Internat. J. Hath. & Math Sci.
Vol. i0 No. 2 (1987) 361-371
361
TWO-POINT BOUNDARY VALUE PROBLEMS INVOLVING
REFLECTION OF THE ARGUMENT
CHAITAN P. GUPTA
Department of Mathematical Sciences
Northern Illinois University
DeKalb, Illinois 60115
(Received April 29, 1986 and in revised form July 28, 1986)
ABSTRACT.
Two point boundary value problems involving reflection of the argument are
The nonlinearity involved is allowed to cross asymptotically any number of
studied.
eigenvalues of the associated linear eigenvalue problem as long as those crossings
take place in subsets of sufficiently small measure.
KEY WORDS AND PHRASES.
Boundary value problems, reflection of the argument,
Wirtinger’s inequality, eigenvalues, asymptotic behaviour, Caratheodory’s condition’s
sets of small measure.
I.
INTRODUCTION.
This paper is devoted to the study of the two point boundary value problems
x"(t) + f(x(t))x’(t) + g(t,x(t),x(-t))
x(1)
x(-1)
O;
e(t), t
[-1,1]
(1.1)
and
x"(t) + g(t,x(t),x(-t))
e(t), t
x’(1)
x’(-l)
O;
where
is a continuous function,
f:
uniformly a.e.
ro(t)
+
rol L
r
rl(t)
[-I,I]
of
E
in
+
r(t)
having positive measure,
with
I
-
It is assumed that
(1.3)
r
r(t)dt < 2.
r
(t)
with strict inequality holding on a subset
LI(-1,1),
r
L(-I,I)
sufficiently small and in the case of equation (1.2)
Ll(-l,l)
is a function
x
is given.
where in the case of equaton (1.1)
to(t)
where
g:[-1,1] x
e
l(t’x’Y)
[-1,11, y
t
(1.2)
LI(-I,I)
r(t)x
satisfying Caratheodory’s conditions and
lim sup
[-I,I]
r(t)
and
[rllL1
is such
and
that
It should be observed that the linear eigen-value problem
x"
+ %x
x(-l)
has
X
n
0
x(1)
(1.4)
0
22
4
n
1,2,... for eigen-values and the linear eigen-value problem
--
362
C.P. GUPTA
x"
2 2
has
of
0
Xx
x’(-l)
x’(1)
(1.5)
0
n
0,1,2,... for eigen-values. Accordingly, the asymptotic behavior
X n---R-4
2
g(t,x,y) is related to
the eigen-value
for the problem (I.I) and the first
x
2
two-eigen-values
g(t,x,y)
lim sup
for the problem (1.2).
and
0
However, the exp.ression
is allowed to cross any number of eigen-values
n2 2
as long as
those crossings take place in subsets of [-I,i] of sufficiently small measure.
Differential equations with reflection of the argument represent a particular
case of functional differential equations whose arguments are involutions.
Important
in their own right, they have applications in the investigation of stability of
differential difference equations.
Initial value problems for equations with
A survey of results in this
involutions have been considered in numerous papers.
direction is given in [I].
However, research on boundary value problems for such
equations is developed yet insufficiently.
study of problem (I.i) in the case that
[-I,I]
in [2].
l x l
Wiener and Aftabizadeh initiated the
g(t,x,y)
and
0
f
is bounded on
The methods used in this paper are similar to the ones used
by Gupta-Mawhin [3] for periodic solutions of Lienard’s differential equations.
C([-I,I]),
mention that in addition to using the classical spaces
Lk(-l,l)
ck([-l,l])
We
and
of continuous, k-times continuously differentiable or measurable real
functions the k-th power of whose absolute value is Lebesgue integrable we use the
space
I(-l,l)
HI(-I,I)
H
defined by
{x"
x’
L
[-I,I]
2
x is abs. cont. on [-I,i] and
I
(-I,I)]
with the usual inner-product and the corresponding norm
2.
-
SOME PRELIMINARY LEMMAS.
LEMMA I.
LI(-I,I)
F
Let
be such that for a.e.
r(t)
6(r) > 0
6
[-1,1]
PROOF.
such that for all
x
of positive measure.
H1(-I,I),
Then,
x(1)=x(-l)=0 one has
r(t)x2(t))dt
([x’(t)12
Br(X)
[-I,I]
(2.i)
with strict inequality holding on a subset of
there exists a
t
I’I H
We first see, using (2.1) and the Wirtinger’s inequaIity, [4],
for all
x
HI (-I,I)
with
x(-l)
BF(x)
Br(x)
Moreover,
A
for some
0
,
0
x(1)
O,
that
(2.2)
0.
if and only if
x(t)
A cos ._t
2
but, then
2
(- r(t))x2(t)dt A2
2
(-
r(t))cos 2
tdt
20
-
TWO-POINT BOUNDARY VALUE PROBLEMS
F(t) <
so that by our assumption that
measure we have
A
x(t)
and hence
0
2
363
[-I,I]
[-I,I].
on a subset of
0
for
t e
Assume, now, that the conclusion of the lemma is not true.
{Xn}
sequence
in
HI(-1,1)
HI(-1, I)
in
IXnl H
<.
for all
i,
x (-I)
n
with
0
and
in
n,
Xn
x (I)
n
<--nl
BF(Xn
x
[(xn’ x)
]2
IXnl
<
H
2
IXIHI
and hence
m
H
n
n
x
n
(2.3)
x
and
(2.4)
1,2
n
HI(-I,I)
Now, we have from Schwarz’s inequality in
2
Then we can find a
such that
C([-l,l]),
in
x
for all
0
for all
HI(-1, i)
of positive
that
n
for all
2
[xl H 1"
Also (2.3), (2.4) imply that
IXn 12H I I_l
(l+F(t))x2(t)dt
and so,
Ixl H2 =<
(l+F(t)) x 2 (t)dt
which gives
Br(x)
x(t)
and hence
for
t e
O,
[-I,I]
by the first part of the proof of lemma.
(2 3)
a contradiction to the first equality in
0
,xn,Hl
0
$
Thus,
Hence the lemma is
proved.
LEMMA 2.
Let
for a.e.
t
F0 + F + F where F e L (-I,I), F e LI(-I,I) and F0(t)
[-I,I] with strict inequality holding on a subset of [-I,I] of
F
in
x(-1)
x(1)
6(F
Let
positive measure.
[6(r O)
We have
Sr(X)
f_
<[x’(t)] z
Using the fact that
Then for all
HI(-1,1)
x
with
Irl
r0()xZ<t))dt f_ r()xZ(t)dt
HI(-1,1)
f-l r(t)x2(t)dt.
and the Wirtinger inequalities, [4],
C([-l,l])
Ix’l L 2 (-1,1)
IxIH
I, we get
as well as lema
Br(X)
"
4
Irxl L
I’1 L 2 (-1,1) Ixl (-1,1)
I1 L 2 (-1,1)
a
6(r o)
2
Irt L -4’too
2
IxIH1 -7 IxlHt
Irl L
[(ro) 7
REMARK I.
be given by lemma i.
0
Br(x) >PROOF.
0)
2
4
4
2
-
Clearly the best value for
we have
2
Br(x)
Z
(-
’-’"
Irl L]
2
Ixlnt
,
2
6(0)
is
rl L
)IXIH
2
so that when
r0
0,
F
0,
C.P. GUPTA
364
for all
x
LEMMA 3.
Let
H(-I,I)
F
F0
x(1)
x(-l)
with
F
F=o
+
O.
Then for all measurable real functions
p
[-I,I]
on
I-l,1], all continuous f: R
I and all
continuous on [-I,I] and x(-l)
x(1)
0
on
-Jl
PROOF
Since
6(F
be as in Lemma 2 and
0)
be given by Lemma I.
F(t) a.e.
p(t)
x’ absolutely
such that
Hl(-l,1)
x
with
we have
x(t)(x"(t) + f(x(t))x’(t) + p(t)x(t))dt
H (-i,i)
x
L
L
absolutely-continuous and
x’
with
x(1)
x(-l)
O,
we get on integrating by parts that
x(t)(x"(t) + f(x(t))x’(t)
I [(x’(t))Z
f
>
P(t)x2(t)]dt
2
[(x’(t))
p(t)x(t))dt
F(t)x2(t)]dt
2
71rl L --Irl Lllxl H
[(ro)
in view of Lemma 2.
We observe that the main ingredient in Lemmas 1,2, and 3 is that the
REMARK 2.
x
HI(-I,I)
satisfy Wirtinger inequalities.
x
H (-I,I)
with
O, x’(1) + kx(1)
x(-l)
(or x’(-l)-hx(-l) =0, where h
>.
k
>.
0
is given,
0) Wirtinger type inequalities
can be obtained.
f E 0,
g(-,x,y)
I x l
exists
t
x, y e
Assume, moreover, that for each
er LI(-I’I)
[-I,I].
.
Ig(t,x,y)l <"
such that
and all
y e
er(t)
We say that such a
for a.e.
is continuous
r > 0
there
[-I,I],
in
t
be
x
g(t,-,’)
and
is measurable for each
for a.e.
[-r,r]
in
g- [-i,I] x
be a continuous function and let
f:
such that
x
where
EXISTENCE THEOREMS
Let
on
O,
is given, x(1)
0
hold, analogues of Lemma I, 2, 3, with
3.
Now, since it is easy to see that for
satisfies Caratheodory’s
g
conditions.
We consider the following boundary value problem
e(t),
x"(t) + f(x(t))x’(t) + g(t,x(t),x(-t))
x(1)
x(-1)
[-l,l]
t
(3.1)
0
We prove the following existence theorem for (3.1).
F LI(-I,I)
g(t,x,y)
< r(t)
lim sup
THEOREM I.
ixl
-
L==(-I
I)
in
4
Irl L
LI(-I,I)
F
x
and y
and
.
Suppose that
1
L (-I,I)
F0
(3.2)
+
]rlJ L
<
6(r0)’
where
6 (F O)
Let
that for a.e.
n
=7 [(ro) -7
t
in
[-I,I]
Irl L
all
w
x
with
F0
+
F
F 2 where
+
Fo(t)
-.
<
for a.e.
[-I,I] of positive measure and
is as determined in Lemma I.
Then (3 I) has at least one solution for each
PROOF
F
are such that
with strict inequality on a subset of
[-I,I]
t
[-I I]
t
uniformly a.e. in
L
such that
Assume that there exists
Irl L
Jx
]-
Z r
e
L (-I,I).
.
Then, there exists
and
y
r
> 0
such
TWO-POINT BOUNDARY VALUE PROBLEMS
g(t,x,y) < F(t)
Define
YI"
[-l,1]
x R x I
365
B
(3.3)
by
l
g(t,x,y)
Ixl
if
x
> r
g(t,rl,Y)
0 < x < r
if
"rl
(t,x,y)
r(t)
x= 0
,if
g(t,-rl,y)
if
-r
-r
< x < O.
Then, by (3.3)
.
(t,x,y) < r(t) + n
for a.e.
t
in
[-I,I]
(t,x,y)
e
x,y
and
[-I,I]
x
in
Now, the function
(t,x,y)x
x
h: [-1,1]
satisfies the Caratheodory’s conditions and the function
defined by
h(t,x,y)
g(t,x,y)
LI(-1,1)
a
is such that there is an
Jh(t,x,y)
for a.e.
t
in
[-1,1]
Yl(t,x,y)x
a(t)
x,y
and all
satisfying
in
.
(3.4)
Now equation (3.1) can be written as
x"(t) + f(x(t))x’(t) +
l(t,x,(t),x(-t))x(t)
x(1)
x(-l)
+
e(t),
h(t,x(t),x(-t))
(3.5)
0
We next apply Theorem IV.5 of Mawhin [5] to (3.5) in the manner applied by GuptaMawhin in [3] (see also Mawhin-Ward [6]).
To do this we need to verify that all
possible solutions of the family of equations
x"(t)
f(x(t))x’(t) +
+
h(t,x(t),x(-t))
x(1)
x(-l)
{(l-x)(r(t)+n)
+
l(t,x(t),x(-t))}x(t)
(3.6)
e(t)
0
are, a priori, bounded by a constant independent of X e [0,1]
Let, now, x(t) be a possible solution of (3.6) for some
(1-x)(r(t)+n)
for a.e.
t
in
[-I,I]
+
%-fl(t,x(t),x(-t))
e
[0,I]. Since, now,
n
we have on integrating by parts the equation obtained by
multiplying the equation in (3.6) by
-x(t)
x(t)[x"(t) + f(x(t))x’(t)
+
[x’ 2 (t)
r(t) +
C1([-1,1]).
in
and applying Lemma 3 with
+
h(t,x(t),x(-t))
{(1-)(r(t)+n)
+
too
replaced
%-(l(t,x(t),x(-t))}x(t)
e(t)]dt
{(1-x)(r(t)+n) + l(t,x(t),x(-t))}x2(t)
-{h(t,x(t),x(-t)) e(t)}x(t)]dt
366
C.P. GUPTA
->
[(r
’’112H
I>
,4-:-Ir L
1/21rl
o)
4
IZH
n]
z,
I1 L 1)lxl,
L
-(lall
L
II L Ix! H
2(IIL
[0,I]
such that
independent of
k
such that
is some constant independent of
X
[0,11.
It follows that there exists a constant
C
independent of
k
Ixlc,
Ixlc
H
L
and from (3.6) there is still another constant
C
L
Ixl
and hence
where
C2’
C
2
Thus equation (3.I) has at least one solution for each
e e
LI(-1,1).
([2], Theorem 3.4) is
The foIlowing result of Wiener-Aftabizadeh
an imediate
corollary of Theorem I.
COROLLARY 1.
I-I,1]
.
g: [-I,I] x
Let
be continuous and bounded on
Then the probIem
x"(t)
g(t,x(t),x(-t))
x(-1)
Xo,
x(1)
x
(3.7)
1
has at least one solution.
PROOF.
Define
h:
[-1,1]
g(t,x
h(t,x,y)
by
/
/
l/t
x
1-t
/
x 2, y
1-t
/
x
I/t
1
/
Xo).
Then (3.7) is equivalent to
x"(t)
h(t,x(t),x(-t))
x(-1)
x(1)
0
.
and (3.8) has at Ieast one solution by Theorem
for a.e.
[-1,1]
in
t
COROLLARY 2.
Let
y
and all
in
that
lim sup
F
in
t
[-I,I].
L (-I,I)
x
x(-l)
x(1)
e(t)
(-1,1).
Existence of solutions for boundary value problems
x(-)
+
e(t),
g(t,x(t),x(-t))
o
x’(1) + kx(1)
O, k
0
t
uniformly
be as in Theorem 1 and be such
0
e < L
0
g: [-I,1]
Then the boundary value problem
has at least one solution for each given
x"(t)
h(tx)
F(t)
f(x(t))x’(t) + g(t,x(t))
x"(t)
or
e
g(tx)
Ixl-
REMARK 3.
lim
be a continuous function and let
f:
satisfy Caratheodory conditions and let
uniformly for a.e.
I, since
[-I,I]
TWO-POINT BOUNDARY VALUE PROBLEMS
x"(t) + g(t,x(t),x(-t))
x’(-l)
hx(-1)
x(1)
e(t),
O, h
367
[-I,I]
t
0
0
can be proved in view of Remark 2; for then Lemmas 1, 2, 3 remain valid (with different
constants involved, of course) and they are the main ingredients in the proof of
and Theorem IV.5 of [5] continues to be applicable.
Theorem
We next study the boundary value problem
x"(t) + g(t,x(t),x(-t))
x’(-1)
Now, for x
x’(-l)
H (-1,1)
and
x’
x
O,
(3.9)
o.
x’(1)
for which
x’(1)
with
e(t)
x’
and
are absolutely-continuous on
[-1,1]
Wirtinger type inequalities are no longer available for
However, in this case Wirtinger type
as are needed in the proofs of Lemma 2.
x’
inequalities are available for
x".
and
x
This fact is exploited next to obtain an
existence theorem for (3.9).
LEt@CA 4.
Let
(-I,I), F
e c
possible solution
x(t)
c
LI(-I,I)
with
x’(1)
x’(-l)
LI(-I,I)
p
O.
Then for every
e(t)
x"(t) + p(t)x(t)
where
(t)dt
F
of the boundary value problem
with
(3.10)
0
I p(t)dtF, p(t) 0
P
[-I,I]
a.e. in
satisfies the
inequality
(IPROOF.
Let
p(t)
2
)Ix"IL 21elLllX"[ +e [e[L lx[e"
1
be as in the statement of lemma above and
solution of (3.10).
x(t)
Then multiplying the equation in (3.10) by
be a possible
x(t)
and integrating
by parts we have
p(t)x2(t)dt
x’ (t)dt +
Since, by our assumptions
p 112 (t)x(t)
and
p
/2 (t)
belong to
by Schwarz’s inequality
(fll- P(t)dt)(fll- P(t)x2(t)dt)
IP(t)x(t)Idt)
p(t) x (t)dt
and hence using (3.10) we have
[e(t)
x"(t)
On the other hand, since
x’(t)
P(t)x2(tldt
Idt)2 g
x’(-1)
x’(1)
1
fl x"(s)ds -ft x"
0
we have
(s)ds
so that
It follows that
fl
Ix’(t)12dt
1
(3.11)
e(t)x(t)dt.
Ix"(s)lds)
L2(-I,I),
we have
C.P. GUPTA
368
Using this in (3.11) we get
-: I="I L2
g -I le(t)
/
x"(t)l 2
(3.12)
L
Now,
x"(t)121L > (lel L I-
le(t)-
Ix’’l L )2
-21el
L
Ix"l IL
"L
2
Ix" ILl 21el L i.
IX"ILl
+
ix,,l
L
Using this in (3.12) we finally get
21
I
Assume that there exists
THEOREM 2.
r(t)
uniformly a.e. in
[-I,I]
t
there exist real numbers a, A, r
t
[-I,i]
in
and
y
,
in
L
L
r(t)
L
L
y
and all
R
and
in
with
g(t,x,y)
L
L (-1,1)
lim sup
such that
.
A when
a
e(t)dt
e
PROOF.
x
by
2a
[-I,I]
x
gl:
Define
and
el:
[-I,I]
r.
LI(-I,I)
2A.
: (a + A)
g(t,x,y)
gl(t,x,y)
and that
when x
e :
Then the boundary value problem (3.9) has at least one solution for each
verifying the relation
< 2
such that for a.e.
and g(t,x,y)
R
x
r
Suppose that
R, r < 0 < R
a
g(t,x,y)
by
so that for a.e.
+ A)
[-I,I]
in
t
: (a
e(t)
el(t)
gl(t,x,y)
:I (A
a)
0
for all
x
R
and
y
in
gl(t,x,y)
(a
A)
0
for all
x
r
and
y
in
(3.13)
and
.< e I
(a- A)
(3.14)
< (A- a).
Clearly the equation in (3.9) is equivalent to the equation
gl(t,x(t),
g1(t,x,Y)
sup
x" +
Moreover
y
ali
we have
and if
in
we also have
Let
0
N
gl(t,x,y
x
[-I,I].
lim
x
Ixl
gl(t,x,Y)
x
[2-]
r(t) +
0
so that
for all
r(t)
(R,-r)
max
so that
r
el(t).
x(-t))
+
< 2
x
with
uniformly a.e. for
then for a.e.
r(t)
0
a.e.
and let
Ixl
rl,
r
1
in
t
[-1,1]
and
and all
y
in
t
in
in
[-1,1]
n [-1,1].
t
> 0
all
t
be such that
y
in
and
a.e.
Proceeding as in the proof of Theorem I we can write the equation (3.15) in
the form
x"(t)
where
+-fl(t,x(t),x(-t))x(t)
+
h(t,x(t),x(-t))
el(t)
(3.16)
TWO-POINT BOUNDARY VALUE PROBLEMS
0 <_-
and there is an
LI(-I,I)
a
< r(t) +
71(t,x,y)
369
n
such that
lh(t,x,y)
.
< (t)
[I,i] and all x, y in
Once again we apply Theorem IV.5 of [5] to conclude existence of a solution for
(3.16) with the boundary conditions x’(-l)
x’(1)
O. To do that we need to show
for a.e.
t
in
that the set of all possible solutions of the family of equations
x"(t)+
[(l-X)(r(t
%h(t,x(t)
x’(-1)
x’(1)
n)
+
lTl(t,x(t),x(-t))]x(t)
lel(t)
+
x(-t))
x(t)
t
in
[0,1]
e
CI[-I,1].
in
be a solution of (3.17) for some
0 < (l-)(r(t) + n) +
for a.e.
[-1,1]
le
(3.17)
0
is a priori bounded independently of
Let, now,
+
%Yl(t,x(t),x(-t))
r +q < 2
h(t,x(t),x(-t))[
with
[0,1].
< r(t) +
Since,
n
and since
L
L
L
we see from Lemma 4 that
[1
L
+
L
L
</)(lel L
L
(3.18)
I1 L )Ixl L
+
Also integrating the equation in (3.17) we have
I_l
x(t)
If, now,
+
II- [gl
(t,x(t),x(-t))
in
[-I,I]
we have using
(l-l)(r(t)+q)x(t)dt
R
for all
t
e1(t)]dt
0.
(3.13), (3.14) that
(1-)(r+n)R < o
which contradicts the assumption that
[-I,I]
leads to a contradiction.
R > 0.
x(t) S r
Similarly,
Hence, there exists a
T
in
r < x(T) < R.
for all
[-I,I]
t
in
such that
(3.19)
Next, it is easy to write explicitly the solution x(t)
with jl x(t)dt
0
x"(t)
x’(1)
0 for y e
y(t), x’(-l)
with
y(t)dt
O. From this it is easy to deduce the existence of a 6 > 0 such
that for every x e clt-l,l, with (t)
x(t)
x’(1)
x(t)dt and x’(-l)
0
LI(-I,I)
of the boundary value problem
Il
Ill
that
I1 L
lx"l L
(3.20)
and
I’1 L -< 1:"1 L
Noting that
x
(t)
+
1
x(t)dt
.
+
(3.21)
x
and inserting (3.20) in (3.18) we get
2
L
L
L
L
L
L
(3.22)
C.P. GUPTA
370
Now, by the fact that there exists a
t
[-I,I]
T
x’(s)ds[
r < x(T) < R
with
we have that
2[x’[
< max (-r R) +
L
<=
using (3.21).
(-r,R)
max
/
2lx"l L
Hence
x(t)dt[
[(x(t)ldt
<
4[x"[
< 2 max(-r R)
(3.23)
L
Finally inserting (3.23) in (3.22) we get that there exists a constant
independent of
[0,1]
%
Ix"l
z
[0,I]
> 0
o
L
Using this in (3.20) we then deduce the existence of a constant
%
01
such that
independent of
0
such that
[x[
<=
C [-1,11
13.
This completes the proof of the theorem.
COROLLARY 3.
g: [-I,I] x
Let
in
[-I,I].
t
a, A, r, R
x
R
with
Suppose that
F
a < A, r < 0 < R
g(t,x)
and
when
a
I
]R
LI(-I,I)
F(t)
that there exists
x
satisfy Caratheodory’s conditions and assume
such that
-I r(t)dt
lim sup
<
ixl
2 and
such that for a.e.
g(t,x) < F(t)
x
uniformly a.e.
that there exist real numbers
in
t
[-I,I], g(t,x)
A
when
r.
Then the boundary value problem
x"(t) + g(t,x(t))
x’(-1)
has at least one solution for each
REMARK 4.
g(t,-)
In case
e(t)
(3.24)
o
x’(1)
e
L2(-I,I,
with
I_
2a
is monotonically-increasing for a.e.
e(t)dt < 2A.
[-I,I]
in
t
see
Mawhin [7] for the boundary value problem (3.24) for a result similar to Corollary 3,
above.
The existence of a solution for the boundary value problem
"(t) +
x’(-l)
e(t)
g(t,x(t),x(-t))
O, x’(1) + hx(1)
O,
can be obtained in a similar manner as for (3.9).
(3.25)
(h > 0
given)
In fact the following theorem is
true, whose proof we omit as it is very similar to the proof of Theorem 2.
Assume that there exists
THEOREM 3.
lim sup
ixlo
that
F
g(t,x,y)
x
<
<. r(t)
LI(-I,I)
F(t)
uniformly a.e. in
t
and that there exist real numbers
such that for a.e.
t
in
[-I,I]
and
y
in
[-I,I]
such that
and all
a, A, r, R
,
g(t,x,y)
with
y
in
a <
A when
I.
Suppose
A, r < o < R
x
R
and
g(t,x,y) <. a when x < r.
Then the boundary value problem (3.25) has at least one solution for each
e
L (-I,I).
ACKNOWLEDGEMENT.
The author thanks the referee for bringing reference (I) to the
author’s attention.
TWO-POINT BOUNDARY VALUE PROBLEMS
371
REFERENCES
I.
SHAH, S.M., and WIENER, J.:
2.
WIENER, J., and AFTABIZADEH, A.R.:
3.
GUPTA, C.P., and MAWHIN, J.: Asymptotic conditions at the first two eigenvalues
for the periodic solutions of Lienard differential equations and an inequality of E. Schmidt, Z. Anal. Anwendungern. 3 (1984), 33-42.
4.
HARDY, G.H., LITTLEWOOD, J.E. and POLYA, G.:
Press, London and New York, 1952.
5.
MAWHIN, J.:
6.
MAWHIN, J. and WARD, J.R.: Nonuniform, nonresonance conditions at the two first
eigenvalues for periodic solutions of forced Lienard and Dulling equations,
Rocky Mountain J. Math. 12 (1982), 643-654.
7.
MAWHIN, J.:
Reducible functional differential equations,
International J. Math. & Math. Sci. 8 (1985), 1-27.
Boundary value problems for differential
equations with reflection of the arguments, International J. Math. & Math.
Sci. 8 (1985), 151-163.
Inequalities, Cambridge University
Topological degree methods in nonlinear boundary valu pr0blms.
NSF-CBMS Regional Conference series in Maths. No. 140, American Mathematical
Society, Providence, R.I. 1979.
Le role de la regularite dans les problemes non-lineaires resonants.
In publications de I’U.E.R. de Mathematiques pures et appliquee; Universite’
de Lille, fascimile 2 Vol 5 (1983).