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

Year 2019, Volume: 48 Issue: 1, 170 - 179, 01.02.2019

George E. Chatzarakis İrena Jadlovska

Abstract

 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

References

• 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.