Oscillation and nonoscillation theorems for some mixed difference equations

Internat. J. Math.

VOL. 15 NO. 3 (1992) 537-542

OSCILLATION AND NONOSCILLATION

THEOREMS

FOR

SOME

MIXED DIFFERENCE

EQUATIONS

B. SMITHandW.E.TAYLOR, JR.

Department

of Mathematics

NabritScienceCenter

Texas

Southern University

Houston, Texas

77004

(Received

October 10, 1989and in revisedformNovember 6,

1990)

ABSTRACT. In

thispaperwe investigatetheoscillatoryandnonoscillatorybehaviorofsolutionsof

certain mixedthird andfourthorder differenceequations. Specificresultsarealsoobtainedforthe constant

coefficient cases.

KEY

WORDS

AND

PHRASES:Mixed differenceequations,thirdandfourth order equations, oscillatory

andnonoscillatorysolutions.

1991

AMS

SUBJECT

CLASSIFICATION CODES.

39A10,39A12.

1. INTRODUCTION.

In

thispaperwestudytheoscillatoryandnonoscillatorybehaviorofthe solutionsofcertain third and

fourthorder differenceequations. Untilrecently,exceptingthe studiesby Cheng

[1],

Hooker and Patula

[2],

and

[4,

5,6,

7],

therehas notbeen much research devotedtothe oscillationtheoryof difference equations ofordergreaterthan two.

For

asequence

U,,,

andafixed realconstanta,wedefine

A.U.

U.

1-aU..

Whena 1weshall

write

AU,

insteadof

AtU,.

Wecan define inductively

AU,,

A,,(A,-1U,)

for k 1. Theoperator

Ao

was introducedby J. Popenda

[3]

in hisstudy ofcertainnonlinearsecondorder

difference

equations.

The objects ofthisstudywillbethe mixed differenceequations

A2(AoU.)+(-lyP.U.-O

i-1,2,

(1.1)

and

A3(AoU,)+(-1)P,U,-0

i-1,2,

(1.2)

where

P,

is asequenceofpositivenumbers havingapositivelimitinferior, that is, thereisapositive constant c 0such that

P,

cforall nsufficientlylarge.

We

consideronlynontrivial solutions.

A

solution

is callednonoscillatoryifit iseventually

Of

constantsign (positiveornegative)otherwise it iscalled oscillatory. The equations

(1.1)

and

(1.2)

arecalledmixedbecause of thetwodifferenceoperators

A

and

ASYMPTOTIC BEHAVIOR OF NONOSCILLATORY SOLUTIONS.

Inthissectionwestudytheasymptoticbehaviorof thenonoscillatorysolutionsof

(1.1)

and

(1.2).

THEOREM1.

Suppose U,,

is anonoscillatorysolutionof

x(,%,)

+

.v.

-o

where 0<a I. Thenfora{{nsufflcicnt{y{argcwchave

sgn

U,,

sgn

A:’U,

,

sgn

AU,

sgn

A3U,,

and

PROOF.

(2.1)

(2.2)

lira

U,,

0.

(2.3)

Fora 1, Popenda andSchmeidel

[4]

have recently shown that

(2.1)

has a solution satisfying

(2.2). A

nonoscillatorysolutionmaynotexist if 0<a<1,but if it does exist we show that it mustsatisfy

(2.2)

and

(2.3).

Asthenegative ofasolutionof equation

(1.1)

isalso a solutionofthe same

equation,itsufficestoprovethat aneventually positivesolution of

(2.1)

satisfies

(2.2). In

thispaper,we willassume that allinequalitiesaboutsequenceshold for all nsufficiently large. Let

U,,

>0be a

nonos-cillatorysolutionof

(2.1).

Set

Z,,

-A,,U,

-V,,+-aU,,,

(2.4)

thenby

(2.1)

A2Z,,

-P,, U,

<0

(2.5)

so

A

Z,,

is(eventually)strictly decreasing. From

(2.5)

itfollows that ifA

Z,,

iseventually negativewe must have

Z,,

_o% however this is contradictory since

Z,

U,

+,

aU,

AU,

+

(1

a

)U,

--+ -oo implies

AU,

-oo,whichforces

U,,

tobeeventually negative. Wemust have

AZ,,

>0

(2.6)

foralllargen. Indeedwe will show that lira

U,,

0.

Writing

(2.1)

as

A2Z,,

-P,U,,

andsumming fromNtom 1,where

N

ischosenlarge enoughsothat

A

Z,

>0 forall n z

N,

weget

AZ.-AZ,,--

P.u..

Thelim

inf

condition on

P.

yields

Letting m weseethat

:U,,

<ooandthereforelira

U,,

0.

Because

U,,

0asn itfollowsthat

Z,,

0asn

.

From

(2.6), Z,,

isincreasing, hence

Z,,

<0eventually.

It

thenfollows fromtheinequality

Z,, -AU,,

+

(1-a)U,

<0that At],<0and from

(2.6)

A

Z,,-

A2U,,

+

(1-a)AU,

>0 and thus

A2U,,

>0.

Finallyfrom

(2.5),

AZ,,

A3U,,

+

(1

a)AU’,,

<0andweget

AaU,,

<0and theproofiscomplete. Ournextresultthoughsimilar tothe previous onerequiresa 1.

TIIEOREM

2. Considerthefollowing equation

where a >1. If

U,,

is anonoscillatorysolution of

(2.7)

thenforall nsufficiently large sgn

U,,

sgn

AU,,

sgn

A’U,,,

and

lim

lull

lim

AU,,I

lim

&:’U,I

.

PROOF. Assumewithout lossofgenerality that

U,,

>0 for all nsufficiently large. Set

Z,

-AoU,

thenby

(2.7)

So

A

Z,,

isincreasing. IfA

Z,,

iseventually positivethen as n

,

Z,,

andsince

Z,,

AU,,

+(1 a)U,, and a 1 itfollowsthat

AU,,

,

which in turnimplies

U,,

oo.To see why

A:U,,

o,note that

U,,

implies

A:’Z,,

oo_andA

Z,,

_becauseof

(2.8).

_{But A}

Z,,

A’U,,

₊

(1

-a)AU,,

_{and the result follows.}

Now,if

A

Z,,

wereeventually negativeandincreasingthen

A

Z,,

wouldhave

a

limit as n

.

However

A

Z,,

havingalimitimpliesthat

EU,,

< and thisimplies

U,,

0.

But

U,,

0 implies

Z,,

0also and

thereforesince

Z,,

isdecreasingtozero,

Z,,

0.

But

Z,,

AU,,

+

(1 -a)U,

0implies

AU,,

0,a eontra-[lictionsince

U,,

0and

AU,,

0is inconsistentwith

U,

0.

Hence

(2.7)

cannot have anonoscillatory

solution with

AZ,

A2Z,

<0 forall nsufficiently large.

Itshould be noted that the condition0<a<1 was crucial in theproofof Theorem 1. Ournextresult

requiresa 1and is similar to one obtained in

[2]

fortheequation

a’U.

-P./U..-O.

THEOREM3. Considertheequation

a’(aou.)-e.u. -o

(z.9)

where a 1. If

U,

isanonoscillatorysolutionof

(2-)

then forall n sufficiently largeeither

sgn

U,,

sgn

&U,,

sgn

A2U,,

sgn

A3U,,

sgn

U,,

sgn

A(A,,U,,)

,

sgn

A,,U,,

sgn

A’(&,,

U,,

).

PROOF. We provetheeasefora 1. Theprooffora 1is similar. Thereisnoloss of generality

inassuming

U,,

isaneventuallypositivesolutionof

(2.9).

Set

Z,

A,U,

U,

aU,.

Thenby

(2.9)

A3Z.-V.U.

>0.

(2.10)

Clearly

,2Z.

isincreasing.

In

case

A2Z.

iseventuallypositivewe willhavelira

&

Z.

lira

Z

,

and since

Z,,

<

U,,

itfollows that

U.

.

Since

P,,

cforlargen

lim

AZ.

lim

A2Z.

Since

Z. AU.

+

(1

a)U.

and a 1itfollows that

AU.

.

Examining

A

Z.

A2U.

+

(1

a)&U.

weseethat

A2U.

asn

.

Continuingin thismanner weseethat

(I)

holdseventually.

Next

we consider the casewhere

AZ.

0 and

AZ.

<0. Then lira

AZ.

existsandsumming

(2.10)

Lettingm ooitthen follows that

U,,

<ooand hence

,,-.(R)lira

U,,

0whichimplies

Z,,

0. Thus if

ASZ,,

>0 and

A:’Z,,

<0 theneventuallywe musthave

ASZ.

>0,

&2Z.

<0,

AZ.

>0, (2.11)

because

A2Z.

<0 andA

Z,,

<0 is inconsistent with

Z,,

0,itthen follows that either(i)

Z,,

>0 or

(ii)

Z,,

<0

eventually. Wewill show that(i)isimpossible. If

(i)

held then since

Z,,

AU,,

+

(1

a)U,,>0 itfollows that

AU,,

>0,infactwehave that

AU,,

>k+

(a

1)U,,

>kfor somepositiveconstantkand so

U,,

ooas

n o,:,. But thisimplies

ASZ,,

---0%so we musthave

A2Z,,

>0eventually, contradicting

(2.11).

So (i)

cannothold, resultingin(ii)holdingeventually.

3. SUFFICIENT CONDITIONS FOR OSCILLATION AND/OR NONOSCILLATION. THEOREM4.

Every

nontrivialboundedsolutionof

A3(AoU,)+P,U,

-0 (3.1)

wherea>1,isoscillatory.

PROOF.

Suppose (3.1)

hasabounded nonoscillatorysolution

U,,

satisfying

U,

>0 forlargen.Letting

Zn-AU,-U,/-aU,,

we see that

Z,, a-aU,,.

From (3.1),

A3Z,--P,U,

<0. Obviously

A2Z,,

is

decreasing,and if

A2Z,,

iseventually negative,wesee that

Z,,

oo. Thisclearlycontradictsthe boundedness

of

U,,.

Thus,we consider the case where

A2Z,,

>0.

In

thiscase lira

A2Z,,

a:0. Usingthefact

P,,

is boundedawayfrom zero for large n,itfollowsthat

A

Z,,

<0andZ,,>0forlargen.Furthermore,lim

U,

0, since

A2Z,,

implies

U,,

<oo. Since a> and

Z,

AU,,

+(1

a)U,,

>O,

AU,,

>0forall nsufficientlylarge. Butthisis acontradiction,since

U,,AU,,

>0isincongruentwith

U,,

O.

EXAMPLE.

Theequation

A(,

U.)

+

U,,

-0.

sequence

U,

-()"

as a solution. Hence equation

(3.1)

mayhave nonoscillatory

hasthe solutions.

Before statingour finalresults consider the constant coefficientcase

P,,

Q, Q

>0. Equation

(1.1)

with

P,,

Q

is

AU.

+(-a)AU.

+(-IQU.-o

Sothe characteristicpolynomialis

(C,)

[(t)-(t-

1) +(1

-a)(t-

1)

+

(-1)’Q

Similarly, equation

(1.2)

canbewrittenas

A’U,,

+

(1

a)ASU,,

+

(-1)QU,,-0

withcharacteristicpolynomial

(C2)

g(t)-(t-

1)

+

(1

a)(t

1)

+

(-1)Q

i--1,2.

(3.2)

i-1,2.

i-1,2

(3.3)

THEOREMS 541 Theproofsofour finalresultsfollow from acarefulexamination ofthe characteristicpolynomials

(C)

and

(c).

THEOREM $. Consider

(2.4)

where 0<a<I,

P,

-

constant and >a, then all nontrivial solutionsof

(2.4)

areoscillatory.

EXAMPLE.

Thegeneralsolutionof

A2(A,/2U,,)

+

6U,

-0

(3.4)

U, K(-I) +K2(11/2y’f2sinnH

+K3(11/2)’/2cosnO

whereO arc

tan(yr"

fT)

itfollows from Theorem or from Theorem5thatall solutions of

(3.4)

are

oscillatory.

R

THEOREM6. Consider

(2.7)

where 0<a<I,

.

constantand0<a<1

jR,

then all solutions

of

(2.7)

arenonoscillatory.

THEOREM

7.Consider

(2.9)

where0<a<I,

P,

constantand >0, then

(2.9)

has oscillatory andnonoscillatorysolutions.

Moreover,

all nonoscillatorysolutions arebounded andconvergetozero.

Notethat when a

I,

P,

constant,equation

(2.7)

becomes

A3U,

U,

0.Clearlythisequation hasoscillatorysolutionsforany >0.Thus, the result ofTheorem 6depends upon0<a<I.Furthermore,

itshould be noted thatTheorem 7isinteresting because,when a>Iand

P,

isconstant

(2.7)

must have an

unbounded nonoscillatory solution. Clearly the boundedness of the nonoscillatory solutions can be attributedtotheparametera in theoperator

Ao.

1.

CHENG,

S.

S.,

OnaClass of Fourth OrderLinear

Recurrence

Equations, lntcrnat. J.Math.andMath.

(1984),

131-149.

2. HOOKER, J. W. and

PATULA,

W.

T.,

Growth and OscillationPropertiesof Solutions ofaFourth OrderLinearDifference Equation,J. Austrol,

Math.

Soc.Ser.,B26

(1985),

No.3, 310-328.

3.

POPENDA, J.,

Oscillation andNonoscillationTheorems for Second OrderDifferenceEquations,

J.

Math. Anal.Appl..123

(1987),

34-38.

4.

POPENDA,

J.and

SCHMEIDEL, E.,

NonoscillatorySolutionsofThirdOrderDifferenceEquations,

preprint.

5. SMITH,

B.

and

TAYLOR,

JR., W.

E.,

OscillatoryandAsymptotic

Behavior

of Certain Fourth Order

DifferenceEquations,Ro:ky Mt. J.Math., 16

(1986),

403-406.

6. SMITH,

B.,

Oscillation and Nonoscillation Theorems for Third Order Quasi-Adjoint Difference

Equations,Port.Math., 45

(1988),

229-243.

7.

TAYLOR,

JR.,

W.

E.,

OscillationProperties of FourthOrderDifferenceEquations,Port.Math.. 45