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Oscillation and asymptotic properties of a class of second-order Emden–Fowler neutral differential equations
SpringerPlus volume 5, Article number: 1956 (2016)
Abstract
We consider a class of second-order Emden–Fowler equations with positive and nonpositve neutral coefficients. By using the Riccati transformation and inequalities, several oscillation and asymptotic results are established. Some examples are given to illustrate the main results.
Background
In this paper, we study a second-order delay differential equation
where \(z(t)=x(t)+ p(t)x(\tau (t))\) and \(\alpha\) is a positive constant. Throughout this paper, we assume that
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(\(H_{1}\)) \(r,\;p\in {\mathrm {C}}([t_0, \infty ),{\mathbb {R}}),\; r(t)>0,\; 0 \le p(t)\le 1,\;\text {and}\; \int _{t_0}^{\infty }r^{-1/\alpha }(t) {\mathrm{d}}t=\infty ;\)
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(\(H_{2}\)) \(r,\;p\in {\mathrm {C}}([t_0, \infty ),{\mathbb {R}}),\; r(t)>0,\; -1< -p_0 \le p(t)\le 0,\;\text {and}\; \int _{t_0}^{\infty }r^{-1/\alpha }(t) {\mathrm{d}}t=\infty ,\) where \(p_0\) is a positive constant;
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(\(H_{3}\)) \(\tau \in {\mathrm {C}}([t_0, \infty ),{\mathbb {R}}), \; \tau (t)\le t, \; \text {and} \; \lim _{t\rightarrow \infty }\tau (t)=\infty ;\)
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(\(H_{4}\))\(\sigma \in {\mathrm{C}}^{1}([t_0, \infty ),{\mathbb {R}}),\; \sigma (t)\le t,\; \sigma^{\prime}(t)>0,\;\text {and} \; \lim _{t\rightarrow \infty }\sigma (t)=\infty .\)
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(\(H_{5}\)) \(f\in {\mathrm {C}}([t_0,\infty )\times {\mathbb {R}},{\mathbb {R}}),\;uf(u)\ge 0\; \text {for all}\; u\ne 0,\) and there exist a positive constant \(\beta \;\) and a function \(\; q(t)\in {\mathrm {C}}([t_0,\infty ),(0,\infty ))\; \text {such that}\; (f(t,u)/u^{\beta })\ge q(t),\) for all \(u\ne 0\), where \(1<\beta \le \alpha .\)
It is recognized that Emden–Fowler equations have a number of applications in physics and engineering; see, e.g., Berkovich (1997). As a result, there has been a great deal of interest in investigating the oscillation or nonoscillation of differential equations; see, e.g., Hale (1977), Džurina and Stavroulakis (2003), Li (2004), Li et al. (2011, 2013, 2015), Li and Rogovchenko (2015), Erbe et al. (2009), Manojlović (1999), Wang and Yang (2004), Wang (2001), Liu et al. (2012), Shi et al. (2016), Baculíková and Džurina (2011), Bohner and Li (2014), Yang et al. (2006), Xu and Meng (2006, 2007). As we known, many results of half-linear or nonlinear equations with positive neutral coefficients were established, see, e.g., Baculíková and Džurina (2011), Erbe et al. (2009), Li et al. (2011, 2013), Li and Rogovchenko (2015), Liu et al. (2012), Shi et al. (2016), Yang et al. (2006), Xu and Meng (2006, 2007). The equations with nonpositive neutral coefficients have been applied to practical life; see, for instance, Brayton (1966) and Kuang (1993, sec. 1.1.7) provided a model about the system with lossless transmission lines. And, there have been a few oscillation and asymptotic results of the equations with nonpositive neutral coefficients, see, e.g., Bohner and Li (2014), Erbe et al. (2009), Li et al. (2015), Yang et al. (2006). In the following, we provide some background details which motivated our research. Manojlović (1999), Wang (2001), and Wang and Yang (2004) considered the half-linear differential equation
and gave some different oscillation results by using an inequality due to Hardy, Littlewood and Ploya and averaging functions. Motivated by these ideas, many scholars extended the results to delay differential equations or neutral delay differential equations. Džurina and Stavroulakis (2003) and Li (2004) expanded the Eq. (2) to the delay differential equation
Xu and Meng (2006, 2007) extended (2) to the neutral delay differential equation
provided that \(0\le p(t)\le 1\). Li et al. (2015) established some oscillation and asymptotic results to (1) in the case where \(-1<p(t)\le 0\). By using Riccati transformation, Erbe et al. (2009) proposed some oscillation and asymptotic results for (1), under the assumptions that \(0\le p(t)<1\) and \(-1< p(t)< 0\). Liu et al. (2012) considered the the following equation
in the case where \(0\le p(t)\le 1\) and \(\alpha \ge \beta >0\). They established some oscillation and asymptotic criteria by employing averaging technique and Riccati transformation. Shi et al. (2016) extended the results of Liu et al. (2012) to dynamic equations on time scales provided that \(0\le p(t)\le 1\) and \(p(t)> 1\).
However, the results of Liu et al. (2012) and Shi et al. (2016) cannot be applied to Eq. (1) due to \(-1<p(t)\le 0\) in (1), but, in Liu et al. (2012), Shi et al. (2016) the assumption is \(0\le p(t)\le 1\) or \(p(t)>1\). Similarly, the results in Erbe et al. (2009) and Li et al. (2015) cannot be applied to Eq. (1) because there is another parameter \(\beta\) and the condition on function f in Li et al. (2015), Erbe et al. (2009) does not satisfy the hypothesis \((H_5)\). In this paper, we will extend the results of Liu et al. (2012), Shi et al. (2016) to the case of \(-1< p(t) \le 0\) and improve the results of Erbe et al. (2009), Li et al. (2015). By employing Riccati transformation, several new oscillation and asymptotic criteria are obtained under the assumptions that \((H_1)-(H_5)\). Throughout this paper, we suppose that all inequalities hold for sufficiently large t. Without loss of generality, we only consider the positive solutions of (1).
In what follows, let \(D=\{(t, s):t_0\le s \le t\}\; \mathrm {and}\;D_0=\{(t, s):t_0\le s < t\}.\) We say a function \(H=H(t,s)\) belongs to a function class P, if it satisfies
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(i)
\(H(t, t)=0, t\ge t_0; H(t,s)>0, (t, s)\in D_0;\)
-
(ii)
H has partial derivatives \(\partial H/\partial t\) and \(\partial H{/}\partial s\) on \(D_0\), such that
$$\begin{aligned} \frac{\partial H(t, s)}{\partial t}=h_1(t, s)\sqrt{H(t, s)} \end{aligned}$$and
$$\begin{aligned} \frac{\partial H(t, s)}{\partial s}=-h_2(t, s)\sqrt{H(t, s)}, \end{aligned}$$where \(h_1\) and \(h_2\) are nonnegative continuous functions on \(D_0\).
Main results
In this section, we discuss the Eq. (1) under the assumptions that \(-1< p(t)\le 0\) and \(0\le p(t)\le 1\), respectively.
Oscillation of Eq. (1) when \(-1<p(t)\le 0\)
Theorem 1
Assume that \((H_2)-(H_5)\) hold. If there exists a function \(\rho \in {\mathrm{C}}^1([t_0, \infty ), (0,\infty ))\) such that, for any constant \(K>0\),
where \(\Phi (t)=[\beta \sigma ^{\prime }(t)(\xi (\sigma (t)))^{\beta -1}]/[K^{(1-\beta /\alpha )}\rho (t)(r(\sigma (t)))^{1/\alpha }]\), then all solutions of Eq. (1) are oscillatory or tend to zero as \(t\rightarrow \infty\).
Proof
Suppose x is a nonoscillatory solution of (1). Without loss of generality, there exists a \(t_1\ge t_0\), such that \(x(t)>0, x(\tau (t))>0,\;\text {and}\;x(\sigma (t))>0,\) for all \(t\ge t_1\). From (1) and the hypothesis \((H_5)\), we get
Therefore, \(r|z^{\prime }|^{\alpha -1}z^{\prime }\) is nonincreasing. We claim that \(z^{\prime }>0\). Otherwise, if \(z^{\prime }<0\), using the fact that \(r|z^{\prime }|^{\alpha -1}z^{\prime }\) is nonincreasing, there exists a positive \(k>0\), such that
That is,
Integrating the above inequality from \(t_1\) to t, we get
It follows from \((H_2)\) that
We consider the following two cases.
Case 1 If x is unbounded, then there exists a sequence \(\{t_m\}\), such that
where \(\{t_m\}\) satisfies \(\lim _{m\rightarrow \infty }t_m=\infty\) and \(x(t_m)=\max _{t_0\le s\le t_m}\{x(s)\}\). By the definition of \(x(t_m)\) and \(\tau (t)\le t\), we have
Then we get
which contradicts (7).
Case 2 If x is bounded, from the definition of z and \(-1< p(t) \le 0\), z is also bounded, which also contradicts (7).
Hence, it is clear from the above discussion that \(z^{\prime }>0\), and then \(z>0\) or \(z<0\). We consider each of two cases separately.
Suppose first that \(z>0\). Considering the definition of z and \(-1<p(t)\le 0\), we get
From \(\sigma (t)\le t\) and the fact that \(r(z^{\prime })^{\alpha }\) is nonincreasing, we obtain
and there exist a positive constant K and a \(t_2\ge t_1\), such that
From the fact that \(r(z^{\prime })^{\alpha }\) is nonincreasing, we get
where \(\xi (t)=\int _{t_1}^t r^{-1/{\alpha }}(s) {\mathrm{d}}s\). Define a function \(\omega\) by
then \(\omega (t)>0\). Differentiating \(\omega\), we get
From (1) and (8), we conclude that
Taking into account (11), the last inequality implies
It follows from (9), (10), and (12) that
where \(\Phi (t)=[\beta \sigma ^{\prime }(t)(\xi (\sigma (t)))^{\beta -1}]/[K^{(1-\beta /\alpha )}\rho (t)(r(\sigma (t)))^{1/\alpha }].\) Integrating (13) from \(t_2\) to t, we get
which contradicts (5).
If \(z<0\), we claim that \(\lim _ {t\rightarrow \infty }{x(t)}=0\). Using \(z<0\) and \(z^{\prime }>0\), we deduce that
where l is a constant. That is, for all sufficiently large t, z is bounded. We can easily prove that x is also bounded. From the fact that x is bounded, we get
where a is a constant. We claim that \(a=0\). Otherwise, if \(a>0\), there exists a sequence \(\{t_n\}\), such that \(\lim _{n\rightarrow \infty }t_n=\infty\) and \(\lim _{n\rightarrow \infty } x(t_n)=a.\) Letting
for large enough n, we obtain
Then, from the definition of z(t) and \(p(t)\ge -p_0\), we have
which contradicts \(z<0\). Thus \(\limsup _{t\rightarrow \infty } x(t)=0.\) By \(x>0\), we get
The proof is complete. \(\square\)
From Theorem 1, letting \(\rho =1\), we get the following corollary.
Corollary 1
Assume that \((H_2)-(H_5)\) hold. If
then all solutions of Eq. (1) are oscillatory or tend to zero as \(t\rightarrow \infty\).
Theorem 2
Assume that \((H_2)-(H_5)\) hold. If there exist two functions \(H\in P\) and \(\rho \in {\mathrm{C}}^1([t_0, \infty ), (0,+\infty ))\), such that
where \(\Phi\) is as in Theorem 1, then the conclusion of Theorem 1 remains intact.
Proof
Suppose that x is a nonoscillatory solution of (1). Proceeding as in the proof of Theorem 1, we get \(z>0\) or \(z<0\).
Firstly, we consider \(z>0\). As the proof above (13) holds. That is
Multiplying this inequality by H(t, s) and integrating it from \(t_2\) to t, we have
By the property of \(\partial H(t, s)/\partial s=-h_2(t, s)\sqrt{H(t, s)}<0\), we conclude that
Adding \(\int _{t_0}^{t_2}\left[H(t,s)\rho (s)q(s)- \frac{\left( \frac{\rho ^{\prime }(s)}{\rho (s)}H(t,s)-h_2(t,s)\sqrt{H(t,s)}\right) ^2}{4H(t,s)\Phi (s)}\right]{\mathrm{d}}s\) to the latter inequality and multiplying this inequality by \(1/H(t,t_0)\), we get
Letting \(t\rightarrow \infty\) in (16), we can get a contradict to (14).
If \(z<0\), repeating the proof of Theorem 1, we have \(\lim _ {t\rightarrow \infty }{x(t)}=0\). This completes the proof. \(\square\)
From Theorem 2, letting \(H(t,s)=(t-s)^{\lambda } (\lambda >0)\) and \(\rho (t)=1\), we may get the following corollary.
Corollary 2
Assume that \((H_2)-(H_5)\) hold. If
then the conclusion of Theorem 1 remains intact.
Theorem 3
Assume that \((H_2)-(H_5)\) hold. If there exist three functions \(H\in P\), \(\rho \in ([t_0, \infty ), (0,\infty ))\), and \(A\in {\mathrm {C}}([t_0,\infty ), {\mathbb {R}})\), such that
and
for all \(T\ge t_0\) and for some constant \(\theta >1\), where \(A_{+}(t)=\max \{A(t),0\}\), \(\Phi\) is as in Theorem 1, and H satisfies
then every solution of Eq. (1) is oscillatory or tends to zero as \(t\rightarrow \infty\).
Proof
Suppose that x is a nonoscillatory solution of Eq. (1). Then, as in the proof of Theorem 1, \(z>0\) or \(z<0\).
We consider \(z>0\) firstly. By virtue of Theorem 2, (15) holds. That is, for all \(T\ge t_2\ge t_1,\)
Thus,
Taking into account (19) and (21), we deduce that
then
and
Now we will prove that
On the contrary, if
By the condition (20), there exists a positive constant c, such that
On the other hand, using (24), for arbitrary positive M, there exists a \(t_3\ge T\), such that
From the property (ii) of H and (25), we have
Taking account into the fact that M is arbitrary positive constant, (26) implies that
which contradicts (23). Thus,
Using (22) and \(\Phi (t)>0\), we have
which contradicts (18).
If \(z<0\), repeating the proof in Theorem 1, we get \(\lim _{t\rightarrow \infty }x(t)=0\). The proof is complete. \(\square\)
Remark 1
Theorem 1–3 and Corollaries 1 and 2 are the oscillation and asymptotic results of (1) under the assumption that \(-1<p(t)\le 0\). However, the results in Liu et al. (2012), Shi et al. (2016) are established in the case where \(0\le p(t)\le 1\). In the hypothesis \((H_5)\), there is another parameter \(\beta\) and the condition on function f in Li et al. (2015), Erbe et al. (2009) does not satisfy \((H_5)\). Therefore, the results of Li et al. (2015), Erbe et al. (2009) can not apply to Eq. (1).
Oscillation of Eq. (1) when \(0\le p(t)\le 1\)
Theorem 4
Assume that \((H_1)\) and \((H_3)-(H_5)\) hold. If there exists a function \(\rho (t)\in {\mathrm{C}}^1([t_0,\infty ),(0,\infty ))\) such that for any positive number M,
where \({\overline{p}}(t)=q(t)(1-p(\sigma (t)))^\beta\) and \(\xi (t)=\int _{t_1}^t r^{-1/{\alpha }}(s) {\mathrm{d}}s\), then the Eq. (1) is oscillatory.
Proof
See “Appendix”. \(\square\)
Letting \(\rho (t)=1\), we can get the following result.
Corollary 3
Assume that \((H_1)\) and \((H_3)-(H_5)\) hold. If
then the Eq. (1) is oscillatory.
Example 1
Consider the second-order nonlinear neutral delay differential equation
where \(\beta =2, \alpha \ge 2, r(t)=1, p(t)=1-t^{-\frac{1}{2}}, \sigma (t)=t,\; \text {and} \;q(t)=\frac{\gamma }{t^2}\), where \(\gamma\) is a positive constant.
Letting \(\rho (t)=t^2\), we have
and
Therefore, if \(\gamma > \frac{1}{2M^{1-(2/{\alpha })}}\), then
It follows from Theorem 4 that all solutions of (28) are oscillatory if \(\gamma > \frac{1}{2M^{1-(2/{\alpha })}}\). However, the Eq. (28) is oscillatory when \(\gamma > \frac{1}{M^{1-(2/{\alpha })}}\) from Theorem 1 in Liu et al. (2012). That is, if \(\frac{1}{2M^{1-(2/{\alpha })}}<\gamma \le \frac{1}{M^{1-(2/{\alpha })}}\), then Theorem 1 in Liu et al. (2012) can not apply to (28).
Remark 2
From Theorem 4, the Eq. (28) is oscillatory if \(\gamma > \frac{1}{2M^{1-(2/{\alpha })}}\). However, from Theorem 1 in Liu et al. (2012), when \(\gamma > \frac{1}{M^{1-(2/{\alpha })}}\), the Eq. (28) is oscillation. Thus, Theorem 4 improves Theorem 1 in Liu et al. (2012) in the case where \(0\le p(t)\le 1\).
Examples
In this section, we will present two examples to illustrate the main results.
Example 2
Consider the second-order nonlinear neutral delay differential equation
where \(r(t)=1, p(t)=-1/2, \tau (t)=t-2, \sigma (t)=t,\; \text {and} \;q(t)=2\left( \frac{e^2}{2}-1\right) ^2\).
We see that
It follows from Corollary 1 that all solutions of (29) are oscillatory or converge to zero. Letting \(\alpha =\beta =2\), we can certify that \(x(t)=e^{-t}\) is an asymptotic solution of (29).
Example 3
Consider the second-order nonlinear neutral delay differential equation
where \(\alpha >\beta \ge 1\), \(0<\lambda _0<1, 0<\sigma _0<1\), \(r(t)=1, p(t)=\lambda _0, \tau (t)\le t, q(t)=\frac{\gamma }{t^2},\) and \(\sigma (t)=\sigma _0t\).
Letting \(\rho (t)=\frac{\gamma }{2} \frac{\beta \sigma _0\left( t-\frac{t_1}{\sigma _0}\right) ^{\beta -1}}{K^{1-\frac{\beta }{\alpha }}}\) and \(H(t,s)=(t-s)^2\), then
and
That is, (17) holds. It follows from Corollary 2 that all solutions of (30) are oscillatory or converge to zero.
Remark 3
If \(\alpha >\beta\), the results of Erbe et al. (2009) and Li et al. (2015) can not apply to (29) and (30). The results of Liu et al. (2012) and Shi et al. (2016) also can not apply to (29) and (30) because \(-1< p(t)\le 0\) which does not satisfy the assumptions in Liu et al. (2012), Shi et al. (2016).
Conclusion
In this paper, we consider the oscillation of a class of second-order differential equations with positive and nonpositive neutral coefficients. It is difficult to study the nonpositive neutral coefficients equations because the sign of z is not explicit. Using Riccati transformation, some oscillation and asymptotic criteria are obtained under the assumptions that \((H_1)-(H_5)\). In Liu et al. (2012), Shi et al. (2016), the results were established for (1) in the case when \(0\le p(t)\le 1\) or \(p(t)>1\). This paper states some oscillation and asymptotic criteria for (1) in the case where \(-1<p(t)<0\) and \(0\le p(t)\le 1\). Erbe et al. (2009), Li et al. (2015) assume that \(\alpha =\beta\), however, in this paper \(\alpha \ne \beta\) is allowed. We give some examples to illustrate our results. There are two interesting questions for future study:
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Authors’ contributions
Both authors contributed equally to this paper. Both authors read and approved the final manuscript.
Acknowledgements
This work is supported by NSF of Shandong Province, China (Grant No: ZR2014FM036). The authors would like to thank the reviewers for the valuable comments.
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Both authors declare that they have no competing interests.
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Appendix
Appendix
In this section, we give the proof of Theorem 4.
Proof
Assume that x is a nonoscillatory solution of the Eq. (1). Without loss of generality, there exists a \(t_1\ge t_0\), such that \(x(t)>0, x(\tau (t))>0,\; \text {and}\; x(\sigma (t))>0\), for all \(t\ge t_1\). Then, by the definition of z, we know
From (1) and the assumption \((H_5)\), we get
So, \(r|z^{\prime }|^{\alpha -1}z^{\prime }\) is a nonincreasing function. We claim that \(z^{\prime }>0\). Otherwise, if \(z^{\prime }<0\), using the fact that \(r|z^{\prime }|^{\alpha -1}z^{\prime }\) is a nonincreasing, there exists a positive \(c>0\), such that
Then, we have
Integrating the above inequality from \(t_1\) to t, we obtain that
Letting \(t\rightarrow \infty\), from \((H_1)\), we have
which contradicts \(z>0\). Thus, \(z^{\prime }>0\).
From (1), (31), and the fact that z is increasing, we conclude that
Using the fact that \(r(z^{\prime })^{\alpha }\) is nonincreasing, (11) holds and there exists a positive constant M and \(t_2\ge t_1\), such that
and form \(\sigma (t)\le t\), we get
Define a function \(\omega\) by
then \(\omega (t)>0\). Differentiating \(\omega\), we get
From (33), we conclude that
where \({\overline{p}}(t)=(1-p(\sigma (t)))^\beta\). Taking into account (11) and \(\alpha \ge \beta \ge 1\), the last inequality implies
It follows from (34) to (36) that
Integrating (37) from \(t_2\) to t, we get
which contradicts (27). This completes the proof. \(\square\)
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Wang, R., Li, Q. Oscillation and asymptotic properties of a class of second-order Emden–Fowler neutral differential equations. SpringerPlus 5, 1956 (2016). https://doi.org/10.1186/s40064-016-3622-2
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Keywords
- Oscillation
- Second-order differential equation
- Emden–Fowler equation
- Nonpositive neutral coefficient
Mathematics Subject Classification
- 34K11
