# Combinatorial Interpretation of General Eulerian Numbers

Tingyao Xiong
<EMAIL_ADDRESS>

Jonathan I. Hall
<EMAIL_ADDRESS>

Hung-ping Tsao
<EMAIL_ADDRESS>Radford UniversityMichigan State UniversityNovato, CA

###### Abstract

Since 1950s, mathematicians have successfully interpreted the traditional
Eulerian numbers and $q-$Eulerian numbers combinatorially. In this paper, the
authors give a combinatorial interpretation to the general Eulerian numbers
defined on general arithmetic progressions $\\{a,a+d,a+2d,\dots\\}$.

Keywords: Traditional Eulerian numbers, general Eulerian numbers, permutation,
weak excedance.

##  1 Introduction

###### Definition 1.1

Given a positive integer $n$, define $\Omega_{n}$ as the set of all
permutations of $[n]=\\{1,\;2,\;3,\;,\ldots,n\\}$. For a permutation
$\pi=p_{1}p_{2}p_{3}\ldots p_{n}\in\Omega_{n}$ , $i$ is called an ascent of
$\pi$ if $p_{i}<p_{i+1}$; $i$ is called a weak excedance of $\pi$ if
$p_{i}\geq i$.

It is well known that a traditional Eulerian number $A_{n,k}$ is the number of
permtations $\pi\in\Omega_{n}$ that have $k$ weak excedances ([1], page 215).
And $A_{n,\;k}$ satisfies the recurrence: $A_{n,1}=1,\;(n\geq
1),\;\;A_{n,k}=0\;\;(k>n)$

$A_{n,k}=kA_{n-1,k}+(n+1-k)A_{n-1,k-1}\;\;(1\leq k\leq n)$ (1)

Besides the recursive formula (1), $A_{n,\;k}$ can be calculated directly by
the following analytic formula ([2], page 8):

$A_{n,k}=\sum_{i=0}^{k-1}(-1)^{i}(k-i)^{n}\left(\begin{array}[]{c}n+1\\\
i\end{array}\right)\hskip 28.45274pt(1\leq k\leq n)$ (2)

###### Definition 1.2

Given a permutation $\pi=p_{1}p_{2}p_{3}\ldots p_{n}\in\Omega_{n}$, define
functions

$maj\;\pi=\sum_{p_{j}>p_{j+1}}j\hskip 17.07182ptand$
$a(n,k,i)=\\#\\{\pi\;|\;maj\;\pi=i\;\;\&\;\;\pi\;\;has\;\;k\;\;ascents\\}$

Since the 1950’s, Carlitz ([3],[4]) and his successors have generalized
Euler’s results to $q-$ sequences $\\{1,q,q^{2},q^{3},\ldots,\\}$. Under
Carlitz’s definition, the $q-$Eulerian numbers $A_{n,k}(q)$ are given by

$A_{n,k}(q)=q^{\frac{(m-k+1)(m-k)}{2}}\sum_{i=0}^{k(n-k-1)}a(n,n-k,i)q^{i}$
(3)

where functions $a(n,k,i)$ are as defined in Definition 1.2.

In [6], instead of studying $q-$sequences, the authors have generalized
Eulerian numbers to any general arithmetic progression

$\\{a,\;a+d,\;a+2d,\;a+3d,\;\ldots\\}$ (4)

Under the new definition, and given an arithmetic progression as defined in
(4), the general Eulerian numbers $A_{n,k}(a,d)$ can be calculated directly by
the following equation ([6], Lemma 2.6)

$A_{n,k}(a,d)=\sum_{i=0}^{k}(-1)^{i}[(k+1-i)d-a]^{n}\left(\begin{array}[]{c}n+1\\\
i\end{array}\right)$ (5)

Interested readers can find more results about the general Eulerian numbers
and even general Eulerian polynomials in [6].

##  2 Combinatorial Interpretation of General Eulerian Numbers

The following concepts and properties will be heavily used in this section.

###### Definition 2.1

Let $W_{n,k}$ be the set of $n-$permutations with $k$ weak excedances. Then
$|W_{n,k}|=A_{n,k}$. Furthermore, given a permutation
$\pi=p_{1}p_{2}p_{3}\ldots p_{n}$, let $Q_{n}(\pi)=i$ where $p_{i}=n$.

Given a permutation $\pi\in\Omega_{n}$, it is known that $\pi$ can be written
as a one line form like $\pi=p_{1}p_{2}p_{3}\ldots p_{n}$. Or $\pi$ can be
written in a disjoint union of distinct cycles. For $\pi$ written in a cycle
form, we can use a _standard representation_ by writing (a) each cycle
starting with its largest element, and (b) the cycles are written in
increasing order of their largest element. Moreover, given a permutation $\pi$
written in a standard representation cycle form, define a function $f$ as
$f(\pi)$ to be the permutation obtained from $\pi$ by erasing the parentheses.
Then $f$ is known as the _fundamental bijection_ from $\Omega_{n}$ to itself
([5], page 30). Indeed, the inverse map $f^{-1}$ of the fundamental bijection
function $f$ is also famous in illustrating the relation between the ascents
and weak excedances as following: ([2], page 98)

###### Proposition 2.2

The function $f^{-1}$ gives a bijection between the set of permutations on
$[n]$ with $k$ ascents and the set $W_{n,k+1}$.

###### Example 1

The standard representation of permutation $\pi=5243716$ is
$(2)(43)(7615)\in\Omega_{7}$, and $f(\pi)=2437615$; $Q_{7}(\pi)=5$;
$\pi=5243716$ has $3$ ascents, while $f^{-1}(\pi)=(5243)(716)=6453271\in
W_{7,4}$ has $3+1=4$ weak excedances because $p_{1}=6>1$, $p_{2}=4>2$,
$p_{3}=5>3$, and $p_{6}=7>6$.

Now suppose we want to construct a sequence consisting of $k$ vertical bars
and the first $n$ positive integers. Then the $k$ vertical bars divide these
$n$ numbers into $k+1$ compartments. In each compartment, there is either no
number or all the numbers are listed in a decreasing order. The following
definition is analogous to the definition of [2], page 8.

###### Definition 2.3

A bar in the above construction is called _extraneous_ if either

  * (a)

it is immediately followed by another bar; or

  * (b)

after removing it each of the rest compartment either is empty or consists of
integers in a decreasing order.

###### Example 2

Suppose $n=7$, $k=4$, then in the following arrangement

$32\;|\;1\;||\;7654\;|$

the $1$st, $2$nd, and the $4$th bars are extraneous.

Now we are ready to give combinatorial interpretations to the general Eulerian
numbers $A_{n,k}(a,d)$. First note that equation (5) implies that
$A_{n,k}(a,d)$ is a homogeneous polynomial of degree $n$ with respect to $a$
and $d$. Indeed,

$\displaystyle A_{n,k}(a,d)$
$\displaystyle=\sum_{i=0}^{k}(-1)^{i}[(k+1-i)d-a]^{n}\left(\begin{array}[]{c}n+1\\\
i\end{array}\right)$ (8)
$\displaystyle=\sum_{i=0}^{k}(-1)^{i}[\;(k+1-i)(d-a)+(k-i)a\;]^{n}\left(\begin{array}[]{c}n+1\\\
i\end{array}\right)$ (11)
$\displaystyle=\sum_{j=0}^{n}\left[\sum_{i=0}^{k}(-1)^{i}(k+1-i)^{n-j}(k-i)^{j}\left(\begin{array}[]{c}n+1\\\
i\end{array}\right)\right]\left(\begin{array}[]{c}n\\\
j\end{array}\right)(d-a)^{n-j}a^{j}$ (16)
$\displaystyle=\sum_{j=0}^{n}c_{n,k}(j)\left(\begin{array}[]{c}n\\\
j\end{array}\right)(d-a)^{n-j}a^{j}$ (19)

where

$c_{n,k}(j)=\sum_{i=0}^{k}(-1)^{i}(k+1-i)^{n-j}(k-i)^{j}\left(\begin{array}[]{c}n+1\\\
i\end{array}\right),0\leq j\leq n.$ (20)

The following Theorem gives combinatorial interpretations to the coefficients
$c_{n,k}(j)$, $0\leq j\leq n$.

###### Theorem 2.4

Let the general Eulerian numbers $A_{n,k}(a,d)$ be written as in equation
$(\ref{Ank in the form of a and d-a})$. Then

$c_{n,k}(j)=\\#\\{\pi\in W_{n,k+1}\text{ and }j<Q_{n}(\pi)\leq
n\\}+\\#\\{\pi\in W_{n,k}\text{ and }1\leq Q_{n}(\pi)\leq j\\}$ (21)

Proof. We can check the result in (21) for two special values $j=0$ and $j=n$
quickly. By equation (2),

  * when $j=0$, $c_{n,k}(0)=\sum_{i=0}^{k}(-1)^{i}(k+1-i)^{n}\left(\begin{array}[]{c}n+1\\\ i\end{array}\right)=A_{n,k+1}$;

  * when $j=n$, $c_{n,k}(n)=\sum_{i=0}^{k}(-1)^{i}(k-i)^{n}\left(\begin{array}[]{c}n+1\\\ i\end{array}\right)=A_{n,k}$. Therefore, (21) is true for $j=0$ and $j=n$.

Generally, for $1\leq j\leq n-1$, we write down $k$ bars with $k+1$
compartments in between. Place each element of $[n]$ in a compartment. If none
of the $k$ bars is extraneous, then the arrangement corresponds to a
permutation with $k$ ascents. Let $B$ be the set of arrangements with at most
one extraneous bar at the end and none of integers $\\{1,2,\ldots,j\\}$
locating in the last compartment. We will show that $c_{n,k}(j)=|B|$.

To achieve that goal, we use the Principle of Inclusion and Exclusion. There
are $(k+1)^{n-j}k^{j}$ ways to put $n$ numbers into $k+1$ compartments with
elements $\\{1,2,\ldots,j\\}$ avoiding the last compartments. Let $B_{i}$ be
the number of arrangements with (1) none of $\\{1,2,\ldots,j\\}$ sitting in
the last compartment; (2) there are at least $i$ “separating” extraneous bars.
Two extraneous bars are separating means that the two bars are not next to
each other. Then the Principle of Inclusion and Exclusion shows that

$|B|=(k+1)^{n-j}k^{j}-B_{1}+B_{2}+\ldots+(-1)^{k}B_{k}$ (22)

Now we consider the value of $B_{i}$, where $1\leq i\leq k$. Suppose that we
have $k+1-i$ compartments with $k-i$ bars in between. There are
$(k+1-i)^{n-j}(k-i)^{j}$ ways to insert $n$ numbers into these $k+1-i$
compartments with first $j$ integers avoiding the last compartment, and list
integers in each component in a decreasing order. Then insert $i$ separating
extraneous bars into $n+1$ positions. So we get

$B_{i}=(k+1-i)^{n-j}(k-i)^{j}\left(\begin{array}[]{c}n+1\\\
i\end{array}\right)$ (23)

Plug the formula (23) into equation (22), we have $c_{n,k}(j)=|B|$.

Given an arrangement $\pi\in B$, if we remove the bars, then we obtain a
permutation $\pi\in\Omega_{n}$. So without confusion, we just use the same
notation $\pi$ to represent both an arrangement in set $B$ and a permutation
on $[n]$. Now for each $\pi\in B$, $\pi$ either

  * case 1

has no extraneous bar and none of $\\{1,2,\ldots,j\\}$ locates in the last
compartment; Or

  * case 2

has only one extraneous bar at the end.

If $\pi$ is in case 1, then $\pi$ has $k$ ascents since each bar is non-
extraneous. And the last compartment of $\pi$ is nonempty. Therefore the last
cycle of $f^{-1}(\pi)$ has to be $(n\ldots p_{g})$. In other words,
$Q_{n}(f^{-1}(\pi))=p_{g}>j$ since none of $\\{1,2,\ldots,j\\}$ locates in the
last compartment. And by Proposition 2.2, $f^{-1}(\pi)\in W_{n,k+1}$.

If $\pi$ is in case 2, then $\pi$ has $k-1$ ascents since only the last bar is
extraneous. Note that in this case, the arrangement with no elements of
$\\{1,2,\ldots,j\\}$ in the compartment second to the last, or the last non-
empty compartment have been removed by the Principle of Inclusion and
Exclusion. Equivalently, at least one number of $\\{1,2,\ldots,j\\}$ has to be
in the compartment second to the last. So the last cycle of $f^{-1}(\pi)$ has
to be $(n\ldots p_{l})$, and $Q_{n}(f^{-1}(\pi))=p_{l}\leq j$. Also by
Proposition 2.2, $f^{-1}(\pi)\in W_{n,k}$.

Combing all the results above, statement (21) is correct.

The next Theorem describes some interesting properties of the coefficients
$c_{n,k}$.

###### Theorem 2.5

Let the coefficients $c_{n,k}$ be as described in Theorem 2.4. Then

  * 1.

$\sum_{k=0}^{n}c_{n,k}(j)=n!$, for any $0\leq j\leq n$;

  * 2.

$c_{n,k}(j)=c_{n,n-k}(n-j)$, for all $0\leq j,k\leq n$

Before we can prove Theorem 2.5, we need the following Lemma which is also
interesting by itself.

###### Lemma 2.6

Given a positive integer $n$, then

$\\#\\{\pi\in W_{n,k}\;\&\;Q_{n}(\pi)=j\\}=\\#\\{\pi\in
W_{n,n+1-k}\;\&\;Q_{n}(\pi)=n+1-j\\}$

for any $1\leq k,j\leq n$.

Proof. First of all, given a positive integer $n$, we define a function
$g:\Omega_{n}\rightarrow\Omega_{n}$ as following:

$\text{for }\pi=p_{1}p_{2}\ldots
p_{n}\in\Omega_{n},\;\;g(\pi)=(n+1-p_{1})(n+1-p_{2})\ldots(n+1-p_{n})$

For instance, for $\pi=53214\in\Omega_{5}$, $g(\pi)=13452$. $g$ is obviously a
bijection of $\Omega_{n}$ to itself.

Now for some fixed $1\leq k,j\leq n$, suppose $S_{k,j}=\\{\pi\in
W_{n,k}\;\&\;Q_{n}(\pi)=j\\}$, and $T_{k,j}=\\{\pi\in
W_{n,n+1-k}\;\&\;Q_{n}(\pi)=n+1-j\\}$. For any $\pi\in S_{k,j}$ written in the
standard representation cycle form. So $\pi=(p_{u}\ldots)\ldots(n\ldots j)$
and $f(\pi)=p_{u}\ldots n\ldots j$ has $k-1$ ascents by Proposition 2.2. Now
we compose $f(\pi)$ with the bijection function $g$ as just defined. Then
$g(f(\pi))=n+1-p_{u}\ldots 1\ldots n+1-j$ has $n-k$ ascents, which implies
that $f^{-1}(g(f(\pi)))$ has $n+1-k$ weak excedances. So $f^{-1}(g(f(\pi)))\in
W_{n,n+1-k}$. Note that the last cycle of $f^{-1}(g(f(\pi)))$ has to be
$(n\ldots n+1-j)$. Therefore $f^{-1}(g(f(\pi)))\in T_{k,j}$. Since both $f$
and $g$ are bijection functions, $f^{-1}gf$ gives a bijection between
$S_{k,j}$ and $T_{k,j}$.

Now we are ready to prove Theorem 2.5.

Proof. (of Theorem 2.5)For part $1$, by Theorem 2.4,

$\displaystyle\sum_{k=0}^{n}c_{n,k}(j)$
$\displaystyle=\sum_{k=0}^{n}\\#\\{\pi\in W_{n,k+1}\text{ and
}j<Q_{n}(\pi)\leq n\\}+\sum_{k=0}^{n}\\#\\{\pi\in W_{n,k}\text{ and }1\leq
Q_{n}(\pi)\leq j\\}$ $\displaystyle=\sum_{k=0}^{n}\\#\\{\pi\in
W_{n,k}\\}=|\Omega_{n}|=n!$

For part $2$, also by Theorem 2.4,

$\displaystyle c_{n,k}(j)$ $\displaystyle=\sum_{i=j+1}^{n}\\#\\{\pi\in
W_{n,k+1}\text{ and }Q_{n}(\pi)=i\\}+\sum_{m=1}^{j}\\#\\{\pi\in W_{n,k}\text{
and }Q_{n}(\pi)=m\\}$ $\displaystyle=\sum_{i=j+1}^{n}\\#\\{\pi\in
W_{n,n-k}\text{ and }Q_{n}(\pi)=n+1-i\\}$
$\displaystyle+\sum_{m=1}^{j}\\#\\{\pi\in W_{n,n+1-k}\text{ and
}Q_{n}(\pi)=n+1-m\\}\hskip 85.35826pt\text{by Lemma }\ref{k and n+1-k}$
$\displaystyle=\\#\\{\pi\in W_{n,k}\text{ and }1\leq Q_{n}(\pi)\leq
n-j\\}+\\#\\{\pi\in W_{n,n+1-k}\text{ and }n-j<Q_{n}(\pi)\leq n\\}$
$\displaystyle=c_{n,n-k}(n-j)\hskip 341.43306pt\hfill\framebox[7.11317pt]{}$

_Remark_ Using the analytic formula of $c_{n,k}(j)$ as in (20), part 2 of
Theorem 2.5 implies the following identity:

$\sum_{i=0}^{k}(-1)^{i}(k+1-i)^{n-j}(k-i)^{j}\left(\begin{array}[]{c}n+1\\\
i\end{array}\right)=\sum_{l=0}^{n-k}(-1)^{l}(n+1-k-l)^{j}(n-k-l)^{n-j}\left(\begin{array}[]{c}n+1\\\
l\end{array}\right),$

where $n$ is a positive integer, and $0\leq j,k\leq n$.

##  3 Another Combinatorial Interpretation of $c_{n,k}(1)$ and $c_{n,k}(n-1)$

In pursuing the combinatorial meanings of the coefficients $c_{n,k}$, the
authors have found some other interesting properties about permutations. The
results in this section will reveal close connections between the traditional
Eulerian numbers $A_{n,k}$ and $c_{n,k}(j)$, where $j=1$ or $j=n-1$.

One fundamental concept of permutation combinatorics is _inversion_. A pair
$(p_{i},p_{j})$ is called an _inversion_ of the permutation
$\pi=p_{1}p_{2}\ldots p_{n}$ if $i<j$ and $p_{i}>p_{j}$ ([5], page 36). The
following definition provides the main concepts of this section.

###### Definition 3.1

For a fixed positive integer $n$, let $AW_{n,k}=\\{\pi=p_{1}p_{2}p_{3}\ldots
p_{n}\;|\;\pi\in W_{n,k}\text{ and }p_{1}<p_{n}\\}$ (or $(p_{1},p_{n})$ is not
an inversion), and $BW_{n,k}=W_{n,k}\backslash AW_{n,k}$ (or $(p_{1},p_{n})$
is an inversion).

It is obvious that $|AW_{n,k}|+|BW_{n,k}|=A_{n,k}$. The following Theorem
interprets coefficients $c_{n,k}(1)$ and $c_{n,k}(n-1)$ in terms of $AW_{n,k}$
and $BW_{n,k}$.

###### Theorem 3.2

Let the coefficients $c_{n,k}$ of the general Eulerian numbers be written as
in equation $(\ref{cnk formula})$. $AW_{n,k}$ and $BW_{n,k}$ are as defined in
Definition 3.1. Then

  * (1)

$c_{n,k}(1)=2|AW_{n,k+1}|$;

  * (2)

$c_{n,k}(n-1)=2|BW_{n,k}|$.

Proof. For part (1). By Theorem 2.4, $c_{n,k}(1)=|S_{1}|+|S_{2}|$, where
$S_{1}=\\{\pi=p_{1}p_{2}\ldots p_{n}|\pi\in W_{n,k+1}\;\&\;p_{1}\not=n\\}$,
$S_{2}=\\{\pi=p_{1}p_{2}\ldots p_{n}|\pi\in W_{n,k}\;\&\;p_{1}=n\\}$. Given a
permutation $\pi=p_{1}p_{2}\ldots p_{n}\in S_{1}$ and $p_{n}\not=n$, then both
$p_{1}p_{2}\ldots p_{n}$ and $p_{n}p_{2}\ldots p_{1}$ belong to $S_{1}$, so
one of them has to be in $AW_{n,k+1}$; If $\pi=p_{1}p_{2}\ldots p_{n}\in
S_{1}$ and $p_{n}=n$, then $\pi\in AW_{n,k+1}$, but $p_{n}p_{2}\ldots p_{1}\in
S_{2}$. Therefore, $\frac{1}{2}c_{n,k}(1)=|AW_{n,k+1}|$.

Part (2) can be proved using exactly the same method. So we leave it to the
readers as an exercise.

$|AW_{n,k}|$ and $|BW_{n,k}|$ are interesting combinatorial concepts by
themselves. Note that generally speaking, $|AW_{n,k}|\not=|BW_{n,k}|$. Indeed,
$|AW_{n,k}|=|BW_{n,n+1-k}|$.

###### Theorem 3.3

For any positive integer $n\geq 2$, the sets $AW_{n,k}$ and $BW_{n,k}$ are
defined in Definition 3.1. Then $|AW_{n,k}|=|BW_{n,n+1-k}|$ for $1\leq k\leq
n$.

Proof. It is an obvious result of part 2 of Theorem 2.5 and Theorem 3.2.

Our last result of this paper is the following Theorem which reveals that both
$|AW_{n,k}|$ and $|BW_{n,k}|$ take exactly the same recursive formula as the
traditional Eulerian numbers $A_{n,k}$ as shown in equation (1).

###### Theorem 3.4

For a fixed positive integer $n$, let $AW_{n,k}$ and $BW_{n,k}$ be as defined
in Definition 3.1, then

$\displaystyle k|AW_{n-1,k}|+(n+1-k)|AW_{n-1,k-1}|$
$\displaystyle=|AW_{n,k}|\hskip 28.45274pt\text{ and }$ (24) $\displaystyle
k|BW_{n-1,k}|+(n+1-k)|BW_{n-1,k-1}|$ $\displaystyle=|BW_{n,k}|$ (25)

Proof. A computational proof can be obtained straightforward by using equation
(20) and Theorem 3.2. But here we provide a proof in a flavor of
combinatorics.

_Idea of the proof_ : For equation (24), given a permutation
$A_{1}=p_{1}p_{2}p_{3}\ldots p_{n-1}\in AW_{n-1,k}$, for each position $i$
with $p_{i}\geq i$, we insert $n$ into a certain place of $A_{1}$, such that
the new permutation $A_{1}^{\prime}$ is in $AW_{n,k}$. There are $k$ such
positions, so we can get $k$ new permutations in $AW_{n,k}$. Similarly, if
$A_{2}=p_{1}p_{2}p_{3}\ldots p_{n-1}\in AW_{n-1,k-1}$, for each position $i$
with $p_{i}<i$, and the position at the end of $A_{2}$, we insert $n$ into a
specific position of $A_{2}$ and the resulting new permutation
$A_{2}^{\prime}$ is in $AW_{n,k}$. There are $n+1-k$ such positions, so we can
get $n+1-k$ new permutations in $AW_{n,k}$. We will show that all the
permutations obtained from the above constructions are distinct, and they have
exhausted all the permutations in $AW_{n,k}$

For any fixed $A^{\prime}=\pi_{1}\pi_{2}\pi_{3}\ldots\pi_{n}\in AW_{n,k}$,
then $\pi_{1}<\pi_{n}$. We classify $A^{\prime}$ into the following disjoint
cases:

  * case a.

$\pi_{i}=n$ with $i<n$. So
$A^{\prime}=\pi_{1}\pi_{2}\ldots\pi_{i-1}n\pi_{i+1}\ldots\pi_{n-1}\pi_{n}$.

    * a.1

$\pi_{1}<\pi_{n-1}$, and $\pi_{n}\geq i$;

    * a.2

$\pi_{1}<\pi_{n-1}$, and $\pi_{n}<i$;

    * a.3

$\pi_{1}>\pi_{n-1}$, $\pi_{n}<n-1$, and $\pi_{n}\geq i$;

    * a.4

$\pi_{1}>\pi_{n-1}$, $\pi_{n}<n-1$, and $\pi_{n}<i$;

    * a.5

$\pi_{1}>\pi_{n-1}$, and $\pi_{n}=n-1$;

  * case b.

$\pi_{n}=n$. So $\pi_{i}=n-1$ for some $i<n$ and
$A^{\prime}=\pi_{1}\pi_{2}\ldots\pi_{i-1}n-1\ldots\pi_{n-1}n$.

    * b.1

$\pi_{1}<\pi_{n-1}$;

    * b.2

$\pi_{n-1}<\pi_{1}<n-1$, and $\pi_{n-1}\geq i$;

    * b.3

$\pi_{n-1}<\pi_{1}<n-1$, and $\pi_{n-1}<i$;

    * b.4

$\pi_{1}=n-1$.

Based on the classifications listed above, we can construct a map
$f:\\{AW_{n-1,k},AW_{n-1,k-1}\\}\rightarrow AW_{n,k}$ by applying the idea of
the proof we have illustrated at the beginning of the proof. To save space,
the map $f$ is demonstrated in Table 1. From Table 1 we can see that in each
case, the positions of inserting $n$ are all different. So all the images
obtained in a certain case are different. Since all the cases are disjoint,
all the images $A^{\prime}\in AW_{n,k}$ are distinct.

$A=p_{1}p_{2}\dots p_{n-1}$ | Position $i$ | Condition | $A^{\prime}\in AW_{n,k}$
---|---|---|---
|  |  |  $A^{\prime}=p_{1}p_{2}\dots p_{i-1}np_{i+1}\dots p_{n-1}p_{i}$,
|  | $p_{i}>p_{1}$ | with $p_{1}<p_{n-1}$ and $p_{1}<p_{i}$.
|  |  | (Case $a.1$)
| $1<i\leq n-1$ |  |  $A^{\prime}=p_{1}p_{2}\dots p_{i-1}np_{i+1}\dots p_{n-2}p_{i}p_{n-1}$,
| and |  $p_{i}<p_{1}$ and | with $p_{i}<p_{1}<p_{n-1}<n-1$.
| $p_{i}\geq i$ |  | (Case $a.3$)
$A\in AW_{n-1,k}$ |  |  |  $A^{\prime}=p_{1}p_{2}\dots p_{i-1}n-1p_{i+1}\dots p_{n-2}p_{i}n$,
|  |  $p_{i}<p_{1}$ and | with $p_{i}<p_{1}$ and $p_{i}\geq i$.
|  |  | (Case $b.2$)
|  |  $p_{i}=n-1$ and |  $A^{\prime}=p_{n-1}p_{2}\dots p_{i-1}np_{i+1}\dots p_{n-2}p_{1}n-1$,
|  | $i<n-1$ | with $p_{1}<p_{n-1}$.
| $i=1$ |  | (Case $a.5$)
|  | $p_{n-1}=n-1$ |  $A^{\prime}=n-1p_{2}\dots p_{n-2}p_{1}n$,
|  |  | (Case $b.4$)
|  |  |  $A^{\prime}=p_{1}p_{2}\dots p_{i-1}np_{i+1}\dots p_{n-1}p_{i}$,
|  | $p_{i}>p_{1}$ | with $p_{1}<p_{n-1}$ and $p_{1}<p_{i}$.
|  |  | (Case $a.2$)
| $1<i\leq n-1$ |  $p_{i}<p_{1}$ and |  $A^{\prime}=p_{1}p_{2}\dots p_{i-1}np_{i+1}\dots p_{n-2}p_{i}p_{n-1}$,
| and | $p_{n-1}<n-1$ | with $p_{i}<p_{1}<p_{n-1}<n-1$.
| $p_{i}<i$ |  | (Case $a.4$)
$A\in AW_{n-1,k-1}$ |  |  $p_{i}<p_{1}$ and |  $A^{\prime}=p_{1}p_{2}\dots p_{i-1}n-1p_{i+1}\dots p_{n-2}p_{i}n$.
|  | $p_{n-1}=n-1$ | with $p_{i}<p_{1}$ and $p_{i}<i$.
|  |  | (Case $b.3$)
| $i=n$ |  |  $A^{\prime}=p_{1}p_{2}\dots p_{n-1}n$.
|  |  | (Case $b.1$)
Table 1: The Map $f:\\{AW_{n-1,k},AW_{n-1,k-1}\\}\rightarrow AW_{n,k}$

Similarly, for each $B^{\prime}=\pi_{1}\pi_{2}\pi_{3}\ldots\pi_{n}\in
BW_{n,k}$, then $\pi_{1}>\pi_{n}$. We classify $B^{\prime}$ into the following
disjoint cases:

  * case c.

$\pi_{i}=n$ with $1<i\leq n-1$. So
$B^{\prime}=\pi_{1}\pi_{2}\ldots\pi_{i-1}n\pi_{i+1}\ldots\pi_{n-1}\pi_{n}$.

    * c.1

$\pi_{1}>\pi_{n-1}$, and $\pi_{n}\geq i$;

    * c.2

$\pi_{1}>\pi_{n-1}$, and $\pi_{n}<i$;

    * c.3

$\pi_{1}<\pi_{n-1}<n-1$, $\pi_{n-1}\geq i$;

    * c.4

$\pi_{1}<\pi_{n-1}<n-1$, $\pi_{n-1}<i$;

    * c.5

$\pi_{n-1}=n-1$;

    * c.6

$\pi_{n-1}=n$;

  * case d.

$\pi_{1}=n$. So $B^{\prime}=n\pi_{2}\ldots\pi_{n-2}\pi_{n-1}$.

    * d.1

$\pi_{n-2}<\pi_{n-1}$;

    * d.2

$\pi_{n-2}>\pi_{n-1}$.

To prove equation (25), we use a similar idea of proof as shown above. If
$B_{1}=p_{1}p_{2}p_{3}\ldots p_{n-1}\in BW_{n-1,k}$, for each position $i$
with $p_{i}\geq i$, we insert $n$ into a certain place of $B_{1}$ to get
$B_{1}^{\prime}\in AW_{n,k}$; If $B_{2}=p_{1}p_{2}p_{3}\ldots p_{n-1}\in
BW_{n-1,k-1}$, for each position $i$ with $p_{i}<i$, and the position $i$
where $p_{i}=n-1$, we insert $n$ into a specific position of $B_{2}$ to obtain
$B_{2}^{\prime}\in AW_{n,k}$. Such a map
$g:\\{BW_{n-1,k},BW_{n-1,k-1}\\}\rightarrow BW_{n,k}$ is illustrated in Table
2. And the distinct images under $g$ exhaust all the permutations in
$BW_{n,k}$.

$B=p_{1}p_{2}\dots p_{n-1}$ | Position $i$ | Condition | $B^{\prime}\in BW_{n,k}$
---|---|---|---
|  |  |  $B^{\prime}=p_{1}\dots p_{i-1}np_{i+1}\dots p_{n-1}p_{i}$,
|  | $p_{1}>p_{i}$ | with $p_{1}>p_{n-1}$ and $p_{i}\geq i$.
|  |  | (Case $c.1$)
| $1<i<n-1$ |  |  $B^{\prime}=p_{1}\dots p_{i-1}np_{i+1}\dots p_{i}p_{n-1}$,
| and | $p_{1}<p_{i}<n-1$ | with $p_{1}<p_{i}<n-1$ and $p_{i}\geq i$.
| $p_{i}\geq i$ |  | (Case $c.3$)
$B\in BW_{n-1,k}$ |  |  |  $B^{\prime}=np_{2}\dots p_{i-1}n-1p_{i+1}\dots p_{1}p_{n-1}$,
|  | $p_{1}<p_{i}=n-1$ | with $p_{i}=n-1$ and $p_{1}>p_{n-1}$.
|  |  | (Case $d.2$)
|  |  |  $B^{\prime}=np_{2}\dots p_{n-1}p_{1}$,
| $i=1$ | $p_{1}\geq 1$ | with $p_{n-1}<p_{1}$.
|  |  | (Case $d.1$)
|  |  |  $B^{\prime}=p_{1}\dots p_{i-1}np_{i+1}\dots p_{n-1}p_{i}$,
|  | $p_{1}>p_{i}$ | with $p_{1}>p_{n-1}$ and $p_{i}<i$.
| $1<i<n-1$ |  | (Case $c.2$)
| and |  |
| $p_{i}<i$ |  |  $B^{\prime}=p_{1}\dots p_{i-1}np_{i+1}\dots p_{i}p_{n-1}$,
|  | $p_{1}<p_{i}$ | with $p_{1}<p_{i}<n-1$ and $p_{i}<i$.
|  |  | (Case $c.4$)
$B\in BW_{n-1,k-1}$ | $i=n-1$ | $p_{1}>p_{i}=p_{n-1}$ |  $B^{\prime}=p_{1}p_{2}\dots p_{n-2}np_{n-1}$.
| $p_{i}<i$ |  | (Case $c.6$)
| $1\leq i<n-1$ |  |  $B^{\prime}=p_{1}p_{2}\dots p_{i-1}np_{i+1}\dots p_{n-2}n-1p_{n-1}$.
| and | $p_{i}=n-1$ | (Case $c.5$)
| $p_{i}=n-1$ |  |
Table 2: The Map $g:\\{BW_{n-1,k},BW_{n-1,k-1}\\}\rightarrow BW_{n,k}$

Here is a concrete example for the constructions illustrated in Table 2:

###### Example 3

Suppose $n=4$, $k=2$. We want to obtain
$BW_{4,2}=\\{3142,3412,3421,4132,4213,4312,4321\\}$ from
$BW_{3,2}=\\{321,231\\}$ and $BW_{3,1}=\\{312\\}$. For $321\in BW_{3,2}$,
$p_{1}=3\geq 1$, then it corresponds to $B^{\prime}=4213$ which is Case d.1 in
Table 2; $p_{2}=2\geq 2$, then it corresponds to $B^{\prime}=3412$ which is
Case c.1 in Table 2. Similarly, we can construct $\\{4312,4321\\}$ from
$231\in BW_{3,2}$, and $\\{3421,3142,4132\\}$ from $312\in BW_{3,1}$ using
Table 2.

## References

  * [1] J. Riordan, “ An Introduction to Combinatorial Analysis”, New York, Wiley, 1958.
  * [2] Miklós Bóna, “ Combinatorics of Permutations”, Chapman $\&$ Hall/CRC, 2004.
  * [3] L. Carlitz, “ $q-$ Bernoulli and Eulerian numbers”, Transaction of American Mathematical Society, Vol. 76, page 332–350, 1954.
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State and parameter learning with  particle Gibbs

$\dag$]Gabriel Cardoso
$\ddag$]Yazid Janati El Idrissi
$\star$]Sylvain Le Corff
$\dag$]Éric Moulines
$\top$]Jimmy Olsson

[$\dag$]CMAP, École Polytechnique, Institut Polytechnique de Paris, Palaiseau.
[$\ddag$]Samovar, Télécom SudParis, département CITI, TIPIC, Institut Polytechnique de Paris, Palaiseau.
[$\star$]LPSM, Sorbonne Université, UMR CNRS 8001, 4 Place Jussieu, 75005 Paris.
[$\top$]Department of Mathematics, KTH Royal Institute of Technology, Stockholm, Sweden.

Non-linear state-space models, also known as general hidden Markov models, are ubiquitous in statistical machine learning, being the most classical generative models for serial data and sequences in general.
The particle-based, rapid incremental smoother () is a sequential Monte Carlo (SMC) technique allowing for efficient online approximation of expectations of additive functionals under the smoothing distribution in these models.
Such expectations appear naturally in several learning contexts, such as likelihood estimation (MLE) and Markov score climbing (MSC).  has linear computational complexity, limited memory requirements and comes with non-asymptotic bounds, convergence results and stability guarantees.
Still, being based on self-normalised importance sampling, the  estimator is biased.
Our first contribution is to design a novel additive smoothing algorithm, the Parisian particle Gibbs () sampler, which can be viewed as a  algorithm driven by conditional SMC moves, resulting in bias-reduced estimates of the targeted quantities. We substantiate the  algorithm with theoretical results, including new bounds on bias and variance as well as deviation inequalities.
Our second contribution is to apply  in a learning framework, covering MLE and MSC as special examples. In this context, we establish, under standard assumptions, non-asymptotic bounds highlighting the value of bias reduction and the implicit Rao–Blackwellization of . These are the first non-asymptotic results of this kind in this setting.
We illustrate our theoretical results with numerical experiments supporting our claims.

§ NUMERICS

§ CONCLUSION

In this section, we develop the theoretical framework necessary to establish <Ref>. We recall the notions of  Feynman–Kac models, many-body Feynman–Kac models, backward interpretations, and conditional dual processes. Our presentation follows closely [19] but with a different and hopefully more transparent definition of the many-body extensions. We restate (in <Ref> below) a duality formula of [19] relating these concepts.
This formula provides a foundation for the particle Gibbs sampler described in <Ref>.

§ LEARNING WITH

This section is divided into three subsections.
<Ref>  establishes, following closely [30], a non-asymptotic bound for stochastic approximation schemes under general assumptions.
<Ref> shows how assumptions <ref> and <ref> imply the assumptions provided in <Ref> and therefore allow to establish  <Ref>.
Finally, <Ref> provides sufficient  assumptions on the model  ensuring that <Ref> holds.

§.§ Non-asymptotic bound

§.§ Application to <Ref>

§.§  Conditions on the model to verify <Ref>

§ LIPSCHITZ PROPERTIES

§ ADDITIONAL NUMERICAL RESULTS

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# EXISTENCE OF TRAVELING WAVES OF LOTKA VOLTERRA TYPE MODELS WITH DELAYED
DIFFUSION TERM AND PARTIAL QUASIMONOTONICITY

William Barker  Department of Mathematics and Statistics, University of
Arkansas at Little Rock, 2801 S University Ave, Little Rock, AR 72204. USA
<EMAIL_ADDRESS>
###### Abstract.

This paper is concerned with the existence of traveling wave solutions for
diffusive two-species Lotka-Volterra systems with delay in both the reaction
and diffusion terms without monotonicity. We extend the partial or cross
monotone iteration method to systems that satisfy the partial quasi-monotone
condition via construction appropriate upper and lower solutions . This is
done by using Schauder’s fixed point theorem.

###### Key words and phrases:

Traveling waves; Reaction-diffusion equations; Delay; Lokta-Volterra Equations

###### 2000 Mathematics Subject Classification:

Primary: 35C07 ; Secondary: 35K57

##  1\. Introduction

Nonlinear parabolic partial differential equations play a vital role in
mathematical modeling for engineering as well as the physical and biological
sciences. The specific type of parabolic equations that we are interested in
are called reaction diffusion equations. When no delay is present there is a
myriad of scholarly work. In particular, there are very elegant and classical
results for traveling wave front solutions of PDE, see [5, 8, 9, 14, 21, 22,
26, 31].

The first work that studied the existence and properties of traveling waves in
diffusion-reaction equations with delay in reaction term is attributed to
Schaaf, [25]. The methods used by Schaaf were phase space analysis and the
maximum principle for parabolic functional differential equations for a Fisher
type nonlinearity.

The seminal work for constructing monotone wave fronts for reaction diffusion
is the paper by Wu and Zou [34]. Wu and Zou studied the existence of traveling
wave solutions with a singular delay in the reaction term. The existence of
such solutions was established under a quasimonotone or exponential
quasimonotone property via an iteration of appropriate upper and lower
solutions. This is the so-called monotone iteration method.

The above results were extended by Ma, [18] by employing the Schauder fixed
point theorem using the decay norm. Moreover, Ma developed so-called super and
sub solutions. This formulation relaxes the requirements of the upper and
lower solutions in Wu and Zou.

Boumenir and Nguyen, [3] introduced the concept of quasi-upper/lower
solutions, which removes the $C^{2}$ requirement for the initial construction
of upper and lower solutions. It was also shown that the quasi solutions
become upper and lower solutions after one iteration. For an overview of
results for traveling waves of reaction diffusion equations with delay, see
[1, 3, 7, 10, 11, 12, 16, 18, 25, 26, 30, 31, 33, 34, 35, 36], their progeny
and references therein.

An important model in Biology is the so called Lotka-Volterra diffusion-
cooperation system. Traveling waves solutions are important in such models.
The literature for traveling diffusion-cooperation models have been concerned
with either no delay or delay in the reaction term. Lotka-Volterra models of
the form

(1.1) $\displaystyle\frac{\partial u(x,t)}{\partial t}$
$\displaystyle=D_{1}\frac{\partial^{2}u(x,t)}{\partial
x^{2}}+\alpha_{1}u(x,t)\left(1-a_{1}u(x,t-\tau_{1})-b_{1}v(x,t-\tau_{2})\right)$
$\displaystyle\frac{\partial v(x,t)}{\partial t}$
$\displaystyle=D_{2}\frac{\partial^{2}v(x,t)}{\partial
x^{2}}+\alpha_{2}v(x,t)\left(1-b_{2}v(x,t-\tau_{3})-a_{2}u(x,t-\tau_{4})\right).$

where, $u(x,t),v(x,t)$ represent the population densities of two competing
species, $D_{1},D_{2}$ are positive diffusion constants,
$\alpha_{1},\alpha_{2}$ is the respective intrinsic population growth for
$u,v,$ and $a_{1},b_{1},a_{2},b_{2}$ are positive constants have been studied
in [13, 15, 17, 23, 28, 29, 32] and references therein.

It is important to mention that the standard monotone iteration and relaxed
exponential monotone iteration methods fail to allow for the construction of
upper/lower solutions. A different iteration method is required, which can be
referred to partial quasi-monotone, cross, or mixed interaction method.

The existence of traveling waves for systems of the form (1.1) was studied in
Li, et al. [15] via constructing weak upper and lower solutions and using the
Schauder fixed point theorem. Feng et al. [28, 29] obtained similar results
for ratio dependent multi-species Lotka-Volterra competition models.

The main purpose of this paper is to establish the existence of traveling wave
solutions of the following Lotka-Volterra competition-cooperation model with
delay in the diffusion term

(1.2) $\displaystyle\frac{\partial u(x,t)}{\partial t}$
$\displaystyle=D_{1}\frac{\partial^{2}u(x,t-\tau_{1})}{\partial
x^{2}}+\alpha_{1}u(x,t)\left(1-a_{1}u(x,t)-b_{1}v(x,t-\tau_{2})\right)$
$\displaystyle\frac{\partial v(x,t)}{\partial t}$
$\displaystyle=D_{2}\frac{\partial^{2}v(x,t-\tau_{3})}{\partial
x^{2}}+\alpha_{2}v(x,t)\left(1-b_{2}v(x,t)-a_{2}u(x,t-\tau_{4})\right).$

Here, $u(x,t),v(x,t)$ represent the population densities of two competing
species, $D_{1},D_{2}$ are positive diffusion constants,
$\alpha_{1},\alpha_{2}$ is the respective intrinsic population growth for
$u,v,.$ The constants $a_{1},b_{1},a_{2},b_{2}$ are positive and the values
$\tau_{i},i=1,2,3,4$ are positive time delays.

In order to use a monotone or partial monotone iteration method the
positiveness of the unique bounded solutions needs to be established. The
question of placing delay in the diffusion term has been difficult to answer.
In Barker and Nguyen, [2] it was shown that the second order delay equation of
the form $x^{\prime\prime}(t)-ax^{\prime}(t+r)-bx(t+r)=f(t)$, where
$a\not=0,b,r>0$ has a unique bounded solution for each given bounded and
continuous $f(t)$. Moreover, if $r>0$ is sufficiently small and $f(t)\geq 0$
for $t\in\mathbb{R}$, then the unique bounded solution $x_{f}(t)\leq 0$ for
all $t\in\mathbb{R}$. This was done by using complex analysis to determine
properties of the characteristic equation, for more information see [4, 6, 11,
20, 19, 24, 27], and references therein. Our question seems to differ from the
standard study where delay in the diffusion term is not considered, see[1, 3,
7, 10, 11, 12, 13, 15, 16, 17, 18, 23, 25, 26, 28, 29, 30, 31, 33, 32, 34, 35,
36].

The paper will be organized as follows: in Section 2 we will put forth
preliminary information that will be utilized throughout the paper. In Section
3, we will show the existence of traveling waves for the system (1.2) via
Schauder’s fixed point theorem in a Banach space equipped with exponential
decay norm. This will be aided by partial monotone conditions that will be
placed on appropriate upper and lower solutions.

It will also be shown that the existence of quasi upper and lower solutions
imply the existence of smooth upper and lower solutions. This will also imply
the existence of quasi upper and lower solutions is a sufficient condition for
the existence of traveling waves. In Section 4, we will construct quasi upper
and lower solutions to an interesting Lotka-Volterra competition model for two
species.

##  2\. Preliminaries

In this paper we will use some standard notations as
$\mathbb{R}^{n},\mathbb{C}^{n}$ for the fields of reals and complex numbers in
$n$ dimensions. We also take standard ordering for $\mathbb{R}^{2}$. This
means $u=(u_{1},u_{2})^{T}$ and $v=(v_{1},v_{2})^{T}.$ We say that $u\leq v$
if $u_{1}\leq v_{1},\ u_{2}\leq v_{2}.$ We also say that $u<v$ if
$u_{1}<v_{1},\ u_{2}<v_{2}.$ We also take $|\cdot|$ to be the Euclidean norm
in $\mathbb{R}^{2}.$

The space of all bounded and continuous functions from
$\mathbb{R}\to\mathbb{R}^{n}$ is denoted by $BC(\mathbb{R},\mathbb{R}^{n})$
which is equipped with the sup-norm $||f||:=\sup_{t\in\mathbb{R}}\|f(t)\|,$
where $f(t)\in C\left(U,\mathbb{R}^{2}\right)$, where $U\subset\mathbb{R}$.
$BC^{k}(\mathbb{R},\mathbb{R}^{n})$ stands for the space of all $k$-time
continuously differentiable functions $\mathbb{R}\to\mathbb{R}^{n}$ such that
all derivatives up to order $k$ are bounded.

If the boundedness is dropped from the above function spaces we will simply
denote them by $C(\mathbb{R},\mathbb{R}^{n})$ and
$C^{k}(\mathbb{R},\mathbb{R}^{n})$. For $f,g\in BC(\mathbb{R},\mathbb{R}^{n})$
we will use the natural order $f\leq g$ if and only if $f(t)\leq g(t)$ for all
$t\in\mathbb{R}$. A constant function $f(t)=\alpha$ for all $t\in\mathbb{R}$
will be denoted by $\hat{\alpha}.$

Moreover, we note that
$f_{ic}(\phi_{\theta},\psi_{\theta}):\mathbb{X}_{c\tau}=C\left([-\tau,0],\mathbb{R}^{2}\right)\to\mathbb{R}$
is defined by

$f_{ic}(\phi,\psi)=f_{i}\left(\phi^{c},\psi^{c}\right),\
\phi^{c}(\theta)=\phi(c\theta),\ \psi^{c}(\theta)=\psi(c\theta),\
\theta\in[\tau,0],\ i=1,2.$

This allows us to see that the system (1.2) under a wave front transformation
can be written as

(2.1) $\displaystyle
D_{1}\phi^{\prime\prime}(t)-c\phi^{\prime}(t+r_{1})+f_{1c}(\phi_{t},\psi_{t})=0$
$\displaystyle
D_{2}\psi^{\prime\prime}(t)-c\psi^{\prime}(t+r_{3})+f_{2c}(\phi_{t},\psi_{t})=0,$

where the delay has been moved out of the higher order term via an invariant
translation. We also implement the following asymptotic boundary conditions

$\displaystyle\lim_{t\to-\infty}\phi(t)=0,\ $
$\displaystyle\lim_{t\to\infty}\phi(t)=k_{1}$
$\displaystyle\lim_{t\to-\infty}\psi(t)=0,\ $
$\displaystyle\lim_{t\to\infty}\psi(t)=k_{2},$

where $0<k_{1}\leq 1,\ 0<k_{2}\leq 1.$ The following conditions for the
reaction terms are employed

  1. (C1)

$f(\hat{0})=f(\hat{K})=0,$ where $K=(k_{1},k_{2})$

  2. (C2)

There are two Lipchitz constants $L_{1}>0,\ L_{2}>0$ such that

$\displaystyle\left|f_{1}(\phi_{1},\psi_{1})-f_{1}(\phi_{2},\psi_{2})\right|\leq
L_{1}\left|\left|\Phi-\Psi\right|\right|$
$\displaystyle\left|f_{2}(\phi_{1},\psi_{1})-f_{2}(\phi_{2},\psi_{2})\right|\leq
L_{2}\left|\left|\Phi-\Psi\right|\right|,$

for $\Phi=(\phi_{1},\psi_{1}),\ \Psi=(\phi_{2},\psi_{2})\in
C\left([\tau,0],\mathbb{R}\right)),$ and $0\leq\Phi(s),\Psi(s)\leq K.$

##  3\. The Case of Partial Quasimonotonicity

###  3.1. Partial Quasimonotonicity Conditions

In this section, we develop sufficient conditions for the existence of
traveling wave solutions for equations of the form (2.1). We invoke the
following cross iteration scheme.

######  Definition 3.1.

Partial Quasi-Monotone Condition (PQM) Fix two constants $\beta_{1},\
\beta_{2}>0$ such that

  1. (P1)

$f_{1}\left(\phi_{1},\psi_{1}\right)-f_{1}\left(\phi_{2},\psi_{1}\right)+\beta_{1}\left[\phi_{1}(0)-\phi_{2}(0)\right]\geq
0,$

  2. (P2)

$f_{1}\left(\phi_{1},\psi_{1}\right)-f_{1}\left(\phi_{1},\psi_{2}\right)\leq
0,$

  3. (P3)

$f_{2}\left(\phi_{1},\psi_{1}\right)-f_{2}\left(\phi_{2},\psi_{2}\right)+\beta_{2}\left[\psi_{1}(0)-\psi_{2}(0)\right]\geq
0.$

  4. (P4)

$f_{2}\left(\phi_{1},\psi_{1}\right)-f_{2}\left(\phi_{2},\psi_{1}\right)\leq
0,$

where $\phi_{1},\ \phi_{2},\ \psi_{1},\ \psi_{2}\in C(\mathbb{R},\mathbb{R}),\
0\leq\phi_{2}\leq\phi_{1}\leq k_{1},\ 0\leq\psi_{2}\leq\psi_{1}\leq k_{2}.$

Let $\beta_{1},\ \beta_{2}$ be the constants in (PQM), define the following
operators

(3.1) $\displaystyle
H_{1}(\phi,\psi)(t)=f_{1c}\left(\phi_{t},\psi_{t}\right)+\beta_{1}\phi(t+r_{1}),\
\phi,\psi\in C(\mathbb{R},\mathbb{R})$ (3.2) $\displaystyle
H_{2}(\phi,\psi)(t)=f_{2c}\left(\phi_{t},\psi_{t}\right)+\beta_{2}\psi(t+r_{3}),\
\phi,\psi\in C(\mathbb{R},\mathbb{R}).$

The operators $H_{1},H_{2}$ satisfy the following properties:

######  Lemma 3.2.

Assume $(C1)$ and (PQM) hold, then

  1. i)

$H_{1}(\phi_{2},\psi_{1})(t)\leq H_{1}(\phi_{1},\psi_{1})(t)$

  2. ii)

$H_{1}(\phi_{1},\psi_{1})(t)\leq H_{1}(\phi_{1},\psi_{2})(t)$

  3. iii)

$H_{2}\left(\phi,\psi\right)(t)\geq 0$

  4. iv)

$H_{2}\left(\phi,\psi\right)(t)$ is non-decreasing for $t\in\mathbb{R}$

  5. v)

$H_{2}\left(\phi_{2},\psi_{2}\right)(t)\leq
H_{2}\left(\phi_{1},\psi_{1}\right)(t).$

###### Proof.

Looking towards $i.).$ By (PQM) it is clear

$\displaystyle
H_{1}(\phi_{2},\psi_{1})(t)-H_{1}(\phi_{1},\psi_{1})(t)=f_{1c}\left(\phi_{2t},\psi_{1t}\right)+\beta_{1}\phi_{2}(t+r_{1})-\left(f_{1c}\left(\phi_{1t},\psi_{1t}\right)+\beta_{1}\phi_{1}(t+r_{1})\right)$
$\displaystyle=f_{1c}\left(\phi_{2t},\psi_{1t}\right)-f_{1c}\left(\phi_{1t},\psi_{1t}\right)+\beta_{1}\left(\phi_{2}(t+r_{1})-\phi_{1}(t+r_{1})\right)$
$\displaystyle=-\left[f_{1c}\left(\phi_{1t},\psi_{1t}\right)-f_{1c}\left(\phi_{2t},\psi_{1t}\right)+\beta_{1}\left(\phi_{1}(t+r_{1})-\phi_{2}(t+r_{1})\right)\right]\leq
0,$

by (P1) pf (PQM). Thus,
$H_{1}(\phi_{2},\psi_{1})(t)-H_{1}(\phi_{1},\psi_{1})(t)\leq 0$. This finishes
$i.)$ We will show $ii.)$ via direct computation as well. Indeed,

$\displaystyle
H_{1}(\phi_{1},\psi_{1})(t)-H_{1}(\phi_{1},\psi_{2})(t)=f_{1c}\left(\phi_{1t},\psi_{1t}\right)+\beta_{1}\phi_{1}(t+r_{1})-\left(f_{1c}\left(\phi_{1t},\psi_{2t}\right)+\beta_{1}\phi_{1}(t+r_{1})\right)$
$\displaystyle=f_{1c}\left(\phi_{1t},\psi_{1t}\right)-f_{1c}\left(\phi_{1t},\psi_{2t}\right)\leq
0$

by (P2) of (PQM). This proves $ii.)$ Part $iii.)$ follows directly from
$(C_{1})$ and each $\phi,\psi,\beta_{2}\geq 0.$ In order to prove $iv.)$ we
fix $t\in\mathbb{R},s>0,$ then

$\displaystyle 0\leq\phi_{t}\leq\phi_{t+s}\leq k_{1}$ $\displaystyle
0\leq\psi_{t}\leq\psi_{t+s}\leq k_{2}$

due to the fact that both $\phi$ and $\psi$ are non-decreasing. From here, we
have

$\displaystyle
H_{2}\left(\phi,\psi\right)(t+s)-H_{2}\left(\phi,\psi\right)(t)=f_{2c}\left(\phi_{t+s},\psi_{t+s}\right)+\beta_{2}\phi(t+s)-\left(f_{2c}\left(\phi_{t},\psi_{t}\right)+\beta_{2}\phi(t)\right)$
$\displaystyle=f_{2c}\left(\phi_{t+s},\psi_{t+s}\right)-f_{2c}\left(\phi_{t},\psi_{t}\right)+\beta_{2}\left[\phi(t+s)-\phi(t)\right]$
$\displaystyle=f_{2c}\left(\phi_{t+s},\psi_{t+s}\right)-f_{2c}\left(\phi_{t},\psi_{t}\right)+\beta_{2}\left[\phi_{t+s}(0)-\phi_{t}(0)\right]\geq
0$

by (P3) of (PQM). This proves $iv.)$ Part $v.)$ follows directly from (P3) of
(PQM). This completes the proof. ∎

We can write the system (2.1) for all $t\in\mathbb{R}$ as

(3.3) $\displaystyle
D_{1}\phi^{\prime\prime}(t)-c\phi^{\prime}(t+r_{1})-\beta_{1}\phi(t+r_{1})+H_{1}\left(\phi,\psi\right)(t)=0$
$\displaystyle
D_{2}\psi^{\prime\prime}(t)-c\psi^{\prime}(t+r_{3})-\beta_{2}\psi(t+r_{3})+H_{2}\left(\phi,\psi\right)(t)=0.$

###  3.2. Existence of Traveling Wave Solutions

In this section we will consider existence of positive solutions for the
system (3.3). In [2], it was shown that equations of the form
$x^{\prime\prime}(t)-ax^{\prime}(t+r)-bx(t+r)=f(t)$ have positive solutions
where $r\in\mathbb{R},\ f\in BC\left(\mathbb{R},\mathbb{R}\right).$ It was
also noted that these results can be extended to higher dimensions.

In order to deal with the nonlinear part, we will show the existence of fixed
point(s) by finding a set of appropriate convolution operators and applying
Schrauder’s fixed point theorem. Defining the set

(3.4) $\displaystyle\Gamma_{1}=\left\\{\left(\phi,\psi\right)\in
C\left(\mathbb{R},\mathbb{R}^{2}\right):\left(0,0\right)\leq\left(\phi,\psi\right)\leq\left(k_{1},k_{2}\right)\right\\}$

we rewrite Eq.(3.3) in the form

(3.5)
$(\phi,\psi)^{T}=-{\mathcal{L}}^{-1}H(\phi,\psi)=\left(-{\mathcal{L}}_{1}^{-1}H_{1}(\phi,\psi),-{\mathcal{L}}_{2}^{-1}H_{2}(\phi,\psi)\right)^{T},\
\phi,\psi\in\Gamma_{1},$

Taking
$F:=-{\mathcal{L}}^{-1}H=-\left(F_{1}H_{1}(\phi,\psi),F_{2}H_{2}(\phi,\psi)\right)^{T}.$
Thus, we can use the following Perron-Lyapunov integral operator

$F=\left(F_{1}\left(\phi,\psi\right),F_{2}\left(\phi,\psi\right)\right):\Gamma_{1}\left(\mathbb{R},\mathbb{R}^{2}\right)\to
C\left(\mathbb{R},\mathbb{R}^{2}\right)$

defined by

(3.6) $\displaystyle F_{1}\left(\phi,\psi\right)$
$\displaystyle=\int^{\infty}_{-\infty}G_{1}(t-s,r)H_{1}(\phi(s),\psi(s))ds$
$\displaystyle F_{2}\left(\phi,\psi\right)$
$\displaystyle=\int^{\infty}_{-\infty}G_{2}(t-s,r)H_{2}(\phi(s),\psi(s))ds.$

Here, we have positive constants $M_{1},M_{2},\delta_{1},\delta_{2}$ that are
uniformly bounded in $r$ for some sufficiently small $r>0$ such that for all
$t\in\mathbb{R}$

$\displaystyle|G_{1}(t,r)|\leq M_{1}e^{-\delta_{1}|t|}$
$\displaystyle|G_{2}(t,r)|\leq M_{2}e^{-\delta_{2}|t|}.$

See Theorem 4.1 [20].

######  Lemma 3.3.

Define
$F=\left(F_{1}\left(\phi,\psi\right),F_{2}\left(\phi,\psi\right)\right)$ as
above, then for any $\left(\phi,\psi\right)\in\Gamma_{1},$ then

  1. i)

$\left(F_{1}\left(\phi,\psi\right),F_{2}\left(\phi,\psi\right)\right):\Gamma_{1}\left(\mathbb{R},\mathbb{R}^{2}\right)\to
C\left(\mathbb{R},\mathbb{R}^{2}\right)$ is well defined.

  2. ii)

$F_{1}\left(\phi,\psi\right),F_{2}\left(\phi,\psi\right)$ satisfy

(3.7) $\displaystyle
D_{1}F_{1}^{\prime\prime}\left(\phi,\psi\right)-cF_{1}^{\prime}\left(\phi,\psi\right)-\beta_{1}F_{1}\left(\phi,\psi\right)+H_{1}\left(\phi,\psi\right)=0$
$\displaystyle
D_{2}F_{2}^{\prime\prime}\left(\phi,\psi\right)-cF_{2}^{\prime}\left(\phi,\psi\right)-\beta_{2}F_{2}\left(\phi,\psi\right)+H_{2}\left(\phi,\psi\right)=0.$

The proof is straight forward, and is hence omitted. Moreover, if
$\left(\phi,\psi\right)$ is a fixed point of
$\left(F_{1}\left(\phi,\psi\right),F_{2}\left(\phi,\psi\right)\right),$ then
the system (3.3) has a traveling wave solution. To this end,
$\left(F_{1}\left(\phi,\psi\right),F_{2}\left(\phi,\psi\right)\right),$ enjoys
similar properties as
$\left(H_{1}\left(\phi,\psi\right),H_{2}\left(\phi,\psi\right)\right),$ in
Lemma (3.2). Indeed,

######  Lemma 3.4.

Assume that $(C1)$ and (PQM) hold, then

  1. i)

$F_{2}\left(\phi,\psi\right)(t)$ is non-decreasing for $t\in\mathbb{R}$

  2. ii)

$F_{1}(\phi_{2},\psi_{1})(t)\leq F_{1}(\phi_{1},\psi_{1})(t)$

  3. iii)

$F_{1}(\phi_{1},\psi_{1})(t)\leq F_{1}(\phi_{1},\psi_{2})(t)$

  4. iv)

$F_{2}\left(\phi_{2},\psi_{2}\right)(t)\leq
F_{2}\left(\phi_{1},\psi_{1}\right)(t),$

when $\phi_{1},\ \phi_{2},\ \psi_{1},\ \psi_{2}\in C(\mathbb{R},\mathbb{R}),\
0\leq\phi_{2}\leq\phi_{1}\leq k_{1},\ 0\leq\psi_{2}\leq\psi_{1}\leq k_{2}.$

The proof follows from the inherited properties of
$\left(H_{1}\left(\phi,\psi\right),H_{2}\left(\phi,\psi\right)\right)$ from
Lemma (3.2).

####  3.2.1. Existence Via Schauder’s Fixed Point Theorem

We define the following set

$\displaystyle\Gamma_{2}\left(\left(\underline{\phi},\underline{\psi}\right),\left(\overline{\phi},\overline{\psi}\right)\right)=\bigl{\\{}\left(\phi,\psi\right)\in
C\left(\mathbb{R},\mathbb{R}^{2}\right):{}$ $\displaystyle i)\ \psi(t)\
\text{is nondecreasing in}\ \mathbb{R}.\ \text{and}$ $\displaystyle ii)\
\underline{\phi}(t)\leq\phi(t)\leq\overline{\phi},\
\underline{\psi}(t)\leq\psi(t)\leq\overline{\psi}(t)\bigr{\\}}.$

Furthermore, we define the exponential decay norm for some $\mu>0$ as

$|\Phi|_{\mu}=\sup_{t\in\mathbb{R}}\ e^{-\mu|t|}||\Phi(t)||.$

Now, define the ball as

$B_{\mu}\left(\mathbb{R},\mathbb{R}^{2}\right)=\left\\{\Phi\in
C\left(\mathbb{R},\mathbb{R}^{2}\right):|\Phi|_{\mu}<\infty\right\\}.$

######  Lemma 3.5.

Define
$\Gamma_{2}\left(\left(\underline{\phi},\underline{\psi}\right),\left(\overline{\phi},\overline{\psi}\right)\right),|\Phi|_{\mu},$
and $B_{\mu}(\mathbb{R},\mathbb{R}^{2})$ as above, then

  1. i)

$\Gamma_{2}$ is nonempty.

  2. ii)

$\Gamma_{2}$ is closed, bounded and convex.

  3. iii)

$\Gamma_{2}\subset B_{\mu}(\mathbb{R},\mathbb{R}^{2}).$

  4. iv)

$\left(B_{\mu}(\mathbb{R},\mathbb{R}^{2}),|\cdot|_{\mu}\right)$ is a Banach
space.

The result is classical, thus the proof is omitted. It is now possible to
complete the requirements of Schauder’s theorem.

######  Lemma 3.6.

Assume (C1), (C2) and (PQM) hold, then

  1. i)

$F:\Gamma_{2}\left(\left(\underline{\phi},\underline{\psi}\right),\left(\overline{\phi},\overline{\psi}\right)\right)\to\Gamma_{2}\left(\left(\underline{\phi},\underline{\psi}\right),\left(\overline{\phi},\overline{\psi}\right)\right).$

  2. ii)

$F=(F_{1},F_{2})$ is continuous with respect to $|\cdot|_{\mu}$ in
$B_{\mu}(\mathbb{R},\mathbb{R}^{2}).$

###### Proof.

For part $i.)$ we only need to show that

$\displaystyle\begin{cases}\underline{\phi}\leq
F_{1}(\underline{\phi},\overline{\psi})\leq
F_{1}(\overline{\phi},\underline{\psi})\leq\overline{\phi}\\\
\underline{\psi}\leq F_{2}(\overline{\phi},\underline{\psi})\leq
F_{2}(\underline{\phi},\overline{\psi})\leq\overline{\psi}.\end{cases}$

This is due to the fact

$\displaystyle\begin{cases}F_{1}(\underline{\phi},\overline{\psi})\leq
F_{1}(\phi,\psi)\leq F_{1}(\overline{\phi},\underline{\psi})\\\
F_{2}(\overline{\phi},\underline{\psi})\leq F_{2}(\phi,\psi)\leq
F_{2}(\underline{\phi},\overline{\psi}).\end{cases}$

We only show the first inequality, because the second follows the same way. We
first show that $\underline{\phi}\leq
F_{1}(\underline{\phi},\overline{\psi})$.

$\displaystyle
F_{1}(\underline{\phi},\overline{\psi})(t)=\int_{-\infty}^{\infty}G_{1}(t-s,r)H_{1}(\underline{\phi},\overline{\psi})(s)ds$
$\displaystyle\geq\int_{-\infty}^{\infty}G_{1}(t-s,r)\left(-D_{1}\underline{\phi}^{\prime\prime}(t)+c\underline{\phi}^{\prime}(t+r_{1})+\beta_{1}\underline{\phi}(t+r_{1})\right)ds=\underline{\phi}(t)$

for all $t\in\mathbb{R}.$ Next, we show
$F_{1}(\overline{\phi},\underline{\psi})\leq\overline{\phi}$.

$\displaystyle
F_{1}(\overline{\phi},\underline{\psi})(t)=\int_{-\infty}^{\infty}G_{1}(t-s,r)H_{1}(\overline{\phi},\underline{\psi})(s)ds$
$\displaystyle\leq\int_{-\infty}^{\infty}G_{1}(t-s,r)\left(-D_{1}\overline{\phi}^{\prime\prime}(t)+c\overline{\phi}^{\prime}(t+r_{1})+\beta_{1}\overline{\phi}(t+r_{1})\right)ds=\overline{\phi}(t)$

for all $t\in\mathbb{R}.$ The proof is complete. The method of proof for part
$ii.)$ is similar to [15], Lemma 3.4 with several modifications. To this end,
we only need to show

$F_{1}:B_{\mu}(\mathbb{R},\mathbb{R}^{2})\to
B_{\mu}(\mathbb{R},\mathbb{R}^{2})$

is continuous with respect to $|\cdot|_{\mu},$ because the proof for
$F_{2}(\phi,\psi)$ is very similar. Take $\mu<\max\\{\delta_{1},\delta_{2}\\}$
and let $\Phi=(\phi_{1},\psi_{1}),\Psi=(\phi_{2},\psi_{2})\in
B_{\mu}(\mathbb{R},\mathbb{R}^{2}).$ It is clear that

$F_{1}=(F_{1}(\Phi),F_{1}(\Psi))\in B_{\mu}(\mathbb{R},\mathbb{R}^{2}).$

We will now turn our attention to the continuity of $F$. Fix $\varepsilon>0,$
and take

$\delta<\min\left\\{\frac{\varepsilon(\delta_{1}-\mu)}{2M_{1}},\frac{\varepsilon}{e^{\mu
c\tau}L_{1}+\beta_{1}}\right\\},$

where $L_{1}$ is the Lipchitz constant from (C2), $\beta_{1}$ is from (PQM),
and $\delta_{1},M_{1}$ are from the bounds of the Green function. We will
first prove that $H_{1}:B_{\mu}(\mathbb{R},\mathbb{R}^{2})\to
B_{\mu}(\mathbb{R},\mathbb{R}^{2})$ is continuous with respect to
$|\cdot|_{\mu}$. We take $\left|\Phi(t)-\Psi(t)\right|_{\mu}<\delta,$ then

$\displaystyle A:=\left|H_{1}(\Phi)(t)-H_{1}(\Psi)(t)\right|_{\mu}$
$\displaystyle=\left|f_{1c}(\phi_{1t},\psi_{1t})+\beta_{1}\phi_{1}(t+r_{1})-\left(f_{2c}(\phi_{2t},\psi_{2t})+\beta_{1}\phi_{2}(t+r_{1})\right)\right|_{\mu}$
$\displaystyle\leq\left|f_{1c}(\phi_{1t},\psi_{1t})-f_{2c}(\phi_{2t},\psi_{2t})\right|_{\mu}+\beta_{1}\left|\phi_{1}(t+r_{1})-\phi_{2}(t+r_{1})\right|_{\mu}$
$\displaystyle\leq
L_{1}||\Phi(t)-\Psi(t)||_{\mathbb{X}_{c\tau}}e^{-\mu|t|}+\beta_{1}\sup_{t\in\mathbb{R}}\left|\phi_{1}(t)-\phi_{2}(t)\right|_{\mu}$
$\displaystyle\leq
L_{1}\sup_{\theta\in(-c\tau,0)}|\Phi(t+\theta)-\Psi(t+\theta)|e^{-\mu|t+\theta|}+\beta_{1}|\Phi(t)-\Psi(t)|_{\mu}$
$\displaystyle\leq e^{\mu
c\tau}L_{1}|\Phi(t)-\Psi(t)|_{\mu}+\beta_{1}\left|\Phi(t)-\Psi(t)\right|_{\mu}<\left(e^{\mu
c\tau}L_{1}+\beta_{1}\right)\delta$ $\displaystyle<\left(e^{\mu
c\tau}L_{1}+\beta_{1}\right)\left(\frac{\varepsilon}{e^{\mu
c\tau}L_{1}+\beta_{1}}\right)<\varepsilon.$

Thus,

$H_{1}:B_{\mu}(\mathbb{R},\mathbb{R}^{2})\to
B_{\mu}(\mathbb{R},\mathbb{R}^{2})$

is continuous. The proof for $H_{2}(\phi,\psi)(t)$ is similar. Therefore,

$H=(H_{1},H_{2}):B_{\mu}(\mathbb{R},\mathbb{R}^{2})\to
B_{\mu}(\mathbb{R},\mathbb{R}^{2})$

is continuous with respect to $|\cdot|_{\mu}.$ We can now prove that $F_{1}$
is continuous in the same manner. Indeed,

$\displaystyle
B:=\left|F_{1}(\Phi)(t)-F_{1}(\Psi)(t)\right|=\left|F_{1}(\phi_{1},\psi_{1})-F_{1}(\phi_{2},\psi_{2})\right|$
$\displaystyle=\left|\int_{-\infty}^{\infty}G_{1}(t-s,r)\left(H_{1}(\phi_{1},\psi_{1})(s)-H_{1}(\phi_{2},\psi_{2})(s)\right)ds\right|$
$\displaystyle\leq\int_{-\infty}^{t}\left|G_{1}(t-s,r)\left(H_{1}(\phi_{1},\psi_{1})(s)-H_{1}(\phi_{2},\psi_{2})(s)\right)\right|ds$
$\displaystyle+\int_{t}^{\infty}\left|G_{1}(t-s,r)\left(H_{1}(\phi_{1},\psi_{1})(s)-H_{1}(\phi_{2},\psi_{2})(s)\right)\right|ds$
$\displaystyle=\int_{-\infty}^{t}\left|G_{1}(t-s,r)\right|\left|\left(H_{1}(\phi_{1},\psi_{1})(s)-H_{1}(\phi_{2},\psi_{2})(s)\right)\right|ds$
$\displaystyle+\int_{t}^{\infty}\left|G_{1}(t-s,r)\right|\left|\left(H_{1}(\phi_{1},\psi_{1})(s)-H_{1}(\phi_{2},\psi_{2})(s)\right)\right|ds.$

We can now use the fact that $(H_{1}(\phi,\psi)(t)$ is continuous with respect
to the exponential decay norm.

$\displaystyle
B=\int_{-\infty}^{t}\left|G_{1}(t-s,r)\right|e^{\mu|s|}\left|\left(H_{1}(\phi_{1},\psi_{1})(s)-H_{1}(\phi_{2},\psi_{2})(s)\right)\right|e^{-\mu|s|}ds$
$\displaystyle+\int_{t}^{\infty}\left|G_{1}(t-s,r)\right|e^{\mu|s|}\left|\left(H_{1}(\phi_{1},\psi_{1})(s)-H_{1}(\phi_{2},\psi_{2})(s)\right)\right|e^{-\mu|s|}ds$
$\displaystyle\leq\left(\int_{-\infty}^{t}\left|G_{1}(t-s,r)\right|e^{\mu|s|}ds+\int_{t}^{\infty}\left|G_{1}(t-s,r)\right|e^{\mu|s|}ds\right)$
$\displaystyle\times\sup_{s\in\mathbb{R}}\left|\left(H_{1}(\phi_{1},\psi_{1})(s)-H_{1}(\phi_{2},\psi_{2})(s)\right)\right|_{\mu}.$

Here, we will use the fact that $G_{1}(t,r)$ is bounded and decays as
$|t|\to\infty$ to see the following:

$\displaystyle
B\leq\delta\left(\int_{-\infty}^{t}\left|G_{1}(t-s,r)\right|e^{\mu|s|}ds+\int_{t}^{\infty}\left|G_{1}(t-s,r)\right|e^{\mu|s|}ds\right)$
$\displaystyle\leq\delta\left(\int_{-\infty}^{t}M_{1}e^{-\delta_{1}|t-s|}e^{\mu|s|}ds+\int_{t}^{\infty}M_{1}e^{-\delta_{1}|t-s|}e^{\mu|s|}ds\right)$
$\displaystyle\leq M_{1}\delta\left(\int_{0}^{t}e^{-\delta_{1}(t-s)+\mu
s}ds+\int_{-\infty}^{0}e^{-\delta_{1}(t-s)-\mu
s}ds+\int_{t}^{\infty}e^{\delta_{1}(t-s)+\mu s}ds\right)$
$\displaystyle<2M_{1}\delta\left(\frac{\delta_{1}e^{\mu t}+\mu
e^{-\delta_{1}t}}{\delta_{1}^{2}-\mu^{2}}\right).$

In the exponential decay norm for $t>0$ we see

$\displaystyle\left|F_{1}(\Phi)(t)-F_{1}(\Psi)(t)\right|_{\mu}=\left|F_{1}(\Phi)(t)-F_{1}(\Psi)(t)\right|e^{-\mu
t}$ $\displaystyle\leq 2M_{1}\delta\left(\frac{\delta_{1}e^{\mu t}+\mu
e^{-\delta_{1}t}}{\delta_{1}^{2}-\mu^{2}}\right)e^{-\mu
t}=2M_{1}\delta\left(\frac{\delta_{1}+\mu
e^{-(\delta_{1}+\mu)t}}{\delta_{1}^{2}-\mu^{2}}\right)$ $\displaystyle\leq
2M_{1}\delta\left(\frac{\delta_{1}+\mu}{\delta_{1}^{2}-\mu^{2}}\right)=2M_{1}\delta\left(\frac{1}{\delta_{1}-\mu}\right)<\varepsilon$

Moreover, when $t\leq 0$ we have the following estimate

$\displaystyle\left|F_{1}(\Phi)(t)-F_{1}(\Psi)(t)\right|=\left|F_{1}(\phi_{1},\psi_{1})-F_{1}(\phi_{2},\psi_{2})\right|$
$\displaystyle\leq\left(\int_{-\infty}^{t}\left|G_{1}(t-s,r)\right|e^{\mu|s|}ds+\int_{t}^{\infty}\left|G_{1}(t-s,r)\right|e^{\mu|s|}ds\right)$
$\displaystyle\times\sup_{s\in\mathbb{R}}\left|\left(H_{1}(\phi_{1},\psi_{1})(s)-H_{1}(\phi_{2},\psi_{2})(s)\right)\right|_{\mu}$
$\displaystyle\leq\delta\left(\int_{-\infty}^{t}\left|G_{1}(t-s,r)\right|e^{\mu|s|}ds+\int_{t}^{\infty}\left|G_{1}(t-s,r)\right|e^{\mu|s|}ds\right)$
$\displaystyle\leq M_{1}\delta\left(\int_{t}^{0}e^{\delta_{1}(t-s)-\mu
s}ds+\int_{-\infty}^{t}e^{-\delta_{1}(t-s)-\mu
s}ds+\int_{0}^{\infty}e^{\delta_{1}(t-s)+\mu s}ds\right)$
$\displaystyle<2M_{1}\delta\left(\frac{\delta e^{-\mu t}+\mu
e^{\delta_{1}t}}{\delta_{1}^{2}-\mu^{2}}\right).$

Similarly as above in the exponential norm for $t\leq 0$ we see

$\displaystyle\left|F_{1}(\Phi)(t)-F_{1}(\Psi)(t)\right|_{\mu}=\left|F_{1}(\Phi)(t)-F_{1}(\Psi)(t)\right|e^{\mu
t}$ $\displaystyle\leq 2M_{1}\delta\left(\frac{\delta_{1}e^{-\mu t}+\mu
e^{\delta_{1}t}}{\delta_{1}^{2}-\mu^{2}}\right)e^{\mu
t}=2M_{1}\delta\left(\frac{\delta_{1}+\mu
e^{(\delta_{1}+\mu)t}}{\delta_{1}^{2}-\mu^{2}}\right)$ $\displaystyle\leq
2M_{1}\delta\left(\frac{\delta_{1}+\mu}{\delta_{1}^{2}-\mu^{2}}\right)=2M_{1}\delta\left(\frac{1}{\delta_{1}-\mu}\right)<\varepsilon.$

Therefore,

$F=(F_{1},F_{2}):B_{\mu}(\mathbb{R},\mathbb{R}^{2})\to
B_{\mu}(\mathbb{R},\mathbb{R}^{2})$

is continuous with respect to $|\cdot|_{\mu}.$ ∎

######  Corollary 3.7.

Assume $(C2)$ holds, then there is exists a constant $C_{1}>0$ such that

$|F(\Phi)-F(\Psi)|_{\mu}\leq C_{1}|\Phi-\Psi|_{\mu}\ \text{for all}\
\phi,\psi\in\Gamma_{2},\ t\in\mathbb{R}.$

######  Lemma 3.8.

Assume $(C1)$ and $(C2)$ hold then $F(\Gamma_{2})\to\Gamma_{2}$ is compact.

###### Proof.

Corollary (3.7) in conjunction with Lemmas (3.5) and (3.6) shows that
$F^{n}(\Gamma_{2})$ is equicontinuous and uniformly bounded on any finite
interval in $\mathbb{R}$. Thus, take $n\in\mathbb{N}$, then in the interval
$[-n,n]$ we can say that $F^{n}$ is compact by Arzela-Ascoli. Now, define

$F(\Phi,\Psi)(t)=\begin{cases}F(\Phi,\Psi)(t),&\ \text{when}\ t\in[-n,n]\\\
F(\Phi,\Psi)(n),&\ \text{when}\ t\in(n,\infty)\\\ F(\Phi,\Psi)(-n),&\
\text{when}\ t\in(-\infty,-n).\end{cases}$

Fix $t\in\mathbb{R},$ then for all $(\phi(t),\psi(t))\in\Gamma_{2}.$

$\displaystyle\sup_{t\in\mathbb{R}}|F^{n}(\Phi,\Psi)(t)-F(\Phi,\Psi)(t)|_{\mu}$
$\displaystyle=\sup_{t\in(-\infty,-n)\bigcup(n,\infty)}|F^{n}(\Phi,\Psi)(t)-F(\Phi,\Psi)(t)|_{\mu}$
$\displaystyle\leq 2C_{1}e^{-\mu n}$

Thus, $\lim_{n\to\infty}2C_{1}e^{-\mu n}=0,$ so $F^{n}\to F$ in $\Gamma_{2}$
as $n\to\infty.$ We can apply Arzela-Ascoli to $F$ as well. The proof is
complete. ∎

We can now state and prove our main result.

######  Definition 3.9.

A pair of functions
$\overline{\Phi}=\left(\overline{\phi},\overline{\psi}\right),\underline{\Phi}=\left(\underline{\phi},\underline{\psi}\right)\in
C^{2}(\mathbb{R}^{2},\mathbb{R}),$ where
$\phi,\phi^{\prime},\phi^{\prime\prime},\psi,\psi^{\prime},\psi^{\prime\prime}$
are bounded on $\mathbb{R}$, is called an upper solution (lower solution,
respectively) for the wave equation (2.1) if it satisfies the following

$\displaystyle
D_{1}\phi^{\prime\prime}(t)-c\phi^{\prime}(t+r_{1})+f_{1c}(\overline{\phi_{t}},\underline{\psi_{t}})\leq
0,$ $\displaystyle
D_{2}\psi^{\prime\prime}(t)-c\psi^{\prime}(t+r_{3})+f_{2c}(\overline{\phi_{t}},\overline{\psi_{t}})\leq
0$

and

$\displaystyle
D_{1}\phi^{\prime\prime}(t)-c\phi^{\prime}(t+r_{1})+f_{1c}(\underline{\phi_{t}},\overline{\psi_{t}})\leq
0,$ $\displaystyle
D_{2}\psi^{\prime\prime}(t)-c\psi^{\prime}(t+r_{3})+f_{2c}(\underline{\phi_{t}},\underline{\psi_{t}})\leq
0.$

######  Theorem 3.10.

Assume $(C1),(C2)$ and $(PQM)$ hold, if there is an upper
$(\overline{\phi},\overline{\psi})\in\Gamma_{2}$ and a lower solution
$(\underline{\phi},\underline{\psi})\in\Gamma_{2}$ of Eq.(2.1) such that for
all $t\in\mathbb{R}$

$0\leq\underline{\psi}(t)\leq\overline{\psi}(t),\
0\leq\underline{\phi}(t)\leq\overline{\phi}(t).$

Then, there exists a monotone traveling wave solution to the system (2.1).

###### Proof.

Lemmas 3.5, 3.6, 3.8 allows us to use Schuader’s fixed point theorem to show
that there exists a fixed point for system(2.1). All we need to show is the
asymptotic boundary conditions hold. If we apply the asymptotic limits and use
the fact that any $(\phi,\psi)\in\Gamma_{2}$ we have

$(\underline{\phi}(t),\underline{\psi}(t))\leq(\phi,\psi)\leq(\overline{\phi},\overline{\psi})\leq(k_{1},k_{2})$

it is easy to see the asymptotic boundary holds. This completes the proof. ∎

It is extremely difficult if not impossible to directly find upper/lower
solutions directly. It is much easier to relax the conditions and construct
upper and/or lower solutions using what are known as quasi-upper/lower
solutions, which are ”rougher” in nature.

######  Definition 3.11.

A pair of functions
$\overline{\Phi}=\left(\overline{\phi},\overline{\psi}\right),\underline{\Phi}=\left(\underline{\phi},\underline{\psi}\right)\in
C^{1}(\mathbb{R},\mathbb{R}^{2}),$ where
$\phi,\phi^{\prime},\psi,\psi^{\prime}$ are bounded on $\mathbb{R}$,
$\phi^{\prime\prime},\psi^{\prime\prime}$ are locally integrable and
essentially bounded on $\mathbb{R}$, is called a quasi- upper solution (quasi-
lower solution, respectively) for the wave equation (2.1) if it satisfies the
following for almost every $t\in\mathbb{R}$

$\displaystyle
D_{1}\phi^{\prime\prime}(t)-c\phi^{\prime}(t+r_{1})+f_{1c}(\overline{\phi_{t}},\underline{\psi_{t}})\leq
0,$ $\displaystyle
D_{2}\psi^{\prime\prime}(t)-c\psi^{\prime}(t+r_{3})+f_{2c}(\overline{\phi_{t}},\overline{\psi_{t}})\leq
0$

and

$\displaystyle
D_{1}\phi^{\prime\prime}(t)-c\phi^{\prime}(t+r_{1})+f_{1c}(\underline{\phi_{t}},\overline{\psi_{t}})\leq
0,$ $\displaystyle
D_{2}\psi^{\prime\prime}(t)-c\psi^{\prime}(t+r_{3})+f_{2c}(\underline{\phi_{t}},\underline{\psi_{t}})\leq
0.$

######  Proposition 3.12.

Let $(\phi,\psi)$ be a quasi- upper solution (quasi-lower solution,
respectively) of Eq. (2.1). Then, $F(\phi,\psi)$ is an upper solution (lower
solution, respectively) of Eq. (2.1).

###### Proof.

This can be done in the same manner for Proposition (4.6) in [2]. Their proof
hinges on the construction of an isomorphism between $W^{1,\infty}$ and
$L^{\infty}$, so if $\varphi\in L^{\infty}$ for
$T:(\varphi,\varphi^{\prime})^{T}\mapsto(0,f).$ This is defined by [20,
Theorem 4.1]. Also, there exists a unique bounded function $w\in
C^{2}(\mathbb{R},\mathbb{R})$ (see e.g. [19, 24]) such that

$Dw^{\prime\prime}(t)-cw^{\prime}(t+r_{1})-\beta
w(t+r_{1})=-\beta\varphi(t+r_{1})-f^{c}(\varphi_{t+r_{1}}).$

The inequalties can be shown using the monotone property. ∎

######  Corollary 3.13.

Let $(\phi,\psi)$ be a quasi- upper solution (quasi-lower solution,
respectively) of Eq. (2.1). Then, there exists a monotone traveling wave
solution of the system (2.1).

##  4\. Applications

We will consider the system of equations (1.2). That is,

(4.1) $\displaystyle\frac{\partial u(x,t)}{\partial t}$
$\displaystyle=D_{1}\frac{\partial^{2}u(x,t-\tau_{1})}{\partial
x^{2}}+\alpha_{1}u(x,t)\left(1-u(x,t)-av(x,t-\tau_{2})\right)$
$\displaystyle\frac{\partial v(x,t)}{\partial t}$
$\displaystyle=D_{2}\frac{\partial^{2}v(x,t-\tau_{3})}{\partial
x^{2}}+\alpha_{2}v(x,t)\left(1-v(x,t)-bu(x,t-\tau_{4})\right).$

######  Lemma 4.1.

Define

$\displaystyle f_{1}$
$\displaystyle(\phi,\psi)=\alpha_{1}\phi(0)\left(1-\phi(0)-a\psi(-r_{2})\right)$
$\displaystyle f_{2}$
$\displaystyle(\phi,\psi)=\alpha_{2}\psi(0)\left(1-\psi(0)-b\phi(-r_{4})\right),$

where, $\phi,\psi\in\ C\left([-c\tau,0],\mathbb{R}^{2}\right),\
\tau=\max\\{\tau_{1},\tau_{3}\\}.$ Then $f_{1}(\phi,\psi),f_{2}(\phi,\psi)$
satisfy $(PQM).$

For the sake of brevity we only need to show that $f_{1}(\phi,\psi)$ satisfies
the conditions, because $f_{2}(\phi,\psi)$ can be shown in the same manner. To
this end take $0\leq\phi_{2}(s)\leq\phi_{1}(s)\leq k_{1},\
0\leq\psi_{2}(s)\leq\psi_{1}(s)\leq k_{2},$ where $\phi_{i},\ \psi_{i}\in
C([-c\tau,0],\mathbb{R}).$ For $i.)$ we see by direct calculation

$\displaystyle
f_{1c}\left(\phi_{1},\psi_{1}\right)-f_{1c}\left(\phi_{2},\psi_{1}\right)$
$\displaystyle=\alpha_{1}\phi_{1}(0)\left(1-\phi_{1}(0)-a\psi_{1}(-r_{2})\right)-\alpha_{1}\phi_{2}(0)\left(1-\phi_{2}(0)-a\psi_{1}(-r_{2})\right)$
$\displaystyle=\alpha_{1}\left[\left(\phi_{1}(0)-\phi_{2}(0)\right)-\left(\phi^{2}_{1}(0)-\phi^{2}_{2}(0)\right)-a\psi_{1}(-r_{2})\right]$
$\displaystyle=\alpha_{1}\left(\phi_{1}(0)-\phi_{2}(0)\right)\left[1-a\psi_{1}(-r_{2})-\left(\phi_{1}(0)+\phi_{2}(0)\right)\right]$
$\displaystyle\geq\alpha_{1}\left(\phi_{1}(0)-\phi_{2}(0)\right)\left[1-ak_{2}-2k_{1}\right].$

Fix $\beta_{1}>0$ such that $-\beta_{1}\leq\alpha_{1}(1-ak_{2}-2k_{1}),$ then

$\displaystyle
f_{1c}\left(\phi_{1},\psi_{1}\right)-f_{1c}\left(\phi_{2},\psi_{1}\right)+\beta_{1}(\phi_{1}(0)-\phi_{2}(0))$
$\displaystyle\geq\left(\phi_{1}(0)-\phi_{2}(0)\right)\left[\alpha_{1}\left(1-ak_{2}-2k_{1}\right)+\beta_{1}\right]\geq
0.$

The proof for part $ii.)$ is also done via direct calculation and rather
straight forward. In fact,

$\displaystyle
f_{1c}\left(\phi_{1},\psi_{1}\right)-f_{1c}\left(\phi_{1},\psi_{2}\right)=\alpha_{1}\phi_{1}(0)\left(1-\phi_{1}(0)-a\psi_{1}(-r_{2})\right)-\alpha_{1}\phi_{1}(0)\left(1-\phi_{1}(0)-a\psi_{2}(-r_{2})\right)$
$\displaystyle=\alpha_{1}a\left(\psi_{2}(-r_{2})-\psi_{1}(-r_{2})\right)\leq
0,$

since $0\leq\psi_{2}(s)\leq\psi_{1}(s)\leq k_{2}$ for all $s\in[-c\tau,0].$ We
define traveling wave solutions as $u(x,t)=\phi(x+ct),\ v(x,t)=\psi(x+ct),$
where $c>0$ is the wave speed. Applying the wave transformation and letting
$\xi=x+ct$ gives

$\displaystyle
u(x,t-\tau_{4})=\phi(x+c(t-\tau_{4})=\phi(x+ct-c\tau_{4})=\phi(\xi-c\tau_{4})$
$\displaystyle\frac{\partial u(x,t)}{\partial
t}=c\phi^{\prime}(x+ct)=c\phi^{\prime}(\xi)$
$\displaystyle\frac{\partial^{2}u(x,t-\tau_{1})}{\partial
x^{2}}=\phi^{\prime\prime}(x+c(t-\tau_{1}))=\phi^{\prime\prime}(\xi-c\tau_{1})$

For the function $v(x,t)$ we see

$\displaystyle
v(x,t-\tau_{2})=\psi(x+c(t-\tau_{2})=\psi(x+ct-c\tau_{2})=\psi(\xi-c\tau_{2})$
$\displaystyle\frac{\partial v(x,t)}{\partial
t}=c\psi^{\prime}(x+ct)=c\psi^{\prime}(\xi)$
$\displaystyle\frac{\partial^{2}v(x,t-\tau_{3})}{\partial
x^{2}}=\psi^{\prime\prime}(x+c(t-\tau_{3}))=\psi^{\prime\prime}(\xi-c\tau_{3}).$

Furthermore, Taking $r_{i}=c\tau_{i},\ i=1,2,3,4$, $D_{1},D_{2}=1$ and for
simplicity we move the delay out of the delay term by taking $t=\xi-
r_{i},\i=1,3$ the system (1.2) becomes

(4.2)
$\displaystyle\phi^{\prime\prime}(t)-c\phi^{\prime}(t+r_{1})+\alpha_{1}\phi(t+r_{1})\left(1-\phi(t+r_{1})-a\psi(t+(r_{1}-r_{2}))\right)$
$\displaystyle\psi^{\prime\prime}(t)-c\psi^{\prime}(t+r_{3})+\alpha_{2}\psi(t+r_{3})\left(1-\psi(t+r_{3})-b\phi(t+(r_{3}-r_{4}))\right).$

The construction of the quasi-upper/lower solutions will be similar to that of
the Belousov-Zhabotinskii Equations in the application section of [2] and
(PQM). In fact, we can use Rouché’s Theorem to proof the following claim.

######  Claim 4.2.

Consider, $P_{1}(\lambda)=\lambda^{2}-c\lambda+\alpha_{1}=0,$ where

$\lambda_{0}=\frac{c+\sqrt{c^{2}-4\alpha_{1}}}{2}$

is the positive root. Let $c>2\sqrt{\alpha_{1}}$ and $U_{1}$ be an open strip
$\\{z\in\mathbb{C}\ |\ \lambda_{0}-\epsilon_{1}<\Re
z<\lambda_{0}+\epsilon_{1}\\}$ so that it does not include the other root of
$P_{1}(\lambda)$ in it. Then, for sufficiently small $r_{1}$ there exists only
a single root $\lambda_{1}(r_{1})$ of the equation

(4.3) $\lambda^{2}-c\lambda e^{r_{1}\lambda}+\alpha_{1}e^{r_{1}\lambda}=0.$

in $U_{1}$ that depends continuously on $r_{1}$. Moreover,
$\lambda_{1}(r_{1})$ is real and

(4.4) $\lim_{s\to 0}\lambda_{1}(r_{1})=\lambda_{0}.$

We also have the following claim by the same reasoning.

######  Claim 4.3.

Consider, $P_{2}(\mu)=\mu^{2}-c\mu+\alpha_{2}=0,$ where

$\mu_{0}=\frac{c+\sqrt{c^{2}-4\alpha_{2}}}{2}$

is the positive root. Let $c>2\sqrt{\alpha_{2}}$ and $U_{2}$ be an open strip
$\\{z\in\mathbb{C}\ |\ \mu_{0}-\epsilon_{2}<\Re z<\mu_{0}+\epsilon_{2}\\}$ so
that it does not include the other root of $P_{2}(\mu)$ in it. Then, for
sufficiently small $r_{1}$ there exists only a single root $\mu_{1}(r_{3})$ of
the equation

(4.5) $\mu^{2}-c\mu e^{r_{3}\mu}+\alpha_{2}e^{r_{3}\mu}=0.$

in $U_{2}$ that depends continuously on $r_{3}$. Moreover, $\mu_{1}(r_{3})$ is
real and

(4.6) $\lim_{s\to 0}\mu_{1}(r_{3})=\mu_{0}.$

We also take $\alpha_{2}\geq\alpha_{1}$ such that $0<\mu_{0}<\lambda_{0}.$
Thus, $0<\mu_{1}<\lambda_{1}.$ Defining $\overline{\phi}_{1}$ and
$\overline{\phi}_{2}$ as follows:

$\overline{\phi}_{1}(t):=\left\\{\begin{array}[c]{l}\frac{1}{2}e^{\lambda_{1}t},\;\;\;\;\;\;\quad
t\leq 0,\\\ 1-\frac{1}{2}e^{-\lambda_{1}t},\quad
t>0\end{array}\right.\;\;\;\;\overline{\phi}_{2}(t):=\left\\{\begin{array}[c]{l}\frac{1}{2}e^{\mu_{1}t},\;\;\;\;\;\quad
t\leq 0,\\\ 1-\frac{1}{2}e^{-\mu_{1}t},\quad t>0,\end{array}\right.$

and we set

$\underline{\varphi_{2}}(t)=0,t\in\mathbb{R},\quad\underline{\varphi_{1}}(t)=\begin{cases}\frac{e^{\lambda_{2}t}}{4},\
t<-T,\\\ f(t),\ -T\leq t\leq T\\\ \frac{1}{2},\ t>T,\end{cases}$

where

$\displaystyle f(t)$ $\displaystyle=a(t-T)^{3}+b(t-T)^{2}+\frac{1}{2},$

and $T$ is a large number. The function $f(t)$ enjoys the following
properties:

  1. i)

This bridges smoothly the function $e^{\lambda_{2}t}/4$ and the constant
function $1/2$

  2. ii)

$f(-T)=(1/4)e^{-\lambda_{2}T}$,
$f^{\prime}(-T)=(\lambda_{2}/4)e^{-\lambda_{2}T}$, $f^{\prime}(T)=0$,
$f(T)=1/2$.

Here, $a$ and $b$ are easily found via simple calculation. In fact,

$\displaystyle a$
$\displaystyle=\frac{\lambda_{2}Te^{-\lambda_{2}T}+e^{-\lambda_{2}T}-2}{16T^{3}}$
$\displaystyle b$
$\displaystyle=\frac{\lambda_{2}Te^{-\lambda_{2}T}+6\left(\frac{e^{-\lambda_{2}T}}{4}-\frac{1}{2}\right)}{8T^{2}}.$

We also have the following claim.

######  Claim 4.4.

Define
$\overline{\phi}_{1},\overline{\phi}_{2},\underline{\phi_{1}},\underline{\phi_{2}}$
as above, then
$\left(\left(\underline{\phi_{1}},\underline{\phi_{2}}\right),\left(\overline{\phi}_{1},\overline{\phi}_{2}\right)\right)\in\Gamma_{2}.$

######  Lemma 4.5.

For sufficiently small $r_{1},r_{2}$ and
$c>\max\\{2\sqrt{\alpha_{1}},2\sqrt{\alpha_{2}}\\}$ where
$\overline{\phi}_{1},\overline{\phi}_{2},\underline{\phi_{1}},\underline{\phi_{2}}$
for all $t\in\mathbb{R}$, then
$\left(\overline{\phi}_{1},\overline{\phi}_{2}\right)^{T}$ is a quasi-upper
solution of Eq. 4.2.

###### Proof.

The proof will be completed in cases. Indeed, fix $r=\max\\{r_{1},r_{3}\\}.$

Case 1: $t\leq-r.$ Direction substitution into the first equation of Eq. 4.2
yields:

$\displaystyle\overline{\phi}_{1}^{\prime\prime}(t)-c\overline{\phi}_{1}^{\prime}(t+r_{1})+\alpha_{1}\overline{\phi}_{1}(t+r_{1})\left(1-\overline{\phi}_{1}(t+r_{1})-a\underline{\phi_{2}}(t+(r_{1}-r_{2})\right)$
$\displaystyle=\frac{\lambda_{1}^{2}}{2}e^{\lambda_{1}t}-c\frac{\lambda_{1}}{2}e^{\lambda_{1}(t+r_{1})}+\frac{\alpha_{1}}{2}e^{\lambda_{1}(t+r_{1})}\left(1-\frac{1}{2}e^{\lambda_{1}(t+r_{1}-r_{2})}\right)$
$\displaystyle=-\frac{\alpha_{1}}{2}e^{\lambda_{1}(t+r_{1})}\left(\frac{1}{2}e^{\lambda_{1}(t+r_{1}-r_{2})}\right)\leq
0.$

For the second equation we have

$\displaystyle\overline{\phi}_{2}^{\prime\prime}(t)-c\overline{\phi}_{2}^{\prime}(t+r_{3})+\alpha_{2}\overline{\phi}_{2}(t+r_{3})\left(1-\overline{\phi}_{2}(t+r_{3})-b\overline{\phi_{1}}(t+(r_{3}-r_{4}))\right)$
$\displaystyle=\frac{\mu_{1}^{2}}{2}e^{\mu_{1}t}-c\frac{\mu_{1}}{2}e^{\mu_{1}(t+r_{3})}+\frac{\alpha_{2}}{2}e^{\mu_{1}(t+r_{3})}\left(1-\frac{1}{2}e^{\mu_{1}(t+r_{3})}-\frac{b}{2}e^{\lambda_{1}(t+(r_{3}-r_{4}))}\right)$
$\displaystyle=e^{\mu_{1}t}\left(\frac{\mu_{1}^{2}}{2}-c\frac{\mu_{1}}{2}e^{\mu_{1}r_{3}}+\frac{\alpha_{2}}{2}e^{\mu_{1}r_{3}}\right)-\frac{\alpha_{2}}{2}e^{\mu_{1}(t+r_{3})}\left(\frac{1}{2}e^{\mu_{1}(t+r_{3})}+\frac{b}{2}e^{\lambda_{1}(t+(r_{3}-r_{4}))}\right)$
$\displaystyle=-\frac{\alpha_{2}}{2}e^{\mu_{1}(t+r_{3})}\left(\frac{1}{2}e^{\mu_{1}(t+r_{3})}+\frac{b}{2}e^{\lambda_{1}(t+(r_{3}-r_{4}))}\right)\leq
0.$

Case 2: $-r\leq t\leq 0.$ Direct substitution into the first equation of Eq.
4.2 yields:

$\displaystyle\overline{\phi}_{1}^{\prime\prime}(t)-c\overline{\phi}_{1}^{\prime}(t+r_{1})+\alpha_{1}\overline{\phi}_{1}(t+r_{1})\left(1-\overline{\phi}_{1}(t+r_{1})-a\underline{\phi_{2}}(t+(r_{1}-r_{2}))\right)$
$\displaystyle=\frac{\lambda_{1}^{2}}{2}e^{\lambda_{1}t}-c\frac{\lambda_{1}}{2}e^{-\lambda_{1}(t+r_{1})}+\alpha_{1}\overline{\phi}_{1}(t+r_{1})\left(1-\overline{\phi}_{1}(t+r_{1})\right)$
$\displaystyle=\frac{\lambda_{1}^{2}}{2}e^{\lambda_{1}t}-c\frac{\lambda_{1}}{2}e^{\lambda_{1}(t+r_{1})}+\frac{\alpha_{1}}{2}e^{\lambda_{1}(t+r_{1})}+c\frac{\lambda_{1}}{2}e^{\lambda_{1}(t+r_{1})}-\frac{\alpha_{1}}{2}e^{\lambda_{1}(t+r_{1})}-c\frac{\lambda_{1}}{2}e^{-\lambda_{1}(t+r_{1})}$
$\displaystyle+\alpha_{1}\overline{\phi}_{1}(t+r_{1})\left(1-\overline{\phi}_{1}(t+r_{1})\right)$
$\displaystyle=c\frac{\lambda_{1}}{2}e^{\lambda_{1}(t+r_{1})}-\frac{\alpha_{1}}{2}e^{\lambda_{1}(t+r_{1})}-c\frac{\lambda_{1}}{2}e^{-\lambda_{1}(t+r_{1})}+\alpha_{1}\left(1-\frac{1}{2}e^{-\lambda_{1}(t+r_{1})}\right)\frac{1}{2}e^{-\lambda_{1}(t+r_{1})}$
$\displaystyle=(c\lambda_{1}-\alpha_{1})\sinh(\lambda_{1}(t+r_{1}))-\frac{\alpha_{1}}{4}e^{-2\lambda_{1}(t+r_{1})}.$

Using the power expansion of $\sinh x$ we have
$\sinh(\lambda_{1}(t+r_{1}))\approx\lambda_{1}(t+r_{1})+o(r^{2}_{1}).$ Thus,
for some small $r_{1}$ we have

$(c\lambda_{1}-\alpha_{1})\sinh(\lambda_{1}(t+r_{1}))-\frac{\alpha_{1}}{4}e^{-2\lambda_{1}(t+r_{1})}=(c\lambda_{1}-\alpha_{1})\left(\lambda_{1}(t+r_{1})+o(r^{2}_{1})\right)-\frac{\alpha_{1}}{4}e^{-2\lambda_{1}(t+r_{1})}\leq
0.$

For the second equation we have

$\displaystyle\overline{\phi}_{2}^{\prime\prime}(t)-c\overline{\phi}_{2}^{\prime}(t+r_{3})+\alpha_{2}\overline{\phi}_{2}(t+r_{3})\left(1-\overline{\phi}_{2}(t+r_{3})-b\overline{\phi_{1}}(t+(r_{3}-r_{4}))\right)$
$\displaystyle=\frac{\mu_{1}^{2}}{2}e^{\mu_{1}t}-\frac{c\mu_{1}}{2}e^{-\mu_{1}(t+r_{3})}+\alpha_{2}\left(1-\frac{1}{2}e^{-\mu_{1}(t+r_{3})}\right)\left[1-\left(1-\frac{1}{2}e^{-\mu_{1}(t+r_{3})}\right)-\frac{b}{2}e^{\lambda_{1}(t+r_{3}-r_{4})}\right]$
$\displaystyle=\frac{\mu_{1}^{2}}{2}e^{\mu_{1}t}-\frac{c\mu_{1}}{2}e^{\mu_{1}(t+r_{3})}+\frac{\alpha_{2}}{2}e^{\mu_{1}(t+r_{3})}+\frac{c\mu_{1}}{2}e^{\mu_{1}(t+r_{3})}-\frac{\alpha_{2}}{2}e^{\mu_{1}(t+r_{3})}-\frac{c\mu_{1}}{2}e^{-\mu_{1}(t+r_{3})}$
$\displaystyle+\alpha_{2}\left(1-\frac{1}{2}e^{-\mu_{1}(t+r_{3})}\right)\left[1-\left(1-\frac{1}{2}e^{-\mu_{1}(t+r_{3})}\right)-\frac{b}{2}e^{\lambda_{1}(t+r_{3}-r_{4})}\right]$
$\displaystyle=c\mu_{1}\sinh(\mu_{1}r_{3})-\frac{\alpha_{2}}{2}\sinh(\mu_{1}r_{3})-\alpha_{2}\left[\frac{1}{4}e^{-2\mu_{1}(t+r_{3})}+\frac{b}{2}e^{\lambda_{1}(t+r_{3}-r_{4})}\left(1-\frac{1}{2}e^{-\mu_{1}(t+r_{3})}\right)\right].$

This is less than zero as $r_{3}\to 0$, so we have the result.

Case 3: $0<t.$ Direction substitution into the first equation of Eq. 4.2
yields:

$\displaystyle\overline{\phi}_{1}^{\prime\prime}(t)-c\overline{\phi}_{1}^{\prime}(t+r_{1})+\alpha_{1}\overline{\phi}_{1}(t+r_{1})\left(1-\overline{\phi}_{1}(t+r_{1})-a\underline{\phi_{2}}(t+(r_{1}-r_{2}))\right)$
$\displaystyle=\frac{-\lambda_{1}^{2}}{2}e^{-\lambda_{1}t}-\frac{c\lambda_{1}}{2}e^{-\lambda_{1}(t+r_{1})}+\alpha_{1}\left(1-\frac{1}{2}e^{-\lambda_{1}(t+r_{1})}\right)\left(1-\left(1-\frac{1}{2}e^{-\lambda_{1}(t+r_{1})}\right)\right)$
$\displaystyle=\frac{-\lambda_{1}^{2}}{2}e^{-\lambda_{1}t}+\frac{c\lambda_{1}}{2}e^{-\lambda_{1}t+\lambda_{1}r_{1}}-\frac{\alpha_{1}}{2}e^{-\lambda_{1}t+\lambda_{1}r_{1}}-\frac{c\lambda_{1}}{2}e^{-\lambda_{1}t+\lambda_{1}r_{1}}+\frac{\alpha_{1}}{2}e^{-\lambda_{1}t+\lambda_{1}r_{1}}$
$\displaystyle-\frac{c\lambda_{1}}{2}e^{-\lambda_{1}(t+r_{1})}+\alpha_{1}\left(1-\frac{1}{2}e^{-\lambda_{1}(t+r_{1})}\right)\left(1-\left(1-\frac{1}{2}e^{-\lambda_{1}(t+r_{1})}\right)\right)$

It is clear that

$-\alpha_{1}\left(1-\frac{e^{-\lambda_{1}t-\lambda_{1}r_{1}}}{2}\right)\left(1-\frac{e^{-\lambda_{1}(t+r_{1}-r_{2})}}{2}\right)\leq
0.$

Furthermore,

$\lim_{r_{1}\to
0}\alpha_{1}-c\lambda_{1}\cosh(\lambda_{1}r_{1})+\alpha_{1}\sinh(\lambda_{1}r_{1})=\alpha_{1}-c\lambda_{0},$

because $\lambda_{1}\to\lambda_{0}.$ A simple calculation shows

$\displaystyle\alpha_{1}-c\lambda_{0}=\alpha_{1}-c\left(\frac{c+\sqrt{c^{2}-4\alpha_{1}}}{2}\right)\leq
0,$

due to the fact that $c\geq 2\sqrt{\alpha_{1}}.$ This means that there exists
some $r^{*}>0$ dependent on $c$ such that $0<r_{1}\leq r^{*}(c)$ and

$\left(\alpha_{1}-c\lambda_{1}\cosh(\lambda_{1}r_{1})+\alpha_{1}\sinh(\lambda_{1}r_{1})\right)e^{-\lambda_{1}t}-\alpha_{1}\left(1-\frac{e^{-\lambda_{1}t-\lambda_{1}r_{1}}}{2}\right)\left(1-\frac{e^{-\lambda_{1}(t+r_{1})}}{2}\right)\leq
0.$

For the second equation we have

$\displaystyle\overline{\phi}_{2}^{\prime\prime}(t)-c\overline{\phi}_{2}^{\prime}(t+r_{3})+\alpha_{2}\overline{\phi}_{2}(t+r_{3})\left(1-\overline{\phi}_{2}(t+r_{3})-b\overline{\phi_{1}}(t+r_{3}-r_{4})\right)$
$\displaystyle=\frac{-\mu_{1}^{2}}{2}e^{-\mu_{1}t}-c\frac{\mu_{1}}{2}e^{-\mu_{1}(t+r_{3})}+\alpha_{2}\left(1-\frac{1}{2}e^{-\mu_{1}(t+r_{3})}\right)$
$\displaystyle\times\left(1-\left(1-\frac{1}{2}e^{-\mu_{1}(t+r_{3})}\right)-b\left(1-\frac{1}{2}e^{-\lambda_{1}(t+r_{3}-r_{4})}\right)\right)$
$\displaystyle=\left(\alpha_{2}-c\mu_{1}\cosh(\mu_{1}r_{3})+\alpha_{2}\sinh(\mu_{1}r_{3})\right)e^{-\mu_{1}t}$
$\displaystyle-\alpha_{2}\left(\left(1-\frac{1}{2}e^{-\mu_{1}(t+r_{3})}\right)+b\left(1-\frac{1}{2}e^{-\lambda_{1}(t+r_{3}-r_{4})}\right)\right)\leq
0$

by the same reasoning as the first equation in this case. This proves the
Lemma. ∎

######  Lemma 4.6.

For sufficiently small $r_{1},r_{2}$ and
$c>\max\\{2\sqrt{\alpha_{1}},2\sqrt{\alpha_{2}}\\}$ where
$\overline{\phi}_{1},\overline{\phi}_{2},\underline{\phi_{1}},\underline{\phi_{2}}$
for all $t\in\mathbb{R}$, then
$\left(\underline{\phi_{1}},\underline{\phi_{2}}\right)^{T}$ is a quasi-lower
solution of Eq. 4.2.

###### Proof.

We will show this Lemma in cases like above. In fact, we only need to show
that the inequality is satisfied for the first equation in the model due to
the fact for all $t\in\mathbb{R}$ the second equation always is zero. This is
shown by

$\displaystyle\underline{\phi}_{2}^{\prime\prime}(t)-c\underline{\phi}_{2}^{\prime}(t+r_{3})+\alpha_{2}\underline{\phi}_{2}(t+r_{3})\left(1-\underline{\phi}_{2}(t+r_{3})-b\underline{\phi_{1}}(t+r_{3}-r_{4})\right)=0.$

Case 1: $t\leq-T-r.$ Direction substitution into the first equation of Eq. 4.2
yields:

$\displaystyle\underline{\phi}_{1}^{\prime\prime}(t)-c\underline{\phi}_{1}^{\prime}(t+r_{1})+\alpha_{1}\underline{\phi}_{1}(t+r_{1})\left(1-\underline{\phi}_{1}(t+r_{1})-a\overline{\phi_{2}}(t+r_{1}-r_{2})\right)$
$\displaystyle=\left[\underline{\phi}_{1}^{\prime\prime}(t)-c\underline{\phi}_{1}^{\prime}(t+r_{1})+\frac{\alpha_{1}}{2}\underline{\phi}_{1}(t+r_{1})\right]+\alpha_{1}\underline{\phi}_{1}(t+r_{1})\left(\frac{1}{2}-\underline{\phi}_{1}(t+r_{1})-a\overline{\phi_{2}}(t+r_{1}-r_{2})\right)$
$\displaystyle=e^{\lambda_{2}t}\left[\lambda_{2}^{2}-c\lambda_{2}e^{\lambda_{2}r_{1}}+\frac{\alpha_{1}}{2}e^{\lambda_{2}r_{1}}\right]+\alpha_{1}\underline{\phi}_{1}(t+r_{1})\left(\frac{1}{2}-\underline{\phi}_{1}(t+r_{1})-a\overline{\phi_{2}}(t+r_{1}-r_{2})\right)$
$\displaystyle=\alpha_{1}\underline{\phi}_{1}(t+r_{1})\left(\frac{1}{2}-\underline{\phi}_{1}(t+r_{1})-a\overline{\phi_{2}}(t+r_{1}-r_{2})\right).$

However, we can take $T$ to be sufficiently large enough such that

$\alpha_{1}\underline{\phi}_{1}(t+r_{1})\left(\frac{1}{2}-\underline{\phi}_{1}(t+r_{1})-a\overline{\phi_{2}}(t+r_{1}-r_{2})\right)\geq
0.$

Case 2: $-r-T<t\leq T.$ Direction substitution into the first equation of Eq.
4.2 yields:

$\displaystyle\underline{\phi}_{1}^{\prime\prime}(t)-c\underline{\phi}_{1}^{\prime}(t+r_{1})+\alpha_{1}\underline{\phi}_{1}(t+r_{1})\left(1-\underline{\phi}_{1}(t+r_{1})-a\overline{\phi_{2}}(t+r_{1}-r_{2})\right)$
$\displaystyle=\frac{1}{4}\left(f^{\prime\prime}(t)-cf^{\prime}(t+r_{1})+\alpha_{1}f(t+r_{1})\left(1-\frac{f(t+r_{1})}{4}-a\overline{\phi_{2}}(t+r_{1}-r_{2})\right)\right).$

Again, we can take $T$ large enough such that

$\alpha_{1}f(t+r_{1})\left(1-\frac{f(t+r_{1})}{4}-a\overline{\phi_{2}}(t+r_{1}-r_{2})\right)\geq
0.$

In [2] it was noted that on this interval

$\displaystyle=\sup_{-T-r_{1}\leq t\leq
T}|\underline{\phi}_{1}^{\prime\prime}(t)|+c|\underline{\phi}_{1}^{\prime}(t+r_{1})|$
$\displaystyle=\sup_{-T\leq t\leq T}|f^{\prime}(t)|+c|f^{\prime\prime}(t)|,$

so we have that

$\frac{1}{4}\left(f^{\prime\prime}(t)-cf^{\prime}(t+r_{1})+\alpha_{1}f(t+r_{1})\left(1-\frac{f(t+r_{1})}{4}-a\overline{\phi_{2}}(t+r_{1}-r_{2})\right)\right)\geq
0.$

Case 3: $t\geq T.$ Direction substitution into the first equation of Eq. 4.2
yields:

$\displaystyle\underline{\phi}_{1}^{\prime\prime}(t)-c\underline{\phi}_{1}^{\prime}(t+r_{1})+\alpha_{1}\underline{\phi}_{1}(t+r_{1})\left(1-\underline{\phi}_{1}(t+r_{1})-a\overline{\phi_{2}}(t+r_{1}-r_{2})\right)$
$\displaystyle=\frac{\alpha_{1}}{2}\left(1-\frac{1}{2}-a\overline{\phi_{2}}(t+r_{1}-r_{2})\right)\geq
0.$

This concludes the proof. ∎

######  Corollary 4.7.

Assume that $c>\max\\{2\sqrt{\alpha_{1}},2\sqrt{\alpha_{2}}\\}$ is given.
Then, the system (4.2) has a traveling wave solution
$(u(x,t),v(x,t))^{T}=\Phi=\left({\phi(x+ct)},{\psi(x+ct)}\right)$ for
sufficiently small delays $\tau_{1},\tau_{2}$.

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