Abstract
We study the existence of nonoscillatory solutions tending to zero of a class of third-order nonlinear neutral dynamic equations on time scales by employing Krasnoselskii’s fixed point theorem. Two examples are given to illustrate the significance of the conclusions.
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1 Introduction
In this paper, we consider the existence of nonoscillatory solutions tending to zero of a class of third-order nonlinear neutral dynamic equations
on a time scale \(\mathbb{T}\) satisfying \(\sup \mathbb{T}=\infty \), where \(t\in [t_{0},\infty )_{\mathbb{T}}=[t_{0},\infty )\cap {\mathbb{T}}\) with \(t_{0}\in \mathbb{T}\). The following conditions are assumed to hold throughout this paper:
- (C1)
\(r_{1}, r_{2}\in \mathrm{ C}_{\mathrm{rd}}([t_{0},\infty )_{\mathbb{T}}, (0, \infty ))\);
- (C2)
\(p\in \mathrm{ C}_{\mathrm{rd}}([t_{0},\infty )_{\mathbb{T}}, \mathbb{R})\) and \(\lim_{t \rightarrow \infty }p(t)=p_{0}\), where \(|p_{0}|<1\);
- (C3)
\(g,h\in \mathrm{ C}_{\mathrm{rd}}([t_{0},\infty )_{\mathbb{T}}, \mathbb{T})\), \(g(t)\leq t\), 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 \(g(c_{k+1})=c_{k}\);
- (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 \(x\neq 0\).
The details of the theory of time scales can be found in [1–4, 8, 9] and hence they are omitted here. In recent years, the existence of nonoscillatory solutions of neutral dynamic equations on time scales has been studied successively in [6, 7, 11, 13–17]. Zhu and Wang [17] were concerned with a first-order neutral dynamic equation
Afterward, Deng and Wang [6] and Gao and Wang [7] investigated a second-order neutral dynamic equation
under the different assumptions \(\int _{t_{0}}^{\infty }1/r(t)\Delta t=\infty \) and \(\int _{t_{0}}^{\infty }1/r(t)\Delta t<\infty \), respectively. Furthermore, Qiu [11] studied (1.1) with \(\int _{t_{0}}^{\infty }1/r_{1}(t)\Delta t=\int _{t_{0}}^{\infty }1/r_{2}(t) \Delta t=\infty \), whereas other cases of the convergence and divergence of \(\int _{t_{0}}^{\infty }1/r_{1}(t)\Delta t\) and \(\int _{t_{0}}^{\infty }1/r_{2}(t)\Delta t\) were considered in [14–16]. Similar sufficient conditions for the existence of nonoscillatory solutions tending to zero of neutral dynamic equations have been presented. However, it is not easy to find a necessary condition for equations to have a nonoscillatory solution tending to zero asymptotically.
Mojsej and Tartal’ová [10] studied the asymptotic behavior of nonoscillatory solutions to a third-order differential equation
They stated some necessary and sufficient conditions ensuring the existence of nonoscillatory solutions tending to zero. Motivated by [10], Qiu [12] studied the existence of nonoscillatory solutions tending to zero of (1.1) under the conditions \(0\leq p_{0}<1\) and \(g(t)\geq t\). The conclusions extend and improve the results reported in the papers [11, 14–16].
The purpose of this paper is to further discuss the same problem of (1.1) with \(|p_{0}|<1\) and \(g(t)\leq t\). The existence of nonoscillatory solutions tending to zero of (1.1) is established by employing Krasnoselskii’s fixed point theorem. Finally, two examples are presented to show the versatility of the conclusions.
2 Auxiliary results
Let \(\mathrm{ BC}[T_{0},\infty )_{\mathbb{T}}\) denote the Banach space of all bounded continuous functions mapping \([T_{0},\infty )_{\mathbb{T}}\) into \(\mathbb{R}\) with the norm \(\|x\|=\sup_{t\in [T_{0},\infty )_{\mathbb{T}}}|x(t)|\). For the sake of convenience, we define
and state the following lemmas which will be used in the sequel.
Lemma 2.1
(see [5, Krasnoselskii’s fixed point theorem])
LetXbe a Banach space andΩbe 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\(x,y\in \varOmega \), whereUis a contraction mapping andVis completely continuous, then\(U+V\)has a fixed point inΩ.
Lemma 2.2
Suppose thatxis an eventually positive solution of (1.1) and there exists a constant\(a\geq 0\)such that\(\lim_{t\rightarrow \infty }z(t)=a\). Then
The proof is similar to those of [6, Lemma 2.3], [7, Theorem 1], and [17, Theorem 7], and thus is omitted.
3 Main results
In this section, our existence criteria for eventually positive solutions tending to zero as \(t\rightarrow \infty \) of (1.1) are established by employing Krasnoselskii’s fixed point theorem.
Theorem 3.1
Assume that
where
which satisfies\(\lim_{t\rightarrow \infty }H_{1}(g(t))/H_{1}(t)=1\). Then (1.1) has an eventually positive solutionxwith\(\lim_{t\rightarrow \infty }x(t)=0\), where\(r_{2}z^{\Delta }\)and\(r_{1} (r_{2}z^{\Delta } )^{\Delta }\)are both eventually negative.
Proof
Suppose that (3.1) holds. There will be two cases to be considered.
Case (i). \(0\leq p_{0}<1\). Take \(p_{1}\) such that \(p_{0}< p_{1}<(1+4p_{0})/5<1\). When \(p_{0}>0\), choose a sufficiently large \(T_{0}\in [t_{0},\infty )_{\mathbb{T}}\) such that
When \(p_{0}=0\), choose \(p_{1}\) such that \(|p(t)|\leq p_{1}\leq 1/13\) for \(t\in [T_{0},\infty )_{\mathbb{T}}\). In view of (C3), there exists a \(T_{1}\in (T_{0},\infty )_{\mathbb{T}}\) such that \(g(t)\geq T_{0}\) and \(h(t)\geq T_{0}\) for \(t\in [T_{1},\infty )_{\mathbb{T}}\).
Define
It is easy to prove that \(\varOmega _{1}\) is a bounded, convex, and closed subset of \(\mathrm{ BC}[T_{0},\infty )_{\mathbb{T}}\). Define \(U_{1}\) and \(V_{1}\): \(\varOmega _{1}\rightarrow \mathrm{ BC}[T_{0},\infty )_{\mathbb{T}}\) as follows:
We can prove that \(U_{1}\) and \(V_{1}\) satisfy all conditions in Lemma 2.1. The proof is expatiatory but similar to those of [6, Theorem 2.5], [7, Theorem 2], [12, Theorem 3.1], and [17, Theorem 8]; so we omit it here. By virtue of Lemma 2.1, there exists an \(x\in \varOmega _{1}\) such that \((U_{1}+V_{1})x=x\). Hence, for \(t\in [T_{1},\infty )_{\mathbb{T}}\), we have
Since
and
we arrive at \(\lim_{t\rightarrow \infty }z(t)=0\), which implies that \(\lim_{t\rightarrow \infty }x(t)=0\) with the help of Lemma 2.2. For \(t\in [T_{1},\infty )_{\mathbb{T}}\), we obtain
and
Case (ii). \(-1< p_{0}<0\). Take \(p_{1}\) satisfying \(-p_{0}< p_{1}<(1-4p_{0})/5<1\). Choose a sufficiently large \(T_{0}\in [t_{0},\infty )_{\mathbb{T}}\) such that (3.2) holds and
Proceeding as in the proof of Case (i), define \(V_{1}\) as in (3.3) and \(U'_{1}\) on \(\varOmega _{1}\) as follows:
Similarly, there exists an \(x\in \varOmega _{1}\) such that \((U'_{1}+V_{1})x=x\). For \(t\in [T_{1},\infty )_{\mathbb{T}}\), we have
and we arrive at the same conclusions as in Case (i). This completes the proof. □
Theorem 3.2
Assume that
Then (1.1) has no eventually positive solutionsxsatisfying that\(r_{2}z^{\Delta }\)and\(r_{1} (r_{2}z^{\Delta } )^{\Delta }\)are both eventually negative.
Proof
Suppose that x is an eventually positive solution of (1.1), and there exists a \(T_{0}\in [t_{0},\infty )_{\mathbb{T}}\) such that
From (C3), there exists a \(T_{1}\in (T_{0},\infty )_{\mathbb{T}}\) such that \(h(t)\geq T_{0}\) for \(t\in [T_{1},\infty )_{\mathbb{T}}\). Integrating (1.1) from \(T_{1}\) to s, \(s\in [\sigma (T_{1}),\infty )_{\mathbb{T}}\), by (C4) we obtain
which yields
Integrating (3.4) from \(T_{1}\) to v, \(v\in [\sigma (T_{1}),\infty )_{\mathbb{T}}\), we get
or
Integrating (3.5) from \(T_{1}\) to t, \(t\in [\sigma (T_{1}),\infty )_{\mathbb{T}}\), we obtain
Letting \(t\rightarrow \infty \), we have \(z(t)\rightarrow -\infty \). From (2.1), it follows that \(p_{0}\in (-1,0]\), and then there exist a \(T_{2}\in [T_{1},\infty )_{\mathbb{T}}\) and a \(p_{1}\) with \(-p_{0}< p_{1}<1\) such that \(z(t)<0\) or
By (C3), choose some positive integer \(k_{0}\) such that \(c_{k}\in [T_{2},\infty )_{\mathbb{T}}\) for all \(k\geq k_{0}\). Then, for any \(k\geq k_{0}+1\), we have
This inequality implies that \(\lim_{k\rightarrow \infty }x(c_{k})=0\). It follows from (2.1) that \(\lim_{k\rightarrow \infty }z(c_{k})=0\) which contradicts \(z(t)\rightarrow -\infty \) as \(t\rightarrow \infty \). The proof is complete. □
Theorem 3.3
Assume that
where
which satisfies\(\lim_{t\rightarrow \infty }H_{2}(g(t))/H_{2}(t)=1\). Then (1.1) has an eventually positive solutionxwith\(\lim_{t\rightarrow \infty }x(t)=0\), where\(r_{2}z^{\Delta }\)is eventually negative and\(r_{1} (r_{2}z^{\Delta } )^{\Delta }\)is eventually positive.
Proof
Suppose that (3.6) holds. There are two cases to be considered.
Case (i). \(0\leq p_{0}<1\). Take \(p_{1}\) as in Case (i) of Theorem 3.1. When \(p_{0}>0\), choose a sufficiently large \(T_{0}\in [t_{0},\infty )_{\mathbb{T}}\) such that
When \(p_{0}=0\), choose \(p_{1}\) such that \(|p(t)|\leq p_{1}\leq 1/13\) for \(t\in [T_{0},\infty )_{\mathbb{T}}\). By virtue of (C3), there exists a \(T_{1}\in (T_{0},\infty )_{\mathbb{T}}\) such that \(g(t)\geq T_{0}\) and \(h(t)\geq T_{0}\) for \(t\in [T_{1},\infty )_{\mathbb{T}}\).
Define
and \(U_{2}\), \(V_{2}\): \(\varOmega _{2}\rightarrow \mathrm{ BC}[T_{0},\infty )_{\mathbb{T}}\) as follows:
The remainder of the proof is similar to that of Theorem 3.1 and so is omitted. By Lemma 2.1, there exists an \(x\in \varOmega _{2}\) such that \((U_{2}+V_{2})x=x\). For \(t\in [T_{1},\infty )_{\mathbb{T}}\), we have
Since
and
we get \(\lim_{t\rightarrow \infty }z(t)=0\), which implies that \(\lim_{t\rightarrow \infty }x(t)=0\) due to Lemma 2.2. For \(t\in [T_{1},\infty )_{\mathbb{T}}\), we obtain
and
Case (ii). \(-1< p_{0}<0\). Introduce \(\mathrm{ BC}[T_{0},\infty )_{\mathbb{T}}\) and its subset \(\varOmega _{2}\) as in (3.7). Define \(V_{2}\) as in (3.8) and \(U'_{2}\) on \(\varOmega _{2}\) as follows:
The following proof is similar to Case (i) and we omit it here. There exists an \(x\in \varOmega _{2}\) such that \((U'_{2}+V_{2})x=x\). For \(t\in [T_{1},\infty )_{\mathbb{T}}\), we have
and obtain the similar results as in Case (i). This completes the proof. □
4 Examples
In this section, two examples are presented to show the applications of our results. The first example is given to illustrate Theorem 3.1.
Example 4.1
Let \(\mathbb{T}=\bigcup_{n=1}^{\infty }[3n-2,3n]\). For \(t\in [4,\infty )_{\mathbb{T}}\), consider
Here, \(r_{1}(t)=t^{4}\), \(r_{2}(t)=t^{2}\), \(p(t)=-(t-1)/(2t)\), \(g(t)=t-3\), \(h(t)=t\), and \(f(t,x)=tx^{3}+x/t^{2}\). It is obvious that the coefficients of (4.1) satisfy (C1)–(C4). Since
and
we obtain
and
By Theorem 3.1, we see that (4.1) has an eventually positive solution x satisfying \(\lim_{t\rightarrow \infty }x(t)=0\), where \(r_{2}z^{\Delta }\) and \(r_{1} (r_{2}z^{\Delta } )^{\Delta }\) are both eventually negative.
Now, we give the second example to demonstrate Theorems 3.2 and 3.3.
Example 4.2
Let \(\mathbb{T}=\bigcup_{n=1}^{\infty }[2^{n}-1,2^{n}]\). For \(t\in [3,\infty )_{\mathbb{T}}\), consider
Here, \(r_{1}(t)=t^{3}\), \(r_{2}(t)=t\), \(p(t)=1/2+1/t\), \(g(t)=t\), \(h(t)=t/2\), and \(f(t,x)=x/t^{2}\). It is obvious that the coefficients of (4.2) satisfy (C1)–(C4). Since
in terms of Theorem 3.2, we deduce that (4.2) has no eventually positive solutions x satisfying \(\lim_{t\rightarrow \infty }x(t)=0\), where \(r_{2}z^{\Delta }\) and \(r_{1} (r_{2}z^{\Delta } )^{\Delta }\) are both eventually negative. However, we have
and
Furthermore, there exists a constant \(M>0\) such that
and
By virtue of Theorem 3.3, we conclude that (4.2) has an eventually positive solution x satisfying \(\lim_{t\rightarrow \infty }x(t)=0\), where \(r_{2}z^{\Delta }\) is eventually negative and \(r_{1} (r_{2}z^{\Delta } )^{\Delta }\) is eventually positive.
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Acknowledgements
The authors express their sincere gratitude to the editors for the careful reading of the original manuscript and useful comments that helped to improve the presentation of the results and accentuate important details.
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Funding
The research of the first author was supported by the National Natural Science Foundation of P. R. China (Grant No. 11671406) and Natural Science Program for Young Creative Talents of Innovation Enhancing College Project of Department of Education of Guangdong Province (Grant Nos. 2017GKQNCX111 and 2018-KJZX039). The research of the second author was supported by the Slovak Research and Development Agency (Grant No. APVV-18-0373). The research of the third author was supported by FGI 10-18 DIUMCE and PGI 03-2020 DIUMCE. The research of the fourth author was supported by the National Natural Science Foundation of P. R. China (Grant No. 61503171), China Postdoctoral Science Foundation (Grant No. 2015M582091), and Natural Science Foundation of Shandong Province (Grant No. ZR2016JL021).
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Qiu, YC., Jadlovská, I., Chiu, KS. et al. Existence of nonoscillatory solutions tending to zero of third-order neutral dynamic equations on time scales. Adv Differ Equ 2020, 231 (2020). https://doi.org/10.1186/s13662-020-02678-x
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DOI: https://doi.org/10.1186/s13662-020-02678-x