# 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 criteria for certain forced second-order nonlinear differential equations. (English) Zbl 1061.34017
Summary: Two new oscillation criteria for forced second-order nonlinear differential equations of the form $$\left(r(t)\Psi\bigl(y(t)\bigr) \bigl|y' (t)\bigr|^{\alpha-1}y'(t)\right)'+q(t)f\bigl(y(t)\bigr)=e(t),\quad t \ge t_0,$$ are established. Our results are based on the information on a sequence of subintervals of $[t_0,\infty)$ only, rather than on the whole half-line. Our methodology is somewhat different from that of previous authors. The results presented here are much more general than a recent result of {\it W. T. Li} and {\it S. S. Cheng} [Appl. Math. Lett. 15, 259--263 (2002; Zbl 1023.34029)].

##### MSC:
 34C10 Qualitative theory of oscillations of ODE: zeros, disconjugacy and comparison theory
Full Text:
##### References:
 [1] Li, W. T.; Cheng, S. S.: An oscillation criterion for nonhomogeneous half linear differential equations. Appl. math. Lett. 15, No. 3, 259-263 (2002) · Zbl 1023.34029 [2] El-Sayed, M. A.: An oscillation criterion for a forced second order linear differential equation. Proc. amer. Math. soc. 118, 813-817 (1993) · Zbl 0777.34023 [3] Hong, H. L.: On the oscillatory behavior of solutions of second order nonlinear differential equations. Publ. math. Debrecen 52, No. 1--2, 55-68 (1998) · Zbl 0908.34020 [4] Kartsatos, A. G.: Maintenance of oscillations under the effect of a periodic forcing term. Proc. amer. Math. soc. 33, 377-383 (1972) · Zbl 0234.34040 [5] Rainkin, S. M.: Oscillation theorems for second order nonhomogeneous linear differential equations. J. math. Anal. appl. 53, 550-553 (1976) · Zbl 0328.34033 [6] Wong, J. S. W: Oscillation criteria for a forced second order linear differential equation. J. math. Anal. appl. 231, 235-240 (1999) · Zbl 0922.34029 [7] Agarwal, R. P.; Grace, S. R.: Second order nonlinear forced oscillations. Dynam. systems appl. 10, 455-464 (2001) · Zbl 1021.34027 [8] Leighton, W.: Comparison theorems for linear differential equations of second order. Proc. amer. Math. soc. 13, 603-610 (1962) · Zbl 0118.08202 [9] Jaros, J.; Kusano, T.: A Picone type identity for second order half-linear differential equation. Acta math. Univ. comenianae 68, No. 1, 137-151 (1999) [10] Wong, J. S. W: On kamenev-type oscillation theorems for second order differential equations with damping. J. math. Anal. appl. 258, 244-257 (2001) · Zbl 0987.34024 [11] Tiryaki, A.; Çakmak, D.; Ayanlar, B.: On the oscillation of certain second order nonlinear differential equations. J. math. Anal. appl. 281, 565-574 (2003) · Zbl 1030.34033 [12] Ayanlar, B.; Tiryaki, A.: Oscillation theorems for nonlinear second order differential equations. Computers math. Applic. 44, No. 3/4, 529-538 (2002) · Zbl 1059.34023 [13] Agarwal, R. P.; Grace, S. R.: On the oscillation of certain second order differential equations. Georgian mathematical journal 15, No. 2, 201-213 (2000) · Zbl 0958.34027