Integral criteria for oscillation of third order nonlinear differential equations

On some classes of nonoscillatory solutions of third-order nonlinear differential equations

“…Since function f is Lipschitz on the interval [3,4] On some classes of nonoscillatory solutions of third-order nonlinear differential equations…”

“…The object of this paper is to continue in study of the nonlinear case and to provide some other results for the equations of the form (N A ). For the sake of brevity, we introduce the following notation z [0] = z, z [1] = 1 p z , z [2] = 1 r 1 p z = 1 r z [1] , z [3] = 1 r 1 p z = z [2] .…”

“…In particular, they have stated conditions for the oscillation, nonoscillation and a specified asymptotic behavior of solutions, existence and nonexistence conditions for oscillatory and nonoscillatory solutions of various types and also many results for the classification of differential equations according to oscillatory and asymptotic properties of their solutions. Among the extensive literature on this area, we mention here [3,7,8,17,20,23,25,27] for the differential equations without deviating argument and [2, 4, 5, 12-15, 26, 28] for those with deviating argument. Especially, we refer to the recent book [24] where exhaustive summary of results regarding to the third-order differential equations is done.…”

On nonoscillatory solutions tending to zero of third-order nonlinear differential equations

“…The case x(t) < 0, x [1] (t) > 0, x [2] (t) > 0 for all t ≥ t 1 (where t 1 ≥ a) can be treated similarly.…”

“…The authors have obtained the sufficient conditions for oscillation and asymptotic behavior of solutions, conditions for existence or nonexistence some types of solutions and also many results for the classification of solutions according to their oscillatory and asymptotic properties. Among the extensive literature on these topics, we mention here [2], [4], [5], [6], [9], [17] for the differential equations without deviating argument and [3], [8], [9], [10], [11], [14], [15], [18], [19] for those with deviating argument.…”