# 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)
Oscillation of second order functional differential equations with damping. (English) Zbl 1105.34326
Summary: Using a class of new test functions $\Phi(t,s,r)$ defined in our recent work [J. Math. Anal. Appl. 291, 341--351 (2004; Zbl 1061.34027)], we establish some new oscillation criteria for the second-order functional-differential equation with damping $$x''(t)+p(t)x'(t)+ \int^b_a q(t,\xi)f\biggl(x\bigl[g_1(t,\xi)\bigr], \dots,x \bigl[g_m(t,\xi)\bigr] \biggr)d\sigma(\xi)=0.$$ Our results are different from most known ones and can be applied to many cases which are not covered by existing results.

##### MSC:
 34K11 Oscillation theory of functional-differential equations
Full Text:
##### References:
 [1] Ayanlar, B.; Tiryaki, T.: Oscillation theorems for nonlinear second order differential equation with damping. Acta. math. Hungar. 89, 1-13 (2000) · Zbl 0973.34021 [2] Baker, J. W.: Oscillation theorems for a second order damped nonlinear equation. SIAM J. Appl. math. 25, 37-40 (1973) · Zbl 0239.34015 [3] Bobisud, L. E.: Oscillation of solutions of damped nonlinear equations. SIAM J. Appl. math. 19, 601-606 (1970) · Zbl 0206.38005 [4] Butler, G. J.: The oscillatory behavior of a second order nonlinear differential equation with damping. J. math. Anal. appl. 57, 273-289 (1977) · Zbl 0348.34022 [5] Grace, S. R.: Oscillation theorems for second order nonlinear differential equations with damping. Math. nachr. 141, 117-127 (1989) · Zbl 0673.34041 [6] Grace, S. R.: Oscillation criteria for second order nonlinear differential equations with damping. J. austral. Math. soc. A 49, 43-54 (1990) · Zbl 0725.34030 [7] Grace, S. R.: Oscillation theorems for nonlinear differential equations of second order. J. math. Anal. appl. 171, 220-241 (1992) · Zbl 0767.34017 [8] Grace, S. R.; Lalli, B. S.: Oscillation theorems for second order superlinear differential equations with damping. J. aust. Math. soc. 53A, 156-165 (1992) · Zbl 0762.34012 [9] Kartsatos, A. G.: Recent results on oscillation of solutions of forced and perturbed nonlinear differential equations of even order. (1977) · Zbl 0361.34031 [10] Kirane, M.; Rogovchenko, Y. V.: Oscillation results for second order damped differential equation with nonmonotonous nonlinearity. J. math. Anal. appl. 250, 118-138 (2000) · Zbl 1008.34029 [11] Kirane, M.; Rogovchenko, Y. V.: On oscillation of nonlinear second order differential equations with damping term. Appl. math. Comput. 117, 177-192 (2001) · Zbl 1035.34019 [12] Li, W. T.; Zhang, M. Y.; Fei, X. L.: Oscillation criteria for second order nonlinear differential equations with damping. Indian J. Pure appl. Math. 30, 1017-1029 (1999) · Zbl 0948.34022 [13] 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 [14] Li, W. T.; Agarwal, R. P.: Interval oscillation criteria for second order forced nonlinear differential equations with damping. Panamer. math. J. 11, 109-117 (2001) · Zbl 1004.34019 [15] Philos, Ch.G.: Oscillation theorems for linear differential equations of second order. Arch. math. 53, 483-492 (1989) · Zbl 0661.34030 [16] Rogovchenko, Y. V.: Oscillation criteria for second order nonlinear perturbed differential equations. J. math. Anal. appl. 215, 334-357 (1997) · Zbl 0892.34031 [17] Rogovchenko, Y. V.: Oscillation theorems for second order differential equations with damping. Nonlinear anal. 41, 1005-1028 (2000) · Zbl 0972.34022 [18] 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 [19] Wong, J. S. W.: Oscillation criteria for second order nonlinear differential equations involving general means. J. math. Anal. appl. 247, 489-505 (2000) · Zbl 0964.34028 [20] Wang, P.; Wu, Y.: Oscillation of certain second order functional differential equations with damping. J. comput. Appl. math. 157, 49-56 (2003) · Zbl 1032.34066 [21] Yan, J.: Oscillation theorems for second order linear differential equations with damping. Proc. amer. Math. soc. 98, 276-282 (1986) · Zbl 0622.34027