Int. J. Math. & Math. Si.
(1978) 187-201
Vol.
187
HAMMERSTEIN INTEGRAL EQUATIONS WITH INDEFINITE KERNEL
ALFONSO CASTRO
University of Cincinnati
(Current Address)
Departamento de Matemtlcas
Centro de Investlgaci6n del IPN
Apartado Postal 14-740
Mxico 14, D. F. Mxico
(Received March 2, 1978)
i. Introduction.
We consider the problem of finding solutions
ueL2(R I)
L2(R)
to
the Hammerstein integral equation
u(t)
R
where
g: I
K(s,t)g(u(s),s)ds,
q,
is a bounded region in l
/
is a continuous function. We let
operator defined by
{i}ieZ
u
K(s,t)
K! (u)(t)
/ K(s,t)u(s)ds.
K(t,s) e L2(RxR), and
KI: L2()
/
We denote by
L2()
be the
{i}ieZ
and
the sequences of eigenvalues and corresponding eigenfunctions of
AK (u)
The equation (i.i) was discussed by Dolph in
(1.2)
[2 ]
and the following
188
A. CASTRO
results were obtained.
If
KI
is positive definite the problem of finding
the solutions of (i.i) can be reduced to the problem of finding the criti-
cal points of certain functional J defined on
L2(). Moreover, if: there
exist two consecutive eigenvalues of (1.2),
bers
, y’
C
and
(g(u,x)
>
with
AN;
/O
AN+I,
and real num-
such that
g(v,x))/(u-v)
u
G(u,x)
and
<_
7’
AN+1
<
g(s,x)ds >_ (7/2)u 2 + C for all
and for every
X
in
+
minimum on the linear manifold
Xx;
n
for all (u,x)eR
span
(u,x) R
{ilAi <_
x
A } J
N
has a unique
then (i.i) has a solution.
Here we prove that even in the case that
K
is indefinite the pro-
blem of finding the solutions of (i.i) can be reduced to the study of the
critical points of certain functional defined on
L2(). We show that (1.3)
alone implies the existence of a solution of (I.I). We also apply a re-
sult on Liusternik-Schnierelmann theory due to Clark
of (i.i) when
g
[ 2,p.71]
to the study
is odd in the first variable, mildly nonlinear and again
not necessari{y positive definite.
In
[5,ch.Vl]
6,ch.V]
and
the equation (I.I) is studied when (1.2)
has a finite number of negative eigenvalues. For a historical account con-
cerning (i.I) see
[3].
2. A Max-min Princip!9.
Throughout this section
inner product
u E H
<,>,
we denote by
H
and f: H /I
is a real separable Hilbert space, with
is a function of class
C
Vf(u) the unique element of H such that
For each
HAMMERSTEIN INTEGRAL EQUATIONS
f(u)
lira
f(u+tv)
tO
t
’for
<Vf(u),v
is differentiable we denote by
189
veil.
D2f(u)
the differential of
Vf at
If
Vf
u.
If {x } is a sequence in a Hilbert space which converges to some element
n
x
x
we denote
/
n
{x } converges weakly to
n
x, if
x we denote
x
11
x.
We say that a function is continuous with respect to weak convergence
(CWC) if it takes weakly convergent sequences into convergent sequences. We
say that a function w: H
x
/
n
x
<Vf(x+y 1)
Assertion:
Yl
e
Y’
which are closed subspaces of
XY and for some m > 0
H
Vf(x+Y2),yl-Y2 >
x e X,
for ever7
X and Y,
Suppose there exist
H, such that
i)
W(Xn).
implies w(x) < lira
Lemma 2.1.
is weakly lower semicontinuous (WLSC) if
l
/
and
Y2
>_
m[[y x-y2[[ 2
Y"
There exists a continuous function
f(x+(x))
:
X
/
Y satisfying:
rain (x+y)
yeY
ii)
The function
: X /l, x
/
iii) l__f, i__n addition, there exist an isomorphism A:H
A(X)=X, A(Y)CY,
Vf
A
F
for some
ml>
Proof: For each
we have
0 <A(y),y > >
i_s continuous with respect
is CWC when either dim Y <=
<Vfx(Yl )
or X and
xX we define
CI
f(x+(x)) is of class
fx:
to weak
yll 2
/
H such that
o= an y, an__d
convergen.ce (CWC),then
Y are orthogonal.
Y/l
by
fx(y)
Vfx(Y2)’Yl -Y2 "> > mll Yl-Y2 ][ 2.
f(x+y). From (2.1)
Thus, for each
A. CASTRO
190
xEX f
x
has a unique critical point (x)
min f (y) -,tin (f(x+y)). Therefore
X
yEY
(see6,p.803). Moreover, f X ((x))
(x) is the only element of Y such
yEY
that
<Vf (%(x)),K>
0
(2.2)
X
<Vf(x+ (x)),K>
Let us see now that
6 > 0
exist
II (xn)
is continuous. Suppose, on the contrary, that there
and a sequence
(x)II
> 6. Since
II
I[ P*(Vf(xn+@(x)))
where P: H
/
Y
KEY.
for all
{Xn} n
in X such that
Vf is continuous, for
x
n
/
n
xeX and
sufficiently large..
< 6m,
is defined by
(2.3)
P(x+y)
y, xEX, yEY, and
P*
is the ad-
joint of P
Because of (2.1) we obtain using Schwarz’s inequality
I] P*(Vf(Xn+b(Xn)))
Since, by (2.2)
--P*(f(xn+(x)))[I
P*(Vf(xn+(Xn)))
>
m[[
.
For t>O
(x+th)
and
heX, we have
(,X)
f (x+th+ (x+th))
f (xd (x))
t
t
(x+t.h.+, (x.))
<
f ( (x))
1
f<Vf(x+(h) + sth),h>ds
0
In a similar manner, we can see that
f(x+th)
t
f(x) >
0
(xn)
[[
>_ m6.
(2.4)
0, the inequality (2.4) contradicts (2.3).
is continuous, and this proves part i).
Thus
(x)
<Vf (x+ (x+th)+sth, h>ds.
HAMMERSTEIN INTEGRAL EUATIONS
Therefore, since
li
(_’th)
t-
are continuous we have
(x)..,,
t
<Vf (x+ (x)) h>
f has a continuous Gateaux derivative and hence is of class
This shows that
C
and
Vf
191
(see ,p.42]). From the above
<Vf(x+(x)),h>
<Vf(x),h>
We prove now part
(2.6)
for all heX.
iii). By (2.1) we have
(2.7)
<Vf(x+(x)),(x)>
Since
II vf (x> II
By
iii), Vf
O, by Schwarz’s inequality we obtain
>-- m II Cx> II
(2.8)
is bounded on bounded sets; hence (2.8) imp.lies that
bounded on bounded sets. In particular, if {x }
x
that
then
n
{(xn) } such that
{(Xn)} is
(Xn.)- ycY.
bounded. Let
Since
F
is a sequence in
is
X
such
{(xn )} be a subsequence of
is CWC, F(x
n.+(xn.)) /F(R+y).
Therefore, for any KeY
0
lim<A(Xn.)
+
A((Xn.))
+ F(xn.+(xn.)), K>
<A(H) + A(y) + F(+y), K>.
Consequently, by the characterization
implies
is
@(xn)(@).
In particular, if
(2.2), we have y
dim Y <
then
(H). This
(Xn)/ ();i.e.
CC.
If dim Y
0
+
<A(x n) +
and
A((Xn))
X and Y orthogonal then by (2.2) and iii)
+ F(x
+(Xn)),(Xn)
n
(2.9)
192
A. CASTRO
<A((Xn) )
+
F(Xn+(Xn)),
(xn)
#(R)>
<A(#(x)) + F(x+#(x+(x)), #(x
n)
Also, 0
-#(x)>. Thereore,
subtracting
the last equation from (2.9) we obtain
A(@()),(xn) -()>
<A((Xn))
<F(x+(x))
F(Xn+(Xn)
(xn) -(R)>.
Xn+(x_)-n +#()’ F(Xn+(Xn))/ F(x+()).
Since
This together with the fact
is bounded on bounded sets and relation (2.10)
that
<A(#(Xn))
ml[l (Xn )
(xn)
A(#(R)),%(Xn)-(x)>/0 as n
()[J2:_< <A((Xn) ) A((x)),(Xn )
/
converges strongly to #(R). Therefore,
imply that
But, according to
iii)
-(x)>. This proves that
is CWC and the lemma is
proved.
Now we are ready to prove our variational, principle.
Theorem 2.2.:
Le___ f,X,Y,H
.s
II xll
then there exists
u0H
f<x)
vf (u
0)
and m be as in lemma 2.1.
If, i__n addition., -f i_.
x x,
such
(2.11)
.that
(2. 2)
0
and
f(u0)
max
Proof: Let
and
and
]I xll
(rain
xX "yEY
be as in lemma 2.1. We proved that
is continuous. Since
/
-f(x)
(2.13)
f(x+y) )
is of class C !
-f(x) --f(x+(x)) >__-f(x) and f(x)
/-
as
then
/
+=asll Xll
/
+’"
(2.14)
HAMMERSTEIN INTEGRAL EQUATIONS
s
Therefore, since- f
-(x O)
that
f(x
0)
-(x).
rain
f(x)
max
XoeX(see6,p.80])
such
Thus,
rain f(x+y)
max
xeX
(2.15)
(yeY
xeX
(o+
(x o) ).
is of class C I, for heX
Also, since
<v (x0) ,h>
(2.16)
0
Therefore, if w
<Vf
NLSC, then there exists
193
XY, then
h+keH
(xo+ (xQ) ) ,h+k>’=Vf (x0+ (x0)) ,h>
+
<Vf
(x0+ (xO) ) ,k>.
(2.17)
Because of (2.2), the second term of the right hand side of (2.17) is equal
to zero. Because of (2.6) and
(2.16), the first term of the right hand side
of (2.17) is also equal to zero. Thus, if u
and, by (2.15), f(u
O)
ax
xeX
) and g:
u()
operators
in
L2()
]R x
/
/I
q,
is a bounded region in l
-
0
K(s,t)
K(t,s) e
is a continuous function such that the Nemytsky
g(u(),) and u()
KI:L2()
0)
Inuations.
and range contained in
we know that
we have Vf(u
in f(x+y)
Throughout this section
2( x
-x0+(Xo),
(yeY
3. Ap___plications to Hammerstein
L
0
L2(),
/
0g
s,)ds are continuous with domain
L2().
From the theory of linear operators
u(t)
fK(s,t)u(s)ds
/
is a compact operator.
We want to consider the problem
Kl(g(u(t),t))
u(t)
Let
{i}i
{Ai;
i
(3.1)
ten.
_+i, +2,... } be the sequence of eigenvalues of u
AKI(U).
Let
be an orthonormal sequence of eigenfuctions corresponding to the se-
quence of eigenvalues
{A.}.. We assume that
A-2 < X-I
< 0 <
XI < 2"’"
A. CASTRO
19
{i}i form
span{l,2,...}.
and Y
L2(). Let X span{_l,_2,... }
KI(X)CX and (Y) cY, and Y is the
complete set in
and that the
It follows that
reover,
orthogonal complement of X.
K
1
restricted to Y is positive defi-
I restricted to X is negative definite.
nite and K
The completeness of
{@i }i
-1 i
implies that for any
ueL 2(G)
i"
-.z
<u’ ,j
y<u,,j>,.+z
3 j=l
3"" J
"’J
where
<, > is the usual inner product in
Now we define the operators
Ql(U)
i
r._7_ <u, @q >@ j.
j=
operators. Suppose u
-1 1
Q(Q(u))
z
Q(
L2().
Q’QI: L2 ()
by Q(u)
It is eily seen that
y
Q and
-i
1
E../,A. <u,j>j
Q1
and
are compact linear
xeX, yeY,
with
LA. <u,j>)j
(3.2)
j=-i
z
=-(R)
r.
_.
i
<u,
j >Q(j)
_-’. <x, j>j
-K (x).
i
Similary
QI (Ql(U))
That is,
QI
(3.3)
Kl(y).
is a square root of K
X. From the definition of
Q and
I
QI
on Y
and Q a square root of -KI on
it is easy to see that Q and
QI
are
selfadjoint.
Next we reduce the problem (3.1) to an operator equation envolving Q
195
HAMMERSTEIN INTEGRAL EQUATIONS
and
QI"
Let P:
PI"
I
P. Clearly PQ
L2(fl)/ L2(fl)
be the orthogonal projection onto X and
Q and
PQI
O.
we
(u)(t)
denote
for u L
2(fl).
Lemma
3.1.: Th___e problem (3.1) has a solution iff
qCCCy)
-x()+y()
+ qCx))()) +
g(u(t),t),
the. problem
q((q(y)
+ q(x))())
(3.4)
has a solution (x,y) XxY.
x+y, xcX, yY satisfy (3.1); therefore x
Proof: Let u
KI(P((u)))
-Q2(P((u))).
Thus, there exists
A similar argument shows the existence of
Y
X
P(u)
such that
such that y
P(KI((u)))
Q()
QI().
x.
There-
fore,
Q()
x
Q(-Q((Q() + Q1 ()) ) )"
Since 0 is not an
eisenvalue of Q
restricted to X and
(3.5)
-Q((Q()+QI()))
belongs to X, we have
-Q((Q()
/
Q1(9)))"
(3.6)
A parallel argument shos that
Q1 ((Q() + Q1 ()) )"
(3.7)
Subtracting (3.6) from (3.7) we obtain
-+
so (3.4)
Q((Q() +
Q1())) + QI(i(Q() + QI()))
has a solution. This proves the necessity of (3.4).
Conversely, suppose (x,y)eXY satisfies (3.4). Let us call
u
+ Q(x). Applying Q
QI(y)
-Q2((u))
Q(x)
QI (y)
K
and
QI
to (3.4) we obtain respectively
l(P((u))
(3.8)
K1 (PI ( (u)).
(3.9)
Adding the last two equations we get
u
K
l(P((u))
+ K l(Pl((u))))
Kl((u))-
Therefore (3.1) has a solution, and the lemma is proved.
A. CASTRO
196
Le___!t g: R
Theorem 3.2.
/
R
be _a
continuous function. If there exist
real numbers y, y’ and C such that for some integer N
’
a)
AN< 7 --<
b)
For ue]R,
if N=-I we let
< 7
--<
ve, te (g(u,t)-g(v,t))/(u-v)
_<
<
AN+l,
l-i
’
<
Ii
y’
and
c)
For
o
ue, te G(u,t)
(y/2)u 2
g(s,t)ds
C
then the proble (3.1) has at least one solution.
J:L2()
Proof: We define
((PlU)2/2)
J(u)
(u) (t)
where
constants
by
((pu)2/2)
(Q(P(u)) +
QI(P(u))),
G(u(t),t). Conditions b) and c) imply the existence of
A,B > 0 such that
Ig(s,t) l<_A +
for any
+ ]R
(s,t)el
BIs
x.
G(s,t)
and
Xl>
A +
Blsl 2
(3.10)
Therefore, J is a functional of class C I. An ele-
ueL2(),
mentary computation shows that if
<VJ(u)
_<
f-x.x
I
(Q(x) +
u
x+y, xeX, yeY
Ql(y))Q(x l)
for
then
XlX
(3.11)
yleY.
(3.12)
and
<VJ(u)’Yl>
]YYl
g(Q(x) +
From (3.11) and (3.12) we see
iff
(PU,Pl(U))
Ql(y))Ql(Yl
that ueL2() is
for
a critical point of J
is a solution of (3.4). Thus, by lemma 3.1, J has a
critical point iff (3.1) has a solution.
Let us prove that J has at least one critical point. First we assume N
-i. Let xeX, yeY, and
<VJ(x+y I)
yleY;
so, by condition b), we have
VJ(x+y),yl-y> (yl-y) 2
T’ (QI(Yl-y)) 2.
Using Parseval’s formula and the definition of
<VJCx+y I)
where, by a),
JCx+y),Yl-Y>
(l-(y’/%l) ) > 0.
->
QI
we infer
(I-(y’I%I))I1Yl-YI[
From (3.11) and (3.12) we see that
(3.13)
2
(3.14)
HAMMERSTEIN INTEGRAL EQUATIONS
197
P(VJ(u)) +
VJ(u)
PI(VJ(u))
Q((Q(u) l(u)))
-P(u)
(3.15)
+
I relaiona (3.14) and (3.15) mply tha
- -
QI((Q(u)QI(U)))"
X,Y,J and (i-(’/I)) atisfy
Pl(U)
th conditiona of i 2.1. Conaequently, there .iata a C function
:X
+
yy
-J (x-l- (x)) can be wtten as
(x)
Since
rain J(x+y).
a(x(x))
such that
[ yn(((x))2/2) + ,(Q(x) +
((1/2)ynx2)
-3(x)
since the function x
/
(I/2)Ix
=ii=
poi.
if J(x)
(3.16)
is convex, and since the expression in
the bracketed integral is CWC,
h.
QI((x)))],
is WLSC. Therefore, by theorem 2.2, J
/-(R)
. llx II
+
"-
L us v=iy hi on-
dition. Since
/R{-(x2()/2)
J(x)
G(Q(x()),)}d,
it follows from c) that
J(x)
<-/fl{(x2()/2) + (y/2)(Q(x())) 2- C}d.
From (3.17) we see that J(x)
/--
then by Parseval’s formula we have
as
[Ix I
/
(3.17)
when y > 0. If y < 0,
-(y/2)/Q2(x())d < (y/%l)/x2()d.
Substituting this in (3.17) we have
II I 2 + Cmeas(fl).
-
J(x) < (1/2) (-l+(y/A I)) x
Therefore, J(x)
/-
AN-O
K
< 0 and
--.
llx II
-i. Let
Suppose now N
Thus,
as
AN+I-0
m- So (3.1) has a solution.
be a real number such that
<
< y.
> 0. Let
2(s,t) ---" (j (s)Oj (t)) (;kj-i)
-I
Therefore, the eigenvalues of the problem
XK(s,t)u(s)ds
u(t)
are
{Aj-0
j=_,
of (3.18) is
where
AN-0
1.3
(3.18)
is as before. The greatest negative eigenvalue
and the smallest positive eigenvalue of (3.18) is
A. CASTRO
198
%N+I-O.
Consequently, by the reasoning for the case N
fl(u’) g(u,) pu, then there
u0(t) 2(s,t)(g(u0(s),s)
u0L 2()
Pu0(s))ds.
exists
-i, if
such that
An elementary computation shows that u 0 is a solution of (3.1). And
the theorem is proved.
Remark: If we change (b) and (c) by
(b’)
For
u, v, t,
For
u, tE, G(u,t)
(g(u, t)-g(b, t))/(u-v) _>
and
(c’)
og(S,t)ds
(y’u 2)/2
C,
respectively, then (3.1) has at least one solution. This comes from
_(-K(s,t))(-g(u(s),s))ds.
applying theorem 3.3 to u(t)
From here on we consider the equation (3.1) assuming g(u,t)
g(u)
-g(-u). In our next theorem we make use of the following result which is
specialization of a theorem due to Clark
Le___t H be
Lemma 3.3.:
a real Hilbert
function defined
whenever {x }
n
sce
[ l,p.71].
and f an even, real valued C 2
has the property that
o._.n H. Suppose that f
= H is a bounded sequence such that f(xn)
> 0 and
then {x } contains a convergent subsequ.e.nce. Suppose that f(0)
ded above, there exists a subspace M
<D2f(0)h,h>
> 0
i__f hM with h
Then there exist at least 2
Theorem 3.4.
+
+
o__f H o_f
> 0 such
dimension
0, an__d f(x) > 0 fo___/_r
[ix II
i solutions of Vf(x)
O.
Suppose g is a function of class C
f(Xn)
there exists a real number 7 with
g’ (u)
0,
0, f is boun-
that
sufficiently
an__d g’ i_s bounded, i_f there
exist two integers N and r,N < r, such that:
i’)
/
_< 7 <
for all ue]R,
199
HAMMRSTEIN INTEGRAL EQUATIONS
IN
ii’)
<
g’(0) <
IN+l’
iii’) there exists real numbers y’ and C, y’ >
such that
%r-i
og(s)ds >_ (y’/2)u 2 + C,
G(u)
then the equation (3.1) has at least 2(r-N)
+ 1
solutions.
Proof: As shown in theorem 3.2 it is sufficient to prove that J has at least
2(r-N) + i
assume
critical points. Also, arguing as in the previous theorem, we can
r
and
i
If X and Y are as in lemma 3.1 then condition
i --i"
i’) implies that for xeX, yeY, and
Vj(x+y),yl_y >
<Vj(x+Yl)
_>
yleY,
we have
(l_(Y/%l)) [lyl_l
So by (3.15) and lemma 2.1 there is a CWC function
2.
(3.20)
:X
+
Y such that for each
xeX
J(x+(x))
min J(x+y)
ysY
Since g is of class C
I and
g
(x).
is bounded it follows that J is of class C 2
Hence, following a reasoning based on the implicit function theorem, as in
[4,theorem i],
is of class C
it can be seen that
and
is of class C 2.
Moreover, for xeX, heX we have
<DZI(x+(x)) (h+’ (x)h),h>.
<D2O(x)h,h>
(3.21)
Now we show that 3’ satisfy the hypothesis of lemma 3.3. Because
g’(0)(Q(h)) 2
l-h 2
<D2j(0) h, h>
for heX.
0. Thus, from (3.21) we obtain
is even, and (0)
is odd. So
From iii’)
(3.23)
_< J(x)
(y’/2)(Q(x))2 +
< /-x 2
Thus, since y’ >
/-
span{_l,_2,...,N+l}.
we have
and the definition of
3(x)
(3.22)
From (3.22) and ii’) it is easy to see that D2j is positive definite
in a subspace of dimension (r-N), namely
3(x)
is even
as
llx I
l-i
-
and
.
i -< --i
Let .{x
n
C.
we infer that
is bounded above and
is a bounded sequence in X such that
A. CASTRO
200
Vf(xn)
/
0. By (2.6) and (3.15) we have
V(xn)
Let {x
n.
Q and
-Xn Q(’(Q(Xn)
Ql((Xn)
"+
O.
(3.24)
} be a weakly convergent subsequence of {x }. In consequence, since
n
are CWC hence the sequence {Q((Q(x
converges strong+ Ql(0(Xn)
n)
ly. Therefore, by (3.24)
at least
+
{Xn.}
converges strongly. Thus, by le 3.3,
J
has
2(r-N)+l critical points. Since by (2.2) and (2.6) every critical
point of J is of the form x+(x), where x is a critical point of ] the theorem
is proved.
Acknowl edg ements.
This paper is part of the author’s Ph.D. dissertation at the University of
Cincinnati under the advisorship of Professor Alan C. Lazer. The author is
greatly indebted to Professor Lazer for many helpful suggestions and permanent encouragement. This research was done while the author was a Taft Fel-
Iow
References.
A variant of the Lusternik-Schnierelman Theory, Indiana Univ.
Math. J., 22(1972), 65-74.
i.
Clark, D.
2.
Dolph, C. Nonlinear integral equations of the Hammerstein type, Trans.
Amer. Math. Soc., 66(1949), 289-307.
3.
Dolph, C. and Minty, G. On nonlinear integral equations of the Hammerstein type, in Nonlinear Integral .Equation, pp.99-154, Univ. of Wisconsin Press, Madison, 1964.
Lazer, A., Landesman, E. and Meyers, D.
On saddle point problems in the
calculus of variations, the Ritz algorithm, and monotone convergence,
J. Math. Anal. AppI., 53 (1975), 594-614.
Krasnosel’skii, A. ,Topological Methods in the Theory of Nonlinear Integral Eq.u.atip.ns Pergamon Press, New York, 1964.
6.
Vainberg, M. Variational Methods for the Study of Nonlinear Operators,
Holden Day, San Francisco, 1964.
HAMMERSTEIN INTEGRAL EQUATIONS
ABSTRACT.
--._rstein
201
This paper deals with the problem of finding solutions of the
integral equation.
It is shown that this problem can be
reduced to the study of the critical points of certain functional defined
on
L2().
is proved.
Existence of a solution of the Hammersteln integral equation
Some other related results of interest are obtained.
KEY WORDS AND PHRASES.
Hammtzin integral equine, max-rain prin6ipl,
v ’prlnplz and indefinite
AMS(MOS) SUBJECT CLASSIFICATION (1970) CODES.
45G99, 45H05.