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Oscillation of solutions of third order nonlinear neutral differential equations
Advances in Difference Equations volume 2020, Article number: 314 (2020)
Abstract
The main objective of this article is to improve and complement some of the oscillation criteria published recently in the literature for third order differential equation of the form
where \(z(t)=x(t)+p(t)x(\tau (t))\) and α is a ratio of odd positive integers in the two cases \(\int _{t_{0}}^{\infty }r^{\frac{-1}{\alpha } }(s)\,\mathrm {d}s<\infty \) and \(\int _{t_{0}}^{\infty }r^{\frac{-1}{\alpha } }(s)\,\mathrm {d}s=\infty \). Some illustrative examples are presented.
1 Introduction
Consider the nonlinear third order differential equation
where \(t\geq t_{0}>0\), \(z(t)=x(t)+p(t)x ( \tau (t) ) \), and α is a ratio of odd positive integers. We assume that the following conditions hold:
- (H1):
\(r(t),p(t),q(t),\tau (t),\sigma (t)\in C ( [t_{0},\infty ) ) \), \(r(t)\), \(q(t)\) are positive and \(0\leq p(t)\leq p_{0}<\infty \);
- (H2):
\(\lim_{t\rightarrow \infty }\tau (t)=\lim_{t \rightarrow \infty }\sigma (t)=\infty \), \(\sigma (t)>0\), and \(\tau (t)\leq t\);
- (H3):
\(f(u)\in C ( \mathbb{R} ) \) and there exists a positive constant k such that \(f(u)/u^{\gamma }\geq k\) for all \(u\neq 0\) and γ is a ratio of odd positive integers;
- (H4):
\(\tau ^{\prime }(t)\geq \tau _{0}>0\) and \(\tau \circ \sigma =\sigma \circ \tau \).
By a solution of (1.1), we mean a nontrivial function \(x(t)\in C ( [T_{x},\infty ) ) \), \(T_{x}\geq t_{0}\), which has the properties \(z(t)\in C^{2} ( [T_{x},\infty ) ) \), \(r(t)(z^{\prime \prime }(t))^{\alpha }\in C^{1} ( [T_{x},\infty ) ) \) and satisfies (1.1) on \([T_{x},\infty )\). Our attention is restricted to those solutions \(x(t)\) of (1.1) satisfying \(\sup \{ \vert x(t) \vert :t\geq T \} >0\) for all \(T\geq T_{x}\). We assume that (1.1) possesses such a solution. A solution of (1.1) is called oscillatory if it has arbitrarily large zeros on \([T_{x},\infty )\); otherwise, it is termed nonoscillatory. Equation (1.1) is said to be oscillatory if all its solutions are oscillatory.
The oscillatory behavior of solutions of various classes of nonlinear differential and dynamic equations on time scales has received much attention, we refer the reader to [1–17] and the references cited therein.
In 2012, Liu et al. [9] established new oscillation criteria for the second order Emden–Fowler equation
under the assumptions
and \(\alpha \geq \gamma >0\). In 2016, Wang et al. [16] studied Eq. (1.2) with condition (1.3),
and \(\sigma ^{\prime }(t)>0\) with \(\alpha \geq \gamma >1\) when the condition \(r^{\prime }(t)\geq 0\) is neglected. Meanwhile, Wu et al. [17] established oscillation criteria for (1.2) in the general case when \(\alpha >0\) and \(\gamma >0\) are constants with conditions (1.3) and (1.4). Baculíková et al. [2] considered (1.2) in the more general case when \(0\leq p(t)\leq p_{0}<\infty \) with condition (1.5) and \(\sigma ^{\prime }(t)\geq 0\). For the case of third order differential equations, Džurina et al. [18] obtained sufficient conditions for the oscillation of solutions of the differential equation
where
with condition (1.5). Meanwhile, Baculíková et al. [1] and Su et al. [19] discussed the oscillatory behavior of third order Eq. (1.6) when \(r^{\prime }(t)\geq 0\), (1.7) and (1.5) hold. Also Thandapani et al. [14] studied Eq. (1.6) when (1.7) holds and
Recently, Jiang et al. [7] established new oscillation criteria for Eq. (1.1), where \(\gamma =\alpha \geq 1\) and (1.5) hold without requiring (1.4).
More recently, Graef et al. [6] discussed the special case of Eq. (1.1) in which \(r=1\) and \(\alpha =\gamma \).
The main goal of this paper is to establish new oscillation criteria motivated by [6, 7], and [17] for Eq. (1.1) under all cases of γ, α (i.e., \(\gamma >\alpha \), \(\gamma =\alpha \), and \(\gamma <\alpha \)), \(\int _{t_{0}}^{\infty }\frac{1}{r^{\frac{1}{\alpha }}(t)}\,\mathrm {d}t<\infty \) and \(\int _{t_{0}}^{\infty }\frac{1}{r^{\frac{1}{\alpha }}(t)}\,\mathrm {d}t=\infty \) without assumption (1.4). We consider the two cases when (H4) holds or not.
In the sequel, we give the following notations:
where \(\tau ^{-1}\) is the inverse of τ, \(m_{*}\) and \(m_{**}\) are functions to be specified later. All functional inequalities considered in this article are assumed to hold eventually, that is, they are satisfied for all t large enough.
2 Some preliminaries
We enlist some known results which will be needed. We first present the following classes of nonoscillatory (let us say positive) solutions of (1.1):
\(z(t)\in N_{I}\Leftrightarrow z^{\prime }(t)>0\), \(z^{\prime \prime }(t)>0\), \(( r(t)(z^{\prime \prime }(t))^{\alpha } ) ^{\prime }<0\),
\(z(t)\in N_{\mathit{II}}\Leftrightarrow z^{\prime }(t)<0\), \(z^{\prime \prime }(t)>0\), \(( r(t)(z^{\prime \prime }(t))^{\alpha } ) ^{\prime }<0\), and
\(z(t)\in N_{\mathit{III}}\Leftrightarrow z^{\prime }(t)>0\), \(z^{\prime \prime }(t)<0\), \(( r(t)(z^{\prime \prime }(t))^{\alpha } ) ^{\prime }<0\), eventually.
The following lemma comes directly from combining Lemma 1 and Lemma 2 in [13] with Lemma 3 and Lemma 4 in [20].
Lemma 2.1
Assume that\(A\geq 0\)and\(B\geq 0\). Then
and
Lemma 2.2
Let\(g>0\). Then
and
Proof
See [21, p. 28]. □
Lemma 2.3
[17]Assume that\(A\geq 0\), \(B>0\), \(U\geq 0\), and\(\lambda >0\). Then
Lemma 2.4
Assume thatxis an eventually positive solution of (1.1). If (1.5) holds, then\(z(t)\in N_{I}\)or\(z(t)\in N_{\mathit{II}}\). While if (1.8) holds, then either\(z(t)\in N_{I}\)or\(z(t)\in N_{\mathit{II}}\)or\(z(t)\in N_{\mathit{III}}\).
Proof
The proof is similar to [22, Theorem 2.1 and Theorem 2.2]. □
Lemma 2.5
Let the function\(f(t)\)satisfy\(f^{(i)}(t)>0\), \(i=0,1,2,\ldots,n\), and\(f^{(n+1)}(t)<0\)eventually, then there exists a constant\(k_{1}\in (0,1)\)such that\(\frac{f(t)}{f^{\prime }(t)}\geq \frac{k_{1}t}{n}\)eventually.
3 Oscillation criteria in the case when (H4) holds
In this section, we establish new oscillation criteria for Eq. (1.1) in the case when (H4) holds.
Theorem 3.1
Assume that (H1)–(H4) hold. If there exists a positive function\(\rho (t)\in C^{1} ( [t_{0},\infty ) ) \)such that
where
holds for some constant\(k>0\), sufficiently large\(t_{1}\geq t_{0}\), and for some\(t_{\ast }>t_{2}>t_{1}\), then there exists no positive solution\(x(t)\)of Eq. (1.1) satisfying\(z(t)\in N_{I}\).
Proof
Assume that \(x(t)\) is a positive solution of Eq. (1.1) satisfying \(z(t)\in N_{I}\) for \(t\geq t_{1}\). Then from (1.1) and (H3) it follows that
Since \(( r(\tau (t))(z^{\prime \prime }(\tau (t)))^{\alpha } ) ^{ \prime }= ( r ( z^{\prime \prime } ) ^{\alpha } ) ^{\prime }(\tau (t))\tau ^{\prime }(t)\), then in view of (H4) there exists \(t_{2}\geq t_{1}\) such that
In the following, we consider the two cases \(\gamma >1\) and \(\gamma \leq 1\). Firstly, assume that \(\gamma >1\). Using (2.2) with (3.5), we get
Define the functions \(\omega (t)\) and \(\nu (t)\) by
and
Then clearly \(\omega (t)\) and \(\nu (t)\) are positive for \(t\geq t_{2}\) and satisfy
and
Now, we consider the two cases \(\gamma \geq \alpha \) and \(\gamma <\alpha \). We first assume that \(\gamma \geq \alpha \). From (3.7), we have
Substituting into (3.9), we get
But since \(z^{\prime }(t)\) is positive and increasing, it follows that there exists a constant \(M>0\) satisfying \(z^{\prime }(t)\geq M\) and
Using inequality (2.5) with \(A=\rho _{+}^{\prime }(t)r(t)\), \(U=\frac{\omega (t)}{\rho (t)r(t)}\), and \(B=\gamma M^{\frac{\gamma }{\alpha } -1}\rho (t)r(t)\), it follows that
In view of (3.8), we have
Substituting into (3.10), we get
Again by inequality (2.5), we get
But since \(z^{\prime \prime }(t)>0\) and \(\tau (t) \leq t\), we obtain
Combining (3.12) and (3.13) and using (3.6), we get
Now, assume that \(\gamma <\alpha \). Then from (3.7) we have
Substituting into (3.9), we get
It is clear that \((r^{\frac{1}{\alpha }}(t)z^{\prime \prime }(t) )^{1- \frac{\alpha }{\gamma }}\) is positive and increasing, and so there exists a positive constant \(m_{1}\) such that
for all sufficiently large t. Using inequality (2.5), we conclude that
But since from (3.8) we have
then, by substituting into (3.10), we get
This with (2.5) leads to
Combining (3.15) and (3.16), using (3.6), we get
Combining (3.14) and (3.17), we obtain for any α, γ ratios of odd positive integers that
Now, we consider the two cases \(\sigma (t)< t\) and \(\sigma (t)\geq t\). We start by considering the case \(\sigma (t)< t\). Since \(r(t)(z^{\prime \prime }(t))^{\alpha }\) is positive and decreasing, we have
i.e.,
But since \(\sigma (t)< t\), then it follows that
Now since by (3.19) we have
which means that
then we have
This with (3.20) leads to
Substituting into (3.18), we get
Now, consider the case \(\sigma (t)\geq t\). Since \(z(t)\) is positive and increasing, it follows from (3.18) that
Since \(( r(t) ( z^{\prime \prime }(t) ) ^{\alpha } ) ^{ \prime }<0\), we get (3.19) and consequently we arrive at (3.21). Then, substituting into (3.25), we have
Combining (3.24) and (3.26), we get
Integrating from \(t_{4}\) (>\(t_{3}\)) to t, we have
which contradicts (3.1). Secondly, assume that \(\gamma \leq 1\). Using (2.1) with (3.5), we obtain
By completing the proof as the above case of \(\gamma >1\), using (3.27) instead of (3.6), the proof is completed. □
Lemma 3.1
Assume that conditions (H1)–(H4) hold. Letxbe an eventually positive solution of Eq. (1.1) and the corresponding\(z(t)\)satisfies\(z(t)\in N_{\mathit{II}}\). If
or
then\(\lim_{t\rightarrow \infty }x(t)=\lim_{t \rightarrow \infty }z(t)=0\).
Proof
Assume that \(x(t)\) is a positive solution of Eq. (1.1) satisfying \(z(t)\in N_{\mathit{II}}\) for \(t\geq t_{1}\). Going through as in the proof of Theorem 3.1, we arrive at (3.5). In the following, we consider the two cases \(\gamma >1\) and \(\gamma \leq 1\). Firstly, assume that \(\gamma >1\). Then we have (3.6). Since \(z(t)\) is positive and decreasing, we have \(\lim_{t\rightarrow \infty }z(t)=l\geq 0\) exists. We claim that \(l=0\). If not, then there exists \(t_{3}\geq t_{2}\) such that \(z(\sigma (t))>l\) for \(t\geq t_{3}\). Substituting into (3.6), we get
Integrating (3.30) from \(t_{3}\) to t and taking into account (3.28), we have
which is a contradiction. Thus \(l=0\) and consequently \(\lim_{t\rightarrow \infty }x(t)=0\). In the following, we obtain the same conclusion in the case when \(\int _{t_{0}}^{\infty }Q(s)\,\mathrm {d}s<\infty \). Integrating (3.30) from t to ∞, we have
But since \(\tau (t)\leq t\), then we can observe that \(r(\tau (t))(z^{\prime \prime }(\tau (t)))^{\alpha }\geq r(t)(z^{ \prime \prime }(t))^{\alpha }\) and consequently we have
i.e.,
Integrating from t to ∞ followed by integrating from \(t_{3}\) to ∞, we obtain
which contradicts (3.29). Thus \(\lim_{t\rightarrow \infty }x(t)=0\). Secondly, assume that \(\gamma \leq 1\). As in the proof of Theorem 3.1, we have (3.27). By completing the proof as in the above case of \(\gamma >1\), using (3.27) instead of (3.6), the proof is completed. □
Theorem 3.2
Assume that (H1)–(H4) hold. If
then there exists no positive solution\(x(t)\)of Eq. (1.1) satisfying\(z(t)\in N_{\mathit{III}}\).
Proof
Let \(x(t)\) be an eventually positive solution of Eq. (1.1) satisfying \(z(t)\in N_{\mathit{III}}\) for all \(t\geq t_{1}\geq t_{0}\). Since \(z^{\prime \prime }(t)<0\) and \(z^{\prime }(t)>0\), then by Lemma 2.5, there exist \(t_{2}\geq t_{1}\) and a constant \(k_{1}\) satisfying \(0< k_{1}<1\) such that \(z(t)\geq k_{1}tz^{\prime }(t)\) for \(t\geq t_{2}\), i.e.,
Going through as in Theorem 3.1, we arrive at (3.5). In the following, we consider the two cases \(\gamma >1\) and \(\gamma \leq 1\). Firstly, assume that \(\gamma >1\). Then we have (3.6), and using (3.32) we get
But since \(v(t)=-r^{\frac{1}{\alpha }}(t)z^{\prime \prime }(t)\) is positive and increasing, then there exists a constant \(g_{1}>0\) such that \(v(t)\geq g_{1}\) for \(t\geq t_{3}\geq t_{2}\). Hence
Substituting into (3.33) and integrating from \(t_{3}\) to t, we get
But since \(\tau (t)\leq t\), then we can conclude that \(r(\tau (t)) ( z^{\prime \prime }(\tau (t)) ) ^{\alpha } \geq r(t) ( z^{\prime \prime }(t) ) ^{\alpha }\). Now since from (3.35) we have
i.e.,
Then integrating from \(t_{4}\) (\(\geq t_{3}\)) to t, we get
which contradicts (3.31). Secondly, assume that \(\gamma \leq 1\). As in the proof of Theorem 3.1, we arrive at (3.27), and then using (3.32) we get
Going through as in the proof of the case \(\gamma >1\), using (3.36) instead of (3.33), this completes the proof. □
The following results are immediate consequences of Lemma 2.4, Lemma 3.1, Theorem 3.1, and Theorem 3.2.
Theorem 3.3
Assume that (1.8) and all the conditions of Lemma3.1, Theorem3.1, and Theorem3.2hold. Then every solution\(x(t)\)of Eq. (1.1) is either oscillatory or satisfies\(\lim_{t\rightarrow \infty }x(t)=0\).
Theorem 3.4
Assume that (1.5) and all the conditions of Lemma3.1and Theorem3.1hold. Then every solution\(x(t)\)of Eq. (1.1) is either oscillatory or satisfies\(\lim_{t\rightarrow \infty }x(t)=0\).
The following results deal with the special case \(\alpha \leq 1\) and \(\gamma \geq 1\) of Eq. (1.1).
Theorem 3.5
Assume that conditions (H1)–(H4), \(\alpha \leq 1\), and\(\gamma \geq 1\)hold. If there exists a positive function\(\rho (t)\in C^{1} ( [t_{0},\infty ) ) \)such that
holds for any positive constantsk, M, sufficiently large\(t_{1}\geq t_{0}\), and for some\(t_{\ast }>t_{2}>t_{1}\), where\(\lambda _{1}(t)\)is defined by (3.3) and
then there exists no positive solution\(x(t)\)of Eq. (1.1) satisfying\(z(t)\in N_{I}\).
Proof
Assume that \(x(t)\) is an eventually positive solution of Eq. (1.1) satisfying \(z(t)\in N_{I}\). As in the proof of Theorem 3.1, we arrive at (3.6). Now define the function \(W(t)\) by
Then \(W(t)>0\) for \(t\geq t_{1}\) and
Since \(z^{\prime }(t)\) and \(z^{\prime \prime }(t)\) are positive, then there exist \(t_{2}\geq t_{1}\) and constant \(M>0\) such that \(z^{\prime }(t)\geq M\) for all \(t\geq t_{2}\). Now, from (3.38) and (2.3), we get
This with (3.39) yields
Now define
As we did for W, we can get
But since \(z^{\prime }\) is increasing and \(\tau (t) \leq t\), then
This with (3.41) leads to
Thus, by (3.6) and (2.4), we get
Now, we consider the two cases \(\sigma (t)< t\) and \(\sigma (t)\geq t\).
First assume that \(\sigma (t)< t\). As in the proof of Theorem 3.1, we get (3.23). Substituting into (3.44), we have
Secondly, assume that \(\sigma (t)\geq t\). Since \(z^{\prime }(t)>0\), it follows from (3.44) that
As in the proof of Theorem 3.1, we arrive at (3.21). Then, substituting into (3.46), we have
This with (3.45) yields
Integrating from \(t_{4}\) (\(>t_{3}\)) to t, we get
This contradicts (3.37) and completes the proof. □
Theorem 3.6
Assume that (1.8) and all the conditions of Lemma3.1, Theorem3.2, and Theorem3.5hold. Then every solution\(x(t)\)of Eq. (1.1) is either oscillatory or satisfies\(\lim_{t\rightarrow \infty }x(t)=0\).
Theorem 3.7
Assume that (1.5) and all the conditions of Lemma3.1and Theorem3.5hold. Then every solution\(x(t)\)of Eq. (1.1) is either oscillatory or satisfies\(\lim_{t\rightarrow \infty }x(t)=0\).
4 Oscillation criteria without condition (H4)
In this section, we study the oscillation of Eq. (1.1) when either of the two conditions \(0\leq p(t)\leq p_{0}<1\) or \(p(t)\geq 1\), \(p(t)\not \equiv 1\) holds for large t. Now, we begin by establishing new oscillation criteria for Eq. (1.1) in the case when \(p(t)\geq 1\), \(p(t)\not \equiv 1\) for large t with the condition \(\tau (t)< t\) and \(\tau (t)\) is strictly increasing.
Theorem 4.1
Assume that (H1)–(H3) hold, \(p(t)\geq 1\), \(p(t)\not \equiv 1\)for sufficiently larget, \(\tau (t)< t\)and\(\tau ^{\prime }(t)>0\). Further assume that there exists a positive function\(m_{*}(t)\in C^{1} ( [t_{0},\infty ) ) \)such that
and\(p_{\ast }(t)>0\)for sufficiently larget. If there exists a positive function\(\rho (t)\in C^{1} ( [t_{0},\infty ) ) \)such that
holds for some constant\(k>0\), sufficiently large\(t_{1}\geq t_{0}\), and for some\(t_{\ast }>t_{2}>t_{1}\), whereλ, m, gare defined by (3.2), (3.3), and
then there exists no positive solution\(x(t)\)of Eq. (1.1) satisfying\(z(t)\in N_{I}\).
Proof
Assume that \(x(t)\) is an eventually positive solution of Eq. (1.1) satisfying \(z(t)\in N_{I}\) for \(t\geq t_{1}\). From the definition of z (see also (2.2) in [6]), we have
Define the function \(\omega (t)\) as in (3.7). Then \(\omega (t)>0\) for \(t\geq t_{1}\) satisfying (3.9). As in the proof of Theorem 3.1, since \(( r(t) ( z^{\prime \prime }(t) ) ^{\alpha } ) ^{ \prime }<0\), we have (3.19) and then
This with (4.1) yields
This means that \(\frac{z(t)}{m_{*}(t)}\) is nonincreasing. But since \(\tau (t)< t\) and \(\tau ^{\prime }(t)>0\), it follows that \(\tau ^{-1}(t)\leq \tau ^{-1}(\tau ^{-1}(t))\), and so
Substituting from (4.6) into (4.4), we get
This in the view of (1.1) leads to
In the following, we consider the two cases \(\gamma \geq \alpha \) and \(\gamma <\alpha \).
First, assume that \(\gamma \geq \alpha \). As in the proof of Theorem 3.1, we have (3.12). Then, substituting from (4.8) into (3.12), we obtain
Now assume that \(\gamma <\alpha \). As in the proof of Theorem 3.1, we have (3.15). Then, substituting from (4.8) into (3.15), we obtain
This with (4.9) yields
Now, consider the two cases \(\sigma (t)<\tau (t)\) and \(\sigma (t)\geq \tau (t)\). First assume that \(\sigma (t)<\tau (t)\). Since \(\tau ^{-1}(\sigma (t))< t\) and \(( \frac{z^{\prime }(t)}{\int _{t_{1}}^{t}\frac{\,\mathrm {d}s}{r^{\frac{1}{\alpha }}(s)}} ) ^{\prime }\leq 0\), then by (4.5) we have
Substituting into (4.11), we get
Secondly, assume that \(\sigma (t)\geq \tau (t)\). Hence since \(z^{\prime }(t)>0\) and \(\tau ^{-1}(\sigma (t))\geq t\), we have \(z(\tau ^{-1}(\sigma (t)))\geq z(t)\). Thus it follows from (4.11) and (4.5) that
Combining (4.12) and (4.13) and then integrating from \(t_{3}\) (\(>t_{2}\)) to t, we get
which contradicts (4.2). This completes the proof. □
Theorem 4.2
Assume that (H1)–(H3) hold, \(p(t)\geq 1\), \(p(t)\not \equiv 1\)for sufficiently larget, \(\tau (t)< t\), \(\tau ^{\prime }(t)>0\), and\(p^{\ast }(t)>0\). If\(x(t)\)is an eventually positive solution of Eq. (1.1) satisfying\(z(t)\in N_{\mathit{II}}\)with
or
then\(\lim_{t\rightarrow \infty }x(t)=0\).
Proof
Let \(x(t)\) be an eventually positive solution of Eq. (1.1) satisfying \(z(t)\in N_{\mathit{II}}\) for \(t\geq t_{1}\). Going through as in the proof of Theorem 4.1, we arrive at (4.4). Since \(z(t)\) is decreasing and \(\tau (t)< t\), then \(z(\tau ^{-1}(t))\geq z(\tau ^{-1}(\tau ^{-1}(t)))\). Substituting into (4.4), we get
This with (1.1) leads to
Since \(z(t)>0\) and \(z^{\prime }(t)<0\), then \(\lim_{t\rightarrow \infty }z(t)=l\geq 0\) exists. We claim that \(l=0\). If not, then there exists \(t_{2}\geq t_{1}\) such that \(\tau ^{-1}(\sigma (t))>t_{1}\) and \(z(\tau ^{-1}(\sigma (t)))\geq l\) for \(t\geq t_{2}\). Substituting into (4.17), we get
Integrating from \(t_{2}\) to t and taking into account (4.14), we have
which is a contradiction. Thus \(l=0\) and \(\lim_{t\rightarrow \infty }x(t)=0\). In the following, we obtain the same conclusion in the case when \(\int _{t_{0}}^{\infty }q(s) ( p^{\ast }(\sigma (s)) ) ^{ \gamma }\,\mathrm {d}s<\infty \). Integrating (4.18) from t to ∞ and dividing both sides by \(r(t)\), we have
Integrating again from t to ∞, we obtain
Moreover, by integrating again from \(t_{3}\) to ∞, we get
which contradicts (4.15). Hence, \(l=0\). So from the fact that \(0< x(t)< z(t) \), it follows that \(\lim_{t\rightarrow \infty }x(t)=0\). □
Theorem 4.3
Assume that (H1)–(H3) hold, \(p(t)\geq 1\), \(p(t)\not \equiv 1\)for sufficiently larget, \(\tau (t)< t\)and\(\tau ^{\prime }(t)>0\). If for some constant\(k_{1}\in (0,1)\)there exists a function\(m_{**}(t)\in C^{1} ( [t_{0},\infty ),(0,\infty ) ) \)such that
\(p_{\ast \ast }(t)>0\)for all sufficiently largetand
then there exists no positive solution\(x(t)\)of Eq. (1.1) satisfying\(z(t)\in N_{\mathit{III}}\).
Proof
Let \(x(t)\) be an eventually positive solution of Eq. (1.1) such that \(x(t)>0\), \(x(\tau (t))>0\), and \(x(\sigma (t))>0\), \(z(t)\) satisfies \(z(t)\in N_{\mathit{III}}\) and \(\tau ^{-1}(\sigma (t))>t_{0}\) for \(t\geq t_{1}\geq t_{0}\). From the definition of z, we have (4.4) as in the proof of Theorem 4.1. Since \(z^{\prime \prime }(t)<0\) and \(z^{\prime }(t)>0\), then by Lemma 2.5 there exists \(t_{2}\geq t_{1}\) such that
This with (4.19) yields
and so \(\frac{z(t)}{m_{**}(t)}\) is nonincreasing. Hence \(z(\tau ^{-1}(\tau ^{-1}(t)))\leq \frac{m_{**}(\tau ^{-1}(\tau ^{-1}(t)))z(\tau ^{-1}(t))}{m_{**}(\tau ^{-1}(t))}\). Now, from (1.1), (4.4), and (4.21), we have
But since \(-r^{\frac{1}{\alpha }}(t)z^{\prime \prime }(t)\) is positive and increasing, then we have \(-r^{\frac{1}{\alpha }}(t)z^{\prime \prime }(t)\geq g_{1}\) for \(t\geq t_{1}\). Hence
Thus
This with (4.22) leads to
Integrating from \(t_{2}\) to t, we get
Integrating again from \(t_{3}\) (\(\geq t_{2}\)) to t, we have
This contradicts (4.20) and completes the proof. □
Theorem 4.4
Assume that (1.8) and all the conditions of Theorem4.1, Theorem4.2, and Theorem4.3hold. Then every solution\(x(t)\)of Eq. (1.1) is either oscillatory or satisfies\(\lim_{t\rightarrow \infty }x(t)=0\).
Theorem 4.5
Assume that (1.5) and all the conditions of Theorem4.1and Theorem4.2hold. Then every solution\(x(t)\)of Eq. (1.1) is either oscillatory or satisfies\(\lim_{t\rightarrow \infty }x(t)=0\).
Remark 1
The assumptions concerning the existence of the two functions \(m_{\ast }(t)\) and \(m_{\ast \ast }(t)\) hold, for example, \(\mu (t)=\xi (t)\), \(\mu (t)= ( \xi (t) ) ^{\eta }\), \(\mu (t)=\xi (t)e^{\xi ^{\epsilon }(t)}\), \(\mu (t)= ( \xi (t) ) ^{\eta }e^{\epsilon \xi (t)}\) with
\(\eta \geq 1\) and \(\epsilon \geq 0\), etc.
Remark 2
From Theorem 4.4 and Theorem 4.5, we can obtain more than one oscillation criterion for Eq. (1.1) in the two theorems with different choices of \(m_{\ast }(t)\) and \(m_{\ast \ast }(t)\) which are mentioned in Remark 1.
In the following, we discuss the oscillatory behavior of solutions of Eq. (1.1) in the case when \(0\leq p(t)\leq p_{0}<1\).
Theorem 4.6
Assume that (H1)–(H3) hold and\(0\leq p(t)\leq p_{0}<1\). If there exists a positive function\(\rho (t)\in C^{1}([t_{0},\infty ))\)such that
holds for some constant\(k>0\), for sufficiently large\(t_{1}\geq t_{0}\), and for some\(t_{\ast }>t_{2}>t_{1}\), whereλ, m, g, \(\lambda _{1}(t)\)are as defined by (3.2) and (3.3), then there exists no positive solution\(x(t)\)of Eq. (1.1) satisfying\(z(t)\in N_{I}\).
Proof
Let \(x(t)\) be an eventually positive solution of Eq. (1.1) satisfying \(z(t)\in N_{I}\). From the definition of z, we have
This with (1.1) yields
Defining \(\omega (t)\) by (3.7), completing the proof as in the proof of Theorem 4.1 by applying (4.25) instead of (4.8), and considering the two cases \(\sigma (t)< t\) and \(\sigma (t)\geq t\) instead of the two cases \(\sigma (t)<\tau (t)\) and \(\sigma (t)\geq \tau (t)\), we get a contradiction to (4.24). □
Theorem 4.7
Assume that (H1)–(H3) hold, \(0\leq p(t)\leq p_{0}<1\), and\(x(t)\)is an eventually positive solution of Eq. (1.1) satisfying\(z(t)\in N_{\mathit{II}}\). If
or
then\(\lim_{t\rightarrow \infty }x(t)=0\).
Proof
Let \(x(t)\) be an eventually positive solution of Eq. (1.1) satisfying \(z(t)\in N_{\mathit{II}}\) for \(t\geq t_{1}\geq t_{0}\). Since \(z(t)\) is positive and decreasing, we have \(\lim_{t\rightarrow \infty }z(t)=l\geq 0\) exists. We claim that \(l=0\). If not, then for any \(\epsilon >0\) we have \(l< z(t)< l+\epsilon \) eventually. Choose \(0<\epsilon <\frac{l(1-p_{0})}{p_{0}}\). It is easy to verify that
where \(k_{2}=\frac{l-p_{0}(l+\epsilon )}{(l+\epsilon )}>0\). Now, it follows from (1.1) that
Going through as in the proof of Theorem 4.2 by applying (4.28) instead of (4.18), we can get a contradiction to (4.26) or (4.27). This completes the proof. □
Using a similar technique to the proof of Theorem 4.3 and using (4.25) with (4.21) instead of (4.22), we can get the following result.
Theorem 4.8
Assume that (H1)–(H3) hold and\(0\leq p(t)\leq p_{0}<1\). If
then there exists no positive solution\(x(t)\)of Eq. (1.1) satisfying\(z(t)\in N_{\mathit{III}}\).
Theorem 4.9
Assume that (1.8) and all the conditions of Theorem4.6, Theorem4.7, and Theorem4.8hold. Then every solution\(x(t)\)of Eq. (1.1) is either oscillatory or satisfies\(\lim_{t\rightarrow \infty }x(t)=0\).
Theorem 4.10
Assume that (1.5) and all the conditions of Theorem4.6and Theorem4.7hold. Then every solution\(x(t)\)of Eq. (1.1) is either oscillatory or satisfies\(\lim_{t\rightarrow \infty }x(t)=0\).
5 Examples
Example 1
Consider the third order differential equation
Here, \(r(t)=1\), \(p=\frac{25}{4}\), \(\tau (t)=\frac{t}{2}\), \(q(t)=\frac{3}{t}\), \(\sigma (t)=t\), and \(1=\gamma >\alpha =\frac{1}{3}\). It is clear that \(\int _{t_{0}}^{\infty }\frac{1}{r^{\frac{1}{\alpha }}(t)}\,\mathrm {d}t= \infty \). Choosing \(\rho (t)=\frac{1}{t}\), then we have \(\rho _{+}^{\prime }(t)=0\), and
Thus, it follows from Theorem 3.4 that every solution \(x(t)\) of Eq. (5.1) is either oscillatory or satisfies \(\lim_{t\rightarrow \infty }x(t)=0\). In fact, \(x(t)=\frac{1}{t}\) is a solution of Eq. (5.1).
Example 2
Consider the third order differential equation
Here, \(r(t)=1\), \(p=p_{0}\), \(\tau (t)=t-\frac{1}{2}\), \(q(t)= ( t-\frac{1}{2} ) ^{\frac{4}{3}}\), \(\sigma (t)=t-\frac{1}{2}\), and \(\frac{1}{3}=\gamma <\alpha =5\). It is clear that \(\int _{t_{0}}^{\infty }\frac{1}{r^{\frac{1}{\alpha }}(t)}\,\mathrm {d}t=\infty \). Choosing \(\rho (t)=1\), we have \(\rho _{+}^{\prime }(t)=0\), and
Thus, by Theorem 3.4, it follows that every solution \(x(t)\) of Eq. (5.2) is either oscillatory or satisfies \(\lim_{t\rightarrow \infty }x(t)=0\).
Example 3
Consider the third order differential equation
Here, \(r(t)=t\), \(p=\frac{1}{3\sqrt{3}}\), \(\tau (t)=\frac{t}{3}\), \(q(t)=\lambda t^{6}\), \(\sigma (t)=\frac{t}{2}\), and \(3=\gamma >\alpha =\frac{1}{3}\). It is clear that \(\int _{t_{0}}^{\infty }\frac{1}{r^{\frac{1}{\alpha }}(t)}\,\mathrm {d}t=\int _{1}^{\infty }\frac{1}{t^{3}}\,\mathrm {d}t=\frac{1}{2}< \infty \). Choosing \(\rho (t)=\frac{1}{t^{10}}\), we have \(\rho _{+}^{\prime }(t)=0\), and
and
Thus, by Theorem 3.3, it follows that every solution \(x(t)\) of Eq. (5.3) is either oscillatory or satisfies \(\lim_{t\rightarrow \infty }x(t)=0\). We may note that, for \(\lambda =\frac{\sqrt[3]{35}}{\sqrt{2^{17}}}\), we have \(x(t)=\frac{1}{t^{\frac{5}{2}}}\) is a solution of Eq. (5.3).
Example 4
Consider the third order neutral delay differential equation
Here, \(r(t)=t^{3}\), \(p(t)=t^{\frac{5}{3}}\frac{5t+6}{t+1}\), \(q(t)=t^{9}\), \(\tau (t)=\frac{t}{2}\), \(\sigma (t)=t-1\), \(f(u)=u^{3}\), \(\alpha =1\), and \(\gamma =3\). It is clear that \(p(t)=t^{\frac{5}{3}} [ 5+\frac{1}{t+1} ] \geq (5)(2^{\frac{5}{3}})\simeq 15.874>1\), \(\tau \circ \sigma \neq \sigma \circ \tau \), \(\sigma (t)\geq \tau (t)\), conditions (H1)–(H3), and (1.8) hold, and
Let \(m_{\ast }(t)=\int _{t_{2}}^{t}\int _{t_{1}}^{s} \frac{1}{r^{\frac{1}{\alpha }}(u)}\,\mathrm {d}u \,\mathrm {d}s\) and \(m_{\ast \ast }(t)=t^{\frac{1}{k_{1}}}\). Thus
where \(\phi (t)=\frac{13t-9}{6t^{2}-13t+6}\). Since \(\phi ^{\prime }(t)=\frac{-78t^{2}+108t-39}{(6t^{2}-13t+6)^{2}}\), which is negative for \(t\geq t_{2}=3>t_{1}=2\). Thus \(\phi (t)\) is positive and decreasing for \(t\geq t_{2}=3\). It follows that \(\phi (t)\leq \frac{10}{7}\). Thus by (5.6) we have
By choosing \(\rho (t)=\frac{1}{t^{8}}\), condition (4.2) becomes
where \(\zeta (s)=\frac{2s-1}{10s-4}\). Then \(\zeta ^{\prime }(s)=\frac{2}{(10s-4)^{2}}>0\), i.e., \(\zeta (s)\) is positive and increasing and \(\zeta (s)\geq \frac{5}{26}\) for \(s\geq t_{2}=3\). Now from (5.7) we have
But since by (5.5) we have
then it follows that condition (4.14) reads
where \(\epsilon _{1}=(0.05868968172)^{3}\). Moreover, since by using (5.5) and letting \(k_{1}=\frac{1}{2}\) we have
then condition (4.20) becomes
Thus, all the conditions of Theorem 4.4 are satisfied, and so every solution \(x(t)\) of Eq. (5.4) is either oscillatory or satisfies \(\lim_{t\rightarrow \infty }x(t)=0\).
6 General remarks
-
(1)
In this paper, several new oscillation criteria for Eq. (1.1) have been presented which complement and improve the existing results introduced in the cited papers. In fact, our results are applicable in the cases either with \(p(t)\) is bounded or unbounded and where the restriction \(r^{\prime }(t)\geq 0\) imposed by the authors in [1, 8, 9, 14, 19], and [17] is dropped in this paper.
-
(2)
It is our belief that the present paper is of significance because it extends most of the cited papers which are concerned with unbounded \(p(t)\) and relaxes some of their conditions. For example, Theorem 4.5 includes Theorem 2.6 and Theorem 2.9 of [15], where the author was only concerned with the special case \(\alpha =\gamma \) with \(\int _{t_{0}}^{\infty }r^{-\frac{1}{\alpha }}(s)\,\mathrm {d}s=\infty \), and with the restriction \(\sigma (t)\) is nonincreasing. Moreover, our results in this paper extend those of [5] in the special case \(r(t)=1\), \(\alpha =1\), and \(f(u)=u^{\gamma }\), where \(\gamma \leq 1\). At the same time it extends those of [4] in the special case \(p(t)=0\), \(\alpha =\gamma \), with \(\sigma (t)\) being strictly increasing.
-
(3)
Our criteria could be extended to the dynamic equation on time scales. In this case, if we consider \(m_{\ast }(t)=\int _{t_{2}}^{t}\int _{t_{1}}^{s} \frac{1}{r^{\frac{1}{\alpha }}(u)}\Delta u \Delta s\) and \(\int _{t_{0}}^{\infty }\frac{1}{r^{\frac{1}{\alpha }}(s)}\Delta s= \infty \), then the obtained results will be more general than those of [10], because one may note that the results of [10] are applicable only in the case \(\gamma \leq \alpha \), \(0\leq p(t)\leq p_{0}<1\), and \(\sigma (t)\) is nondecreasing, while our results are applicable in the case \(\gamma >\alpha \) and \(p(t)\geq 1\).
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Sallam, R.A., Salem, S. & El-Sheikh, M.M.A. Oscillation of solutions of third order nonlinear neutral differential equations. Adv Differ Equ 2020, 314 (2020). https://doi.org/10.1186/s13662-020-02777-9
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DOI: https://doi.org/10.1186/s13662-020-02777-9
MSC
- 34C10
- 34K11
Keywords
- Oscillation
- Third order differential equation
- Nonlinear neutral equation
- Nonoscillation