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Oscillation criteria for second order EmdenFowler functional differential equations of neutral type
Journal of Inequalities and Applications volume 2016, Article number: 328 (2016)
Abstract
In this article, some new oscillation criterion for the second order EmdenFowler functional differential equation of neutral type
where \(z(t)=x(t)+p(t)x(\tau(t))\), \(\alpha>0\) and \(\beta>0\) are established. Our results improve some wellknown results which were published recently in the literature. Some illustrating examples are also provided to show the importance of our results.
Introduction
In this article we are concerned with the second order EmdenFowler functional differential equation of neutral type of the form
where \(z(t)=x(t)+p(t)x(\tau(t))\), \(\alpha>0\) and \(\beta>0\) are constants.
In the following we assume that
 \((A_{1})\) :

\(r(t)\in C^{1}([t_{0},\infty),R)\), \(r(t)>0\), \(r^{\prime }(t)\geq0\);
 \((A_{2})\) :

\(p(t),q(t)\in C([t_{0},\infty),R)\), \(0\leq p(t)\leq 1\), \(q(t)\geq0\);
 \((A_{3})\) :

\(\tau(t)\in C([t_{0},\infty),R)\), \(\tau(t)\leq t\), \(\lim_{t\rightarrow\infty}\tau(t)=\infty\);
 \((A_{4})\) :

\(\sigma(t)\in C^{1}([t_{0},\infty),R)\), \(\sigma (t)>0\), \(\sigma^{\prime}(t)>0\), \(\sigma(t)\leq t\), \(\lim_{t\rightarrow \infty}\sigma(t)=\infty\).
A function \(x(t)\in C^{1}([t_{0},\infty),R)\), \(T_{x}\geq t_{0}\), is called a solution of equation (1) if it satisfies the property \(r(t)\vert z^{\prime}(t)\vert ^{\alpha1}z^{\prime}(t)\in C^{1}([T_{x},\infty),R)\) and equation (1) on \([T_{x},\infty)\). In this article we only consider the nontrivial solutions of equation (1), which ensure \(\sup{ \{\vert x(t)\vert :t\geq T \}}>0\) for all \(T\geq T_{x}\). A solution of equation (1) is said to be oscillatory if it has arbitrarily large zero point on \([T_{0},\infty)\); otherwise, it is called nonoscillatory. Moreover, equation (1) is said to be oscillatory if all its solutions are oscillatory.
Recently, there were a large number of papers devoted to the oscillation of the delay and neutral differential equations. We refer the reader to [1–20].
Dzurina and Stavroulakis [1] studied the oscillation for the second order halflinear differential equations
and established some sufficient conditions for oscillation of (2).
Sun and Meng [2] examined further the oscillation of (2). Their results hold for the condition
or
which improves the results of Dzurina and Stavroulakis [1].
In 2008, Erbe et al. [3] studied the oscillatory behavior of the following second order neutral EmdenFowler differential equation:
where \(\int^{\infty}_{t_{0}}\frac{1}{a(t)}\,dt=\infty\) and \(\alpha>1\). Some new oscillation criteria of Philos type were established for equation (5).
In 2011, Li et al. [4] considered further the oscillation criteria for equation (5), where \(\int^{\infty}_{t_{0}}\frac{1}{a(t)}\,dt<\infty\) and \(\alpha\geq1\). In fact, equations (2) and (5) cannot be contained in each other. So in 2012, Liu et al. [5] considered the oscillation criteria for second order generalized EmdenFowler equation (1) for the condition \(\alpha\geq\beta>0\).
In 2015, Zeng et al. [6] used the Riccati transformation technique to get some new oscillation criterion for equation (1) under the condition \(\alpha\geq\beta>0\) or \(\beta\geq \alpha>0\), which improves the related results reported in [5].
Now in this article we shall apply the generalized Riccati inequality to study of the oscillation criteria of equation (1) under a more general case, namely, for all \(\alpha> 0\) and \(\beta> 0\).
Results and proofs
Theorem 1
Suppose that \((A_{1})\)\((A_{4})\) and (3) hold. If there exists a function \(\rho(t)\in C^{1}([t_{0},\infty), (0,\infty))\) such that
where
Then equation (1) is oscillatory for all \(\alpha> 0\) and \(\beta> 0\).
Proof
Suppose that equation (1) has a nonoscillatory solution \(x(t)\). Without loss of generality, we assume that \(x(t)>0\) for all large t. The case of \(x(t)<0\) can be treated by the same method. In view of \((A_{3})\) and \((A_{4})\), there exists \(t_{1}\geq t_{0}\) such that \(x(t)>0\), \(x(\tau(t))>0\), \(x(\delta(t))>0\) on \([t_{1},\infty)\). It follows that \(z(t)=x(t)+p(t)x(\tau(t))\geq x(t)>0\). It follows from (1) that
Hence, \(r(t)\vert z^{\prime}(t)\vert ^{\alpha1}z^{\prime}(t)\) is nonincreasing on \([t_{1},\infty)\).
We now claim that
If not, then there exists \(t_{3}\in[t_{2},\infty)\) such that \(z^{\prime}(t_{3})<0\). Hence
which implies that
Integrating (11) from \(t_{3}\) to t, we find from (3) that
which implies that \(z(t)\) is eventually negative. This contradicts \(z(t)>0\). Hence our claim is true.
Now we have
This inequality together with (1) and (10) suggest
We set
Then \(w(t)>0\). By (13) and (14) we have
In the following we consider three cases for (15):
Case (i): \(\alpha=\beta\). In view of the inequality \(r^{\frac{1}{\alpha }}(t)z^{\prime}(t)\leq r^{\frac{1}{\alpha}} (\sigma(t) )z^{\prime } (\sigma(t) )\) and (15) we see that
Case (ii): \(\alpha<\beta\). Noting that \(z(\sigma(t))\) is increasing on \([T,\infty)\), then there exists a constant \(m_{1}>0\) such that
Case (iii): \(\alpha>\beta\). From \((r(t) (z^{\prime}(t) )^{\alpha} )^{\prime}\leq0\) and \(r^{\prime}(t)\geq0\), we get \(z^{\prime\prime}(t)\leq0\), then \(z^{\prime}(t)\) is nonincreasing. Thus, there exists a positive constant \(m_{2}\), such that
Combining (16)(18), we obtain for any \(\alpha> 0\), \(\beta>0\),
Multiplying (19) by \(\rho(t)\) and integrating it from T to t, we obtain
By the inequality
where \(A\geq0\), \(B>0\), \(w\geq0\), and \(\lambda>0\), we now can rewrite inequality (20) as
Letting \(t\rightarrow\infty\) in the above inequality, we get a contradiction with (6). Hence the theorem is proved. □
Remark 1
Theorems 15 of [1], Theorem 1 of [2] and [7] hold only for equation (1) with \(p(t)=0\) and \(\alpha =\beta\). Theorem 2.1 of [5] (or [6]) holds only for equation (1) with \(\alpha\geq\beta\), and Theorem 3.1 of [6] holds only for equation (1) with \(\beta\geq\alpha\). Hence our theorem improves and unifies the above results.
In the following, we shall use the generalized Riccati technique and the integral averaging technique to show a new Philos type oscillation criterion for equation (1).
For this purpose, we first define the sets \(D_{0}=(t,s)\): \(t>s\geq t_{0}\) and \(D=(t,s)\): \(t\geq s\geq t_{0}\). We introduce a general class of parameter functions \(H:D\rightarrow R\), which have continuous partial derivatives on D with respect to the second variable and satisfy
 \((H_{1})\)::

\(H(t,t)=0\) for \(t\geq t_{0}\) and \(H(t,s)>0\) for all \((t,s)\in D_{0}\),
 \((H_{2})\)::

\(\frac{\partial H(t,s)}{\partial s}\geq0\) for all \((t,s)\in D\).
Suppose that \(h:D_{0}\rightarrow R\) is a continuous function and \(\rho \in C^{1}([t_{0},\infty),R^{+})\), such that
 \((H_{3})\)::

\(\frac{\partial H(t,s)}{\partial s}+\frac{\rho^{\prime }(s)}{\rho(s)}H(t,s)=h(t,s)H^{\frac{\lambda}{\lambda+1}}(t,s)\) for all \((t,s)\in D_{0}\).
Theorem 2
Suppose that \((A_{1})\)\((A_{4})\) and (3) hold. Suppose there exist functions H, h, and ρ, such that \((H_{1})\), \((H_{2})\), and \((H_{3})\) hold. Further assume for all sufficiently large T,
where λ, m, \(Q_{1}(t)\), and \(\lambda(t)\) are given in (7) and (8). Then equation (1) is oscillatory for all \(\alpha >0\) and \(\beta>0\).
Proof
Similar to Theorem 1, we assume that there exists a solution x of equation (1) such that \(x(t)>0\) on \([t_{1},\infty )\) for some \(t_{1}\geq t_{0}\). Multiplying both sides of (19) by \(H(t,s)\rho(s)\) and integrating from T to t, we have, for all \(t\geq T\geq t_{1}\),
where w is defined by (14) and
Applying integration by parts, from (\(H_{3}\)) and (24) we have
Using the inequality (21), combining (26) and (25), we get
It follows that
which contradicts the assumption (23). Therefore, equation (1) is oscillatory. Now we finish the proof of this theorem. □
Corollary 1
Theorem 2 remains true if the condition (23) is replaced by
and
Notice that by choosing specific functions ρ and H, it is possible to derive several oscillation criteria for equation (1) and its special cases, the halflinear equation (2) and the EmdenFowler equation (5).
Remark 2
Theorem 2.1 of [3] holds only for equation (1) with \(\alpha=1\) and \(\beta>1\), Theorem 2.2 of [5] holds only for equation (1) with \(\alpha\geq\beta\), Theorem 5 of [7] holds only for equation (1) with \(\beta\geq\alpha\). Hence, Theorem 2 improves and unifies above oscillation criteria.
Note that the theorems above hold for the condition (3), now we consider the case for (4). In order to do this we first define
and
Then we have the following.
Theorem 3
Suppose that \((A_{1})\)\((A_{4})\) and (4) hold. Suppose
and (6) are satisfied. Further assume there exists a constant \(K>0\) such that
where \(\mu=\max\{\alpha,\beta\}\). Then equation (1) is oscillatory for all \(\alpha>0\) and \(\beta>0\).
Proof
As in Theorem 1 we assume that there exists a solution x of equation (1) such that \(x(t)>0\) on \([t_{1},\infty)\) for some \(t_{1}\geq t_{0}\). Then we have
from which we see that there exist two possible cases of the sign of \(z^{\prime}(t)\). If \(z^{\prime}(t)>0\), then we come back to the proof of Theorem 1, and we can get a contradiction with (6). If \(z^{\prime}(t)<0\), we have
Therefore, \(x^{\prime}(t)<0\) and
from which together with (32) we have
Then equation (1) becomes
Now we define a function v by
Obviously, \(v(t)>0\) for \(t\geq t_{1}\). It follows from (34) that \(r(t)\vert z^{\prime}(t)\vert ^{\alpha1}z^{\prime}(t)\) is nonincreasing. Hence we get
Dividing the above inequality by \(r^{\frac{1}{\alpha}}(t)\) and integrating it from t to l, we have
It follows that
Moreover, we have
By (38) and the fact \(z^{\prime}(t)<0\) we find that there exists a constant \(c_{1}>0\) such that
On the other hand, from (39) we get
Hence we have
Since \(r^{\frac{1}{\alpha}}(t)(z^{\prime}(t))\) is nondecreasing, then there exists a constant \(c_{2}>0\) such that
Next, differentiating (38) yields
We consider the following three cases:
Case (i): \(\alpha> \beta\). In this case, since \(z(t)\) is decreasing, it follows from (43) that
where \(c_{1}= \beta [z(t_{1}) ]^{\frac{\beta\alpha}{\alpha}}\).
Case (ii): \(\alpha=\beta\). In this case, we see that \([z(t)]^{\frac{\beta\alpha}{\alpha}}=1\), then (43) becomes
Case (iii): \(\alpha<\beta\). By the inequality (37) we have \((r(t)(z^{\prime}(t))^{\alpha})^{\prime}\geq0\), from which together with \(r^{\prime}(t)\geq0\) we find that \(z^{\prime\prime}(t)\leq0\). Hence we get \(z^{\prime}(t)\leq z^{\prime}(t_{2})\) for \(t\geq t_{2}\). Now the inequality (43) suggests that
where \(c_{2}=\beta[z^{\prime}(t_{2})]^{\frac{\beta\alpha}{\beta}}\).
Combining (44)(46), we obtain
where \(\mu=\max\{{\alpha,\beta\}}\) and
Multiplying (47) by \(\pi^{\mu}(t)\) and integrating it from \(t_{2}\) to t, we have
Using integration by parts, the inequality (48) yields
By the inequality (21), we get
Substituting in (49), we obtain
where \(K=\frac{\mu^{2\mu+1} }{c^{\mu}(\mu+1)^{\mu+1}}\).
In view of (41) and (42), we have
which contradicts condition (33). Then equation (1) is oscillatory for all \(\alpha>0\) and \(\beta>0\). Hence the theorem is proved. □
Remark 3
Theorem 2.2 of [2] holds only for equation (1) with \(p(t)=0\) and \(\alpha=\beta\), Theorem 2.12.3 of [4] hold only for \(\alpha=1\) and \(\beta\geq1\), Theorem 2.5 of [5] and Theorem 2.3 of [6] hold only for \(\alpha\geq\beta\). Our Theorem 3 holds for equation (1) with all \(\alpha>0\) and \(\beta>0\).
Examples
Now in this section we shall give two examples to illustrate our results.
Example 1
Consider the differential equation
where \(z(t)=x(t)+\frac{1}{2}x(t1)\), \(\alpha>0\), \(\beta>0\), and \(\lambda=\min \{\alpha,\beta\}\). Noticing that \(r(t)=1\), \(p(t)=\frac{1}{2}\), \(q(t)=\frac{1}{t^{1+\frac {\lambda}{2}}}\), \(\tau(t)=t1\), \(\sigma(t)=t2\) and
then (3) is satisfied. Since \(Q_{1}(t)=(\frac{1}{2})^{\beta}\frac {1}{t^{1+\frac{\lambda}{2}}}\), \(\lambda>0\), then \(\int^{\infty }_{t_{0}}Q_{1}(t)\,dt<\infty\). In order to apply Theorem 1, it remains to discuss condition (6). If we choose \(\rho(t)=t^{\frac{\lambda}{2}}\), we have
Then by Theorem 1, every solution of (52) is oscillatory for all \(\alpha>0\) and \(\beta>0\).
Example 2
Consider the differential equation
where \(z(t)=x(t)+\frac{1}{3}x(t2)\), \(\alpha>0\), \(\beta>0\). Observe \(r(t)=t^{2\alpha}\), \(p(t)=\frac{1}{3}\), \(q(t)=t^{2\alpha+\beta}\), \(\tau(t)=t2\), \(\sigma(t)=t3\) and
Then (4) is satisfied. It is clear that (32) is satisfied. Since \(p(t)=\frac{1}{3}\), we have
If we choose \(\rho(t)=1\), then condition (6) is satisfied. To apply Theorem 3, it remains to discuss the condition (33); in view of \(\pi(t)=\frac{1}{t}\), we have
Then by Theorem 3, (53) is oscillatory for all \(\alpha >0\) and \(\beta>0\).
Remark 4
We note that the results obtained for those equations in [1–20] cannot deal with (52) and (53).
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Acknowledgements
The first author is supported by the Guangdong Engineering Technology Research Center of Cloud Robot (Grant 2015B090903084), sponsored by Science and technology project of Guangdong Province, P.R. China. The fourth author is supported by the National Natural Science Foundation of China (Grant 11501131) and the Training Project for Young Teachers in Higher Education of Guangdong, China (Grant YQ2015117).
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Wu, Y., Yu, Y., Zhang, J. et al. Oscillation criteria for second order EmdenFowler functional differential equations of neutral type. J Inequal Appl 2016, 328 (2016). https://doi.org/10.1186/s1366001612689
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MSC
 34C10
 34K11
Keywords
 EmdenFowler equation
 oscillation criterion
 Riccati method