Oscillation criteria for second order EmdenFowler functional differential equations of neutral type
Abstract
Keywords
EmdenFowler equation oscillation criterion Riccati methodMSC
34C10 34K111 Introduction
 \((A_{1})\)

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

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

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

\(\sigma(t)\in C^{1}([t_{0},\infty),R)\), \(\sigma (t)>0\), \(\sigma^{\prime}(t)>0\), \(\sigma(t)\leq t\), \(\lim_{t\rightarrow \infty}\sigma(t)=\infty\).
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, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20].
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 \(\beta> 0\).
2 Results and proofs
Theorem 1
Proof
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).
 \((H_{1})\):

\(H(t,t)=0\) 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 \((t,s)\in D\).
 \((H_{3})\):

\(\frac{\partial H(t,s)}{\partial s}+\frac{\rho^{\prime }(s)}{\rho(s)}H(t,s)=h(t,s)H^{\frac{\lambda}{\lambda+1}}(t,s)\) for all \((t,s)\in D_{0}\).
Theorem 2
Proof
Corollary 1
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.
Theorem 3
Proof
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 \(\beta>0\).
3 Examples
Now in this section we shall give two examples to illustrate our results.
Example 1
Example 2
Notes
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).
References
 1.Dzurina, J, Stavroulakis, IP: Oscillation criteria for second order delay differential equations. Appl. Math. Comput. 140, 445453 (2003) MathSciNetMATHGoogle Scholar
 2.Sun, YG, Meng, FW: Note on the paper of Dgurina and Stavroulakis. Appl. Math. Comput. 174, 16341641 (2006) MathSciNetMATHGoogle Scholar
 3.Erbe, L, Hassan, TS, Peterson, A: Oscillation of second order neutral delay differential equations. Adv. Dyn. Syst. Appl. 3, 5371 (2008) MathSciNetGoogle Scholar
 4.Li, TX, Han, ZL, Zhang, CH, Sun, SR: On the oscillation of second order EmdenFowler neutral differential equations. J. Appl. Math. Comput., Int. J. 37, 601610 (2011) MathSciNetCrossRefMATHGoogle Scholar
 5.Liu, HD, Meng, FW, Liu, PH: Oscillation and asymptotic analysis on a new generalized EmdenFowler equation. Appl. Math. Comput. 219, 27392748 (2012) MathSciNetMATHGoogle Scholar
 6.Zeng, YH, Lou, LP, Yu, YH: Oscillation for EmdenFowler delay differential equations of neutral type. Acta Math. Sci. 35A, 803814 (2015) MathSciNetMATHGoogle Scholar
 7.Tiryaki, A: Oscillation criteria for a certain second order nonlinear differential equations with deviating arguments. Electron. J. Qual. Theory Differ. Equ. 2009, 61 (2009) MathSciNetMATHGoogle Scholar
 8.Baculikova, B, Dgurina, J: Oscillation theorems for second order nonlinear neutral differential equations. Comput. Math. Appl. 62, 44724478 (2011) MathSciNetCrossRefMATHGoogle Scholar
 9.Baculikova, B, Li, T, Dzurina, J: Oscillation theorems for second order superlinear neutral differential equations. Math. Slovaca 63, 123134 (2013) MathSciNetCrossRefMATHGoogle Scholar
 10.Hasanbulli, M, Rogovchenko, YV: Oscillation criteria for second order nonlinear neutral differential equations. Appl. Math. Comput. 215, 43924399 (2010) MathSciNetMATHGoogle Scholar
 11.Dong, JG: Oscillation behavior of secondorder nonlinear neutral differential equations with deviating arguments. Comput. Math. Appl. 59, 37103717 (2010) MathSciNetCrossRefMATHGoogle Scholar
 12.Karpug, B, Manojlovic, JV, Ocalan, O, Shoukaku, Y: Oscillation criteria for a class of second order neutral delay differential equations. Appl. Math. Comput. 210, 303312 (2009) MathSciNetMATHGoogle Scholar
 13.Hasanbulli, M, Rogovchenko, YV: Oscillation of nonlinear neutral functional differential equations. Dyn. Contin. Discrete Impuls. Syst. 16, 227233 (2009) MathSciNetMATHGoogle Scholar
 14.Li, T, Rogovchenko, YV, Zhang, C: Oscillation of second order neutral differential equations. Funkc. Ekvacioj 56, 111120 (2013) MathSciNetCrossRefMATHGoogle Scholar
 15.Liu, L, Bai, Y: New oscillation criteria for second order nonlinear delay neutral differential equations. J. Comput. Appl. Math. 231, 657663 (2009) MathSciNetCrossRefMATHGoogle Scholar
 16.Qin, H, Shang, N, Lu, Y: A note on oscillation criteria of second order nonlinear neutral delay differential equations. Comput. Math. Appl. 56, 29872992 (2008) MathSciNetCrossRefMATHGoogle Scholar
 17.Rogovchenko, YV, Tuncay, F: Oscillation criteria for second order nonlinear differential equations with damping. Nonlinear Anal. 69, 208221 (2008) MathSciNetCrossRefMATHGoogle Scholar
 18.Wang, XL, Meng, FW: Oscillation criteria of second order quasilinear neutral delay differential equations. Math. Comput. Model. 46, 415421 (2007) MathSciNetCrossRefMATHGoogle Scholar
 19.Xu, R, Meng, FW: Oscillation criteria for second order quasilinear neutral delay differential equations. Appl. Math. Comput. 192, 216222 (2007) MathSciNetMATHGoogle Scholar
 20.Ye, L, Xu, Z: Oscillation criteria for second order quasilinear neutral delay differential equations. Appl. Math. Comput. 207, 388396 (2009) MathSciNetMATHGoogle Scholar
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