Internat. J. Math. & Math. Sci.
(1985) 37-48
37
Vol. 8 No.
BIFURCATION FOR THE SOLUTIONS OF EOUATIONS
INVOLVING SET VALUED MAPPINGS
E.U. TARAFDAR
Department of Mathematical Sciences,
McMaster University,
Hamilton. ONTARIO.
CANADA. L8S 4KI.
H.B. THOMPSON
Department of Mathematics,
University of Queensland,
ST. LUCIA. 4067. QUEENSLAND.
AUSTRALIA.
(Received December 27, 1983)
ABSTRACT.
This paper is devoted to a generalization of the bifurcation theorem of
snosel’skii
and Rabinowitz to the set valued situation.
KEY WORDS AND PHRASES.
Compact set valued mapping, isolated zero, index, character-
istic value, multiplicity, bifurcation point.
1980 MATHEMATICS SUBJECT CLASSIFICATION CODE.
Primary 47H15, 47A50; Seconda.y 47H10,
47A55, 54H25.
1.
INTRODUCTION.
In the Leray-Schauder degree theory the study of the index of an isolated
fixed point of a compact mapping is a basic tool for the calculation of the degree.
A fundamental and well-known result of Leray and Schauder
[I]
commonly known as
the Leray-Schauder principle, is that the topological index at zero of an invertible
linear perturbation of the identity I-A in a Banach space can be expressed in terms
of multiplicity of the characteristic values of A lying in the open interval (0,1).
Krasnosel’skii [2] employed this result as a basic tool to develop his bifurcation
theory for equations of the form:
uAx
x
in a real Banach space X, where A:X
compact mapping,
R(x,v)
is of o(
Ixl I)
R(x,u)
(1.1)
0
X is a linear compact mapping,
uniformly
a bounded open neighbourhood of zero in X.
in
Rabinowitz
R:
x R
in compact intervals and
[3]
X is
is
studied the global char-
acter of the solution set of such equations and applied his results in many directions.
Because of
t|e
wide scope of applicability of the bifurcation theorem of :<rasnosel’skii
and Rabinowitz a tremendous research interest in this field has been evidenced
E. U. TARAFDAR AND H. B. THOMPSON
38
currently (see Crandall and Rabinowitz [] and the literature cited there).
The aim of this paper is to generalize the theorems of Krasnowel’skii and Rabinowitz to the set valued situation, i.e., to equations of the form:
B:
where A is above,
CK(X) is
x R
in compact intervals
uniformly in
(1.2)
Ax + R(x,)
x
a compact set valued mapping and
Cfor
definitions see Section
and CK(X) being the set of all compact convex subsets of X.
tions of the form (1.2)
pAx +
is a linear mapping, A:X
Z
Ixl I)
being as above
From our result for equa-
have dedu ed the same result for equations of the form:
nx
where L:X
2),
B is of o(
a compact mapping and of
o(I IxlI)
(1.3)
R(x,)
Z a compact linear mapping,
uniformly in
B:
in compact intervals.
x R
[5] (see
the equation when R is single valued was first obtained by Laloux and Mawhin
also Gaines and Mawhin [6]
).
CK(Z)
The result for
even when R is single valued our result is
However,
more general than that of Laloux and Mawhin [5]
(see remark 4.3).
As an application of our results, we have included a two point boundary value
problem for a generalized ordinary differential equation,
although we hope that these
results will find application to problems in control theory, mathematical economics
and related problems.
We have made the paper self-contained as far as is practically
possible.
2.
a real Banach space and
Throughout the paper X will denote
A
set of all compact convex subsets of X.
CK(X) will denote the
set valued mapping N:Q
X
CK(X)
is said
to be compact if N is upper semicontinuous and maps bounded subsets of Q into
elatively compact subsets of X.
to Ma
The degree theory which will be used here is due
[?] although all the results obtained for compact mapDins will equally hold
for more general mappings known as ultimately compact mappings by applying the degree
theory for such mappings due to Petryshyn and Fitzpatrick [8].
INDEX.
mapping.
be an open neighbourhood of a
Let
We assume that a
(I-N)-I(0)
Beo
N(a) and there exists e o
is Identity mapping on X.
(a)
Be(a)
DEFINITION 2.1.
N is the integer
0 #
Be(a).
(0,e o) by the
Bee (a)
eo
(l-N)(Be(a))
Thus the degree
d(l-N,Be(a),0
is defined
excision property of the degree.
for any e
(O,eo)
and will be denoted by
i(l-N,a).
If N is single valued, then the index of a fixed point a of N as
defined above coincides with the Leray-Schauder index.
PROPOSITION 2.1. Let N:
CK(X) be n compact-mapping such that 0
and
(l-N)-l(0)
is f in ite.
and
Under the above assumptions the index of the fixed point a of
d(l-N,Be(a),O)
REMARK 2.1.
0 such that
CK(X) be a compact
This implies that for every
denotes the boundary of
and is independent of e
X:II-all
{
(0,o)
where
N:
{a:" i.e., a is an isolated zero of I-N, where
(a)
Beo
and
X and
is a finite set, say,
(al,a2,...,an};
(I-N)()
i.e., the set of fixed points of N
BIFURCATION FOR THE SOLUTIONS OF EQUATIONS
39
Then
n
d(l-N, O)
=j__Z I
(l-N,a.).
i
3
We can find positive numbers
PROOF.
Bej (aj)
o
Bek (a k)
1,2,...,n such that
j
ej,
,n}
e{l,2
for any j,k
with j # k.
By the additivity and excision properties of degree we have
n
n
d(I-N,,0)
Let
DEFINITION 2.2,
Ixl
I)
set valued mapping.
B is said to be of order
if
-I
Ixl
sup
{I lylI:y
B(x)}
0 as
o(llxll)
coincides with the usual definition of
Let N
THEOREM 2.1.
{0} and
lxll
O.
It is clear from the definition that B(O)
REMARK 2.2.
o(I Ixll)
(I-A)
(I-N,aj)o
i
X be a bounded open neighbourhood of the origin and
CK(X) 5e an upper semicontinuous
B:
(I
d(l-N,jlJ.=l Bej(aj )’0) =j=II
B:
0 and the definition of
when B is single valued.
X is a linear compact mapping with :’er
A + B where A:X
CK(X) is a compact mapping of order o(
Ixl I)
where
X is a
Then 0 is an isolated zero of (I-N) and
bounded open neighbourhood of the origin.
i(l-N,O)
i(I-A,O)
where i(I-A,O) is the usual Leray-Schauder index of the fixed point 0 of A.
Since B is of order
PROOF.
Ixll)
o(
and A is linear, N(O)
O.
We can easily verify that for each x
(I-A)[I- (l-A)
(l-N)(x)
That (I-A)
-I
B](x)
I)
it follows that there exists
(2.1)
exists follows from Riesz theory.
Now from the fact that B is of order o(
such that
-i
eo(O)
and for each
sup{llYll:
Hence for each e
(O,e o]
x
Y
and every
Ix
Be0(0)
(I-A)-IB(x)}
-<
1/2
Be(0)
x
[0,13
(x,k)
lxll.
we have
%(I-A)-IB(x)} _> inf{[lx[I
llxll -sup {IIyII:y
Ilxll
inf{llx-yIl:y
%(I-A)-IB(x)}
%(I-A)-IB(x)}
[[y[l:y
(2.2)
(2.1) and (22) together imply that the only fixed point of N in
’0).
(I-N) (0)
eo(O)=
By the homotopy invariance proFerty
d(I-k(I-A)-IB,Be(O),
O)
d(I,Be(O),O)=
for every e
Now from
(2.1), (2.3)
1
(O,e o]
and
[0, i]
and the product theorem of degree we have
Bo(O)
is
O; i.e.,
E. U. TARAFDAR AND H. B. THOMPSON
40
d(I-A,Be(0),0)’l
.i([ -,’..,Be(0),0)
for each e
(0,eo];
i.e.,
i(I-A, 0).
i(I-N, 0)
3
CHARACTERISTIC VALUE AND MULTIPLICITY.
(r(A)
Each
{: -I
X be a linear compact mapping and r(A)
Let A:’"
is an eigenvalue of
A}.
called a characteristic value of A.
is
r(A)
The multiplicity of b
is the integer
8()
dim ker
I-A]
n(u),
where n() is the smallest non-negative integer n such that
ker[ I-A]
Since A is compact,
n
ker[ I- A]
n+l
8() is finite.
-i
exists and is continuous.
A real number U is said to be a regular value of A if (I-A)
0 is an isolated zero of I-A and
R is not a characteristic value of A, x
If
iLS(I-A,0),
the Leray-Schauder index of
I-A
at zero will be simply denoted by i().
We write the following well-known Leray-Schauder principle as a lemma.
X is a compact linear mapping and
If A:X
LEMMA 3.1.
characteristic values of A, then i(
I)
(-I)8i(2)
BIFURCATION.
with
are not
2
i
where 8 is the sum of multiplicities
of the characteristic values of A lying in the interval
4.
Ul,2
[I,2 ].
Throughout the rest of the paper we will assume
to be an open
bounded neighbourhood of the origin in X.
DEFINITION 4.1.
(o)
Let
N:
CK(X) be a set valued mapping satisfying
N is upper semicontinuous and compact on bounded subsets of
(oo)
x R;
R,
for each
N(0,).
0
R, x
Thus for each
0 is a solution of the equation
x e
N(x,u).
(4ol)
A point (0,U o) will be said to be a bifurcation point for the solution of the equation
(4.1),
or simply a bifurcation point of N if every neighbourhood of
tains at least one solution
(x,u)
of the equation (4.1) with x
we will sometimes refer to Uo as the bifurcation point.
LEMMA 4.1.
Let N be as above satisfying (o) and (OO)
contains no bifurcation point of
[I,2
and x
,
N, then there
Let S
such that
con-
If the interval
[Ul,U23
0 such that, for each
exists a
{(x,)
N(x,)x
Oo
[l,2]:x
N(x,)}.
follows that S is a compact subset of X
not true.
(0, o)
By abuse of notation
m(0),
x
PROOF,
# O.
From the compactness of N, it
R. We suppose, if possible, that the lemma is
Then for each positive integer n, there exist
n
[i,2 ], Xn
Bl/n(0)
BIFURCATION FOR THE SOLUTIONS OF EQUATIONS
N(Xn, n)
xn
(Xn, n)
Now
to
(XO,n).
Let N be as above satisfying (o) and (oo) and let
<
PROOF.
[(I-N(,j))
As
-I (0)]
i(j),
(0,() s
sch that
63
1,2 are defined, there
j
0 such that for each
0
I,2
R with
e:,ist
O,
j
I)
@
N.
1,2 such that
B6j(O)
[i,2
min(l,62,3)"
and x
N(x,u
x
We set
which con-
a bifurcation point of
Suppose that the conclusion of the theorem is false.
exists
[i,I2]
are defined and i(
1,2
i[l-N(’,j),O],
2" Further suppose that i(j)
i(2). Then there exists 0 [I,
l
is a bifurcation point of N in
Hence the lemma is proved.
tradicts the hypothesis.
THEOREM 4.1.
(0, O)
O and
Clearly x 0
@ O.
has a convergent subsequence converging
(xn, n)
Hence
S which is compact.
and xn
41
x
(0,60]
Then for each
N(x,% I + (i-)2)
x
Hence it follows that for each 6
x
.
B63(0
(0,60]
e
each
,n
% e
[0,I],
x e
B6(O)
O.
=> x
[0,i]
Then by the above lemma there
and x
B6(O).
N(x,I I + (i-)2)
Therefore by Homotopy Invariance Theorem
i(l)
which contradicts our hypothesis.
THEOREM 4.2.
Let
N:
R
d[l-N(-,Ul) B6(0),O]
d[l-N(-,2) B6(O),O]
Hence the theorem is proved.
CK(X) be such that
N(x,u)
where A:X
=i( 2)
wAx + B(x,u),
X is a linear compact (single valued) mapping and
continuous and compact on bounded subsets of
on compact intervals.
B:
R with B(x,)
Then for each bifurcation point
(0,o)
R
o(
of N,
X is upper semi-
Ix I)
o
uniformly in
is a characteristic
value of A.
PROOF.
Again we prove this theorem by contradiction.
bifurcation point of N and
o
is not a characteristic value.
Suppose that (O,u o) is a
Then A
(l-oA) -I exists
o
and is continuous.
Let
F:
R
X and
G:
R
CK(X) be defined as follows:
F(x,u)
(-o)AoAX
G(x,u)
AoB(X,U ).
and
Clearly F and G are compact on bounded subsets of
R. Also from the assumption
o( Ixl I) and the continuity of A it follows that AoB(X,)
o(
I)
o
in compact intervals.
uniformly in
1
_<
Let
0 be such that 6
and 9
0 be such that whenever
that
B(x,)
Ixl
AoAII
E. U. TARAFDAR AND H. B. THOMPSON
42
[-6,Uo+61
B0(O
(x,)
lxl
Then for each x
we have
-I
{I
sup
IYlI:Y
-<
AoB(X,D)
[Bp(0) n ] \ {0} and
infI !Aox_Yl I:y AoN#X,l
inf{l ,Ao[(l-oA)X+UoAxJ-
_>_
>_
AoAX-V l’v
infl ix+(o-)AoAX-vlI:v
Ixll I-oI IAoAI lxll
[1- 611AoAI IJ llxll ->
Hence there exist
(x,u)
sup
0 and
> 0 such that for each
{[B; (0)
n ] "{O}}
AoB(x,l)}
{I Ivll
[Uo-,Uo+4]
x
v
AoB(X,)}
N(x,u).
is not a bifurcation point, which is a contradiction.
Sut this implies that
Thus the
theorem is proved.
The following theorem in the single valued case is due to Krasnosel’skii
THEOREM 4.3.
Let
N:
CK(X) be given by N(x,)
R
and B are as in Theorem 4.2.
If
Uo
Ax + B(x,)
[J.
where A
is a characteristic value of A of odd multiplicity
5o, then (0,u o) is a bifurcation point of N.
PROOF.
Since A is compact,
there exists
0 such that
Thus from Lemma 3.
o
o
is an isolated characteristic value of A.
is the only characteristic value of A in
Hence
[Uo-e,o+e].
we have
i(o-e
i[l-(o-e)A,0]
(-I)8O i(o+e).
(-I)8O i[l-(o+e)A,O]
Hence from the above equality and Theorem 2.1 we have
i[I-N(.,o-e),O]
(_l)SO i[I-N(-,o+e),O].
Now since 8 o is odd, we have
i[l-N(’,o-e),O]
i[l-N(’,o+e),O].
Hence by Theorem 4.1 there exists a bifurcation point (0,) of N with
[Uo-e,o+e].
But since
is the only characteristic value in [Uc-e,Uo+e], Theorem 4.2 implies that
o
(O,u o) is a bifurcation point of N.
We note that all Theorems proved above hold with the same proof if we replace
Uo"
Thus we have proved that
by X.
Let E denote the space X
R or
R where
is a bounded open neighbourhood of
the origin in X.
The following global version in the single valued case is due to Rabinowitz [].
CK(X) be such that N(x,u)
THEOREM 4.4 Let N:E
uAx+B(x,) where A and B are as
R being replaced by E. Let Uo be a characteristic value of A of
Let S denote the closure of all nontrivial solutions of (4.1). Then
S contains a component C (i.e., a maximal closed an@ connected subset) which contains
(O,u o) and either is unbounded or contains (O,u) where u is a characteristic value of A
in Theorem
4.2 with
odd multiplicity.
43
BIFURCATION FOR THE SOLUTIONS OF EQUATIONS
# o"
and
The proof of this theorem is exactly similar to that given by Rabinowitz [8J
for the single valued case. However we in(also see Crandall and Rabinowitz [4]
We will need the following two lemmas.
of
sake
completeness
clude the proof for the
Let K be a compact metric space and A and B two
LEMMA 42. (Whyburn (1958)).
disjoint closed subsets of K. Then either there is a subcontinuum (i.e., a closed and
connected subset)of K meeting both A and B, or there exist disjoint compact subsets
KA u KB.
B such that K
and KB
Under the hypothesis of Theorem 4.4 assume that there exists no
LEMMA 4.3.
subcontinuum C of S u {(0,o)} such that either (i) C is unbounded, or (ii) C contains
KA
A
(0,)
with
in E such that
Then there exists a bounded open set
r(L).
o #
e, e n
(0, o)
S
and
for some positive
R)
(Bo-,o+6)
{0}
{(0,o)}.
and compactness of N,
Then
Co
r(L)by
the fact that for
Co
is
Let
U
is an isolated solution of (4.1) and
contains no solution (0,)
< e o sufficiently small
U S is compact
Now since S is locally compact, it follows that K
U
from (ii) it follows that for 0 <
I-oI
with
> 6.
in the relative topology induced from E.
(U)
S.
component.
K
A u B.
and B.
Now from
Co.
be a -neighbourhood of
2.1,(0,%)
Theorem
be the component of
Hence by the upper semicontinuity
bounded,by (i).
is a compact.
Co
Let
The proof is similar to that of Lemma 1.2.
PROOF.
(0, o) in S u
.
e n ({o}
.
U
Co
Also
and K 2
will not be a
Co
Let K
Co
There is no subcontinuum meeting K and K 2 for otherwise
Hence by Lemma 4.2 there exist compact subsets A
K] and B K 2 such that
the distance between A
than
less
is
where
e
A
of
be an e-neighbourhood
Let
We can easily see that 0 fulfills the demand of the lemma.
be the component of (0, o) in S
Let
{0,o)}.
CO
PROOF OF THEOREM 4.4.
by
Then
that the theorem is false.
Co
Assume
Lemma 4.3 there is a bounded open set 0 such that
O, 0 n S
and
0 n
R)
(o-,o+)
{0}
I-oI
6,(0,)
6}. Now for
satisfying 0 <
X:(x,%)
(by construction of O and Theorem 2.1). Hence there
{%}. For
0 such that (00,%) is the only solution of (4.1) in (%)
0.
for some 6
({o}
Let
{x
e%
is an isolated solution of (4.1)
0(%)
+
o 6
exists
0(o+/-6)
>
we choose
sufficiently small,
Thus for
therefore d(l-N(’,%),
variance
tion in
0(o+6)
0(A)
#
and for
we can assume that
o
Do-6
we choose
(%) %
0%-0(%),0)
is well defined
or
all
d(l-(-,), O-p(),O)
0(o-6).
for
there is no solution of (4.1)in
d(l-N(’,%),%-O(%),0) constant for
%-0(%)" Hence for % o we have
(A)
<
each %
o"
O.
I0%-0(%)I
o"
By choosing
satisfying
{} and
By homotopy in-
However (4.1) hss no sol-
E.U. TARAFDAR AND H. B. THOMPSON
44
imilarly (A) holds for I
d(l-N(’,l),
(B)
for
ll-oi
6
Moreover, by homotopy invariance we have
<-o"
-
01, 0)
Let
o
<
6
’u
Then from
O
u
BO()
constant
o
(0
+ 6.
Bp(G))
and by additivity of degree we have
d(I-N(’,),
0
d(I-N(-,),Bo(),O)
,0)
d(I-N(-,),
+
d(l-N(,), Bp(),O)
0(),0)
by (A).
Similarly, we can show that
d(I-N(.,),Bp(),0).
d(l-N(,), 0,0)
Hence by using (B) we obtain
(C)
0)= d(I-N(,),
d(I-N(-,_), Bp(u)
Bp() ,0).
We now define the homotopy
IAx + tB(x,X), 0 <- t -< I.
N(x,X,t)
As B(x,l) is of o(
Ixl I),
we can choose
p()
sufficiently small to obtain
(I-) (x,,t)
0
for any
(x,t)
e
LO,I]
Bo()
Thus by homotopy invariance
d(l-N(’,), Bp(), O)=
d(l-(’,,t), Bp(),O)
d(l-(-,,O), Bp(),O)
By repeating the same argument for
d(l-A,
and using (C) we obtain
d(I-__A, Bp(),O)
i(I-A,0)
d(I-A, Bp(),O)
i(I-A,0)
which is a contradiction in view of the fact that
multiplicity and in view of Lemma 3.1.
REMARK 4.1.
[]
Bp(),O)o
o
is a characteristic value of odd
Thus the theorem is proved.
Results similar to Theorems 1.16, 1.25, 1.27, 1.40 in Rabinowitz
can also be proved in the set valued case.
We have omitted these as their proofs
are only repetition of the arguments given by Rabinowitz.
Let X and Z be real Banach spaces.
Lx
Let us now consider the equation
IAx + B(x,l)
N(x,)
(4.2)
BIFURCATION FOR THE SOLUTIONS OF EQUATIONS
Z is a continuous linear mapping, A:X
where L:X
B:
45
Z is a compact linear mapping and
CK(Z) is a set valued mapping which is upper semicontinuous and is compact on
R. We also assume that 0 E B(0,k) for all % E R.
bounded subset of
A point (0, o) is said to be a bifurcation point of the equation (4.2) if every
R
neighbourhood of (0, o) contains at least one solution (x,) of (4.2) with x
-
o(IlxlI)
Let B(x,k) be
Let L,A and B be as above.
THEORE 4.5.
O.
uniformly in
% in compact intervals; that is,
!1II
uniformly in
ullyll:
Let
(Po-)
PROOF.
Ilxll
o
R such that
(L-A) -1
be a characteristic value of the compact operator
(L-A)-IA
Ao
of odd multiplicity
0 a
Assume that there exists
in compact intervals.
exists and is continuous.
B(x,X)
y
Then Po is a bifurcation point of the equation (4.2).
Let us consider the equation
B o.
x
kAox +
(L-A)-IB(x,+)
(x,).
(4.3)
Now (x,p) is a solution of (4.2) if and only if (x,u-) is a solution of (4.3).
Lx e N(x,)
FAx
Lx
c
pAx
+ B(x,p)
Indeed,
<=>
(p-)Ax + B(x,-+)
<-
(-)(L-A)-IAx + (e-A)-iB(x,_+)
(p-)Aox + (e-A)-iB(x,-+)
x
x
(x,p-).
(4.4)
-I
Noting that (L-A)
B(x,+) is of o(I
I) uniformly in % in compact intervals, by applying Theorem 4.3 we conclude that
is a bifurcation point of (4.3).
Hence it follows from (4.4) that Po is a bifurcation point of (4.2).
REMARK 4.2
The corresponding global version of Theorem 4.5 also holds.
REtARK 4.3.
It is clear that in Theorem 4.5 instead of assuming A and
Ixl
o-
(L-A)-lA
B(x,)
(L-A)-lB
to be compact it would suffice to assume A
and
to be compact.
o
If
we do this, then even when B is single valued our Theorem will
be more general than that
proved in Laloux and Mawhin [5]
in the sense that we are not assuming L to be a Fredholm
mapping of index zero. Note that all the conditions in Theorem 4.5 have also
been assumed
implicitly by Laloux and Mawhin.
5.
APPLICATION.
Let CK(X) be as defined in Section 2.
F:[O,]
be upper semicontinuous and q
sup{
t c
We
lulI:u
R
Rn
LI[0,]
F(t,%,y,z), some
Let
Rn -CK(Rn)
(abbreviated as L I) be such that
%
R, y,z
[0,].
consider the two point boundary value problem
Rn} <_ q(t)
(5.1)
E. U. TARAFDAR AND H. B. THOMPSON
46
(t)
y(0)
where a solution
(5.2)
y(t) + F(t,X,y(t),(t)), a.e. t [0,]
Rn satisfies
y:[O,]
(5.3)
y()
0
is absolutely continuous on
[0,]
and y sat-
sfies (5.2).(5.3).
[0,],
We assume that for t
sup{[lull:u
uniformly for t
(abbreviated as
[0,] and
(C)n)
when z
t
Rn
R, y,z
F(t,k,y,z,)}
lYll
% bounded, as
(5.4)
0
lzll
+
For y,z e
O.
(C[O,])n
let
Rnmeasura|le, f(t) e F(t,%,y(t),
{f:f:[O,]
B(y,z,%)
k e
B(y,%).
we abbreviate this to
z(t))};
For
(C) n
R,y,z
let
F(t,l,y(t),z(t))
H(t)
then
CK(Rn)
H:[0,]
is upper semicontinuous as a composition of an upper semicontinuous function with a continuous function and hence
is closed and hence measurable.
in Rockafeller
[9]
Rn
[0,]
{(t,u,(t))
u(t) e H(t)
t
[0,]}
Thus the set B(y,z,%) is non empty by the results stated
Also B is convex valued as F is convex valued.
f is measurable and
llf(t) ll
so f
-< q(t)
e[O,]
t
(el) n.
Let
[0,]
G:[O,]
R
be given by
(r-x) t
0 <- t
-<x -<
x
t <
G(x, t)
x(-t)
and let K:
(LI) n
X
0
be given by
K(x)
fG(x,t)(t)
o
dt
where
(t)
(l(t),...,n(t)),i
L
I,
i <_ i <- n,
Since f
B(y,z,%),
BIFURCATION FOR THE SOLUTIONS OF EQUATIONS
47
and
(CI[o,])n.
X
(LI)
n
For S
let
X
For y
S}.
{K
KS
let
N(y,X)
(v
+ KB(y,l).
Thus
N(y,%)
CK(X)
(5.2), (5.3)
and finding a solution to problems
is equivalent to finding a solution to
%Ky + KB(y,%).
y e N(y,l)
Now
KB(y,%)
is upper semicontinuous.
a.e.
[0,]
t
-
f
-< q(t)
[0,],
t/ t
fj
sequence h i
(el) n.
(LI)n
llf(t) ll
If
fi
and
-< q(t),
B(y,X) and Kf i
f in
Thus there exists
hi
(el p,
h i convex combinations of the
(LI)
n
Kf
g in X then
-< q(t)
Thus h i
a e
B(y,) and hjK
f
Thus as F is upper semicontinuous f
B(y,X) and
hi
g
t
[0 ]
some sub-
KB(y,X)
A similar argument using the upper semicontinuity of F with
X
R
CK(X) is upper semicontinuous. Now K:X
X is
is closed and hence compact.
respect to
As B is convex valued and K is
II (’:)(t)l
so KB(y,X) is equicontinuous in X.
weakly in
fi
such that
CK(X)
R
B(y,X) implies f
Since f
[[fi(t)[]
so
X
This can be seen as follows.
linear KB is convex valued.
lf(t)ll
CK(X) and KB
(X,y,z) shows KB
completely continuous with eigenvalues
i
n
1,2,3,... all of multiplicity one.
From (5.4) we see that for
R and y
sup
as y
I
0 in X; here
][u[l
and
{]
X
]u]]:u_. KB(Y,) )
[[Y[I
are the X norms of u and y respectively.
conditions of Theorem 4.2 are satisfied and the points
the problem
(5o2), (53)
REMARK 5.1.
0
(0,n 2) are
has nontrivial solutions (y,X) near
Thus the
bifurcation points and
(0,n’).
The upper semicontinuity of F can be relaxed to
E. U. TARAFDAR AND H. B. THOMPSON
48
(i)
F(t,’,’,’)
(ii)
F(’,%,y,z,)
(iii)
is
upper semicontinuous for almost every t e [0,]
is measurable for all
(%,y,z) e R
Rn
Rn
F is closed convex valued
and
(iv)
there exists
Rn
for each (%,y,z) e R
f:[O,]
Rn
Rn
measurable such that
f(t) e F(t,%,y,z)
for all t e
[0,]
IIf(t) II
for almost every t
Note.
L 1 such that
and there exists fixed q
_<
q(t)
[0,].
The permanent address of the first author is: Department of Mathematics,
University of Queensland, St. Lucia, Queensland, Australia, 4067.
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