Oscillation and nonoscillation criteria for a half-linear difference equation of the second order and extended discrete Hardy inequality
UDC 517.9 We establish the oscillatory properties of a half-linear difference equation of the second order by using a suitable extension of the weighted discrete Hardy inequality.  
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| author | Kalybay, A. Karatayeva, D. Kalybay, A. Karatayeva, D. D. |
| author_facet | Kalybay, A. Karatayeva, D. Kalybay, A. Karatayeva, D. D. |
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| description | UDC 517.9
We establish the oscillatory properties of a half-linear difference equation of the second order by using a suitable extension of the weighted discrete Hardy inequality.
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| doi_str_mv | 10.37863/umzh.v74i1.2298 |
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DOI: 10.37863/umzh.v74i1.2298
UDC 517.9
A. Kalybay (KIMEP Univ., Almaty, Kazakhstan),
D. Karatayeva (L. N. Gumilyov Eurasian Nat. Univ., Nur-Sultan, Kazakhstan)
OSCILLATION AND NONOSCILLATION CRITERIA
FOR A HALF-LINEAR DIFFERENCE EQUATION OF THE SECOND ORDER
AND EXTENDED DISCRETE HARDY INEQUALITY*
КРИТЕРIЇ КОЛИВАННЯ ТА НЕКОЛИВАННЯ
ДЛЯ НАПIВЛIНIЙНОГО РIЗНИЦЕВОГО РIВНЯННЯ ДРУГОГО ПОРЯДКУ
ТА РОЗШИРЕННЯ ДИСКРЕТНОЇ НЕРIВНОСТI ГАРДI
We establish the oscillatory properties of a half-linear difference equation of the second order by using a suitable extension
of the weighted discrete Hardy inequality.
За допомогою вiдповiдного розширення дискретної нерiвностi Гардi встановлено коливнi властивостi напiвлiнiйного
рiзницевого рiвняння другого порядку.
1. Introduction. We consider the following second order half-linear difference equation:
\Delta
\bigl(
\rho i| \Delta yi| p - 2\Delta yi
\bigr)
+ vi| yi+1| p - 2yi+1 = 0, i = 0, 1, 2, . . . , (1.1)
where 1 < p < \infty , \Delta yi = yi+1 - yi. The coefficients \rho = \{ \rho i\} and v = \{ vi\} of equation (1.1) are
sequences of real numbers. Moreover, \rho i > 0 for any i = 0, 1, 2, . . . .
Let us list notions and statements required for this paper. Let m \geq 0 and n \geq 0 be integer
numbers. For simplicity we will use the term “interval” meaning “discrete interval”.
If there exists a nontrivial solution y = \{ yi\} of equation (1.1) such that ym \not = 0 and ymym+1 < 0,
then we say that the solution y has a generalized zero on the interval (m,m+ 1].
A nontrivial solution y of equation (1.1) is called oscillatory if it has an infinite number of
generalized zeros, otherwise it is called nonoscillatory.
Equation (1.1) is called oscillatory if all its nontrivial solutions are oscillatory, otherwise it is
called nonoscillatory.
By Sturm’s separation theorem [18] (Theorem 3), equation (1.1) is oscillatory if one of its non-
trivial solutions is oscillatory.
Equation (1.1) is called disconjugate on the interval [m,n], 0 \leq m < n, if its any nontrivial
solution has no more than one generalized zero on the interval (m,n+ 1] and its nontrivial solution
\~y with the initial condition \~ym = 0 has not a generalized zero on the interval (m,n+ 1], otherwise
it is called conjugate on the interval [m,n].
Equation (1.1) is called disconjugate on the interval [m,\infty ) if for any n > m it is disconjugate
on the interval [m,n].
The investigation of the oscillatory properties of (1.1) is a subject of many works (see, e.g., the
papers [1, 6 – 11, 16 – 19, 21 – 23] and references given there). This problem was firstly studied for
p = 2, when equation (1.1) is the following linear difference equation:
* This paper was supported by the Ministry of Education and Science of the Republic of Kazakhstan (Grant
No. AP08856100, in area “Scientific research in the field of natural sciences”).
c\bigcirc A. KALYBAY, D. KARATAYEVA, 2022
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 1 45
46 A. KALYBAY, D. KARATAYEVA
\Delta (\rho i\Delta yi) + viyi+1 = 0, i = 0, 1, 2, . . . , (1.2)
that is often written in the form
\rho n - 1yn - 1 + \rho nyn+1 + unyn = 0, un = vn - \rho n - \rho n - 1, n = 1, 2, 3, . . . .
Moreover, there are many works devoted to the differential analogues of equations (1.1) and (1.2):\bigl(
\rho (t)| y\prime (t)| p - 2y\prime (t)
\bigr) \prime
+ v(t)| y(t)| p - 2y(t) = 0, t > 0, (1.3)\bigl(
\rho (t)y\prime (t)
\bigr) \prime
+ v(t)y(t) = 0, t > 0,
respectively. The last equation is the famous Sturm equation, the investigation of which was started
in 1836 in the work [20] and has been continued up to the present days.
One of the known methods to study the oscillatory properties of equation (1.1) is the “variational
method”. This method is based on the following (see [18]).
Theorem A. Let 0 \leq m < n < \infty . Equation (1.1) is disconjugate on the interval [m,n] if and
only if
n\sum
i=m
(\rho i| \Delta yi| p - vi| yi+1| p) \geq 0 (1.4)
holds for all nontrivial y = \{ yi\} n+1
i=m, ym = 0 and yn+1 = 0.
Here, we use a statement equivalent to Theorem A, the proof of which is given in [1]. To
introduce this equivalent statement we need the definition of the set
\circ
Y (m,n) for 0 \leq m < n \leq \infty .
Denote by
\circ
Y (m,n) the set of all nontrivial sequences of real numbers y = \{ yi\} \infty i=0 such that
\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p} y \subset [m + 1, n], n < \infty , where \mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p} y := \{ i \geq 0 : yi \not = 0\} . When n = \infty , we suppose that
for any y there exists an integer k = k(y), m < k < \infty , such that \mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p} y \subset [m+ 1, k].
Theorem B. Let 0 \leq m < n \leq \infty . Equation (1.1) is disconjugate on the interval [m,n]
([m,n] = [m,\infty ) for n = \infty ) if and only if
n\sum
i=m
vi - 1| yi| p \leq
n\sum
i=m
\rho i| \Delta yi| p, y \in
\circ
Y (m,n), (1.5)
holds, where v - 1 = 0.
Let w = \{ wi\} be a fixed sequence of nonnegative real numbers. Let \rho = \{ \rho i\} , as before, be
a fixed sequence of positive real numbers. For an arbitrary sequence a = \{ ai\} we consider the
inequality \left( \infty \sum
i=1
wi
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
i\sum
j=1
aj
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
q\right)
1
q
\leq C
\Biggl( \infty \sum
i=1
\rho i| ai| p
\Biggr) 1
p
, (1.6)
that is, the known (now classical) weighted discrete Hardy inequality. In the papers [2 – 5] there are
criteria for the validity of inequality (1.6) for all relations between p and q. In addition, the work
[13] presents the history of the development of this inequality and relative results.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 1
OSCILLATION AND NONOSCILLATION CRITERIA FOR A HALF-LINEAR DIFFERENCE EQUATION . . . 47
For yi+1 =
\sum i
j=1
aj and \Delta yi = yi+1 - yi, i = 1, 2, . . . , inequality (1.6) can be rewritten in the
difference form \Biggl( \infty \sum
i=1
wi| yi+1| q
\Biggr) 1
q
\leq C
\Biggl( \infty \sum
i=1
\rho i| \Delta yi| p
\Biggr) 1
p
, (1.7)
where y = \{ yi\} is an arbitrary sequence of real numbers with y1 = 0.
In view of Theorem B it is easy to see the connection between the difference equation
\Delta (\rho i| \Delta yi| p - 2\Delta yi) + wi| yi+1| p - 2yi+1 = 0, i = 0, 1, 2, . . . , (1.8)
and the Hardy inequality (1.7) for a sequence y from the set
\circ
Y (m,n), given in the form\Biggl(
n\sum
i=m
wi - 1| yi| p
\Biggr) 1
p
\leq C
\Biggl(
n\sum
i=m
\rho i| \Delta yi| p
\Biggr) 1
p
, y \in
\circ
Y (m,n). (1.9)
Indeed, in the paper [12] the oscillation of equation (1.8) was established on the basis of inequa-
lity (1.9).
The difference between equations (1.1) and (1.8) is the fact that in (1.8) the sequence w consists
of nonnegative real numbers, while in (1.1) the sequence v consists of any real numbers. If in
(1.1) the sequence v consists of negative real numbers, then it is obvious that (1.4) holds for all
0 \leq m < n < \infty . Therefore, equation (1.1) is nonoscillatory. It is naturally to pose a question about
the influence of the positive part of the sequence v on the oscillation of equation (1.1). Assume
that v+i = \mathrm{m}\mathrm{a}\mathrm{x}(0; vi) and v - i = \mathrm{m}\mathrm{a}\mathrm{x}(0; - vi), then vi = v+i - v - i for all i = 0, 1, 2, . . . . Hence,
equation (1.1) and inequality (1.4) have the forms
\Delta (\rho i| \Delta yi| p - 2\Delta yi) - v - i | yi+1| p - 2yi+1 + v+i | yi+1| p - 2yi+1 = 0, i = 0, 1, 2, . . . ,
n\sum
i=m
(\rho i| \Delta yi| p + v - i | yi+1| p - v+i | yi+1| p) \geq 0,
(1.10)
respectively, i.e., the positive part \{ v+i \} of the sequence \{ vi\} can be considered as a perturbation of
the nonoscillation equation
\Delta (\rho i| \Delta yi| p - 2\Delta yi) - v - i | yi+1| p - 2yi+1 = 0, i = 0, 1, 2, . . . .
To investigate this problem in a more general situation, we consider the following difference equation:
\Delta (\rho i| \Delta yi| p - 2\Delta yi) + wi| yi+1| p - 2yi+1 - ri| yi+1| p - 2yi+1 = 0, i = 0, 1, 2, . . . , (1.11)
where r = \{ ri\} , as w = \{ wi\} , is a sequence of nonnegative real numbers. If we assume that
wi = v+i and ri = v - i for all i = 0, 1, 2, . . . , then equation (1.11) turns to equation (1.10), i.e.,
equation (1.1) is a partial case of equation (1.11). For equation (1.11), inequality (1.5) is equivalent
to the inequality
n\sum
i=m
wi - 1| yi| p \leq
n\sum
i=m
(\rho i| \Delta yi| p + ri - 1| yi| p) , y \in
\circ
Y (m,n). (1.12)
From Theorem B and the last inequality it follows theorem.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 1
48 A. KALYBAY, D. KARATAYEVA
Theorem 1.1. Let 0 \leq m < n \leq \infty . Equation (1.11) is disconjugate on the interval [m,n]
([m,n] = [m,\infty ) for n = \infty ) if and only if inequality (1.12) holds.
Summing up, the difference between the previous oscillation results of equation (1.1) and the
presented results is that here we study the case when the sequence v is not ultimately positive: we
drop this restriction and allow arbitrary real numbers vi. Moreover, in view of Theorem 1.1, the
proofs of the main results are based on the fulfillment of the extended Hardy inequality
n\sum
i=m
wi - 1| yi| p \leq C
n\sum
i=m
(\rho i| \Delta yi| p + ri - 1| yi| p) , y \in
\circ
Y (m,n), (1.13)
and estimates of its constant C. The results concerning inequality (1.13) are of independent interest.
This paper is organized as follows. In Section 2, we state and prove necessary and sufficient
conditions for the fulfillment of the extended discrete Hardy inequality (1.13). In Section 3, first
we present the results devoted to the oscillatory properties of the auxiliary equation (1.11), then we
present the oscillatory properties of the main equation (1.1).
We note that the similar problem for the differential equation (1.3) was considered in the pa-
per [15].
2. Extended Hardy inequality. In this section, we consider the extended discrete Hardy
inequality (1.13). This inequality has been already considered in [14]. However, in order to apply
inequality (1.13) to equation (1.11), it is important not only to find its characterizations, but also to
estimate its constant C, which is the purpose of this section.
In the sequel, the sums
\sum m
i=k
for m < k and
\sum
i\in \Omega
for empty \Omega are equal to zero. Moreover,
1
p
+
1
p\prime
= 1. The numbers m, n, \alpha , \beta , c, d, t, s, x, and z with and without indexes are integers.
We need the following lemma proved in [15]. Here, we give both its statement and proof for a
more complete presentation.
Lemma 2.1. Let 1 < p < \infty be a real number. Let g be a function defined as g(\lambda ) =
=
\lambda p
\lambda p - 1
- 1
(\lambda - 1)p
on (1,\infty ) \subset R. Then there exists a number \lambda 0 := \lambda 0(p) such that 1 < \lambda 0 < 2
and
1
(\lambda 0 - 1)p
=
\lambda p
0
\lambda p
0 - 1
, satisfying the conditions g(\lambda ) > 0 for \lambda > \lambda 0 and g(\lambda ) < 0 for
1 < \lambda < \lambda 0.
Proof. It is obvious that g(2) > 0 and \mathrm{l}\mathrm{i}\mathrm{m}\lambda \rightarrow 1+
\lambda p(\lambda - 1)p
\lambda p - 1
= 0. Using the definition of limit
there exists a number \delta > 0 for \varepsilon = 1 such that
\widetilde \lambda p(\widetilde \lambda - 1)p\widetilde \lambda p - 1
< 1 or
\widetilde \lambda p\widetilde \lambda p - 1
<
1
(\widetilde \lambda - 1)p
for every
\widetilde \lambda \in (1, 1 + \delta ). Thus, g(\widetilde \lambda ) < 0. Since the function g is continuous on (1,\infty ) there exists a number
\lambda 0 \in (1, 2) such that g(\lambda 0) = 0, i.e.,
1
(\lambda 0 - 1)p
=
\lambda p
0
\lambda p
0 - 1
or \lambda p
0(\lambda 0 - 1)p = \lambda p
0 - 1.
We define the functions g1(\lambda ) :=
1
(\lambda - 1)p
and g2(\lambda ) :=
\lambda p
\lambda p - 1
, which are strongly decreasing
on (1,\infty ). Then g2(\lambda ) > g1(\lambda ) for \lambda > \lambda 0 and g1(\lambda ) > g2(\lambda ) for 1 < \lambda < \lambda 0.
Lemma 2.1 is proved.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 1
OSCILLATION AND NONOSCILLATION CRITERIA FOR A HALF-LINEAR DIFFERENCE EQUATION . . . 49
Assume that
\varphi -
r (m, d) = \mathrm{i}\mathrm{n}\mathrm{f}
m<c\leq d
\left\{
\Biggl(
d\sum
i=c
\rho 1 - p\prime
i
\Biggr) 1 - p
+
d - 1\sum
i=c
ri
\right\} ,
\varphi +
r (d, n) = \mathrm{i}\mathrm{n}\mathrm{f}
d\leq c<n
\left\{
\Biggl(
c\sum
i=d
\rho 1 - p\prime
i
\Biggr) 1 - p
+
c - 1\sum
i=d
ri
\right\} ,
Br,w := Br,w(m,n) = \mathrm{s}\mathrm{u}\mathrm{p}
m<t\leq s<n
\Biggl(
s - 1\sum
i=t
wi
\Biggr) \Biggl(
\varphi -
r (m, t) +
s - 1\sum
i=t
ri + \varphi +
r (s, n)
\Biggr) - 1
.
Theorem 2.1. Let \lambda 0 and \lambda be defined as in Lemma 2.1. Let 0 \leq m < n \leq \infty and 1 < p < \infty .
Inequality (1.13) holds if and only if Br,w(m,n) < \infty . Moreover, the least constant in (1.13) satisfies
Br,w \leq C \leq 2\gamma pBr,w, (2.1)
where
\gamma p = \mathrm{i}\mathrm{n}\mathrm{f}
1<\lambda <\lambda 0
\lambda p(\lambda p - 1)
(\lambda - 1)p
.
Proof. Necessity. Suppose that inequality (1.13) holds for all sequences y \in
\circ
Y (m,n) with the
least constant C > 0. Let \alpha , t, s and \beta be integers satisfying the condition m < \alpha \leq t \leq s \leq \beta < n.
We introduce a test sequence y = \{ yk\} as the following:
yk =
\left\{
\sum k - 1
i=\alpha - 1
\rho 1 - p\prime
i
\Bigl( \sum t - 1
i=\alpha - 1
\rho 1 - p\prime
i
\Bigr) - 1
, \alpha \leq k \leq t,
1, t \leq k \leq s,
\sum \beta
i=k
\rho 1 - p\prime
i
\biggl( \sum \beta
i=s
\rho 1 - p\prime
i
\biggr) - 1
, s \leq k \leq \beta ,
0, m \leq k < \alpha or \beta < k \leq n.
It is obvious that y \in
\circ
Y (m,n).
Let us calculate \Delta yk :
\Delta yk =
\left\{
\rho 1 - p\prime
k
\Bigl( \sum t - 1
i=\alpha - 1
\rho 1 - p\prime
i
\Bigr) - 1
, \alpha - 1 \leq k \leq t - 1,
0, t \leq k \leq s,
- \rho 1 - p\prime
k
\biggl( \sum \beta
i=s
\rho 1 - p\prime
i
\biggr) - 1
, s \leq k \leq \beta ,
0, m \leq k < \alpha - 1 or \beta < k \leq n.
Then we have
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 1
50 A. KALYBAY, D. KARATAYEVA
n\sum
i=m
\rho i| \Delta yi| p =
\Biggl(
t - 1\sum
i=\alpha - 1
\rho 1 - p\prime
i
\Biggr) 1 - p
+
\Biggl(
\beta \sum
i=s
\rho 1 - p\prime
i
\Biggr) 1 - p
. (2.2)
Moreover, simple calculations give that
n\sum
i=m
wi - 1| yi| p \geq
s\sum
i=t
wi - 1 =
s - 1\sum
i=t - 1
wi (2.3)
and
n\sum
i=m
ri - 1| yi| p \leq
s\sum
i=t
ri - 1 =
s - 1\sum
i=t - 1
ri. (2.4)
From (1.13), (2.2), (2.3) and (2.4), we get
s - 1\sum
i=t - 1
wi \leq C
\left( \Biggl( t - 1\sum
i=\alpha - 1
\rho 1 - p\prime
i
\Biggr) 1 - p
+
\Biggl(
\beta \sum
i=s
\rho 1 - p\prime
i
\Biggr) 1 - p
+
\beta - 1\sum
i=\alpha - 1
ri
\right) =
= C
\left( \Biggl( t - 1\sum
i=\alpha - 1
\rho 1 - p\prime
i
\Biggr) 1 - p
+
\Biggl(
\beta \sum
i=s
\rho 1 - p\prime
i
\Biggr) 1 - p
+
t - 2\sum
i=\alpha - 1
ri +
s - 1\sum
i=t - 1
ri +
\beta - 1\sum
i=s
ri
\right) \leq
\leq C
\left( \left( \Biggl( t - 1\sum
i=\alpha - 1
\rho 1 - p\prime
i
\Biggr) 1 - p
+
t - 2\sum
i=\alpha - 1
ri
\right) +
s - 1\sum
i=t - 1
ri +
+
\left( \Biggl( \beta \sum
i=s
\rho 1 - p\prime
i
\Biggr) 1 - p
+
\beta - 1\sum
i=s
ri
\right) \right)
or
s\sum
i=t
wi \leq C
\Biggl(
\varphi -
r (\alpha - 1, t - 1) +
s\sum
i=t
ri + \varphi +
r (s, \beta )
\Biggr)
. (2.5)
Since the left-hand side of (2.5) is independent of \alpha , m < \alpha \leq t, and \beta , s \leq \beta < n, and the
constant C is independent of t, s, m < t \leq s < n, we have\Biggl(
s - 1\sum
i=t - 1
wi
\Biggr) \Biggl(
\varphi -
r (m, t - 1) +
s - 1\sum
i=t - 1
ri + \varphi +
r (s, n)
\Biggr) - 1
\leq C
or
Br,w \leq C. (2.6)
Sufficiency. Let Br,w < \infty . Without loss of generality, we assume that y = \{ yi\} \in
\circ
Y (m,n)
and yi \geq 0 for i = 0, 1, 2, . . . . Let \lambda > 1. For any integer k we define the set Tk := \{ i :
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 1
OSCILLATION AND NONOSCILLATION CRITERIA FOR A HALF-LINEAR DIFFERENCE EQUATION . . . 51
yi > \lambda k\} . Since the set \{ yi\} is bounded, there exists an integer number \tau = \tau (y, \lambda ) such that
T\tau \not = \varnothing and T\tau +1 = \varnothing . Let \Delta Tk = Tk\setminus Tk+1. Then
[m,n] =
\tau \bigcup
k= - \infty
Tk =
\tau \bigcup
k= - \infty
\Delta Tk. (2.7)
Remember that [m,n] = [m,\infty ) for n = \infty . From the definition of Tk and the condition T\tau \not = \varnothing ,
we have Tk \not = \varnothing for all k \leq \tau . Let k < \tau . We present the set Tk as Tk =
\bigcup
j
[tjk, s
j
k], where
[tjk, s
j
k]
\bigcap
[tik, s
i
k] = \varnothing for i \not = j. Let M j
k = Tk+1
\bigcap
[tjk, s
j
k] and \Omega k = \{ j : M j
k \not = \varnothing \} . For j \in \Omega k
we define xjk = \mathrm{m}\mathrm{i}\mathrm{n}M j
k and zjk = \mathrm{m}\mathrm{a}\mathrm{x}M j
k . It is obvious that tjk \leq xjk and zjk \leq sjk. Moreover,
Tk+1 \subset
\bigcup
j\in \Omega k
[xjk, z
j
k] and \Delta Tk \supset
\bigcup
j\in \Omega k
\Bigl(
[tjk, x
i
k - 1]
\bigcup
[zjk + 1, sjk]
\Bigr)
. (2.8)
Let tjk < xjk. Then
y
tjk - 1
\leq \lambda k and y
xj
k
> \lambda k+1.
Hence,
\lambda k(\lambda - 1) = \lambda k+1 - \lambda k \leq y
xj
k
- y
tjk - 1
=
xj
k - 1\sum
i=tjk
\Delta yi \leq
\left( xj
k - 1\sum
i=tjk
\rho 1 - p\prime
i
\right)
1
p\prime
\left( xj
k - 1\sum
i=tjk
\rho i| \Delta yi| p
\right)
1
p
. (2.9)
From (2.9), we obtain
\lambda pk
\left( xj
k - 1\sum
i=tjk
\rho 1 - p\prime
i
\right)
1 - p
\leq 1
(\lambda - 1)p
xj
k - 1\sum
i=tjk
\rho i| \Delta yi| p. (2.10)
Similarly, for zjk < sjk, we have
\lambda pk
\left( sjk\sum
i=zjk
\rho 1 - p\prime
i
\right)
1 - p
\leq 1
(\lambda - 1)p
sjk\sum
i=zjk
\rho i| \Delta yi| p. (2.11)
Let zjk = sjk. Since y
sjk
= y
zjk
> \lambda k+1 and y
sjk+1
= y
zjk+1
\leq \lambda k, then
\lambda k(\lambda - 1) \leq y
zjk
- y
zjk+1
= - \Delta y
zjk
=
sjk\sum
i=zjk
( - \Delta yi). (2.12)
From (2.12) it follows that
\lambda pk\rho
zjk
= \lambda pk
\left( sjk\sum
i=zjk
\rho 1 - p\prime
i
\right)
1 - p
\leq
\rho
zjk
(\lambda - 1)p
| \Delta y
zjk
| p = 1
(\lambda - 1)p
sjk\sum
i=zjk
\rho i| \Delta yi| p. (2.13)
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52 A. KALYBAY, D. KARATAYEVA
Again similarly, for tjk = xjk, we have
\lambda pk\rho
xj
k - 1
= \lambda pk
\left( xj
k - 1\sum
i=tjk - 1
\rho 1 - p\prime
i
\right)
1 - p
\leq
\rho
xj
k - 1
(\lambda - 1)p
| \Delta y
xj
k - 1
| p = 1
(\lambda - 1)p
xj
k - 1\sum
i=tjk - 1
\rho i| \Delta yi| p. (2.14)
If we join inequalities (2.10) and (2.14), we get
\lambda pk
\left( xj
k - 1\sum
i=\~tjk
\rho 1 - p\prime
i
\right)
1 - p
\leq 1
(\lambda - 1)p
xj
k - 1\sum
i=\~tjk
\rho i| \Delta yi| p, (2.15)
where \~tjk = tjk for tjk < xjk and \~tjk = tjk - 1 for tjk = xjk.
If we join inequalities (2.11) and (2.13), we obtain
\lambda pk
\left( sjk\sum
i=zjk
\rho 1 - p\prime
i
\right)
1 - p
\leq 1
(\lambda - 1)p
sjk\sum
i=zjk
\rho i| \Delta yi| p. (2.16)
Let us estimate the left-hand side of inequality (1.13). For \Delta Tk+1 \not = \varnothing , we have\sum
i\in \Delta Tk+1
wi - 1| yi| p \leq \lambda p(k+2)
\sum
i\in \Delta Tk+1
wi - 1. (2.17)
In view of the assumption that any sum with respect to an empty set is equal to zero, inequality (2.17)
is valid also for \Delta Tk+1 = \varnothing .
We need the following equality:
\lambda pk = (1 - \lambda - p)
k\sum
t= - \infty
\lambda pt. (2.18)
By using (2.7), (2.8), (2.17) and (2.18), we get
F \equiv
n\sum
k=m
wk - 1| yk| p =
\tau - 1\sum
k= - \infty
\sum
i\in \Delta Tk+1
wi - 1| yi| p \leq
\tau - 1\sum
k= - \infty
\lambda p(k+2)
\sum
i\in \Delta Tk+1
wi - 1 =
= \lambda 2p
\tau - 1\sum
k= - \infty
\lambda pk
\sum
i\in \Delta Tk+1
wi - 1 = \lambda 2p(1 - \lambda - p)
\tau - 1\sum
k= - \infty
\sum
i\in \Delta Tk+1
wi - 1
k\sum
t= - \infty
\lambda pt \leq
\leq \lambda p(\lambda p - 1)
\tau - 1\sum
t= - \infty
\lambda pt
\sum
k\geq t
\sum
i\in \Delta Tk+1
wi - 1 = \lambda p(\lambda p - 1)
\tau - 1\sum
t= - \infty
\lambda pt
\sum
i\in Tt+1
wi - 1 \leq
\leq \lambda p(\lambda p - 1)
\tau - 1\sum
k= - \infty
\lambda pk
\sum
j\in \Omega k
zjk - 1\sum
i=xj
k - 1
wi. (2.19)
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OSCILLATION AND NONOSCILLATION CRITERIA FOR A HALF-LINEAR DIFFERENCE EQUATION . . . 53
By the condition Bp < \infty , we obtain
zjk - 1\sum
i=xj
k - 1
wi \leq Bp
\left( \varphi -
r (m,xjk - 1) +
zjk - 1\sum
i=xj
k - 1
ri + \varphi +
r (z
j
k, n)
\right) . (2.20)
It is obvious that
\varphi -
r (m,xjk - 1) \leq
\left( xj
k - 1\sum
i=\widetilde tjk
\rho 1 - p\prime
i
\right)
1 - p
+
xj
k - 2\sum
i=\widetilde tjk - 1
ri (2.21)
and
\varphi +
r (z
j
k, n) \leq
\left( sjk\sum
i=zjk
\rho 1 - p\prime
i
\right)
1 - p
+
sjk - 1\sum
i=zjk
ri. (2.22)
By using (2.20), (2.21) and (2.22), we have
zjk - 1\sum
i=xj
k - 1
wi \leq Bp
\left(
\left( xj
k - 1\sum
i=\widetilde tjk
\rho 1 - p\prime
i
\right)
1 - p
+
sjk - 1\sum
i=\widetilde tjk - 1
ri +
\left( sjk\sum
i=zjk
\rho 1 - p\prime
i
\right)
1 - p\right) . (2.23)
If we substitute (2.23) into (2.19), we get
F \leq \lambda p(\lambda p - 1)Bp
\tau - 1\sum
k= - \infty
\lambda pk
\sum
j\in \Omega k
\left(
\left( xj
k - 1\sum
i=\widetilde tjk
\rho 1 - p\prime
i
\right)
1 - p
+
sjk - 1\sum
i=\widetilde tjk - 1
ri +
\left( sjk\sum
i=zjk
\rho 1 - p\prime
i
\right)
1 - p\right) \leq
\leq \lambda p(\lambda p - 1)Bp
\left[ \tau - 1\sum
k= - \infty
\sum
j\in \Omega k
\left( \lambda pk
\left( xj
k - 1\sum
i=\widetilde tjk
\rho 1 - p\prime
i
\right)
1 - p
+ \lambda pk
\left( sjk\sum
i=zjk
\rho 1 - p\prime
i
\right)
1 - p\right) +
+
\tau - 1\sum
k= - \infty
\lambda pk
\sum
j\in \Omega k
sjk\sum
i=\widetilde tjk
ri - 1
\right] = \lambda p(\lambda p - 1)Bp[F1 + F2]. (2.24)
By using (2.15) and (2.16), we obtain
F1 =
\tau - 1\sum
k= - \infty
\sum
j\in \Omega k
\left( \lambda pk
\left( xj
k - 1\sum
i=\widetilde tjk
\rho 1 - p\prime
i
\right)
1 - p
+ \lambda pk
\left( sjk\sum
i=zjk
\rho 1 - p\prime
i
\right)
1 - p\right) \leq
\leq 1
(\lambda - 1)p
\tau - 1\sum
k= - \infty
\sum
j\in \Omega k
\left( xj
k - 1\sum
i=\~tjk
\rho i| \Delta yi| p +
sjk\sum
i=zjk
\rho i| \Delta yi| p
\right) . (2.25)
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54 A. KALYBAY, D. KARATAYEVA
Assume that
\omega +
k =
\Bigl\{
j \in \Omega k : tjk < xjk, zjk < sjk
\Bigr\}
, \omega k,1 =
\Bigl\{
j \in \Omega k : tjk = xjk, zjk < sjk
\Bigr\}
,
\omega k,2 =
\Bigl\{
j \in \Omega k : tjk < xjk, zjk = sjk
\Bigr\}
, \omega -
k =
\Bigl\{
j \in \Omega k : tjk = xjk, zjk = sjk
\Bigr\}
.
Moreover,
\Delta +
k,1 = \omega +
k
\bigcup
\omega k,2, \Delta +
k,2 = \omega +
k
\bigcup
\omega k,1, \Delta -
k,1 = \omega -
k
\bigcup
\omega k,1, \Delta -
k,2 = \omega -
k
\bigcup
\omega k,2.
It is obvious that \Omega k = \omega +
k
\bigcup
\omega k,1
\bigcup
\omega k,2
\bigcup
\omega -
k . By the relation for \Delta Tk, from (2.8) we have
\Delta Tk \supset
\left( \bigcup
j\in \Delta +
k,1
[tjk, x
j
k - 1]
\right) \bigcup
\left( \bigcup
j\in \Delta +
k,2
[zjk + 1, sjk]
\right) . (2.26)
Since
\sum
j\in \Omega k
\left( xj
k - 1\sum
i=\~tjk
\rho i| \Delta yi| p +
sjk\sum
i=zjk
\rho i| \Delta yi| p
\right) =
=
\sum
j\in \omega +
k
\left( xj
k - 1\sum
i=tjk
\rho i| \Delta yi| p +
sjk\sum
i=zjk+1
\rho i| \Delta yi| p + \rho
zjk
| \Delta y
zjk
| p
\right) +
+
\sum
j\in \omega k,1
\left( \rho
xj
k - 1
| \Delta y
xj
k - 1
| p +
sjk\sum
i=zjk+1
\rho i| \Delta yi| p + \rho
zjk
| \Delta y
zjk
| p
\right) +
+
\sum
j\in \omega k,2
\left( xj
k - 1\sum
i=tjk
\rho i| \Delta yi| p + \rho
zjk
| \Delta y
zjk
| p
\right) +
\sum
j\in \omega -
k
\Bigl(
\rho
xj
k - 1
| \Delta y
xj
k - 1
| p + \rho
zjk
| \Delta y
zjk
| p
\Bigr)
=
=
\left( \sum
j\in \Delta +
k,1
xj
k - 1\sum
i=tjk
\rho i| \Delta yi| p +
\sum
j\in \Delta +
k,2
sjk\sum
i=zjk+1
\rho i| \Delta yi| p
\right) +
+
\left( \sum
j\in \Delta -
k,1
\rho
xj
k - 1
| \Delta y
xj
k - 1
| p +
\sum
j\in \Delta -
k,2
\bigcup
\Delta +
k,2
\rho
zjk
| \Delta y
zjk
| p
\right) = Fk,1 + Fk,2,
from (2.25), we have
F1 \leq
1
(\lambda - 1)p
\tau \sum
k= - \infty
(Fk,1 + Fk,2). (2.27)
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OSCILLATION AND NONOSCILLATION CRITERIA FOR A HALF-LINEAR DIFFERENCE EQUATION . . . 55
On the basis of (2.26) we have that Fk,1 \leq
\sum
i\in \Delta Tk
\rho i| \Delta yi| p. Hence,
\tau \sum
k= - \infty
Fk,1 \leq
\tau \sum
k= - \infty
\sum
i\in \Delta Tk
\rho i| \Delta yi| p =
n\sum
i=m
\rho i| \Delta yi| p. (2.28)
Since tjk - 1 = xjk \leq \lambda k for j \in \Delta -
k,1 and zjk > \lambda k+1 for j \in \Delta -
k,2
\bigcup
\Delta +
k,2, then there exist
integers k1 = k1(k, j) < k and k2 = k2(k, j) > k such that xjk - 1 \in \Delta Tk1 for j \in \Delta -
k,1 and
zjk \in \Delta Tk2 for j \in \Delta -
k,2
\bigcup
\Delta +
k,2. We note that \Delta T\tau = T\tau . Therefore,
\tau \sum
k= - \infty
Fk,2 \leq
\tau \sum
k= - \infty
\sum
i\in \Delta Tk
\rho i| \Delta yi| p =
n\sum
i=m
\rho i| \Delta yi| p. (2.29)
Thus, from (2.27), (2.28) and (2.29), we have
F1 \leq
2
(\lambda - 1)p
n\sum
i=m
\rho i| \Delta yi| p. (2.30)
Let us estimate F2 :
F2 \leq
\tau \sum
k= - \infty
\lambda pk
\sum
j\in \Omega k
sjk\sum
i=\widetilde tjk
ri - 1 \leq 2
\tau \sum
k= - \infty
\lambda pk
\sum
i\in Tk
ri - 1 = 2
\tau \sum
k= - \infty
\lambda pk
\tau \sum
t= - \infty
\sum
i=\Delta Tt
ri - 1 =
= 2
\tau \sum
t= - \infty
\sum
i=\Delta Tt
ri - 1
t\sum
k= - \infty
\lambda pt =
2
1 - \lambda - p
\tau \sum
t= - \infty
\lambda pt
\sum
i=\Delta Tt
ri - 1 \leq
\leq 2\lambda p
\lambda p - 1
\tau \sum
t= - \infty
\sum
i=\Delta Tt
ri - 1| yi| p =
2\lambda p
\lambda p - 1
n\sum
i=m
ri - 1| yi| p. (2.31)
If we combine (2.30) and (2.31) with (2.24), we get
F \leq 2\lambda p(\lambda p - 1)Br,w \mathrm{m}\mathrm{a}\mathrm{x}
\biggl\{
1
(\lambda - 1)p
,
\lambda p
\lambda p - 1
\biggr\} n\sum
i=m
(\rho i| \Delta yi| p + ri - 1| yi| p). (2.32)
Since the left-hand side of inequality (2.32) is independent of \lambda > 1, by Lemma 2.1 we have
n\sum
i=m
wk| yk| p \leq
\leq 2Br,w \mathrm{i}\mathrm{n}\mathrm{f}
\lambda >1
\biggl\{
\lambda p(\lambda p - 1)\mathrm{m}\mathrm{a}\mathrm{x}
\biggl\{
1
(\lambda - 1)p
,
\lambda p
\lambda p - 1
\biggr\} \biggr\} n\sum
i=m
(\rho i| \Delta yi| p + ri - 1| yi| p) =
= 2Br,w \mathrm{m}\mathrm{i}\mathrm{n}
\biggl\{
\mathrm{i}\mathrm{n}\mathrm{f}
1<\lambda <\lambda 0
\lambda p(\lambda p - 1)
(\lambda - 1)p
, \mathrm{i}\mathrm{n}\mathrm{f}
\lambda \geq \lambda 0
\lambda 2p
\biggr\} n\sum
i=m
(\rho i| \Delta yi| p + ri - 1| yi| p) =
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56 A. KALYBAY, D. KARATAYEVA
= \gamma pBr,w
n\sum
i=m
(\rho i| \Delta yi| p + ri - 1| yi| p).
Thus, inequality (1.13) is valid with the estimate
C \leq 2\gamma pBr,w, (2.33)
where C is the least constant in (1.13). Inequalities (2.6) and (2.33) give (2.1).
Theorem 2.1 is proved.
Let us turn to inequality (1.9). In the case ri = 0, i = 1, 2, . . . , we have the following theorem.
Theorem 2.2. Let 0 \leq m < n \leq \infty and 1 < p < \infty . Inequality (1.9) holds if and only if
Bw(m,n) < \infty . Moreover, the least constant in (1.9) satisfies
Bw \leq C \leq 2\widetilde \gamma p Bw,
where
Bw := Bw(m,n) = \mathrm{s}\mathrm{u}\mathrm{p}
m<t\leq s<n
\Biggl(
s - 1\sum
i=t
wi
\Biggr) \left( \Biggl( t\sum
i=m
\rho 1 - p\prime
i
\Biggr) 1 - p
+
\Biggl(
n\sum
i=s
\rho 1 - p\prime
i
\Biggr) 1 - p
\right) - 1
and
\widetilde \gamma p = \mathrm{i}\mathrm{n}\mathrm{f}
1<\lambda
\lambda p(\lambda p - 1)
(\lambda - 1)p
.
3. Main results. First, we study the oscillatory properties of equation (1.11) that follow from
Theorems 1.1 and 2.1. Relation (2.1) obviously gives the following corollary.
Corollary 3.1. Let 0 \leq m < n \leq \infty and 1 < p < \infty . If (1.12) holds, then Br,w \leq 1, and if
2\gamma pBr,w \leq 1, then (1.12) holds.
Applying Corollary 3.1 and Theorem 1.1 to the problem of the conjugacy and disconjugacy of
equation (1.11) on the interval [m,n], we get the following theorem.
Theorem 3.1. Let 0 \leq m < n \leq \infty and 1 < p < \infty . Then:
(i) for the disconjugacy of equation (1.11) on the interval [m,n] the condition Br,w \leq 1 is
necessary and the condition 2\gamma pBr,w \leq 1 is sufficient;
(ii) for the conjugacy of equation (1.11) on the interval [m,n] the condition 2\gamma pBr,w > 1 is
necessary and the condition Br,w > 1 is sufficient.
Proof. The statements (i) and (ii) are equivalent. Thus, we will prove only the statement (i).
If equation (1.11) is disconjugate on the interval [m,n], then by Theorem 1.1 inequality (1.12)
holds. Hence, by Corollary 3.1 we have Br,w \leq 1.
Conversely, if 2\gamma pBr,w \geq 1, then by Corollary 3.1 inequality (1.12) holds. Hence, by Theo-
rem 1.1 equation (1.11) is disconjugate on the interval [m,n].
The statement (i) is proved. Thus, Theorem 3.1 is also proved.
Corollary 3.2. Let 0 \leq m < n \leq \infty and 1 < p < \infty . Then:
(i) if there exist integers t and s, m \leq t < s < n, such that
s - 1\sum
i=t
wi > \varphi -
r (m, t) +
s - 1\sum
i=t
ri + \varphi +
r (s, n)
holds, then equation (1.11) is conjugate on the interval [m,n];
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OSCILLATION AND NONOSCILLATION CRITERIA FOR A HALF-LINEAR DIFFERENCE EQUATION . . . 57
(ii) if equation (1.11) is conjugate on the interval [m,n], then there exist integers t and s,
m \leq t < s < n, such that
s - 1\sum
i=t
wi >
1
2\gamma p
\Biggl[
\varphi -
r (m, t) +
s - 1\sum
i=t
ri + \varphi +
r (s, n)
\Biggr]
holds;
(iii) if equation (1.11) is disconjugate on the interval [m,n], then there exist integers t and s,
m \leq t < s < n, such that
s - 1\sum
i=t
wi \leq \varphi -
r (m, t) +
s - 1\sum
i=t
ri + \varphi +
r (s, n)
holds.
We will present oscillation and nonoscillation results of equation (1.11).
Theorem 3.2. Let 1 < p < \infty .
(i) For equation (1.11) to be nonoscillatory the condition Br,w(m,\infty ) \leq 1 is necessary and the
condition 2\gamma pBr,w(m,\infty ) \leq 1 is sufficient for some m \geq 0.
(ii) For equation (1.11) to be oscillatory the condition 2\gamma p \mathrm{l}\mathrm{i}\mathrm{m}m\rightarrow \infty \mathrm{s}\mathrm{u}\mathrm{p}Br,w(m,\infty ) > 1 is
necessary and the condition \mathrm{l}\mathrm{i}\mathrm{m}m\rightarrow \infty \mathrm{s}\mathrm{u}\mathrm{p}Br,w(m,\infty ) > 1 is sufficient.
Proof. The statement (i) directly follows from the statement (i) of Theorem 3.1. We will prove
the statement (ii).
Let equation (1.11) be oscillatory. Then there exists an integer k, 0 \leq k < \infty , such that for all
m > k equation (1.11) is conjugate on the interval [m,\infty ). Therefore, by Theorem 3.1 we have that
2\gamma pBr,w(m,\infty ) > 1 for all m > k. Hence, it follows that 2\gamma p \mathrm{l}\mathrm{i}\mathrm{m}m\rightarrow \infty \mathrm{s}\mathrm{u}\mathrm{p}Br,w(m,\infty ) > 1.
Conversely, let \mathrm{l}\mathrm{i}\mathrm{m}m\rightarrow \infty \mathrm{s}\mathrm{u}\mathrm{p}Br,w(m,\infty ) > 1. Then there exists an increasing sequence of
natural numbers \{ mk\} \infty k=1 such that mk \rightarrow \infty for k \rightarrow \infty and Br,w(mk,\infty ) > 1 for all k \geq 1.
Then by Theorem 3.1 equation (1.11) is conjugate on the interval [mk,\infty ) for all k \geq 1, i.e., for
all k \geq 1 there exists a nontrivial solution of equation (1.11) that has at least two generalized zeros
on the interval [mk,\infty ). Hence, there exists a sequence \{ \~mk\} \subset \{ mk\} such that on all intervals
[ \~mk, \~mk+1 - 1] some nontrivial solution of equation (1.11) has two zeros. Then by Sturm’s separation
theorem [18] (Theorem 3) there exists a nontrivial solution of equation (1.11) that has at least one
generalized zero on each interval [mk,mk+1 - 1], k \geq 1. Thus, this solution of equation (1.11) is
oscillatory.
Theorem 3.2 is proved.
From Theorem 3.2 we have the following corollary.
Corollary 3.3. Let 1 < p < \infty .
(i) If there exist sequences of integers mk, tk, and sk, k \geq 1, such that 0 < mk \leq tk < sk,
mk \rightarrow \infty for k \rightarrow \infty and
sk - 1\sum
i=tk
wi > \varphi -
r (mk, tk) +
sk - 1\sum
i=tk
ri + \varphi +
r (sk,\infty )
holds for a sufficiently large k, then equation (1.11) is oscillatory.
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58 A. KALYBAY, D. KARATAYEVA
(ii) If equation (1.11) is oscillatory, then there exist sequences of integers mk, tk, and sk, k \geq 1,
such that 0 < mk \leq tk < sk, mk \rightarrow \infty for k \rightarrow \infty and
sk - 1\sum
i=tk
wi >
1
2\gamma p
\left( \varphi -
r (mk, tk) +
sk - 1\sum
i=tk
ri + \varphi +
r (sk,\infty )
\right)
holds.
Now we turn to equation (1.1). We will remind that equation (1.1) is a partial case of equa-
tion (1.11). If we replace w by v+ and r by v - , we get the following theorems and corollaries.
Theorem 3.3. Let 0 \leq m < n \leq \infty and 1 < p < \infty . Then:
(i) for the disconjugacy of equation (1.1) on the interval [m,n] the condition Bv - ,v+ \leq 1 is
necessary and the condition 2\gamma pBv - ,v+ \leq 1 is sufficient;
(ii) for the conjugacy of equation (1.1) on the interval [m,n] the condition 2\gamma pBv - ,v+ > 1 is
necessary and the condition Bv - ,v+ > 1 is sufficient.
Corollary 3.4. Let 0 \leq m < n \leq \infty and 1 < p < \infty . Then:
(i) if there exist integers t and s, m \leq t < s < n, such that
s - 1\sum
i=t
v+i > \varphi -
v - (m, t) +
s - 1\sum
i=t
v - i + \varphi +
v - (s, n)
or
s - 1\sum
i=t
vi > \varphi -
v - (m, t) + \varphi +
v - (s, n)
holds, then equation (1.1) is conjugate on the interval [m,n];
(ii) if equation (1.1) is conjugate on the interval [m,n], then there exist integers t and s,
m \leq t < s < n, such that
s - 1\sum
i=t
v+i >
1
2\gamma p
\Biggl[
\varphi -
v - (m, t) +
s - 1\sum
i=t
v - i + \varphi +
v - (s, n)
\Biggr]
holds;
(iii) if equation (1.1) is disconjugate on the interval [m,n], then there exist integers t and s,
m \leq t < s < n, such that
s - 1\sum
i=t
vi \leq \varphi -
v - (m, t) + \varphi +
v - (s, n)
holds.
Theorem 3.4. Let 1 < p < \infty .
(i) For equation (1.1) to be nonoscillatory the condition Bv - ,v+(m,\infty ) \leq 1 is necessary and
the condition 2\gamma pBv - ,v+(m,\infty ) \leq 1 is sufficient for some m \geq 0.
(ii) For equation (1.1) to be oscillatory the condition 2\gamma p \mathrm{l}\mathrm{i}\mathrm{m}m\rightarrow \infty \mathrm{s}\mathrm{u}\mathrm{p}Bv - ,v+(m,\infty ) > 1 is
necessary and the condition \mathrm{l}\mathrm{i}\mathrm{m}m\rightarrow \infty \mathrm{s}\mathrm{u}\mathrm{p}Bv - ,v+(m,\infty ) > 1 is sufficient.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 1
OSCILLATION AND NONOSCILLATION CRITERIA FOR A HALF-LINEAR DIFFERENCE EQUATION . . . 59
Corollary 3.5. Let 1 < p < \infty .
(i) If there exist sequences of integers mk, tk, and sk, k \geq 1, such that 0 < mk \leq tk < sk,
mk \rightarrow \infty for k \rightarrow \infty and
sk - 1\sum
i=tk
vi > \varphi -
v - (mk, tk) + \varphi +
v - (sk,\infty )
holds for a sufficiently large k, then equation (1.1) is oscillatory.
(ii) If equation (1.1) is oscillatory, then there exist sequences of integers mk, tk, and sk, k \geq 1,
such that 0 < mk \leq tk < sk, mk \rightarrow \infty for k \rightarrow \infty and
sk - 1\sum
i=tk
v+i >
1
2\gamma p
\left( \varphi -
v - (mk, tk) +
sk - 1\sum
i=tk
v - i + \varphi +
v - (sk,\infty )
\right)
holds.
If starting from some number n > 1 we have v+i = 0 for any i > n, then equation (1.1) is
nonoscillatory. Therefore, for equation (1.1) to be oscillatory, it is necessary that for each natural
n > 1 there exists an index in > n such that v+in \not = 0. Since \mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p} v+ \cap \mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p} v - = \varnothing , then from
Corollary 3.5 we get a statement that answers the question about the influence of the positive part of
the sequence v on the oscillation of equation (1.1) posed in Introduction.
Corollary 3.6. Let 1 < p < \infty . Let \mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p} v+ = \{ i1, i2, . . . , in, . . .\} and \mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow \infty in = \infty . If
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
\mathrm{s}\mathrm{u}\mathrm{p}
v+in
\rho in + \varphi +
v - (in + 1,\infty )
> 1,
then equation (1.1) is oscillatory.
Corollary 3.6 follows from the first part (i) of Corollary 3.5 for mk = tk = in and sk = tk + 1.
From Theorems 2.2, 3.1 and 3.2 we get the results for equation (1.8). We note that the values
Bw and \widetilde \gamma p used below are defined in Theorem 2.2.
Theorem 3.5. Let 0 \leq m < n \leq \infty and 1 < p < \infty . Then:
(i) for the disconjugacy of equation (1.8) on the interval [m,n] the condition Bwv \leq 1 is
necessary and the condition 2\widetilde \gamma pBw \leq 1 is sufficient;
(ii) for the conjugacy of equation (1.8) on the interval [m,n] the condition 2\widetilde \gamma pBw > 1 is
necessary and the condition Bw > 1 is sufficient.
Theorem 3.6. Let 1 < p < \infty .
(i) For equation (1.8) to be nonoscillatory the condition Bw(m,\infty ) \leq 1 is necessary and the
condition 2\widetilde \gamma pBw(m,\infty ) \leq 1 is sufficient for some m \geq 0.
(ii) For equation (1.8) to be oscillatory the condition 2\widetilde \gamma p \mathrm{l}\mathrm{i}\mathrm{m}m\rightarrow \infty \mathrm{s}\mathrm{u}\mathrm{p}Bw(m,\infty ) > 1 is neces-
sary and the condition \mathrm{l}\mathrm{i}\mathrm{m}m\rightarrow \infty \mathrm{s}\mathrm{u}\mathrm{p}Bw(m,\infty ) > 1 is sufficient.
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Received 09.01.20
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 1
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| id | umjimathkievua-article-2298 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:21:57Z |
| publishDate | 2022 |
| publisher | Institute of Mathematics, NAS of Ukraine |
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| resource_txt_mv | umjimathkievua/9d/7dd64056835cb3ff15641b6dd65ff09d.pdf |
| spelling | umjimathkievua-article-22982022-03-27T15:39:11Z Oscillation and nonoscillation criteria for a half-linear difference equation of the second order and extended discrete Hardy inequality Oscillation and nonoscillation criteria for a half-linear difference equation of the second order and extended discrete Hardy inequality Kalybay, A. Karatayeva, D. Kalybay, A. Karatayeva, D. D. Generalized zeros, oscillatory and non-oscillatory properties, conjugate, disconjugate, discrete weighted inequality. UDC 517.9 We establish the oscillatory properties of a half-linear difference equation of the second order by using a suitable extension of the weighted discrete Hardy inequality. &nbsp; UDC 517.9 КРИТЕРІЇ КОЛИВАННЯ ТА НЕКОЛИВАННЯ ДЛЯ НАПІВЛІНІЙНОГО РІЗНИЦЕВОГО РІВНЯННЯ ДРУГОГО ПОРЯДКУ ТА РОЗШИРЕННЯ ДИСКРЕТНОЇ НЕРІВНОСТІ ГАРДІ За допомогою відповідного розширення дискретної нерівності Гарді встановлено коливні властивості напівлінійного різницевого рівняння другого порядку. Institute of Mathematics, NAS of Ukraine 2022-01-24 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2298 10.37863/umzh.v74i1.2298 Ukrains’kyi Matematychnyi Zhurnal; Vol. 74 No. 1 (2022); 45 - 60 Український математичний журнал; Том 74 № 1 (2022); 45 - 60 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2298/9175 Copyright (c) 2022 Aigerim Kalybay |
| spellingShingle | Kalybay, A. Karatayeva, D. Kalybay, A. Karatayeva, D. D. Oscillation and nonoscillation criteria for a half-linear difference equation of the second order and extended discrete Hardy inequality |
| title | Oscillation and nonoscillation criteria for a half-linear difference equation of the second order and extended discrete Hardy inequality |
| title_alt | Oscillation and nonoscillation criteria for a half-linear difference equation of the second order and extended discrete Hardy inequality |
| title_full | Oscillation and nonoscillation criteria for a half-linear difference equation of the second order and extended discrete Hardy inequality |
| title_fullStr | Oscillation and nonoscillation criteria for a half-linear difference equation of the second order and extended discrete Hardy inequality |
| title_full_unstemmed | Oscillation and nonoscillation criteria for a half-linear difference equation of the second order and extended discrete Hardy inequality |
| title_short | Oscillation and nonoscillation criteria for a half-linear difference equation of the second order and extended discrete Hardy inequality |
| title_sort | oscillation and nonoscillation criteria for a half-linear difference equation of the second order and extended discrete hardy inequality |
| topic_facet | Generalized zeros oscillatory and non-oscillatory properties conjugate disconjugate discrete weighted inequality. |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2298 |
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