In this section, we shall present the existence of traveling wave solutions for any \(c\ge c^{*}\). When the wave speed is large, there exists a positive traveling wave solution.
Theorem 3.1
If
\(c>c^{*}\), then (2.1) has a positive solution
\((\phi_{1}( \xi),\phi_{2}(\xi))\)
such that
$$ 0< \phi_{1}(\xi)< 1,\qquad 0< \phi_{2}(\xi)< 1+c_{2},\quad \xi \in \mathbb{R} $$
(3.1)
and
$$\lim_{\xi \rightarrow -\infty }\bigl(\phi_{1}(\xi), \phi_{2}(\xi)\bigr)=(0,0),\qquad \lim_{\xi \rightarrow -\infty }\bigl( \phi_{1}(\xi)e^{-\lambda_{1}^{c} \xi }, \phi_{2}(\xi)e^{-\lambda_{2}^{c}\xi }\bigr)=(1,1). $$
Proof
We shall prove it by Lemma 2.4, and first construct generalized upper and lower solutions. For convenience, we denote \(\lambda_{i} ^{c}\) by \(\lambda_{i}\) for simplicity, and we prove the result for any fixed \(c>c^{*}\).
Define continuous functions
$$\underline{\phi}_{1}(\xi)=\max \bigl\{ e^{\lambda_{1}\xi }-qe^{\eta \lambda_{1}\xi },0 \bigr\} ,\qquad \underline{\phi}_{2}(\xi)=\max \bigl\{ e^{\lambda _{2}\xi }-qe^{\eta \lambda_{2}\xi },0 \bigr\} $$
and
$$\overline{\phi}_{1}(\xi)=\min \bigl\{ e^{\lambda_{1}\xi },1\bigr\} ,\qquad \overline{ \phi }_{2}(\xi)=\min \bigl\{ e^{\lambda_{2}\xi }+pe^{\eta \lambda_{2} \xi },1+c_{2} \bigr\} , $$
where
$$\eta \in \biggl( 1,\min \biggl\{ \frac{\lambda_{3}}{\lambda_{1}},\frac{ \lambda_{4}}{\lambda_{2}}, \frac{\lambda_{1}+\lambda_{2}}{\lambda_{1}},\frac{ \lambda_{1}+\lambda_{2}}{\lambda_{2}} \biggr\} \biggr) $$
and \(p>1\), \(q>1\) are constants, of which the definitions will be clarified later. We now show these functions satisfy (2.5)–(2.8) if they are differentiable.
If \(\overline{\phi}_{1}(\xi)=1< e^{\lambda_{1}\xi }\), then \(H_{1}(\overline{ \phi }_{1},\underline{\phi}_{1},\underline{\phi}_{2})(\xi)\leq 0 \) such that (2.5) is clear. Otherwise, \(\overline{\phi}_{1}( \xi)=e^{\lambda_{1}\xi }<1\) implies that
$$\begin{aligned}& d_{1}[J_{1}\ast \overline{\phi}_{1}](\xi)-c\overline{\phi}_{1} ^{\prime }(\xi)+r_{1} \overline{\phi}_{1}(\xi)H_{1}(\overline{ \phi }_{1},\underline{\phi}_{1},\underline{\phi}_{2}) (\xi) \\& \quad \leq d_{1}[J_{1}\ast \overline{\phi}_{1}](\xi)-c\overline{\phi} _{1}^{\prime }(\xi)+r_{1}\overline{\phi}_{1}(\xi) \\& \quad =d_{1} \biggl[ \int_{\mathbb{R}}J_{1}(y)\overline{\phi}_{1}( \xi -y)\,dy-e ^{\lambda_{1}\xi } \biggr] -c\lambda_{1}e^{\lambda_{1}\xi }+r_{1}e^{ \lambda_{1}\xi } \\& \quad \leq d_{1} \biggl[ \int_{\mathbb{R}}J_{1}(y)e^{\lambda_{1}(\xi -y)}\,dy-e ^{\lambda_{1}\xi } \biggr] -c\lambda_{1}e^{\lambda_{1}\xi }+r_{1}e^{ \lambda_{1}\xi } \\& \quad =e^{\lambda_{1}\xi } \biggl\{ d_{1} \biggl[ \int_{\mathbb{R}}J_{1}(y)e ^{\lambda_{1}y}\,dy-1 \biggr] -c \lambda_{1}+r_{1} \biggr\} \\& \quad =0, \end{aligned}$$
which implies what we wanted.
If \(\overline{\phi}_{2}(\xi)=1+c_{2}< e^{\lambda_{2}\xi }+pe^{\eta \lambda_{2}\xi }\), then \(H_{2}(\overline{\phi}_{1},\underline{\phi} _{2},\overline{\phi}_{2})(\xi)\leq 0\) such that (2.7) is clear. Otherwise, \(\overline{\phi}_{2}(\xi)=e^{\lambda_{2}\xi }+pe^{\eta \lambda_{2}\xi }<1+c_{2}\) such that
$$\begin{aligned}& r_{2}\overline{\phi}_{2}(\xi)H_{2}(\overline{ \phi }_{1},\underline{ \phi }_{2},\overline{\phi}_{2}) (\xi) \\& \quad =r_{2}\overline{\phi}_{2}(\xi) \biggl[ 1-\overline{\phi}_{2}( \xi)-b_{2} \int_{-\tau }^{0}\underline{\phi}_{2}(\xi +cs)\,d\eta_{22}(s)+c _{2} \int_{-\tau }^{0}\overline{\phi}_{1}(\xi +cs)\,d\eta_{21}(s) \biggr] \\& \quad \leq r_{2}\overline{\phi}_{2}(\xi) \biggl[ 1+c_{2} \int_{-\tau }^{0}\overline{ \phi }_{1}(\xi +cs)\,d\eta_{21}(s) \biggr] \\& \quad \leq r_{2}\overline{\phi}_{2}(\xi) \bigl[ 1+c_{2}e^{\lambda_{1} \xi } \bigr] \\& \quad =r_{2}\bigl[e^{\lambda_{2}\xi }+pe^{\eta \lambda_{2}\xi }\bigr] \bigl[ 1+c_{2}e ^{\lambda_{1}\xi } \bigr] \\& \quad =r_{2}\bigl[e^{\lambda_{2}\xi }+pe^{\eta \lambda_{2}\xi }\bigr]+r_{2}c_{2}e ^{\lambda_{1}\xi }\bigl[e^{\lambda_{2}\xi }+pe^{\eta \lambda_{2}\xi }\bigr] \end{aligned}$$
and
$$\begin{aligned}& d_{2}[J_{2}\ast \overline{\phi}_{2}](\xi)-c \overline{\phi}_{2} ^{\prime }(\xi)+r_{2}\overline{\phi}_{2}(\xi)H_{2}(\overline{ \phi }_{1}, \underline{\phi}_{2},\overline{\phi}_{2}) (\xi) \\& \quad =d_{2} \biggl[ \int_{\mathbb{R}}J_{2}(y)\overline{\phi}_{2}(\xi -y)\,dy- \bigl( e^{\lambda_{2}\xi }+pe^{\eta \lambda_{2}\xi } \bigr) \biggr] \\& \qquad {}-c \bigl( \lambda_{2}e^{\lambda_{2}\xi }+p\eta \lambda_{2}e^{\eta \lambda_{2}\xi } \bigr) +r_{2}\overline{\phi}_{2}(\xi)H_{2}( \overline{ \phi }_{1},\underline{\phi}_{2},\overline{\phi}_{2}) (\xi) \\& \quad \leq d_{2} \biggl[ \int_{\mathbb{R}}J_{2}(y)\bigl[e^{\lambda_{2}(\xi -y)}+pe ^{\eta \lambda_{2}(\xi -y)}\bigr]\,dy- \bigl( e^{\lambda_{2}\xi }+pe^{\eta \lambda_{2}\xi } \bigr) \biggr] \\& \qquad {}-c \bigl( \lambda_{2}e^{\lambda_{2}\xi }+p\eta \lambda_{2}e^{\eta \lambda_{2}\xi } \bigr) +r_{2}\overline{\phi}_{2}(\xi)H_{2}( \overline{ \phi }_{1},\underline{\phi}_{2},\overline{\phi}_{2}) (\xi) \\& \quad \leq d_{2} \biggl[ \int_{\mathbb{R}}J_{2}(y)\bigl[e^{\lambda_{2}(\xi -y)}+pe ^{\eta \lambda_{2}(\xi -y)}\bigr]\,dy- \bigl( e^{\lambda_{2}\xi }+pe^{\eta \lambda_{2}\xi } \bigr) \biggr] \\& \qquad {}-c \bigl( \lambda_{2}e^{\lambda_{2}\xi }+p\eta \lambda_{2}e^{\eta \lambda_{2}\xi } \bigr) +r_{2}\bigl[e^{\lambda_{2}\xi }+pe^{\eta \lambda_{2} \xi } \bigr]+r_{2}c_{2}e^{\lambda_{1}\xi }\bigl[e^{\lambda_{2}\xi }+pe^{\eta \lambda_{2}\xi } \bigr] \\& \quad =p \biggl\{ d_{2} \biggl[ \int_{\mathbb{R}}J_{2}(y)e^{\eta \lambda_{2}( \xi -y)}\,dy-e^{\eta \lambda_{2}\xi } \biggr] -c\eta \lambda_{2}e^{ \eta \lambda_{2}\xi }+r_{2}e^{\eta \lambda_{2}\xi } \biggr\} \\& \qquad {}+r_{2}c_{2}e^{\lambda_{1}\xi }\bigl[e^{\lambda_{2}\xi }+pe^{\eta \lambda_{2} \xi } \bigr] \\& \quad =p\Theta_{2}(\eta \lambda_{2},c)e^{\eta \lambda_{2}\xi }+r_{2}c_{2}e ^{\lambda_{1}\xi }\bigl[e^{\lambda_{2}\xi }+pe^{\eta \lambda_{2}\xi }\bigr] \\& \quad =e^{\eta \lambda_{2}\xi } \bigl[ p\Theta_{2}(\eta \lambda_{2},c)/2+r _{2}c_{2}e^{(\lambda_{1}+\lambda_{2}-\eta \lambda_{2})\xi } \bigr] +pe ^{\eta \lambda_{2}\xi } \bigl[ \Theta_{2}(\eta \lambda_{2},c)/2+r_{2}c _{2}e^{\lambda_{1}\xi } \bigr] . \end{aligned}$$
Note that
$$\eta \lambda_{2}\xi < \ln \frac{1+c_{2}}{p}, $$
then there exists \(p_{1}>1+c_{2}\) such that \(p= p_{1}\) leads to
$$p\Theta_{2}(\eta \lambda_{2},c)/2+r_{2}c_{2}e^{(\lambda_{1}+\lambda_{2}- \eta \lambda_{2})\xi }< 0,\qquad \Theta_{2}(\eta \lambda_{2},c)/2+r_{2}c_{2}e ^{\lambda_{1}\xi }< 0 $$
since \(\lambda_{1}+\lambda_{2}-\eta \lambda_{2}>0\), \(\xi <0\) and \(\Theta_{2}(\eta \lambda_{2},c)<0\) is a constant.
When \(\underline{\phi}_{1}(\xi)=0>e^{\lambda_{1}\xi }-qe^{\eta \lambda_{1}\xi }\), then \(H_{1}(\underline{\phi}_{1},\overline{\phi} _{1},\overline{\phi}_{2})(\xi)=0\) such that (2.6) is clear. Otherwise, \(\underline{\phi}_{1}(\xi)=e^{\lambda_{1}\xi }-qe^{ \eta \lambda_{1}\xi }>0\). Firstly, let \(q>q_{1}>1\) such that \(e^{\lambda_{1}\xi }-q_{1}e^{\eta \lambda_{1}\xi }>0\) implies \(\xi <0\) and
$$\overline{\phi}_{2}(\xi)< 2e^{\lambda_{2}\xi }, $$
which is admissible once p is fixed. Therefore, the monotonicity and \(q>q_{1}\) indicate
$$\begin{aligned}& r_{1}\underline{\phi}_{1}(\xi)H_{1}( \phi_{1},\psi_{1},\phi_{2}) ( \xi) \\& \quad =r_{1}\underline{\phi}_{1}(\xi) \biggl[ 1- \underline{\phi}_{1}( \xi)-b_{1} \int_{-\tau }^{0}\overline{\phi}_{1}(\xi +cs)\,d\eta_{11}(s)-c _{1} \int_{-\tau }^{0}\overline{\phi}_{2}(\xi +cs)\,d\eta_{12}(s) \biggr] \\& \quad \geq r_{1}\underline{\phi}_{1}(\xi)-r_{1}\underline{\phi}_{1} ^{2}(\xi)-r_{1}b_{1}\underline{\phi}_{1}(\xi) \overline{\phi}_{1}( \xi)-2r_{1}c_{1}e^{\lambda_{2}\xi } \underline{\phi}_{1}(\xi) \\& \quad \geq r_{1}\underline{\phi}_{1}(\xi)-r_{1}(1+b_{1})e^{2\lambda_{1} \xi }-2r_{1}c_{1}e^{(\lambda_{1}+\lambda_{2})\xi } \\& \quad =r_{1}e^{\lambda_{1}\xi }-r_{1}q_{1}e^{\eta \lambda_{1}\xi }-r_{1}(1+b _{1})e^{2\lambda_{1}\xi }-2r_{1}c_{1}e^{(\lambda_{1}+\lambda_{2}) \xi }. \end{aligned}$$
By what we have done, (2.6) is true once
$$\begin{aligned}& d_{1}[J_{1}\ast \underline{\phi}_{1}](\xi)-c\underline{\phi}_{1} ^{\prime }(\xi)+r_{1}e^{\lambda_{1}\xi }-r_{1}q_{1}e^{\eta \lambda_{1} \xi }-r_{1}(1+b_{1})e^{2\lambda_{1}\xi }-2r_{1}c_{1}e^{(\lambda_{1}+ \lambda_{2})\xi } \\& \quad \geq d_{1} \biggl[ \int_{\mathbb{R}}J_{1}(y) \bigl( e^{\lambda_{1}( \xi -y)}-qe^{\eta \lambda_{1}(\xi -y)} \bigr)\,dy- \bigl( e^{\lambda _{1}\xi }-qe^{\eta \lambda_{1}\xi } \bigr) \biggr] \\& \qquad {}- \bigl( c\lambda_{1}e^{\lambda_{1}\xi }-cq\eta \lambda_{1}e^{\eta \lambda_{1}\xi } \bigr) +r_{1}e^{\lambda_{1}\xi }-r_{1}qe^{\eta \lambda_{1}\xi } \\& \qquad {}-r_{1}(1+b_{1})e^{2\lambda_{1}\xi }-2r_{1}c_{1}e^{(\lambda_{1}+ \lambda_{2})\xi } \\& \quad =-qe^{\eta \lambda_{1}\xi } \biggl\{ d_{1} \biggl[ \int_{\mathbb{R}}J _{1}(y)e^{\eta \lambda_{1}y}\,dy-1 \biggr] -c \eta \lambda_{1}+r_{1} \biggr\} \\& \qquad {}-r_{1}(1+b_{1})e^{2\lambda_{1}\xi }-2r_{1}c_{1}e^{(\lambda_{1}+ \lambda_{2})\xi } \\& \quad =-q\Theta_{1}(\eta \lambda_{1},c)e^{\eta \lambda_{1}\xi }-r_{1}(1+b _{1})e^{2\lambda_{1}\xi }-2r_{1}c_{1}e^{(\lambda_{1}+\lambda_{2}) \xi } \\& \quad \geq 0. \end{aligned}$$
(3.2)
Let
$$q>-\frac{r_{1}(1+b_{1})+2r_{1}c_{1}}{\Theta_{1}(\eta \lambda_{1},c)}+q _{1}:=q_{2}, $$
then (3.2) holds since \(\xi <0\) and
$$e^{\eta \lambda_{1}\xi }>e^{2\lambda_{1}\xi }>0,\qquad e^{\eta \lambda_{1} \xi }>e^{(\lambda_{1}+\lambda_{2})\xi }>0. $$
The verification of (2.6) is finished.
We now consider (2.8), which is clear if \(\underline{\phi}_{2}( \xi)=0>e^{\lambda_{2}\xi }-qe^{\eta \lambda_{2}\xi }\). If \(\underline{ \phi }_{2}(\xi)=e^{\lambda_{2}\xi }-qe^{\eta \lambda_{2}\xi }>0\), we first select \(q_{3}\geq q_{2}\) implies
$$\overline{\phi}_{2}(\xi)< 2e^{\lambda_{2}\xi } $$
for any \(q\geq q_{3}\), which is admissible for fixed \(p=p_{1}\). Then
$$\begin{aligned}& r_{2}\underline{\phi}_{2}(\xi)H_{2}( \underline{\phi}_{1},\overline{ \phi }_{2},\underline{\phi}_{2}) (\xi) \\& \quad =r_{2}\underline{\phi}_{2}(\xi) \biggl[ 1-\underline{ \phi }_{2}( \xi)-b_{2} \int_{-\tau }^{0}\overline{\phi}_{2}(\xi +cs)\,d\eta_{22}(s)+c _{2} \int_{-\tau }^{0}\underline{\phi}_{1}(\xi +cs)\,d\eta_{21}(s) \biggr] \\& \quad \geq r_{2}\underline{\phi}_{2}(\xi) \biggl[ 1- \underline{\phi}_{2}( \xi)-b_{2} \int_{-\tau }^{0}\overline{\phi}_{2}(\xi +cs)\,d\eta_{22}(s) \biggr] \\& \quad \geq r_{2}\underline{\phi}_{2}(\xi) \bigl[ 1-e^{\lambda_{2}\xi }-2b _{2}e^{\lambda_{2}\xi } \bigr] \\& \quad =r_{2}\underline{\phi}_{2}(\xi)-r_{2} ( 1+2b_{2} ) \underline{ \phi }_{2}(\xi)e^{\lambda_{2}\xi } \\& \quad \geq r_{2} \bigl( e^{\lambda_{2}\xi }-qe^{\eta \lambda_{2}\xi } \bigr) -r_{2} ( 1+2b_{2} ) e^{2\lambda_{2}\xi }. \end{aligned}$$
Therefore, if
$$q>q_{3}-\frac{r_{2} ( 1+2b_{2} ) }{\Theta_{2}(\eta \lambda _{2},c)}:=q_{4}, $$
then (2.8) holds since
$$\begin{aligned}& d_{2}[J_{2}\ast \underline{\phi}_{2}](\xi)-c \underline{\phi}_{2} ^{\prime }(\xi)+r_{2}\underline{ \phi }_{2}(\xi)H_{2}(\underline{ \phi }_{1}, \overline{\phi}_{2},\underline{\phi}_{2}) (\xi) \\& \quad \geq d_{2} \biggl[ \int_{\mathbb{R}}J_{2}(y)\bigl[e^{\lambda_{2}(\xi -y)}-qe ^{\eta \lambda_{2}(\xi -y)}\bigr]\,dy- \bigl( e^{\lambda_{2}\xi }-qe^{\eta \lambda_{2}\xi } \bigr) \biggr] \\& \qquad {}-c \bigl( \lambda_{2}e^{\lambda_{2}\xi }-q\eta \lambda_{2}e^{\eta \lambda_{2}\xi } \bigr) +r_{2} \bigl( e^{\lambda_{2}\xi }-qe^{\eta \lambda_{2}\xi } \bigr) -r_{2}(1+2b_{2})e^{2\lambda_{2}\xi } \\& \quad =-qe^{\eta \lambda_{2}\xi } \biggl\{ d_{2} \biggl[ \int_{\mathbb{R}}J _{2}(y)e^{\eta \lambda_{2}y}\,dy-1 \biggr] -c \eta \lambda_{2}+r_{2} \biggr\} -r_{2}(1+2b_{2})e^{2\lambda_{2}\xi } \\& \quad =-q\Theta_{2}(\eta \lambda_{2},c) e^{\eta \lambda_{2}\xi }-r_{2}(1+2b _{2})e^{2\lambda_{2}\xi } \\& \quad \geq 0,\quad \xi < 0. \end{aligned}$$
Summarizing what we have done, it suffices to verify that (3.1) is true. We now show \(\phi_{1}(\xi)>0\), \(\xi \in \mathbb{R}\). If \(\phi_{1}(\xi_{0})=0\), then it arrives the minimal and so \(\phi_{1}'( \xi_{0})=0\), which further implies that
$$\int_{\mathbb{R}}J_{1}(y){\phi}_{1}( \xi_{0} -y)\,dy=0. $$
Therefore, \(\phi_{1}(\xi)=0\) on an interval. Repeating the process, we see that \(\phi_{1}(\xi)=0\), \(\xi \in \mathbb{R}\). A contradiction occurs since \(\underline{\phi}_{1}(\xi)>0\) if −ξ is large. Similarly, we can verify (3.1). The proof is complete. □
Theorem 3.2
Assume that
\(c^{\ast }=c_{1}^{\ast }>c_{2}^{\ast }\). Further suppose that
\(k_{1}(y)\)
admits compact support. Then (2.1) with
\(c=c^{*}\)
has a positive solution
\((\phi_{1}(\xi),\phi_{2}(\xi))\)
such that
$$0< \phi_{1}(\xi)< 1,0< \phi_{2}(\xi)< 1+c_{2},\xi \in \mathbb{R}, \quad \lim_{\xi \rightarrow -\infty }\bigl(\phi_{1}(\xi), \phi_{2}(\xi)\bigr)=(0,0) $$
and
$$\phi_{1}(\xi)\sim \mathcal{O}\bigl(-\xi e^{\lambda_{1}^{\ast }\xi } \bigr),\qquad \phi_{2}(\xi)\sim \mathcal{O} \bigl(e^{\lambda_{2}\xi }\bigr),\quad \xi \rightarrow -\infty. $$
Proof
By Lemma 2.3, \(\Theta_{1}(\lambda,c^{\ast })\) arrives at its minimum when \(\lambda =\lambda_{1}^{\ast }\), and so
$$d_{1} \int_{\mathbb{R}}J_{1}(y)ye^{\lambda_{1}^{\ast }y}\,dy=c^{\ast }. $$
Let \(S>0\) be a constant such that \(k_{1}(y)=0\), \(\vert y\vert >S\). Moreover, let \(\eta >1\) such that
$$\lambda_{1}^{\ast }/2+\lambda_{2}-\eta \lambda_{2}>0, \qquad \Theta_{2}\bigl( \eta \lambda_{2},c^{\ast } \bigr)< 0. $$
Consider the continuous function \(-L\xi e^{\lambda_{1}^{\ast }\xi }\), \(\xi <0\), where \(L>0\) is a constant. Clearly, if \(L>1\) is large, then
$$ \max_{\xi < 0} \bigl\{ -L\xi e^{\lambda_{1}^{\ast }\xi } \bigr\} >1,\qquad \xi_{2}- \xi_{1}>2S+c^{\ast }\tau, $$
(3.3)
where \(\xi_{2}\), \(\xi_{1}\) with \(\xi_{2}-\xi_{1}>0\) are two roots of \(-L\xi e^{\lambda_{1}^{\ast }\xi }=1\). Moreover, let \(q>L\) be a constant clarified later, then there exists \(\xi_{3}=-q^{2}/L^{2}<-1\) such that
$$( -L\xi -q\sqrt{-\xi } ) e^{\lambda_{1}^{\ast }\xi }>0, \quad \xi < \xi_{3}. $$
By the above constants, define the continuous functions
$$\begin{aligned}& \underline{\phi}_{1}(\xi)=\textstyle\begin{cases} ( -L\xi -q\sqrt{-\xi } ) e^{\lambda_{1}^{\ast }\xi },& \xi < \xi_{3}, \\ 0,& \xi \geq \xi _{3}, \end{cases}\displaystyle \\& \overline{\phi}_{1}(\xi)=\textstyle\begin{cases} -L\xi e^{\lambda_{1}^{\ast }\xi },&\xi < \xi_{1}, \\ 1,& \xi \geq \xi_{1}, \end{cases}\displaystyle \end{aligned}$$
and
$$\underline{\phi}_{2}(\xi)=\max \bigl\{ e^{\lambda_{2}\xi }-qe^{\eta \lambda_{2}\xi },0 \bigr\} ,\qquad \overline{\phi}_{2}(\xi)=\min \bigl\{ e^{\lambda_{2} \xi }+pe^{\eta \lambda_{2}\xi },1+c_{2} \bigr\} , $$
where \(p>1\), \(q>1\) are constants, of which the definition will be further illustrated later. We now show these functions satisfy (2.5)–(2.8) if they are differentiable.
If \(\overline{\phi}_{1}(\xi)=1\), then \(H_{1}(\overline{\phi}_{1},\underline{ \phi }_{1},\underline{\phi}_{2})(\xi)\leq 0\) such that (2.5) is clear. Otherwise, \(\overline{\phi}_{1}(\xi)=-L\xi e^{\lambda_{1} ^{\ast }\xi }<1\) implies that
$$r_{1}\overline{\phi}_{1}(\xi)H_{1}( \overline{\phi}_{1},\underline{ \phi }_{1},\underline{ \phi }_{2}) (\xi)\leq r_{1}\overline{\phi}_{1}( \xi)=-r_{1}L\xi e^{\lambda_{1}^{\ast }\xi }, \quad \xi < \xi_{1}, $$
and (3.3) indicates that
$$\begin{aligned}& d_{1}[J_{1}\ast \overline{\phi}_{1}](\xi)-c^{\ast }\overline{ \phi }_{1}^{\prime }(\xi)+r_{1}\overline{\phi}_{1}(\xi)H_{1}( \overline{ \phi }_{1},\underline{\phi}_{1},\underline{ \phi }_{2}) (\xi) \\& \quad \leq d_{1}[J_{1}\ast \overline{\phi}_{1}](\xi)-c^{\ast }\overline{ \phi }_{1}^{\prime }( \xi)+r_{1}\overline{\phi}_{1}(\xi) \\& \quad \leq -d_{1}L \biggl[ \int_{\mathbb{R}}J_{1}(y) (\xi -y)e^{\lambda_{1} ^{\ast }(\xi -y)}\,dy-\xi e^{\lambda_{1}^{\ast }\xi } \biggr] \\& \qquad {}+c^{\ast }Le^{\lambda_{1}^{\ast }\xi }+c^{\ast }\lambda_{1}^{\ast }L \xi e^{\lambda_{1}^{\ast }\xi }-r_{1}L\xi e^{\lambda_{1}^{\ast } \xi } \\& \quad =-d_{1}L \biggl[ \xi \int_{\mathbb{R}}J_{1}(y)e^{\lambda_{1}^{\ast }( \xi -y)}\,dy-\xi e^{\lambda_{1}^{\ast }\xi }- \int_{\mathbb{R}}J_{1}(y)ye ^{\lambda_{1}^{\ast }(\xi -y)}\,dy \biggr] \\& \qquad {}+c^{\ast }Le^{\lambda_{1}^{\ast }\xi }+c^{\ast }\lambda_{1}^{\ast }L \xi e^{\lambda_{1}^{\ast }\xi }-r_{1}L\xi e^{\lambda_{1}^{\ast } \xi } \\& \quad =-L\xi e^{\lambda_{1}^{\ast }\xi } \biggl\{ d_{1} \biggl[ \int_{\mathbb{R}}J_{1}(y)e^{\lambda_{1}^{\ast }y}\,dy-1 \biggr] -c^{ \ast }\lambda_{1}^{\ast }+r_{1} \biggr\} \\& \qquad {}+d_{1}Le^{\lambda_{1}^{\ast }\xi } \biggl[ \int_{\mathbb{R}}J_{1}(y)ye ^{-\lambda_{1}^{\ast }y}\,dy \biggr] +c^{\ast }Le^{\lambda_{1}^{\ast } \xi } \\& \quad =0, \end{aligned}$$
which implies what we wanted.
If \(\overline{\phi}_{2}(\xi)=1+c_{2}< e^{\lambda_{2}\xi }+pe^{\eta \lambda_{2}\xi }\), then \(H_{2}(\overline{\phi}_{1},\underline{\phi} _{2},\overline{\phi}_{2})(\xi)\leq 0\) such that (2.7) is clear. Otherwise, let \(p_{2}>0\) such that \(\overline{\phi}_{2}(\xi)=e^{ \lambda_{2}\xi }+pe^{\eta \lambda_{2}\xi }<1+c_{2}\) with \(p\geq p_{2}\) implies that
$$\overline{\phi}_{1}(\xi)< e^{\lambda_{1}^{\ast }\xi /2}, $$
which is evident by simple limit analysis. Thus, the monotonicity implies
$$\begin{aligned}& r_{2}\overline{\phi}_{2}(\xi)H_{2}(\overline{ \phi }_{1},\underline{ \phi }_{2},\overline{\phi}_{2}) (\xi) \\& \quad =r_{2}\overline{\phi}_{2}(\xi) \biggl[ 1-\overline{\phi}_{2}( \xi)-b_{2} \int_{-\tau }^{0}\underline{\phi}_{2}\bigl( \xi +c^{\ast }s\bigr)\,d\eta_{22}(s)+c_{2} \int_{-\tau }^{0}\overline{\phi}_{1}\bigl( \xi +c^{ \ast }s\bigr)\,d\eta_{21}(s) \biggr] \\& \quad \leq r_{2}\overline{\phi}_{2}(\xi) \biggl[ 1+c_{2} \int_{-\tau }^{0}\overline{ \phi }_{1}\bigl( \xi +c^{\ast }s\bigr)\,d\eta_{21}(s) \biggr] \\& \quad \leq r_{2}\overline{\phi}_{2}(\xi) \bigl[ 1+c_{2}e^{\lambda_{1} ^{\ast }\xi /2} \bigr] \\& \quad =r_{2}\bigl[e^{\lambda_{2}\xi }+pe^{\eta \lambda_{2}\xi }\bigr] \bigl[ 1+c_{2}e ^{\lambda_{1}^{\ast }\xi /2} \bigr] \\& \quad =r_{2}\bigl[e^{\lambda_{2}\xi }+pe^{\eta \lambda_{2}\xi }\bigr]+r_{2}c_{2}e ^{\lambda_{1}^{\ast }\xi /2}\bigl[e^{\lambda_{2}\xi }+pe^{\eta \lambda_{2} \xi }\bigr] \end{aligned}$$
and
$$\begin{aligned}& d_{2}[J_{2}\ast \overline{\phi}_{2}](\xi)-c^{\ast }\overline{ \phi }_{2}^{\prime }(\xi)+r_{2}\overline{\phi}_{2}(\xi)H_{2}(\overline{ \phi }_{1},\underline{\phi}_{2},\overline{\phi}_{2}) (\xi) \\& \quad =d_{2} \biggl[ \int_{\mathbb{R}}J_{2}(y)\overline{\phi}_{2}(\xi -y)\,dy- \bigl( e^{\lambda_{2}\xi }+pe^{\eta \lambda_{2}\xi } \bigr) \biggr] \\& \qquad {}-c^{\ast } \bigl( \lambda_{2}e^{\lambda_{2}\xi }+p\eta \lambda_{2}e ^{\eta \lambda_{2}\xi } \bigr) +r_{2}\overline{\phi}_{2}(\xi)H_{2}(\overline{ \phi }_{1}, \underline{\phi}_{2},\overline{\phi}_{2}) (\xi) \\& \quad \leq d_{2} \biggl[ \int_{\mathbb{R}}J_{2}(y)\bigl[e^{\lambda_{2}(\xi -y)}+pe ^{\eta \lambda_{2}(\xi -y)}\bigr]\,dy- \bigl( e^{\lambda_{2}\xi }+pe^{\eta \lambda_{2}\xi } \bigr) \biggr] \\& \qquad {}-c^{\ast } \bigl( \lambda_{2}e^{\lambda_{2}\xi }+p\eta \lambda_{2}e ^{\eta \lambda_{2}\xi } \bigr) +r_{2}\overline{\phi}_{2}(\xi)H_{2}(\overline{ \phi }_{1}, \underline{\phi}_{2},\overline{\phi}_{2}) (\xi) \\& \quad \leq d_{2} \biggl[ \int_{\mathbb{R}}J_{2}(y)\bigl[e^{\lambda_{2}(\xi -y)}+pe ^{\eta \lambda_{2}(\xi -y)}\bigr]\,dy- \bigl( e^{\lambda_{2}\xi }+pe^{\eta \lambda_{2}\xi } \bigr) \biggr] \\& \qquad {}-c^{\ast } \bigl( \lambda_{2}e^{\lambda_{2}\xi }+p\eta \lambda_{2}e ^{\eta \lambda_{2}\xi } \bigr) +r_{2} \bigl[e^{\lambda_{2}\xi }+pe^{\eta \lambda_{2}\xi }\bigr]+r_{2}c_{2}^{\ast }e^{\lambda_{1}^{\ast }\xi /2} \bigl[e ^{\lambda_{2}\xi }+pe^{\eta \lambda_{2}\xi }\bigr] \\& \quad =pe^{\eta \lambda_{2}\xi } \biggl\{ d_{2} \biggl[ \int_{\mathbb{R}}J _{2}(y)e^{\eta \lambda_{2}y}\,dy-1 \biggr] -c^{\ast }\eta \lambda_{2}+r _{2} \biggr\} +r_{2}c_{2}e^{\lambda_{1}^{\ast }\xi /2}\bigl[e^{\lambda_{2} \xi }+pe^{\eta \lambda_{2}\xi } \bigr] \\& \quad =p\Theta_{2}\bigl(\eta \lambda_{2},c^{\ast } \bigr)e^{\eta \lambda_{2}\xi }+r _{2}c_{2}e^{\lambda_{1}^{\ast }\xi /2} \bigl[e^{\lambda_{2}\xi }+pe^{\eta \lambda_{2}\xi }\bigr] \\& \quad =e^{\eta \lambda_{2}\xi } \bigl[ p\Theta_{2}\bigl(\eta \lambda_{2},c^{ \ast } \bigr)/2+r_{2}c_{2}e^{(\lambda_{1}^{\ast }/2+\lambda_{2}-\eta \lambda _{2})\xi } \bigr] \\& \qquad {}+pe^{\eta \lambda_{2}\xi } \bigl[ \Theta_{2}\bigl(\eta \lambda_{2},c^{ \ast } \bigr)/2+r_{2}c_{2}e^{\lambda_{1}^{\ast }\xi /2} \bigr] . \end{aligned}$$
Note that
$$\eta \lambda_{2}\xi < \ln \frac{1+c_{2}}{p}, $$
then there exists \(p_{3}>p_{2}+1+c_{2}\) such that \(p\ge p_{3}\) leads to
$$\begin{aligned}& p\Theta_{2}\bigl(\eta \lambda_{2},c^{\ast } \bigr)/2+r_{2}c_{2}e^{(\lambda_{1}/2+ \lambda_{2}-\eta \lambda_{2})\xi }< 0, \\& \Theta_{2} \bigl(\eta \lambda_{2},c ^{\ast }\bigr)/2+r_{2}c_{2}e^{\lambda_{1}^{\ast }\xi /2}< 0 \end{aligned}$$
since \(\lambda_{1}^{\ast }/2+\lambda_{2}-\eta \lambda_{2}>0\), \(\xi <0\) and \(\Theta_{2}(\eta \lambda_{2},c^{\ast })<0\) is a constant. Now, we fix it by \(p=p_{3}\).
When \(\underline{\phi}_{1}(\xi)=0\) with \(\xi <\xi_{3}\), then \(H_{1}(\underline{\phi}_{1},\overline{\phi}_{1},\overline{\phi} _{2})(\xi)=0\) such that (2.6) is clear. Otherwise, if \(\xi \ge \xi_{3}\), then \(\underline{\phi}_{1}(\xi)= ( -L\xi -q\sqrt{- \xi } ) e^{\lambda_{1}^{\ast }\xi }>0\). Firstly, let \(q>q_{1}>1\) such that \(-L\xi -q\sqrt{-\xi }>0\) implies \(\xi <0\) and
$$\overline{\phi}_{2}(\xi)< 2e^{\lambda_{2}\xi },\qquad \underline{\phi}_{1}( \xi)\leq \overline{\phi}_{1}(\xi)< e^{\theta \lambda_{1}^{\ast } \xi } $$
for some \(\theta \in {}[ \frac{2}{3},1)\) with \(\theta \lambda_{1} ^{\ast }+\lambda_{2}>\lambda_{1}^{\ast }\), which is admissible once p is fixed. Therefore, \(q>q_{1}\) indicates
$$\begin{aligned}& r_{1}\underline{\phi}_{1}(\xi)H_{1}( \underline{\phi}_{1},\overline{ \phi }_{1},\overline{\phi}_{2}) (\xi) \\& \quad =r_{1}\underline{\phi}_{1}(\xi) \biggl[ 1- \underline{\phi}_{1}( \xi)-b_{1} \int_{-\tau }^{0}\overline{\phi}_{1}\bigl( \xi +c^{\ast }s\bigr)\,d\eta_{11}(s)-c_{1} \int_{-\tau }^{0}\overline{\phi}_{2}\bigl(\xi +c^{ \ast }s\bigr)\,d\eta_{12}(s) \biggr] \\& \quad \geq r_{1}\underline{\phi}_{1}(\xi)-r_{1}\underline{\phi}_{1} ^{2}(\xi)-r_{1}b_{1}\underline{\phi}_{1}(\xi) \overline{\phi}_{1}( \xi)-2r_{1}c_{1}e^{\lambda_{2}\xi } \underline{\phi}_{1}(\xi) \\& \quad \geq r_{1}\underline{\phi}_{1}(\xi)-r_{1}(1+b_{1})e^{2\theta \lambda_{1}^{\ast }\xi }-2r_{1}c_{1}e^{(\theta \lambda_{1}^{\ast }+ \lambda_{2})\xi } \\& \quad =r_{1} ( -L\xi -q\sqrt{-\xi } ) e^{\lambda_{1}^{\ast } \xi }-r_{1}(1+b_{1})e^{2\theta \lambda_{1}^{\ast }\xi }-2r_{1}c_{1}e ^{(\theta \lambda_{1}^{\ast }+\lambda_{2})\xi }. \end{aligned}$$
Moreover, (3.3) leads to
$$\begin{aligned}& d_{1}[J_{1}\ast \underline{\phi}_{1}](\xi)-c^{\ast }\underline{ \phi }_{1}^{\prime }(\xi) \\& \quad =d_{1} \biggl[ \int_{\mathbb{R}}J_{1}(y)\underline{\phi}_{1}(\xi -y)\,dy-\underline{ \phi }_{1}(\xi) \biggr] -c^{\ast }\underline{\phi}_{1}^{\prime }( \xi) \\& \quad \geq d_{1} \biggl\{ \int_{\mathbb{R}}J_{1}(y) \bigl[ \bigl( -L(\xi -y)-q \sqrt{-( \xi -y)} \bigr) e^{\lambda_{1}^{\ast }(\xi -y)} \bigr]\,dy \\& \qquad {} - ( -L\xi -q\sqrt{-\xi } ) e^{\lambda_{1}^{\ast } \xi } \biggr\} -c^{\ast } \bigl[ ( -L\xi -q\sqrt{-\xi } ) e ^{\lambda_{1}^{\ast }\xi } \bigr] ^{\prime } \\& \quad =d_{1}e^{\lambda_{1}^{\ast }\xi } \biggl[ \int_{\mathbb{R}}J_{1}(y) \bigl[ \bigl( -L(\xi -y) \bigr) e^{-\lambda_{1}^{\ast }y} \bigr]\,dy+L \xi \biggr] \\& \qquad {}-qd_{1}e^{\lambda_{1}^{\ast }\xi } \biggl[ \int_{\mathbb{R}}J_{1}(y)\sqrt{-( \xi -y)}e^{-\lambda_{1}^{\ast }y}\,dy- \sqrt{-\xi } \biggr] \\& \qquad {}+c^{\ast }L \bigl( 1+\lambda_{1}^{\ast }\xi \bigr) e^{\lambda_{1} ^{\ast }\xi }+c^{\ast }q \biggl( \lambda_{1}^{\ast } \sqrt{-\xi }-\frac{1}{2\sqrt{- \xi }} \biggr) e^{\lambda_{1}^{\ast }\xi }. \end{aligned}$$
By what we have done, (2.6) is true if
$$\begin{aligned}& d_{1}[J_{1}\ast \underline{\phi}_{1}](\xi)-c^{\ast }\underline{ \phi }_{1}^{\prime }(\xi)+r_{1}\underline{\phi}_{1}(\xi)H_{1}( \underline{ \phi }_{1},\overline{\phi}_{1},\overline{\phi}_{2}) (\xi) \\& \quad \geq d_{1}e^{\lambda_{1}^{\ast }\xi } \biggl[ \int_{\mathbb{R}}J_{1}(y) \bigl[ \bigl( -L(\xi -y) \bigr) e^{-\lambda_{1}^{\ast }y} \bigr]\,dy+L \xi \biggr] \\& \qquad {}-qd_{1}e^{\lambda_{1}^{\ast }\xi } \biggl[ \int_{\mathbb{R}}J_{1}(y)\sqrt{-( \xi -y)}e^{-\lambda_{1}^{\ast }y}\,dy- \sqrt{-\xi } \biggr] \\& \qquad {}+c^{\ast }L \bigl( 1+\lambda_{1}^{\ast }\xi \bigr) e^{\lambda_{1} ^{\ast }\xi }+c^{\ast }q \biggl( \lambda_{1}^{\ast } \sqrt{-\xi }-\frac{1}{2\sqrt{- \xi }} \biggr) e^{\lambda_{1}^{\ast }\xi } \\& \qquad {}+r_{1} ( -L\xi -q\sqrt{-\xi } ) e^{\lambda_{1}^{\ast } \xi }-r_{1}(1+b_{1})e^{2\theta \lambda_{1}^{\ast }\xi }-2r_{1}c_{1}e ^{(\theta \lambda_{1}^{\ast }+\lambda_{2})\xi } \\& \quad =-L\xi e^{\lambda_{1}^{\ast }\xi } \biggl\{ d_{1} \biggl[ \int_{\mathbb{R}}J_{1}(y)e^{-\lambda_{1}^{\ast }y}\,dy-1 \biggr] -c^{ \ast }\lambda_{1}^{\ast }+r_{1} \biggr\} \\& \qquad {}+d_{1}Le^{\lambda_{1}^{\ast }\xi } \biggl[ \int_{\mathbb{R}}J_{1}(y)ye ^{-\lambda_{1}^{\ast }y}\,dy+c^{\ast } \biggr] -qd_{1}e^{\lambda_{1}^{ \ast }\xi } \biggl[ \int_{\mathbb{R}}J_{1}(y)\sqrt{-(\xi -y)}e^{- \lambda_{1}^{\ast }y}\,dy- \sqrt{-\xi } \biggr] \\& \qquad {}+c^{\ast }q \biggl( \lambda_{1}^{\ast }\sqrt{-\xi }- \frac{1}{2\sqrt{- \xi }} \biggr) e^{\lambda_{1}^{\ast }\xi }-r_{1}q\sqrt{-\xi }e^{ \lambda_{1}^{\ast }\xi }-r_{1}(1+b_{1})e^{2\theta \lambda_{1}^{\ast } \xi }-2r_{1}c_{1}e^{(\theta \lambda_{1}^{\ast }+\lambda_{2})\xi } \\& \quad =-qd_{1}e^{\lambda_{1}^{\ast }\xi } \biggl[ \int_{\mathbb{R}}J_{1}(y)\sqrt{-( \xi -y)}e^{-\lambda_{1}^{\ast }y}\,dy- \sqrt{-\xi } \biggr] +c^{\ast }q \biggl( \lambda_{1}^{\ast } \sqrt{-\xi }-\frac{1}{2\sqrt{-\xi }} \biggr) e ^{\lambda_{1}^{\ast }\xi } \\& \qquad {}-r_{1}q\sqrt{-\xi }e^{\lambda_{1}^{\ast }\xi }-r_{1}(1+b_{1})e ^{2\theta \lambda_{1}^{\ast }\xi }-2r_{1}c_{1}e^{(\theta \lambda_{1} ^{\ast }+\lambda_{2})\xi } \\& \quad =e^{\lambda_{1}^{\ast }\xi } \biggl\{ -qd_{1} \biggl[ \int_{\mathbb{R}}J _{1}(y)\sqrt{-(\xi -y)}e^{-\lambda_{1}^{\ast }y}\,dy- \sqrt{-\xi } \biggr] +c^{\ast }q \biggl( \lambda_{1}^{\ast } \sqrt{-\xi }-\frac{1}{2\sqrt{- \xi }} \biggr) \\& \qquad {} -r_{1}q\sqrt{-\xi }-r_{1}(1+b_{1})e^{(2\theta -1)\lambda _{1}^{\ast }\xi }-2r_{1}c_{1}e^{(\theta \lambda_{1}^{\ast }+\lambda _{2}-\lambda_{1}^{\ast })\xi } \biggr\} \\& \quad \geq 0 \end{aligned}$$
or
$$\begin{aligned}& q \biggl\{ -d_{1} \biggl[ \int_{\mathbb{R}}J_{1}(y)\sqrt{-(\xi -y)}e ^{-\lambda_{1}^{\ast }y}\,dy-\sqrt{-\xi } \biggr] +c^{\ast } \biggl( \lambda_{1}^{\ast }\sqrt{-\xi }-\frac{1}{2\sqrt{-\xi }} \biggr) -r _{1}\sqrt{-\xi } \biggr\} \\& \quad \geq r_{1}(1+b_{1})e^{(2\theta -1)\lambda_{1}^{\ast }\xi }+2r_{1}c _{1}e^{(\theta \lambda_{1}^{\ast }+\lambda_{2}-\lambda_{1}^{\ast }) \xi }. \end{aligned}$$
We first analyze the left of the above inequality
$$\begin{aligned}& -d_{1} \biggl[ \int_{\mathbb{R}}J_{1}(y)\sqrt{-(\xi -y)}e^{-\lambda_{1} ^{\ast }y}\,dy- \sqrt{-\xi } \biggr] +c^{\ast } \biggl( \lambda_{1}^{ \ast } \sqrt{-\xi }-\frac{1}{2\sqrt{-\xi }} \biggr) -r_{1}\sqrt{- \xi } \\& \quad =-d_{1} \biggl\{ \int_{\mathbb{R}}J_{1}(y) \bigl[ \sqrt{-\xi }+\sqrt{-( \xi -y)}-\sqrt{-\xi } \bigr] e^{-\lambda_{1}^{\ast }y}\,dy-\sqrt{- \xi } \biggr\} \\& \qquad {}+c^{\ast } \biggl( \lambda_{1}^{\ast }\sqrt{-\xi }- \frac{1}{2\sqrt{- \xi }} \biggr) -r_{1}\sqrt{-\xi } \\& \quad =-d_{1} \biggl\{ \int_{\mathbb{R}}J_{1}(y) \bigl[ \sqrt{-(\xi -y)}-\sqrt{- \xi } \bigr] e^{-\lambda_{1}^{\ast }y}\,dy \biggr\} -c^{\ast }\frac{1}{2\sqrt{- \xi }} \\& \quad =d_{1} \biggl\{ \int_{\mathbb{R}}J_{1}(y) \bigl[ \sqrt{-\xi }-\sqrt{-( \xi -y)} \bigr] e^{-\lambda_{1}^{\ast }y}\,dy \biggr\} -\frac{d_{1}}{2\sqrt{- \xi }} \int_{\mathbb{R}}J_{1}(y)ye^{\lambda_{1}^{\ast }y}\,dy \\& \quad =d_{1} \int_{\mathbb{R}}J_{1}(y) \biggl[ \frac{y}{2\sqrt{-\xi }}+ \sqrt{- \xi }-\sqrt{-(\xi -y)} \biggr] e^{-\lambda_{1}^{\ast }y}\,dy \\& \quad =d_{1} \int_{\mathbb{R}}J_{1}(y) \biggl[ \frac{y}{2\sqrt{-\xi }}- \frac{y}{\sqrt{- \xi }+\sqrt{-(\xi -y)}} \biggr] e^{-\lambda_{1}^{\ast }y}\,dy \\& \quad =d_{1} \int_{\mathbb{R}}J_{1}(y) \biggl[ \frac{y}{2\sqrt{-\xi }}- \frac{y}{\sqrt{- \xi }+\sqrt{-(\xi -y)}} \biggr] e^{-\lambda_{1}^{\ast }y}\,dy \\& \quad =d_{1} \int_{\mathbb{R}}J_{1}(y) \biggl[ \frac{y [ \sqrt{-( \xi -y)}-\sqrt{-\xi } ] }{2\sqrt{-\xi } [ \sqrt{- \xi }+\sqrt{-(\xi -y)} ] } \biggr] e^{-\lambda_{1}^{\ast }y}\,dy \\& \quad =d_{1} \int_{\mathbb{R}}J_{1}(y) \biggl[ \frac{y^{2}}{2\sqrt{-\xi } [ \sqrt{-\xi }+\sqrt{-(\xi -y)} ] ^{2}} \biggr] e^{- \lambda_{1}^{\ast }y}\,dy \\& \quad \geq d_{1} \int_{\mathbb{R}}J_{1}(y) \biggl[ \frac{y^{2}}{2\sqrt{-( \xi -S)} [ \sqrt{-(\xi -S)}+\sqrt{-(\xi -S)} ] ^{2}} \biggr] e^{-\lambda_{1}^{\ast }y}\,dy \\& \quad =\frac{d_{1}}{8 [ -(\xi -S) ] ^{3/2}} \int_{\mathbb{R}}J _{1}(y)y^{2}e^{-\lambda_{1}^{\ast }y}\,dy. \end{aligned}$$
Let
$$q\geq \frac{\max_{\xi < 0} \{ 8 [ -(\xi -S) ] ^{3/2} [ r_{1}(1+b_{1})e^{(2\theta -1)\lambda_{1}^{\ast }\xi }+2r_{1}c_{1}e ^{(\theta \lambda_{1}+\lambda_{2}-\lambda_{1}^{\ast })\xi } ] \} }{d_{1}\int_{\mathbb{R}}J_{1}(y)y^{2}e^{-\lambda_{1}^{\ast }y}\,dy}+q _{1}:=q_{2}, $$
then (3.2) holds since \(\xi <0\) and
$$(2\theta -1)\lambda_{1}^{\ast }>0,\qquad \theta \lambda_{1}^{\ast }+ \lambda_{2}- \lambda_{1}^{\ast }>0. $$
The verification of (2.7) is finished.
We now consider (2.8), which is clear if \(\underline{\phi}_{2}( \xi)=0>e^{\lambda_{2}\xi }-qe^{\eta \lambda_{2}\xi }\). If \(\underline{ \phi }_{2}(\xi)=e^{\lambda_{2}\xi }-qe^{\eta \lambda_{2}\xi }>0\), we first select \(q_{3}\geq q_{2}\) such that \(\underline{\phi}_{2}( \xi)>0\) implies
$$\overline{\phi}_{2}(\xi)< 2e^{\lambda_{2}\xi } $$
for any \(q\geq q_{3}\), which is admissible for fixed \(p=p_{1}\). Then
$$\begin{aligned}& r_{2}\underline{\phi}_{2}(\xi)H_{2}( \underline{\phi}_{1},\overline{ \phi }_{2},\underline{\phi}_{2}) (\xi) \\& \quad =r_{2}\underline{\phi}_{2}(\xi) \biggl[ 1-\underline{ \phi }_{2}( \xi)-b_{2} \int_{-\tau }^{0}\overline{\phi}_{2}\bigl(\xi +c^{\ast }s\bigr)\,d\eta_{22}(s)+c_{2} \int_{-\tau }^{0}\underline{\phi}_{1}\bigl( \xi +c^{ \ast }s\bigr)\,d\eta_{21}(s) \biggr] \\& \quad \geq r_{2}\underline{\phi}_{2}(\xi) \biggl[ 1- \underline{\phi}_{2}( \xi)-b_{2} \int_{-\tau }^{0}\overline{\phi}_{2}\bigl(\xi +c^{\ast }s\bigr)\,d\eta_{22}(s) \biggr] \\& \quad \geq r_{2}\underline{\phi}_{2}(\xi) \bigl[ 1-e^{\lambda_{2}\xi }-2b _{2}e^{\lambda_{2}\xi } \bigr] \\& \quad =r_{2}\underline{\phi}_{2}(\xi)-r_{2} \underline{\phi}_{2}(\xi) \bigl[ e^{\lambda_{2}\xi }+2b_{2}e^{\lambda_{2}\xi } \bigr] \\& \quad \geq r_{2} \bigl( e^{\lambda_{2}\xi }-qe^{\eta \lambda_{2}\xi } \bigr) -r_{2}e^{\lambda_{2}\xi } \bigl[ e^{\lambda_{2}\xi }+2b_{2}e^{\lambda_{2} \xi } \bigr] . \end{aligned}$$
Therefore, if
$$q>q_{3}-\frac{r_{2} ( 1+2b_{2} ) }{\Theta_{2}(\eta \lambda _{2},c^{*})}:=q_{4}, $$
then (2.8) holds since
$$\begin{aligned}& d_{2}[J_{2}\ast \underline{\phi}_{2}](\xi)-c^{\ast }\underline{ \phi }_{2}^{\prime }(\xi)+r_{2}\underline{\phi}_{2}(\xi)H_{2}(\underline{ \phi }_{1},\overline{\phi}_{2},\underline{\phi}_{2}) (\xi) \\& \quad \geq d_{2} \biggl[ \int_{\mathbb{R}}J_{2}(y)\bigl[e^{\lambda_{2}(\xi -y)}-qe ^{\eta \lambda_{2}(\xi -y)}\bigr]\,dy- \bigl( e^{\lambda_{2}\xi }-qe^{\eta \lambda_{2}\xi } \bigr) \biggr] \\& \qquad {}-c^{\ast } \bigl( \lambda_{2}e^{\lambda_{2}\xi }-q\eta \lambda_{2}e ^{\eta \lambda_{2}\xi } \bigr) +r_{2} \bigl( e^{\lambda_{2}\xi }-qe ^{\eta \lambda_{2}\xi } \bigr) -r_{2}e^{\lambda_{2}\xi } \bigl[ e^{ \lambda_{2}\xi }+2b_{2}e^{\lambda_{2}\xi } \bigr] \\& \quad =-qe^{\eta \lambda_{2}\xi } \biggl\{ d_{2} \biggl[ \int_{\mathbb{R}}J _{2}(y)e^{\eta \lambda_{2}y}\,dy-1 \biggr] -c^{\ast }\eta \lambda_{2}+r _{2} \biggr\} -r_{2}e^{\lambda_{2}\xi } \bigl[ e^{\lambda_{2}\xi }+2b _{2}e^{\lambda_{2}\xi } \bigr] \\& \quad \geq 0,\quad \xi < 0. \end{aligned}$$
By Lemma 2.4 and a discussion similar to (3.1), we complete the proof. □
Theorem 3.3
If
\(c^{\ast }=c_{2}^{\ast }>c_{1}^{\ast }\). Further suppose that
\(k_{2}(y)\)
admits compact support. Then (2.1) with
\(c=c^{\ast }\)
has a positive solution
\((\phi_{1}(\xi),\phi_{2}(\xi))\)
such that
$$0< \phi_{1}(\xi)< 1,0< \phi_{2}(\xi)< 1+c_{2},\xi \in \mathbb{R},\quad \lim_{\xi \rightarrow -\infty }\bigl(\phi_{1}(\xi), \phi_{2}(\xi)\bigr)=(0,0), $$
and
$$\phi_{1}(\xi)\sim \mathcal{O}\bigl(e^{\lambda_{1}\xi }\bigr),\qquad \phi_{2}(\xi) \sim \mathcal{O}\bigl(-\xi e^{\lambda_{2}^{\ast }\xi }\bigr),\quad \xi \rightarrow - \infty. $$
Proof
Under the assumption, we see that
$$d_{2} \int_{\mathbb{R}}J_{2}(y)ye^{\lambda_{2}^{\ast }y}\,dy=c^{\ast } $$
by Lemma 2.3. Let \(S>0\) be a constant such that \(k_{2}(y)=0\), \(\vert y\vert >S\). Select a constant \(\eta >1\) such that
$$\lambda_{2}^{\ast }/2+\lambda_{1}-\eta \lambda_{1}>0, \qquad \Theta_{1}\bigl( \eta \lambda_{1},c^{\ast } \bigr)< 0. $$
Let \(L>1\) be large enough such that
$$-L\xi e^{\lambda_{2}^{\ast }\xi }=1+c_{2} $$
has two real roots \(\xi_{5}< \xi_{6}\) and \(\xi_{6}-\xi_{5}>2S\).
We now define
$$\underline{\phi}_{1}(\xi)=\max \bigl\{ e^{\lambda_{1}\xi }-qe^{\eta \lambda_{1}\xi },0 \bigr\} ,\qquad \overline{\phi}_{1}(\xi)=\min \bigl\{ e^{\lambda_{1} \xi },1\bigr\} $$
and
$$\begin{aligned}& \underline{\phi}_{2}(\xi)=\textstyle\begin{cases} ( -L\xi -q\sqrt{-\xi } ) e^{\lambda_{2}^{\ast }\xi },& \xi < \xi_{3}, \\ 0, &\xi \geq \xi_{3}, \end{cases}\displaystyle \\& \overline{\phi}_{2}(\xi)= \textstyle\begin{cases} ( -L\xi +p\sqrt{-\xi } ) e^{\lambda_{2}^{\ast }\xi },& \xi < \xi_{4}, \\ 1+c_{2},& \xi \geq \xi_{4}, \end{cases}\displaystyle \end{aligned}$$
where \(\xi_{3}=L^{2}/q^{2}\) and \(\xi_{4}<\xi_{5}\) such that \(\overline{\phi}_{2}(\xi)\) is continuous.
For \(\overline{\phi}_{1}(\xi)\), the verification is similar to that in Theorem 3.1 and we omit it here. If \(\overline{\phi}_{2}( \xi)=1+c_{2}\), then \(H_{2}(\overline{\phi}_{1},\underline{\phi} _{2},\overline{\phi}_{2})(\xi)\leq 0\) such that (2.7) is clear. Otherwise, let \(p_{2}>0\) such that
$$\overline{\phi}_{2}(\xi)\ge \underline{\phi}_{2} (\xi),\quad \xi \in \mathbb{R}. $$
Thus,
$$\begin{aligned}& r_{2}\overline{\phi}_{2}(\xi)H_{2}(\overline{ \phi }_{1},\underline{ \phi }_{2},\overline{\phi}_{2}) (\xi) \\& \quad =r_{2}\overline{\phi}_{2}(\xi) \biggl[ 1-\overline{\phi}_{2}( \xi)-b_{2} \int_{-\tau }^{0}\underline{\phi}_{2}\bigl( \xi +c^{\ast }s\bigr)\,d\eta_{22}(s)+c_{2} \int_{-\tau }^{0}\overline{\phi}_{1}\bigl( \xi +c^{ \ast }s\bigr)\,d\eta_{21}(s) \biggr] \\& \quad \leq r_{2}\overline{\phi}_{2}(\xi) \biggl[ 1+c_{2} \int_{-\tau }^{0}\overline{ \phi }_{1}\bigl( \xi +c^{\ast }s\bigr)\,d\eta_{21}(s) \biggr] \\& \quad \leq r_{2}\overline{\phi}_{2}(\xi) \bigl[ 1+c_{2}e^{\lambda_{1} \xi } \bigr] \\& \quad =r_{2}e^{\lambda_{2}^{\ast }\xi } ( -L\xi +p\sqrt{-\xi } ) \bigl[ 1+c_{2}e^{\lambda_{1}\xi } \bigr] \end{aligned}$$
and
$$\begin{aligned}& d_{2}[J_{2}\ast \overline{\phi}_{2}](\xi)-c^{\ast }\overline{ \phi }_{2}^{\prime }(\xi)+r_{2}\overline{\phi}_{2}(\xi)H_{2}(\overline{ \phi }_{1},\underline{\phi}_{2},\overline{\phi}_{2}) (\xi) \\& \quad \leq d_{2} \biggl\{ \int_{\mathbb{R}}J_{2}(y) \bigl[ \bigl( -L(\xi -y)+p\sqrt{-( \xi -y)} \bigr) e^{\lambda_{2}^{\ast }(\xi -y)} \bigr]\,dy \\& \qquad {} - \bigl[ ( -L\xi +p\sqrt{-\xi } ) e^{\lambda_{2} ^{\ast }\xi } \bigr] \biggr\} \\& \qquad {}-c^{\ast }\lambda_{2}^{\ast } ( -L\xi +p\sqrt{-\xi } ) e ^{\lambda_{2}^{\ast }\xi }-c^{\ast } \biggl( -L-\frac{p}{2\sqrt{- \xi }} \biggr) e^{\lambda_{2}^{\ast }\xi } \\& \qquad {}+r_{2}e^{\lambda_{2}^{\ast }\xi } ( -L\xi +p\sqrt{-\xi } ) +r_{2}c_{2}e^{\lambda_{2}^{\ast }\xi }e^{\lambda_{1}\xi } ( -L \xi +p\sqrt{-\xi } ) \\& \quad =-L\xi e^{\lambda_{2}^{\ast }\xi } \biggl\{ d_{2} \biggl[ \int_{\mathbb{R}}J_{2}(y)e^{-\lambda_{2}^{\ast }y}\,dy-1 \biggr] -c^{ \ast }\lambda_{2}^{\ast }+r_{2} \biggr\} \\& \qquad {}+d_{2}L \biggl[ \int_{\mathbb{R}}J_{2}(y)ye^{-\lambda_{2}^{\ast }y}\,dy+c ^{\ast } \biggr] \\& \qquad {}+d_{2}pe^{\lambda_{2}^{\ast }\xi } \biggl\{ \biggl[ \int_{\mathbb{R}}J _{2}(y)\sqrt{-(\xi -y)}e^{-\lambda_{2}^{\ast }y}\,dy- \sqrt{-\xi } \biggr] -c^{\ast }\lambda_{2}^{\ast }\sqrt{-\xi }+r\sqrt{-\xi } \biggr\} \\& \qquad {}+\frac{c^{\ast }p}{2\sqrt{-\xi }}e^{\lambda_{2}^{\ast }\xi }+r _{2}c_{2}e^{\lambda_{2}^{\ast }\xi }e^{\lambda_{1}\xi } ( -L \xi +p\sqrt{-\xi } ) \\& \quad =d_{2}pe^{\lambda_{2}^{\ast }\xi } \biggl\{ \biggl[ \int_{\mathbb{R}}J _{2}(y)\sqrt{-(\xi -y)}e^{-\lambda_{2}^{\ast }y}\,dy- \sqrt{-\xi } \biggr] -c^{\ast }\lambda_{2}^{\ast }\sqrt{-\xi }+r\sqrt{-\xi } \biggr\} \\& \qquad {}+\frac{c^{\ast }p}{2\sqrt{-\xi }}e^{\lambda_{2}^{\ast }\xi }+r _{2}c_{2}e^{\lambda_{2}^{\ast }\xi }e^{\lambda_{1}\xi } ( -L \xi +p\sqrt{-\xi } ) \\& \quad =d_{2}pe^{\lambda_{2}^{\ast }\xi } \int_{\mathbb{R}}J_{2}(y) \bigl[ \sqrt{-( \xi -y)}-\sqrt{- \xi } \bigr] e^{-\lambda_{2}^{\ast }y}\,dy \\& \qquad {}+\frac{c^{\ast }p}{2\sqrt{-\xi }}e^{\lambda_{2}^{\ast }\xi }+r _{2}c_{2}e^{\lambda_{2}^{\ast }\xi }e^{\lambda_{1}\xi } ( -L \xi +p\sqrt{-\xi } ) \\& \quad =d_{2}pe^{\lambda_{2}^{\ast }\xi } \int_{\mathbb{R}}J_{2}(y) \biggl[ \frac{y}{\sqrt{-( \xi -y)}+\sqrt{-\xi }} \biggr] e^{-\lambda_{2}^{\ast }y}\,dy \\& \qquad {}+\frac{c^{\ast }p}{2\sqrt{-\xi }}e^{\lambda_{2}^{\ast }\xi }+r _{2}c_{2}e^{\lambda_{2}^{\ast }\xi }e^{\lambda_{1}\xi } ( -L \xi +p\sqrt{-\xi } ) \\& \quad =d_{2}pe^{\lambda_{2}^{\ast }\xi } \int_{\mathbb{R}}J_{2}(y) \biggl[ \frac{y}{\sqrt{-( \xi -y)}+\sqrt{-\xi }}- \frac{y}{2\sqrt{-\xi }} \biggr] e^{-\lambda_{2} ^{\ast }y}\,dy \\& \qquad {}+r_{2}c_{2}e^{\lambda_{2}^{\ast }\xi }e^{\lambda_{1}\xi } ( -L \xi +p \sqrt{-\xi } ) \\& \quad =d_{2}pe^{\lambda_{2}^{\ast }\xi } \int_{\mathbb{R}}J_{2}(y)\frac{y [ \sqrt{-\xi }-\sqrt{-(\xi -y)} ] }{2\sqrt{-\xi } [ \sqrt{-(\xi -y)}+\sqrt{-\xi } ] }e^{-\lambda_{2}^{ \ast }y}\,dy \\& \qquad {}+r_{2}c_{2}e^{\lambda_{2}^{\ast }\xi }e^{\lambda_{1}\xi } ( -L \xi +p \sqrt{-\xi } ) \\& \quad =d_{2}pe^{\lambda_{2}^{\ast }\xi } \int_{\mathbb{R}}J_{2}(y)\frac{-y ^{2}}{2\sqrt{-\xi } [ \sqrt{-(\xi -y)}+\sqrt{-\xi } ] ^{2}}e^{-\lambda_{2}^{\ast }y}\,dy \\& \qquad {}+r_{2}c_{2}e^{\lambda_{2}^{\ast }\xi }e^{\lambda_{1}\xi } ( -L \xi +p \sqrt{-\xi } ) \\& \quad \leq \frac{-d_{2}pe^{\lambda_{2}^{\ast }\xi }}{8 ( \vert \xi \vert +S ) ^{\frac{3}{2}}} \int_{\mathbb{R}}J_{2}(y)y^{2}e^{-\lambda_{2}^{\ast }y}\,dy+r _{2}c_{2}e^{\lambda_{2}^{\ast }\xi }e^{\lambda_{1}\xi } ( -L \xi +p\sqrt{- \xi } ) \\& \quad \leq 0 \end{aligned}$$
if
$$p\geq \frac{\max_{\xi < 0} \{ 8r_{2}c_{2} ( \vert \xi \vert +S ) ^{\frac{3}{2}}e^{\lambda_{1}\xi } ( -L\xi +p\sqrt{-\xi } ) \} }{d_{2}\int_{\mathbb{R}}J_{2}(y)y^{2}e^{-\lambda_{2}^{\ast }y}\,dy}. $$
When \(\underline{\phi}_{1}(\xi)=0\) with \(\xi <\xi_{3}\), then \(H_{1}(\underline{\phi}_{1},\overline{\phi}_{1},\overline{\phi} _{2})(\xi)=0\) such that (2.6) is clear. Otherwise, if \(\xi \geq \xi_{3}\), then \(\underline{\phi}_{1}(\xi)=e^{\lambda_{1} \xi }-qe^{\eta \lambda_{1}\xi }>0\). Firstly, let \(q>q_{1}>1\) such that \(e^{\lambda_{1}\xi }-qe^{\eta \lambda_{1}\xi }>0\) implies \(\xi <0\) and
$$\overline{\phi}_{2}(\xi)< 2e^{\lambda_{2}^{\ast }\xi },\qquad \underline{ \phi }_{1}(\xi)\leq \overline{\phi}_{1}(\xi)\le e^{\lambda_{1} \xi }, $$
which is admissible once p is fixed. Therefore, \(q>q_{1}\) indicates
$$\begin{aligned}& r_{1}\underline{\phi}_{1}(\xi)H_{1}( \underline{\phi}_{1},\overline{ \phi }_{1},\overline{\phi}_{2}) (\xi) \\& \quad =r_{1}\underline{\phi}_{1}(\xi) \biggl[ 1- \underline{\phi}_{1}( \xi)-b_{1} \int_{-\tau }^{0}\overline{\phi}_{1}\bigl( \xi +c^{\ast }s\bigr)\,d\eta_{11}(s)-c_{1} \int_{-\tau }^{0}\overline{\phi}_{2}\bigl(\xi +c^{ \ast }s\bigr)\,d\eta_{12}(s) \biggr] \\& \quad \geq r_{1}\underline{\phi}_{1}(\xi)-r_{1}\underline{\phi}_{1} ^{2}(\xi)-r_{1}b_{1}\underline{\phi}_{1}(\xi) \overline{\phi}_{1}( \xi)-2r_{1}c_{1}e^{\lambda_{2}^{\ast }\xi } \underline{\phi}_{1}( \xi) \\& \quad \geq r_{1}\underline{\phi}_{1}(\xi)-r_{1}(1+b_{1})e^{2\lambda_{1} \xi }-2r_{1}c_{1}e^{(\lambda_{1}+\lambda_{2}^{\ast })\xi } \\& \quad =r_{1} \bigl( e^{\lambda_{1}\xi }-qe^{\eta \lambda_{1}\xi } \bigr) -r _{1}(1+b_{1})e^{2\lambda_{1}\xi }-2r_{1}c_{1}e^{(\lambda_{1}+\lambda_{2} ^{\ast })\xi }. \end{aligned}$$
Moreover, (3.3) leads to
$$\begin{aligned}& d_{1}[J_{1}\ast \underline{\phi}_{1}](\xi)-c^{\ast }\underline{ \phi }_{1}^{\prime }(\xi)+r_{1}\underline{\phi}_{1}(\xi)H_{1}( \underline{ \phi }_{1},\overline{\phi}_{1},\overline{\phi}_{2}) (\xi) \\& \quad \geq d_{1} \biggl[ \int_{\mathbb{R}}J_{1}(y)\underline{\phi}_{1}( \xi -y)\,dy-\underline{\phi}_{1}(\xi) \biggr] -c^{\ast }\underline{ \phi }_{1}^{\prime }(\xi) \\& \qquad {}+r_{1} \bigl( e^{\lambda_{1}\xi }-qe^{\eta \lambda_{1}\xi } \bigr) -r _{1}(1+b_{1})e^{2\lambda_{1}\xi }-2r_{1}c_{1}e^{(\lambda_{1}+\lambda_{2} ^{\ast })\xi } \\& \quad \geq d_{1} \biggl[ \int_{\mathbb{R}}J_{1}(y) \bigl[ \bigl( e^{\lambda_{1}( \xi -y)}-qe^{\eta \lambda_{1}(\xi -y)} \bigr) \bigr]\,dy- \bigl( e^{ \lambda_{1}\xi }-qe^{\eta \lambda_{1}\xi } \bigr) \biggr] \\& \qquad {}-c^{\ast } \bigl( \lambda_{1}e^{\lambda_{1}\xi }-q\eta \lambda_{1}e ^{\eta \lambda_{1}\xi } \bigr) +r_{1} \bigl( e^{\lambda_{1}\xi }-qe ^{\eta \lambda_{1}\xi } \bigr) \\& \qquad {} -r_{1}(1+b_{1})e^{2\lambda_{1}\xi }-2r_{1}c_{1}e^{(\lambda_{1}+ \lambda_{2}^{\ast })\xi } \\& \quad =-q\Theta_{1} \bigl( \eta \lambda_{1},c^{\ast } \bigr) e^{\eta \lambda_{1}\xi }-r_{1}(1+b_{1})e^{2\lambda_{1}\xi }-2r_{1}c_{1}e^{( \lambda_{1}+\lambda_{2}^{\ast })\xi } \\& \quad \geq 0 \end{aligned}$$
provided that
$$q>\frac{r_{1}(1+b_{1})-2r_{1}c_{1}}{-\Theta_{1} ( \eta \lambda_{1},c ^{\ast } ) }+q_{1}:=q_{2}. $$
Let \(q_{3}\ge q_{2}\) such that \(q>q_{3}\) indicates
$$\underline{\phi}_{2}(\xi) < \overline{\phi}_{2}(\xi),\quad \xi \in \mathbb{R}, $$
and \(q>q_{3}\), \(( -L\xi -q\sqrt{-\xi } ) >0\), imply
$$( -L\xi +q\sqrt{-\xi } ) e^{\lambda_{2}^{\ast }\xi }< e ^{2\lambda_{2}^{\ast }\xi /3} $$
and so
$$\begin{aligned}& r_{2}\underline{\phi}_{2}(\xi)H_{2}( \underline{\phi}_{1},\overline{ \phi }_{2},\underline{\phi}_{2}) (\xi) \\& \quad =r_{2}\underline{\phi}_{2}(\xi) \biggl[ 1-\underline{ \phi }_{2}( \xi)-b_{2} \int_{-\tau }^{0}\overline{\phi}_{2}\bigl(\xi +c^{\ast }s\bigr)\,d\eta_{22}(s)+c_{2} \int_{-\tau }^{0}\underline{\phi}_{1}\bigl( \xi +c^{ \ast }s\bigr)\,d\eta_{21}(s) \biggr] \\& \quad \geq r_{2}\underline{\phi}_{2}(\xi) \biggl[ 1- \underline{\phi}_{2}( \xi)-b_{2} \int_{-\tau }^{0}\overline{\phi}_{2}\bigl(\xi +c^{\ast }s\bigr)\,d\eta_{22}(s) \biggr] \\& \quad \geq r_{2}\underline{\phi}_{2}(\xi) \bigl[ 1-(1+b_{2})e^{2\lambda_{2} ^{\ast }\xi /3} \bigr] \\& \quad =r_{2} ( -L\xi -q\sqrt{-\xi } ) e^{\lambda_{2}^{\ast } \xi }-r_{2}(1+b_{2})e^{4\lambda_{2}^{\ast }\xi /3}. \end{aligned}$$
By direct calculations, we see
$$\begin{aligned}& d_{2}[J_{2}\ast \underline{\phi}_{2}](\xi)-c^{\ast }\underline{ \phi }_{2}^{\prime }(\xi)+r_{2}\underline{\phi}_{2}(\xi)H_{2}(\underline{ \phi }_{1},\overline{\phi}_{2},\underline{\phi}_{2}) (\xi) \\& \quad \geq d_{2} \biggl[ \int_{\mathbb{R}}J_{2}(y) \bigl[ \bigl( -L(\xi -y)-q\sqrt{-( \xi -y)} \bigr) e^{\lambda_{2}^{\ast }(\xi -y)} \bigr]\,dy- ( -L \xi -q\sqrt{-\xi } ) e^{\lambda_{2}^{\ast }\xi } \biggr] \\& \qquad {}+c^{\ast }L \bigl( 1+\lambda_{2}^{\ast }\xi \bigr) e^{\lambda_{2} ^{\ast }\xi }+c^{\ast }q \biggl( \lambda_{2}^{\ast } \sqrt{-\xi }-\frac{1}{2\sqrt{- \xi }} \biggr) e^{\lambda_{2}^{\ast }\xi } \\& \qquad {}+r_{2} ( -L\xi -q\sqrt{-\xi } ) e^{\lambda_{2}^{\ast } \xi }-r_{2}(1+b_{2})e^{4\lambda_{2}^{\ast }\xi /3} \\& \quad =d_{2} \biggl[ \int_{\mathbb{R}}J_{2}(y) \bigl( -L(\xi -y) \bigr) e ^{\lambda_{2}^{\ast }(\xi -y)}\,dy+L\xi e^{\lambda_{2}^{\ast }\xi } \biggr] +c^{\ast }L \bigl( 1+ \lambda_{2}^{\ast }\xi \bigr) e^{\lambda_{2}^{ \ast }\xi }-r_{2}L\xi e^{\lambda_{2}^{\ast }\xi } \\& \qquad {}-q\sqrt{-\xi }d_{2} \int_{\mathbb{R}}J_{2}(y)e^{\lambda_{2}^{ \ast }(\xi -y)}\,dy+q\sqrt{-\xi }e^{\lambda_{2}^{\ast }\xi } \\& \qquad {}+c^{\ast }q\lambda_{2}^{\ast }\sqrt{-\xi }e^{\lambda_{2}^{\ast } \xi }+r_{2} ( -L\xi -q\sqrt{-\xi } ) e^{\lambda_{2}^{ \ast }\xi } \\& \qquad {}+d_{2}q \int_{\mathbb{R}}J_{2}(y) \bigl[ \sqrt{-\xi }-\sqrt{-( \xi -y)} \bigr] e^{\lambda_{2}^{\ast }(\xi -y)}\,dy-\frac{c^{\ast }q}{2\sqrt{- \xi }}e^{\lambda_{2}^{\ast }\xi }-r_{2}(1+b_{2})e^{4\lambda_{2}^{ \ast }\xi /3} \\& \quad =d_{2}q \int_{\mathbb{R}}J_{2}(y) \bigl[ \sqrt{-\xi }-\sqrt{-( \xi -y)} \bigr] e^{\lambda_{2}^{\ast }(\xi -y)}\,dy-\frac{c^{\ast }q}{2\sqrt{- \xi }}e^{\lambda_{2}^{\ast }\xi }-r_{2}(1+b_{2})e^{4\lambda_{2}^{ \ast }\xi /3} \\& \quad =d_{2}q \int_{\mathbb{R}}J_{2}(y) \biggl[ \frac{-y}{\sqrt{-\xi }-\sqrt{-( \xi -y)}} \biggr] e^{\lambda_{2}^{\ast }(\xi -y)}\,dy-\frac{c^{\ast }q}{2\sqrt{- \xi }}e^{\lambda_{2}^{\ast }\xi }-r_{2}(1+b_{2})e^{4\lambda_{2}^{ \ast }\xi /3} \\& \quad =d_{2}q \int_{\mathbb{R}}J_{2}(y) \biggl[ \frac{y}{2\sqrt{-\xi }}- \frac{y}{\sqrt{- \xi }+\sqrt{-(\xi -y)}} \biggr] e^{\lambda_{2}^{\ast }(\xi -y)}\,dy-r _{2}(1+b_{2})e^{4\lambda_{2}^{\ast }\xi /3} \\& \quad =d_{2}q \int_{\mathbb{R}}J_{2}(y) \biggl[ \frac{y^{2}}{2\sqrt{- \xi } [ \sqrt{-\xi }+\sqrt{-(\xi -y)} ] } \biggr] e^{ \lambda_{2}^{\ast }(\xi -y)}\,dy-r_{2}(1+b_{2})e^{4\lambda_{2}^{\ast } \xi /3} \\& \quad =d_{2}q \int_{\mathbb{R}}J_{2}(y)\frac{y^{2}e^{\lambda_{2}^{\ast }( \xi -y)}}{2\sqrt{-\xi } [ \sqrt{-\xi }+\sqrt{-(\xi -y)} ] ^{2}}\,dy-r_{2}(1+b_{2})e^{4\lambda_{2}^{\ast }\xi /3} \\& \quad =e^{\lambda_{2}^{\ast }\xi } \biggl\{ d_{2}q \int_{\mathbb{R}}J_{2}(y)\frac{y ^{2}e^{\lambda_{2}^{\ast }y}}{2\sqrt{-\xi } [ \sqrt{-\xi }+\sqrt{-( \xi -y)} ] ^{2}}\,dy-r_{2}(1+b_{2})e^{\lambda_{2}^{\ast }\xi /3} \biggr\} \\& \quad \geq e^{\lambda_{2}^{\ast }\xi } \biggl\{ d_{2}q \int_{\mathbb{R}}J _{2}(y)\frac{y^{2}e^{\lambda_{2}^{\ast }y}}{8(\vert \xi \vert +S)^{3/2}}\,dy-r _{2}(1+b_{2})e^{\lambda_{2}^{\ast }\xi /3} \biggr\} \\& \quad \geq 0 \end{aligned}$$
if
$$q\ge \sup_{\xi < 0}\frac{8e^{\lambda_{2}^{\ast }\xi /3}(S+\vert \xi \vert )^{3/2}r _{2}(1+b_{2})}{d_{2}\int_{\mathbb{R}}J_{2}(y)y^{2}e^{\lambda_{2}^{ \ast }y}\,dy}+q_{4}:=q_{5}. $$
Fix \(q=q_{5}\), we complete the proof by Lemma 2.4 and a discussion similar to (3.1). □
Theorem 3.4
Assume that
\(c_{1}^{*}=c_{2}^{*}\). Further suppose that
\(k_{1}\), \(k_{2}\)
have compact supports. Then (2.1) with
\(c=c^{\ast }\)
has a positive solution
\((\phi_{1}(\xi),\phi_{2}(\xi))\)
such that
$$0< \phi_{1}(\xi)< 1,0< \phi_{2}(\xi)< 1+c_{2},\xi \in \mathbb{R}, \quad \lim_{\xi \rightarrow -\infty }\bigl(\phi_{1}(\xi), \phi_{2}(\xi)\bigr)=(0,0) $$
and
$$\phi_{1}(\xi)\sim \mathcal{O}\bigl(-\xi e^{\lambda_{1}^{\ast }\xi }\bigr),\qquad \phi _{2}(\xi)\sim \mathcal{O}\bigl(-\xi e^{\lambda_{2}^{\ast }\xi }\bigr),\quad \xi \rightarrow -\infty. $$
Proof
Using the notation in Theorems 3.2–3.3, we define
$$\begin{aligned}& \underline{\phi}_{1}(\xi)= \textstyle\begin{cases} ( -L\xi -q\sqrt{-\xi } ) e^{\lambda_{1}^{\ast }\xi },& \xi < \xi_{1}, \\ 0, &\xi \geq \xi _{1}, \end{cases}\displaystyle \\& \overline{\phi}_{1}(\xi)= \textstyle\begin{cases} -L\xi e^{\lambda_{1}^{\ast }\xi },&\xi < \xi_{2}, \\ 1, &\xi \geq \xi_{2}, \end{cases}\displaystyle \end{aligned}$$
and
$$\begin{aligned}& \underline{\phi}_{2}(\xi)= \textstyle\begin{cases} ( -L\xi -q\sqrt{-\xi } ) e^{\lambda_{2}^{\ast }\xi }, &\xi < \xi_{3}, \\ 0,& \xi \geq \xi_{3}, \end{cases}\displaystyle \\& \overline{\phi}_{2}(\xi)=\textstyle\begin{cases} ( -L\xi +p\sqrt{-\xi } ) e^{\lambda_{2}^{\ast }\xi },& \xi < \xi_{4}, \\ 1+c_{2},& \xi \geq \xi_{4}, \end{cases}\displaystyle \end{aligned}$$
where \(p,q>1\) are large enough, \(\xi_{1}\), \(\xi_{2}\), \(\xi_{3}\), \(\xi_{4}\) are similar to above. Then we can obtain a pair of upper and lower solutions. Since the verification is similar to those in Theorems 3.2–3.3, we omit it here. □