Oscillation theorems for 𝑛th order nonlinear differential equations with deviating arguments

Grace, S. R.; Lalli, B. S.

doi:10.1090/S0002-9939-1984-0722416-7

Oscillation theorems for $n$th order nonlinear differential equations with deviating arguments
HTML articles powered by AMS MathViewer

by S. R. Grace and B. S. Lalli PDF

Proc. Amer. Math. Soc. 90 (1984), 65-70 Request permission

Abstract:

In this note we study the oscillatory behavior of solutions of the $n$ th order nonlinear functional differential equation \[ {x^{(n)}}(t) + q(t)f(x\left [ {g(t)} \right ]) = 0,\quad n\;{\text {even,}}\] without assuming that the deviating argument is retarded or advanced. Sufficient conditions are established for all solutions of the equation to be oscillatory.

References

M. K. Grammatikopoulos, Y. G. Sficas, and V. A. Staïkos, Oscillatory properties of strongly superlinear differential equations with deviating arguments, J. Math. Anal. Appl. 67 (1979), no. 1, 171–187. MR 524471, DOI 10.1016/0022-247X(79)90015-5
Athanassios G. Kartsatos, Recent results on oscillation of solutions of forced and perturbed nonlinear differential equations of even order, Stability of dynamical systems, theory and applications (Proc. Regional Conf., Mississippi State Univ., Mississippi State, Miss., 1975) Lecture Notes in Pure and Appl. Math., Vol. 28, Dekker, New York, 1977, pp. 17–72. MR 0594954
Yuichi Kitamura and Takaŝi Kusano, Oscillation of first-order nonlinear differential equations with deviating arguments, Proc. Amer. Math. Soc. 78 (1980), no. 1, 64–68. MR 548086, DOI 10.1090/S0002-9939-1980-0548086-5
Takaŝi Kusano and Hiroshi Onose, Oscillatory and asymptotic behavior of sublinear retarded differential equations, Hiroshima Math. J. 4 (1974), 343–355. MR 350154
Imrich Ličko and Marko Švec, Le caractère oscillatoire des solutions de l’équation $y^{(n)}+f(x)y^{\alpha }=0,\,n>1$, Czechoslovak Math. J. 13(88) (1963), 481–491 (French, with Russian summary). MR 161001
Z. Opial, Sur une critère d’oscillation des intégrales de l’équation différentielle $(Q(t)x’)’+f(t)x=0$, Ann. Polon. Math. 6 (1959/60), 99–104 (French). MR 104876, DOI 10.4064/ap-6-1-99-104
Y. G. Sficas and V. A. Staïkos, Oscillations of retarded differential equations, Proc. Cambridge Philos. Soc. 75 (1974), 95–101. MR 333402, DOI 10.1017/s0305004100048283
Urszula Sztaba, Note on: “On oscillatory behavior of even order delay equations” [J. Math. Anal. Appl. 42 (1973), 183–190; MR 48 #2524] by R. S. Dahiya and B. Singh, J. Math. Anal. Appl. 63 (1978), no. 2, 313–318. MR 489226, DOI 10.1016/0022-247X(78)90075-6
James S. W. Wong, Second order oscillation with retarded arguments, Ordinary differential equations (Proc. NRL-MRC Conf., Math. Res. Center, Naval Res. Lab., Washington, D.C., 1971) Academic Press, New York, 1972, pp. 581–596. MR 0425310