Asymptotic behavior of positive solutions of fourth-order nonlinear difference equations
We consider a class of fourth-order nonlinear difference equations of the form $$ \Delta^2(p_n(\Delta^2y_n)^{\alpha})+q_n y^{\beta}_{n+3}=0, \quad n\in {\mathbb N} $$ where $\alpha, \beta$ are the ratios of odd positive integers, and $\{p_n\}, \{q_n\}$ are positive real sequences defined for all $n...
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| author | Agarwal, P. Manojlović, J. V. Агарвал, Р. П. Манойловіч, Дж В. |
| author_facet | Agarwal, P. Manojlović, J. V. Агарвал, Р. П. Манойловіч, Дж В. |
| author_sort | Agarwal, P. |
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| datestamp_date | 2020-03-18T19:46:36Z |
| description | We consider a class of fourth-order nonlinear difference equations of the form
$$ \Delta^2(p_n(\Delta^2y_n)^{\alpha})+q_n y^{\beta}_{n+3}=0, \quad n\in {\mathbb N} $$
where $\alpha, \beta$ are the ratios of odd positive integers, and $\{p_n\}, \{q_n\}$ are positive real sequences defined for all $n\in {\mathbb N} $.
We establish necessary and sufficient conditions for the existence of nonoscillatory solutions with specific asymptotic behavior under suitable combinations of the convergence or divergence conditions of the sums
$$ \sum\limits_{n=n_0}^{\infty}\frac n{p_n^{1/\alpha}}\quad \text{and}\quad \sum\limits_{n=n_0}^{\infty}\left(\frac n{p_n}\right)^{1/\alpha}.$$ |
| first_indexed | 2026-03-24T02:36:51Z |
| format | Article |
| fulltext |
UDС 517.9
R. P. Agarwal (Florida Inst. Technol., USA),
J. V. Manojlović* (Univ. Niš, Serbia)
ASYMPTOTIC BEHAVIOR
OF POSITIVE SOLUTIONS OF FOURTH-ORDER
NONLINEAR DIFFERENCE EQUATIONS
АСИМПТОТИЧНА ПОВЕДIНКА ДОДАТНИХ РОЗВ’ЯЗКIВ
НЕЛIНIЙНИХ РIЗНИЦЕВИХ РIВНЯНЬ
ЧЕТВЕРТОГО ПОРЯДКУ
We consider a class of fourth-order nonlinear difference equations of the form
∆2(pn(∆2yn)α) + qnyβ
n+3 = 0, n ∈ N,
where α, β are the ratios of odd positive integers, and {pn}, {qn} are positive real sequences defined
for all n ∈ N(n0). We establish necessary and sufficient conditions for the existence of nonoscillatory
solutions with specific asymptotic behavior under suitable combinations of the convergence or divergence
conditions of the sums
∞∑
n=n0
n
p
1/α
n
and
∞∑
n=n0
(
n
pn
)1/α
.
Розглянуто клас нелiнiйних рiзницевих рiвнянь четвертого порядку, що мають вигляд
∆2(pn(∆2yn)α) + qnyβ
n+3 = 0, n ∈ N,
де α, β є спiввiдношеннями непарних додатних цiлих чисел, а {pn}, {qn} — додатними дiйсними
послiдовностями, визначеними для всiх n ∈ N(n0). Встановлено необхiднi i достатнi умови iсну-
вання неколивних розв’язкiв iз специфiчною асимптотичною поведiнкою у випадку прийнятних
комбiнацiй умов збiжностi або розбiжностi сум
∞∑
n=n0
n
p
1/α
n
та
∞∑
n=n0
(
n
pn
)1/α
.
1. Introduction. In the last few years, there has been an increasing interest in the
study of oscillatory and asymptotic behavior of solutions of difference equations (see
monographs [1, 2] and the references therein). Compared to second-order difference
equations, the study of higher-order equations (see [3 – 8]) and, in particular, fourth-
order difference equations (see [9 – 14]) has received considerably less attention.
In this paper we are concerned with the fourth-order quasilinear difference equation
∆2(pn(∆2yn)α) + qnyβ
n+3 = 0, n ∈ N(n0), (1.1)
*Supported by Ministry of Science of Republic of Serbia (Grant 144003).
c© R. P. AGARWAL, J. V. MANOJLOVIĆ, 2008
8 ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 1
ASYMPTOTIC BEHAVIOR OF POSITIVE SOLUTIONS OF FOURTH-ORDER ... 9
where N(n0) = {n0, n0 + 1, n0 + 2, . . .}, n0 is a positive integer, ∆ is the forward
difference operator defined by ∆yn = yn+1 − yn, α and β are ratios of odd positive
integers and {pn} and {qn} are positive real sequence defined for all n ∈ N(n0).
By a solution of (1.1) we mean a real sequence {yn} satisfying (1.1) for n ∈ N(n0).
A nontrivial solution {yn} of equation (1.1) is called oscillatory if for every M ∈ N
there exist m, n ∈ N, M ≤ m < n such that xmxn < 0, otherwise, it is nonoscillatory.
Equation (1.1) is called oscillatory if all its solutions are oscillatory.
Oscillatory and nonoscillatory behavior of solutions of (1.1) under the condition
∞∑
n=n0
n
p
1/α
n
= ∞ and
∞∑
n=n0
(
n
pn
)1/α
= ∞ (1.2)
have been considered by Thandapani and Selvaraj in [13] and Agarwal and Manojlović
in [15]. The aim of this paper is to proceed further in this direction and to obtain a more
detailed information on the asymptotic behavior of nonoscillatory solutions of (1.1),
under the assumptions which was not yet considered. Namely, we will investigate the
structure of the set of positive solutions of (1.1) under each of the following conditions:
∞∑
n=n0
n
p
1/α
n
< ∞ and
∞∑
n=n0
(
n
pn
)1/α
= ∞, (1.3)
∞∑
n=n0
n
p
1/α
n
= ∞ and
∞∑
n=n0
(
n
pn
)1/α
< ∞, (1.4)
∞∑
n=n0
n
p
1/α
n
< ∞ and
∞∑
n=n0
(
n
pn
)1/α
< ∞. (1.5)
We emphasize that if (1.3) holds, then α < 1 and if (1.4) holds, then α ≥ 1.
Under assumptions (1.2) – (1.5), the following four sequences play a special role in
the set of positive solutions of (1.1):
αn = 1, γn =
n−1∑
s=n0
(n− s− 1)
(
s
ps
)1
α
,
βn = n, δn =
∞∑
s=n
(s− n + 1)
(
1
ps
)1
α
.
Under the condition (1.2), Thandapani, Selvaraj in [13] established necessary and suffi-
cient conditions for the existence of positive solutions of the following two types:
yn ∼ c αn as n →∞, 0 < c < ∞, (1.6)
yn ∼ c γn as n →∞, 0 < c < ∞. (1.7)
Namely, they proved the following two theorems:
Theorem A. Suppose that (1.2) holds. A necessary and sufficient condition for
the equation (1.1) to have a positive solution {yn} which satisfies (1.6) is that
ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 1
10 R. P. AGARWAL, J. V. MANOJLOVIĆ
∞∑
n=n0
n
p
1/α
n
( ∞∑
s=n
(s− n) qs
)1
α
< ∞.
Theorem B. Suppose that (1.2) holds. A necessary and sufficient condition for
the equation (1.1) to have a positive solution {yn} which satisfies (1.7) is that
∞∑
n=n0
qnγβ
n+3 < ∞.
Moreover, a solution {yn} satisfying (1.6) is minimal in the set of eventually positive
solution of (1.1), while a solution {yn} satisfying (1.7) is maximal in the set of eventually
positive solution of (1.1). Namely, there exists positive constants k1, k2 such that
k1 ≤ yn ≤ k2γn for all large n.
In this paper, we are going to investigate asymptotic behavior of positive solutions
as n → ∞ under the other three conditions (1.3) – (1.5). If (1.3) is satisfied, we give
necessary and sufficient conditions for the existence of positive solutions satisfying (1.7)
and
yn ∼ c δn as n →∞, 0 < c < ∞. (1.8)
It is observed that a solution satisfying (1.8) is “minimal” in the set of all eventually
positive solution of (1.1) , while a solution satisfying (1.7) is “maximal” in the set of all
eventually positive solution of (1.1).
If (1.4) holds, in the set of all eventually positive solution of (1.1), a solution
satisfying (1.6) may be a “minimal” solution, while a solution satisfying
yn ∼ c βn as n →∞, 0 < c < ∞, (1.9)
may be a “maximal” solution. We will establish necessary and sufficient conditions for
the existence of this types of positive solutions.
If (1.5) holds, a solution {yn} of (1.1) having the asymptotic property (1.9) may be
expected as a “minimal” solution in the set of all eventually positive solutions of (1.1).
Moreover, a solution {yn} of (1.1) having the asymptotic property (1.8) is a “maximal”
solution in the set of all eventually positive solutions of (1.1). Under the assumption
(1.5), necessary and sufficient conditions are established for the existence of “minimal”
and “maximal” positive solution.
2. Classification of positive solutions. We first have to classify the positive solutions
in term of the signs of their differences, i.e., of the signs of
∆yn, ∆2yn, ∆(pn (∆2yn)α).
For a positive solution {yn} the next eight cases can occur:
ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 1
ASYMPTOTIC BEHAVIOR OF POSITIVE SOLUTIONS OF FOURTH-ORDER ... 11
Case ∆(pn (∆2yn)α) ∆2yn ∆yn Case ∆(pn (∆2yn)α) ∆2yn ∆yn
(a) + + + (e) – + +
(b) + + – (f) – + –
(c) + – + (g) – – +
(d) + – – (h) – – –
The cases (d) and (h) never hold, because if ∆yn < 0 and ∆2yn < 0 for all large n,
we would have that limn→∞ yn = −∞, which contradicts the positivity of solution
{yn}. Similarly, if ∆(pn (∆2yn)α) < 0, taking into account that from the equation
∆(pn (∆2yn)α) is decreasing sequence, we would have that limn→∞ pn (∆2yn)α =
= −∞, that is ∆2yn < 0 for all large n, which eliminates cases (e) and (f).
Accordingly, for a positive solution {yn}, the one of the following four cases holds:
(I) : ∆(pn (∆2yn)α) > 0, ∆2yn > 0, ∆yn > 0 for all large n,
(II) : ∆(pn (∆2yn)α) > 0, ∆2yn < 0, ∆yn > 0 for all large n,
(III) : ∆(pn (∆2yn)α) > 0, ∆2yn > 0, ∆yn < 0 for all large n,
(IV) : ∆(pn (∆2yn)α) < 0, ∆2yn < 0, ∆yn > 0 for all large n.
Moreover, we have the following two lemmas:
Lemma 2.1 (Lemma 2.1 [13]). Let {yn} be a positive solution of (1.1). If
∞∑
n=n0
(
n
pn
)1/α
= ∞
holds, then
∆(pn (∆2yn)α) > 0 for all large n.
Lemma 2.2. Let {yn} be a positive solution of (1.1) such that
∆(pn (∆2yn)α) > 0, ∆2yn > 0 for all large n. (2.1)
If
∞∑
n=n0
n
p
1/α
n
= ∞ (2.2)
holds, then ∆yn > 0 for all large n.
Proof. From (2.1) we have
pn (∆2yn)α ≥ pN (∆2yN )α = c > 0, n ≥ N,
or
ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 1
12 R. P. AGARWAL, J. V. MANOJLOVIĆ
∆2yn ≥
(
c
pn
)1/α
, n ≥ N.
Multiplying by n previous inequality and summing from N to n− 1, we obtain
n∆yn −N∆yN − yn+1 + yN+1 ≥ c1/α
n−1∑
s=N
s
p
1/α
s
, n ≥ N,
which implies that
n∆yn ≥ k + c1/α
n−1∑
s=N
s
p
1/α
s
, n ≥ N.
Then, it follows from (2.2) that n∆yn →∞ as n →∞ and consequently ∆yn > 0 for
all large n.
The lemma is proved.
Therefore, by all previous discussion we make the following conclusions:
Lemma 2.3. Let {yn} be a positive solution of (1.1).
(i) If (1.3) holds, then (I) or (II) or (III) holds;
(ii) If (1.4) holds, then (I) or (II) or (IV) holds;
(iii) If (1.5) holds, then (I) or (II) or (III) or (IV) holds.
3. Auxiliary lemmas. In this section we collect some lemmas which will be used
in order to prove the main results. We will use the following fixed point theorem, which
was proved in [14] and which can be considered as a discrete analog of Schauder’s fixed
point theorem.
Lemma 3.1. Suppose X is a Banach space and K is closed, bounded and convex
subset of X . If F : K → X is a continuous mapping such that F(K) ⊂ K and F(K)
is uniformly Cauchy, then F has a fixed point in K.
Lemma 3.2. (i) If {ϕn} is eventually negative sequence such that ∆ϕn > 0 and
∆2ϕn < 0 for all large n, then lim
n→∞
∆ϕn = 0.
(ii) If {ϕn} is eventually positive sequence such that ∆ϕn < 0 and ∆2ϕn > 0 for
all large n, then lim
n→∞
∆ϕn = 0.
Proof. (i) Since {∆ϕn} is positive and decreasing sequence, there exists lim
n→∞
∆ϕn =
= ϕ, 0 ≤ ϕ < ∞. If we suppose that ϕ > 0, from ∆ϕn ≥ ϕ, we get
ϕn ≥ ϕN + ϕ (n−N), n ≥ N,
which obviously implies that lim
n→∞
ϕn = ∞, contradiction negativity of the sequence
{ϕn}. Consequently, ϕ = 0.
(ii) Since {∆ϕn} is negative and increasing sequence, there exists lim
n→∞
∆ϕn = ϕ,
−∞ < ϕ ≤ 0. If we suppose that ϕ < 0, from ∆ϕn ≤ ϕ, we get
ϕn ≤ ϕN + ϕ (n−N), n ≥ N,
which obviously implies that lim
n→∞
ϕn = −∞, contradiction positivity of the sequence
{ϕn}. Therefore, ϕ = 0.
The lemma is proved.
ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 1
ASYMPTOTIC BEHAVIOR OF POSITIVE SOLUTIONS OF FOURTH-ORDER ... 13
As a direct consequence of the previous lemma we have the following results for
the (1.1).
Lemma 3.3. Let {yn} be a positive solution of (1.1).
(i) If a solution {yn} is of type (II), then lim
n→∞
∆(pn (∆2yn)α) = 0.
(ii) If a solution {yn} is of type (III), then lim
n→∞
∆yn = 0.
Next lemma gives some useful properties of positive solution of (1.1).
Lemma 3.4. Let {yn} be a positive solution of (1.1).
(i) If {yn} is of type (I), then lim
n→∞
yn
n
> 0.
(ii) Let (1.3) or (1.5) holds and {yn} is a positive solution of (1.1) which satisfi-
es (1.8). Then {yn} must be of type (III).
(iii) Let (1.4) holds and {yn} is a positive solution of (1.1) which satisfies (1.6).
Then {yn} must be of type (II).
Proof. (i) Since {yn} is of type (I), there exists some N ≥ n0 such that (I) holds
for all n ≥ N . Then {∆yn} is the increasing sequence, so that ∆yn ≥ ∆ynN
> 0 for
all n ≥ N . Therefore, yn ≥ yN + ∆yN (n−N), n ≥ N, or
yn
n
≥ yN
n
+ ∆yN
(
1− N
n
)
, n ≥ N.
Accordingly, lim
n→∞
yn
n
≥ ∆yN > 0.
(ii) Let {yn} be a positive solution of (1.1) which satisfies (1.8). If (1.5) holds, by
Lemma 2.3, positive solution {yn} could be of type (I), (II), (III) or (IV). If we suppose
that {yn} is of type (I), (II) or (IV), then lim
n→∞
yn = w0 ∈ (0,∞]. Moreover, (1.5)
implies that lim
n→∞
δn = 0. But, then we would have that lim
n→∞
yn
δn
= ∞, contradicting
the assumption that {yn} satisfies (1.8). Therefore, the solution {yn} must be of
type (III).
On the other hand, if (1.3) holds, by Lemma 2.3 positive solution {yn} is of type (I),
(II) or (III) and therefore, using that (1.3) also implies that lim
n→∞
δn = 0, by the same
arguments as in the previous case, we prove that {yn} is neither of type (I) nor (II), so
it must be of type (III).
(iii) Let {yn} be a positive solution of (1.1) which satisfies (1.6). Then,
lim
n→∞
yn
n
= 0. (3.1)
If (1.4) holds, by Lemma 2.3, positive solution {yn} is of type (I), (II), or (IV).
(a) If we suppose that {yn} is of type (I), then by (i) we have that lim
n→∞
yn
n
> 0
contradicting (3.1).
(b) If we suppose that {yn} is of type (IV), then
{
pn(∆2yn)α
}
is decreasing, so
that
pn(∆2yn)α ≤ pN (∆2yN )α = −K < 0, n ≥ N.
Then,
ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 1
14 R. P. AGARWAL, J. V. MANOJLOVIĆ
n ∆2yn ≤ −K
1
α
n
p
1/α
n
, n ≥ N,
and by summing obtained inequality from N to n− 1 we get
n−1∑
k=N
k ∆2yk ≤ −K
1
α
n−1∑
k=N
k
p
1/α
k
, n ≥ N,
or
n−1∑
k=N
k ∆2yk = n ∆yn −N∆yN − yn+1 + yN+1 ≤ −K
1
α
n−1∑
k=N
k
p
1/α
k
, n ≥ N.
Accordingly, taking into account that ∆yn > 0, n ≥ N, we have
yn+1 ≥ M + K
1
α
n−1∑
k=N
k
p
1/α
k
, n ≥ N,
where M = yN+1 −N∆yN . Therefore, (1.4) implies that lim
n→∞
yn = ∞, contradicting
that {yn} satisfies (1.6).
Finally, the solution {yn} must be of type (II).
The lemma is proved.
We will also need the following lemma.
Lemma 3.5. Let {yn} be the positive solution of (1.1) such that ∆(pn (∆2yn)α) >
> 0 for all large n, then
lim
n→∞
1
n
n−1∑
k=N
∞∑
j=k
qj yβ
j+3 = 0. (3.2)
Proof. Summing (1.1) from N to n− 1, we get
ξ3 −∆(pn (∆2yn)α) =
n−1∑
k=N
qk yβ
k+3, n ≥ N + 1, (3.3)
where ξ3 = ∆(pN (∆2yN )α). Since
{
∆(pn (∆2yn)α)
}
is positive and decreasing
sequence, it tends to a finite limit w3 ≥ 0 as n →∞. Now, letting n →∞ in (3.3) we
have that
∞∑
k=N
qk yβ
k+3 < ∞.
Therefore, by Stolz’s theorem we get
lim
n→∞
1
n
n−1∑
k=N
∞∑
j=k
qj yβ
j+3 = lim
n→∞
∞∑
j=n
qj yβ
j+3 = 0.
The lemma is proved.
4. “Maximal” and “minimal” positive solutions of (1.1). Next three result gives a
growth and decaying estimate of all positive solutions of (1.1) under the condition (1.3),
(1.4) or (1.5).
ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 1
ASYMPTOTIC BEHAVIOR OF POSITIVE SOLUTIONS OF FOURTH-ORDER ... 15
Theorem 4.1. Let (1.3) holds. If {yn} is an eventually positive solution of (1.1),
then there are positive constants c1 and c2 such that
c1δn ≤ yn ≤ c2 γn for all large n. (4.1)
Proof. In order to prove the first inequality, notice that if {yn} is a solution of
type (I) or (II), it is an eventually increasing solution. Then, clearly yn ≥ c δn for some
c > 0 and for all large n. Therefore, taking into account Lemma 2.3, let us prove this
inequality for a solution of type (III). Since {pn (∆2yn)α} is positive and increasing
sequence, we get
∆2yn ≥
(
c
pn
)1/α
, n ≥ N, (4.2)
where c = pN (∆2yN )α. Also, by Lemma 3.3 (ii), lim
n→∞
∆yn = ω1 = 0. Then,
summing (4.2) from n to m and letting m →∞, we obtain
−∆yn ≥ c1/α
∞∑
s=n
1
p
1/α
s
, n ≥ N.
Summing once again obtained inequality from n to m, letting m → ∞ and using that
lim
n→∞
yn = w0 ≥ 0 , we get
yn ≥ ω0 + c1/α
∞∑
s=n
∞∑
k=s
1
p
1/α
k
≥ c1/αδn, n ≥ N.
Next, let us prove the second inequality. We consider two cases, either {yn} is a
solution of type (II) i.e., ∆2yn < 0 for all large n or it is a solution of type (I), or (III),
i.e., ∆2yn > 0 for all large n. In the first case, we have ∆yn ≤ ∆yN = λ1, for all
n ≥ N ≥ n0 and summing from N to n− 1, we get yn ≤ yN + ∆yN (n−N), n ≥ N,
from where we conclude that {yn/n} is bounded sequence. Then, yn ≤ c2γn for some
c2 > 0, since we have by (1.3) that
lim
n→∞
γn
n
= lim
n→∞
1
n
n−1∑
s=n0
s−1∑
k=n0
(
k
pk
)1/α
= lim
n→∞
n−1∑
s=n0
(
s
ps
)1/α
= ∞.
In the second case, when ∆2yn > 0 for all large n, by Lemma 2.1 we have that{
∆(pn(∆2yn)α)
}
is positive and decreasing sequence, so that
∆(pn(∆2yn)α) ≤ ∆(pN (∆2yN )α) = λ3, n ≥ N ≥ n0. (4.3)
Summing this inequality repeatedly from N to n− 1, we obtain
yn ≤ yN + λ1(n−N) +
n−1∑
s=N
s−1∑
k=N
1
p
1/α
k
(
λ2 + λ3(k −N)
)1/α
, n ≥ N, (4.4)
where λ1 = ∆yN , λ2 = pN (∆2yN )α. Now, it is easy to verify that (4.4) implies that
yn ≤ c2γn, with c2 > λ
1/α
3 > 0.
The theorem is proved.
ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 1
16 R. P. AGARWAL, J. V. MANOJLOVIĆ
Therefore, under the condition (1.3), in the set of all eventually positive solutions
of (1.1), a solution {yn} of type (1.8) may be regard as a “minimal” solution, while a
solution {yn} of type (1.7) may be regard as a “maximal” solution.
Moreover, under the condition (1.4), in the set of all eventually positive solutions
of (1.1), a solution {yn} of type (1.6) may be regard as a “minimal” solution, while a
solution {yn} of type (1.9) may be regard as a “maximal” solution. Namely, we have
the following theorem:
Theorem 4.2. Let (1.4) holds. If {yn} is an eventually positive solution of (1.1),
then there are positive constants c1 and c2 such that
c1αn ≤ yn ≤ c2 βn for all large n. (4.5)
Proof. Let {yn} be an eventually positive solution of (1.1). By Lemma 2.3 we have
∆yn > 0 for all large n, so clearly there is c1 > 0 such that yn ≥ c1 for all large n.
Next, we will prove that {yn/n} is bounded sequence, so that there is c2 > 0 such
that yn ≤ c2 n for all large n. We consider two cases, either ∆2yn < 0 or ∆2yn > 0,
for all large n. In the first case, as in the proof of Theorem 4.1 we may prove that
{yn/n} is bounded sequence. In the second case, when ∆2yn > 0 for all large n, i.e.,
a solution {yn} is of type (I) and accordingly pn (∆2yn)α > 0 for all large n, as in the
proof of Theorem 4.1 we get (4.3). Summing (4.3) from N to n− 1, we obtain
pn(∆2yn)α ≤ λ2 + λ3(n−N), n ≥ N,
or
∆2yn ≤ λ
(
n
pn
)1/α
, n ≥ N,
where λ = (λ2 + λ3)1/α > 0. Summing previous inequality from N to n− 1 and using
the condition (1.4), we conclude that {∆yn} is bounded sequence. Accordingly, there is
some c2 > 0 such that ∆yn ≤ c2 for n ≥ N1 ≥ N, or we get that yn ≤ yN1 +c2(n−N1)
for n ≥ N1. Therefore, we have that {yn/n} is bounded sequence.
The theorem is proved.
Under the condition (1.5), a solution {yn} of type (1.8) is a “minimal” solution in
the set of all eventually positive solutions of (1.1), and a solution {yn} of type (1.9) is
a “maximal” solution in the set of all eventually positive solutions of (1.1).
Theorem 4.3. Let (1.5) holds. If {yn} is an eventually positive solution of (1.1),
then there are positive constants c1 and c2 such that
c1δn ≤ yn ≤ c2 βn for all large n. (4.6)
Proof. The first inequality may be proved as in the proof of Theorem 4.1. In order
to prove the second inequality, we will again prove that {yn/n} is bounded sequence.
If {yn} is a solution of type (II) or (IV), then ∆2yn < 0 for all large n, so that as in
the proof of Theorem 4.1 we may prove that {yn/n} is bounded sequence. If {yn} is
a solution of type (I), i.e., ∆(pn (∆2yn)α) > 0 and ∆2yn > 0 for all large n, as in
the proof of Theorem 4.2 we may prove that {yn/n} is bounded sequence. Finally, if
{yn} is a solution of type (III), it is obviously bounded sequence and accordingly, we
conclude that {yn/n} is bounded sequence.
The theorem is proved.
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ASYMPTOTIC BEHAVIOR OF POSITIVE SOLUTIONS OF FOURTH-ORDER ... 17
5. Existence of positive solutions. In this section we give necessary and sufficient
conditions for the existence of specific kinds of positive solutions.
5.1. Existence of positive solutions under the condition (1.3). Necessary and
sufficient conditions for the existence of positive solutions of (1.1) satisfying (1.7) or
(1.8) are given in the following two theorems.
Theorem 5.1. Suppose that (1.3) holds. Equation (1.1) has a positive solution of
type (1.7) if and only if
∞∑
s=n0
qs γβ
s+3 < ∞. (5.1)
Theorem 5.2. Suppose that (1.3) holds. Equation (1.1) has a positive solution of
type (1.8) if and only if
∞∑
s=n0
s qs δβ
s+3 < ∞. (5.2)
The statement and the proof of Theorem 5.1 is the same as of Theorem B (Theorem 1
in [13]). Consequently, we here prove only Theorem 5.2.
Proof of Theorem 5.2. Necessity. Let {yn} be a positive solution of (1.1) of
type (1.8). Then there is N ≥ n0
c
2
δn ≤ yn ≤ c δn, n ≥ N. (5.3)
Then, by Lemma 3.4 (ii) the solution {yn} is of the type (III), so that by Lemma 3.3 (ii)
lim
n→∞
∆yn = 0. Moreover, {pn(∆2yn)α} is increasing, so we find that
−∆yn =
∞∑
s=n
∆2ys =
∞∑
s=n
p
1/α
s ∆2ys
p
1/α
s
≥ p1/α
n ∆2yn
∞∑
s=n
1
p
1/α
s
, n ≥ N.
Summing this inequality from n to m, letting m →∞ and using that yn → w0 ∈ [0,∞),
as n →∞, we obtain
yn ≥
∞∑
s=n
p1/α
s ∆2ys
∞∑
k=s
1
p
1/α
k
≥ p1/α
n ∆2yn δn, n ≥ N.
Accordingly,
pn(∆2yn)α ≤
(
yn
δn
)α
, n ≥ N,
which combined with (5.3) implies that
{
pn(∆2yn)α
}
is bounded.
Multiplying (1.1) by n and summing the resulting equation from N to n − 1, we
have
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18 R. P. AGARWAL, J. V. MANOJLOVIĆ
n ∆(pn (∆2yn)α) +
n−1∑
s=N
s qs yβ
s+3 = K + pn+1(∆2yn+1)α, n ≥ N, (5.4)
where K is a constant. Since ∆(pn (∆2yn)α) > 0 and pn(∆2yn)α is bounded, letting
n →∞ in (5.4), we conclude that
∞∑
s=N
s qs yβ
s+3 < ∞.
This, together with (5.3) implies (5.2).
Sufficiency. We assume that (5.2) holds and let c > 0 be an arbitrary number. Then,
there is N ≥ n0 such that
∞∑
s=N
(s−N) qsδ
β
s+3 <
cα − (c/2)α
cβ
. (5.5)
Consider the Banach space AN of all real sequences y = {yn} with norm
‖y‖ = sup
n≥N
|yn|
δn
,
and define the set
Y1 =
{
y ∈ AN
∣∣∣∣ c
2
δn ≤ yn ≤ c δn
}
,
which is clearly bounded, closed and convex subset of AN . We will define the operator
G1 : Y1 → AN by
(
G1y
)
n
=
∞∑
s=n
s− n + 1
p
1/α
s
(
cα −
∞∑
k=s
(k − s + 1)qk yβ
k+3
)1
α
, n ≥ N. (5.6)
The operator G1 has the following properties:
(i) G1 maps Y1 to Y1. For y ∈ Y1, obviously
(G1y)n ≤ c δn, n ≥ N,
and using (5.5), we have
(
G1y
)
n
≥
∞∑
s=n
s− n + 1
p
1/α
s
(
cα − cβ
∞∑
k=s
(k − s + 1)qk δβ
k+3
)1
α
≥
≥
(
cα − cβ cα − (c/2)α
cβ
) 1
α
∞∑
s=n
s− n + 1
p
1/α
s
=
c
2
δn, n ≥ N.
Therefore, G1y ∈ Y1 for all y ∈ Y1, i.e., G1(Y1) ⊂ Y1.
(ii) G1 is continuous on Y1. Let ε > 0 and let
{
y(m) =
(
y
(m)
1 , y
(m)
2 , . . .
)}
be a
sequence in Y1, such that limm→∞ ‖y(m) − y‖ = 0. Since Y1 is closed, y ∈ Y1. We
can choose M ≥ N so large that
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ASYMPTOTIC BEHAVIOR OF POSITIVE SOLUTIONS OF FOURTH-ORDER ... 19
∞∑
s=M
(s−M)qsδ
β
s+3 < ε. (5.7)
For all m ∈ N, n > N we have that
∣∣∣(G1y
(m)
)
n
− (G1y)n
∣∣∣ ≤ ∞∑
s=n
(s− n + 1)
1
p
1/α
s
×
×
∣∣∣∣∣∣
(
cα −
∞∑
k=s
(k − s)qk
(
y
(m)
k+3
)β
)1
α
−
(
cα −
∞∑
k=s
(k − s)qk yβ
k+3
)1
α
∣∣∣∣∣∣ =
=
∞∑
s=n
s− n + 1
p
1/α
s
∣∣∣F (m)
s − Fs
∣∣∣ , (5.8)
where
F (m)
s =
(
cα −
∞∑
k=s
(k − s)qk
(
y
(m)
k+3
)β
)1
α
, Fs =
(
cα −
∞∑
k=s
(k − s)qk yβ
k+3
)1
α
.
Since, using (5.7), we have for all large m ∈ N and all s ≥ N that∣∣∣∣∣
∞∑
k=s
(k − s) qk
(
y
(m)
k+3
)
−
∞∑
k=s
(k − s) qk yβ
k+3
∣∣∣∣∣ ≤
≤
∣∣∣∣∣
M∑
k=s
(k − s) qk
(
y
(m)
k+3
)β
−
M∑
k=s
(k − s) qk yβ
k+3
∣∣∣∣∣+
+
∣∣∣∣∣
∞∑
k=M
(k −M) qk
(
y
(m)
k+3
)β
∣∣∣∣∣+
∣∣∣∣∣
∞∑
k=M
(k −M) qk yβ
k+3
∣∣∣∣∣ ≤
≤
∣∣∣∣∣
M∑
k=s
(k − s) qk
(
y
(m)
k+3
)β
−
M∑
k=s
(k − s) qk yβ
k+3
∣∣∣∣∣+
+2 cβ
∞∑
k=M
(k −M) qk δβ
k+3 < 3 ε.
Therefore, limm→∞
∣∣F (m)
s −Fs
∣∣ = 0, for each s ≥ N . Now, from (5.8) we get that for
all large m ∈ N
∣∣∣(G1y
(m)
)
n
− (G1y)n
∣∣∣ < ε
∞∑
s=n
s− n + 1
p
1/α
s
= ε δn, n > N,
which shows that
∥∥G1y
(m) − G1y
∥∥→ 0 as m →∞, i.e., that G1 is continuous on Y1.
(iii) G1(Y1) is uniformly Cauchy. To see this, we have to show that for any given
ε > 0, there exists an integer M1 such that for m > n ≥ M1
ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 1
20 R. P. AGARWAL, J. V. MANOJLOVIĆ∣∣∣∣ (G1y)m
δm
− (G1y)n
δn
∣∣∣∣ < ε,
for any y ∈ Y1. Indeed, by (5.6), for some y ∈ Y1 and m > n ≥ N, we have
∣∣∣∣ (G1y)m
δm
− (G1y)n
δn
∣∣∣∣ =
=
∣∣∣∣∣ 1
δm
∞∑
s=m
(s−m + 1)
1
p
1/α
s
(
cα −
∞∑
k=s
(k − s)qk yβ
k+3
)1
α
−
− 1
δn
∞∑
s=n
(s− n + 1)
1
p
1/α
s
(
cα −
∞∑
k=s
(k − s)qk yβ
k+3
)1
α
∣∣∣∣∣ ≤
≤ 1
δn
∣∣∣∣∣
∞∑
s=n
(s− n + 1)
1
p
1/α
s
(
cα −
∞∑
k=s
(k − s)qk yβ
k+3
)1
α
−
−
∞∑
s=m
(s−m + 1)
1
p
1/α
s
(
cα −
∞∑
k=s
(k − s)qk yβ
k+3
)1
α
∣∣∣∣∣+
+
∣∣∣∣∣ 1
δm
− 1
δn
∣∣∣∣∣
∞∑
s=m
(s−m + 1)
1
p
1/α
s
(
cα −
∞∑
k=s
(k − s)qk yβ
k+3
)1
α
≤
≤ 1
δn
m−1∑
s=n
(s− n + 1)
1
p
1/α
s
(
cα −
∞∑
k=s
(k − s)qk yβ
k+3
)1
α
+
+
2
δm
∞∑
s=m
(s−m + 1)
1
p
1/α
s
(
cα −
∞∑
k=s
(k − s)qk yβ
k+3
)1
α
≤
≤ 1
δn
∞∑
s=n
(s− n + 1)
1
p
1/α
s
(
cα −
∞∑
k=s
(k − s)qk yβ
k+3
) 1
α
+
+
2
δm
∞∑
s=m
(s−m + 1)
1
p
1/α
s
(
cα −
∞∑
k=s
(k − s)qk yβ
k+3
)1
α
≤
≤ 1
δn
δn
(
cα −
∞∑
k=N
(k −N)qk yβ
k+3
)1
α
+
+
2
δm
δm
(
cα −
∞∑
k=N
(k −N)qk yβ
k+3
)1
α
≤
≤ 3
(
cα +
∞∑
k=N
(k −N)qk yβ
k+3
)1
α
≤ 3
(
cα + cβ
∞∑
k=N
(k −N)qk δβ
k+3
)1
α
.
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ASYMPTOTIC BEHAVIOR OF POSITIVE SOLUTIONS OF FOURTH-ORDER ... 21
Using the condition (5.2), it is clear that G1(Y1) is uniformly Cauchy.
Accordingly, by Lemma 3.1, we conclude that there exists an ỹ ∈ Y1 such that
G1ỹ = ỹ. It is easy to check that ỹ = {ỹn} is a positive solution of (1.1). Moreover,
clearly
lim
n→∞
ỹn
δn
≤ c,
and using (5.6) we have that
lim
n→∞
ỹn
δn
≥ lim
n→∞
∑∞
s=n
s− n + 1
p
1/α
s
(
cα −
∑∞
k=n
(k − n)qk ỹ β
k+3
)1
α
∑∞
s=n
(s− n + 1)
1
p
1/α
s
=
= lim
n→∞
(
cα −
∞∑
k=n
(k − n)qk ỹ β
k+3
)1
α
= c.
Therefore, we have
lim
n→∞
ỹn
δn
= c.
The theorem is proved.
5.2. Existence of positive solutions under the condition (1.4). Now, we present
necessary and sufficient condition for the existence of positive solutions satisfying (1.6).
Theorem 5.3. Suppose that (1.4) holds. Equation (1.1) has a positive solution
which satisfies (1.6) if and only if
∞∑
n=n0
n
p
1/α
n
( ∞∑
s=n
(s− n) qs
)1
α
< ∞. (5.9)
The statement of Theorem 5.3 is the same as of Theorem A (Theorem 2 in [13]),
except that instead of the assumption (1.2) it is assumed that (1.4) holds. If (1.4) holds
and {yn} is a positive solution which satisfies (1.6), by Lemma 3.4 (iii) the solution {yn}
is of type (II). Therefore, the proof of necessity part of Theorem 5.3 and Theorem A is the
same. Moreover, as in the proof of Theorem A we may prove that the condition (5.9)
is sufficient for the existence of solution of type (1.6). Consequently, the proof of
Theorem 5.3 is essentially given in [13] (Theorem 2).
Now, we turn to the existence of positive solutions of type (1.9). We will consider
the Banach space BN of all real sequences y = {yn} with norm
‖y‖ = sup
n≥N
|yn|
n
,
and define sets
H1 =
{
y ∈ BN
∣∣∣ c
2
(n−N) ≤ yn ≤ c (n−N), n ≥ N
}
,
and
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22 R. P. AGARWAL, J. V. MANOJLOVIĆ
H2 =
{
y ∈ BN
∣∣∣ c(n−N) ≤ yn ≤ 2c (n−N), n ≥ N
}
,
which are both clearly bounded, closed and convex subsets of BN .
Theorem 5.4. Suppose that (1.4) holds. Equation (1.1) has a positive solution of
type (I) which satisfies (1.9) if and only if
∞∑
n=n0
(n + 3)βqn < ∞. (5.10)
Proof. Necessity. Let {yn} be a positive solution of (1.1) of type (I) which satisfies
(1.9). Then there is N ≥ n0 such that (I) holds for all n ≥ N and
c n ≤ yn ≤ 2c n, n ≥ N. (5.11)
Summing (1.1) from n to k − 1 we get
∆(pn (∆2yn)α) = ∆(pk (∆2yk)α) +
k−1∑
i=n
qi yβ
i+3, k > n ≤ N. (5.12)
Since {∆(pk (∆2yk)α)} is positive and decreasing sequence, it tends to a finite limit
w3 ≥ 0 as k →∞. Then, letting k →∞ in (5.12) we have
∆(pn (∆2yn)α) = w3 +
∞∑
i=n
qi yβ
i+3 ≥
∞∑
i=n
qi yβ
i+3, n ≥ N. (5.13)
Then, using (5.11) from (5.13) we get
∆(pN (∆2yN )α) ≥ cβ
∞∑
k=N
(k + 3)β qk,
which proves that (5.10) holds.
Sufficiency. Let c > 0 be an arbitrary number. We assume that (1.4) and (5.10) hold.
Then, there is N ≥ n0 such that
Q =
∞∑
k=N
(k + 3)βqk, 2Q
1
α
∞∑
k=N
(
k −N
pk
) 1
α
≤ c1− β
α . (5.14)
We will define the operator H1 : H1 → BN by
(
H1y
)
n
= c(n−N)−
n−1∑
k=N
∞∑
j=k
1
p
1/α
j
(
j−1∑
i=N
∞∑
s=i
qs yβ
s+3
)1
α
, n ≥ N. (5.15)
For y ∈ H1, using (5.14), we have
c(n−N) ≥
(
H1y
)
n
≥ c(n−N)− c
β
α
n−1∑
k=N
∞∑
j=k
1
p
1/α
j
(
j−1∑
i=N
∞∑
s=i
(s + 3)β qs
)1
α
≥
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ASYMPTOTIC BEHAVIOR OF POSITIVE SOLUTIONS OF FOURTH-ORDER ... 23
≥ c(n−N)− c
β
α Q
1
α
n−1∑
k=N
∞∑
j=k
(
j −N
pj
) 1
α
≥
≥
(
c− 1
2
c
β
α c1− β
α
)
(n−N) =
c
2
(n−N), n ≥ N.
Therefore, H1y ∈ H1 for all y ∈ H1, i.e., H1 maps H1 to H1. Moreover, in the similar
way as in the proof of Theorem 5.2, we may show that the operator H1 is continuous on
H1 and H1(H1) is uniformly Cauchy. In view of Lemma 3.1, we see that there exists
an ŷ ∈ H1 such that H1ŷ = ŷ. It is easy to see that ŷ = {ŷn} is a positive solution of
(1.1) of type (I). Furthermore, from (5.15), using the condition (1.4), we have that
c ≥ lim
n→∞
ŷn
n
= lim
n→∞
(
H1 ŷ
)
n
n
≥ c− cβ/αQ1/α lim
n→∞
1
n
n−1∑
k=N
∞∑
j=k
(
j −N
pj
)1
α
=
= c− cβ/αQ1/α lim
n→∞
∞∑
j=n
(
j −N
pj
)1
α
= c,
which shows that the solution ŷ is of type (1.9).
The theorem is proved.
Theorem 5.5. Suppose that (1.4) holds. Equation (1.1) has a positive solution of
type (II) which satisfies (1.9) if and only if
∞∑
n=n0
n (n + 3)βqn < ∞. (5.16)
Proof. Necessity. Let {yn} be a positive solution of (1.1) of type (II) which sati-
sfies (1.9), so that there is N ≥ n0 such that (II) and (5.11) hold for all n ≥ N .
Then, since {∆(pn (∆2yn)α)} is again positive and decreasing sequence, as in the
proof of Theorem 5.4 we get (5.13) for all n ≥ N . Moreover, by Lemma 3.3 (i),
lim
n→∞
∆(pn (∆2yn)α) = w3 = 0 and (5.13) becomes
∆(pn (∆2yn)α) =
∞∑
k=n
qk yβ
k+3, n ≥ N.
Summing this inequality from N to n− 1 and using (5.11) we get
pn (∆2yn)α = ξ2 +
n−1∑
k=N
∞∑
i=k
qi yβ
i+3 ≥ ξ2 + cβ
n−1∑
k=N
∞∑
i=k
(i + 3)β qi, n ≥ N, (5.17)
where ξ2 = pN (∆2yN )α < 0. Since {pn (∆2yn)α} is negative and increasing sequence,
there exist lim
n→∞
pn (∆2yn)α = w2 ≤ 0. Accordingly, letting n →∞ in (5.17) we have
∞∑
k=N
(k −N + 1)(k + 3)β qk < ∞,
which proves that (5.16) is satisfied.
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24 R. P. AGARWAL, J. V. MANOJLOVIĆ
Sufficiency. We assume that (5.16) holds. Moreover, the condition (1.4) implies that
∞∑
n=n0
1
p
1/α
n
< ∞.
Therefore, for an arbitrary positive constant c there is N ≥ n0 such that
∞∑
k=N
1
p
1/α
k
≤ 1, 2β
∞∑
k=N
(k −N)(k + 3)βqk ≤ cα−β . (5.18)
A solution of (1.1) satisfying (II) and (1.9) may be obtained as a fixed point of the
operator H2 : H2 → BN defined by
(
H2y
)
n
= c(n−N) +
n−1∑
k=N
∞∑
j=k
1
p
1/α
j
(
cα −
j−1∑
i=N
∞∑
s=i
qs yβ
s+3
)1
α
, n ≥ N. (5.19)
The operator H2 satisfies the assumptions of Lemma 3.1. Indeed, for all y ∈ H2, using
(5.18), we have
(
H2y
)
n
≥ c(n−N) +
n−1∑
k=N
∞∑
j=k
1
p
1/α
j
(
cα − (2c)β
j−1∑
i=N
∞∑
s=i
(s + 3)β qs
) 1
α
≥
≥ c(n−N) +
n−1∑
k=N
∞∑
j=k
1
p
1/α
j
(
cα − (2c)β
∞∑
i=N
∞∑
s=i
(s + 3)β qs
) 1
α
≥
≥ c(n−N) +
n−1∑
k=N
∞∑
j=k
1
p
1/α
j
(
cα − (2c)β
∞∑
i=N
(i−N)(i + 3)β qi
) 1
α
≥
≥ c(n−N) +
n−1∑
k=N
∞∑
j=k
1
p
1/α
j
(
cα − cβ cα−β
) 1
α = c(n−N), n ≥ N,
and
(
H2y
)
n
≤
n−1∑
k=N
c + c
∞∑
j=k
1
p
1/α
j
≤ 2c(n−N), n ≥ N.
Therefore, H2 maps H2 to H2. We may verify that the operator H2 is continuous on
H2 as well as that H2(H2) is uniformly Cauchy. Therefore, by Lemma 3.1 we conclude
that there exists an ŷ ∈ H2 such that H2ŷ = ŷ. It is easy to see that ŷ = {ŷn} is a
positive solution of (1.1) of type (II). Furthermore, by application of Stolz’s theorem,
we have that
c ≤ lim
n→∞
ŷn
n
= lim
n→∞
(
H2 ŷ
)
n
n
≤ c + c lim
n→∞
∞∑
j=n
1
p
1/α
j
= c
which shows that the solution ŷ is of type (1.9).
The theorem is proved.
ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 1
ASYMPTOTIC BEHAVIOR OF POSITIVE SOLUTIONS OF FOURTH-ORDER ... 25
Theorem 5.6. Equation (1.1) has a positive solution of type (IV) satisfying (1.9)
if and only if
∞∑
n=n0
1
p
1/α
n
(
n−1∑
k=n0
(n− k − 1)(k + 3)βqk
)1
α
< ∞. (5.20)
Proof. Necessity. Let {yn} be a positive solution of (1.1) of type (IV) which
satisfies (1.9). Then there is N ≥ n0 such that (IV) and (5.11) hold for all n ≥ N .
Summing (1.1) twice from N to n− 1 we have
pn (∆2yn)α = ξ2 + ξ3(n−N)−
n−1∑
k=N
k−1∑
i=N
qi yβ
i+3, n ≥ N, (5.21)
where ξ2 = pN (∆2yN )α < 0 and ξ3 = ∆(pN (∆2yN )α) < 0. Accordingly,
pn (∆2yn)α ≤ −
n−1∑
k=N
(n− k − 1) qk yβ
k+3, n ≥ N,
implying that
−∆2yn ≥
1
p
1/α
n
(
n−1∑
k=N
(n− k − 1) qk yβ
k+3
)1/α
, n ≥ N.
For the solution {yn} of type (IV), {∆yn} is positive and decreasing sequence, so there
exits lim
n→∞
∆yn = w1, 0 ≤ w1 < ∞. Therefore, summing the previous inequality from
N to r − 1, letting r →∞ and using (5.11) we get
∆yN ≥ cβ
∞∑
k=N
1
p
1/α
k
(
k−1∑
i=N
(k − i− 1) (i + 3)β qi
)1/α
.
Accordingly, we conclude that (5.20) is satisfied.
Sufficiency. We assume that (5.20) holds and let c > 0 be an arbitrary number. Then,
there is N ≥ n0 such that
2
β
α
∞∑
k=N
1
p
1/α
k
(
k−1∑
i=N
(k − i− 1)(i + 3)βqi
)1
α
≤ c1− β
α . (5.22)
We will define the operator H3 : H2 → BN by
(
H3y
)
n
= c(n−N) +
n−1∑
k=N
∞∑
i=k
1
p
1/α
i
i−1∑
j=N
(i− j − 1)qjy
β
j+3
1
α
, n ≥ N. (5.23)
By Lemma 3.1, we may conclude that there exists an ŷ ∈ H2 such that H3ŷ = ŷ. The
operatorH3 satisfies the assumptions of Lemma 3.1, sinceH3 is the continuous operator
on H2, H3(H2) is uniformly Cauchy and for all y ∈ H2, using (5.22), we have
ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 1
26 R. P. AGARWAL, J. V. MANOJLOVIĆ
c(n−N) ≤
(
H3y
)
n
≤
≤ c(n−N) + (2c)
β
α
n−1∑
k=N
∞∑
i=k
1
p
1/α
i
i−1∑
j=N
(i− j − 1)(j + 3)β qj
1
α
≤
≤ c(n−N) + c
β
α c1− β
α (n−N) = 2c(n−N), n ≥ N,
so that H3 maps H2 to H2.
It is easy to see that ŷ = {ŷn} is a positive solution of (1.1) of type (IV) and by
Stolz’s theorem, taking into account the assumption (5.20), we have that
lim
n→∞
ŷn
n
= lim
n→∞
(
H3 ŷ
)
n
n
= c
which shows that the solution ŷ is of type (1.9).
The theorem is proved.
Nothing that under the condition (1.4), we have that
(5.16) ⇒ (5.10) ⇒ (5.20)
we have the following result on the existence of the positive solution of type (1.9), under
the assumptio (1.4).
Theorem 5.7. Suppose that (1.4) holds. Equation (1.1) has a positive solution
which satisfies (1.9) if and only if (5.20) holds.
5.3. Existence of positive solutions under the condition (1.5). If we suppose that
(1.5) holds and {yn} is a positive solution which satisfies (1.8), by Lemma 3.4 (ii) the
solution {yn} is of type (III). Therefore, under the condition (1.5), as in the proof of
Theorem 5.2 we may prove that the condition (5.2) is necessary for the existence of
positive solution of type (1.8). On the other hand, in the sufficiently part of the proof
of Theorems 5.2, only the first part of the condition (1.3) has been used. Therefore, the
statement of Theorem 5.2 remains to hold if the condition (1.5) is assumed to hold and
we have the following result on the existence of solution of type (1.8):
Theorem 5.8. Suppose that (1.5) holds. The condition (5.2) is a necessary and
sufficient condition for the equation (1.1) to have a positive solution {yn} which satisfi-
es (1.8).
Notice that in the sufficiently part of the proof of Theorems 5.4 – 5.6, we used only
the second part of the condition (1.4), i.e., that
∞∑
n=n0
(
n
pn
)1/α
< ∞.
Moreover, if (1.5) holds, by Lemma 2.3, (I) or (II) or (III) or (IV) holds. If {yn}
is a positive solution which satisfies (1.9), it can not be of type (III), because if we
suppose on the contrary that {yn} is a positive and decreasing sequence, we would
have that lim
n→∞
yn/n = 0. Accordingly, if (1.5) holds we can prove in the same way
Theorems 5.4 – 5.6, so that we have the following results:
ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 1
ASYMPTOTIC BEHAVIOR OF POSITIVE SOLUTIONS OF FOURTH-ORDER ... 27
Theorem 5.9. Suppose that (1.5) holds. The condition (5.20) is a necessary
and sufficient condition for the equation (1.1) to have a positive solution {yn} which
satisfies (1.9).
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Received 26.06.07
ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 1
|
| id | umjimathkievua-article-3134 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:36:51Z |
| publishDate | 2008 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/57/ed1bceb55d9f44c45d64566de93e4d57.pdf |
| spelling | umjimathkievua-article-31342020-03-18T19:46:36Z Asymptotic behavior of positive solutions of fourth-order nonlinear difference equations Асимптотична поведінка додатних розв'язків нелінійних різницевих рiвнянь четвертого порядку Agarwal, P. Manojlović, J. V. Агарвал, Р. П. Манойловіч, Дж В. We consider a class of fourth-order nonlinear difference equations of the form $$ \Delta^2(p_n(\Delta^2y_n)^{\alpha})+q_n y^{\beta}_{n+3}=0, \quad n\in {\mathbb N} $$ where $\alpha, \beta$ are the ratios of odd positive integers, and $\{p_n\}, \{q_n\}$ are positive real sequences defined for all $n\in {\mathbb N} $. We establish necessary and sufficient conditions for the existence of nonoscillatory solutions with specific asymptotic behavior under suitable combinations of the convergence or divergence conditions of the sums $$ \sum\limits_{n=n_0}^{\infty}\frac n{p_n^{1/\alpha}}\quad \text{and}\quad \sum\limits_{n=n_0}^{\infty}\left(\frac n{p_n}\right)^{1/\alpha}.$$ Розглянуто клас нелінійних ріницевих рівнянь четвертого порядку, що мають вигляд $$ \Delta^2(p_n(\Delta^2y_n)^{\alpha})+q_n y^{\beta}_{n+3}=0, \quad n\in {\mathbb N} $$ де $\alpha, \beta$ є співвідношеннями непарних додатних цілих чисел, а $\{p_n\}, \{q_n\}$ — додатними дійсними послідовностями, визначеними для всіх $n\in {\mathbb N} $. Встановлено необхідні і достатні умови існування неколивних розв'язків із специфічною асимптотичною поведінкою у випадку прийнятних комбінацій умов збіжності або розбіжності сум $$ \sum\limits_{n=n_0}^{\infty}\frac n{p_n^{1/\alpha}}\quad \text{and}\quad \sum\limits_{n=n_0}^{\infty}\left(\frac n{p_n}\right)^{1/\alpha}.$$ Institute of Mathematics, NAS of Ukraine 2008-01-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3134 Ukrains’kyi Matematychnyi Zhurnal; Vol. 60 No. 1 (2008); 8–27 Український математичний журнал; Том 60 № 1 (2008); 8–27 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/3134/3016 https://umj.imath.kiev.ua/index.php/umj/article/view/3134/3017 Copyright (c) 2008 Agarwal P.; Manojlović J. V. |
| spellingShingle | Agarwal, P. Manojlović, J. V. Агарвал, Р. П. Манойловіч, Дж В. Asymptotic behavior of positive solutions of fourth-order nonlinear difference equations |
| title | Asymptotic behavior of positive solutions of fourth-order nonlinear difference equations |
| title_alt | Асимптотична поведінка додатних розв'язків нелінійних різницевих рiвнянь четвертого порядку |
| title_full | Asymptotic behavior of positive solutions of fourth-order nonlinear difference equations |
| title_fullStr | Asymptotic behavior of positive solutions of fourth-order nonlinear difference equations |
| title_full_unstemmed | Asymptotic behavior of positive solutions of fourth-order nonlinear difference equations |
| title_short | Asymptotic behavior of positive solutions of fourth-order nonlinear difference equations |
| title_sort | asymptotic behavior of positive solutions of fourth-order nonlinear difference equations |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3134 |
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