In this paper, a survey of the most interesting conditions for the oscillation of all solutions to first-order linear differential equations with a retarded argument is presented in chronological ...order, especially in the case when well-known oscillation conditions are not satisfied. The essential improvement and the importance of these oscillation conditions is also indicated.
Consider the first-order linear delay differential equation(1)x′(t)+p(t)x(τ(t))=0,t⩾t0,where p,τ∈C(t0,∞),R+), τ(t)<t for t⩾t0 and limt→∞τ(t)=∞, and the (discrete analogue) difference ...equation(1′)Δx(n)+p(n)x(τ(n))=0,n=0,1,2,…,where Δ denotes the forward difference operator Δx(n)=x(n+1)-x(n),p(n) is a sequence of nonnegative real numbers and τ(n) is a sequence of integers such that τ(n)⩽n-1 for all n⩾0 and limn→∞τ(n)=∞. The state-of-the-art on the oscillation of all solutions to these equations are established especially in the case of non-monotone arguments. Examples illustrating the results are given.
Consider the first-order delay difference equation with a constant argument Δxn+pnxn-k=0, n=0,1,2,…, and the delay difference equation with a variable argument Δxn+pnxτn=0, n=0,1,2,…, where p(n) is ...a sequence of nonnegative real numbers, k is a positive integer, Δx(n)=x(n+1)-x(n), and τ(n) is a sequence of integers such that τ(n)≤n-1 for all n≥0 and limn→∞τ(n)=∞. A survey on the oscillation of all solutions to these equations is presented. Examples illustrating the results are given.
This paper presents new sufficient conditions for the oscillation of all proper solutions of the first order linear difference equation with delay argument
Δ
u
(
k
)
+
p
(
k
)
u
(
τ
(
k
)
)
=
0
,
k
∈
...N
,
where
Δ
u
(
k
)
=
u
(
k
+
1
)
−
u
(
k
)
,
p
:
N
→
R
+
,
τ
:
N
→
N
and
lim
k
→
+
∞
τ
(
k
)
=
+
∞
. Examples illustrating the results are given. It is to be pointed out that this is the first paper dealing with the oscillatory behaviour of the equation in the case of a general delay argument
τ
(
k
)
.
This paper deals with the oscillation of the first-order differential equation with several delay arguments x′t+∑i=1mpitxτit=0,t≥t0, where the functions pi,τi∈Ct0,∞,R+, for every i=1,2,…,m,τit≤t for ...t≥t0 and limt→∞τit=∞. In this paper, the state-of-the-art on the sharp oscillation conditions are presented. In particular, several sufficient oscillation conditions are presented and it is shown that, under additional hypotheses dealing with slowly varying at infinity functions, some of the “liminf” oscillation conditions can be essentially improved replacing “liminf” by “limsup”. The importance of the slowly varying hypothesis and the essential improvement of the sufficient oscillation conditions are illustrated by examples.
We consider difference equations with several non-monotone deviating arguments and non-negative coefficients. The deviations (delays and advances) are, generally, unbounded. Sufficient oscillation ...conditions are obtained in an explicit iterative form. Additional results in terms of
are obtained for bounded deviations. Examples illustrating the oscillation tests are presented.
It is known that all solutions of the difference equation
Δ
x
(
n
)
+
p
(
n
)
x
(
n
−
k
)
=
0
,
n
≥
0
,
where
{
p
(
n
)
}
n
=
0
∞
is a nonnegative sequence of reals and
k
is a natural number, ...oscillate if
lim inf
n
→
∞
∑
i
=
n
−
k
n
−
1
p
(
i
)
>
(
k
k
+
1
)
k
+
1
. In the case that
∑
i
=
n
−
k
n
−
1
p
(
i
)
is slowly varying at infinity, it is proved that the above result can be essentially improved by replacing the above condition with
lim sup
n
→
∞
∑
i
=
n
−
k
n
−
1
p
(
i
)
>
(
k
k
+
1
)
k
+
1
. An example illustrating the applicability and importance of the result is presented.
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