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

Abstract

It has been shown that under certain conditions on coefficient functions, the nonoscillatory solutions of this equation tends to either zero or ~oo as t-* (~.

Oscillation criteria for third-order nonlinear functional differential equations

“…(3.12) Defineα(t) = 1 H (t, t 1 ) t t 1 H(t, s)A(s)ω 2 (s) ds; β(t) = 1 H (t, t 1 ) t t 1H(t, s)Q (t, s)ω(s)ds.It follows from (3.12) that lim inf t→∞ [α(t) + β(t)] < ∞. The remainder of the proof of this case is similar to that of the proofs of the corresponding theorems in[26][27][28][29]13,18,30] and hence is omitted.If y(t) > 0 and L 1 y(t) < 0 holds, the proof is similar to that of the proof of Lemma 2.4 and hence is omitted. Thus, the proof is complete.…”

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

“…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.…”

Oscillation criteria for third order nonlinear delay differential equations with damping

“…Compared to second order differential equations, the study of oscillation and asymptotic behavior 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 Cecchi and Marini [3,4], Parhi and Das [10,11], Parhi and Padhi [12], Skerlik [13], Tiryaki and Yaman [14], Aktas and Tiryaki [1]. It is interesting to note that there are third order delay differential equations which have only oscillatory solutions or have both oscillatory and nonoscillatory solutions.…”

“…But the corresponding ordinary differential equation y (t) + y(t) = 0, admits a nonoscillatory solution y 1 (t) = e −t and oscillatory solutions y 2 (t) = e t/2 sin √ 3/2t and y 3 (t) = e t/2 cos √ 3/2t . In the literature there are some papers and books, for example Agarwal et al [2], Grace and Lalli [5], Parhi and Das [10,11], Parhi and Padhi [12], Skerlik [13], and Tiryaki and Yaman [14], which deal with the oscillatory and asymptotic behavior of solutions of functional differential equations. In [1,15], the authors used a generalized Riccati transformation and an integral averaging technique for establishing some sufficient conditions which insure that any solution of equation (1.1) oscillates or converges to zero.…”

“…It is easy that to check that all hypotheses of Theorem 2.9 are satisfied and hence every solution y of equation (2.24) is oscillatory or y is oscillatory. One such solution is y(t) = sin t. We note that none of the results in [3,8,[10][11][12][13][14][15] are applicable to equation (2.24).…”