Oscillatory and Asymptotic Behavior of Solutions of Third Order Delay ...

Funkcialaj Ekvacioj, 42

(1999) 355-369

Oscillatory and Asymptotic Behavior of Solutions of Third Order Delay Difference Equations By John R. GRAEF* and E. THANDAPANI (Mississippi State Univ., USA and Periyar Univ., India)

1.

Introduction In this paper, we study the oscillatory and nonoscillatory behavior of

solutions of the delay difference equation

(1)

$¥Delta(a_{n}¥Delta(b_{n}¥Delta y_{n}))+q_{n}f(y_{n-m+1})=h_{n}$

,

,

,

$n¥in N_{0}=¥{0,1,2, ¥ldots¥}$

are real sequences, : is continuous, ¥ ¥ $u ¥ neq 0$ , and $m$ is $a_{n}>0$ , $b_{n}>0$ , and $q_{n}>0$ for all , $uf(u)>0$ for a positive integer. A solution of (1) is a real sequence defined for all $n_{0}-m+1$ and satisfying (1) for all $n>n_{0}$ . In what follows, we assume that equation (1) has solutions which are nontrivial and defined for all large . A nontrivial solution of (1) is said to be oscillatory if for any $N¥geq n_{0}$ there exists $n>N$ such that $y_{n+1}y_{n}¥leq 0$ . Otherwise, the solution is said to be nonoscillatory. Equation (1) is said to be oscillatory if every solution of (1) is oscillatory, and it is said to be almost oscillatory if every solution is either oscillatory or satisfies for $i=0,1,2$ . Determining oscillation criteria for third order nonlinear difference equations has not received a great deal of attention in the literature even though such equations arise in the study of economics, mathematical biology, and other areas of mathematics in which discrete models are used (see, for example, [2]). Some recent results on third order equations can be found in [1, 4?9]. In Section 2, we obtain sufficient conditions for the oscillation or almost oscillation of equation (1). A necessary and sufficient condition for almost oscillation is obtained for a special case of (1). Section 3 contains a result giving sufficient conditions for equation (1) to have a bounded nonoscillatory solution converging to a nonzero constant. Examples to illustrate the results are included. where

$¥{a_{n}¥}$

$¥{b_{n}¥}$

$¥{q_{n}¥}$

, and

$f$

$¥{h_{n}¥}$

$R¥rightarrow R$

$n geq n_{0} in N_{0}$

$¥{y_{n}¥}$

$ n¥geq$

$n$

$¥{y_{n}¥}$

$¥{y_{n}¥}$

$¥lim_{n¥rightarrow¥infty}¥Delta^{i}y_{n}=0$

*

Research supported by the Mississippi State University Biological and Physical Sciences Research Institute.

356

2.

John R. GRAEF and E. THANDAPANI

Oscillation theorems In this section, we also assume that

(2)

$¥Delta a_{n}¥geq 0$

$¥sum_{n=n_{0}}^{¥infty}¥frac{1}{a_{n}}=¥sum_{n=n_{0}}^{¥infty}¥frac{1}{b_{n}}=¥infty$

for all

$h_{n}¥equiv 0$

and

.

Our first two results are for the unforced equation, i.e., Theorem 1. Let $f(u)=u$ and . $H:N_{0}¥times N_{0}¥rightarrow R$ such that , functions

$n¥geq n_{0}$

for all

$h_{n}¥equiv 0$

$n¥in N_{0}$

.

Assume that there exist real valued

$h$

$H(n, n)=0$

$H(n,s)>0$ $¥Delta {}_{2}H(n,s)¥leq 0$

for for for

$n¥geq n_{0}¥geq 0$

$n>s¥geq n_{0}$

,

$n>s¥geq n_{0}$

,

where $¥Delta_{2}H(n,s)=H(n,s+1)-H(n,s)$ .

and

for

$-¥Delta {}_{2}H(n,s)=h(n,s)¥sqrt{H(n,s)}$

(3)

,

$n>s¥geq n_{0}$

$alf$

If for every

,

$n_{1}>n_{0}+m$

$¥mathrm{l}¥mathrm{i}¥mathrm{m}n¥rightarrow¥infty¥sup¥frac{1}{H(n,n_{1})}¥sum_{s=n_{1}}^{n-1}[H(n,s)q_{s}-¥frac{a_{s}b_{s-m}h^{2}(n,s)}{4(s-m-n_{0})}]=¥infty$

,

and

(4)

$¥sum_{i=n}^{n+m-1}q_{i}[¥sum_{j=n}^{i}¥frac{1}{b_{j}}(¥sum_{k=j}^{i}¥frac{1}{a_{k}})]>1$

then every solution

of equation (1)

,

is oscillatory.

Let be a nonoscillatory solution of equation (1). Without loss of generality, we may assume that is eventually positive. Then, there exists an integer such that $y_{n}>0$ and $y_{n-m}>0$ for all . From equation (1), we have

Proof.

$¥{y_{n}¥}$

$¥{y_{n}¥}$

$n¥geq n_{1}$

$n_{1}¥geq n_{0}$

$¥Delta(a_{n}¥Delta(b_{n}¥Delta y_{n}))=-q_{n}y_{n-m+1}$

so that

(5)

$¥Delta(a_{n}¥Delta(b_{n}¥Delta y_{n}))0$

We

mird Order Delay

for

Difference

357

Equations

To prove this, suppose, to the contrary, that and $a_{n}>0$ , it is clear that there is an integer . Then, for ¥ , we have

$n¥geq n_{2}$

$q_{n}>0$

.

Since such that

$¥Delta(b_{n}¥Delta y_{n})¥leq 0$

$n_{3}¥geq n_{2}$

.

$n geq n_{3}$

$a_{n_{3}}¥Delta(b_{n_{3}}¥Delta y_{n_{3}})N+m+1=M$ .

From

(9) and (12), we obtain

$¥Delta z_{n}¥leq-q_{n}-¥frac{(n-m-N)a_{n}¥Delta(b_{n}¥Delta y_{n})}{y_{n-m}a_{n}b_{n-m}}z_{n+1}$

Also, from (5) and the fact that

$¥Delta y_{n}¥geq 0$

,

(13) yields

$¥Delta z_{n}¥leq-q_{n}-¥frac{(n-m-N)}{a_{n}b_{n-m}}z_{n+1}^{2}$

.

.

$b_{n}¥Delta y_{n}=$

John R. GRAEF and E. THANDAPANI

358

Thus, for all

$n¥geq M$

,

$¥sum_{n=M}^{n-1}H(n,s)q_{s}¥leq H(n, M)z_{M}-¥sum_{s=M}^{n-1}[z_{s+1}(-¥Delta {}_{2}H(n, s))+¥frac{(s-m-N)H(n,s)}{a_{s}b_{s-m}}z_{s+1}^{2}]$

$=H(n, M)z_{M}-¥sum_{s=M}^{n-1}[¥frac{(s-m-N)}{a_{s}b_{s-m}}H(n,s)z_{s+1}^{2}+¥sqrt{H(n,s)}h(n,s)z_{s+1}]$

$¥leq H(n, M)z_{M}+¥sum_{s=M}^{n-1}¥frac{a_{s}b_{s-m}h^{2}(n,s)}{4(s-m-N)}$

Hence, for all

$n¥geq M$

.

, we have

$¥mathrm{l}¥mathrm{i}¥mathrm{m}n¥rightarrow¥infty¥sup¥frac{1}{H(n,M)}¥sum_{s=M}^{n-1}[H(n,s)q_{s}-¥frac{a_{s}b_{s-m}h(n,s)^{2}}{4(s-m-N)}]¥leq z_{M}$

which contradicts (3). for all Case 2: ¥ have $ Delta y_{n}n_{0}+m$

$7f$

and

there exists a positive sequence

$ g(u, v)¥geq¥mu$

such that

$¥{p_{n}¥}$

for

every

,

,

(17)

$¥sum_{n=n_{1}}^{¥infty}[p_{n}q_{n}-¥frac{a_{n}b_{n-m}(¥Delta p_{n})^{2}}{4¥mu¥rho_{n}(n-m-n_{0})}]=¥infty$

and

(18)

$¥sum_{i=n}^{n+m-1}q_{i}[¥sum_{j=n}^{i}¥frac{1}{b_{j}}(¥sum_{k=j}^{i}¥frac{1}{a_{k}})]=¥infty$ $¥lim_{n¥rightarrow}¥sup_{¥infty}$

then every solution if

of equation (1)

is oscillatory.

Proceeding exactly as in Theorem 1, we see that is eventually positive, then we set

Proof. $¥{¥Delta y_{n}¥}$

$z_{n}=¥frac{a_{n}¥Delta(b_{n}¥Delta y_{n})¥rho_{n}}{f(y_{n})}$

for

,

$n¥geq n_{2}$

.

It is easy to see that

$z_{n}>0$

(6) holds. Now,

,

and

$¥Delta z_{n}¥leq-¥rho_{n}q_{n}+¥frac{¥Delta p_{n}}{¥rho_{n+1}}Z_{n+1}-¥frac{g(y_{n-m+1},y_{n-m})¥Delta y_{n-m}¥rho_{n}}{f(y_{n-m})p_{n+1}}Z_{n+1}$

Using (12) and the fact that

$¥{a_{n}¥Delta(b_{n}¥Delta y_{n})¥}$

.

is nonincreasing, and

$¥{y_{n}¥}$

is

360

John R. GRAEF and E. THANDAPANI

nondecreasing, we see that $¥Delta z_{n}¥leq-p_{n}q_{n}+¥frac{¥Delta p_{n}}{p_{n+1}}z_{n+1}-¥frac{¥mu(n-m-N)p_{n}}{a_{n}b_{n-m}p_{n+1}}z_{n+1}^{2}$

,

for $n¥geq N+m+1=M>n_{2}$ , and completing the square, we obtain

,

$¥Delta z_{n}¥leq-p_{n}q_{n}+¥frac{a_{n}b_{n-m}(¥Delta p_{n})^{2}}{4¥mu p_{n}(n-m-N)}$

$n¥geq M$ .

Summing the last inequality from $M$ to and letting , we see, in view of condition (17), that . This contradicts the fact that is eventually positive. Now assume that is eventually negative. By summing equation (1) three times as in the proof of Theorem 1, we see that $ n¥rightarrow¥infty$

$n$

$¥lim_{n¥rightarrow¥infty}z_{n}=-¥infty$

$¥{z_{n}¥}$

$¥{¥Delta y_{n}¥}$

(19)

$¥sum_{i=n}^{¥infty}[¥sum_{j=n}^{i}¥frac{1}{b_{j}}(¥sum_{k=j}^{i}¥frac{1}{a_{k}})]q_{i}f(y_{i-m+1})¥leq y_{n}$

Since

$¥{y_{n}¥}$

is decreasing and

(20)

$f(u)$

is increasing, it follows from (19) that

$¥sum_{i=n}^{n+m-1}[¥sum_{j=n}^{i}¥frac{1}{b_{j}}(¥sum_{j=k}^{i}¥frac{1}{a_{k}})]q_{i}¥leq¥frac{y_{n}}{f(y_{n})}$

Clearly, possible.

$¥lim_{n¥rightarrow¥infty}y_{n}=b¥geq 0$

.

.

In view of (18),

If $b=0$ , then

.

(20) implies that

$¥lim_{n¥rightarrow¥infty}¥frac{y_{n}}{f(y_{n})}=¥lim¥underline{1}¥leq¥underline{1}$ $n¥rightarrow¥infty g(y_{n+1}, y_{n})$

$b>0$

is not

’

$¥mu$

which of course contradicts (18). The proof for the case $y_{n}^{¥vee}n_{0}+m+1$

,

$¥sum_{n=n_{1}}^{¥infty}(p_{n}q_{n}-¥frac{a_{n}b_{m-n}(¥Delta p_{n})^{2}}{4¥mu¥lambda(n-m-n_{0})p_{n}})=¥infty$

Then equation

(1)

.

is almost oscillatory.

Proof.

Suppose there is a nonoscillatory solution such that is eventually positive and . Consider the function defined by $¥{y_{n}¥}$

$¥{y_{n}¥}$

$¥lim_{n¥rightarrow¥infty}y_{n}=0$

(23)

$x_{n}$

$x_{n}=y_{n}-¥phi_{n}$

.

It is easy to see that is eventually positive, for otherwise we would have , and this would contradict the oscillatory nature of . From equation (1), we see that $x_{n}$

$y_{n}0$

$¥Delta^{2}(b_{n}¥Delta x_{n}))¥leq 0$

for ¥ . Now assume that is eventually positive. Since eventually positive and increasing, and as , it follows from that there exists an integer such that $n geq n_{1}$

$¥{¥Delta x_{n}¥}$

$¥{x_{n}¥}$

$¥phi_{n}¥rightarrow 0$

$ n¥rightarrow¥infty$

$n_{2}¥geq n_{1}$

(23)

for

$ y_{n-m+1}¥geq¥lambda x_{nm+1}¥_$

$n¥geq n_{2}$

.

Therefore,

(26)

$f(y_{n-m+1})¥geq f(¥lambda x_{n-m+1})$

.

Define $z_{n}=¥frac{a_{n}¥Delta(b_{n}¥Delta x_{n})}{f(¥lambda x_{n-m})}p_{n}$

and observe that

$z_{n}>0$

for

$n¥geq n_{2}$

,

$n¥geq n_{2}$

,

and .

$¥Delta z_{n}¥leq-q_{n}p_{n}¥frac{f(y_{n-m+1})}{f(¥lambda x_{n-m+1})}+¥frac{¥Delta p_{n}}{p_{n+1}}Z_{n+1}-¥frac{¥lambda g(¥lambda x_{n-m+1},¥lambda x_{n-m})¥Delta x_{n-m}p_{n}}{f(¥lambda x_{n-m})p_{n+1}}z_{n+1}$

By

(26) and (16), we have $¥Delta z_{n}¥leq-q_{n}p_{n}+¥frac{¥Delta p_{n}}{p_{n+1}}z_{n+1}-¥frac{¥lambda¥mu¥Delta x_{n-m}p_{n}}{f(¥lambda x_{n-m})p_{n+1}}z_{n+1}$

.

is

(23)

362

John R. GRAEF and E. THANDAPANI

We now proceed as in the proof of Theorem 2 and obtain a contradiction must be eventually negative. In this case, decreases to (22). Thus, to a nonnegative constant . Since , from (23) we have that . Summing equation (1) three times as we did in the previous theorems, we have $¥{x_{n}¥}$

$¥{x_{n}¥}$

$¥lim_{n¥rightarrow¥infty}¥phi_{n}=0$

$c$

$¥lim_{n¥rightarrow¥infty}y_{n}=c$

.

$¥sum_{i=n}^{¥infty}[¥sum_{j=n}^{i}¥frac{1}{b_{j}}(¥sum_{k=j}^{¥infty}¥frac{1}{a_{k}})]q_{i}f(y_{i-m+1})¥leq x_{n}$

is monotonic, we have , and since Hence, $c=0$ for , and by (21) and (23), we have Thus, This completes the proof of the theorem. $¥lim¥inf_{n¥rightarrow¥infty}y_{n}=0$

$¥lim_{n¥rightarrow¥infty}y_{n}=0$

$¥{y_{n}¥}$

$¥lim_{n¥rightarrow¥infty}¥Delta^{i}y_{n}=0$

Example 3.

.

$i=0,1,2$ .

The difference equation

$¥Delta[(n+1)¥Delta((n+1)¥Delta y_{n})]+n^{4}(y_{n-3}^{3}+y_{n-3})$

$(¥mathrm{E}_{3})$

$=¥frac{(-1)^{n+1}[8n^{3}+24n^{2}+18n+1]}{n(n+1)}$

,

$n¥geq 1$

,

and $¥{¥emptyset_{n}¥}=¥{(-1)^{n}/n¥}$ . satisfies all conditions of Theorem 3 for are almost oscillatory. all solutions of equation $p_{n}¥equiv 1$

Hence,

$(¥mathrm{E}_{3})$

Remark. If $f(u)=u$ , then condition (18) of Theorems 2 and 3 can be replaced by condition (4) of Theorem 1. . Next, we study the almost oscillation of equation (1) when $a_{n}¥equiv 1$

$¥{p_{n}¥}$

Theorem 4. Assume that (16) holds and that there exists a positive sequence such that and an oscillatory sequence $¥{¥phi_{n}¥}$

$¥Delta^{2}(b_{n}¥Delta¥phi_{n}))=h_{n}$

$¥Delta p_{n}¥leq 0$

(27)

, and

$¥Delta^{2}p_{n}¥geq 0$

for

,

$¥lim_{n¥rightarrow¥infty}¥Delta^{i}¥phi_{n}=0$

$n¥geq n_{0}$

.

,

$i=0,1,2$ ,

for

If

$¥sum_{n=n_{0}}^{¥infty}q_{n}p_{n}=¥infty$

and

(28)

$¥sum_{n=n_{0}}^{¥infty}¥frac{1}{b_{n}p_{n}}¥sum_{s=n}^{¥infty}(s-n+1)q_{s}p_{s+2}=¥infty$

then equation (1) is almost oscillatory.

,

Third Order Delay

Difference

363

Equations

Similar to the proof of Theorem 3, we obtain

Proof.

$¥Delta z_{n}¥leq-q_{n}p_{n}+¥frac{¥Delta p_{n}}{p_{n+1}}Z_{n+1}-¥frac{g(¥lambda x_{n-m+1},¥lambda x_{n-m})¥Delta x_{n-m}p_{n}}{f(¥lambda x_{n-m})p_{n+1}}Z_{n+1}$

for

$n¥geq N$

for some

$N¥geq n_{0}$

.

The hypotheses on $¥Delta z_{n}¥leq-q_{n}p_{n}$

then yield

$¥{p_{n}¥}$

.

and letting , we obtain a Summing the last inequality from $N$ to must be eventually negative, and so contradiction to (27). Thus, . We , (23) implies that decreases to $c¥geq 0$ . Since will prove that $c=0$ , so suppose $c>0$ . Then there is an integer $N_{1}¥geq N>0$ such that $ n¥rightarrow¥infty$

$n$

$¥{x_{n}¥}$

$¥{¥Delta x_{n}¥}$

$¥lim_{n¥sim¥infty}¥phi_{n}=0$

$¥lim_{n¥rightarrow¥infty}y_{n}=c$

$y_{n-m+1}¥geq¥frac{c}{2}$

for all

$n¥geq N_{1}$

.

Let

$w_{n}=b_{n}p_{n}¥Delta x_{n}$

.

Then

$¥Delta^{2}w_{n}=-q_{n}f(y_{n-m+1})p_{n+2}+2¥Delta p_{n+1}¥Delta(b_{n}¥Delta x_{n})+(b_{n}¥Delta x_{n})¥Delta^{2}p_{n}$

Since

$¥Delta p_{n}0$

.

, we have

$¥Delta^{2}w_{n}+q_{n}p_{n+2}f(c/2)¥leq 0$

for

Summing the last inequality from follows that it ,

$n¥geq N_{1}$

$¥Delta w_{j}>0$

Letting

.

$ j¥rightarrow¥infty$

$n$

to

$j$

$-¥Delta w_{n}+f(¥frac{c}{2})¥sum_{s=n}^{j}q_{s}p_{s+2}¥leq 0$

.

$-¥Delta w_{n}+f(¥frac{c}{2})¥sum_{s=n}^{¥infty}q_{s}p_{s+2}¥leq 0$

.

and using the fact that

, we have

Summing again and now using the fact that

$w_{j}0$ and Now if eventually is becomes unbounded, which is a contradiction. Thus, negative, and so there exists a nonnegative number such that . To complete the proof, it suffices to show that $c=0$ , so , we assume $c>0$ . Summing equation (1) and using the fact that have

Proof.

$¥{x_{n}¥}$

$¥{¥Delta x_{n}¥}$

$¥{¥Delta^{2}x_{n}¥}$

$¥{¥Delta^{2}x_{n}¥}$

$¥{¥Delta x_{n}¥}$

$¥{¥Delta^{2}x_{¥dot{n}}¥}$

$¥{x_{n}¥}$

$¥Delta^{2}x_{n}¥geq 0$

$¥Delta^{2}x_{n}¥geq 0$

$¥{¥Delta x_{n}¥}$

$¥{x_{n}¥}$

$¥{¥Delta x_{n}¥}$

$c$

$¥lim_{n¥rightarrow¥infty}y_{n}=$

$¥lim_{n¥rightarrow¥infty}x_{n}=c$

$¥Delta^{2}x_{n}>0$

$-¥Delta^{2}x_{n}+¥sum_{s=n}^{¥infty}q_{s}f(y_{s-m+1})¥leq 0$

.

Third Order Delay

Difference

365

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Repeating the above procedure, we have $¥Delta x_{n}+¥sum_{s=n}^{¥infty}(s-n+1)q_{s}f(y_{s-m+1})¥leq 0$

and summing one last time from

(30)

$N$

to

$¥infty$

,

, we obtain

$c-x_{N}+¥sum_{s=N}^{¥infty}¥frac{(s-N+1)(s-N+2)}{2}q_{s}f(y_{s-m+1})¥leq 0$ .

By (29), it follows from

(30) that

$¥lim_{n¥rightarrow}¥inf_{¥infty}f(y_{n-m+1})=¥lim_{n¥rightarrow¥infty}x_{n}=0$

.

This contradiction completes the proof of the theorem. Example 6.

The difference equation

$¥Delta^{3}y_{n}+¥frac{2^{2n-3}}{27000}(30-27(-1)^{n})y_{n}^{3}=¥frac{27(-1)^{n+1}}{2^{n+3}}$

$(¥mathrm{E}_{6})$

,

$n¥geq 2$

,

satisfies all conditions of Theorem 5 with $¥{¥phi_{n}¥}=¥{(-1)^{n}/2^{n}¥}$ . Hence, every is almost oscillatory, and $¥{y_{n}¥}=¥{30/2^{n}¥}$ is bounded solution of equation one such solution. $(¥mathrm{E}_{6})$

In our next theorem, we do not require that Theorem 6.

Let

$a_{n}=b_{n}¥equiv 1$

$q_{n}$

be positive.

and $f$ be nondecreasing.

If

$¥sum_{n=n_{0}}^{¥infty}n^{2}|q_{n}|n ¥ geq N_{1}$ such that for all , we have $|Ty_{n}-Ty_{k}|0$

$N_{1}$

$ T¥Psi$

$¥mathrm{Y}=¥{y_{n}¥}$

$Y¥in¥ovalbox{¥tt¥small REJECT}$

Example 7. $(¥mathrm{E}_{7})$

Consider the difference equation

$¥Delta^{3}y_{n}+¥frac{2^{-n}[(1/8)-(-1)^{n}2^{-n}]}{(1+2^{-n})}y_{n}=(-1)^{n+1}2^{-2n}$

,

$n¥geq 3$

.

With $¥{¥phi_{n}¥}=¥{(4/5)^{3}(-1)^{n}/4^{n}¥}$ , all conditions of Theorem 6 are satisfied. Hence, equation (E7) has a nonosciUatory solution that approaches a non-zero real number. In fact, $¥{y_{n}¥}=¥{1+2^{-n}¥}$ is such a solution of equation (E7).

Third Order Delay

$DifJ¥dot{e}$

rence Equations

367

By combining Theorem 5 and 6, we have the following necessary and sufficient condition for the almost oscillation of equation (1).

be nondecreasing, and conditions (21) and (31) hold. Then every bounded solution of equation (1) is almost oscillatory if and only if (29) holds. Theorem 7.

3.

Let

$a_{n}=b_{n}=1$ ,

$f$

Asymptotic behavior

In this section, we obtain a sufficient condition for the asymptotic behavior of solutions of equation (1). We do not require $q_{n}>0$ here. Let , , and be defined by $A_{n}$

$B_{n}$

$C_{n}$

$A_{n}=¥sum_{s=n_{0}}^{n-1}¥frac{1}{a_{s}}$

Theorem 8.

that

$a_{n}¥geq d$

for

Let $afl$

,

$B_{n}=¥sum_{s=n_{0}}^{n-1}¥frac{1}{b_{s}}$

and

$C_{n}=¥sum_{s=n_{0}}^{n-1}¥frac{A_{s}}{b_{s}}$

.

be nondecreasing and let $d>0$ be a constant such . Suppose that

$f(u)$

$n¥geq n_{0}$

(34)

$¥sum_{n=n_{0}}^{¥infty}[C_{n+1}+A_{n+1}B_{n+1}]|h_{n}|