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On the Oscillation, the Convergence, and the Boundedness of Solutions for a Neutral Difference Equation Dinh Cong Huong* Dept.. Keywork: Neutral difference equation, oscillation, nonos

On the Oscillation, the Convergence, and the Boundedness of

Solutions for a Neutral Difference Equation

Dinh Cong Huong*

Dept of Math, Quy Nhon University 170 An Duong Vuong, Quynhon, Binhdinh, Vietnam

Received 14 April 2009

Abstract In this paper, the oscillation, convergence and boundedness for neutral difference

equations

r

A(&p + OpXpn—7) + » œ;(n)F (n—my¿) =0, n=0,1, -

z=l

are investigated

Keywork: Neutral difference equation, oscillation, nonoscillation, convergence, boundedness

1 Introduction

Recently there has been a considerable interest in the oscillation of the solutions of difference equations of the form

A(@n + 6%p_-7) + A(N)an—o = 0,

where n € N, the operator A is defined as Av, = %,41 — %p, the function a(n) is defined on N, 6 is

a constant, 7 is a positive integer and o is a nonnegative integer, (see for example the work in [1-7] and the references cited therein)

In [2], the author obtained some sufficient criterions for the oscillation and convergence of solutions of the difference equation

A(x + ð#„ +) + » a;(n) F(an—m;) = 9,

i=1

forn € N,n > a for some a € N, the operator A is defined as Ax, = a%41 — Yn, 6 is a constant,

T,7,™M1,™M,+++,™m, are fixed positive integers, and the functions a;(n) are defined on N and the function F' is defined on R

Motivated by the work above, in this paper, we aim to study the oscillation and asymptotic behavior for neutral difference equation

r

i=l where 6,, 7” € N is not zero for infinitely many values of nm and F’ : R —— R is continuous

* Corresponding author Tel.: 0984769741

E-mail: dinhconghuong@qnu.edu.vn

155

Put A = max{r,m,, -,m,} Then, bya solution of (1) we mean a function which is defined for n > —A and sastisfies the equation (1) for n € N Clearly, if

In=An, N= —A,-A+t+1, -,-1,0

are given, then (1) has a unique solution, and it can be constructed recursively

A nontrivial solution {a} no of (1) is called oscillatory if for any n; > no there exists

nz Sn, such that x,,.%p.+1 < 0 The difference equation (1) is called oscillatory if all its solutions are

oscillatory If the solution {x}, n, is not oscillatory then it is said to be nonoscillatory Equivalently, the solution {2}, ø¿ 1s nonoseillatory 1Ý it is eventually positive or negative, ie there exists an

integer n, > no such that #„#„+¡ > 0 for all ø > m1

2 Main results

To begin with, we assume that

xF (x) > 0 for x A 0 (2)

By an argument analogous to that used for the proof of Lemma 3, Theorem 6 and Theorem 7 in [2],

we get the following results

Lemma 1 Let {x,,} be a nonoscillatory solution of (1) Put zy, = Lp + On Xp—r-

(i) If {x} is eventually positive (negative), then {z,} is eventually nonincreasing (nondecreas- ing)

(ii) If {ap} is eventually positive (negative) and there exists a constant y such that

-l<y<on, WneN

then eventually zy, > 0 (z„ < 0)

Theorem 1 Suppose there exist positive constants a;(i = 1,2, -,7) and M such that

a(n) >a;, Vn eN,

|F(x)| 2 Mx], Va,

on 20, Yn EN

Then, every nonoscillatory solution of (1) tend to 0 as n — oo

Theorem 2 Assume that

€=1 i=1

and there exists a constant n such that

—-l<n<o, <0, YWneEN (4) Suppose further that, if |x| > c then |F(x)| > c, where c and c, are positive constants Then, every nonoscillatory solution of (1) tends to 0 as n — oo

Theorem 3 Assume that the given hypothese in Theorem 2 are satisfied If F is a nondecreasing function such that

dt

mm <% am | — > -o foralla> 0, 5

then the equation (1) is oscillatory

Proof Suppose that (1) has a nonoscillatoty solution {2,,} If z, > 0 for n > no, then by Lemma 1

there exists an, > no such that z, > 0,¢%p-m, >0 (1 <i<r),z, >Oand Az, <0 forn > n4 Put 2, = %p, + 6,%,_, and m, = max m,; We note that (4) implies that z,, < 2, and from (1), we

lir

Azn + » œ;(n)F (Z„—m„) < Ö

Azn + S/ ai(n)F (zn) <0 forn >ng=n, +m,

So ax(n) <— ^Z› forn > nạ — mỊ + Tạ

Now for zn41 <t < Zp, we have F(t) < F(z,), and so

Pain < [7 a for n > nz

Ỷ —

Summing both sides of the above inequality from nz to n and taking the limit as n — oo, we get

y Naty < [™ as < |” eo

l=ng i=1 a na E(t) o_ #Œ) , which contradicts (3) The proof for the case {x,,} eventually negative is similar

Example 1 Consider the difference equation

2

It is clear that this equation is a particular case of (1), where 6, = 12 a;(n) = aap wn EN,ti=

1,i=2and F(x) = 23

It is easy to verify that all conditions of Theorem 3 hold Hence, the equation (6) is oscillatory Theorem 4 Assume that the first and the third condition in Theorem 2 are satisfied and there exists constants o, pu such that

Suppose further that, T > m= max m; and F is a nondecreasing function such that

mm <% mg | —~ <co foralle>0, (8)

then the equation (1) is oscillatory

Proof Suppose that (1) has a nonoscillatoty solution {x,}, v, > 0 for n > no From Lemma | there

exists any > no such that v,_- > 0,%p-m, >0 A <i<r), zn < Oand Az, < 0 forn > n, Then

from (7) we have

ứừn—~ < On&n—r <Zn < 0

and hence

Zntr

ụu Ũ< <0, form >mị

Thus, it follows that

Z —n,

p(n) < F(®p-m,;) fồrn >nạ >m +my¿L<¡<r

Lb Sinee ?ø -L 7 — m¿ > n + l, Ì < ¿ < r the above inequality gives

p(1)< p(t) < F(®n-m), L<i<r

Hence, from (1) we find

— n1

AZz„ + >- œ;(n F( (0P )< 0

or

Now for am <t< an we have P (224) > F(t), and so

Zn-L1

dt

for n > nạ (10)

1

Using (10) in (9) and summing both sides from nz to n and taking the limit as n — oo, we get

» 3 a() < -u fi Fi for n > nz

H

l=nz i=1

But this in view of (8) contradicts (7) The proof for the case {x,,} eventually negative is similar Example 2 Consider the difference equation

2

It is clear that this equation is a particular case of (1), where 6, = _ 2, a;(n) = aa ,Vn CÑ,?—

1,¢=2 and F(x) = 2°

It can be verified that all conditions of Theorem 4 hold Hence, the equation (11) is oscillatory Theorem 5 Suppose that 6, >0, nm €N Then, all unbounded solutions of the equation (1) are oscillatory

Proof Suppose the contrary Without loss of generality, let {z,,} be an unbounded and eventually

positive solution of (1) By Lemma 1, we have z, > 0 and Az, < 0 eventually Hence, there exists lim z, Put lim z, = @ We have

Now, in view of 6, >0, néN we have z, > x,» and (12) show that {,,} is bounded, which is a contradiction

From now we always assume that

xP (x) <0 for x £0 (13)

ϡ r

Theorem 6 Assume that 6, >0, néEN, >> So aj(€) < co and F is nonincreasing Suppose

| Ap and [=o jor all c > 0 (14) Then, all nonoscillatory solutions of the equation (1) are bounded

further that

Proof Let {x,} be a nonoscillatory solution of (1), and let m9 € Ñ be such that |z„| Z 0 for all

nm => no Assume that x, > O for all n > no Put m, = max and nj = m9 +7+ m, We have KG > 0O for all ø 3> mị¡ and 1l < ¿ <r Put z, = 2, +6,%,_, We have z„ > 0 and

=— » œ;(n)F (na—m¿) > 0 for all n > n1 Hence, {z,,} is nondecreasing and satisfies z, > vn

for all n S mị Therefore, we find

Am = - » œ(n)F(„ my) <Ấ — » oœ(n)F(2z—m,)

or

Azn "

————< Flin) | 2 › œ;(n), ; Vn > mị >1Ị (15)

Since ý € [z„, z„ +1], P() < F{za) By (15) we obtain

Summing the inequality (16) from nm; to n — 1 and taking the limit as n — 00, we have

From (17) and the hypothese of Theorem 6 we find that {z,,} is bounded from above Since 0 < %, < Zn; {%n} is also bounded from above The proof is similar when {2,,} is eventually negative

Example 3 Consider the difference equation

2 Lo

A(an +2 tn-2) + » ứt pet œ5 ,)=0, neil (18)

It is clear that this equation is a particular case of (1), where 6, = 2”, a;(n) = Yn EN t=

It can be verified that all conditions of Theorem 6 hold Hence, all nonoscillatory solutions of the equation (18) are bounded

a

Corollary Suppose that the assumptions of Theorem 6 hold Further, suppose that {6,,} tends to 0

as n — oo Then, every nonoscillatory solution of (1) tends to 0 as n — oo

Proof Let {x,} be an eventually positive solution of (1) By Theorem 6, {z,,} is eventually positive, nondecreasing and bounded above Thus, there exists a constant C’ > 0 such that

Ôy#n_+x < Z„ < Ơ for sufficiently large n Hence,

#„_x~ < — HW O0an— ow

On

Theorem 7 Assume that

œœ› r

@=1 #=1 and there exists a constant 6 > 0 such that

Suppose further that, if |x| > c then |F(œ)| 3 c, where c and c, are positive constants Then, for every bounded nonoscillatory solution {x,,} of (1) we have

lim inf |z,,| = 0

NICO

Proof Assume that, {2,,} is a bounded nonoscillatory solution of (1) Then, there exists constants c,C > 0 such that c < #„ < C for all n > no EN It implies that

Put m, = max and ny = nro +7 + my We have v, -~m, 2 ¢ for alln > nm, and 1 <i <r By the

tr

hypothese of Theorem 7, there exists a constant c; > 0 such that |F'(¢@,—m,| > c for all n > ø¡ and

1<ic<ryr Thus,

r

Ain = — » œ;(n)F (8n—m,) 2y œ;(n)c,, Wn > ny (22)

i=l Summing the inequality (22) from n; to n — 1, we obtain

n-l r

ln = ln, $1 ) ) a;(£) > 00 as n > 00,

l=n, i=1

which contradicts (22) The proof is complete

Example 4 Consider the difference equation

2

where @ is an odd integer It is clear that this equation is a particular case of (1), where 6, = a;(n) = aap wn €Ñ,¿= 1,¿= 2 and (+) = —z2

It can be verified that all conditions of Theorem 7 hold

Theorem 8 Assume that the conditions (3), (7) hold and F is a nonincreasing function such that

mm <% am | —~ >-oco foralla> 0

Further, suppose thatm; >7, V1 <i<vr Then, every nonoscillatory solution {x,} of (1) satisfies

|2n| 3 o© as n > 00

Proof Let {x,} be a nonoscillatory solution of (1) Assume that {z,} is eventually positive Then,

there exists m9 € N such that z„ ;_„„ > 0 for all ø > øọ and | <i<r Put z, = &% + Ôn» +

— a(n) (ama)

Therefore, z, — L > —coasn — o0 If L

Then, since Az,, > 0 for all n > no, {z,} is nondecreasing for n > no

< 0 then z„ < 0 for all n > 0 and hence O> 22 = Ay 4+ on@n_-7 > NXyn-7, NSNo

en +7

It implies 2,4, > 9%, n Sno Or %, > n7

is nonincreasing, we have

n > no Now since m; 27, Vl <i<rand F

r

Ain =r So au(n) (2) > — »Ắ

or

An "

"=>

F(#) + =1

Now for “+ <t< “ we have -4 >- 7 1 F(t) (=) , and so

tệ

or

n —

4

Summing both sides of the inequality (24) from mp to n and taking the limit as n — oo, we get

which contradicts (3) Thus, L > 0 Now let rn; 2 no be such that 0 < z, < a@,+oe,_, forn > ny Then, x, > —o#,,_, and by induction, we have x,4;, > (—o)Fan—7 for each positive integer 7 This

implies that x, — oo as n — oo The proof is similar when {2} is eventually negative

Example 5 Consider the difference equation

2

A(EsT— ST 1) x yy ent — œ5 x? )=0, ,) =0, n m>1 (25) 2

It is clear that this equation is a particular case of (1), where 6, = —258n a;(n) = ai ,Vn CÑ,?—

1,¿—= 2 and Ƒ(œ) = —z5

It can be verified that all conditions of Theorem 8 hold

Acknowledgement The authors would like to thank the referees for the careful reading and helpful suggestions to improve this paper

References

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