On the global solvability of a class of fourth order nonlinear boundary value problems

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Internat. J. Math. & Math. Sci. VOL. 20 NO. 2 (1997) 257-262

257

ON

THE GLOBAL SOLVABILITY OF A CLASS OF

FOURTH-ORDER

NONLINEAR BOUNDARY VALUE PROBLEMS

M.B.M. ELGINDI

Department

ofMathematics

University ofWisconsin-EauClaire EauClaire,Wisconsin 54702U.S.A. elgindmb@uwecedu

ZHENGYUAN GUAN

Department

ofMathematics

University ofWisconsin-EauClaire EauClaire, Wisconsin54702U.S.A guanz(uwec.edu

(ReceivedMay16, 1995)

ABSTRACT. Thispaperis concernedwiththeglobalsolvability ofaclass of fourth-ordernonlinear

boundaryvalueproblemsthatgovern thedeformation ofan elastic beam which is actedupon byaxial

compression, lateralforces and is in contact with a semi-infinite medium acting as a foundation For

certainranges oftheacting axialcompression force, thesolvability of theequations follows from the

coerciVity oftheir linearparts. Beyondthese rangesthis coercivityis lost Itis shown herethatthe

coercivitywhichensurestheglobalsolvabilitycanbegenerated bythenonlinear partsof the equations for

acertaintype of foundation.

KEY WORDS AND PHRASES: Global solvability, fourth-ordernonlinearboundaryvalueproblems, homogeneous nonlinearity, Leray-Schauderfixedpointtheorem, coercivity

1991AMSSUBJECTCLASSIFICATION CODES: 49G99, 73H05, 73K15.

1. INTRODUCTION

Inthispaperwe areconcernedwiththe global solvability of the fourth-ordernonlinearboundary

valueproblemswhichgoverntheequilibriumstatesofabeam-column. The source ofthenonlinearity

comes from a nonlinear lateral constraint (foundation). The equilibrium equationis formulated as a

fourth-order nonlinear differential equation. Differentboundary conditions, corresponding tovarious

waysinwhichthe endsof the beam maybesupported,willbeconsidered. Theprogfofthe existence of solution isbased uponacorollary ofLeray-SchauderFixed PointTheorem,which we will state in Section 2ofthispaper,togetherwithanidea whichoriginatedin[2].

Existenceof solutions of theboundary valueproblemsconsidered in thispaperhasbeen thesubject of severalrecent papers. The reader is referredto [1], [3] and [4] and the referencestherein foran extensive account onthe subject. Inalltheseworks the necessary coercivity condition,whichensuresthe existenceof solutions, was derivedfrom thelinearparts of the equations. Since thiscoercivityislost beyondcertain critical valueofthecompressive force, thesepapersfailedtoobtainanyexistence theorem

of globalnature. Inthispaper,onthe other hand,thecoercivityisgenerated bythe nonlinear partofthe equationsand the existenceresultswe obtain areglobalin nature.

Following the differential equationwhichgoverns the lateral displacementy(x)is

Y’’

+

,kS/"

+

ky

-+-

C(x,

y,

y’,

y")

f(x),

(1.1)

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M B M ELGLND! AND Z. C-UAN

u(o)

u"(o)

u()

u"()

o;

u(o)

u"(o)

u()

u’()

o;

y(o)

y"(o)

y’(1)

y’"(1)

o;

y(O)

y’(O)

y(1)

y’(1)

O;

(o)

’(o)

"()

’"()

o;

(o)

’(o)

’(1)

’"()

o,

(1 2) (13) (1 4) (1

5)

(16) (1 7)

whichrepresentthefollowingcases: bothends aresimply-supported,oneendissimply-supportedand

theother isfixed;oneendissimply-supported and the otherisslidingclamped;both endsarefixed;one

endisfixedand the otherisfree and one endisfixedand the otherisslidingclamped.

Therestof thispaperis organizedinthree sections. InSection2 we statethe conditions and the

Lemmas,

on which the proofofthe main result of this paper will be based, and we obtain some

preliminary results In Section 3 we state andprovethe main result of the paper. In Section 4some resultsconcerning the uniqueness of thesolutions are obtained.

2. ASSUMPTIONS AND PRELIMINARY RESULTS

Throughout therestofthispaperwewill usethe following notations

Wk

(y:

[0,1]

R:

yO)

e

AC[O, 1],

j O,l,...,k 1and

y(k)

e

L2(O,

1)},

L2’

YEWk’

3=0

D(L,)

{y

W4:

y satisfies the ithboundaryconditions(1.i), 2,3

,7,

L,

D(L,)

L2(0,1)

isdefinedbyL,(y)

y’"’.

Wemake the following assumptions.

H().

I

e

L(0,

H(2). G(x,y,y,y

")

g(y)

/h(x,y,

ff,y"),

whereboth g and h arecontinuous, and themap H

[0,1]

W --,

L2(0,1)

defined by H(x,y) h(x,y,y,y

")

is continuous Furthermore, we assume:

a. thereexists p

>

1such thatg(rx) rPg(x),forr, x E

R

withr

>

0; b. for any y W

2,

fg(y)ydx

>_

0; and

f0

g(y)ydx

0iff y 0;

.

]h(,,’,")4

>_

0,

e W

.

Theproofofourmainresult of thenextsectionconsistsof verifyingthe conditionsofacorollaryof

Leray-SchauderFixed Point Theorem which we statehereasthe

foil?wing

lemma.

LEMMA

2.1.

Let

B

beaBanach xpace and

K B

--

B

beacompact operator.

Suppose

that

there exists a priori bourn1m

>

0 such that every solutmn

of

y tKy O,

for

[0,1], satisfies

[[YI[

<-

m. Then

K

has afixedpoint ywith

[lY]] <

m.

Wecollect some preliminary resultswhich we will use in Section 3 in thefollowinglemma.

LEMMA

2.2. Foreach

L

3,j 2, 3, 7, the followingaretrue:

A.

La,

as anoperatoron

L

(0,1),

isdensely

defined

andself-adjoint;

2. 7r2

B.

IIilz,

-<

CXllff’ll,

for

D(L,),

U

C --7Y;

C3-

-;

C4-

,

C

--7Y;

C-"

and

CT

"

c.

/or

any

u

D(L,),

L,U

0

ff

U

O:

D. there existunique

3"

L2(

O,

1)

W such that

L3(b(h))

h

for

any h

L2(0,1),

and

3

:L2(

0,1)

--

W isbounded;

E.

K’--

L2(0,1)

W

defined

by

-3

i.

q23,

where

D(L3)

-,W denotes theidenutymap, iscompact.

The proofsof

(A)-(E)

are directandarethereforeomitted. Forsome of theestimates in(B),one

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GLOBAL SOLVABILITYOFFOURTH-ORDER NONLIENARBVP 2 59

3. GLOBAL EXISTENCE OF

SOLUTIONS

Inthis section we consider theglobal solvability ofthe sixboundaryvalueproblems consisting ofthe differential equation(1.1) and one ofthe six sets of boundaryconditions

(1.2)-(1

7) in thefollowing theorem.

THEOREM3.1. Under the assumptzonH(1)andH(2), theboundaryvalueproblem conszstmg

of

(1.1)and(l.j),j 2,3, 7,hasatleast one solution

for

each k

>_

0 and eachA

>_

O.

PROOF. Theboundaryvalueproblem (1.1), (1.j),2

_<

<

7,canbewrittenas

y--

Ky

(3 l)

where

K.y

K--:[Ay" +

ky

+

a(x,

y,

y’,

y")

f(x)],

K

W W is compact,and isas inLemma2.2 We provetheexistenceofa solutionof(3 l) by verifying theconditionsofLemma2.1.

Assume that the solutions of y-

tKy

are not uniformly bounded with respectto

[0,1]

Thenthereexistsequence

{. }

C

(0, 1)

and

{y,

}

CW such that

y,

,K.yr,,

n

>_

1 (3 2)

and

Ilyll2

ooasn oo.

From (3.2),itfollowsthateach y,satisfies

"

+

a(,

,

’,

)

f(/,

withYnE

D(Lj),

which in turnimplies(uponmultiplying bothsidesof the

e.luation

byyn,integrating by parts and using the boundaryconditions)

II"ll

" az

")dz

,

laz.

Set z, then

{z,}

CW is a bounded sequence, and since a bounded set ofW isweakly

-II’

relatively compact,itfollows that there existsasubscquenceof

{

z

},

which wecall

{ z },

thatconverges

weaklyinW

.

Bythe factthat the imbeddingi"

D(L)

CW

C[0,1]

iscompact,itfollowsthat thereexists asubsequence of

{ z,},

whichwecall

{

z,

}

again,thatconverges stronglyinC

[0,1]

tosome

0

c

[0,

]

From (3.3)andassumptionH(2)weobtain

A"

"

(3 4)

<

t,,C IlY,llL,

+

Using(3.4)and homogeneityof g we obtain(sincep

>

1)

,Xll’

IIz,

0

<

()zd <

Cll.llT

+

11.llT

0 ( )

asn oo Since g is continuous, itfollows from

(3.5)

that

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M.BM ELGINDI AND Z GUAN

which, in viewof assumptionH(2)(b),impliesthatz0 0, and

z

--

0 inC

[0,

1].

Onthe otherhand, from(3.3)wehave

IIfIIL21IYIIL2

whichimplies that(bythefact

z

--,_{0 in}_C

[0,1])

0. (3 6)

However, (frompart(B)ofLemma2.2and

[lY’IIL_

[lYI[L2

for yE

D(L))

wehave

Ilyllg

Ilyll

+

Ily’ll

<

(2

+

c

-2)

IIIIL,

"

and this contradicts(3.6). Thiscompletestheproof

4. UNIQUENESS

Assuming that

G(z,

y,

y’,

y")satisfiesthecondition

H(3).

fd[G(x,y,y’,y")-

G(x,z,z’,z")](y- z)dx >

0, forall y, z

e

W

2,

we obtainthe followingresult on theuniqueness ofthe solution.

TitEOREM 4.1. Assume H(3), the solution

of

the boundary value problem (1.1) and (l.j),

k

V/o

2

<

j

<

7,tsunique,provided thatk

<

C2

andA

<

C+

ork

>

C

andA

<

2

PROOF, Let and zbetwo solutionsof theboundaryvalue problem. Setw zand assume thatw

:

0 wsatisfiesthe equation

w"

+

aw"

+

w

+

G(z,

u,

u’, u")

G(z,

z,

z’, z")

0 (4 )

and theboundarycondition(1.j). Let

A

Ilw"llL

wandintegrating by parts, usingtheboundary conditions, Htlder’s inequality andH(3),weobtain

A

2-AAB+kB

<0

If k

<

C

and

A

<

C

a+

,

wehave

A

AAB

+

kB

>

A

C

+

AB

+

kB

A(A

CaB

(A-

C,B)(A-and

B

[IWllL2.

Upon

multiplying equation(4.1)by

>0,

sincew

D(L),

A >

CaB,

andk

<

C.

This contradictsthe inequality(4.2)

If k

>

C

and

A <

2

V/,

wehave

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GLOBAL SOLVABILITYOFFOURTH-ORDER NONLIENARBVP 261

A

AAB

+

kB

>0,

sinceA

<

2X/

impliesk

()2

>

0 Thisagaincontradicts

(4.2).

Thusw 0 Thisprovesthetheorem

REMARKS. 1. Fromtheproof of Theorem3 1,wecan seethat theassumptionsk

_>

0 andA

_>

0 arenotneeded. Itisduetothephysicsnatureoftheproblem,weassume k

>

0 andA

_>

0.

2. When the foundationofbeam isnotuniform,kcoulddependonx, say,k

k(x).

Assume

is continuous. Theorem 3 is stilltrue. Let

km

min(k(x),

z E

[0, 1]},

Theorem4.1 isalso true,with kreplaced byk,.

3. With a trivialmodification oftheproof,we canreplace assumptionH(2)

()

by the following

condition

H(2) c’

f

G(z,

y,

y’,

y")ydx

>_

0and

f

(,

,

’)/d

0 as

Ilyll

IIII

4. It is quite clear that most ofthe functions G which are of interest physically satisfy our

assumption

H(2)

(a), (b) and (c) or

(c’),

and

H(3).

For example, G(x,y,y’,y")=

y3

satisfies all these assumptions. More generally, G(x,y,y,y

")

c3y

+

csy

+ %y’: +

+

c2,+y’/ with

G(x,

y,

!/’)y

_>

0and c_,,+

>

0,(care constants, 3, 5, 2n

+

1)satisfies ourassumptionsH(2) (a), (b) and

(d),

if we takeg(y) c,,+y

+.

Ifwefurtherassume

c _>

0for 3, 5, 2n 1,G also satisfiesH(3).

REFERENCES

[1 ELGINDI, M.B.M.and

YEN, D.H.Y.,

Ontheexistenceof equilibriumstatesofanelastic beam on a nonlinearfoundation,Internat.J.Math.

&

Math.Sci. 16,1(1993), 193-198.

[2] GUAN, Z., Solvability of semilinear equations with compact perturbations of operators of

monotonetype,Proc. Amer.Math.Soc.121, 1(1994),93-102.

[3]

GUPTA, C.P.,

Existence and uniqueness theorems for the bending of anelastic beam equation, Applicable Analysis26(1988),289-304.

[4] GUPTA, C.P., Existence and uniqueness results for the bending ofan elastic beam equation at resonance,J.Math. Anal.Appl.135(1988),208-225.