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On the oscillation of fourth-order delay differential equations. (English) Zbl 1459.34152

Summary: In the paper, fourth-order delay differential equations of the form \[ \bigl(r_{3} \bigl(r_{2} \bigl(r_{1}y' \bigr)' \bigr)' \bigr)'(t) + q(t) y \bigl( \tau (t) \bigr) = 0 \] under the assumption \[\int _{t_{0}}^{\infty }\frac{\text{d}t}{r_{i}(t)} < \infty , \quad i = 1,2,3, \] are investigated. Our newly proposed approach allows us to greatly reduce a number of conditions ensuring that all solutions of the studied equation oscillate. An example is also presented to test the strength and applicability of the results obtained.

MSC:

34K11 Oscillation theory of functional-differential equations
34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
34A30 Linear ordinary differential equations and systems
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[1] Agarwal, R.P., Grace, S.R.: The oscillation of higher-order differential equations with deviating arguments. Comput. Math. Appl. 38(3-4), 185-199 (1999) · Zbl 0935.34058 · doi:10.1016/S0898-1221(99)00193-5
[2] Agarwal, R.P., Grace, S.R., Manojlovic, J.V.: Oscillation criteria for certain fourth order nonlinear functional differential equations. Math. Comput. Model. 44(1-2), 163-187 (2006) · Zbl 1137.34031 · doi:10.1016/j.mcm.2005.11.015
[3] Baculíková, B., Džurina, J., Graef, J.R.: On the oscillation of higher-order delay differential equations. J. Math. Sci. 187(4), 387-400 (2012) · Zbl 1267.34121
[4] Dennis, S.C.R., Walker, J.D.A.: Calculation of the steady flow past a sphere at low and moderate Reynolds numbers. J. Fluid Mech. 48(4), 771-789 (1971) · Zbl 0266.76023 · doi:10.1017/S0022112071001848
[5] Džurina, J., Baculíková, B.: Oscillation of the even-order delay linear differential equation. Carpath. J. Math. 31(1), 69-76 (2015) · Zbl 1349.34265
[6] Džurina, J., Jadlovská, I.: Oscillation of third-order differential equations with noncanonical operators. Appl. Math. Comput. 336, 394-402 (2018) · Zbl 1384.90058 · doi:10.1016/j.cam.2017.12.052
[7] Elias, U.: Oscillation Theory of Two-Term Differential Equations, vol. 396. Springer, Berlin (1997) · Zbl 0878.34022 · doi:10.1007/978-94-017-2517-0
[8] Esmailzadeh, E., Ghorashi, M.: Vibration analysis of beams traversed by uniform partially distributed moving masses. J. Sound Vib. 184(1), 9-17 (1995) · Zbl 1055.74530 · doi:10.1006/jsvi.1995.0301
[9] Grace, S.R., Agarwal, R.P., Graef, J.R.: Oscillation theorems for fourth order functional differential equations. J. Appl. Math. Comput. 30(1-2), 75-88 (2009) · Zbl 1188.34085 · doi:10.1007/s12190-008-0158-9
[10] Grace, S.R., Bohner, M., Liu, A.: Oscillation criteria for fourth-order functional differential equations. Math. Slovaca 63(6), 1303-1320 (2013) · Zbl 1340.34242 · doi:10.2478/s12175-013-0172-8
[11] Graef, J.R., Grace, S.R., Tunç, E.: Oscillation of even-order advanced functional differential equations. Publ. Math. (Debr.) 93(3-4), 445-455 (2018) · Zbl 1424.34225 · doi:10.5486/PMD.2018.8205
[12] Graef, J.R., Tunç, E.: Oscillation of fourth-order nonlinear dynamic equations on time scales. Panam. Math. J. 25(4), 16-34 (2015) · Zbl 1336.34131
[13] Kiguradze, I.T., Chanturia, T.A.: Asymptotic Properties of Solutions of Nonautonomous Ordinary Differential Equations. Mathematics and Its Applications (Soviet Series), vol. 89. Kluwer Academic, Dordrecht (1993). Translated from the 1985 Russian original · Zbl 0782.34002
[14] Ladde, G.S., Lakshmikantham, V., Zhang, B.G.: Oscillation Theory of Differential Equations with Deviating Arguments. Monographs and Textbooks in Pure and Applied Mathematics, vol. 110. Dekker, New York (1987) · Zbl 0622.34071
[15] Lega, J., Moloney, J.V., Newell, A.C.: Swift-Hohenberg equation for lasers. Phys. Rev. Lett. 73(22), 2978-2981 (1994) · doi:10.1103/PhysRevLett.73.2978
[16] Li, T., Rogovchenko, Yu.V.: Oscillation criteria for even-order neutral differential equations. Appl. Math. Lett. 61, 35-41 (2016) · Zbl 1347.34106 · doi:10.1016/j.aml.2016.04.012
[17] Mahfoud, W.E.: Comparison theorems for delay differential equations. Pac. J. Math. 83(1), 187-197 (1979) · Zbl 0441.34053 · doi:10.2140/pjm.1979.83.187
[18] McKenna, P.J., Walter, W.: Nonlinear oscillations in a suspension bridge. Arch. Ration. Mech. Anal. 98(2), 167-177 (1987) · Zbl 0676.35003 · doi:10.1007/BF00251232
[19] McKenna, P.J., Walter, W.: Travelling waves in a suspension bridge. SIAM J. Appl. Math. 50(3), 703-715 (1990) · Zbl 0699.73038 · doi:10.1137/0150041
[20] Oǧuztöreli, M.N., Stein, R.B.: An analysis of oscillations in neuro-muscular systems. J. Math. Biol. 2(2), 87-105 (1975) · Zbl 0309.92007 · doi:10.1007/BF00275922
[21] Swanson, C.A.: Comparison and Oscillation Theory of Linear Differential Equations, vol. 48. Elsevier, Amsterdam (2000)
[22] Trench, W.F.: Canonical forms and principal systems for general disconjugate equations. Trans. Am. Math. Soc. 189, 319-327 (1974) · Zbl 0289.34051 · doi:10.1090/S0002-9947-1974-0330632-X
[23] Truesdell, C.: Rational Mechanics. Academic Press, New York (1983) · Zbl 1225.70002
[24] Tunç, E.: Oscillation results for even order functional dynamic equations on time scales. Electron. J. Qual. Theory Differ. Equ. 2014, 27 (2014) · Zbl 1343.34031 · doi:10.1186/1687-1847-2014-27
[25] Zhang, C., Agarwal, R.P., Bohner, M., Li, T.: Oscillation of fourth-order delay dynamic equations. Sci. China Math. 58(1), 143-160 (2015) · Zbl 1321.34121 · doi:10.1007/s11425-014-4917-9
[26] Zhang, C., Agarwal, R.P., Li, T.: Oscillation and asymptotic behavior of higher-order delay differential equations with p-Laplacian like operators. J. Math. Anal. Appl. 409(2), 1093-1106 (2014) · Zbl 1314.34141 · doi:10.1016/j.jmaa.2013.07.066
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