# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
New oscillation criteria of second-order nonlinear differential equations. (English) Zbl 1172.34322

Summary: By employing a class of new functions $𝛷=𝛷\left(t,s,l\right)$ and a generalized Riccati technique, some new oscillation and interval oscillation criteria are established for the second-order nonlinear differential equation

$\left(r\left(t\right)y{}^{\text{'}}\left(t\right)\right){}^{\text{'}}+Q\left(t,y\left(t\right),y{}^{\text{'}}\left(t\right)\right)=0·$

The obtained interval oscillation criteria can be applied to equations with forcing term. Two examples are also included to show the significance of our results.

##### MSC:
 34C10 Qualitative theory of oscillations of ODE: zeros, disconjugacy and comparison theory
##### References:
 [1] Agarwal, R. P.; Wang, Q. R.: Oscillation and asymptotic behavior for second-order nonlinear perturbed differential equations, Math. comput. Modell. 39, 1477-1490 (2004) · Zbl 1079.34020 · doi:10.1016/j.mcm.2004.07.007 [2] Kamenev, I. V.: An integral criterion for oscillation of linear differential equations of second order, Mat. zametki 23, 249-251 (1978) · Zbl 0386.34032 [3] Kong, Q.: Interval criteria for oscillation of second-order linear ordinary differential equations, J. math. Anal. appl. 229, 258-270 (1999) · Zbl 0924.34026 · doi:10.1006/jmaa.1998.6159 [4] Li, W. T.; Agarwal, R. P.: Interval oscillation criteria for second-order nonlinear differential equations with damping, Comput. math. Appl. 40, 217-230 (2000) · Zbl 0959.34026 · doi:10.1016/S0898-1221(00)00155-3 [5] Li, W. T.; Agarwal, R. P.: Interval oscillation criteria related to integral averaging technique for certain nonlinear differential equations, J. math. Anal. appl. 245, 171-188 (2000) · Zbl 0983.34020 · doi:10.1006/jmaa.2000.6749 [6] Philos, Ch.G.: Oscillation theorems for linear differential equations of second order, Arch. math. (Basel) 53, 483-492 (1989) · Zbl 0661.34030 · doi:10.1007/BF01324723 [7] Sun, Y. G.: New kamenev-type oscillation criteria for second-order nonlinear differential equations with damping, J. math. Anal. appl. 291, 341-351 (2004) · Zbl 1039.34027 · doi:10.1016/j.jmaa.2003.11.008 [8] Wang, Q. R.: Oscillation criteria for nonlinear second order damped differential equations, Acta math. Hungar. 102, 117-139 (2004) · Zbl 1052.34040 · doi:10.1023/B:AMHU.0000023211.53752.03 [9] Wang, Q. R.: Interval criteria for oscillation of certain second order nonlinear differential equations, Dynam. cont. Discrete impulsive syst. Series A: math. Anal. 12, 769-781 (2005) · Zbl 1087.34015 [10] Yang, Q. G.: Interval oscillation criteria for a forced second order nonlinear ordinary differential equations with oscillatory potential, Appl. math. Comput. 135, 49-64 (2003) · Zbl 1030.34034 · doi:10.1016/S0096-3003(01)00307-1