty}^{t}A\epsilon
e^{-a_{L}(t-s)}ds+2A\|\phi\|\sum_{t_{k}<t}\int_{\\{s:|s-\tau_{i}(s+\omega)-t_{k}|<\epsilon,s\leq
t\\}}e^{-a_{L}(t-s)}ds.$

It should be noted that, by (A4), $t-\tau_{i}(t)$, $i\in\underline{m}$, are
strictly increasing functions, and thus, there exist the inverse functions
$\tau^{*}_{i}(t)$ of $t-\tau_{i}(t)$. For each $t\in\mathbb{R}$, denote
$\bar{t}=t-\epsilon-\tau_{i}(t+\omega)$ then

(3.16) $t+\omega=\tau^{*}_{i}(\bar{t}+\omega+\epsilon).$

Let $\underline{\lambda}_{i}=\inf_{s\in\mathbb{R}}\dot{\tau}^{*}_{i}(s)$,
$\overline{\lambda}_{i}=\sup_{s\in\mathbb{R}}\dot{\tau}^{*}_{i}(s),i\in\underline{m}$,
then, by (A4), $0<\underline{\lambda}_{i},\overline{\lambda}_{i}<\infty.$
Therefore,

(3.17)
$\tau^{*}_{i}(\bar{t}+\omega+\epsilon)-\tau^{*}_{i}(t_{k}+\omega+\epsilon)\geq\underline{\lambda}_{i}(\bar{t}-t_{k}),\
t_{k}<\bar{t},$

and hence, from equations (3.15)-(3.17), we have

(3.18) $\displaystyle C_{i3}$
$\displaystyle\leq\frac{A\epsilon}{a_{L}}+2A\|\phi\|\sum_{t_{k}<\bar{t}}\int_{\tau^{*}_{i}(t_{k}+\omega-\epsilon)-\omega}^{\tau^{*}_{i}(t_{k}+\omega+\epsilon)-\omega}e^{-a_{L}(t-s)}ds$
$\displaystyle\leq\frac{A\epsilon}{a_{L}}+2A\|\phi\|\sum_{t_{k}<\bar{t}}e^{-a_{L}[\tau^{*}_{i}(\bar{t}+\omega+\epsilon)-\tau^{*}_{i}(t_{k}+\omega+\epsilon)]}\left[\tau^{*}_{i}(t_{k}+\omega+\epsilon)-\tau^{*}_{i}(t_{k}+\omega-\epsilon)\right]$
$\displaystyle\leq\frac{A\epsilon}{a_{L}}+4A\|\phi\|\overline{\lambda}_{i}\epsilon\sum_{t_{k}<\bar{t}}e^{-a_{L}{\underline{\lambda}}_{i}(\bar{t}-t_{k})}\leq\bigg{(}\frac{1}{a_{L}}+\frac{4\overline{\lambda}_{i}\|\phi\|}{1-e^{-a_{L}{\underline{\lambda}}_{i}\eta}}\bigg{)}A\epsilon.$

By the same arguments used in deriving (3.18), we obtain

(3.19) $C_{i4}\leq
A\epsilon\int_{-\infty}^{t}e^{-a_{L}(t-s)}ds+2A\|\phi\|\sum_{t_{k}<t}\int_{\tau^{*}_{i}(t_{k}-\epsilon)}^{\tau^{*}_{i}(t_{k}+\epsilon)}e^{-a_{L}(t-s)}ds\leq\bigg{(}\frac{1}{a_{L}}+\frac{4\overline{\lambda}_{i}\|\phi\|}{1-e^{-a_{L}{\underline{\lambda}}_{i}\eta}}\bigg{)}A\epsilon.$

Combining (3.13)-(3.15), (3.18) and (3.19), we readily obtain

(3.20)
$C_{i}\leq\left[\frac{A}{a_{L}}+\frac{2b_{iM}(M+AK_{i})}{a_{L}}+Ab_{iM}\bigg{(}\frac{4e^{\frac{1}{2}a_{L}\eta}}{1-e^{-a_{L}\eta}}+\frac{8K_{i}\overline{\lambda}_{i}\|\phi\|}{1-e^{-a_{L}{\underline{\lambda}}_{i}\eta}}\bigg{)}\right]\epsilon.$

Next, let us set

(3.21) $\displaystyle D_{i1}$
$\displaystyle=\int_{-\infty}^{t}|H(t+\omega,s+\omega)-H(t,s)|\int_{0}^{T}\frac{v_{i}(r)}{1+\phi^{\beta_{i}}(s+\omega-r)}drds,$
$\displaystyle D_{i2}$
$\displaystyle=\int_{-\infty}^{t}H(t,s)|c_{i}(s+\omega)-c_{i}(s)|\int_{0}^{T}\frac{v_{i}(r)}{1+\phi^{\beta_{i}}(s+\omega-r)}drds,$
$\displaystyle D_{i3}$
$\displaystyle=\int_{-\infty}^{t}H(t,s)\int_{0}^{T}v_{i}(r)\left|\frac{1}{1+\phi^{\beta_{i}}(s+\omega-r)}-\frac{1}{1+\phi^{\beta_{i}}(s-r)}\right|drds.$

It follows from (3.10) and (3.21) that

(3.22) $D_{i}\leq c_{iM}D_{i1}+D_{i2}+c_{iM}D_{i3},\ i\in\underline{m}.$

By Lemmas 2.5 and 2.6, from (3.8) we have

(3.23) $\displaystyle D_{i1}$ $\displaystyle\leq\int_{-\infty}^{t}M\epsilon
e^{-\frac{1}{2}a_{L}(t-s)}ds+\sum_{t_{k}<t}2A\int_{t_{k}-\epsilon}^{t_{k}+\epsilon}e^{-a_{L}(t-s)}\leq\bigg{(}\frac{2M}{a_{L}}+\frac{4Ae^{\frac{1}{2}a_{L}\eta}}{1-e^{-a_{L}\eta}}\bigg{)}\epsilon,$
$\displaystyle D_{i2}$ $\displaystyle\leq\int_{-\infty}^{t}A\epsilon
e^{-a_{L}(t-s)}ds=\frac{A}{a_{L}}\epsilon,$

and

(3.24) $\displaystyle D_{i3}$
$\displaystyle\leq\int_{-\infty}^{t}Ae^{-a_{L}(t-s)}\int_{0}^{T}G_{i}v_{i}(r)|\phi(s+\omega-r)-\phi(s-r)|drds$
$\displaystyle\leq\int_{-\infty}^{t}AG_{i}\epsilon
e^{-a_{L}(t-s)}ds+2AG_{i}\|\phi\|\int_{0}^{T}v_{i}(r)\bigg{(}\sum_{t_{k}+r<t}\int_{t_{k}-r-\epsilon}^{t_{k}+r+\epsilon}e^{-a_{L}(t-s)}ds\bigg{)}dr$
$\displaystyle\leq\frac{AG_{i}\epsilon}{a_{L}}+{4AG_{i}\|\phi\|\epsilon}\int_{0}^{T}v_{i}(r)\bigg{(}\sum_{t_{k}+r<t}e^{-a_{L}(t-t_{k}-r-\epsilon)}\bigg{)}dr$
$\displaystyle\leq\bigg{(}\frac{1}{a_{L}}+\frac{4e^{\frac{1}{2}a_{L}\eta}\|\phi\|}{1-e^{-a_{L}\eta}}\bigg{)}AG_{i}\epsilon,$

where $G_{i}=\sup_{\phi_{L}\leq x\leq\|\phi\|}\beta_{i}x^{\beta_{i}-1}$. From
(3.22)-(3.24), we readily obtain

(3.25)
$D_{i}\leq\left[\frac{A+2Mc_{iM}+Ac_{iM}G_{i}}{a_{L}}+\frac{4Ac_{iM}e^{\frac{1}{2}a_{L}\eta}(1+G_{i}\|\phi\|)}{1-e^{-a_{L}\eta}}\right]\epsilon.$

Now, we define

(3.26) $\displaystyle E_{i1}$
$\displaystyle=\int_{-\infty}^{t}|H(t+\omega,s+\omega)-H(t,s)|H_{i}(s+\omega,\phi(s+\omega-\sigma_{i}(s+\omega)))ds,$
$\displaystyle E_{i2}$
$\displaystyle=\int_{-\infty}^{t}H(t,s)|H_{i}(s+\omega,\phi(s+\omega-\sigma_{i}(s+\omega)))-H_{i}(s+\omega,\phi(s-\sigma_{i}(s+\omega)))|ds,$
$\displaystyle E_{i3}$
$\displaystyle=\int_{-\infty}^{t}H(t,s)|H_{i}(s+\omega,\phi(s-\sigma_{i}(s+\omega)))-H_{i}(s+\omega,\phi(s-\sigma_{i}(s)))|ds,$
$\displaystyle E_{i4}$
$\displaystyle=\int_{-\infty}^{t}H(t,s)|H_{i}(s+\omega,\phi(s-\sigma_{i}(s)))-H_{i}(s,\phi(s-\sigma_{i}(s)))|ds,$

then, from (3.10) and (3.26), we have

(3.27) $E_{i}\leq E_{i1}+E_{i2}+E_{i3}+E_{i4},\ i\in\underline{m}.$

Also using Lemma 2.5 and Lemma 2.6, from (3.8) and the fact that
$\int_{-\infty}^{t}e^{-\frac{1}{2}a_{L}(t-s)}ds=2/a_{L}$, we obtain

(3.28) $\displaystyle E_{i1}$
$\displaystyle\leq\frac{2H_{iM}M\epsilon}{a_{L}}+\sum_{t_{k}<t}2AH_{iM}\int_{t_{k}-\epsilon}^{t_{k}+\epsilon}e^{-a_{L}(t-s)}ds\leq
2\left(\frac{M}{a_{L}}+\frac{2Ae^{\frac{1}{2}a_{L}\eta}}{1-e^{-a_{L}\eta}}\right)H_{iM}\epsilon,$
$\displaystyle E_{i4}$ $\displaystyle\leq\int_{-\infty}^{t}A\epsilon
e^{-a_{L}(t-s)}ds\leq\frac{A}{a_{L}}\epsilon.$

Let
$\underline{\xi}_{i}=\inf_{t\in\mathbb{R}}\dot{\sigma}^{*}_{i}(t),\overline{\xi}_{i}=\sup_{t\in\mathbb{R}}\dot{\sigma}^{*}_{i}(t),i\in\underline{m}$.
Similarly to (3.18) and (3.19), we readily obtain

(3.29) $\displaystyle E_{i2}$ $\displaystyle\leq
L_{i}\int_{-\infty}^{t}H(t,s)|\phi(s+\omega-\sigma_{i}(s+\omega))-\phi(s-\sigma_{i}(s+\omega))|ds$
$\displaystyle\leq\left(\frac{1}{a_{L}}+\frac{4\overline{\xi}_{i}\|\phi\|}{1-e^{-a_{L}{\underline{\xi}}_{i}\eta}}\right)AL_{i}\epsilon,$
$\displaystyle E_{i3}$ $\displaystyle\leq
L_{i}\int_{-\infty}^{t}H(t,s)|\phi(s-\sigma_{i}(s+\omega))-\phi(s-\sigma_{i}(s))|ds$
$\displaystyle\leq\left(\frac{1}{a_{L}}+\frac{4\overline{\xi}_{i}\|\phi\|}{1-e^{-a_{L}{\underline{\xi}}_{i}\eta}}\right)AL_{i}\epsilon.$

Inequalities (3.27)-(3.29) yield

(3.30)
$E_{i}\leq\bigg{(}\frac{A(2L_{i}+1)+2H_{iM}M}{a_{L}}+\frac{4Ae^{\frac{1}{2}a_{L}\eta}H_{iM}}{1-e^{-a_{L}\eta}}+\frac{8AL_{i}\overline{\xi}_{i}\|\phi\|}{1-e^{-a_{L}\underline{\xi}_{i}\eta}}\bigg{)}\epsilon.$

Let us set

(3.31) $G_{1}=\sum_{t_{k}<t}|H(t+\omega,t_{k+p})-H(t,t_{k})|,\quad
G_{2}=\sum_{t_{k}<t}|H(t,t_{k})|$

then

(3.32)
$G\leq\sum_{t_{k}<t}|H(t+\omega,t_{k+p})-H(t,t_{k})||\delta_{k+p}|+\sum_{t_{k}<t}|H(t,t_{k})||\delta_{k+p}-\delta_{k}|\leq\bar{\delta}G_{1}+\epsilon
G_{2}.$

For each $t\geq t_{k}$, there exists a unique integer $l=l(t)$ such that
$t_{l}<t\leq t_{l+1}$. By Lemma 2.3 we have $t_{l+p}<t+\omega<t_{l+p+1}$. Thus

(3.33) $\displaystyle G_{1}$
$\displaystyle\leq\sum_{t_{k}<t}\bigg{(}\Gamma_{M}e^{-a_{L}(t-t_{k}-\epsilon)}\left|\int_{t_{k+p}}^{t+\omega}a(s)ds-\int_{t_{k}}^{t}a(s)ds\right|$
$\displaystyle\quad+e^{-a_{L}(t-t_{k})}\left|\Gamma(k+p,l+p)-\Gamma(k,l)\right|\bigg{)}$
$\displaystyle\leq\Gamma_{M}I_{1}+I_{2},$

where

(3.34) $\displaystyle
I_{1}=\sum_{t_{k}<t}e^{-a_{L}(t-t_{k}-\epsilon)}\left|\int_{t_{k+p}}^{t+\omega}a(s)ds-\int_{t_{k}}^{t}a(s)ds\right|,$
$\displaystyle
I_{2}=\sum_{t_{k}<t}e^{-a_{L}(t-t_{k})}\left|\Gamma(k+p,l+p)-\Gamma(k,l)\right|.$

Note that $\epsilon<\eta/2,\
\dfrac{1}{2}a_{L}(t-t_{k})e^{-\frac{1}{2}a_{L}(t-t_{k})}<1$ and
$|t^{p}_{k}-\omega|<\epsilon_{1}<\epsilon$, from (3.8) and Lemma 2.5, we have

(3.35) $\displaystyle I_{1}$
$\displaystyle\leq\sum_{t_{k}<t}e^{-a_{L}(t-t_{k}-\epsilon)}\bigg{(}\int_{t_{k}}^{t}|a(s+\omega)-a(s)|ds+\bigg{|}\int_{t_{k+p}-\omega}^{t_{k}}a(s)ds\bigg{|}\bigg{)}$
$\displaystyle\leq\epsilon\sum_{t_{k}<t}e^{-a_{L}(t-t_{k}-\epsilon)}(t-t_{k}+a_{M})$
$\displaystyle\leq\left(a_{M}\epsilon
e^{\frac{1}{2}a_{L}\eta}+\frac{2\epsilon}{a_{L}}e^{\frac{1}{2}a_{L}\eta}\right)\sum_{t_{k}<t}e^{-\frac{1}{2}a_{L}(t-t_{k})}$
$\displaystyle\leq\frac{e^{\frac{1}{2}a_{L}\eta}}{1-e^{-\frac{1}{2}a_{L}\eta}}\bigg{(}a_{M}+\frac{2}{a_{L}}\bigg{)}\epsilon.$

Similarly, we obtain

(3.36) $\displaystyle I_{2}$
$\displaystyle\leq\frac{\Gamma_{M}}{1+\gamma_{L}}\sum_{t_{k}<t}e^{-a_{L}(t-t_{k})}(l-k+1)\epsilon\leq\frac{\Gamma_{M}}{1+\gamma_{L}}\sum_{t_{k}<t}e^{-a_{L}(t-t_{k})}\bigg{(}\frac{t-t_{k}}{\eta}+1\bigg{)}\epsilon$
$\displaystyle\leq\frac{\Gamma_{M}}{(1+\gamma_{L})(1-e^{-\frac{1}{2}a_{L}\eta})}\bigg{(}\frac{2}{a_{L}\eta}+1\bigg{)}\epsilon.$

It follows from (3.33)-(3.36) that

(3.37)
$G_{1}\leq\frac{\Gamma_{M}}{1-e^{-\frac{1}{2}a_{L}\eta}}\left[e^{\frac{1}{2}a_{L}\eta}\left(a_{M}+\frac{2}{a_{L}}\right)+\frac{1}{1+\gamma_{L}}\left(\frac{2}{a_{L}\eta}+1\right)\right]\epsilon.$

We also have

(3.38) $G_{2}\leq
A\sum_{t_{k}<t}e^{-a_{L}(t-t_{k})}\leq\frac{A}{1-e^{-a_{L}\eta}}.$

Therefore

(3.39)
$G\leq\frac{\epsilon\overline{\delta}\Gamma_{M}}{1-e^{-\frac{1}{2}a_{L}\eta}}\left[e^{\frac{1}{2}a_{L}\eta}\bigg{(}a_{M}+\frac{2}{a_{L}}\bigg{)}+\frac{1}{1+\gamma_{L}}\bigg{(}\frac{2}{a_{L}\eta}+1\bigg{)}\right]+\frac{A\epsilon}{1-e^{-\frac{1}{2}a_{L}\eta}}.$

We can see clearly from (3.11), (3.20), (3.25), (3.30) and (3.39) that there
exists a positive constant $\Lambda$ such that
$|F(\phi(t+\omega)-F\phi(t)|\leq\Lambda\epsilon$, for all
$t\in\mathbb{R},|t-t_{k}|>\epsilon$, $k\in\mathbb{Z}$. This shows that
$F\phi(t)$ is almost periodic. The proof is completed. ∎

######  Theorem 3.2.

Let Assumptions (A1)-(A7) hold. If $M_{2}$, defined in (3.2), is positive and

(3.40)
$\frac{A}{a_{L}}\sum_{i=1}^{m}\left(b_{iM}K^{*}_{i}+c_{iM}G^{*}_{i}+L_{i}\right)<1,$

where $K^{*}_{i}=\sup_{M_{2}\leq x\leq M_{1}}\alpha_{i}x^{\alpha_{i}-1},\
G^{*}_{i}=\sup_{M_{2}\leq x\leq M_{1}}\beta_{i}x^{\beta_{i}-1}$, then equation
(1.6) has a unique positive almost periodic solution.

###### Proof.

We define
$\mathcal{D}_{2}=\big{\\{}\phi\in\mathcal{D}_{1}:M_{2}\leq\phi(t)\leq M_{1},\
t\in\mathbb{R}\big{\\}}$. It is worth noting that, from Lemma 3.1, Theorem 3.1
and the assumption $M_{2}>0$, we have
$F(\mathcal{D}_{2})\subset{\mathcal{D}}_{2}$. For any
$\phi,\psi\in\mathcal{D}_{2}$, applying Lemma 2.6 we obtain

$\displaystyle|F\phi(t)-F\psi(t)|\leq\;$
$\displaystyle\int_{-\infty}^{t}H(t,s)\sum_{i=1}^{m}\bigg{\\{}L_{i}|\phi(s-\sigma_{i}(s))-\psi(s-\sigma_{i}(s))|$
$\displaystyle+c_{iM}\int_{0}^{T}v_{i}(r)\left|\frac{1}{1+\phi^{\beta_{i}}(s-r)}-\frac{1}{1+\psi^{\beta_{i}}(s-r)}\right|dr$
$\displaystyle+b_{iM}\left|\frac{1}{1+\phi^{\alpha_{i}}(s-\tau_{i}(s))}-\frac{1}{1+\psi^{\alpha_{i}}(s-\tau_{i}(s))}\right|\bigg{\\}}ds$
$\displaystyle\leq\;$
$\displaystyle\frac{A}{a_{L}}\sum_{i=1}^{m}\left(b_{iM}K^{*}_{i}+c_{iM}G^{*}_{i}+L_{i}\right)\|\phi-\psi\|.$

Therefore

$\|F\phi-F\psi\|\leq\frac{A}{a_{L}}\sum_{i=1}^{m}\left(b_{iM}K^{*}_{i}+c_{iM}G^{*}_{i}+L_{i}\right)\|\phi-\psi\|$

which yields $F$ is a contraction mapping on ${\mathcal{D}}_{2}$ by condition
(3.40). Then $F$ has a unique fixed point in $\mathcal{D}_{2}$, namely
$\phi_{0}$. It should be noted that,
$F(\mathcal{D}_{1})\subset\mathcal{D}_{2}$, and hence, $F$ also has a unique
fixed point $\phi_{0}$ in $\mathcal{D}_{1}$. This shows that (1.6) has a
unique positive almost periodic solution $x^{*}(t)=\phi_{0}(t)$. The proof is
completed. ∎

######  Theorem 3.3.

Let Assumptions (A1)-(A7) hold. If $M_{2}>0,\sigma\leq\eta$ and

(3.41)
$\frac{1}{a_{L}}\max\left\\{A,(\gamma_{L}+1)^{-1}\right\\}\sum_{i=1}^{m}\left(b_{iM}K^{*}_{i}+c_{iM}G^{*}_{i}+L_{i}\right)<1,$

then (1.6) has a unique positive almost periodic solution $x^{*}(t)$.
Moreover, every solution $x(t)=x(t,\alpha,\xi)$ of (1.6) converges
exponentially to $x^{*}(t)$ as $t\to\infty$.

###### Proof.

By Theorem 3.2, (1.6) has a unique positive almost periodic solution
$x^{*}(t)$. Let $x(t)=x(t,\alpha,\xi)$ be a solution of (1.6) and (2.1). We
define $V(t)=|x(t)-x^{*}(t)|$ then

$\displaystyle D^{+}V(t)$ $\displaystyle\leq-
a_{L}|x(t)-x^{*}(t)|+\sum_{i=1}^{m}\bigg{[}b_{iM}K^{*}_{i}|x(t-\tau_{i}(t))-x^{*}(t-\tau_{i}(t))|$
$\displaystyle\quad+c_{iM}G^{*}_{i}\int_{0}^{T}v_{i}(s)|x(t-s)-x^{*}(t-s)|ds+L_{i}|x(t-\sigma_{i}(t))-x^{*}(t-\sigma_{i}(t))|\bigg{]}$
$\displaystyle\leq-
a_{L}V(t)+\sum_{i=1}^{m}\left(b_{iM}K^{*}_{i}+c_{iM}G^{*}_{i}+L_{i}\right)\overline{V}(t),\;t\neq
t_{k},\ t\geq\alpha,$ $\displaystyle\Delta V(t_{k})$
$\displaystyle=\gamma_{k}V(t^{-}_{k}),\;t_{k}\geq\alpha,k\in\mathbb{Z},$

where $\overline{V}(t)=\sup_{t-\sigma\leq s\leq t}V(s)$. By Lemma 2.8, there
exists a positive constant $\lambda$ such that

$V(t)\leq\overline{V}(\alpha)\prod_{\alpha<t_{k}\leq
t}(\gamma_{k}+1)e^{-\lambda(t-\alpha)}\leq\Gamma_{M}\overline{V}(\alpha)e^{-\lambda(t-\alpha)},\;t\geq\alpha.$

This shows that $x(t)$ converges exponentially to $x^{*}(t)$ as $t\to\infty$.
The proof is completed. ∎

The existence and exponential stability of positive almost periodic solution
of (1.4) is presented in the following corollary as an application of our
obtained results with $m=1$.

######  Corollary 3.4.

Under the Assumptions (A1), (A2) (with $c(t)=0$) and (A5)-(A7), if
$M_{2}>0,\tau\leq\eta$ and

(3.42)
$\frac{1}{a_{L}}\max\left\\{A,(\gamma_{L}+1)^{-1}\right\\}b_{M}K^{*}<1,$

where $K^{*}=\sup_{M_{2}\leq x\leq M_{1}}\alpha x^{\alpha-1}$, then (1.4) has
a unique positive almost periodic solution which is exponentially stable.

##  4\. An illustrative example

In this section we give a numerical example to illustrate the effectiveness of
our conditions. For illustrating purpose, let us consider the following
equation

(4.1)
$\begin{split}\dot{x}(t)=\;&-a(t)x(t)+\frac{b(t)}{1+x^{2}(t-\tau(t))}+\int_{0}^{1}\frac{c(t)ds}{1+x^{2}(t-s)}\\\
&-d(t)\frac{|x(t-\sigma(t))|}{10+|x(t-\sigma(t))|},\quad t\neq k,\\\ \Delta
x(k)=\;&\gamma_{k}x(k-0)+\delta_{k},\quad k\in\mathbb{Z},\end{split}$

where

$\displaystyle
a(t)=5+|\sin(t\sqrt{2})|,\;b(t)=\frac{1}{10}(1+|\sin(t\sqrt{3})|),\;c(t)=\frac{1}{10}(1+|\cos(t\sqrt{3})|),$
$\displaystyle
d(t)=\frac{1}{20}\sin^{2}(t\sqrt{3}),\;\tau(t)=\sin^{2}(\frac{\sqrt{3}}{2}t),\;\sigma(t)=\cos^{2}(\frac{\sqrt{3}}{2}t),$
$\displaystyle\gamma_{2m}=-\frac{1}{2},\ \gamma_{2m+1}=1,\ \delta_{2m}=1,\
\delta_{2m+1}=\frac{1}{2},\ m\in\mathbb{Z}.$

It should be noted that, the functions $a(t),b(t),c(t),\tau(t)$ and
$\sigma(t)$ are almost periodic in the sense of Bohr,
$H(t,x)=d(t)\dfrac{|x|}{10+|x|}$ is almost periodic in $t\in\mathbb{R}$
uniformly in $x\in\mathbb{R}_{+}$, $|H(t,x)-H(t,y)|\leq\frac{1}{2}|x-y|$.
Therefore, Assumptions (A1)-(A5) and (A7) are satisfied. On the other hand,
$\Gamma(q,p)=\prod_{i=q}^{p}(1+\gamma_{i})\in\\{\frac{1}{2},1,2\\}$ for any
$p,q\in\mathbb{Z},p\geq q$. Thus, $\Gamma_{M}=2,\Gamma_{L}=\dfrac{1}{2}$ and
Assumption (A6) is satisfied. Taking some computations we obtain

$\displaystyle a_{L}=5,\ a_{M}=6,\ b_{L}=c_{L}=0.1,\ b_{M}=c_{M}=0.2,L\leq
0.5,H_{M}=\frac{1}{20},H_{L}=0,$ $\displaystyle A=2,\ B=0.5,\ M_{1}=2.1736,\
M_{2}=0.0027,\ K^{*}=G^{*}=2M_{1}$

and $\dfrac{A}{a_{L}}(b_{M}K^{*}+c_{M}G^{*}+L)\leq 0.8956$. By Theorem 3.2,
equation (4.1) has a unique positive almost periodic solution $x^{*}(t)$.
Furthermore, it can be seen that $\gamma_{L}=-\frac{1}{2}$, and hence,
$\dfrac{1}{a_{L}}\max\\{A,(\gamma_{L}+1)^{-1}\\}(b_{M}K^{*}+c_{M}G^{*}+L)\leq
0.8956$. By Theorem 3.3, every solution $x(t,\alpha,\xi)$ of (4.1) converges
exponentially to $x^{*}(t)$ as $t$ tends to infinity. As presented in figure
1, state trajectories of (4.1) with different initial conditions converge to
the unique positive almost periodic solution of (4.1).

Figure 1. State trajectories of (4.1) converge to the unique positive almost
periodic solution

##  5\. Conclusion

This paper has dealt with the existence and exponential attractivity of a
unique positive almost periodic solution for a generalized model of
hematopoiesis with delays and impulses. Using the contraction mapping
principle and a novel type of impulsive delay inequality, new sufficient
conditions have been derived ensuring that all solutions of the model converge
exponentially to the unique positive almost periodic solution.

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having inhomogeneous as well as homogeneous subspaces.) The hyperplane of a
half edge $e$ at $v_{i}$ is $x_{i}=-\varphi(e)$. The “honorary hyperplane” of
a loose edge is given by the equation $0=-\varphi(e)$, which is the
_degenerate hyperplane_ $\mathbf{h}(e)=\mathbb{A}^{n}(\mathbf{K})$ if
$\varphi(e)=0$ and the _phantom hyperplane_ $\varnothing$ if $\varphi(e)\neq
0$. (That makes sense in the projectivization.) The honorary hyperplane of the
extra point $e_{\infty}$ is the phantom hyperplane. Thus from a gain signed
graph $\Upsilon$ with gain group $\mathbf{K}^{+}$ we get the hyperplane
arrangement

$\mathcal{A}[\Upsilon]:=\\{\mathbf{h}(e):e\in E\\}.$

The intersection semilattice of this arrangement is

$\mathcal{L}(\mathcal{A}[\Upsilon]):=\\{\textstyle\bigcap\mathcal{S}:\mathcal{S}\subseteq\mathcal{A}[\Upsilon],\
\bigcap\mathcal{S}\neq\varnothing\\}.$

We list some arrangements of the type gain signed graphs are intended for,
which we call _(signed) affinographic_ since they are affine deformations of
graphic or signed-graphic hyperplane arrangements. By $\varepsilon e_{ij}$ we
denote an edge with endpoints $v_{i}$ and $v_{j}$ and sign $\varepsilon$,
while by $\pm e_{ij}$ we denote two edges with both signs, a positive edge
$+e_{ij}$ and a negative edge $-e_{ij}$. A simple graph $\Gamma$ thus gives
rise to a signed graph $\varepsilon\Gamma$, in which all edges have the same
sign $\varepsilon$, and to $\pm\Gamma=(+\Gamma)\cup(-\Gamma)$, in which all
edges are doubled with both signs.

  1. (Af1)

Affine arrangements whose edge sets have forms like $\\{(+e_{ij},g):i<j,\
-k\leq g\leq l\\}$, known as deformations of the (all-positive) complete-
graphic arrangement $\mathcal{A}_{n-1}$ with edge set $\\{(+e_{ij},0):i<j\\}$.

  2. (Af2)

Affine arrangements whose edge sets have forms like $\\{(\pm e_{ij},g):i<j,\
-k\leq g\leq l\\}\cup\\{(-e_{ii},g),\ -k\leq g\leq l\\}$, known as
deformations of the complete signed-graphic arrangement $\mathcal{B}_{n}$ with
edge set $\\{(\pm e_{ij},0):i<j\\}\cup\\{(-e_{ii},0)\\}$.

  3. (Af3)

Affine deformations whose edge sets have forms like $\\{(\pm e_{ij},g):i<j,\
-k\leq g\leq l\\}$, of the signed-graphic arrangement $\mathcal{D}_{n}$ with
edge set $\\{(\pm e_{ij},0):i<j\\}$.

  4. (Af4)

An affinographic arrangement of Coxeter type, whose edge set is as in (Af2) or
(Af3) with $k=0$ [2, Theorem 3.5].

  5. (Af5)

The sign-symmetric Shi arrangement, whose edge set is $\\{(\pm e_{ij},g):i<j,\
g=0,1\\}$, and a variety of similar arrangements [2, Section 3].

  6. (Af6)

The Shi threshold arrangement, whose edge set is $\\{(-e_{ij},g):i<j,\
g=0,1\\}$ [14], and the similar arrangement in [2, Theorem 5.4].

  7. (Af7)

The Linial threshold arrangement with coordinate and shifted hyperplanes,
whose edge set is $\\{(-e_{ij},1):i<j\\}\cup\\{(-e_{i},g):g=0,1\\}$ [16, 17,
18, 19].

  8. (Af8)

The Catalan threshold arrangement, whose edge set is $\\{(-e_{ij},g):i<j,\
g=0,\pm 1\\}$ [15].

  9. (Af9)

A generalized threshold arrangement, whose edge set is $\\{(-e_{ij},g):i<j,\
-k\leq g\leq l\\}$ [3].

###  10.2. Projective hyperplanes

The link between this affine arrangement and the vector model in linear space
is through the _projectivization_ ,

$\mathcal{A}_{\mathbb{P}}[\Upsilon]:=\\{\mathbf{h}(e)_{\mathbb{P}}:e\in
E\\}\cup\\{h_{\infty}\\},$

where $h_{\infty}$ denotes the ideal or infinite hyperplane while
$\mathbf{h}(e)_{\mathbb{P}}$ is the extension of $\mathbf{h}(e)$ into the
projective space $\mathbb{P}^{n}(\mathbf{K})$ if $\mathbf{h}(e)$ is not the
phantom hyperplane and $\mathbf{h}(e)_{\mathbb{P}}=h_{\infty}$ if
$\mathbf{h}(e)$ is the phantom hyperplane. (The affine part of $h_{\infty}$ is
$\varnothing$, which should explain the name “phantom hyperplane”.) We define
$\mathbf{h}_{\mathbb{P}}(e):=\mathbf{h}(e)_{\mathbb{P}}$ in order to have the
function $h_{\mathbb{P}}$ that maps edges to projective hyperplanes. The
addition of the ideal hyperplane in the projectivization ensures that, in the
real case, the regions remain the same, and in the complex case, the
complement of the arrangement remains the same.

The intersection lattice of the projective arrangement is

$\mathcal{L}(\mathcal{A}_{\mathbb{P}}[\Upsilon]):=\\{\textstyle\bigcap\mathcal{S}:\mathcal{S}\subseteq\mathcal{A}_{\mathbb{P}}[\Upsilon]\\}.$

This is the lattice of closed sets of a matroid
$\mathbf{M}(\mathcal{A}_{\mathbb{P}})$ whose ground set is the set of
hyperplanes and whose rank function is
$\operatorname{rk}\mathcal{S}=\operatorname{codim}\bigcap\mathcal{S}$. The
affine intersection semilattice $\mathcal{L}(\mathcal{A}[\Upsilon])$, which is
a meet subsemilattice of $\mathcal{L}(\mathcal{A}_{\mathbb{P}}[\Upsilon])$, is
a _geometric semilattice_ , as defined by Wachs and Walker [20]. One
definition is that it consists of the flats of a geometric lattice that do not
lie above a fixed atom; in our case the lattice is
$\mathcal{L}(\mathcal{A}_{\mathbb{P}}[\Upsilon])$ and the atom is
$h_{\infty}$. The corresponding matroid-like structure is a _semimatroid_ ,
defined subsequently by Ardila [1].

The coordinates in projective space are homogeneous coordinates
$[x_{0},x_{1},\ldots,x_{n}]$ (not all zero). Projecting the vector model in
$\mathbf{K}^{1+n}$ to homogeneous coordinates in $\mathbb{P}^{n}(\mathbf{K})$,
the hyperplane $\mathbf{h}_{\mathbb{P}}(e)$ is the dual space of the projected
vector $[\mathbf{z}(e)]$; that is,
$\mathbf{h}_{\mathbb{P}}(e)=\\{[\mathbf{y}]\in\mathbb{P}^{n}(\mathbf{K}):[\mathbf{z}(e)]\cdot[\mathbf{y}]=0\\}$.

######  Theorem 10.1 (Hyperplane Representations).

The matroid $\mathbf{M}(\mathcal{A}_{\mathbb{P}}[\Upsilon])$ is isomorphic to
$\mathbf{M}_{\infty}(\Upsilon)$ by the mapping $\mathbf{h}_{\mathbb{P}}$,
which induces a lattice isomorphism
$\mathcal{L}(\mathcal{A}_{\mathbb{P}}[\Upsilon])\cong\operatorname{Lat}\mathbf{M}_{\infty}(\Upsilon)$
and a semilattice isomorphism
$\mathcal{L}(\mathcal{A}[\Upsilon])\cong\operatorname{\operatorname{Lat}^{\text{\rm
b}}}\Upsilon$.

###### Proof.

This follows from Theorem 4.2 by vector-space duality. ∎

###  10.3. Regions and polynomials

Consider a hyperplane arrangement $\mathcal{A}$ in
$\mathbb{A}^{d}(\mathbf{K})$. The _characteristic polynomial_ of $\mathcal{A}$
is

$\displaystyle p(\mathcal{A};\lambda)$
$\displaystyle:=\sum_{{\mathcal{S}\subseteq\mathcal{A}:}\,{\bigcap\mathcal{S}\neq\varnothing}}(-1)^{|\mathcal{S}|}\lambda^{\dim\bigcap\mathcal{S}}=\sum_{s\in\mathcal{L}(\mathcal{A})}\mu(\hat{0},s)\lambda^{\dim
s},$

where $\hat{0}=\mathbb{A}^{d}(\mathbf{K})$, the bottom element of
$\mathcal{L}(\mathcal{A})$, and $\mu$ is the Möbius function.

We define the _chromatic polynomial_ of $\Upsilon$:

$\displaystyle\chi_{\Upsilon}(\lambda)$ $\displaystyle:=\sum_{S\subseteq
E:}(-1)^{|S|}\lambda^{n-\operatorname{rk}_{\Upsilon}S}=\sum_{S\in\operatorname{Lat}\Upsilon}\mu(\varnothing,S)\lambda^{n-\operatorname{rk}_{\Upsilon}S},$

which is interpreted as $0$ if $\Upsilon$ has any neutral loose edges. A
variant is $\chi_{\Upsilon_{\infty}}(\lambda)$, which is
$\chi_{\Upsilon\cup\\{e_{\infty}\\}}(\lambda)$ where $e_{\infty}$ is
interpreted as a non-neutral loose edge. The _balanced chromatic polynomial_
of $\Upsilon$ is

$\displaystyle\chi^{\text{\rm b}}_{\Upsilon}(\lambda)$
$\displaystyle:=\sum_{{S\subseteq
E:}\,\text{hyperbalanced}}(-1)^{|S|}\lambda^{n-\operatorname{rk}_{\Upsilon}S}=\sum_{S\in\operatorname{\operatorname{Lat}^{\text{\rm
b}}}\Upsilon}\mu(\varnothing,S)\lambda^{n-\operatorname{rk}_{\Upsilon}S},$

which also is $0$ if $\Upsilon$ has neutral loose edges. (Note that a graph
has a chromatic polynomial, while an arrangement (or matroid) has a
characteristic polynomial. One expects a chromatic polynomial to count
colorings, as is the case with signed graphs and gain graphs; we hope to
present such an interpretation separately.)

######  Lemma 10.2.

Assuming that $\Upsilon$ has no neutral loose edges, we have
$p(\mathcal{A}[\Upsilon];\lambda)=\chi^{\text{\rm b}}_{\Upsilon}(\lambda)$ and
$p(\mathcal{A}_{\infty}[\Upsilon];\lambda)=\chi_{\Upsilon_{\infty}}(\lambda)$.

###### Proof.

This is a consequence of Theorem 10.1 and the semilattice expressions for the
polynomials. ∎

We are finally ready to count the regions of a real hyperplane arrangement
described by a gain signed graph.

######  Theorem 10.3.

Consider the arrangements $\mathcal{A}[\Upsilon]$ in
$\mathbb{A}^{n}(\mathbb{R})$ and $\mathcal{A}_{\infty}[\Upsilon]$ in
$\mathbb{R}^{1+n}$.

The number of regions of $\mathcal{A}[\Upsilon]$ equals
$(-1)^{n}\chi^{\text{\rm b}}_{\Upsilon}(-1)$. The number of bounded regions
equals $(-1)^{n}\chi^{\text{\rm b}}_{\Upsilon}(1)$.

The number of regions of $\mathcal{A}_{\infty}[\Upsilon]$ equals
$(-1)^{n+1}\chi_{\Upsilon_{\infty}}(-1)$.

###### Proof.

Special cases of [21, Theorem A], which applies to all real affine hyperplane
arrangements, are that $\mathcal{A}_{\infty}[\Upsilon]$ has
$(-1)^{n+1}p(\mathcal{A}_{\infty}[\Upsilon];-1)$ regions and
$\mathcal{A}[\Upsilon]$ has $(-1)^{n}p(\mathcal{A}[\Upsilon];-1)$ regions. A
special case of [21, Theorem C] is that $\mathcal{A}[\Upsilon]$ has
$(-1)^{n}p(\mathcal{A}[\Upsilon];1)$ bounded regions. Applying Lemma 10.2
gives the theorem. ∎

##  11\. Abstract abelian gains

The theory we developed for an additive group of a field largely applies to
any abelian gain group, except of course for the vector representation. We
demonstrate that here by presenting a purely combinatorial proof that
$\mathbf{M}(\Upsilon)$ and $\mathbf{M}_{\infty}(\Upsilon)$ are matroids. All
the analysis in the preceding sections applies without change except that
$\mathfrak{G}^{n}$ may not be a vector space.

We had hoped to treat general groups, not necessarily abelian, in which the
additive definitions become multiplicative and multipliers become exponents,
but the failure of commutativity led to the failure of so many properties from
Section 2 that we were unable even to establish that neutrality of sign
circuits was independent of the method of calculating the gain. Thus, we leave
open the problem of dealing with nonabelian gain groups.

An _abelian gain signed graph_ is a triple $\Upsilon=(\Gamma,\sigma,\varphi)$
where $\Gamma=(V,E)$ is a graph, the _signature_ or sign function $\sigma$
gives each edge an element of the sign set $\\{+1,-1\\}$, and the _gain
function_ $\varphi:\vec{E}\to\mathfrak{G}$ is an oriented labelling of edges
from an abelian group $\mathfrak{G}$, which we write additively so that, for
instance, $\varphi(e^{-1})=-\varphi(e)$. All the formulas and results of
Section 2 remain valid if we replace the group $\mathbf{K}^{+}$ by
$\mathfrak{G}$.

We define rank exactly as in Equation (4.1). The task is to prove this is a
matroid rank function without using a representation. Instead, we use the fact
that $\mathbf{F}(\Sigma)$ is known to be a matroid. (A side effect of this
proof is that Theorem 4.2 becomes a proof that $\mathbf{z}$ is a module
representation of $\mathbf{M}(\Upsilon)$ even when $\mathfrak{G}$ is not the
additive group of a field.) As before, we can treat the extra point
$e_{\infty}$ as if it were a non-neutral half edge, so we are simultaneously
proving that $\mathbf{M}_{\infty}(\Upsilon)$ is a matroid.

######  Theorem 11.1.

The function $\operatorname{rk}_{\Upsilon}$ is a matroid rank function on
$E(\Upsilon)$ and $E_{\infty}(\Upsilon)$.

###### Proof.

We establish the fundamental properties of a matroid rank function on $E$,
which are:

  1. R1.

_Normalization_ : $\operatorname{rk}(\varnothing)=0$.

  2. R2.

_Unit Increase_ : If $e\notin S\subset E$, then
$\operatorname{rk}S\leq\operatorname{rk}(S\cup\\{e\\})\leq\operatorname{rk}S+1$.

  3. R3.

_Submodularity_ : If $e,f\notin S\subset E$, then

$\operatorname{rk}(S\cup\\{e\\})-\operatorname{rk}S\geq\operatorname{rk}(S\cup\\{e,f\\})-\operatorname{rk}(S\cup\\{f\\}).$

R1 is obvious.

For R2, what needs to be proved is that

$\operatorname{rk}_{\Sigma}S+\delta_{\Upsilon}(S)\leq\operatorname{rk}_{\Sigma}(S\cup\\{e\\})+\delta_{\Upsilon}(S\cup\\{e\\})\leq\operatorname{rk}_{\Sigma}S+\delta_{\Upsilon}(S)+1.$

The left-hand inequality is true because $\operatorname{rk}_{\Sigma}$ and
$\delta_{\Upsilon}$ are weakly increasing. The right-hand inequality is true
by property R2 of $\operatorname{rk}_{\Sigma}$ if
$\delta_{\Upsilon}(S)=\delta_{\Upsilon}(S\cup\\{e\\})$, so assume $S$ is
hyperbalanced and $S\cup\\{e\\}$ is hyperfrustrated. Then
$\operatorname{rk}_{\Sigma}(S\cup\\{e\\})=\operatorname{rk}_{\Sigma}S$ by
Lemma 11.2. That proves R2.

######  Lemma 11.2.

Suppose $S\subset E$ is hyperbalanced, $e\in E$, and $S\cup\\{e\\}$ is
hyperfrustrated. Then $e\in\operatorname{clos}_{\Sigma}S$,
$rk_{\Sigma}(S\cup\\{e\\})=\operatorname{rk}_{\Sigma}S$, and
$rk_{\Upsilon}(S\cup\\{e\\})=\operatorname{rk}_{\Upsilon}S+1$.

###### Proof.

There is a non-neutral sign circuit $C\subseteq S\cup\\{e\\}$ that contains
$e$. Since $C$ exists, $e\in\operatorname{clos}_{\Sigma}S$, so
$\operatorname{rk}_{\Sigma}(S\cup\\{e\\})=\operatorname{rk}_{\Sigma}S$. Then
$\operatorname{rk}_{\Upsilon}(S\cup\\{e\\})$ follows from Definition 4.1. ∎

To prove R3 we use a simpler equivalent form [13, Theorem 1.4.14]:

  1. R3′.

If $e,f\notin S\subset E$ and
$\operatorname{rk}(S\cup\\{e\\})=\operatorname{rk}(S\cup\\{f\\})=\operatorname{rk}S$,
then $\operatorname{rk}(S\cup\\{e,f\\})=\operatorname{rk}S.$

Applying this to $\operatorname{rk}_{\Upsilon}$, if all the sets in this
formula are hyperbalanced, or all are hyperfrustrated, then
$\delta_{\Upsilon}$ drops out and we have a known property of
$\operatorname{rk}_{\Sigma}$. If $S$ is hyperbalanced, by Theorem 2.13 we may
assume every edge in $S$ is neutral. Lemma 11.2 implies that
$e,f\in\operatorname{clos}_{\Sigma}S$ and also that $S\cup\\{e\\}$ and
$S\cup\\{f\\}$ are hyperbalanced. Thus, there exist sign circuits $C_{e}$ and
$C_{f}$ such that $e\in C_{e}\subseteq S\cup\\{e\\}$ and $f\in C_{f}\subseteq
S\cup\\{f\\}$, both of which are neutral. Therefore $e$ and $f$ are neutral
edges, so $S\cup\\{e,f\\}$ is hyperbalanced and R3′ is proved. ∎

As we proved the matroid properties subsequent to Section 4 without reference
to the vector representation, they are all true in general with suitably
adapted notation and the same proofs.

## Data statement

There are no associated data.

## References

  * [1] Federico Ardila, Semimatroids and their Tutte polynomials. _Rev. Colombiana Mat._ 41 (2007), no. 1, 39–66. MR 2355665 (2008j:05082). Zbl 1136.05008.
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