In this section, we firstly present some sufficient conditions for the existence of each type of eventually positive solutions to (1).

### Theorem 3.1

*If there exists some constant*
\(K>0\)*such that*

$$ \int _{t_{0}}^{\infty } \int _{t_{0}}^{u_{3}} \int _{t_{0}}^{u_{2}} \int _{t_{0}}^{u_{1}} \frac{f(u_{0},K)}{r_{1}(u_{1})r_{2}(u_{2})r_{3}(u _{3})}\Delta u_{0}\Delta u_{1}\Delta u_{2}\Delta u_{3}< \infty, $$

(8)

*then* (1) *has an eventually positive solution**x**with*
\(\lim_{t\rightarrow \infty }x(t)=b\), *where**b**is a positive constant*.

### Proof

Suppose that there exists some constant \(K>0\) satisfying (8). For \(0\leq p_{0}<1\), there are two cases \(p_{0}>0\) and \(p_{0}=0\). If \(p_{0}>0\), then we choose a constant \(p_{1}\) with \(p_{0}< p_{1}<(1+4p_{0})/5<1\), and thus there exists \(T_{0}\in [t_{0},\infty )_{\mathbb{T}}\) such that for \(t\in [T_{0}, \infty )_{\mathbb{T}}\), we have \(p(t)>0\), \((5p_{1}-1)/4\leq p(t) \leq p_{1}<1\), and

$$\begin{aligned} \int _{T_{0}}^{\infty } \int _{T_{0}}^{u_{3}} \int _{T_{0}}^{u_{2}} \int _{T_{0}}^{u_{1}} \frac{f(u_{0},K)}{r_{1}(u_{1})r_{2}(u_{2})r_{3}(u _{3})}\Delta u_{0}\Delta u_{1}\Delta u_{2}\Delta u_{3}\leq \frac{(1-p _{1})K}{8}. \end{aligned}$$

If \(p_{0}=0\), then we take \(p_{1}\) satisfying that \(|p(t)|\leq p_{1} \leq 1/13\) for \(t\in [T_{0},\infty )_{\mathbb{T}}\).

Choose \(T_{1}\in (T_{0},\infty )_{\mathbb{T}}\) such that \(g(t)\geq T _{0}\) and \(h(t)\geq T_{0}\) for \(t\in [T_{1},\infty )_{\mathbb{T}}\). Let \(\varOmega _{1}=\{x\in {\mathrm{BC}}_{0}[T_{0},\infty )_{\mathbb{T}}: K/2 \leq x(t)\leq K\}\), where \(\mathrm{BC}_{0}[T_{0},\infty )_{\mathbb{T}}\) is defined as (3) when \(\lambda =0\), and \(U_{1},V_{1}: \varOmega _{1}\rightarrow {\mathrm{BC}}_{0}[T_{0},\infty )_{\mathbb{T}}\) as follows:

$$\begin{aligned} &(U_{1}x) (t)=\textstyle\begin{cases} (U_{1}x)(T_{1}),& t\in [T_{0},T_{1})_{\mathbb{T}}, \\ 3Kp_{1}/4-p(t)x(g(t)), & t\in [T_{1},\infty )_{\mathbb{T}}, \end{cases}\displaystyle \\ &(V_{1}x) (t)=\textstyle\begin{cases} (V_{1}x)(T_{1}),& t\in [T_{0},T_{1})_{\mathbb{T}}, \\ 3K/4 +\int _{t}^{\infty }\int _{T_{1}}^{u_{3}}\int _{T_{1}}^{u_{2}}\int _{T _{1}}^{u_{1}}\frac{f(u_{0},x(h(u_{0})))}{r_{1}(u_{1})r_{2}(u_{2})r _{3}(u_{3})}\Delta u_{0}\Delta u_{1}\Delta u_{2}\Delta u_{3}, & t \in [T_{1},\infty )_{\mathbb{T}}. \end{cases}\displaystyle \end{aligned}$$

Similar to the proofs in [4, Theorem 2.5], [5, Theorem 2], [12, Theorem 3.1], [15, Theorem 3.1], and [19, Theorem 8], we omit the explanation that \(U_{1}\) and \(V_{1}\) satisfy the conditions in Lemma 2.1. Then there exists \(x\in \varOmega _{1}\) such that \((U_{1}+V_{1})x=x\), which means that, for \(t\in [T_{1},\infty )_{\mathbb{T}}\), we obtain

$$\begin{aligned} x(t)={}&\frac{3(1+p_{1})K}{4}-p(t)x\bigl(g(t)\bigr) \\ &{} + \int _{t}^{\infty } \int _{T_{1}}^{u_{3}} \int _{T_{1}}^{u_{2}} \int _{T _{1}}^{u_{1}}\frac{f(u_{0},x(h(u_{0})))}{r_{1}(u_{1})r_{2}(u_{2})r _{3}(u_{3})}\Delta u_{0}\Delta u_{1}\Delta u_{2}\Delta u_{3}. \end{aligned}$$

Letting \(t\rightarrow \infty \), from (C4) and Lemma 2.2, we deduce

$$\begin{aligned} \lim_{t\rightarrow \infty }z(t)=\frac{3(1+p_{1})K}{4} \quad\text{and} \quad\lim _{t\rightarrow \infty }x(t)=\frac{3(1+p_{1})K}{4(1+p _{0})}>0. \end{aligned}$$

For \(-1< p_{0}<0\), change \(p_{1}\) to satisfy \(-p_{0}< p_{1}<(1-4p_{0})/5<1\) and \((5p_{1}-1)/4\leq -p(t)\leq p_{1}<1\) for \(t\in [T_{0},\infty )_{\mathbb{T}}\). Let

$$\begin{aligned} (\overline{U}_{1}x) (t)=\textstyle\begin{cases} (\overline{U}_{1}x)(T_{1}),& t\in [T_{0},T_{1})_{\mathbb{T}}, \\ -3Kp_{1}/4-p(t)x(g(t)),& t\in [T_{1},\infty )_{\mathbb{T}}. \end{cases}\displaystyle \end{aligned}$$

Similarly, there also exists \(x\in \varOmega _{1}\) such that \(( \overline{U}_{1}+V_{1})x=x\), and we obtain

$$\begin{aligned} \lim_{t\rightarrow \infty }z(t)=\frac{3(1-p_{1})K}{4} \quad\text{and}\quad \lim _{t\rightarrow \infty }x(t)=\frac{3(1-p_{1})K}{4(1+p _{0})}>0. \end{aligned}$$

This completes the proof. □

### Theorem 3.2

*Assume that* (4) *holds*. *If there exists some constant*
\(K>0\)*such that*

$$ \int _{t_{0}}^{\infty } \int _{t_{0}}^{u_{1}}\frac{f(u_{0},KR(h(u_{0})))}{r _{1}(u_{1})}\Delta u_{0}\Delta u_{1}< \infty, $$

(9)

*then* (1) *has an eventually positive solution*
\(x\in A(b)\), *where**b**is a positive constant*.

### Proof

Suppose that there exists some constant \(K>0\) such that (9) holds. Proceed as in the proof of Theorem 3.1, except that, for \(p_{0}>0\), take \(T_{0}\in [t_{0},\infty )_{ \mathbb{T}}\) satisfying \(p(t)>0\), \((5p_{1}-1)/4\leq p(t)\leq p_{1}<1\), \(p(t)R(g(t))/R(t)\geq (5p_{1}-1)\eta /4\) for \(t\in [T_{0},\infty )_{ \mathbb{T}}\), and

$$\begin{aligned} \int _{T_{0}}^{\infty } \int _{T_{0}}^{u_{1}}\frac{f(u_{0},KR(h(u_{0})))}{r _{1}(u_{1})}\Delta u_{0}\Delta u_{1}\leq \frac{(1-p_{1}\eta )K}{8}. \end{aligned}$$

Let \(\varOmega _{2}= \{ x\in {\mathrm{BC}}_{1}[T_{0},\infty )_{ \mathbb{T}}:KR(t)/2\leq x(t)\leq KR(t) \} \), where \(\mathrm{BC}_{1}[T_{0},\infty )_{\mathbb{T}}\) is defined as (3) when \(\lambda =1\), and \(U_{2},V_{2}: \varOmega _{2} \rightarrow {\mathrm{BC}}_{1}[T_{0},\infty )_{\mathbb{T}}\) as follows:

$$\begin{aligned} &(U_{2}x) (t)=\textstyle\begin{cases} 3Kp_{1}\eta R(t)/4-p(T_{1})x(g(T_{1}))R(t)/R(T_{1}),& t\in [T_{0},T _{1})_{\mathbb{T}}, \\ 3Kp_{1}\eta R(t)/4-p(t)x(g(t)),& t\in [T_{1},\infty )_{\mathbb{T}}, \end{cases}\displaystyle \\ &(V_{2}x) (t)=\textstyle\begin{cases} 3KR(t)/4,& t\in [T_{0},T_{1})_{\mathbb{T}}, \\ 3KR(t)/4 \\ \quad{}+\int _{T_{1}}^{t}\int _{T_{1}}^{u_{3}}\int _{u_{2}}^{\infty }\int _{T _{1}}^{u_{1}} \frac{f(u_{0},x(h(u_{0})))}{r_{1}(u_{1})r_{2}(u_{2})r _{3}(u_{3})}\Delta u_{0}\Delta u_{1}\Delta u_{2}\Delta u_{3}, & t \in [T_{1},\infty )_{\mathbb{T}}, \end{cases}\displaystyle \end{aligned}$$

where \(T_{1}\) is defined as in Theorem 3.1. Similarly, there exists \(x\in \varOmega _{2}\) such that \((U_{2}+V_{2})x=x\). For \(t\in [T_{1}, \infty )_{\mathbb{T}}\), it follows that

$$\begin{aligned} x(t)={}&\frac{3(1+p_{1}\eta )KR(t)}{4}-p(t)x\bigl(g(t)\bigr) \\ &{} + \int _{T_{1}}^{t} \int _{T_{1}}^{u_{3}} \int _{u_{2}}^{\infty } \int _{T _{1}}^{u_{1}} \frac{f(u_{0},x(h(u_{0})))}{r_{1}(u_{1})r_{2}(u_{2})r _{3}(u_{3})}\Delta u_{0}\Delta u_{1}\Delta u_{2}\Delta u_{3}. \end{aligned}$$

Letting \(t\rightarrow \infty \), we derive

$$\begin{aligned} \lim_{t\rightarrow \infty }\frac{z(t)}{R(t)}= \frac{3(1+p_{1}\eta )K}{4} \quad\text{and}\quad \lim_{t\rightarrow \infty }\frac{x(t)}{R(t)}= \frac{3(1+p_{1}\eta )K}{4(1+p _{0}\eta )}>0. \end{aligned}$$

For \(-1< p_{0}<0\), similar to the proof in Theorem 3.1, we have

$$\begin{aligned} \lim_{t\rightarrow \infty }\frac{z(t)}{R(t)}= \frac{3(1-p_{1}\eta )K}{4} \quad\text{and}\quad \lim_{t\rightarrow \infty }\frac{x(t)}{R(t)}= \frac{3(1-p_{1}\eta )K}{4(1+p _{0}\eta )}>0. \end{aligned}$$

Moreover, it is clear that \(\lim_{t\rightarrow \infty }x(t)=\infty \). The proof is complete. □

### Theorem 3.3

*Assume that* (4) *holds*. *If there exists a positive constant**M**satisfying that*
\(|p(t)R(t)| \leq M\)*for*
\(t\in [t_{0},\infty )_{\mathbb{T}}\),

$$ \int _{t_{0}}^{\infty } \int _{t_{0}}^{u_{1}}\frac{f(u_{0},R(h(u_{0})))}{r _{1}(u_{1})}\Delta u_{0}\Delta u_{1}< \infty, $$

(10)

*and*

$$ \int _{t_{0}}^{\infty } \int _{t_{0}}^{u_{3}} \int _{u_{2}}^{\infty } \int _{t_{0}}^{u_{1}} \frac{f(u_{0},M+3/4)}{r_{1}(u_{1})r_{2}(u_{2})r _{3}(u_{3})}\Delta u_{0}\Delta u_{1}\Delta u_{2}\Delta u_{3}=\infty, $$

(11)

*then* (1) *has an eventually positive solution*
\(x\in A(0)\).

### Proof

Suppose that there exists a constant \(M>0\) such that \(|p(t)R(t)|\leq M\) for \(t\in [t_{0},\infty )_{\mathbb{T}}\), and both of (10) and (11) hold. It is easy to see that \(p_{0}=0\). There exist \(T_{0}\in [t_{0},\infty )_{\mathbb{T}}\) and \(p_{1}\) with \(0< p_{1}<1\) such that, for \(t\in [T_{0},\infty )_{ \mathbb{T}}\), we have \(|p(t)|\leq p_{1}<1\), \(2M+3/2\leq R(t)/4\), and

$$\begin{aligned} \int _{T_{0}}^{\infty } \int _{T_{0}}^{u_{1}}\frac{f(u_{0},R(h(u_{0})))}{r _{1}(u_{1})}\Delta u_{0}\Delta u_{1}\leq \frac{1-p_{1}}{8}. \end{aligned}$$

Let \(\varOmega _{3}= \{ x\in {\mathrm{BC}}_{1}[T_{0},\infty )_{ \mathbb{T}}: M+3/4\leq x(t)\leq R(t) \} \) and \(U_{3},V_{3}: \varOmega _{3}\rightarrow {\mathrm{BC}}_{1}[T_{0},\infty )_{\mathbb{T}}\) as follows:

$$\begin{aligned} &(U_{3}x) (t)=\textstyle\begin{cases} M+3/4-p(T_{1})x(g(T_{1}))R(t)/R(T_{1}), &t\in [T_{0},T_{1})_{ \mathbb{T}}, \\ M+3/4-p(t)x(g(t)),& t\in [T_{1},\infty )_{\mathbb{T}}, \end{cases}\displaystyle \\ &(V_{3}x) (t)=\textstyle\begin{cases} M+3/4,& t\in [T_{0},T_{1})_{\mathbb{T}}, \\ M+3/4 +\int _{T_{1}}^{t}\int _{T_{1}}^{u_{3}}\int _{u_{2}}^{\infty }\int _{T _{1}}^{u_{1}} \frac{f(u_{0},x(h(u_{0})))}{r_{1}(u_{1})r_{2}(u_{2})r _{3}(u_{3})}\Delta u_{0}\Delta u_{1}\Delta u_{2}\Delta u_{3},& t \in [T_{1},\infty )_{\mathbb{T}}, \end{cases}\displaystyle \end{aligned}$$

where \(T_{1}\) is defined as in Theorem 3.1. Similarly, there exists \(x\in \varOmega _{3}\) such that, for \(t\in [T_{1},\infty )_{\mathbb{T}}\), we have

$$\begin{aligned} x(t)={}&2M+\frac{3}{2}-p(t)x\bigl(g(t)\bigr) \\ &{} + \int _{T_{1}}^{t} \int _{T_{1}}^{u_{3}} \int _{u_{2}}^{\infty } \int _{T _{1}}^{u_{1}} \frac{f(u_{0},x(h(u_{0})))}{r_{1}(u_{1})r_{2}(u_{2})r _{3}(u_{3})}\Delta u_{0}\Delta u_{1}\Delta u_{2}\Delta u_{3}. \end{aligned}$$

Letting \(t\rightarrow \infty \), it is not difficult to see that

$$\begin{aligned} \lim_{t\rightarrow \infty }z(t)=\infty \quad\text{and}\quad \lim _{t\rightarrow \infty }\frac{z(t)}{R(t)}=0, \end{aligned}$$

which implies that

$$\begin{aligned} \lim_{t\rightarrow \infty }x(t)=\infty \quad\text{and}\quad \lim _{t\rightarrow \infty }\frac{x(t)}{R(t)}=0 \end{aligned}$$

since \(|p(t)x(g(t))|\leq |p(t)R(t)|\leq M\) for \(t\in [T_{1},\infty )_{ \mathbb{T}}\). This completes the proof. □

### Remark 3.4

It is not easy to find a sufficient condition for the existence of nonoscillatory solutions tending to zero to (1) since their asymptotic behaviors are more complex than those of other solutions. However, we refer the reader to [4, Theorem 2.8 and Remark 2.9], [5, Theorem 3], [12, Theorems 3.5 and 3.6], [15, Theorems 3.2 and 3.3], and [19, Theorems 9 and 10], where some instructive results are presented.