Geodesic
Oscillatory behavior of higher order neutral differential equation with multiple functional delays under derivative operator
Archivum mathematicum, Tome 58 (2022) no. 2, pp. 65-84 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

In this article, we obtain sufficient conditions so that every solution of neutral delay differential equation \[ \big (y(t)- \sum _{i=1}^k p_i(t) y(r_i(t))\big )^{(n)}+ v(t)G( y(g(t)))-u(t)H(y(h(t))) = f(t) \] oscillates or tends to zero as $t\rightarrow \infty $, where, $n \ge 1$ is any positive integer, $p_i$, $r_i\in C^{(n)}([0,\infty ),\mathbb{R})$  and $p_i$ are bounded for each $i=1,2,\dots ,k$. Further, $f\in C([0, \infty ), \mathbb{R})$, $g$, $h$, $v$, $u \in C([0, \infty ), [0, \infty ))$, $G$ and $H \in C(\mathbb{R},\mathbb{R})$. The functional delays $r_i(t)\le t$, $g(t)\le t$ and $h(t)\le t$ and all of them approach $\infty $ as $t\rightarrow \infty $. The results hold when $u\equiv 0$ and $f(t)\equiv 0$. This article extends, generalizes and improves some recent results, and further answers some unanswered questions from the literature.
In this article, we obtain sufficient conditions so that every solution of neutral delay differential equation \[ \big (y(t)- \sum _{i=1}^k p_i(t) y(r_i(t))\big )^{(n)}+ v(t)G( y(g(t)))-u(t)H(y(h(t))) = f(t) \] oscillates or tends to zero as $t\rightarrow \infty $, where, $n \ge 1$ is any positive integer, $p_i$, $r_i\in C^{(n)}([0,\infty ),\mathbb{R})$  and $p_i$ are bounded for each $i=1,2,\dots ,k$. Further, $f\in C([0, \infty ), \mathbb{R})$, $g$, $h$, $v$, $u \in C([0, \infty ), [0, \infty ))$, $G$ and $H \in C(\mathbb{R},\mathbb{R})$. The functional delays $r_i(t)\le t$, $g(t)\le t$ and $h(t)\le t$ and all of them approach $\infty $ as $t\rightarrow \infty $. The results hold when $u\equiv 0$ and $f(t)\equiv 0$. This article extends, generalizes and improves some recent results, and further answers some unanswered questions from the literature.
DOI : 10.5817/AM2022-2-65
Classification : 34C10, 34C15, 34K40
Keywords: oscillation; non-oscillation; neutral equation; asymptotic behaviour 
@article{10_5817_AM2022_2_65,
     author = {Rath, R.N. and Panda, K.C. and Rath, S.K.},
     title = {Oscillatory behavior of higher order neutral differential equation with multiple functional delays under derivative operator},
     journal = {Archivum mathematicum},
     pages = {65--84},
     year = {2022},
     volume = {58},
     number = {2},
     doi = {10.5817/AM2022-2-65},
     mrnumber = {4448484},
     zbl = {07547202},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.5817/AM2022-2-65/}
}
TY  - JOUR
AU  - Rath, R.N.
AU  - Panda, K.C.
AU  - Rath, S.K.
TI  - Oscillatory behavior of higher order neutral differential equation with multiple functional delays under derivative operator
JO  - Archivum mathematicum
PY  - 2022
SP  - 65
EP  - 84
VL  - 58
IS  - 2
UR  - http://geodesic.mathdoc.fr/articles/10.5817/AM2022-2-65/
DO  - 10.5817/AM2022-2-65
LA  - en
ID  - 10_5817_AM2022_2_65
ER  - 
%0 Journal Article
%A Rath, R.N.
%A Panda, K.C.
%A Rath, S.K.
%T Oscillatory behavior of higher order neutral differential equation with multiple functional delays under derivative operator
%J Archivum mathematicum
%D 2022
%P 65-84
%V 58
%N 2
%U http://geodesic.mathdoc.fr/articles/10.5817/AM2022-2-65/
%R 10.5817/AM2022-2-65
%G en
%F 10_5817_AM2022_2_65
Rath, R.N.; Panda, K.C.; Rath, S.K. Oscillatory behavior of higher order neutral differential equation with multiple functional delays under derivative operator. Archivum mathematicum, Tome 58 (2022) no. 2, pp. 65-84. doi: 10.5817/AM2022-2-65

[1] Dix, J.G., Misra, N., Padhy, L.N., Rath, R.N.: On oscillation and asymptotic behaviour of a neutral differential equations of first order with oscillating coefficients. Electron. J. Qual. Theory Differ. Equ. 19 (2008), 1–10. | DOI | MR

[2] Gyori, I., Ladas, G.: Oscillation Theory of Delay-Differential Equations with Applications. Clarendon Press, Oxford, 1991. | MR

[3] Hilderbrandt, T.H.: Introduction to the Theory of Integration. Academic Press, New York, 1963. | MR

[4] Karpuz, B., Rath, R.N., Padhy, L.N.: On oscillation and asymptotic behaviour of a higher order neutral differential equation with positive and negative coefficients. Electron. J. Differential Equations 2008 (113) (2008), 1–15, MR2430910. | MR

[5] Ladde, G.S., Lakshmikantham, V., Zhang, B.G.: Oscillation Theory of Differential Equations with Deviating Arguments. Marcel Dekker Inc., New York, 1987. | MR | Zbl

[6] Parhi, N., Rath, R.N.: On oscillation criteria for a forced neutral differential equation. Bull. Inst. Math. Acad. Sinica 28 (2000), 59–70. | MR

[7] Parhi, N., Rath, R.N.: On oscillation and asymptotic behaviour of solutions of forced first order neutral differential equations. Proceedings of Indian Acad. Sci. Math. Sci., vol. 111, 2001, pp. 337–350. | DOI | MR | Zbl

[8] Parhi, N., Rath, R.N.: Oscillation criteria for forced first order neutral differential equations with variable coefficients. J. Math. Anal. Appl. 256 (2001), 525–541. | DOI | MR | Zbl

[9] Parhi, N., Rath, R.N.: On oscillation of solutions of forced non linear neutral differential equations of higher order. Czechoslovak Math. J. 53 (2003), 805–825, MR2018832(2005g:34163). | DOI | MR

[10] Parhi, N., Rath, R.N.: On oscillation of solutions of forced nonlinear neutral differential equations of higher order II. Ann. Polon. Math. 81 (20033), 101–110. | DOI | MR | Zbl

[11] Parhi, N., Rath, R.N.: Oscillation of solutions of a class of first order neutral differential equations. J. Indian Math. Soc. 71 (2004), 175–188. | MR

[12] Royden, H.L.: Real Analysis. 3rd ed., MacMilan Publ. Co., New York, 1988. | MR | Zbl

[13] Sahiner, Y., Zafer, A.: Bounded oscillation of non-linear neutral differential equations of arbitrary order. Czechoslovak Math. J. 51 (2001), 185–195. | DOI | MR

Cité par Sources :