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Nonoscillatory solutions to fourth-order neutral dynamic equations on time scales

Abstract

In this paper, we present some sufficient conditions and necessary conditions for the existence of nonoscillatory solutions to a class of fourth-order nonlinear neutral dynamic equations on time scales by employing Banach spaces and Krasnoselskii’s fixed point theorem. Two examples are given to illustrate the applications of the results.

Introduction

In this paper, we consider the existence of nonoscillatory solutions to fourth-order nonlinear neutral dynamic equations of the form

$$ \bigl(r_{1}(t) \bigl(r_{2}(t) \bigl(r_{3}(t) \bigl(x(t)+p(t)x\bigl(g(t)\bigr) \bigr) ^{\Delta } \bigr)^{\Delta } \bigr)^{\Delta } \bigr)^{\Delta }+f\bigl(t,x\bigl(h(t)\bigr)\bigr)=0 $$
(1)

on a time scale \(\mathbb{T}\) satisfying \(\sup \mathbb{T}=\infty \), where \(t\in [t_{0},\infty )_{\mathbb{T}}\) with \(t_{0}\in \mathbb{T}\).

The oscillation and nonoscillation of nonlinear differential and difference equations have been developed rapidly in the recent decades. Afterwards, the theory of time scale united the differential and difference ones, and since then many researchers have investigated the oscillation and nonoscillation criteria of nonlinear dynamic equations on time scales; see, for instance, the papers [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19] and the references cited therein. The majority of the scholars obtained the sufficient conditions to ensure that the solutions of the equations oscillate or tend to zero by using the generalized Riccati transformation and integral averaging technique. The correlative research has made a great achievement. However, we note that there is not much research into the field of existence of nonoscillatory solutions to dynamic equations on time scales. Generally, a functional space and a fixed point theorem would be employed to analyze it. We refer the reader to [20,21,22,23,24,25] for details of the theory of time scale and to [4,5,6,7,8,9,10,11,12,13,14,15,16, 19] for the studies on the existence of nonoscillatory solutions to nonlinear neutral dynamic equations on time scales.

Definition 1.1

A solution x to (1) is defined to be eventually positive (or eventually negative) if there exists \(c\in \mathbb{T}\) such that \(x(t)>0\) (or \(x(t)<0\)) on \([c,\infty )_{\mathbb{T}}\). If a solution is either eventually positive or eventually negative, then we say it is nonoscillatory.

For a class of nth-order nonlinear neutral dynamic equations as follows:

$$ R_{n}\bigl(t,x(t)\bigr)+f\bigl(t,x\bigl(h(t)\bigr) \bigr)=0, $$
(2)

where

$$\begin{aligned} R_{k}\bigl(t,x(t)\bigr)= \textstyle\begin{cases} z(t)=x(t)+p(t)x(g(t)),& k=0, \\ r_{n-k}(t)R_{k-1}^{\Delta }(t,x(t)),& 1\leq k \leq n-1, \\ R_{n-1}^{\Delta }(t,x(t)),& k=n, \end{cases}\displaystyle \end{aligned}$$

some scholars had been concerned with the existence of nonoscillatory solutions to (2), successively. Without loss of generality, only the eventually positive solutions were discussed. Zhu and Wang [19] considered (2) for \(n=1\). The authors introduced a Banach space

$$\begin{aligned} {\mathrm{BC}}[T_{0},\infty )_{\mathbb{T}}= \Bigl\{ x\in { \mathrm{C}}\bigl([T _{0},\infty)_{\mathbb{T}},\mathbb{R}\bigr): \sup_{t\in [T_{0},\infty )_{\mathbb{T}}} \bigl\vert x(t) \bigr\vert < \infty \Bigr\} \end{aligned}$$

with the norm \(\|x\|=\sup_{t\in [T_{0},\infty )_{\mathbb{T}}} \vert x(t) \vert \), where \(\mathrm{C}([T_{0},\infty )_{\mathbb{T}},\mathbb{R})\) denotes all continuous functions mapping \([T_{0},\infty )_{\mathbb{T}}\) into \(\mathbb{R}\), and established the existence of nonoscillatory solutions to (2) by Krasnoselskii’s fixed point theorem. Note that there exist only two cases for every eventually positive solution x to (2): \(\lim_{t\rightarrow \infty }x(t)=a>0\) or \(\lim_{t\rightarrow \infty }x(t)=0\).

Gao and Wang [5] and Deng and Wang [4] investigated (2) for \(n=2\) under different conditions \(\int _{t_{0}} ^{\infty }1/r_{1}(t)\Delta t<\infty \) and \(\int _{t_{0}}^{\infty }1/r _{1}(t)\Delta t=\infty \), respectively. Gao and Wang [5] introduced the same Banach space as in [19] and concluded that all the eventually positive solutions to (2) converge to a positive constant or zero. Deng and Wang [4] defined an improved Banach space

$$ {\mathrm{BC}}_{\lambda }[T_{0},\infty )_{\mathbb{T}}= \biggl\{ x \in {\mathrm{C}}\bigl([T_{0},\infty )_{\mathbb{T}},\mathbb{R}\bigr): \sup_{t\in [T_{0},\infty )_{\mathbb{T}}} \biggl\vert \frac{x(t)}{R^{2\lambda }(t)} \biggr\vert < \infty \biggr\} $$
(3)

with the norm \(\|x\|_{\lambda }=\sup_{t\in [T_{0},\infty )_{ \mathbb{T}}} \vert x(t)/R^{2\lambda }(t) \vert \), where \(\lambda =0,1\) and \(R(t)=1+\int _{t_{0}}^{t}1/r_{1}(s)\Delta s\), and presented four cases for the eventually positive solution x to (2): (i) \(x \in A(0,0)\), (ii) \(x\in A(b,0)\) for some positive constant b, (iii) \(x \in A(\infty,b)\) for some positive constant b, (iv) \(\limsup_{t\rightarrow \infty }x(t)=\infty \) and \(\lim_{t\rightarrow \infty }x(t)/R(t)=0\), where

$$\begin{aligned} A(\alpha, \beta )= \biggl\{ x: \lim_{t\rightarrow \infty }x(t)=\alpha \text{ and } \lim_{t\rightarrow \infty }\frac{x(t)}{R(t)}= \beta \biggr\} . \end{aligned}$$

Therefore, it is not difficult to see that the existences of nonoscillatory solutions to (2) are greatly different for diverse cases of convergence and divergence of the integrals \(\int _{t_{0}}^{\infty }1/r_{i}(t)\Delta t\), \(i=1,2,\dots,n-1\).

For \(n=3\), Qiu [12] studied (2) under the assumption \(\int _{t_{0}}^{\infty }1/r_{1}(t)\Delta t=\int _{t_{0}}^{\infty }1/r _{2}(t)\Delta t=\infty \) and summarized five cases for the eventually positive solutions to (2). To obtain the existence and asymptotic behavior of nonoscillatory solutions to (2) for other cases of \(\int _{t_{0}}^{\infty }1/r_{i}(t)\Delta t\), \(i=1,2\), Qiu et al. [16] were concerned with (2) under \(\int _{t_{0}}^{\infty }1/r_{1}(t)\Delta t=\infty \) and \(\int _{t_{0}} ^{\infty }1/r_{2}(t)\Delta t<\infty \), while the case that \(\int _{t _{0}}^{\infty }1/r_{1}(t)\Delta t<\infty \) and \(\int _{t_{0}}^{\infty }1/r_{2}(t)\Delta t=\infty \) was considered in Qiu et al. [14]. There are four cases for the eventually positive solutions to (2). However, the existence and asymptotic behavior of the solutions are evidently different between [14] and [16].

Furthermore, for \(n\geq 3\), Qiu and Wang [15] investigated (2) under the condition \(\int _{t_{0}}^{\infty }1/r_{i}(t) \Delta t<\infty \), \(i=1,2,\dots,n-1\), and deduced that every eventually positive solution converges to a positive constant or zero, which complements and unites the results in [5, 19]. Qiu et al. [13] continued to study (2) with \(\int _{t_{0}} ^{\infty }1/r_{1}(t)\Delta t=\infty \) and \(\int _{t_{0}}^{\infty }1/r _{i}(t)\Delta t<\infty \), \(i=2,3,\dots,n-1\). Four cases for the eventually positive solutions have also been presented, and the results are consistent with those in [16] when \(n=3\).

In this paper, we consider the existence of nonoscillatory solutions to (1), which is (2) for \(n=4\), under the following conditions:

  1. (C1)

    \(r_{1}, r_{2}, r_{3}\in {\mathrm{C_{rd}}}([t_{0},\infty )_{ \mathbb{T}}, (0,\infty ))\) and there exist two positive constants \(M_{1}\) and \(M_{3}\) such that

    $$\begin{aligned} \int _{t_{0}}^{\infty }\frac{\Delta t}{r_{i}(t)}=M_{i}< \infty,\quad i=1,3\quad \text{and}\quad \int _{t_{0}}^{\infty }\frac{\Delta t}{r _{2}(t)}=\infty; \end{aligned}$$
  2. (C2)

    \(p\in {\mathrm{C_{rd}}}([t_{0},\infty )_{\mathbb{T}}, \mathbb{R})\) and \(-1<\lim_{t \rightarrow \infty }p(t)=p_{0}<1\);

  3. (C3)

    \(g,h\in {\mathrm{C_{rd}}}([t_{0},\infty )_{\mathbb{T}}, \mathbb{T})\), \(g(t)\leq t\), and \(\lim_{t\rightarrow \infty }g(t)= \lim_{t \rightarrow \infty }h(t)=\infty \); if \(p_{0}\in (-1,0]\), then there exists a sequence \(\{c_{k}\}_{k\geq 0}\) such that \(\lim_{k \rightarrow \infty }c_{k}=\infty \) and \(g(c_{k+1})=c_{k}\);

  4. (C4)

    \(f\in {\mathrm{C}}([t_{0},\infty )_{\mathbb{T}}\times \mathbb{R}, \mathbb{R})\), \(f(t,x)\) is nondecreasing in x, and \(xf(t,x)>0\) for \(x\neq 0\);

  5. (C5)

    if

    $$ \int _{t_{0}}^{\infty } \int _{t_{0}}^{u_{3}}\frac{\Delta u_{2}\Delta u _{3}}{r_{2}(u_{2})r_{3}(u_{3})}=\infty, $$
    (4)

    then we define

    $$\begin{aligned} R(t)=1+ \int _{t_{0}}^{t} \int _{t_{0}}^{u_{3}}\frac{\Delta u_{2}\Delta u _{3}}{r_{2}(u_{2})r_{3}(u_{3})}, \end{aligned}$$

    where \(\lim_{t\rightarrow \infty }R(g(t))/R(t)=\eta \in (0,1]\) is satisfied.

In view of (C1), it is clear that the results in the references are not available for (1). The conclusions in this paper can bring a deeper understanding of the existence and asymptotic behavior of nonoscillatory solutions to (2). In addition, we provide two significant examples to illustrate our results.

Auxiliary results

Firstly, we state Krasnoselskii’s fixed point theorem (see [3]) as follows, which will be used in the sequel.

Lemma 2.1

Suppose thatXis a Banach space andΩis a bounded, convex, and closed subset ofX. If there exist two operators \(U,V:\varOmega \rightarrow X\)such that \(Ux+Vy\in \varOmega \)for all \(x,y\in \varOmega \), Uis a contraction mapping, andVis completely continuous, then \(U+V\)has a fixed point inΩ.

Then, we have another lemma to show the relationship between the functions z and x. The proof is omitted since it is similar to those in [4, Lemma 2.3], [5, Theorem 1], [15, Lemma 2.3], and [19, Theorem 7].

Lemma 2.2

Suppose thatxis an eventually positive solution to (1) and

$$\begin{aligned} \lim_{t\rightarrow \infty }\frac{z(t)}{R^{\lambda }(t)}=a, \quad\lambda =0,1, \end{aligned}$$

where \(\lambda =1\)only if (4) holds. Ifais finite, then we have

$$\begin{aligned} \lim_{t\rightarrow \infty }\frac{x(t)}{R^{\lambda }(t)}=\frac{a}{1+p _{0}\eta ^{\lambda }}, \end{aligned}$$

or \(\limsup_{t\rightarrow \infty }x(t)/R^{\lambda }(t)=\infty \).

Finally, we need to divide all the eventually positive solutions to (1) into four groups for the sake of simplicity. In addition, define

$$\begin{aligned} A(\alpha )= \biggl\{ x\in S: \lim_{t\rightarrow \infty }x(t)=\infty \text{ and } \lim_{t\rightarrow \infty }\frac{x(t)}{R(t)}= \alpha \biggr\} , \end{aligned}$$

where S is the set of all eventually positive solutions to (1).

Theorem 2.3

Suppose thatxis an eventually positive solution to (1). Then the solutionxbelongs to one of the following four cases:

  1. (A1)

    \(\lim_{t\rightarrow \infty }x(t)=0\);

  2. (A2)

    \(\lim_{t\rightarrow \infty }x(t)=b\), wherebis a positive constant;

  3. (A3)

    \(x\in A(b)\), wherebis a positive constant;

  4. (A4)

    \(\limsup_{t\rightarrow \infty }x(t)=\infty \)and \(\lim_{t\rightarrow \infty }x(t)/R(t)=0\).

Proof

In view of (C2) and (C3), for any eventually positive solution x to (1), there always exist \(t_{1}\in [t_{0}, \infty )_{\mathbb{T}}\) and \(p_{1}\) with \(|p_{0}|< p_{1}<1\) such that \(x(t)>0\), \(x(g(t))>0\), \(x(h(t))>0\), and \(|p(t)|\leq p_{1}\), \(t\in [t_{1},\infty )_{\mathbb{T}}\). From (1) and (C4), for \(t\in [t_{1},\infty )_{\mathbb{T}}\), we have

$$\begin{aligned} R_{4}\bigl(t,x(t)\bigr)=R_{3}^{\Delta } \bigl(t,x(t)\bigr)=-f\bigl(t,x\bigl(h(t)\bigr)\bigr)< 0, \end{aligned}$$

which means that \(R_{3}\) is strictly decreasing on \([t_{1},\infty )_{ \mathbb{T}}\). Hence, it follows that

$$ R_{2}^{\Delta }\bigl(t,x(t)\bigr)\leq \frac{r_{1}(t_{1})R_{2}^{\Delta }(t_{1},x(t _{1}))}{r_{1}(t)},\quad t\in [t_{1},\infty )_{\mathbb{T}}. $$
(5)

From (5) we deduce that there are two cases for \(R_{2}\). Assume that there exists \(T\in [t_{1},\infty )_{\mathbb{T}}\) satisfying \(R_{2}^{\Delta }(T,x(T))\leq 0\), then it is clear that \(R_{2}^{\Delta }\) is eventually negative. If not, then we get \(R_{2}^{\Delta }(t,x(t))>0\), \(t\in [t_{1},\infty )_{\mathbb{T}}\). That is, \(R_{2}\) is always eventually monotonic. Substituting s for t in (5) and integrating (5) from \(t_{1}\) to t, where \(t\in [\sigma (t_{1}),\infty )_{\mathbb{T}}\), we obtain

$$\begin{aligned} R_{2}\bigl(t,x(t)\bigr)-R_{2}\bigl(t_{1},x(t_{1}) \bigr) &\leq r_{1}(t_{1})R_{2}^{\Delta } \bigl(t_{1},x(t_{1})\bigr) \int _{t_{1}}^{t}\frac{\Delta s}{r_{1}(s)} \\ & < r_{1}(t_{1}) \bigl\vert R_{2}^{\Delta } \bigl(t_{1},x(t_{1})\bigr) \bigr\vert \cdot M_{1}, \end{aligned}$$

which implies that \(R_{2}\) is upper bounded. Therefore, \(R_{2}\) (or \(r_{2}R_{1}^{\Delta }\)) is eventually monotonic and upper bounded. There are two cases to be discussed.

Case 1. \(r_{2}R_{1}^{\Delta }\) is eventually decreasing. It follows that

$$\begin{aligned} -\infty \leq \lim_{t\rightarrow \infty }r_{2}(t)R_{1}^{\Delta } \bigl(t,x(t)\bigr)=L _{1}< \infty. \end{aligned}$$

(a) If \(-\infty \leq L_{1}<0\), then there exist \(t_{2}\in [t_{1}, \infty )_{\mathbb{T}}\) and a constant \(a_{1}<0\) such that \(r_{2}(t)R _{1}^{\Delta }(t,x(t))\leq a_{1}\) or

$$ R_{1}^{\Delta }\bigl(t,x(t)\bigr)\leq \frac{a_{1}}{r_{2}(t)},\quad t\in [t_{2}, \infty )_{\mathbb{T}}. $$
(6)

Letting t be replaced by s in (6) and integrating (6) from \(t_{2}\) to t, where \(t\in [\sigma (t_{2}), \infty )_{\mathbb{T}}\), we derive

$$\begin{aligned} r_{3}(t)z^{\Delta }(t)=R_{1}\bigl(t,x(t)\bigr) \leq R_{1}\bigl(t_{2},x(t_{2}) \bigr)+a_{1} \int _{t_{2}}^{t}\frac{\Delta s}{r_{2}(s)}\rightarrow - \infty,\quad t \rightarrow \infty, \end{aligned}$$

which means that \(z^{\Delta }\) is eventually negative, and thus z is eventually strictly decreasing. It is easy to see that z is eventually positive or eventually negative, but we can claim that z is eventually positive in terms of (C3). Assume not; then we have \(-1< p_{0}\leq 0\). Moreover, there exists \(t_{3}\in [t_{2},\infty )_{\mathbb{T}}\) such that \(x(t)<-p(t)x(g(t))\leq p_{1}x(g(t))\) for \(t\in [t_{3},\infty )_{ \mathbb{T}}\). Choose a positive integer N satisfying \(c_{k}\in [t _{3},\infty )_{\mathbb{T}}\) for all \(k\geq N\). For any \(k>N\), we always have

$$\begin{aligned} x(c_{k})< p_{1}x(c_{k-1})< p_{1}^{2}x(c_{k-2})< \cdots < p_{1}^{k-N}x(c _{N}), \end{aligned}$$

which implies that \(\lim_{k\rightarrow \infty }x(c_{k})= \lim_{k\rightarrow \infty }z(c_{k})=0\). It causes a contradiction, since z is eventually strictly decreasing and eventually negative. Therefore, we deduce

$$\begin{aligned} 0\leq \lim_{t\rightarrow \infty }z(t)=L_{0}< \infty. \end{aligned}$$

By virtue of Lemma 2.2, it follows that case (A1) or (A2) holds.

(b) If \(0< L_{1}<\infty \), then there exist \(t_{4}\in [t_{1},\infty )_{ \mathbb{T}}\) and a constant \(a_{2}>0\) such that \(r_{2}(t)R_{1}^{ \Delta }(t,x(t))\geq a_{2}\) or

$$\begin{aligned} R_{1}^{\Delta }\bigl(t,x(t)\bigr)\geq \frac{a_{2}}{r_{2}(t)},\quad t\in [t_{4}, \infty )_{\mathbb{T}}. \end{aligned}$$

Similarly, we have

$$\begin{aligned} r_{3}(t)z^{\Delta }(t)=R_{1}\bigl(t,x(t)\bigr) \geq R_{1}\bigl(t_{4},x(t_{4}) \bigr)+a_{2} \int _{t_{4}}^{t}\frac{\Delta s}{r_{2}(s)}\rightarrow \infty,\quad t \rightarrow \infty. \end{aligned}$$

It means that \(z^{\Delta }\) is eventually positive and thus z is eventually strictly increasing. Similar to the proof in (a), we derive

$$ 0\leq \lim_{t\rightarrow \infty }z(t)=L_{0}\leq \infty. $$
(7)

If \(L_{0}=\infty \), then we deduce

$$\begin{aligned} \lim_{t\rightarrow \infty }r_{2}(t)R_{1}^{\Delta } \bigl(t,x(t)\bigr)= \lim_{t\rightarrow \infty }\frac{z(t)}{R(t)}=L_{1} \end{aligned}$$

according to L’Hôpital’s rule (see [24, Theorem 1.120]). By Lemma 2.2 we know that one of cases (A1)–(A3) holds.

(c) If \(L_{1}=0\), since \(r_{2}R_{1}^{\Delta }\) is eventually strictly decreasing, then \(r_{2}R_{1}^{\Delta }\) and \(R_{1}^{\Delta }\) are both eventually positive, and so \(R_{1}\) (or \(r_{3}z^{\Delta }\)) is eventually strictly increasing. It follows that \(r_{3}z^{\Delta }\) and \(z^{\Delta }\) are both eventually positive or eventually negative, and z is always eventually monotonic, which implies that z is eventually positive or eventually negative. Similarly, we also obtain (7). In view of (C3), we see that one of cases (A1), (A2), and (A4) holds.

Case 2. \(r_{2}R_{1}^{\Delta }\) is eventually increasing. It follows that

$$\begin{aligned} -\infty < \lim_{t\rightarrow \infty }r_{2}(t)R_{1}^{\Delta } \bigl(t,x(t)\bigr)=L _{1}< \infty. \end{aligned}$$

(a) If \(-\infty < L_{1}<0\), then there also exist \(t_{5}\in [t_{1}, \infty )_{\mathbb{T}}\) and a constant \(a_{3}<0\) such that \(r_{2}(t)R _{1}^{\Delta }(t,x(t))\leq a_{3}\) for \(t\in [t_{5},\infty )_{ \mathbb{T}}\). Similar to the proof in Case 1, we get one of cases (A1) or (A2) holds.

(b) If \(0< L_{1}<\infty \), similarly, then one of cases (A1)–(A3) holds.

(c) If \(L_{1}=0\), since \(r_{2}R_{1}^{\Delta }\) is eventually strictly increasing, then \(r_{2}R_{1}^{\Delta }\) and \(R_{1}^{\Delta }\) are both eventually negative. Hence, \(R_{1}\) (or \(r_{3}z^{\Delta }\)) is eventually strictly decreasing. Similarly, we always have (7), and one of cases (A1), (A2), and (A4) holds.

The proof is complete. □

Sufficient conditions

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 solutionxwith \(\lim_{t\rightarrow \infty }x(t)=b\), wherebis 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 \(x\in A(b)\), wherebis 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 constantMsatisfying 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.

Necessary conditions

Some necessary conditions for the existence of eventually positive solutions to (1) are provided in this section, where an additional assumption is needed as follows:

$$ \int _{t_{0}}^{\infty } \int _{t_{0}}^{u_{3}} \int _{t_{0}}^{u_{2}} \frac{ \Delta u_{1}\Delta u_{2}\Delta u_{3}}{r_{1}(u_{1})r_{2}(u_{2})r_{3}(u _{3})}< \infty. $$
(12)

Theorem 4.1

Assume that (12) holds and

$$ \int _{t_{0}}^{\infty } \int _{t_{0}}^{u_{3}}\frac{\Delta u_{2}\Delta u _{3}}{r_{2}(u_{2})r_{3}(u_{3})}< \infty. $$
(13)

If (1) has an eventually positive solutionxwith \(\lim_{t\rightarrow \infty }x(t)=b\), wherebis a positive constant, then there exists some constant \(K>0\)satisfying (8).

Proof

Suppose that (12) and (13) hold, and (1) has an eventually positive solution x with \(\lim_{t\rightarrow \infty }x(t)=b>0\). It follows that \(\lim_{t\rightarrow \infty }z(t)=(1+p_{0})b\), and there exists \(t_{1}\in [t_{0},\infty )_{\mathbb{T}}\) such that \(x(t)>0\), \(x(g(t))>0\), and \(x(h(t))\geq b/2\) for \(t\in [t_{1},\infty )_{ \mathbb{T}}\). Letting t be replaced by \(u_{0}\) in (1) and integrating (1) from \(t_{1}\) to \(u_{1}\), where \(u_{1} \in [\sigma (t_{1}),\infty )_{\mathbb{T}}\), we have

$$\begin{aligned} R_{3}\bigl(u_{1},x(u_{1}) \bigr)=R_{3}\bigl(t_{1},x(t_{1})\bigr)- \int _{t_{1}}^{u_{1}}f\bigl(u _{0},x \bigl(h(u_{0})\bigr)\bigr)\Delta u_{0} \end{aligned}$$

and thus

$$ R_{2}^{\Delta }\bigl(u_{1},x(u_{1}) \bigr)=\frac{R_{3}(t_{1},x(t_{1}))}{r_{1}(u _{1})}- \int _{t_{1}}^{u_{1}}\frac{f(u_{0},x(h(u_{0})))}{r_{1}(u_{1})} \Delta u_{0}. $$
(14)

Integrating (14) with respect to \(u_{1}\) from \(t_{1}\) to \(u_{2}\), where \(u_{2}\in [\sigma (t_{1}),\infty )_{\mathbb{T}}\), we have

$$\begin{aligned} R_{2}\bigl(u_{2},x(u_{2}) \bigr)={}&R_{2}\bigl(t_{1},x(t_{1}) \bigr)+R_{3}\bigl(t_{1},x(t_{1})\bigr) \int _{t_{1}}^{u_{2}}\frac{\Delta u_{1}}{r_{1}(u_{1})} \\ &{} - \int _{t_{1}}^{u_{2}} \int _{t_{1}}^{u_{1}}\frac{f(u_{0},x(h(u_{0})))}{r _{1}(u_{1})}\Delta u_{0}\Delta u_{1}. \end{aligned}$$
(15)

By analogy, we deduce

$$\begin{aligned} z(t)={}&z(t_{1})+R_{1} \bigl(t_{1},x(t_{1})\bigr) \int _{t_{1}}^{t}\frac{\Delta u _{3}}{r_{3}(u_{3})} +R_{2}\bigl(t_{1},x(t_{1})\bigr) \int _{t_{1}}^{t} \int _{t_{1}} ^{u_{3}}\frac{\Delta u_{2}\Delta u_{3}}{r_{2}(u_{2})r_{3}(u_{3})} \\ &{} +R_{3}\bigl(t_{1},x(t_{1})\bigr) \int _{t_{1}}^{t} \int _{t_{1}}^{u_{3}} \int _{t_{1}} ^{u_{2}}\frac{\Delta u_{1}\Delta u_{2}\Delta u_{3}}{r_{1}(u_{1})r_{2}(u _{2})r_{3}(u_{3})} \\ &{}- \int _{t_{1}}^{t} \int _{t_{1}}^{u_{3}} \int _{t_{1}}^{u_{2}} \int _{t_{1}} ^{u_{1}}\frac{f(u_{0},x(h(u_{0})))}{r_{1}(u_{1})r_{2}(u_{2})r_{3}(u _{3})}\Delta u_{0}\Delta u_{1}\Delta u_{2}\Delta u_{3}. \end{aligned}$$
(16)

Letting \(t\rightarrow \infty \), in terms of (C1), (C4), (12), (13), and the fact that \(x(h(t)) \geq b/2\) for \(t\in [t_{1},\infty )_{\mathbb{T}}\), we conclude

$$\begin{aligned} \int _{t_{1}}^{\infty } \int _{t_{1}}^{u_{3}} \int _{t_{1}}^{u_{2}} \int _{t_{1}}^{u_{1}} \frac{f(u_{0},b/2)}{r_{1}(u_{1})r_{2}(u_{2})r_{3}(u _{3})}\Delta u_{0}\Delta u_{1}\Delta u_{2}\Delta u_{3}< \infty, \end{aligned}$$

which means that (8) holds. The proof is complete. □

Here is a lemma to present a sufficient condition to ensure that (4) is satisfied, and then Theorems 4.3 and 4.4 follow.

Lemma 4.2

Assume that (12) holds. If (1) has an eventually positive solutionxwith \(\lim_{t\rightarrow \infty }x(t)=\infty \), then (4) holds.

Proof

If (1) has an eventually positive solution x with \(\lim_{t\rightarrow \infty }x(t)=\infty \), then we claim that \(\lim_{t\rightarrow \infty }z(t)=\infty \). Assume not; by Theorem 2.3 we have \(0\leq \lim_{t\rightarrow \infty }z(t)<\infty \), which implies that \(0\leq \lim_{t\rightarrow \infty }x(t)<\infty \) in view of Lemma 2.2. Hence, we deduce \(\lim_{t\rightarrow \infty }z(t)=\infty \). Similar to the proof in Theorem 4.1, we arrive at (16). Letting \(t\rightarrow \infty \), from (C1) and (12) we obtain (4) and complete the proof. □

Theorem 4.3

Assume that (12) holds. If (1) has an eventually positive solution \(x\in A(b)\), wherebis a positive constant, then there exists some constant \(K>0\)satisfying (9).

Proof

Suppose that (12) holds and (1) has an eventually positive solution \(x\in A(b)\), where b is a positive constant, then (4) holds according to Lemma 4.2. Define R as in (C5), by Lemma 2.2 and Theorem 2.3 we obtain

$$\begin{aligned} \lim_{t\rightarrow \infty }z(t)=\infty \quad\text{and}\quad \lim _{t\rightarrow \infty }R_{2}\bigl(t,x(t)\bigr)=\lim _{t\rightarrow \infty } \frac{z(t)}{R(t)}=(1+p_{0}\eta )b. \end{aligned}$$

Then there exists \(t_{1}\in [t_{0},\infty )_{\mathbb{T}}\) such that \(x(t)>0\), \(x(g(t))>0\), and \(x(h(t))\geq bR(h(t))/2\) for \(t\in [t_{1}, \infty )_{\mathbb{T}}\). Proceeding as the proof in Theorem 4.1, we get (15). Letting \(u_{2}\rightarrow \infty \), by (C1) and (C4) we conclude

$$\begin{aligned} \int _{t_{1}}^{\infty } \int _{t_{1}}^{u_{1}}\frac{f(u_{0},bR(h(u_{0}))/2)}{r _{1}(u_{1})}\Delta u_{0}\Delta u_{1}< \infty, \end{aligned}$$

which implies that (9) holds. The proof is complete. □

Theorem 4.4

Assume that (12) holds. If (1) has an eventually positive solution \(x\in A(0)\), then we have

$$ \int _{t_{0}}^{\infty } \int _{t_{0}}^{u_{1}}\frac{f(u_{0},3/4)}{r_{1}(u _{1})}\Delta u_{0}\Delta u_{1}< \infty $$
(17)

and

$$ \int _{t_{0}}^{\infty } \int _{t_{0}}^{u_{3}} \int _{u_{2}}^{\infty } \int _{t_{0}}^{u_{1}} \frac{f(u_{0},R(h(u_{0})))}{r_{1}(u_{1})r_{2}(u _{2})r_{3}(u_{3})}\Delta u_{0}\Delta u_{1}\Delta u_{2}\Delta u_{3}= \infty. $$
(18)

Proof

Suppose that (12) holds and (1) has an eventually positive solution \(x\in A(0)\). It follows that (4) holds, and we define R as in (C5), then we obtain

$$\begin{aligned} \lim_{t\rightarrow \infty }z(t)=\infty \quad\text{and}\quad \lim _{t\rightarrow \infty }R_{2}\bigl(t,x(t)\bigr)=\lim _{t\rightarrow \infty } \frac{z(t)}{R(t)}=0. \end{aligned}$$

There exists \(t_{1}\in [t_{0},\infty )_{\mathbb{T}}\) such that \(3/4\leq x(t)\leq R(t)\), \(3/4\leq x(g(t))\leq R(g(t))\), and \(3/4\leq x(h(t))\leq R(h(t))\) for \(t\in [t_{1},\infty )_{\mathbb{T}}\). Similar to the proof in Theorem 4.1, we get (14). Integrating (14) with respect to \(u_{1}\) from \(t_{2}\) to t, where \(t_{2}\in [\sigma (t_{1}),\infty )_{\mathbb{T}}\) and \(t\in [\sigma (t_{2}),\infty )_{\mathbb{T}}\), we have

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

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

$$\begin{aligned} R_{2}\bigl(t_{2},x(t_{2}) \bigr)={}&{-}R_{3}\bigl(t_{1},x(t_{1})\bigr) \int _{t_{2}}^{\infty }\frac{ \Delta u_{1}}{r_{1}(u_{1})} \\ &{} + \int _{t_{2}}^{\infty } \int _{t_{1}}^{u_{1}}\frac{f(u_{0},x(h(u_{0})))}{r _{1}(u_{1})}\Delta u_{0}\Delta u_{1}. \end{aligned}$$
(19)

According to (C1) and (C4), it follows that

$$\begin{aligned} \int _{t_{2}}^{\infty } \int _{t_{1}}^{u_{1}}\frac{f(u_{0},3/4)}{r_{1}(u _{1})}\Delta u_{0}\Delta u_{1}< \infty, \end{aligned}$$

which implies that (17) holds. Substituting \(u_{2}\) for \(t_{2}\) in (19) and integrating (19) with respect to \(u_{2}\) from \(t_{1}\) to \(u_{3}\), where \(u_{3}\in [\sigma (t _{1}),\infty )_{\mathbb{T}}\), we have

$$\begin{aligned} R_{1}\bigl(u_{3},x(u_{3}) \bigr)={}&R_{1}\bigl(t_{1},x(t_{1}) \bigr)-R_{3}\bigl(t_{1},x(t_{1})\bigr) \int _{t_{1}}^{u_{3}} \int _{u_{2}}^{\infty }\frac{\Delta u_{1}\Delta u_{2}}{r _{1}(u_{1})r_{2}(u_{2})} \\ &{} + \int _{t_{1}}^{u_{3}} \int _{u_{2}}^{\infty } \int _{t_{1}}^{u_{1}}\frac{f(u _{0},x(h(u_{0})))}{r_{1}(u_{1})r_{2}(u_{2})}\Delta u_{0}\Delta u_{1} \Delta u_{2}. \end{aligned}$$

Similarly, it follows that

$$\begin{aligned} z(t)={}&z(t_{1})+R_{1}\bigl(t_{1},x(t_{1}) \bigr) \int _{t_{1}}^{t}\frac{\Delta u _{3}}{r_{3}(u_{3})} \\ &{} -R_{3}\bigl(t_{1},x(t_{1})\bigr) \int _{t_{1}}^{t} \int _{t_{1}}^{u_{3}} \int _{u_{2}} ^{\infty } \frac{\Delta u_{1}\Delta u_{2}\Delta u_{3}}{r_{1}(u_{1})r _{2}(u_{2})r_{3}(u_{3})} \\ &{}+ \int _{t_{1}}^{t} \int _{t_{1}}^{u_{3}} \int _{u_{2}}^{\infty } \int _{t _{1}}^{u_{1}}\frac{f(u_{0},x(h(u_{0})))}{r_{1}(u_{1})r_{2}(u_{2})r _{3}(u_{3})}\Delta u_{0}\Delta u_{1}\Delta u_{2}\Delta u_{3}. \end{aligned}$$

Letting \(t\rightarrow \infty \), we conclude that (18) holds and complete the proof. □

Now, we can present some necessary and sufficient conditions for the existence of eventually positive solutions to (1). In fact, according to Theorems 3.1 and 4.1, we obtain the following corollary.

Corollary 4.5

Assume that (12) and (13) hold. Then (1) has an eventually positive solutionxwith \(\lim_{t\rightarrow \infty }x(t)=b>0\)if and only if there exists some constant \(K>0\)satisfying (8).

Similarly, we get another corollary in terms of Theorems 3.2 and 4.3.

Corollary 4.6

Assume that (4) holds. Then (1) has an eventually positive solution \(x\in A(b)\)if and only if there exists some constant \(K>0\)satisfying (9), wherebis a positive constant.

Examples

To illustrate the applications of the conclusions in this paper, two interesting examples are presented as follows.

Example 5.1

Let \(\mathbb{T}=\bigcup_{n=0}^{\infty }[3^{n},2 \cdot 3^{n}]\). For \(t\in [3,\infty )_{\mathbb{T}}\), consider

$$ \biggl(t^{3} \biggl(\frac{1}{t^{\alpha }} \biggl(t^{\beta } \biggl(x(t)-\frac{t+ \cos t}{3t}x \biggl( \frac{t}{3} \biggr) \biggr)^{\Delta } \biggr)^{ \Delta } \biggr)^{\Delta } \biggr)^{\Delta } +t^{2}x(3t)=0. $$
(20)

Here, \(r_{1}(t)=t^{3}\), \(r_{2}(t)=1/t^{\alpha }\), \(r_{3}(t)=t^{\beta }\), \(p(t)=-(t+\cos t)/(3t)\), \(g(t)=t/3\), \(h(t)=3t\), \(f(t,x)=t^{2}x\), \(t_{0}=3\), and \(p_{0}=-1/3\), where \(\alpha \geq 0\) and \(\beta >\alpha +3\). It is clear that conditions (C1)–(C4) are satisfied. Moreover, we have

$$\begin{aligned} & \int _{3}^{\infty } \int _{3}^{u_{3}} \int _{3}^{u_{2}} \int _{3} ^{u_{1}} \frac{f(u_{0},1)}{r_{1}(u_{1})r_{2}(u_{2})r_{3}(u_{3})} \Delta u_{0}\Delta u_{1}\Delta u_{2}\Delta u_{3} \\ &\quad= \int _{3}^{\infty } \int _{3}^{u_{3}} \int _{3}^{u_{2}} \int _{3}^{u_{1}} \frac{u_{0}^{2}u_{2}^{\alpha }}{u_{1}^{3}u_{3}^{\beta }}\Delta u_{0} \Delta u_{1}\Delta u_{2}\Delta u_{3} \\ &\quad< \frac{1}{3} \int _{3}^{\infty } \int _{3}^{u_{3}} \int _{3}^{u_{2}}\frac{u _{2}^{\alpha }}{u_{3}^{\beta }}\Delta u_{1}\Delta u_{2}\Delta u_{3}< \frac{1}{3} \int _{3}^{\infty } \int _{3}^{u_{3}}\frac{u_{2}^{\alpha +1}}{u _{3}^{\beta }}\Delta u_{2}\Delta u_{3} \\ &\quad< \frac{1}{3(\alpha +2)} \int _{3}^{\infty }\frac{\Delta u_{3}}{u_{3} ^{\beta -\alpha -2}}< \infty. \end{aligned}$$

According to Theorem 3.1, we conclude that (20) has an eventually positive solution x with \(\lim_{t\rightarrow \infty }x(t)=b>0\).

Example 5.2

Let \(\mathbb{T}=[1,\infty )_{\mathbb{R}}\). For \(t\in [2,\infty )_{\mathbb{T}}\), consider

$$ \biggl(t^{\lambda } \biggl(\frac{1}{t} \biggl(t^{2} \biggl(x(t)+ \frac{1}{2t}x(t-1) \biggr)^{\prime } \biggr)^{\prime } \biggr)^{\prime } \biggr)^{\prime } +\frac{x(t)}{t ^{2}}=0. $$
(21)

Here, we have \(r_{1}(t)=t^{\lambda }\), \(r_{2}(t)=1/t\), \(r_{3}(t)=t ^{2}\), \(p(t)=1/(2t)\), \(g(t)=t-1\), \(h(t)=t\), \(f(t,x)=x/t^{2}\), \(t_{0}=2\), and \(p_{0}=0\), where \(1<\lambda \leq 2\). It is obvious that conditions (C1)–(C4) and (4) are satisfied. From (C5), we have

$$\begin{aligned} R(t)=1+ \int _{2}^{t} \int _{2}^{u_{3}}\frac{du_{2}\,du_{3}}{r_{2}(u_{2})r _{3}(u_{3})} =1+ \int _{2}^{t} \int _{2}^{u_{3}}\frac{u_{2}}{u_{3}^{2}}\,du _{2}\,du_{3}=\frac{t}{2}-1+ \frac{2}{t}, \end{aligned}$$

which satisfies that \(\eta =\lim_{t \rightarrow \infty }R(g(t))/R(t)=1\). Moreover, we deduce

$$\begin{aligned} &\bigl\vert p(t)R(t) \bigr\vert =\frac{1}{2t}\cdot \biggl( \frac{t}{2}-1+\frac{2}{t} \biggr) \leq M=\frac{1}{4},\quad t\in [2,\infty )_{\mathbb{T}}, \\ & \int _{2}^{\infty } \int _{2}^{u_{1}}\frac{f(u_{0},R(h(u_{0})))}{r _{1}(u_{1})} \,du_{0}\,du_{1}\\ &\quad = \int _{2}^{\infty } \int _{2}^{u_{1}}\frac{u _{0}/2-1+2/u_{0}}{u_{0}^{2}u_{1}^{\lambda }} \,du_{0}\,du_{1} \\ &\quad < \frac{1}{2} \int _{2}^{\infty } \int _{2}^{u_{1}}\frac{1}{u_{0}u_{1} ^{\lambda }} \,du_{0}\,du_{1} < \frac{1}{2} \int _{2}^{\infty }\frac{\ln u _{1}}{u_{1}^{\lambda }} \,du_{1}< \infty, \end{aligned}$$

and

$$\begin{aligned} & \int _{2}^{\infty } \int _{2}^{u_{3}} \int _{u_{2}}^{\infty } \int _{2}^{u_{1}} \frac{f(u_{0},M+3/4)}{r_{1}(u_{1})r_{2}(u_{2})r_{3}(u _{3})} \,du_{0}\,du_{1}\,du_{2} \,du_{3} \\ &\quad = \int _{2}^{\infty } \int _{2}^{u_{3}} \int _{u_{2}}^{\infty } \int _{2} ^{u_{1}} \frac{u_{2}}{u_{0}^{2}u_{1}^{\lambda }u_{3}^{2}} \,du_{0}\,du_{1}\,du _{2} \,du_{3}=\infty. \end{aligned}$$

By virtue of Theorems 3.2 and 3.3, we conclude that (21) has two eventually positive solutions \(x_{1}\in A(b)\) for some positive constant b and \(x_{2}\in A(0)\).

On the other hand, if we take \(r_{3}(t)=t^{4}\) and \(f(t,x)=t\cdot x\), but other functions remain unchanged in (21), then (C1)–(C4) are still satisfied. In addition, we derive

$$\begin{aligned} \int _{2}^{\infty } \int _{2}^{u_{3}} \int _{2}^{u_{2}} \frac{u_{2}}{u_{1} ^{\lambda }u_{3}^{4}} \,du_{1}\,du_{2}\,du_{3}< \infty \quad\text{and}\quad \int _{2}^{\infty } \int _{2}^{u_{3}}\frac{u_{2}}{u_{3}^{4}} \,du_{2}\,du _{3}< \infty, \end{aligned}$$

which means that both of (12) and (13) hold. However, for all \(K>0\), we always have

$$\begin{aligned} & \int _{2}^{\infty } \int _{2}^{u_{3}} \int _{2}^{u_{2}} \int _{2} ^{u_{1}} \frac{f(u_{0},K)}{r_{1}(u_{1})r_{2}(u_{2})r_{3}(u_{3})}\,du _{0}\,du_{1}\,du_{2}\,du_{3} \\ &\quad =K \int _{2}^{\infty } \int _{2}^{u_{3}} \int _{2}^{u_{2}} \int _{2}^{u_{1}} \frac{u_{0}u_{2}}{u_{1}^{\lambda }u_{3}^{4}} \,du_{0}\,du_{1}\,du_{2} \,du_{3}= \infty, \end{aligned}$$

which implies that (8) is not satisfied. Therefore, we deduce that (21) has no eventually positive solution tending to a positive constant in terms of Theorem 4.1 or Corollary 4.5.

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Acknowledgements

The author thanks the anonymous referees for their valuable suggestions.

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This project was supported by the National Natural Science Foundation of China (11671406) and the Natural Science Program for Young Creative Talents of Innovation Enhancing College Project of Department of Education of Guangdong Province (2017GKQNCX111 and 2018-KJZX039).

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Qiu, YC. Nonoscillatory solutions to fourth-order neutral dynamic equations on time scales. Adv Differ Equ 2019, 512 (2019). https://doi.org/10.1186/s13662-019-2451-3

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MSC

  • 34N05
  • 34C10
  • 39A13

Keywords

  • Nonoscillatory solution
  • Neutral dynamic equation
  • Fourth-order
  • Time scale