GENERAL BOUNDEDNESS THEOREMS TO SOME SECOND ORDER NONLINEAR DIFFERENTIAL

Internat. J. ^Math. & Math. Sci.

VOL. 18 NO. 4 (1995) 823-824

RESEARCH NOTES

GENERAL BOUNDEDNESS THEOREMS TO SOME SECOND ORDER NONLINEAR DIFFERENTIAL

^EQUATION

WITH INTEGRABLE FORCING TERM

823

ALLAN KROOPNICK

Office of Retirement

&

SurvivorsInsurance SocialSecurity Administration

3-D-21Operations Building 6401Security Boulevard Baltimore,

MD

21235 USA

(Received November 1,1991 and in revised form January 12, 1995)

ABSTRACT.

In

this note we present a boundedness theorem to the equation

x"+ c(t,x,x’)+ a(t)b(x)= e(t)where e(t)

is a continuous absolutely integrable function over the nonnegative real line.

We

then extend the resulttothe equation

x" + ^c(t,

^x,

:r’) + a(t, x) e(t).

^The

firsttheorem providesthe motivation for thesecondtheorem. Also,anexample illustrating the theoryis thengiven.

KEY

WOROS AND PHRASES. Integrableforcing term,bounded,nonlineardifferential equation.

1990AMSSUBJECT CLASSIFICATIONCODE. 34C11.

1.

INTRODUCTION.

In this article we shall discuss using standard methods the boundedness properties ofasecond ordernonlineardifferential equationwithintegrable forcingterm, i.e.the equation,

" ^{+ (,, ’) + ()b() () (.l)}

Our purpose here is to simplify some of the previous proofs tothis well-known equation aswell as extendingsome of the previous results. For example,wearereplacingthe condition

c(t,

x,

y)y >

0for y

:/:

0with

c(t,

x,y)y

_>

0andletting

a(t)

^benon-increasing

(see

^and

[2]

for details,especially

[2]

^for its excellent bibliography of previous

work).

Also, as in

[2]

we shall not need to make use of any Liapunovfunction. Finally, theresult will be of sucha naturethatitcovers the ease whennodamping factorappears,^{i.e.itcoversthe}equation,

x" + a()b(x) e() (1.2)

Laterweshall brieflymentionhow this result carries overtothemoregeneral nonlinearequation,

" ^{+ (, , ’) + (, ) () (.3)}

However,

this ease requires a more delicate discussion.

We

now state and prove the boundedness theorem. Without lossofgenerality,weshall assume t

_>

0.

2. MAIN RESULTS.

THEOREM L Given the differential equationin

(1.1). Suppose c(g,z,t/)

is continuous on

[0,03) ^R R, c(,x,t/)t/_>

^0and

e(o)

^{iscontinuouson}

[0, c)

^with

f le<)ld <

^oo. Furthermore, if

a(o)>_ >

0 for some and continuous on

[0, oo), a’(o)_<

0,

b(o)

continuous on

R,

and

B(x) fb()du

^{approachescas}

Iz[

^cthen all solutions as well as their derivativesare bounded as

---

^{PROOF. Byoo. standardexistencetheory,there isasolutionto}

⁽¹⁾

which exists on

[0, T)

for some

T >

O for anyinitial conditions

x(0)

^and

x’ _(0).

Multiplyequation

(1)

^{byz and}

perform

anintegration by partsonthelasttermfrom0 to

< ^T

ⁱⁿorder to obtain,

824 A.KROOPNICK

x’ _(t) /2 + ^{c(, z(s),} x’ (s))z’ (s)d + a(t)B(x(t)) a’ (s)B(x(s))ds

’(o/ + (/:’(/a _< z’(o// + I(l’(lld

^(.l

Nowif

x(t)

becomes unbounded thenwe musthavethat allterms ontheLHSof

(2.1)

becomepositive fromourhypotheses.

By

the mean valuetheorem, equation

(2.

l) maybe rewritten as,

x’(t)2/2 + c(a,x(s),x’(s))ds + a(t)B(x(t)) a’(a)B(x(s))ds

(/o )

< x’(0)2/2 + ^Ix’()lg

^g

le(t)ldt,

⁰

< <

^t

(2.2)

Nowfrom(2.2) wesee that if’

Ixl

^{approachesoo thenso must}

^Ix’(t)l.

Otherwise, theLHS of’

(2.2)

becomes unboundedwhiletheRHSstays boundedwhich isimpossible. Also,as

la ^(t)l

^{approachesooso}

must

Iz()l.

Nowonanycompact subinterval choose t where

z(t)

isa maximum.

Integrate

equation

(1.1)

asbefore from0 totanddivideby :d

(t) (assume x(t) >

^{0, a}similar argument works forz

(t) <

⁰

onlythe inequalityisreversed)inordertoobtain,

x’ (t)/2 + 1/x’ (t) ( fot ^{C(S, x(s), x’ () )d + a(t)B (x(t)} fot ^{a’ (s)B(x() )d)}

< (x’(0)2/2 + Ix’()lK)/x’(t) ^(2.3)

Nowif

x’(t)

approachesoo then the LHS of

(2.3)

becomes unbounded while theRHS of

(2.3)

^stays boundedwhich isacontradiction. Thus,

Izl

^and

I’1

^muststay bounded on

[0, T).

^Astandardargument ([3, pp.

17-18])

now permits thesolution tobe emended on all of

[0, oo).

Asforequation

(1.3)

wemay multiplyitby

x’

andintegrateas beforeobtainingthefollowing,

’(t)/ + (,(),e())’()a + .’

_Jz(o)

^{(t, )d _’}

_J(o)

^Oa(s,

^dd

x’ (0) 2/2 + e(s)x’(s)ds. (2.4)

f0

Weseehere thataslongas

a(t, u)du

^oouniformlyint andx

- ^{a(t, x) <_}

^{0thenwemayusethe}

sameargumentas in our firsttheorem. Wenowstatethis finalresult.

TIIEOREM IL

Given equation

(1.3). Suppose c(t,x,y)

is continuouson

[0,oo)

x

R

x

R, c(t,

x,

y)y > O, a(t, x)

^continuous on

[0, oo)

^x

R

with x

a(t, x) <

O. Furthermore, if

fo:att ^u)du

^{oouniformly in t and}

^e(.)

is continuous on

[0, oo)

^with

fo ^{le(t)ldt <}

^{oo, thenall}

solutionstoequation

(1.3)

as well as their derivatives areboundedas oo.

EXAMPLE. Considerthenonlineardifferential equation,

x" + cx2’-lx +

^bx2n-

^exp( t) (2.5)

where t

>

0, c,b arepositiveand m,narepositive integers.

By

Theorem we seethat all solutionto equation

(2.5)

arebounded.

[]

[2]

[3]

REFERENCES

ANTOSIEWlCZ,

H.A.,

Onnonlineardifferential equations of second order with integrable forcing term,J.Lond. Math.Soc.,30(1955),64-67.

ATHANASOV, Z.S., Boundedness criteria for solutions of certain second order nonlinear differentialequations,

J.

Math. Anal.Appl., 123

(1987),

^461-479.

HALE, J.,

Ordinary

Differential

Equations, Interscience, New York, ^1969.