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)y0(t)α0+q(t)yα(τ(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.

PDF

___

R. P. Agarwal, M. Bohner and W.-T. Li, Nonoscillation and oscillation: theory for functional di erential 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 di erence and functional di erential equations, Springer Science & Business Media, 2013.
R. P. Agarwal, C. Zhang and T. Li, Some remarks on oscillation of second order neutral di erential equations, Appl. Math. Comput. 274, 178–181, 2016.
O. Došlý and P. Rehák, Half-linear di erential 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 di erential equations, Appl. Math. Lett. 69, 126–132, 2017.
J. Džurina and I. P. Stavroulakis, Oscillation criteria for second-order delay di er-ential equations, Appl. Math. Comput. 140 (2-3), 445–453, 2003.
L. Erbe, A. Peterson and S. H. Saker, Kamenev-type oscillation criteria for second-order linear delay dynamic equations, Dynam. Systems Appl. 15 (1), 65–78, 2006.
S. Fišnarová and R. Mařík, Oscillation of half-linear di erential equations with delay, Abstr. Appl. Anal. 2013, Article ID: 583147, 1–6, 2013.
I. Győri and G. Ladas, Oscillation theory of delay di erential 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 di erential equations, Funct. Diﬀer. Equ. 7 (1-2), 121–145, 2000.
R. G. Koplatadze, Criteria for the oscillation of solutions of di erential 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-di erential 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 di er-ential equations, Funkcial. Ekvac. 56 (1), 111–120, 2013.
T. Li, Y. V. Rogovchenko and C. Zhang, Oscillation results for second-order nonlinear neutral di erential equations, Adv. Diﬀerence Equ. 2013 (336), 1–13, 2013.
W. E. Mahfoud, Comparison theorems for delay di erential equations, Pacific J. Math. 83 (1), 187–197, 1979.
Y. G. Sun and F. W. Meng, Note on the paper of Džurina and Stavroulakis: “Oscilla-tion criteria for second-order delay di erential 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 di erential equa-tions with deviating arguments, Electron. J. Qual. Theory Diﬀer. Equ. 2009 (61), 1–11, 2009.
J. J. Wei, Oscillation of second order delay di erential equation, Ann. Diﬀerential Equations, 4 (4), 473–478, 1988.
H. Wu, L. Erbe and A. Peterson, Oscillation of solution to second-order half-linear de-lay 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 di erential equations, Appl. Math. Comput. 182 (1), 797–803, 2006.
L. Ye and Z. Xu, Oscillation criteria for second order quasilinear neutral delay dif-ferential 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.