Internat. J. Math. & Math. Sci.
VOL. 11 NO. 2 (1988) 275-284
275
SOLVABILITY OF A FOURTH ORDER BOUNDARY VALUE PROBLEM
WITH PERIODIC BOUNDARY CONDITIONS
CHAITAN P. GUPTA
Department of Mathematical Sciences
Northern Illinois University
DeKalb, IL
60115
(Received May 28, 1987 and in revised form July 16, 1987)
ABSTRACT.
Fourth order boundary value problems arise in the study of the equilibrium
The author earlier investigated the exis-
of an elastaic beam under an external load.
tence and uniqueness of the solutions of the nonlinear analogues of fourth order boundary
value problems that arise in the equilibrium ofanelastic beam depending on how the ends
of the beam are supported.
This paper concerns the existence and uniqueness of solu-
tions of the fourth order boundary value problems with periodic boundary conditions.
KEY WORDS AND PHRASES.
fourth order boundary value problem, periodic boundary conditions,
linear eigenvalue problem, Leray-Schauder continuation theorem, equilibrium of an
elastic beam, non-trivial kernel.
AMS SUBJECT CLASSIFICATION:
34B15, 34C25
I. INTRODUCTION
Fourth order boundary value problems arise in the study of the equilibrium of an
elastic beam under an external load, (e.g., see
[I], [2], [3]) where the existence,
uniqueness and iterative methods to construct the solutions have been studied extensively.
The purpose of this paper is to study the fourth order boundary value problem
with periodic boundary conditions:
---+
dx
d4u
f(u)u’
u(O)
u(2)
g(x,u)
e(x),
u’(O)
u’(2)
u’"(O)
where
-->
f-
is continuous and
-
conditions with
e
[0, 2],
x
u"(O)
u’"(2n)
g"
u"(2)
(1.1)
O,
[0,2n]
x R
-->
satisfies Caratheodory’s
1
L [0,2].
We note that the fourth order linear eigenvalue problem
d4u --u,
(1.2)
dx
u(0)
has
n
4
n
u(2;[)
u’(0)
O, 1, 2
since the linear operator
u’(2n)
hu
d4u
=--.
dx
with
u"(0)
u"(2n)
u"(0)
u’" (2n)
O,
Now the problem (1.1) is at resonance
D(L)
{u H 3 (0,2) u(O) u(2),
as eigenvalues.
276
C.P. GUPTA
u"’(2)l
u"(2), u’" (0)
u’(2), u"(O)
u’(O)
has a non-tirival kernel.
H3(0,2).)
of this introduction for the definition of
x)dx
value problem (1.1) has at Ieast one solution if
stant
0
>
0
g(x,u)u_.>
such that
0
lul >_. 0-
for
(See end
We shall prove that the boundary
0, and there exists a con-
To prove the existence of a
solution for the boundary value problem
d4u
dx
u’(O)
u(2n)
u(O)
u’" (0)
x [0 2,t]
e(x)
cu’ + g(x u)
4
u’(2n)
u’" (2n)
u"(O)
(.3)
u"(2n)
O,
we also need to assume that
g(x,u)
lim sup
B
< I,
uniformly for
[0,2].
x
a.e.
of the linear eigenvalue problem (1.2)
This is because the second eigenvalue
g(x,u) in (1.3). The question of asymptotic cong(x,u) in (1.3) can interact with infinitely many
interferes with the non-linearity
ditions in which non-linearity
eigenvalues of the eigenvalue problem (1.2) will be studied in a forthcoming paper [4].
To obtain the existence of solutions for (I.I) and (1.3), we use Mawhin’s version
of Leray Schauder continuation theorem as given in [5], [6], [7].
a,
f
case
We also show that in
is a constant, any two solutions of the boundary value problem
a
where
(I.I), (respectively, (1.3)), differ by a constant and have a unique solution when, for
example, g(x,u) is strictly increasing in u for a.e. x in [0,2n].
We note that in addition to using the classical spaces C([0,2]), ck([o,2n]), and
Lk(0,2)
and
L(0,2)
of continuous, k-times continuously differentiable, measurable
real-valued functions whose k-th power of the absolute value is
Lebesgue integrable or
measurable functions which are essentially-bounded on [0,2] we shall use the Sobolev
H3(O,2n) defined by
H3(0,2) {u:
space
[0,2]
u’"
Also for
LI(0,2)
u
u, u’, u" aDs. cont. on [0,2],
@
L2(0,2)}.
we define
12
I
u
u(x)dx.
0
2.
MAIN RESULTS
Let
and let
X, Y
H
CI[0,2], Y LI(0,2) with usual
X
L2(0,2). Let Y2 be the subspace of Y
denote the Banach spaces
denote the Hilbert space
norms
defined by
Y2
and let
sum.)
YI
{u
Ylu
Y
be the subspace of
We note that for
u(x)
Y
u
a.e.
constant
such that
(u(x)
t)dt)
/
1
for
u
Y.
Clearly,
-f
12
Q
I
1
Y
[0,2]I,
(
YIY2.
denotes the direct
we can write
We define the canonicaI projection operators
Q(u)
on
/--.
P:
u(t)dt, x
[0,21.
0
-"1;
Q:
2
as folIows
2
u(t)dt,
u(t)dt,
0
and the projection operators
P,
P
where
and
Q
I
denotes the identity mapping on
are continous.
Now let
X
2
X
Y
Y2"
SOLVABILITY OF A FOURTH ORDER BOUNDARY VALUE PROBLEM
X
Clearly
)
X
X
such that
X.
is a closed subspace of
2
X
We note that
2.
HI H 2
H
Similarly, we obtain
Let
X
PIX"
X
277
be the closed subspace of
QIX"
-> X I,
X
and continuous projectlons
->
X
PII’H
X
are continuous.
2
-.>
QIIDH
H|,
-> H
2
X, Y, H, P, Q, etc. Wl!l refe to Banach spaces, H11bort space and
!n the tollowing,
the projections as defined above and we shall not distinguish between
P,
PIX PIH
QIH) and depend on tle context fo, poper meaning.
2
u(x)v(x)dx denote the duality
Y let (u,v)
Also for u
X, v
0
Qu,
Y where u Pu
X, v
pairing between X and Y. We note that for u
(esp. Q, Q]X,
/
v
Qv,
Pv
we have
(Pu,Pv)
(u,v)
{u
H
3
by setting
u’(2),
u(2), u’(O)
u"(2), u’" (0)
(2)}
u’"
(2.)
D(L),
u
d4u
Lu
dx
Now, for
([8])
X ->Y
(0,2)lu(O)
u"(O)
and for
D(L)
L-
Define a linear operator
D(L)
(Qu,Qv).
--
we see using integration by parts and Wirtinger’s inequality
D(L)
u
that
42.2)
4
d4u
(Lu,u)
0
LEMMA 2 I"
[2 u ,,2 dx
udx
]0
dx
For a given
h
and
a
2
[(Pu)(x)]2dx
>
--30
YI
i e
h
> O.
I
L (0,2)
42.3)
with
h
O,
Qh
the linear boundary value problem
d4u
+
au’
[0,21],
h(x), x
(2.4)
dx
u(x)
has a unique solution
with
2
i Sin
Cos--
PROOF’- Let us set co
2 I/3
(
are the three cube roots of
co
I
(x)
v3(x)
Then
u(x)
C2e
C
O.
Qu
u
2n
a
]R.
x
h(t)dt,
v2(x)
e
-a
-al/3JxlX" v2(t)ea
and the condition
u
For
i/3
1/3 2
co
[0,2],
x
cox
I
tdt,
so that
x
(t)ea
-a
1/3
-
1/3
x
cox
C3e-n
+C4e
-a
CI, C2, C3,
I
f2
j0
C
4
u(x)dx
a
2
C
3
C4 C2e-(X
I/3
2
1/3
t’td t,
v(x)= e
-1/3x
1/3 2
co x + v(x)
I
x
e
l/3t
0
is such that
v3(t)dt.
Rl(u(x))
using the boundary conditions in (2.4) and
O.
C2, C3,
C
4
are computed uniquely from the
three linearly indpendent equations
C
coa
define
we
1/3
1/3
0
0
1
is a general solution of the equation (2.4).
Next, we compute
/Z]--,
i
0
e
u’" (2n),
u"(2n), u’" (0)
u’(2n), u"(O)
u(2n), u’(O)
u(O)
1/
C e -(
3
32co
1/
C4e
-(
32rco2
v(2)
278
C.P. GUPTA
C
L0C
2
C2 2C 3
C
The constant
oC
YI’
h
C2e-@
4
toC3 e-a
II 32
2C3.e
- ].I
3202 -a-I/3v’(2),
o2C4e-a
2t0
32
].13
0C4e
is computed uniquely using the condition
1
i.e.
LI(0,2)
h
with
I
h
12
2w
z
+
a
-2/3
v,,( 2).
-02u(x)dx
1 [
u
as the unique solution for (2.4).
R1 u(x)
In this way we get
For
32
2C4 C2e-a
+
3
O.
H
h(x)dx
O; let
Kh
u
be
0
the unique solution of the problem
d4u
dx
h(x)
4
such that
->
YI
K"
u
/
u(t)dt
0
Now define
a
X.
u
r
g
for each
[0,2].
such that
,
-->
[0,2]
g:
[0,2hi
(x,u) --> g(x,u)
g(x,.) is
and
u
Assume, moreoever, that for each
Ig(x u) < (x) for a.e. x
[0,2]
r
f(u(x)) u’(x)
g(x,u(x)), x e [0,2n],
It follows easily from Arzela-Ascoli theorem that
e(x)
(2.5)
will be said to satisfy Caratheodory’s conditions.
a well-defined compact mapping and
For
(KP(u), P(u)) > O.
-> Y by setting
X
N:
el(0,2)
Such a
(Nu)(x)
for
x c
for almost each
[-r,r].
u
P(u), and
is measurable on
there exists an
and all
Y,
u
be continuous and let
g(.,u)
continuous on
>
It is immediate that the linear mapping
D(L), LK P(u)
-->
f:
be such that
r
O.
0
is bounded and for
XI
u"’(2n),
u"(2n), u’"(O)
2
KP(u)
Let
u’(2n), u"(O)
u(2n), u’(O)
u(O)
1
[0 2n]
x
Ll(0,2n),
Y
QN:
X
-> X 2
KPN"
X
->
X
1
is
is bounded.
the boundary value probelm (1.1) now reduces to the
functional equation
Lu + Nu
in
X
with
(2.6)
e,
Y, given.
e e
THEOREM 2.2"-
Let
->
f"
be continuous and let
and
R
with
a
< A
and
r
<
< R
0
g"
->
[0,2n]
Assume that there exist real numbers
satisfy Caratheodory’s conditions.
a, A, r
such that
g(x,u) > A,
for
a.e.
x
[0,2]
(2.7)
and all
u
< R;
and
g(x,u) < a,
for a.e.
x
[0,2n]
(2.8)
and all
<
u
least one solution for each given
a
<
PROOF’- Define
e
I
L (0,2)
by
e
<
Then the boundary value problem (1.1) has at
e
1
L (0,2n)
with
A.
gl:
el(x)
-
r.
[0,2]
x ]R->
e(x)
I (a
by
A),
gl(x,u)
so that, for
g(x,u)
a.e.
x
(2.9)
(a
A)
[0,2],
and
SOLVABILITY OF A FOURTH ORDER BOUNDARY VALUE PROBLEM
(2.7), (2.8), (2.9)
using
gl(x’u)
we have
I (A
>
1
gl(x,u) _<
279
(a
a)
>
0
if
u
>_
R,
(2.1o)
A)
_<
0
if
u
_<
r,
(2.11)
and
I
<71 <- (A
A)
a
(2.12)
a).
Clearly, the boundary value problem (I.I) is equivalent to
d4u
dx
Y
X
N"
for
X.
u
QN"
l(x)
[0,2]
x
(2.13)
u"’(2n).
u"(2n), u’"(O)
be defined by
(Nu)(x)
’mapping.
e
u’(2n), u"(O)
u(2), u’(O)
u(O)
Let
u(x))
gl(x
f(u)u’
4
gl(x,u(x)),
f(u(x))u’(x)
KPN"
We then see, as above, that
X
->
X
->
(2.14)
[0,2I],
x
X
is a well-defined compact
1
is bounded and the boundary value problem (2.13) is
X
2
equivalent to the functional equation,
Lu
Nu
(2.15)
el,
Y. Setting, e
KPe, we see that to solve the functional
with e
in X
I
I
equation (2.15) it suffices to solve the system of equations
Pu + KPNu
eI
QNu
u
X.
Indeed, if
is a solution of (2.16) then
X
u
(2.16)
el,
LKPNu
Lu
PNu
QNu
e
Qel,
which gives on adding that
Lu
LPu
I
Le
Nu
D(L)
u
and
Pe I,
I
e
I
Now, (2.16) is clearly equivalent to the single equation
Pu
QNu
KPNu
e
I
+
(2.17)
el,
which has the form of a compact perturbation of the Fredholm operator
zero.
We can therefore apply the version given in [6]
[5] (Theorem IV.4)
or [7]
P
of index
(Theorem I, Corollary I) or
of the Leray-Schauder Continuation theorem which ensures
the existence of a solution for (2.17) if the set of solutions of the family of
equations,
Pu
(I-lu
is, a priori, bounded in
X
IQNu
%KPNu
I I’
e
by a constant independent of
(0,I),
I.
Notice that
(2.18)
(2.18) is
equivalent to the system of equations,
Pu
%e I,
%KPNu
(l-%)Qu + %QNu
Let for
(0,I),
u%
X
(2.19)
l
el,
(0,I).
be a solution of
(2.19)
so that
280
C.P. GUPTA
Pu
AKPNu
A
(l-),)Qu
Ael,
A
AQNu
A
Ae
A
(2.20)
1
The second equation in (2.20) can now be written as
(l-)t)
uX(x) _.>
So, if
"
R for
0
we have, using (2.10), (2.12) that
[0,2n]
x
gl(X’u/t(x))dx
u)t(x)dx
0
0
< (1
A) R
/’(A- a)_!-ff
0
< (1
A) R
< O,
(A- a),
i.e.
uA(x)
Similarly if
exists a
T
A
r
Now, for
for
r
[0,2]
x
<
a contradiction.
[0,2]
x
leads to a contradiction.
Hence, there
such that
<
uk(T A)
(2.21)
R.
[0,2]
we have
u),(x)
u(),)
x
/ u
(s)ds,
so that
(R,-r)
max
1/z
(u (s))Zds)
(2)1/2(
0
(R,-r)
max
(u(s))2ds) 1/2
(2)I/2(
0
since
u%
D(L),
Wirtinger’s inequality applies.
CI,
for some constants
C
2
independent of
Thus,
(2.22)
.
Next, the first equation in (2.20) gives that
LPux ALKPNu%
%Lel,
i.e.
Lu
A
+
APNu
A
APe
(2.23)
1
From (2.23) and the second equation in
(LUA,PU A)
(2.20), we get
A(PNUA,PU A) A(PeI,PUA)
(I-A)(QUA,QU A) + A(QNUA,QU A) A(’I, QUA).
+
We next note that our assumptions on
constant
C3,
indpendent of
(Nu,u)
and
(LuA,Pu A)
_>
A
gl
and
such that for
(2.24)
(2.10), (2.12) imply that
u
there is a
X,
-C 3,
(LuA,uA)
(u)2 llull H
0
get on adding the equations in (2.24) that
2
since
(2.3) holds.
Using this we
281
SOLVABILITY OF A FOURTH ORDER BOUNDARY VALUE PROBLEM
-< (Lu%,u%) (l-%)(Qu,Qu%) k(Nu%,u%)
3
%(Pel,PU %)
_<
C
where
C4C lllull H C4C 2,
is a constant independent of
4
independent of
%,
%(el,QU %)
+
Accordingly, there is a constant
%.
C
5,
such that
% H--
5’
which implies, using (2.22) that
c2
Iluxllx ClC fi
m C.
We have thus proved that the set of solutions of the family of equations (2.18) is bounded
in
X
REMARK 2.3:-
If we take
(0,I).
X
by a constant independent of
A
a
Hence the theorem.
H
in Theorem 2.2, then we immediately obtain the
0
assertion made in the introduction concernlng the boundary value problem (I.I).
Now, to study the boundary value problem (1.3) we define, for a given
’linear
D(L
L
operator
H3(0,2)
{u
D(L
a
X
Y
a
u"(2),
u’(2), u"(O)
u(2), u’(0)
u(0)
(2.25)
u’"(2))
u’"(O)
and for
,
a
by setting
D(L ),
u
d4u
L u
a
dx
+
4
(2.26)
au’
It follows, using integration by parts and Wirtinger’s inequality, ([8]), that
12 --d4u
u)
(L u
a
0
2
dx
[2n
u’u dx
0
(u")2dx
>
12 (d4u)2
dx
0
0
>-
a
udx
4
4
(2.27)
dx
2
IILull.
We, next, use lemma 2.1 to define a bounded linear mapping
u
K h
for a given
h
where
Yl’
u
X
(so that
u
K
Qu
"YI
X
by setting
O) is the unique
solution of the boundary value problem
d4u
dx
eu’
4
[0, 2],
(2.28)
u(2n), u’(O)
u(O)
h(x), x
The bounded linear mapping
u’(2n), u"(0)
Ka Y1
X
u"(2n), u’"(O)
u’"(Zn).
defined in this way has the following
properties"
(i)
for
u
Y,
KaP(u)
D(La), LaKaP(u)
(K P(u), P(u))
a
(ii)
if
g
[0, 2]
is defined by setting
>
,,,,lleull H2
P(u)
and
(in view of (2.27))"
satisfies Caratheodory’s conditions and
(2.29)
N
X
Y
C.P. GUPTA
282
g(x,u(x)), x
(Nu)(x)
KPN
then
X
Theorem 2.4-
<
< R
0
is a well-defined compact mapping and
e
Let
be given and
[0, 2]
g
QN
X
X
is
2
bounded.
satisfy Caratheodory’s
x
a, A, r, R
Assume that there exist real numbers
conditions.
r
X
[0, 2l
with
a
< A,
and
such that
g(x, u) > A,
for a.e.
[0, 2],
x
(2.30)
and all
_>
u
R;
nd
g(x, u) < a,
for a.e.
[0, 2],
x
and all
uniformly for a.e.
least one solution for each given
Pr.oof--
-
<
<
e
Su-ppose, further, that
r.
8 <
(2.32)
Then the boundary value problem (1.3) has at
[0, 2].
x
<
u
’’Ig(x’u)l
lim sup
a
(2.31)
2
L [0, 2]
e
(2.33)
A.
As in the proof of Theorem 2.2, define
(a
g(x,u)
x
[0,2],
A)
and
L2(O,
e
with
2w)
[0, Z]
gl
gl(x,u)
by
x
(a +A).
e(x)
el(x)
by
Then for a.e.
gl(x,u) >
(A
a)
>
0
if
u
_>
R,
(2.34)
gl(x,u) _<
(a
A)
_<
0
if
u_<
r,
(2.35)
lim sup
gl(x’u)
lul*=
u
8 < I,
(2.36)
a)
(2 .37)
uniformly, and
1
(a
A)
71
<
<
(A
Also the boundary value problem (1.3)
d4u
dx
(u’
4
N
u(2n), u’(O)
X
Y
for
u E X.
[0, 2],
u’(2), u"(0)
u"(2), u’"(0)
gl(x,u(x)),
>
Choosing, now,
0
x e
C(e)
>
0
such that
8
X.
is bounded.
Also,
< I, we see, using the fact that
(2.35), (2.36), that there exists
such that
(Nu,u)
u
u’"(2n).
[0,
satisfies Caratheodory’s conditions and (2.34),
a constant
for
x e
be defined by
Nu(x)
g,
el(x)
(2.38)
u(O)
Next, let
gl(x,u)
is equivalent to
K PN
2
I --e IINull H
X
X,
(2.39)
C(E),
is a well-defined compact mapping and
QN
X
X
2
SOLVABILITY OF A FOURTH ORDER BOUNDARY VALUE PROBLEM
283
Again, we see as in tae proof of Theorem 2.2, that the boundary value problem
(2.38) is equivalent to the system of equations
Pu
$
K PNu
&
KaPe
(2.40)
"’1"
QNu
Further, it suffices to prove that the set of solutions of the family of equations
%K PNu
a
Pu
(I -%)Qu
is, a priori, bounded in
Let, now, for
X
Pu
(I-
(2.41)
I’
%QNu
(0,I)
by a constant independent of
(0,I),
%
he
e X
u%
kKaPNu
I
X)Qu%
(0,I).
%
be a solution of (2.41)
so that
kel,
(2.42)
-I"
%QNu%
It, now, follows from the second equation in (2.42), in a manner similar to deriving
the estimate (2.22) in the proof of Theorem 2.2, that
lieu xll+/-llu xll x_<
for some constants
C I, C
C
IlLau.JJ
c2
(2.43)
(0,I).
independent of
2
.
Also, we have from (2.42) that
(Pu
k,
x)
PNu
k)
(i, PNuk )’
X(KaPNuk, PNu%)
I qu x II 2
k(71, qu k).
x(qu x, QNu k)
These equations then give us, in view of (2.29) and
S+--7
(2.39), that
2
c(e) _< (Nux,u x)
IINuXlIH2 IIPNuxlIH-
--< (I’
IIPvll H _< llvll H
Using, now, the facts that
these exist constants
C
3,
C
4
for
independent of
%(K(PNux, PNu%)
PNux)
X,
v
(0,i)
%
1/2
IlNu xll _< c 3 Ilq x II
C
(-I’
and
QuA )"
+ e
< I,
we see that
such that
(2.44)
4
Now, the first equation in (2.42) gives that
Lauk
PNu
kPel,
k
so that
IILaux II H i X IlPe
!
i
Iletl] H
3llquxll
PNu
x I1 +/-
IIPe
]l H
IIPNux II H
IINuxll H
1/2
C
4
IIPelll H"
(2.43) and (2.45) now imply that there exist a constant
such that
C5,
independent of
(0,I),
C.P. GUPTA
284
and
llul[I X < CIC 3
CIC 4
C
II[Pe III H C2
C.
This completes the proof of the theorem //.
in (1.3) is replaced by
The analogue of Theorem 2.4 when
Remark 2.5:-
f(u),
where
is a given continuous function will be treated in a forthcoming paper [4].
f
Remark 2.6:for a.e
If
,
f(u)
[0, 2]
x
exactly one solution.
is a
B < I,
in
x
g(x,u)
is strictly increasing in
u
Similarly if
g(x,u)
is strictly increasing in
u
and
there
such that
(g(x,u
for a.e.
given and
then it is easy to see that the boundary value problem (1.1) has
Remark 2.7:-
I)
u2 >
g(x,u2))(Ul
B(g(x,u I)
g(x,u2))2,
[0, 2], then the boundary value problem (1.3) has exactly
If we take
a
A
0
one solution.
in Theorem 2.4, we immediately obtain the assertion
concerning the boundary value problem (1.3) in the introduction.
cknowledgement:references
The author is grateful to the referee for his suggestions to add
[I], [2], [3]
to the bibliography and for editorial improvement of this paper.
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I.
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2.
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3.
USMANI, R.A.
4.
GUPTA, C.P. Asymptotic conditions for the solvability of a fourth order boundary
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5.
MAWHIN, J.
6.
MAWHIN, J.
7.
8.
9.
I0.
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