Oscillatory and Asymptotic Behaviour of a Class of Nonlinear Differential Equations of Third Order

Asymptotic behaviour of a class of nonlinear functional differential equations of third order

“…For some results for this equation, we refer in particular to Lazer [1]. Later his results were improved by several authors (see, e.g., [2][3][4]). Some of these results are also generalized to delay differential equations [5][6][7].…”

“…The motivation for our work comes chiefly from the articles of Parhi [6,7] and Dzurina [5]. In the special case F(y(g(t))) =-y(g(t)), Dzurina obtained sufficient conditions under which every nonoscillatory solution y of (1) has the property that limt-,oo y(t) = A E R. Unfortunately, he did not have any results about the possibility that limt--,oo y(t) = 0.…”

“…In the special case F(y(g(t))) =-y(g(t)), Dzurina obtained sufficient conditions under which every nonoscillatory solution y of (1) has the property that limt-,oo y(t) = A E R. Unfortunately, he did not have any results about the possibility that limt--,oo y(t) = 0. Parhi [6,7] investigated a more general equation and established conditions which ensure that every nonoscillatory solution y satisfies either limt__,~ y(t) = 0, limt-,~ y(t) = A ~ 0, or limt--,oo [y(t)] = c~.…”

Oscillation criteria of a certain class of third order nonlinear delay differential equations with damping

“…Compared to second order differential equations, the study of oscillation and asymptotic behaviour of third order differential equations has received considerably less attention in the literature. In the ordinary case for some recent results on third order equations the reader can refer to Bartusek [3], Cecchi and Marini [4,5], Parhi and Das [19][20][21]23,24], Skerlik [32][33][34], Tiryaki and Yaman [38].…”

“…On the other hand, all solutions of y + y(t − τ ) = 0, τ > 0, are oscillatory if and only if τ e > 3 [16]. But the corresponding ordinary differential equation y + y = 0 admits a nonoscillatory solution y 1 (t) = e −t and oscillatory solutions y 2 (t) = e t/2 cos √ 3 2 t and y 3 (t) = e t/2 sin √ 3 2 t. In the literature there are some papers and books, for example Agarwal et al [1], Dzurina [8,9], Erbe et al [10], Grace and Lalli [6], Gyori and Ladas [11], Kartsatos and Manougian [13], Kusano and Onose [14,15], Ladde et al [17], Parhi and Das [22,26], Parhi and Padhi [25,27], Saker [31], Tiryaki and Yaman [37] which deal with the oscillatory and asymptotic behaviour of solutions of functional differential equations. In this paper, by using a generalized Riccati transformation and an integral averaging technique, we establish some new sufficient conditions which insure that every solution of (1.1) oscillates or converges to zero.…”