Applied Mathematics and Computation 362 (2019) 124530
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Applied Mathematics and Computation
journal homepage: www.elsevier.com/locate/amc
On oscillation of third-order noncanonical delay differential
equations
Said R. Grace
a
, Irena Jadlovská
b
, Agacik Zafer
c,∗
a
Department of Engineering Mathematics, Faculty of Engineering, Cairo University, Orman, Giza 12221, Egypt
b
Department of Mathematics and Theoretical Informatics, Faculty of Electrical Engineering and Informatics, Technical University of
Košice, Nˇ emcovej 32, Košice 042 00, Slovakia
c
Department of Mathematics, College of Engineering and Technology, American University of the Middle East, Kuwait
a r t i c l e i n f o
MSC:
34C10
34K11
Keywords:
Linear
Differential equation
Delay
Third-order
Noncanonical operators
Oscillation
a b s t r a c t
In the paper, we give new criteria for the oscillation of third-order delay differential equa-
tions with noncanonical operators. Our results essentially improve and simplify the ones in
Dzurina et al. (2018). The results are illustrated by a Euler-type delay differential equation.
© 2019 Elsevier Inc. All rights reserved.
1. Introduction
The purpose of this work is to improve the oscillation results given in [12] for the third-order delay differential equation
of the form
r
2
(t )
(
r
1
(t )y
′
(t )
)
′
′
+ q(t )y(τ (t )) = 0, t ≥ t
0
, (1.1)
where it is assumed without further mention that
(H
1
) the operator Ly =
r
2
(
r
1
y
′
)
′
′
is of noncanonical type, that is,
∞
t
0
ds
r
1
(s)
< ∞ and
∞
t
0
ds
r
2
(s)
< ∞;
(H
2
) q ∈ C ([t
0
, ∞), [0, ∞)) does not vanish eventually, and r
1
, r
2
∈ C ([t
0
, ∞), (0, ∞));
(H
3
) τ ∈ C
1
([t
0
, ∞), (0, ∞)), τ (t) ≤ t, τ
′
(t) > 0, and lim
t →∞
τ (t ) = ∞.
With regard to many indications of the importance of third-order differential equations in the applications as well as
the number of mathematical problems involved [15], the subject of the qualitative theory for such equations has undergone
rapid development in the last three decades.
∗
Corresponding author.
E-mail addresses: srgrace@eng.cu.edu.eg (S.R. Grace), irena.jadlovska@tuke.sk (I. Jadlovská), agacik.zafer@aum.edu.kw (A. Zafer).
https://doi.org/10.1016/j.amc.2019.06.044
0096-3003/© 2019 Elsevier Inc. All rights reserved.