Abstract
The aim of this paper is to investigate oscillatory properties of a class of second-order nonlinear differential equations with damping. Employing the generalized Riccati transformation and a class of functions, several oscillation criteria are presented that improve the results obtained in the literature. Two examples are presented to demonstrate the main results.
Similar content being viewed by others
1 Introduction
During the past few decades, the oscillation of differential equations has attracted a great deal of interest in various fields due to its theoretical and practical applications in natural sciences and technology. For instance, the oscillation of a building or a machine, the beam vibration in a synchrotron accelerator, the complicated oscillation in a chemical reaction, and so on; see, e.g., [1, 2]. For some related contributions on the oscillation of various classes of differential equations, we refer the reader to [3–27] and the references cited therein. In particular, the oscillatory behavior of second-order damped differential equations has been studied by many authors due to the fact that such equations arise in the study of noise, vibration, and harshness (NVH) of vehicles, see, e.g., the paper by Fu et al. [4].
In this paper, we are concerned with the oscillation of a nonlinear second-order damped differential equation
where \(h, q\in{C([t_{0},\infty),\mathbb{R})}\), \(\psi, f, g\in{C(\mathbb{R},\mathbb{R})}\), and \(H\in{C([t_{0},\infty )\times{\mathbb{R}}^{2}, \mathbb{R})}\). We suppose also that the following hypotheses are satisfied:
- (H1):
-
\(r\in{C([t_{0},\infty),(0,\infty))}\);
- (H2):
-
\(0< k_{1}\leq{\psi(x)}\leq{k_{2}}\) for all \(x\neq0\);
- (H3):
-
there exists a constant \(m>0\) such that \(f^{2}(y)\leq {myf(y)}\) for all \(y\in{\mathbb{R}}\);
- (H4):
-
\(H(t,y,x)/g(x)\leq{p(t)}\) for \(t\in{[t_{0},\infty)}\), \(x, y\in{\mathbb{R}}\), \(x\neq{0}\), and \(p\in{C([t_{0},\infty),\mathbb {R})}\).
As usual, a solution x of (1.1) is called oscillatory if the set of its zeros is unbounded from above; otherwise, it is said to be nonoscillatory. Equation (1.1) is termed oscillatory if all its solutions are oscillatory.
In what follows, we present some background details that motivate the study of this paper. Grace and Lalli [6], Kirane and Rogovchenko [7], Li and Agarwal [8], Rogovchenko [9], Rogovchenko and Tuncay [10], and Sun [23] considered a particular case of (1.1), namely, the second-order damped equation
In particular, Sun [23] used a class of functions Y defined in the sequel to establish several Kamenev-type (see [25]) oscillation criteria for (1.2). Grace [11], Grace and Lalli [17], Kirane and Rogovchenko [18], Manojlović [19], Rogovchenko and Tuncay [20], and Tunç and Avci [21] investigated the oscillation of the nonlinear damped differential equations
and
Very recently, Salhin et al. [22] established several oscillation criteria for (1.1) by using a generalized Riccati transformation, some of which we present below for convenience of the reader.
Theorem 1.1
([22], Corollary 2.1)
Let assumptions (H1)-(H4) be fulfilled and assume
- (H5):
-
\(g'(x)\) exists and \(g'(x)\geq{k}>0\) for all \(x\neq {0}\).
If there exist functions \(\delta, \tilde{\phi}\in C([t_{0},\infty),\mathbb{R})\) such that \((r\delta)\in C^{1}([t_{0},\infty),\mathbb{R})\),
and
for any \(T\geq{t_{0}}\) and \(\alpha>{1}\), where
and \(\tilde{\phi}_{+}(t)=\max\{\tilde{\phi}(t),0\}\), then (1.1) is oscillatory.
Theorem 1.2
([22], Corollary 2.2)
Let assumptions (H1)-(H5) hold. Suppose that there exists a function \(\delta\in C([t_{0},\infty),\mathbb{R})\) such that \((r\delta)\in C^{1}([t_{0},\infty), \mathbb{R})\) and
for some integer \(n>2\) and \(\alpha\geq{1}\), where
and Q̃ is defined as in Theorem 1.1. Then (1.1) is oscillatory.
Equations (1.2), (1.3), and (1.4) are special cases of (1.1). Note that Theorems 1.1 and 1.2 are Kamenev-type or Philos-type (see [26]) criteria for (1.1). The natural question now is: Can one apply methods reported in [23] to (1.1) and improve Theorems 1.1 and 1.2? The objective of this paper is to give an affirmative answer to this question.
Now, we introduce a class of functions Y. Let \(\mathbb{E}=\{(t,s,l): t_{0}\leq{l}\leq{s}\leq{t}<\infty\}\). We say that a function \(\Phi\in C(\mathbb{E},\mathbb{R})\) belongs to Y, denoted by \(\Phi\in{Y}\), if
-
(i)
\(\Phi(t,t,l)=0\), \(\Phi(t,l,l)=0\), and \(\Phi(t,s,l)\neq{0}\) for \(l< s< t\);
-
(ii)
Φ has the partial derivative \(\partial{\Phi}/\partial{s}\) in \(\mathbb{E}\) such that \(\partial{\Phi}/\partial{s}\) is locally integrable with respect to s in \(\mathbb{E}\) and satisfies
$$ \frac{\partial{\Phi(t,s,l)}}{\partial{s}}=\phi(t,s,l)\Phi (t,s,l). $$(1.5)
Next, we define the operator \(A[\cdot;l,t]\) by
It is obvious that \(A[\cdot;l,t]\) is a linear operator and satisfies
In what follows, all functional inequalities are assumed to hold for all t large enough, unless mentioned otherwise.
2 Oscillation criteria for increasing g
Theorem 2.1
Assume conditions (H1)-(H5) hold. Equation (1.1) is oscillatory provided that, for each \(l\geq{t_{0}}\), there exist three functions \(\Phi\in{Y}\), \(\rho\in{C^{1}([t_{0},\infty ), (0,\infty))}\), and \(b\in{C^{1}([t_{0},\infty),\mathbb{R})}\) such that
where
\(\phi=\phi(t,s,l)\) and A are defined by (1.5) and (1.6), respectively.
Proof
Let x be a nonoscillatory solution of (1.1). Without loss of generality, we may suppose that \(x(t)>0\) for \(t\geq{t_{1}}\geq{t_{0}}\). A similar argument holds for the case when x is eventually negative. Define a generalized Riccati transformation w by
Differentiating (2.2) and using (1.1), we have
By virtue of (H1) and (H3)-(H5),
From (H2) and (2.2), we conclude that
Using the inequality
we get, for \(t\geq{t_{1}}\),
Applying \(A[\cdot;l,t]\) (\(t\geq{l}\geq{t_{1}}\)) to (2.3), we obtain
Combining (1.7) and the latter inequality, we have, for \(t\geq{l}\geq{t_{1}}\),
Hence
for \(t\geq{l}\geq{t_{1}}\), which contradicts (2.1). Therefore, all solutions of (1.1) are oscillatory. The proof is complete. □
With an appropriate choice of the functions Φ, we can obtain a number of oscillation criteria for (1.1) by Theorem 2.1. For example, assume that \(\Phi(t,s,l)=(R(t)-R(s))^{\gamma}(R(s)-R(l))^{\beta}\) for \(\gamma, \beta>1/2\) and \(R\in{C^{1}([t_{0},\infty), \mathbb{R})}\) satisfying \((R(t)-R(s))(R(s)-R(l))\neq0\) for \(l< s< t\). By a simple calculation,
Thus, we derive the following oscillation result.
Corollary 2.1
Let conditions (H1)-(H5) be satisfied. If there exist three functions \(\rho\in C^{1}([t_{0},\infty),(0,\infty))\), \(b\in{C^{1}([t_{0},\infty), \mathbb{R})}\), \(R\in{C^{1}([t_{0},\infty), \mathbb{R})}\), and two constants \(\gamma, \beta>1/2\) such that \((R(t)-R(s))(R(s)-R(l))\neq0\) for \(l< s< t\) and, for all \(l\geq t_{0}\),
where the functions Q and a are the same as in Theorem 2.1, then (1.1) is oscillatory.
Letting \(r(t)=1\), \(\rho(t)=1\), and \(\Phi(t,s,l)=(t-s)^{\gamma}(s-l)\), we have the following criterion.
Corollary 2.2
Let conditions (H1)-(H5) hold. Equation (1.1) with \(r(t)=1\) is oscillatory provided that, for each \(l\geq t_{0}\), there exist a function \(b\in{C^{1}([t_{0},\infty), \mathbb{R})}\) and a constant \(\gamma>1/2\) such that
where the functions Q and a are as in Theorem 2.1.
Proof
Note that
It follows from (2.5) that
Thus, by (2.4) and (2.6), we have
Consequently, (1.1) with \(r(t)=1\) is oscillatory by Corollary 2.1. This completes the proof. □
Similarly, the following result can be obtained with the choice of \(r(t)=\rho(t)=1\) and \(\Phi(t,s,l)=(t-s)(s-l)^{\beta}\).
Corollary 2.3
Let conditions (H1)-(H5) hold. Equation (1.1) with \(r(t)=1\) is oscillatory provided that, for each \(l\geq t_{0}\), there exist a function \(b\in{C^{1}([t_{0},\infty), \mathbb{R})}\) and a constant \(\beta>1/2\) such that
where the functions Q and a are as in Theorem 2.1.
Let
A function \(H=H(t,s)\in C(\mathbb{D}, [0,\infty))\) is said to belong to the function class P, if \(H(t,t)=0\) for \(t\geq{t_{0}}\), \(H(t,s)>0\) for \(t>s\), and H has partial derivatives \({\partial{H}}/{\partial{s}}\) and \({\partial {H}}/{\partial{t}}\) on \(\mathbb{D}_{0}\) satisfying
where \(h_{1}\) and \(h_{2}\) are locally integrable with respect to t and s, respectively, in \(\mathbb{D}_{0}\).
Set \(\Phi(t,s,l)=\sqrt{H_{1}(s,l)H_{2}(t,s)}\), \(H_{1}, H_{2}\in{P}\). It follows from (1.5) that
where \(h^{(1)}_{1}\) and \(h^{(2)}_{2}\) are defined by
After a simple computation, we have the following result when using Theorem 2.1.
Theorem 2.2
Suppose assumptions (H1)-(H5) are satisfied. If there exist four functions \(H_{1}, H_{2}\in{P}\), \(\rho\in{C^{1}([t_{0},\infty), (0,\infty))}\), and \(b\in {C^{1}([t_{0},\infty),\mathbb{R})}\) such that, for each \(l\geq{t_{0}}\),
where the functions Q and a are defined as in Theorem 2.1, \(h^{(1)}_{1}\) and \(h^{(2)}_{2}\) are defined as in (2.7), then (1.1) is oscillatory.
3 Oscillation results for nonmonotonic g
Theorem 3.1
Assume conditions (H1)-(H4) and
- (H6):
-
g satisfies \(g(x)/x\geq{k}>0\) for all \(x\neq{0}\) and \(q(t)-p(t)\geq{0}\) for \(t\geq{t_{0}}\).
Equation (1.1) is oscillatory provided that, for each \(l\geq{t_{0}}\), there exist three functions \(\Phi\in{Y}\), \(\rho\in{C^{1}([t_{0},\infty ), (0,\infty))}\), and \(b\in{C^{1}([t_{0},\infty), \mathbb{R})}\) such that
where
\(\phi=\phi(t,s,l)\) and A are defined by (1.5) and (1.6), respectively.
Proof
As in the proof of Theorem 2.1, suppose x is a nonoscillatory solution of (1.1) which satisfies \(x(t)>0\) for \(t\geq {t_{1}}\geq{t_{0}}\) since the case \(x<0\) can be treated similarly. We introduce a generalized Riccati transformation by
Differentiating (3.2) and using (1.1), we have, for \(t\geq{t_{1}}\),
From (H1)-(H4) and (H6), we obtain
The rest of the proof is similar to that of Theorem 2.1 and one can get a contradiction to (3.1). This completes the proof. □
In what follows, we derive some corollaries from Theorem 3.1 by choosing different \(\Phi(t,s,l)\). If we choose \(r(t)=1\), \(\rho(t)=1\), and \(\Phi(t,s,l)=(t-s)^{\gamma}(s-l)\), then the following oscillation result can be obtained.
Corollary 3.1
Let conditions (H1)-(H4) and (H6) hold. Equation (1.1) with \(r(t)=1\) is oscillatory provided that, for each \(l\geq t_{0}\), there exist a function \(b\in{C^{1}([t_{0},\infty), \mathbb{R})}\) and a constant \(\gamma>1/2\) such that
where the functions Q̅ and a̅ are as in Theorem 3.1.
Proof
The proof of this corollary is similar to that of Corollary 2.2, and hence it is omitted. □
Similarly, letting \(r(t)=1\), \(\rho(t)=1\), and \(\Phi (t,s,l)=(t-s)(s-l)^{\beta}\), we have the following result.
Corollary 3.2
Let conditions (H1)-(H4) and (H6) be satisfied. Equation (1.1) with \(r(t)=1\) is oscillatory provided that, for each \(l\geq t_{0}\), there exist a function \(b\in{C^{1}([t_{0},\infty), \mathbb{R})}\) and a constant \(\beta>1/2\) such that
where the functions Q̅ and a̅ are as in Theorem 3.1.
As discussion in Section 2, we choose \(\Phi(t,s,l)=\sqrt {H_{1}(s,l)H_{2}(t,s)}\), then we get the following result.
Theorem 3.2
Suppose (H1)-(H4) and (H6) are satisfied. If there exist four functions \(H_{1}, H_{2}\in{P}\), \(\rho\in{C^{1}([t_{0},\infty), (0,\infty))}\), and \(b\in {C^{1}([t_{0},\infty),\mathbb{R})}\) such that, for each \(l\geq{t_{0}}\),
where the functions Q̅ and a̅ are defined as in Theorem 3.1, \(h^{(1)}_{1}\) and \(h^{(2)}_{2}\) are defined by (2.7), then (1.1) is oscillatory.
4 Interval oscillation criteria
Our purpose in this section is to establish some interval oscillation criteria for (1.1). First of all, we consider the case where g is an increasing function.
Theorem 4.1
Let conditions (H1)-(H5) hold. Equation (1.1) is oscillatory provided that, for each \(T\geq{t_{0}}\), there exist three functions \(\Phi\in{Y}\), \(\rho\in{C^{1}([t_{0},\infty), (0,\infty))}\), \(b\in{C^{1}([t_{0},\infty ),\mathbb{R})}\), and two constants \(d>c\geq{T}\) such that
where ϕ, Q, a, and A are defined as in Theorem 2.1.
Proof
The proof is similar to that of Theorem 2.1, where t and l are replaced by d and c, respectively. Then it can be seen that every solution of (1.1) has at least one zero in \((c, d)\), i.e., every solution of (1.1) has arbitrarily large zeros on \([t_{0},\infty)\). The proof is complete. □
From Theorem 4.1, we have the following corollaries by choosing \(\Phi (d,s,c)=(d-s)^{\gamma}(s-c)^{\beta}\) and \(\Phi(d,s,c)=\sqrt {H_{1}(s,c)H_{2}(d,s)}\), respectively.
Corollary 4.1
Let conditions (H1)-(H5) be satisfied. Assume there exist two functions \(\rho\in C^{1}([t_{0},\infty),(0,\infty))\), \(b\in{C^{1}([t_{0},\infty),\mathbb{R})}\), two constants \(\gamma, \beta >1/2\), and two constants \(d>c\geq{T}\) such that, for all \(T\geq t_{0}\),
where the functions Q and a are as in Theorem 2.1. Then (1.1) is oscillatory.
Corollary 4.2
Let conditions (H1)-(H5) hold. Equation (1.1) is oscillatory provided that, for each \(T\geq{t_{0}}\), there exist four functions \(H_{1}, H_{2}\in{P}\), \(\rho\in C^{1}([t_{0},\infty),(0,\infty))\), \(b\in{C^{1}([t_{0},\infty),\mathbb{R})}\), and two constants \(d>c\geq{T}\) such that
where the functions Q and a are as in Theorem 2.1, \(h^{(1)}_{1}\) and \(h^{(2)}_{2}\) are defined as in (2.7).
Similarly, we can obtain the following oscillation results when g is a nonmonotonic function.
Theorem 4.2
Assume that conditions (H1)-(H4) and (H6) are satisfied. Equation (1.1) is oscillatory provided that, for each \(T\geq{t_{0}}\), there exist three functions \(\Phi\in{Y}\), \(\rho\in{C^{1}([t_{0},\infty), (0,\infty))}\), \(b\in{C^{1}([t_{0},\infty ),\mathbb{R})}\), and two constants \(d>c\geq{T}\) such that
where ϕ, Q̅, a̅, and A are the same as in Theorem 3.1.
Corollary 4.3
Let (H1)-(H4) and (H6) be satisfied. Assume there exist two functions \(\rho\in C^{1}([t_{0},\infty ),(0,\infty))\), \(b\in{C^{1}([t_{0},\infty),\mathbb{R})}\), two constants \(d>c\geq{T}\), and two constants \(\gamma, \beta>1/2\) such that, for all \(T\geq t_{0}\),
where the functions Q̅ and a̅ are as in Theorem 3.1. Then (1.1) is oscillatory.
Corollary 4.4
Let (H1)-(H4) and (H6) hold. Equation (1.1) is oscillatory provided that, for each \(T\geq{t_{0}}\), there exist four functions \(H_{1}, H_{2}\in{P}\), \(\rho\in{C^{1}([t_{0},\infty), (0,\infty))}\), \(b\in{C^{1}([t_{0},\infty ),\mathbb{R})}\), and two constants \(d>c\geq{T}\) such that
where the functions Q̅ and a̅ are as in Theorem 3.1, \(h^{(1)}_{1}\) and \(h^{(2)}_{2}\) are defined by (2.7).
5 Examples
The following examples illustrate some applications of the results presented in this paper.
Example 5.1
For \(t\geq{t_{0}}=1\) and \(\gamma>0\), consider the equation
Let \(r(t)=1\), \(h(t)=\sqrt{\gamma}/{t}\), and \(q(t)={\gamma}/{t^{2}}\). It is not difficult to see that
Therefore, (5.1) satisfies conditions (H1)-(H5). Next, we consider the following two cases separately.
(1) If we choose \(\rho(t)=1\), \(\Phi(t,s,l)=(t-s)(s-l)^{\beta}\), and \(b(t)=0\), then \(Q(t)=\gamma/(2t^{2})\) and by virtue of
and
we conclude that
Hence, by Corollary 2.3, (5.1) is oscillatory if \(\gamma>8\beta^{2}\) for some \(\beta\in(1/2,\infty)\).
On the other hand, if we take \(\alpha\geq{1}\), \(n=3\), and \(\delta (t)=0\), then \(\widetilde{Q}(t)=3\gamma/(4t^{2})\) and
Furthermore, we have
Thus, Theorem 1.2 cannot be applied to (5.1).
(2) If we choose \(\rho(t)=t^{\frac{\sqrt{\gamma}}{2}}\), \(\Phi (t,s,l)=(t-s)(s-l)^{\beta}\), and \(b(t)=0\), then \(Q(t)=3\gamma/(4t^{2})\) and
Consequently, by Corollary 2.3, (5.1) is oscillatory if \(\gamma>16\beta ^{2}/3\) for some \(\beta\in(1/2,\infty)\). By the above discussion, we observe that (5.1) is oscillatory if \(\gamma >64/27\approx2.38\) (by letting \(\beta=2/3\)).
On the other hand, if we take \(\alpha={2}\), \(n=3\), and \(\delta(t)=0\), then \(\widetilde{Q}(t)=3\gamma{t^{\frac{\sqrt{\gamma}}{2}-2}}/4\), \(\tilde{\rho}(t)=t^{\frac{\sqrt{\gamma}}{2}}\), and
By a direct computation, we obtain the following results. When \(0<\gamma<4\),
whereas
Therefore, Theorem 1.1 cannot be applied to (5.1) in this case where \(0<\gamma<4\). When \(\gamma\geq{4}\),
Define ϕ̃ by \(\tilde{\phi}(t)=t^{\frac{\sqrt{\gamma}}{4}}\). We have
From Theorem 1.1, we conclude that (5.1) is oscillatory for \(\gamma\geq {4}\).
Example 5.2
For \(t\geq1\), consider the equation
where \(r(t)=1\), \(h(t)=\beta\sin^{2}t\), \(q(t)={3{\beta}^{2} \sin ^{4}t}/4+3\), and \(\beta>8/3\). Note that
Let \(c=0\) and \(d=\pi\). Choosing \(\rho(t)=1\), \(b(t)=1\), and \(\Phi(d,s,c)=\sin s\), we have \(\overline{Q}(s)=\beta\sin^{2}s\), \(\Phi^{2}(d,s,c)\phi^{2}(d,s,c)=\cos ^{2} s\), and
Using Theorem 4.2, we conclude that (5.2) is oscillatory if \(\beta >8/3\).
References
Agarwal, RP, Avramescu, C, Mustafa, OG: On the oscillation theory of a second order strictly sublinear differential equation. Can. Math. Bull. 53, 193-203 (2010)
Zhang, QX, Wang, L: Oscillatory behavior of solutions for a class of second order nonlinear differential equation with perturbation. Acta Appl. Math. 110, 885-893 (2010)
Agarwal, RP, Grace, SR, O’Regan, D: Oscillation Theory for Difference and Functional Differential Equations. Kluwer Academic, Dordrecht (2000)
Fu, XL, Li, TX, Zhang, CH: Oscillation of second-order damped differential equations. Adv. Differ. Equ. 2013, 326 (2013)
Bohner, M, Saker, SH: Oscillation of damped second order nonlinear delay differential equations of Emden-Fowler type. Adv. Dyn. Syst. Appl. 1, 163-182 (2006)
Grace, SR, Lalli, BS: Oscillation theorems for second order superlinear differential equations with damping. J. Aust. Math. Soc. 53, 156-165 (1992)
Kirane, M, Rogovchenko, YuV: On oscillation of nonlinear second order differential equation with damping term. Appl. Math. Comput. 117, 177-192 (2001)
Li, W-T, Agarwal, RP: Interval oscillation criteria for second-order nonlinear differential equations with damping. Comput. Math. Appl. 40, 217-230 (2000)
Rogovchenko, YuV: Oscillation theorems for second-order equations with damping. Nonlinear Anal. 41, 1005-1028 (2000)
Rogovchenko, YuV, Tuncay, F: Oscillation criteria for second-order nonlinear differential equations with damping. Nonlinear Anal. 69, 208-221 (2008)
Grace, SR: Oscillation theorems for second order nonlinear differential equations with damping. Math. Nachr. 141, 117-127 (1989)
Li, TX, Rogovchenko, YuV, Tang, SH: Oscillation of second-order nonlinear differential equations with damping. Math. Slovaca 64, 1227-1236 (2014)
Wang, PG, Cai, H: Oscillatory criteria for higher order functional differential equations with damping. J. Funct. Spaces Appl. 2013, 1-5 (2013)
Zhang, CH, Agarwal, RP, Li, TX: Oscillation and asymptotic behavior of higher-order delay differential equations with p-Laplacian like operators. J. Math. Anal. Appl. 409, 1093-1106 (2014)
Pas̆ić, M: Fite-Wintner-Leighton-type oscillation criteria for second-order differential equations with nonlinear damping. Abstr. Appl. Anal. 2013, Article ID 852180 (2013)
Pas̆ić, M: New interval oscillation criteria for forced second-order differential equations with nonlinear damping. Int. J. Math. Anal. 7, 1239-1255 (2013)
Grace, SR, Lalli, BS: Oscillation theorems for nonlinear second order differential equations with a damping term. Comment. Math. Univ. Carol. 30, 691-697 (1989)
Kirane, M, Rogovchenko, YuV: Oscillation results for a second order damped differential equation with nonmonotonous nonlinearity. J. Math. Anal. Appl. 250, 118-138 (2000)
Manojlović, JV: Oscillation criteria for sublinear differential equations with damping. Acta Math. Hung. 104, 153-169 (2004)
Rogovchenko, YuV, Tuncay, F: Oscillation theorems for a class of second order nonlinear differential equations with damping. Taiwan. J. Math. 13, 1909-1928 (2009)
Tunç, E, Avci, H: Oscillation criteria for a class of second order nonlinear differential equations with damping. Bull. Math. Anal. Appl. 4, 40-50 (2012)
Salhin, AA, Din, UKS, Ahmad, RR, Noorani, MSM: Some oscillation criteria for a class of second order nonlinear damped differential equations. Appl. Math. Comput. 247, 962-968 (2014)
Sun, YG: New Kamenev-type oscillation criteria for second-order nonlinear differential equations with damping. J. Math. Anal. Appl. 291, 341-351 (2004)
Wang, XJ, Song, GH: Oscillation criteria for a second-order nonlinear damped differential equation. Int. J. Inf. Syst. Sci. 7, 73-82 (2011)
Kamenev, IV: Integral criterion of linear differential equations of second order. Mat. Zametki 23, 249-251 (1978)
Philos, ChG: Oscillation theorems for linear differential equations of second order. Arch. Math. 53, 482-492 (1989)
Agarwal, RP, Wang, QR: Oscillation and asymptotic behavior for second-order nonlinear perturbed differential equations. Math. Comput. Model. 39, 1477-1490 (2004)
Acknowledgements
The authors express their sincere gratitude to the editors and two anonymous referees for a very thorough reading of the manuscript and for pointing out several inaccuracies. This work was supported by the Natural Science Foundation of Shandong Province under Grant Nos. ZR2012FL06 and JQ201119, and the National Natural Science Foundation of China under Grant Nos. 61503171, 61403061, 61174217, 61374074, and 61473133.
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All four authors contributed equally to this work. They all read and approved the final version of the manuscript.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Jiang, C., Tian, Y., Jiang, Y. et al. Some oscillation results for nonlinear second-order differential equations with damping. Adv Differ Equ 2015, 354 (2015). https://doi.org/10.1186/s13662-015-0688-z
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13662-015-0688-z