Improved oscillation results for second-order half-linear delay differential equations

In this paper, we study the second-order half-linear delay differential equation of the form\[(r(t)(y'(t))^\alpha)'+q(t)y^\alpha(\tau(t))= 0.\:\:\:(E)\]We establish new oscillation criteria for (E), which improve a number of related ones in the literature. Our approach essentially involves establishing sharper estimates for the positive solutions of (E) than those presented in known works and a comparison principle with first-order delay differential inequalities. We illustrate the improvement over the known results by applying and comparing our method with the other known methods on the particular example of Euler-type equations.

Keywords:

half-linear differential equation, delay second-order, oscillation,

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R. P. Agarwal, M. Bohner and W.-T. Li, Nonoscillation and oscillation: theory for functional differential equations, Monographs and Textbooks in Pure and Applied Mathematics 267, Marcel Dekker, Inc., New York, 2004.
R. P. Agarwal, S. R. Grace and D. O’Regan, Oscillation theory for second order linear, half-linear, superlinear and sublinear dynamic equations, Kluwer Academic Publishers, Dordrecht, 2002.
R. P. Agarwal, S. R. Grace and D. O’Regan, Oscillation theory for second order dynamic equations, Series in Mathematical Analysis and Applications 5, Taylor & Francis, Ltd., London, 2003.
R. P. Agarwal, S. R. Grace and D. O’Regan, Oscillation theory for difference and functional differential equations, Springer Science & Business Media, 2013.
R. P. Agarwal, C. Zhang and T. Li, Some remarks on oscillation of second order neutral differential equations, Appl. Math. Comput. 274, 178–181, 2016.
O. Došlý and P. Rehák, Half-linear differential equations, North-Holland Mathematics Studies 202, Elsevier Science B.V., Amsterdam, 2005.
J. Džurina and I. Jadlovská, A note on oscillation of second-order delay differential equations, Appl. Math. Lett. 69, 126–132, 2017.
J. Džurina and I. P. Stavroulakis, Oscillation criteria for second-order delay differential equations, Appl. Math. Comput. 140 (2-3), 445–453, 2003.
L. Erbe, A. Peterson and S. H. Saker, Kamenev-type oscillation criteria for secondorder linear delay dynamic equations, Dynam. Systems Appl. 15 (1), 65–78, 2006.
S. Fišnarová and R. Marík, Oscillation of half-linear differential equations with delay, Abstr. Appl. Anal. 2013, Article ID: 583147, 1–6, 2013.
I. Gyori and G. Ladas, Oscillation theory of delay differential equations, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1991.
Z. Han, T. Li, S. Sun and Y. Sun, Remarks on the paper [Appl. Math. Comput. 207 (2009) 388–396] [mr2489110], Appl. Math. Comput. 215 (11), 3998–4007, 2010.
J. Jaroš and I. P. Stavroulakis, Oscillation tests for delay equations, Rocky Mountain J. Math. 29 (1), 197–207, 1999.
R. Koplatadze, G. Kvinikadze and I. P. Stavroulakis, Oscillation of second order linear delay differential equations, Funct. Differ. Equ. 7 (1-2), 121–145, 2000.
R. G. Koplatadze, Criteria for the oscillation of solutions of differential inequalities and second-order equations with retarded argument, Tbiliss. Gos. Univ. Inst. Prikl. Mat. Trudy, 17, 104–121, 1986.
T. Kusano and J. Wang, Oscillation properties of half-linear functional-differential equations of the second order, Hiroshima Math. J. 25 (2), 371–385, 1995.
T. Li, Y. V. Rogovchenko and C. Zhang, Oscillation of second-order neutral differential equations, Funkcial. Ekvac. 56 (1), 111–120, 2013.
T. Li, Y. V. Rogovchenko and C. Zhang, Oscillation results for second-order nonlinear neutral differential equations, Adv. Difference Equ. 2013 (336), 1–13, 2013.
W. E. Mahfoud, Comparison theorems for delay differential equations, Pacific J. Math. 83 (1), 187–197, 1979.
Y. G. Sun and F. W. Meng, Note on the paper of Džurina and Stavroulakis: “Oscillation criteria for second-order delay differential equations” [Appl. Math. Comput. 140 (2003), no. 2-3, 445–453; mr1953915], Appl. Math. Comput. 174 (2), 1634–1641, 2006.
A. Tiryaki, Oscillation criteria for a certain second-order nonlinear differential equations with deviating arguments, Electron. J. Qual. Theory Differ. Equ. 2009 (61), 1–11, 2009.
J. J. Wei, Oscillation of second order delay differential equation, Ann. Differential Equations, 4 (4), 473–478, 1988.
H.Wu, L. Erbe and A. Peterson, Oscillation of solution to second-order half-linear delay dynamic equations on time scales, Electron. J. Differential Equations, 2016 (71), 1–15, 2016.
R. Xu and F. Meng, Some new oscillation criteria for second order quasi-linear neutral delay differential equations, Appl. Math. Comput. 182 (1), 797–803, 2006.
L. Ye and Z. Xu, Oscillation criteria for second order quasilinear neutral delay differential equations, Appl. Math. Comput. 207 (2), 388–396, 2009.
C. Zhang, R. P. Agarwal, M. Bohner and T. Li, Oscillation of second-order nonlinear neutral dynamic equations with noncanonical operators, Bull. Malays. Math. Sci. Soc. 38 (2), 761–778, 2015.