297
Internat. J. Math. & Math. Sci.
VOL. 12 NO. 2 (1989) 297-304
ON THE NUMBER OF SOLUTIONS OF SOME INTEGRAL
ECIUATIONS ARISING IN RADIATIVE TRANSFER
IOANNIS K. ARGYROS
Department of Mathematics
New Mexico State University
Las Cruces, NM 88003
(Received on March I0, 1988 and in revised form on August 4, 1988)
We discuss the number of solutions of some nonlinear integral equations
ABSTRACT.
neutron transport and in the kinetic
arising in the theories of radiative transfer,
theory of gases.
KEYWORDS AND PHRASES.
Radiative transfer, integral equation.
65R20, 45LI0.
A.M.S. 1980 CLASSIFICATION CODES:
I.
In
INTRODUCTION.
the
theories
[3], [4] an important role
transport
of
radiative
transfer
[I]
[2]
and
neutron
is played by nonlinear integral equations of the
form
H(x)
The known
+
function
xH(x)/
is
(t)H(t)
{
assumed
(I.I)
dt.
to be
nonegative,
and measurable on
bounded,
[0,I], and
a positive, continuous solution H of (I.I) is sought.
Chandrasekhar’s treatment of (I.I) can be found in [2]. The first proof however
of the existence of a solution of (I.I) was given by M. Crum,
equation in the complex plane
[5].
Crum also showed that if
who considered the
f
(t)dt
I/2
then
(I.I) has at most two solutions which are bounded in [0,I] and in case
0
(t)dt
=I/2
there is only one such solution.
in order to prove existence of solutions of
are
not
necessarily solutions
of
(I.I).
(I.I)[I].
C.
C. Fox [6] solved simpler equations
But the solution of Fox’s equation
Stuart
[7] gave a nonconstructive
existence proof for (I.I) using the Leray-Schauder degree theory but did not discuss
the number or location of solutions.
Darbo for a
set
B. Cahlon and M. Eskin [3] used a theorem of
contraction map to prove
a
nonconstructive
existence
theorem for
(1.1).
Finally, C. Kelley [8] had solved some interesting generalizations of (I.I) using
the solutions of finite rank approximations of solutions of (I.I).
I.K. ARGYROS
298
Here we consider the generalized equation:
xH(x)o
+
H(x)
The known kernal function
(a) 0
<
k(x,t)
<
k(x,t)
(1.2)
k(x,t)(t)H(t)dt.
is a measurable function on
[0,I] x [0,I] satisfying
[0,I],
for all x, t
and
(b) k(x,t) + k(t,x)
for all x, t
I/2
(t)dt
We show that whenever
a minimal solution H can be found using a
specific iteration.
Flnally,
assumptlon,
same
the
under
we
provide
a
way
of
constructing
new
nonmlnlmal solutions H of (I.I) in terms of the minimal solution.
2.
BASIC RESULTS.
We denote by C[0,1] the Banach space of all real continuous functions on [0,I]
with the maximum norm
I"
u(t)
max
Otl
We now llst the following well-known theorem whose proof can be found in
u
[2, pp. 106-107].
If H is a solution of (1.2), then either
THEOREM I.
or
A
condition that (Io2) has a solution is tt
necessa
A function H
e
C[0,1] satisfies the equation
If and only if H satisfies (1.2) and (2.1).
Chandrasekr in [2], after proving tt a solution H of (I.I) satisfies either
(2.1) or (2.2), clal tt, in fact, H st satisfy (2.1). is claim is not true
because as we sh, there always exists a solution H satisfying (2.1), but In ny
cases there exists a second solution H satisfying (2.2) and not (2.1).
t
C[O,I], tt is, if
(P2(X) for all x [0,I] and
be the natural partial ordering on
C[0,1], then pl
PI’ P2
P2
if
Pl(X)
follong:
d-
D
tz-
zS
{p C[0,1llp(x)
,(t)dt
Y
d, x
[0,11 },
the operator
R:D
R(p(x)
and for d
>
+ p(x)
0, define the operator F
f
D
C[0,1] by
k(x,t)(t)p(t)dt, p
C[0,1] by
D
define
the
INTEGRAL EQUATIONS ARISING IN RADIATIVE TRANSFER
/0 k(t’x)(t)(p(t))-Idt’
d +
F(p(x))
It is routine to verify that R is isotone, that is if
and F is antltone, that is if
Finally,
denote
Pl
then
P2
F(P2)
(respectively,
by
d)
p
D.
Pl
P2
299
then
R(Pl)
R(P2)
F(Pl).
the
function with constant
value
(respectively, d).
We can now prove the proposition:
PROPOSITION.
Asstne that the kernel function k(x,t) is as in the introduction and
satisfies the condition
and some b
PROOF.
A
{p
>
Ix- ,I
0.
I, 2,
n
is equlcontinuous.
Let H be a solution satisfying (1.2) and (2.1) and
H}.
C[0,1]/I
p
C[O,I] by
A
Define Q
/0 k(x,t)(t)p(t)dt,
Q(p(x))
>
Let
a
Rn(l),
Then the sequence
(t)H(t)dt
0
<
then there exists a, 0
<
and
<
+
>
@(t)A(t)dt
0.
a
<
[0,11, p
x
A.
I, such that
[0,I],
Then for x, y
=
se
A) is
equicontinuous.
Let p
A and x
[0,1], then
<
(t)HCt)dt
Therefore there exists c, 0
x
c
<
1.
<
I, such that Q(p(x))
c for all p
A and
[o,].
For any
Ig(x)- g(y)l
The
<
1- d
function
exists 0
<
0
>
60
0, there exists
-1
(I- c) g0 if
IH[[
R(1)
61 60
is
continuous
>
0 such that for every g (Q(A),
Ix- Yl
and
<
hence
6
(since
0
uniformly
Q(A)
continuous,
such that
R((x))
-R((Y))I < 0
f
x
is
-yl <
,
equlcontlnuous).
therefore
there
I.K. ARGYROS
300
We shall show that the same
61
works for
sk(l(x)) -sk (I(Y))I <
Rn(1).
Set p=
]R(p(x))
Then if
if
0
and
Rn+l(1)
Ix- y[ < 61
if
I, 2,
for k
n.
Ix- y] < 61
p(y)Q(p(y))]
[p(x)Q(p(x))
R(p(y))]
0
Ip(x)Q(p(x))
p(x)Q(p(y))]
p(x)[Q(p(x))
Q(P(Y))!
+
]p(x)Q(p(y))
+
p(y)Q(p(y)))
P(Y)I
Q(p(y))]p(x)
that is,
<
,{
o
which completes the induction and the proof of the proposition.
TTEOREM.
2. Assume that the kernel function k(x,t) is as in the proposition.
Then the
following are true:
(a) equation (1.2) has exactly one solution H satisfying (2.1) if and only if
converges to
(2.3) holds.
0, I, 2
Moreover, the increasing sequence Rn(1), n
and
H;
0, I, 2
(b) if inequality holds in (2.3), the sequence Fn(d), n
-I
H
converges to
and
]tt-l(x)
IFn
Fn(d(x))]
F n+l
(d(x))
(d(x))],
fO
If (1.2) has a solution H, then by Theorem I,
(A).
CASE I.
PROOF.
(t)dt
Assume
I/2
It
can easily be
[0,I].
x
$(t)dt
I/2
that
verified
(2.5)
since
F
is
antitone:
F7(d) F5(d) F3(d)
F2(d) F4(d) F6(d)
d
F(d).
Working as in the proposition we can easily show that the bounded set
{F(p)/d
N
is equlcontinuous.
and
F2n+1(d),
n
0, I, 2,
have
functions v and w respectively.
p
F2n(d),
Then the sequences
n
convergent
F(d)}
I, 2,
subsequences
converging
to
the
From the monotonlcity of the above sequences and the
continuity of F we obtain
F2n(d)
F
2n+l
v,
(d)
w,
dvw,
F(v)
w
F(w)
v.
and
The
function v has
largest ntunber q, 0
<
q
minimum value
I,
with qw
v.
greater than
If q
zero,
I, then
that
so
w
v
there exists
w, that is v
w.
a
INTEGRAL EQUATIONS ARISING IN RADIATIVE TRANSFER
<
If q
I, define
on the domain of F the operator F
F(p)
F l(p)
301
by
d.
Then
d +
v
d +
Fl(W)
Fl(q-lv)
(I -q)d + q(d +
d +
qFl(v)
(I -q)d + qw
Fl(V))
ew + qw-- (e +q)w,
for some e
>
O.
But this contradicts the maxlmallty of q.
F(v)
Therefore,
w,
v
-I
H _=v
(1.2), satisfying (2.1), and the sequence
is a solution of
converges to H
-I
Fn(d),
0, 1, 2,
n
Inequality (2.5) follows from the fact that
F2k(d)
-I
H
CASE 2.
Assume that
F2k+l(d),
’0
I, 2, 3
for k
(t)dt
=I/2
.....
let {c
},
I, 2
n
be a strictly increasing
’sequence of positive numbers converging to 1, and consider the functions
c
n
,
n
I, 2, 3,
Si nce
Cn(t)dt =1/2 Cn < 1/2’
fO
has a solution
Hn for
n
it follows from Case
+
H(x)
H(x)fo k(x,t)Cn(t)H(t)dt
I, 2, 3
n
1, 2, 3,
Set M= {p
Define F: M
>
C[0,1)/r p(x)
C[0,1] by
F(p(x))
fo
+
Cnf01 k(t,x)(t)dt clf
Therefore, there exists r
[0,I]
Then for each x
Cn
n
that the equation
k(t,x)
and
Cn(t)Hn(t)dt
0 k(t,x)(t)dt
0 such that (H (x))
n
I, x
hn(X)
[0,I]}.
-1
r for each x
Then H
-I
k(t,x)(t)(p(t)) -Idt, P
each
I, 2
M, n
n
[0,I] and
M.
It is easy to verify that the set F(M) is bounded and equlcontlnuous.
Also, for each n,
H
Since
n
F(HI)
n
[I-2fO Cn(t)dt]I/2+ fO k(t,x)
(x)
F(M) for each n, some subsequence
F(Hn
), j
I, 2
....
converges to
-I
F(Hnj
HO
F(H 1)
converges in C[0,1] to some point
I, 2,
sequence
-1
c
n
(t)Hn(t)dt
I, 2,
),
so that H
and H
-I
n.
of
H
0
that is,
F(Hnl),
Then the
302
I.K. ARGYROS
F(H
HO
Then H satisfies (1.2), (2.1) and (2.2).
0
Therefore there exists a positive
(B).
(2.3)
Assume
R(1),
H and
Since
holds,
H
satisfying
suppose
H
satisfies
(1.2)
(2.1)
and
Rn(1),
n
R2(1) R3(1)
I, 2,
n
is a convergent subsequence, say R
continuity of R that R(h)
k
h
H, and, since the sequence
-I
(2.1).
Rn(1),
f k(t,x)0(t)H(t)dt
n
0, I,
It follows from the
h.
(t)dt]I/2+
2f
[I-
and
H.
Now h must satisfy either (2.1)
H, h must satisfy (2.1). Therefore, for x
[0,I],
h
that is, h
(1.2)
is uniformly bounded and equicontlnuous there
the entire sequence converges to h.
is nondecreasing,
since 0
and
it follows from the fact that R is isotone that
I R(1)
Since the sequences
2,
function
satisfies (2.3).
whenever
or
(2.2), and
H-l(x),
-I
H
Together with the inequality h
H, this implies h H.
We have proved that H is the only function satisfying both (1.2) and (2.1), and
that the increasing sequence Rn(1), n
0, I, 2,
converges to H which completes
the proof of the theorem.
COROLLARY.
[0,I] such that
f
(t)dt
I and 2 are
Suppose that
l(t)
I/2,
2(t)
I, 2.
i
corresponding to
$i’
nonnegatlve, bounded, measurable functions on
almost everywhere in
Let H
i
[0,I] and such that
be the unique solution of equations (1.2) and (2.1)
I, 2.
i
The n,
H
PROOF.
C[0,1]
C[0,1],
Ri(p(x))
+ p(x)
Define R
i
If
Pl
and
H
2
I, 2, by
f k(x,t)i(t)p(t)dt
are nonnegatlve functions in
P2
RI(P 1) R2(P2).
i--
I-nce
R(1) R(1). Since
follows that
HI H2.
RI(I
the
R2(1),
C[0,1]
R21(1) R(1),
increasing sequence
C[0,1].
p
with
Pl
P2’
then
and in general,
R(1),
converges to H
i
=I/2
i
I, 2,
it
Note that if
it follows from the previous results that the
(t)dt
function H satisfying (1.2) and (2.1) is the unique solution of (1.2), since, in this
case
(2.1)
and
(2.2)
reduce
the
to
same
equation.
However,
if
0
(t)dt
<I/2
THEOREM 3.
As sume:
(a)
(t)dt
equation (1.2) may have two distinct solutions.
<I/2and
H is the unique solution of (1.2) and (2.1);
INTEGRAL EQUATIONS ARISING IN RADIATIVE TRANSFER
(b)
.
303
the following estimate is true
(t)
>
H(t)dt
l-t
(2.6)
and
(c)
k(x,t)[
there exists functions
2(x)(1
(1 +
-kt)
WI(X)
+ kx
I’
+
l(t))]
>
C[0,1] such that
W2’ 3
O3(x)
--0, for all x, t
(0,I],
O, for all x
i(0)
(2.7)
[0,1]
(2.8)
0
and
+l(X))[ @2(x)(H(x)-
(1
(HI(x)
I)(I
+O3(x)H(x)]
1)
kx) for all x
[0,I]
6
(2.9)
where k is the unique number in (0,I) for which
1,
and the function H
is given by
l(x)
1+
H (x)
Then H
1-kx
is a solution of
(2.11)
[0,1].
H(x), x
(1.2) and (2.2) and
Hi(x) >
PROOF.
(2.10)
H(t)dt
l-kt
H(x), x
HI(0)
(0,1],
H(0).
By the monotone convergence theorem
$(t)
1-k t
lira
k+l
since (I
kt)
-1
f
H(t)dt
increases monotonleally with k, 0
(t
1-t
<
k
<
H t )d t
I,
If (2.6) holds, since
I-0.?
and since the function f
H(t)dt--
(0,I)
deflned by
f(k)
is
strictly increasing,
)dt] I/2
[I
/01
(t)
l-kt
H(t)dt
there exists a unique k
Let H
be defined as in (2.11).
each x
e [0,I]
(0,i), for which (2.10) holds.
Applying a trick used in [5], [9], we find that for
( t )d t
304
I.K. ARGYROS
02(x)
2 (x) [I
k(x,t)(t)H(t)dt +
----x-’] +
H’x
3
O3(x)
(x)
(by (2.7) and (2.9)) that is, H satisfies (1.2). Since H must satisfy either (2.1)
(2.2) and since Hl(X) > H(x), x
[0,I] (by (2.8)), H satisfies (2.2) and the
or
proof of the theorem is completed.
REMARK.
By choosing the kernel function k(x,t) to be
x
k(x, t)
x+t
x, t
[0,I]
we observe that the conditions (a) and (b) in the introduction are satisfied and that
the equaion (1.2) reduces to equation (I.I).
Moreover the conditions (2.7),
(2.8), and (2.9) can then be satisfied if we
choose
I (x)
kx,
2 (x)
l-kx
l+kx
3 (x)
2kx
l+kx
and
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I.
BUSBRIDGE, L.W.
The Mathematics of Radiative Transfe r.
Cambridge
Publ.
Cambridge, England, 1960.
2.
3.
4.
5.
CHANDRASEKHAR, S. Radiative Transfer, Oxford Univ. Press, London, 1950.
Existence Theorems for an Integral Equation of the
CAHLON, B. AND ESKIN, M.
Chandrasekhar H-equation with Perturbatlon. J. Math. Anal. and Appllc..
83(1981), 159-17 I.
Qadratic Equations and Applications to Chandrasekhar’s and
ARGYROS, I.K.
Related Equations. Bull. Ans..t.ral. math.: Sco:.Voi..32 2(1985), 275-292.
CRUM, M. On an Integrl Equation of Chandrasekhar. Quart. J. Math., 18(1947),
244-252.
A Solution of Chandrasekhar’s Integral Equation. Trans. Amer. Math.
FOX, C.
Soc., 99(1961), 285-291.
7. STUART, C.A.
Existence Theorems for a Class of Nonlinear Integral Equations.
Math. Z. 137(1974), 49-66.
8. KELLEY, C.T. Approximation of Solutions of Some Quadratic Integral Equations in
Transport Theory. J. Int.egr. Eq, 4_(1982), 221-237.
A New Approach to the H-equation of Chandrasekhar. S.I.A.M. J.
9. LEGGETT, R.W.
Math. 7, 4_(197 6), 542-550.
10. CASE, K.M. & ZWEIFEL, P.F.
Linear Transport Theory, Addison-Wesley Publ., MA,
6.
1967.
I. KANTOROVICH, L.V. AND AKILOV, G.P.
Functional Analysis in Normed
Pergamon Pres, London, 1964.
12. KURATCSKI, C. Sur les Espaces Complets. Fund. Math. 15(1930), 301-309.
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