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Interval oscillation criteria for second order super-half linear functional differential equations with delay and advanced arguments. (English) Zbl 1180.34070
Summary: Sufficient conditions are established for oscillation of second order super half linear equations containing both delay and advanced arguments of the form
\[ \big(\varphi_\alpha(k(t)x'(t))\big)'+ p(t) \varphi_\beta(x(\tau(t)))+ q(t) \varphi_\gamma(x(\sigma(t)))= e(t), \quad t\geq 0, \]
where \(\varphi_\delta(u)= |u|^{\delta-1}u\); \(\alpha>0\), \(\beta\geq \alpha\), and \(\gamma\geq \alpha\) are real numbers; \(k,p,q,e,\tau,\sigma\) are continuous real-valued functions; \(\tau(t)\leq t\) and \(\sigma(t)\geq t\) with \(\lim_{t\to\infty}\tau(t)=\infty\). The functions \(p(t)\), \(q(t)\), and \(e(t)\) are allowed to change sign, provided that \(p(t)\) and \(q(t)\) are nonnegative on a sequence of intervals on which \(e(t)\) alternates sign.
As an illustrative example we show that every solution of
\[ \big(\varphi_\alpha(x'(t))\big)'+m_1\sin t\varphi_\beta(x(t-\pi/5))+ m_2\cos t\varphi_\gamma(x(t+\pi/20))= r_0\cos 2t \]
is oscillatory provided that either \(m_1\) or \(m_2\) or \(r_0\) is sufficiently large.

MSC:
34K11 Oscillation theory of functional-differential equations
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References:
[1] R. P. Agarwal, S. R. Grace, and D. O’Reagan, Oscillation Theory for Second Order Linear, Half Linear, Superlinear and Sublinear Dynamic Equations (Kluwer Academic Publishers, Dordrecht, 2002).
[2] O. Dosly, and P. Rehak, Half Linear Differential Equations (Elsevier, North-Holland, 2005).
[3] Dzurina, Oscillation of second-order differential equations with mixed argument, J. Math. Anal. Appl. 190 pp 821– (1995)
[4] El-Sayed, An oscillation criterion for a forced second order linear differential equation, Proc. Amer. Math. Soc. 118 pp 813– (1993) · Zbl 0777.34023
[5] Á. Elbert, A half linear differential equation, in: Proceedings of the International Conference on Qualitative Theory of Differential Equations, Szeged 1979, Colloquia Mathematica Societatis János Bolyai Vol. 30 (North-Holland, Amsterdam -New York, 1979), pp. 153-180.
[6] L. H. Erbe, Q. Kong, and B. G. Zhang, Oscillation Theory for Functional Differential Equations (Marcel Dekker, New York, 1995). · Zbl 0821.34067
[7] Gopalsamy, Nonoscillatory differential equations with retarded and advanced arguments, Quart. Appl. Math. 43 pp 211– (1985) · Zbl 0589.34053
[8] Güvenilir, Second order oscillation of functional differential equations with oscillatory potentials, Comput. Math. Appl. 51 pp 1395– (2006)
[9] I. Gyori, and G. Ladas, Oscillation Theory of Delay Differential Equation with Applications (Clarendon, Oxford, 1991).
[10] J. Hale, Theory of Functional Differential Equations (Springer, New York -Heidelberg - Berlin, 1977). · Zbl 0352.34001
[11] G. H. Hardy, J. E. Littlewood, and G. Polya, Inequalities, 2nd edition (Cambridge Univ. Press, Cambridge, 1988).
[12] Kartsatos, On the maintenance of oscillation of n-th order equations under the effect of small forcing term, J. Differential Equations 10 pp 355– (1971) · Zbl 0216.11504
[13] Kartsatos, On the maintenance of oscillation under the effect of periodic forcing term, Proc. Amer. Math. Soc. 10 pp 377– (1972) · Zbl 0223.34031
[14] Ladas, Oscillations caused by several retarded and advanced arguments, J. Differential Equations 44 pp 134– (1982) · Zbl 0452.34058
[15] Li, Oscillation and nonoscillation of advanced differential equations with variable coefficients, J. Math. Anal. Appl. 269 pp 462– (2002) · Zbl 1013.34067
[16] Li, Interval oscillation of second-order half linear functional differential equations, Applied Math. Comput. 155 pp 451– (2004) · Zbl 1061.34048
[17] Li, An oscillation criteria for nonhomogeneous half linear differential equations, Appl. Math. Lett. 15 pp 259– (2002)
[18] Manojlovic, Oscillation criteria for second-order half linear differential equations, Math. Comput. Model. Dyn. Syst. 30 pp 109– (1999)
[19] Mirzov, On some analogs of Sturm’s and Kneser’s theorems for nonlinear systems, J. Math. Anal. Appl. 53 pp 418– (1976) · Zbl 0327.34027
[20] Nasr, Sufficient conditions for the oscillation of forced superlinear second order differential equations with oscillatory potential, Proc. Amer. Math. Soc. 126 pp 123– (1998) · Zbl 0891.34038
[21] Nasr, Necessary and sufficient conditions for the oscillation of forced nonlinear second order differential equations with delayed argument, J. Math. Anal. Appl. 212 pp 363– (1997) · Zbl 0884.34075
[22] Precup, Some existence results for differential equations with both retarded and advanced arguments, Matematika 44 pp 31– (2002) · Zbl 1084.34542
[23] Skidmore, Oscillatory behavior of solutions of y ” + p (x)y = f (x), J. Math. Anal. Appl. 49 pp 317– (1975) · Zbl 0312.34025
[24] Sun, A note on Nasr’s and Wong’s papers, J. Math. Anal. Appl. 286 pp 363– (2003) · Zbl 1042.34096
[25] Tiryaki, Oscillation criteria for certain forced second order nonlinear differential equations with delayed argument, Comput. Math. Appl. 49 pp 1647– (2005) · Zbl 1093.34552
[26] Yu, Oscillations caused by several retarded and advanced arguments, Acta Math. Appl. Sin., Engl. Ser. 6 pp 67– (1990) · Zbl 0701.34078
[27] Wang, An oscillation criteria for nonhomogeneous half linear differential equations, J. Math. Anal. Appl. 291 pp 224– (2004)
[28] Wong, Oscillation criteria for a forced second order linear differential equation, J. Math. Anal. Appl. 231 pp 235– (1999) · Zbl 0922.34029
[29] Z. X. Zheng, Theory of Functional Differential Equations (Anhui Education Press, Hefei, 1994).
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