Oscillation of second order nonlinear difference equations with super-linear neutral term

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https://doi.org/10.26637/MJM0703/0001

Oscillation of second order nonlinear difference

equations with super-linear neutral term

Govindasamy Ayyappan

1

* and Gunasekaran Nithyakala

2

Abstract

In this paper, the authors establish some new conditions for the oscillation of second order nonlinear difference equation of the form

∆ an∆ yn+pnyαn−k

+qnyβ_n+1−`=0,

whereα>1andβ are ratio of odd positive integers. Examples are provided to illustrate the importance of the

main results.

Keywords

Superlinear neutral term, nonlinear difference equation, oscillation.

AMS Subject Classification 39A11, 39A21

1_{Department of Mathematics, Periyar University College of Arts and Science, Pappireddipatti -636905, Tamilnadu, India.} 2_{Department of Mathematics, Periyar University, Salem-636011, Tamilnadu, India.}

*Corresponding author:1_{[email protected];}2_{[email protected]}

Article History: Received6April2019; Accepted29May2019 c2019 MJM.

1. Introduction

In this paper, we study the oscillatory behavior of all solutions of nonlinear second order difference equations with a superlinear neutral term of the form

∆ an∆ yn+pnyαn−k

+qnyβ_n+1−`=0, n≥n0 (1.1)

wheren0is a positive integer, subject to the following

condi-tions:

(H1) α>1 andβ>0 are quotient of odd positive integers;

(H2) {an},{pn},{qn}forn≥n0are positive real sequences

such that limn→∞pn=0, and∑∞n=n0

1 an <∞;

(H3) k∈Nand`∈N0.

Letη=max{k, `−1}. By a solution of equation (1.1), we mean a real sequence{yn}defined forn≥n0−η, that

satisfies equation (1.1) for alln≥n0. A nontrivial solution of

equation (1.1). A nontrivial solution of equation (1.1) is said to be oscillatory if its terms are neither eventually positive nor eventually negative, and otherwise it is called nonoscillatory. Equation (1.1) is said to be oscillatory if all its solutions are oscillatory.

Recently, there has been a lot of interest in studying the oscillatory and asymptotic behavior of solutions of various classes of second order nonlinear difference equations, see [1, 2,10, 12] and the references contained therein. In [4,

6,15–18], the authors considered equation of the form (1.1) and obtained criteria for the oscillation of (1.1) under the conditions either

∞

∑

n=n0 1

an

=∞ (1.2)

or

∞

∑

n=n0 1

an

<∞ (1.3)

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In view of the above observations our aim in this paper is to present some new sufficient conditions which ensure that all solutions of equation (1.1) when the condition (1.3) holds and our results are different from that of [7,9]. For related results concerning second order differential equations with sublinear or superlinear neutral terms, we refer the reader to [3,8,13,14].

2. Main Results

In the following for convenience we denote

zn = yn+pnyα

n−k, An =

∞

∑

s=n

1

as ,

and

Pn = 1−Mα−1pn Aα

n−k Aα

n ≥0

for every constantM>0 andn≥n1≥n0. For convenience,

for some 0<γ≤1 andn≥n1, we get

Qn=qnA(1+γ)(β

−1)

n+1 P

β

n+1−`.

We begin with a new oscillation result for equation (1.1) when β≥1.

Theorem 2.1. Let assumptions (H1)−(H3)andβ ≥1 be hold. If

lim

n→∞sup

n

∑

s=n1

QsAs+1−

1 4asAs+1

=∞, (2.1)

then equation(1.1)is oscillatory.

Proof. Let{yn}be a nonoscillatory solution of equation (1.1),

say yn>0, yn+1−`>0 andyn−k>0 forn≥n2 for some n2≥n1. It is easy to see thatzn>0 forn≥n2, and equation

(1.1) becomes

∆(an∆zn) +qnyβn+1−`=0. (2.2)

Hence∆(an∆zn)≤0 for alln≥n2, which implies thatan∆zn

is decreasing for alln≥n2. Thus (I)∆zn>0 or (II)∆zn<0

for alln≥n2.

Now we consider Case (I). Since{zn}is a positive

increas-ing sequence, we have

lim

n→∞zn=nlim→∞(yn+pny α

n−k) =_nlim_→_∞yn

since limn→∞pn=0. Further there existsn3≥n2and a con-stantd>0 such thatzn>dandyn>dfor alln≥n3. Using the last inequality in (1.1), we obtain

∆(an∆zn) +dβqn≤0, n≥n3. (2.3)

Summing (2.3) fromn3ton−1, we have

an∆zn−an3∆zn3+d β

n−1

∑

s=n3

qs≤0, n≥n3. (2.4)

But (2.1) implies that

∞

∑

n=n3

qn=∞,

which together with (2.4) yields

lim

n→∞an∆zn =−∞.

This contradicts the eventual positivity ofan∆zn.

Next, we consider Case (II). Define the sequence{wn}by

wn= an∆zn

zn

, n≥n2. (2.5)

Thenwn<0 forn≥n2, and the decreasing nature ofan∆zn

implies that

∆zs≤ an∆zn

as for s≥n≥n2. (2.6)

Summing (2.6) fromnto j−1≥n, we obtain

zj−zn≤an∆zn j−1

∑

s=n

1

as

!

,

which by letting j→∞, leads to

an∆zn zn

An≥ −1 for n≥n2, (2.7)

or

wnAn≥ −1 for n≥n2. (2.8)

Now from (2.7) that

∆

zn An

=An∆zn−zn∆An

AnAn+1

=Anan∆zn+zn

anAnAn+1 ≥0

forn≥n2, and thus

zn An≥

zn−k An−k

for n≥n2+k. (2.9)

From the definition ofzn, we have

yn=zn−pnyαn−k≥zn−pnzαn−k, n≥n2+k

and using (2.9), we obtain

yn≥zn−pn Aα

n−k Aα

n zα

n =

1−pn

Aα

n−k Aα

n zα−1

n

zn. (2.10)

Sinceznis positive and decreasing, we get

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Using (2.11) in (2.10), we obtain

yn≥

1−Mα−1_p n

Aα

n−k Aα

n

zn=Pnzn for n≥n2. (2.12)

Sincen_Anznois positive and increasing, we get

zn An

≥ zn2

An2

=d>0 for n≥n2. (2.13)

Further,{An}is positive and converging to zero, there exists

n3≥n2+ksuch that

0<Aγ

n≤d for n≥n3. (2.14)

Hence by (2.13) and (2.14)

zn≥A1_n+γ, n≥n3. (2.15)

By (2.12), from (2.2), we obtain

∆(an∆zn) = −qnyβ_n₊₁−`

≤ −qnP_nβ+1−`z

β

n+1−`

≤ −qnPnβ+1−`z

β

n+1, n≥n3, (2.16)

where, we also used the decreasing nature ofznin the last estimate. Now (2.16), in view of (2.15) leads to

∆(an∆zn) ≤ −qnA

(1+γ)(β−1)

n+1 P

β

n+1−`zn+1

= −Qnzn+1 for n≥n3. (2.17)

From (2.5) one obtains

∆wn = ∆(an∆zn)

zn+1

− zn anzn+1

w2_n

≤ ∆(an∆zn) zn+1

−w 2 n an

, (2.18)

where we have used again the decreasing nature of{zn}.

Com-bining (2.18) and (2.17), we have

∆wn≤ −Qn− w2_n

an

, n≥n3. (2.19)

Using (2.19), we get

∆(Anwn) = wn∆An+An+1∆wn

= −wn an

+An+1∆wn

≤ −wn an

−An+1Qn− An+1w2n

an

≤ −An+1Qn+

1 4anAn+1

,

and summing this resulting inequality fromn3tonand using

(2.8) yields

n

∑

s=n3

QsAs+1−

1 4asAs+1

≤ An3wn3−An+1wn+1

≤ 1+An3wn3<∞

forn≥n3, contradicting (2.1). This completes the proof.

Whenβ=1, we have the following immediate corollary from Theorem2.1.

Corollary 2.2. Let assumptions(H1)−(H3)andβ =1 be hold. If

lim

n→∞ sup

n

∑

s=n1

qsPs+1−`As+1−

1 4asAs+1

=∞, (2.20)

then equation(1.1)is oscillatory.

Next, we present an oscillation criteria for the case 0<

β<1.

Theorem 2.3. Let assumptions(H1)−(H3)and0<β <1

be hold. If

lim

n→∞sup

n

∑

s=n1

LqsPsβ+1−`As+1−

1 4asAs+1

=∞, (2.21)

for some constant L>0, then equation(1.1)is oscillatory.

Proof. Let{yn}be a nonoscillatory solution of equation (1.1),

sayyn>0,yn+1−`>0 andyn−k>0 for alln≥n1for some n1≥n0. Proceeding as in the proof of Theorem2.1, we

obtain two cases (I)∆zn>0 of (II)∆zn<0 for alln≥n1.

The proof of Case (I) is similar to that of Theorem2.1. For Case (II), we see that{z_n}is positive and decreasing with limn→∞zn=limn→∞yn. Then we have either limn→∞zn=

d1>0 or limn→∞zn=0. The first case implies that limn→∞yn=

d1. Thus, there existd2>0 andn2∈Nsuch thatyn≥d2for

alln≥n2. Then using this estimate in equation (1.1) and

arguing as in Case (I), we obtain a contradiction. For the other case, there existsn3∈Nsuch that

0<zn<k (2.22)

for alln≥n3andK=L

1

β−1 _>_{0. Now proceeding as in the}

proof of Theorem2.1(Case (II)), we obtain (2.16) and then using (2.22) yields

∆(an∆zn) ≤ −qnPnβ+1−`z

β

n+1

= −qnP_nβ+1−`

zn+1

z1_n−₊β₁ ≤ −qnPβ

n+1−`

zn+1

K1−β

= −LqnP_nβ₊₁−`zn+1, n≥n3.

The rest of the proof is similar to that of Theorem2.1and hence is omitted. This completes the proof.

3. Example

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Example 3.1. Consider the second order neutral type differ-ence equation

∆

n(n+1)∆

yn+ yα

n−k n2

+ (n+1)5y3_n₊₁₋_`=0, n≥1.

(3.1)

Hereα>1is a quotient of odd integers,β=3, the delays

k≥1and`≥0, and an=n(n+1), pn=_n12 and qn= (n+1)5. We setγ=1. It is easy to see that(H2)holds. Further An=1_n

and Pn=1− M

α−1

n2−α₍_n₋_k₎α. Also

QnAn+1 −

1 4anAn+1

= (n+1)

1− M

α−1

(n+1−`)2−α₍_n₊₁_−`₋_k₎α 3

− 1

4n.

Moreover

lim

n→∞

QnAn+1−

1 4anAn+1

=∞,

and therefore condition(2.1)of Theorem2.1is satisfied. Hence the equation(3.1)is oscillatory.

Next we give an example for the caseβ=1.

∆



n(n+1)∆ 

yn+ y

5 3

n−k n53



 

+

n+1

n yn+1−`=0, n≥1.

(3.2)

Hereα =5₃, β =1, the delays k≥1 and`≥0, and an= n(n+1), pn= 1

n53

and qn=n+_n1. We set γ=12. It is easy

to see that(H2)holds. Also An= 1_n and Pn=1− M

2 3

(n−k)53

.

Further

qnPn+1−`An+1−

1 4anAn+1

= 3 4n−

M23

n(n+1−`−k)53

.

Therefore condition(2.20)of Corollary2.2is satisfied and hence equation(3.2)is oscillatory.

Finally, we provide an example for the case 0<β<1.

∆

2n∆

yn+ yα

n−k

2n

+n2ny

1 3

n+1−`=0, n≥1. (3.3)

Hereα>1and β=₃1and the delays k≥1and`≥0, and

an=2n, pn=₂1n and qn=n2n. We setγ=1. It is easy to see

that condition(H2)holds. Also An= 1

2n−1, and Pn=1−

Mα−1

2n−αk. Further

LqnP_nβ₊₁−`An+1−

1 4anAn+1

=Ln

1− M

α−1

2n+1−`−αk

−1

4,

and this tends to infinity for any constant L>0. Therefore con-dition(2.21)of Theorem2.3is satisfied and hence equation

(3.3)is oscillatory.

4. Conclusion

In this paper, we present some new oscillation criteria for the equation (1.1). The obtained results simplifies the known results in the literature in the sense that we need only one condition instead of two conditions required in [7,9,11] for the oscillation of all solutions of equation (1.1). Further the results presented in this paper extend and generalize the results known [4–6,15–18] for the case 0<α<1 to the case α>1.

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ISSN(P):2319−3786 Malaya Journal of Matematik

ISSN(O):2321−5666

Oscillation of second order nonlinear difference equations with super-linear neutral term