Necessary and Sufficient Condition for Solvability of a Partial Integral Equation
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| Cite this: | Necessary and sufficient condition for solvability of a partial integral equation / Yu.Kh. Eshkabilov // Methods of Functional Analysis and Topology. — 2009. — Т. 15, № 1. — С. 67-73. — Бібліогр.: 14 назв. — англ. |
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| citation_txt | Necessary and sufficient condition for solvability of a partial integral equation / Yu.Kh. Eshkabilov // Methods of Functional Analysis and Topology. — 2009. — Т. 15, № 1. — С. 67-73. — Бібліогр.: 14 назв. — англ. |
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Methods of Functional Analysis and Topology
Vol. 15 (2009), no. 1, pp. 67–73
NECESSARY AND SUFFICIENT CONDITION FOR SOLVABILITY
OF A PARTIAL INTEGRAL EQUATION
YU. KH. ESHKABILOV
Abstract. Let T1 : L2(Ω2) → L2(Ω2) be a partial integral operator [4,7] with the
kernel from C(Ω3) where Ω = [a, b]ν , ν ∈ N is fixed. In this paper we investigate
solvability of the partial integral equation f − �T1f = g0 in the space L2(Ω2) in
the case where � is a characteristic number. We prove a the theorem that gives
a necessary and sufficient condition for solvability of the partial integral equation
f − �T1f = g0.
In models of solid state physics [1] and also in the lattice field theory [2], there ap-
pear so-called discrete Schrodinger operators, which are lattice analogues of the usual
Schrodinger operators in a continuous space. The study of spectra of lattice Hamilto-
nians (that is, discrete Schrodinger operators) is an important topic in mathematical
physics. Nevertheless, when studying spectral properties of discrete Schrodinger opera-
tors there appear partial integral equations in a Hilbert space of multi-variable functions
[1,3]. Therefore, to investigate spectra of Hamiltonians considered on a lattice, a study
of the solvability problem for a partial integral equations in L2 is essential (and even
interesting from the point of view of functional analysis).
A question on existence of a solution of partial integral equation (PIE) for functions
of two variables was considered in [4–8] and other works. In the work by the author
[9], the PIE f − κT1f = g0 was studied in the space L2(Ω
2), where Ω = [a, b]ν , for a
partial integral operator (PIO) T1 : L2(Ω
2) → L2(Ω
2) with the kernel k(x, s, y) being a
continuous function in three variables on Ω3. The concept of a determinant for the PIE
as a continuous function on Ω and the concepts of a regular number, a singular number,
a characteristic number, and an essential number for a PIE are given. Theorems on
solvability of the PIE are proved in the case where κ is a regular and essential number
[9]. In this paper we study solvability of the PIE f−κT1f = g0 when κ is a characteristic
number, i.e., the paper continues the work by the author [9].
Let L0 = L0(Ω) be a space of classes of complex-valued measurable functions b =
b(y) on Ω. We denote by L2,0(Ω
2) the totality of classes of complex-valued measurable
functions f(x, y) on Ω × Ω satisfying the condition:
∫
|f(x, y)|2dx exists for almost all
y ∈ Ω. It is easy to note that L2,0(Ω
2) is a linear space over C and L2(Ω
2) ⊂ L2,0(Ω
2).
For each b(y) ∈ L0 and f(x, y) ∈ L2,0(Ω
2), we define the function b ◦ f by the formula
(b ◦ f)(x, y) = b(y)f(x, y). Then for any b ∈ L0 we have b ◦ f ∈ L2,0(Ω
2), where f ∈
L2,0(Ω
2). For any f, g ∈ L2,0(Ω
2), the integral
∫
f(x, t)g(x, t) dx exists for almost all
t ∈ Ω and ϕ(t) =
∫
f(x, t)g(x, t) dx ∈ L0.
Let ∇ be the Boolean algebra of idempotents in L0. A system {f1, f2, . . . , fn} ⊂
L2,0(Ω
2) is called ∇-linearly independent, if for all π ∈ ∇ and b1(y), b2(y), . . . , bn(y) ∈ L0
from
∑n
k=1 π ◦ (bk ◦ fk) = θ it follows that π · b1 = π · b2 = · · · = π · bn = θ [10,11].
2000 Mathematics Subject Classification. 45A05, 45B05, 45C05, 45P05.
Key words and phrases. Partial integral operator, partial integral equation, the Fredholm integral
equation, L0-valued internal product.
67
68 YU. KH. ESHKABILOV
Consider the mapping 〈·, ·〉 : L2,0(Ω
2)× L2,0(Ω
2)→ L0 acting by the rule
〈f, g〉 =
∫
f(s, y)g(s, y)ds, f, g ∈ L2,0(Ω
2).
For every b ∈ L0, we have 〈b◦f, g〉 = b · 〈f, g〉, where f, g ∈ L2,0(Ω
2), i.e., the mapping
〈·, ·〉 satisfies the condition of L0-valued internal product [12].
In the space H = L2,0(Ω
2), we consider a partial integral operator (PIO) S defined by
Sf =
∫
Ω
q(x, s, y)f(s, y) ds, f ∈ H,
where q(x, s, y) ∈ L2(Ω
3). The function q(x, s, y) is called kernel of the PIO S.
The kernel q(s, x, y) corresponds to the adjoint operator S∗, i.e.,
S∗f =
∫
Ω
q(s, x, y)f(s, y) ds, f ∈ H.
Let Ω′ = {α ∈ Ω : q(x, s, α) ∈ L2(Ω
2)}. Consider a family of compact operators
{Sα}α∈Ω′ in L2(Ω) associated to S by the following formula
Sαϕ =
∫
Ω
q(x, s, α)ϕ(s) ds, ϕ ∈ L2(Ω) (α ∈ Ω′),
where q(x, s, y) is the kernel of S.
Further, if no set of integration is indicated, we mean integration over the set Ω.
Now we consider the equation
(1) f − κSf = g0
on the space H where f is an unknown function from H, g0 ∈ H is a given function,
κ ∈ C is a parameter of the equation.
For each n ∈ N, we define a measurable function
Π(n) = Π(n)(x1, . . . , xn, s1, . . . , sn, α)
on Ωn × Ωn × Ω by means of the order n determinant,
Π(n)(x1, . . . , xn, s1, . . . , sn, α) =
∣
∣
∣
∣
∣
∣
∣
q(x1, s1, α) . . . q(x1, sn, α)
...
...
...
q(xn, s1, α) . . . q(xn, sn, α)
∣
∣
∣
∣
∣
∣
∣
.
Now, for every κ ∈ C we ”formally” define functions D1(y) = D1(y; κ) on Ω and
M1(x, s, y) = M1(x, s, y; κ) on Ω3 by means of the sum of measurable functional series
composed from sequences of measurable functions dn(y) on Ω and qn(x, s, y) on Ω3,
respectively, by the following rules
(a) D1(α) = D1(α; κ) = 1 +
∑
n∈N
(−κ)n
n!
dn(α), α ∈ Ω,
and
(b) M1(x, s, α) = M1(x, s, α; κ) = q(x, s, α) +
∑
n∈N
(−κ)n
n!
qn(x, s, α), (x, s, α) ∈ Ω3,
where
dk(α) =
∫
. . .
∫
Π(k)(ξ1, . . . , ξk, ξ1, . . . , ξk, α) dµ(ξ1) . . . dµ(ξk),
qk(x, s, α) =
∫
. . .
∫
Π(k+1)(x, ξ1, . . . , ξk, s, ξ1, . . . , ξk, α) dµ(ξ1) . . . dµ(ξk).
THE NECESSARY AND SUFFICIENT CONDITION FOR SOLVABILITY . . . 69
Lemma 1. For each κ ∈ C the functions D1(y) = D1(y; κ) (a) and M1(x, s, y) =
M1(x, s, y; κ) (b) are measurable on Ω and Ω3, respectively. Moreover, for almost all
α ∈ Ω, there exists the integral
∫ ∫
|M1(x, s, α)|2dxds.
Proof. Let κ ∈ C be an arbitrary fixed number. We respectively denote by ∆
(1)
α (κ) and
M
(1)
α (x, s; κ) the Fredholm determinant and the Fredholm minor of the operator I−κSα
for α ∈ Ω′, where I is the identity operator in L2(Ω). Let ϕn(y) and ψn(x, s, y) be the
partial sums of the functional series (a) and (b), respectively. We have the sequences
of measurable functions ϕn(y) on Ω and ψn(x, s, y) on Ω3 such that limn→∞ ϕn(y) =
∆
(1)
y (κ) = D1(y; κ) for almost all y ∈ Ω and limn→∞ ψn(x, s, y) = M
(1)
y (x, s; κ) =
M1(x, s, y; κ) for almost all (x, s, y) ∈ Ω3. Therefore, the function D1(y) = D1(y; κ)
and the function M1(x, s, y) = M1(x, s, y; κ) are measurable on Ω and Ω3, respectively.
It is known that if the kernel h(x, s) of the integral operator Aϕ =
∫
h(x, s)ϕ(s) ds,
ϕ ∈ L2(Ω), is an element of the space L2(Ω
2), then the minor M(x, s; κ) of the operator
I − κA is also an element of the space L2(Ω
2). Hence we have
∫ ∫
|M1(x, s, α)|2dxds <∞ for almost all α ∈ Ω. �
The measurable functions D1(y) = D1(y; κ) and M1(x, s, y) = M1(x, s, y; κ) are,
respectively, called the determinant and the minor of the operator E−κS, κ ∈ C, where
E is the identity operator in L2,0(Ω
2).
Lemma 2. Let S : L2,0(Ω
2) → L2,0(Ω
2) be a PIO with a kernel q ∈ L2(Ω
3). If the
homogeneous equation ϕ − κSαϕ = θ, κ ∈ C, has only the trivial solution in L2(Ω) for
almost all α ∈ Ω′, then PIE (1) is solvable in the space L2,0(Ω
2) for every g0 ∈ L2,0(Ω
2).
Proof. Let κ ∈ C, g0(x, y) be an arbitrary function from the space L2,0(Ω
2). Let the
homogeneous equation ϕ − κSαϕ = θ have only the trivial solution in the space L2(Ω)
for almost all α ∈ Ω′. Then D1(α) = D1(α; κ) �= 0 for almost all α ∈ Ω and the equation
ϕ(x) − κ(Sαϕ)(x) = hα(x) has a solution ϕα(x) ∈ L2(Ω) for almost all α ∈ Ω′ where
hα(x) = g0(x, α) ∈ L2(Ω). Moreover, the solution ϕα(x) has the form [13]
ϕα(x) = hα(x) + κ
∫
M1(x, s, α; κ)
D1(α; κ)
hα(s) ds.
We have
∫ ∫
∣
∣
∣
∣
M1(x, s, α; κ)
D1(α; κ)
∣
∣
∣
∣
2
dxds <∞ for almost all α ∈ Ω.
This means that we can define a PIO W = W (κ) : L2,0(Ω
2) → L2,0(Ω
2) with the
kernel [14]
M1(x, s, α; κ)
D1(α; κ)
.
Therefore we have f0(x, y) = g0(x, y) + κ(Wg0)(x, y) ∈ L2,0(Ω
2) and ϕα(x) = f0(x, α)
for almost all α ∈ Ω. So the function f0(x, y) is a solution of the equation (1). �
The following two propositions are proved analogously to Propositions 1 and 2 from [9].
Proposition 1. Let S : L2,0(Ω
2)→ L2,0(Ω
2) be a PIO with the kernel q ∈ L2(Ω
3). Then
the following two conditions are equivalent:
(i) a number λ ∈ C is an eigenvalue of the operator S;
(ii) a number λ ∈ C is an eigenvalue of operators {Sα}α∈Ω0
, where Ω0 is a subset of
Ω such that µ(Ω0) > 0.
Proposition 2. If λ ∈ C is an eigenvalue of a PIO S : L2,0(Ω
2) → L2,0(Ω
2) with a
kernel q(x, s, y) ∈ L2(Ω
3), then the number λ is an eigenvalue of the operator S∗.
70 YU. KH. ESHKABILOV
Theorem 1. Let S : L2,0(Ω
2) → L2,0(Ω
2) be a PIO with a kernel q(x, s, y) ∈ L2(Ω
3).
Then every eigenvalue of the PIO S corresponds only to a finite number of ∇-linearly
independent eigenfunctions.
Proof. Let λ ∈ C be an eigenvalue of the PIO S and
(2) f1, f2, . . . , fm
be some ∇-linearly independent eigenfunctions, i.e.,
(3) λfj(x, y) = (Sfj)(x, y), j = 1, 2, . . . , m.
Since any linear combination of the eigenfunctions (2) of the operator S with coeffi-
cients from L0 is also an eigenfunction, we can apply to the functions (2) the process
of L0-orthogonalization [12]. Thus, we can assume that the functions (2) are mutually
orthogonal and normed in the sense of L0-valued internal products, i.e.,
〈fi, fj〉 = 0, i �= j and 〈fi, fi〉 = 1.
Therefore we can rewrite (3) in the following form:
λ · fj(x, y) =
∫
q(x, s, y) · fj(s, y)ds.
From here, it is easy to see that for almost all x ∈ Ω the left hand-side of this equality
is an L0-valued Fourier coefficient of the function q(x, s, y) and it is a function of (s, y)
with respect to the orthogonal normed system (2). By the Bessel inequality [12], one can
write
|λ|2
m
∑
j=1
|fj(x, y)|2 ≤
∫
|q(s, x, y)|2ds for almost all x ∈ Ω.
If we integrate both parts of this inequality with respect to x and y, we obtain
m ≤ |λ|−2
∫ ∫ ∫
|q(x, s, y)|2dxdsdy <∞.
Hence, the number of ∇-linearly independent functions corresponding to the eigenvalue
λ is finite. �
Let S be a PIO with a kernel q(x, s, y) ∈ L2(Ω
2). A number κ0 ∈ C is called a
characteristic value of the PIE f−κ0Sf = g0 if the homogeneous equation f−κ0Sf = 0
has a non-trivial solution. From here, it is clear that any characteristic value κ0 of the
PIE f − κSf = g0 is non-zero.
Corollary 1. Let S : L2,0(Ω
2) → L2,0(Ω
2) be a PIO with a kernel q(x, s, y) ∈ L2(Ω
3).
Then any characteristic value of the PIE f − κSf = g0 corresponds only to a finite
number of ∇-linearly independent eigenfunctions.
Theorem 2. Let κ be a characteristic number of the PIE (1). Then the homogeneous
PIE
(4) f − κSf = 0
and the adjoint homogeneous PIE
(4′) f − κS∗f = 0
have the same number of ∇-linearly independent solutions.
Proof. Let f1, . . . , fm and g1, . . . , gn be ∇-linearly independent solutions of the the ho-
mogeneous equations (4) and (4′), respectively. Assume that m < n. We can suppose
that f1, . . . , fm and g1, . . . , gn are orthonormal systems in the sense of L0-valued internal
product.
THE NECESSARY AND SUFFICIENT CONDITION FOR SOLVABILITY . . . 71
Define the function
p(x, s, y) = q(x, s, y)−
m
∑
j=1
fj(s, y)gj(x, y).
We have p(x, s, y) ∈ L2(Ω
3) since fj, gk ∈ L2,0(Ω
2). Consider two homogeneous PIE,
(5) f − κWf = 0
and
(5′) f − κW ∗f = 0,
where W is the PIO with the kernel p(x, s, y).
Let h(x, y) be a solution of the equation (5). Then we have
〈h, gj〉 = 〈κWh, gj〉 = 〈h, κS∗gj〉 − κ 〈h, fj〉 = 〈h, gj〉 − κ 〈h, fj〉 , j = 1, 2, . . . , m.
Hence, since κ �= 0,
(6) 〈h, fj〉 = 0, j = 1, 2, . . . , m.
Thus, any solution of the equation (5) satisfies the conditions (6). But by virtue of this
conditions, one can rewrite the equation (5) in the form f − κSf = 0, i.e., any solution
of the equation (5) satisfies the equation (4), too. We obtain that a solution h(x, y) of
the equation (5) is in the form
h(x, y) =
m
∑
j=1
(bj ◦ fj) (x, y), bj ∈ L0, j = 1, 2, . . . , m.
But we have 0 = 〈h, fk〉 =
∑m
j=1 〈bj ◦ fj , fk〉 =
∑m
j=1 bj · 〈fj, fk〉 = bk, k = 1, 2, . . . , m.
Thus, we have h(x, y) = θ, i.e., the homogeneous PIE (5) has only the trivial solution. We
show that the adjoint equation (5′) has non-trivial solutions. If we substitute g(x, y) =
gk(x, y), where k > m, in the equation (5′) then we obtain gk = κ∗W ∗gk. Thus, we obtain
the contradiction to Proposition 2: the equation (5) has only the trivial solution, but the
adjoint equation (5′) has a non-trivial solution. Hence the case m < n is impossible. One
can prove similarly that the case m > n is also impossible and we obtain that m = n. �
Theorem 3. Let κ0 be a characteristic number of the PIE (1). Then
a) the homogeneous equation f − κ0Sf = 0 has a non-trivial solution, moreover,
the set of all solutions of the homogeneous equation is an infinite dimensional
subspace of H;
b) PIE (1) is solvable if and only if the given function g0 satisfies the condition
(I) 〈g0, g〉 = 0,
where g ∈ H is an arbitrary solution of the adjoint homogeneous equation f −
κ0S
∗f = 0.
Proof. The proof of the property a) follows immediately from Proposition 1 and Propo-
sition 3 from [9]. We prove the property b).
i) (”if-part”) Let κ0 be a characteristic number of the PIE (1) and f0 ∈ H be a
solution of the PIE (1) and g ∈ H be an arbitrary solution of the adjoint homogeneous
equation f − κ0S
∗f = 0. Then
〈f0, g〉 = 〈g0 + κ0Sf0, g〉 = 〈g0, g〉+ 〈κ0Sf0, g〉 = 〈g0, g〉+ 〈f0, κ0S
∗g〉
= 〈g0, g〉+ 〈f0, g〉.
Therefore we have 〈g0, g〉 = 0.
72 YU. KH. ESHKABILOV
ii) (”only if”-part) Let κ0 be a characteristic number of the PIE (1). Suppose that
g0 satisfies the condition (I), i.e., 〈g0, g〉 = 0 for every solution g ∈ H of the equation
f − κ0S
∗f = 0.
Consider the function p(x, s, y) ∈ L2(Ω
3) given by the equality
p(x, s, y) = q(x, s, y)−
m
∑
j=1
fj(s, y)gj(x, y),
where f1, f2, . . . , fm and g1, g2, . . . , gm are orthonormal systems of solutions of the equa-
tions (4) and (4′), respectively, in the sense of L0-valued internal product. Then for
almost all α ∈ Ω the homogeneous Fredholm equation ϕ − κ0Wαϕ = 0 has in L2(Ω)
only the trivial solution [13], where Wα is an integral operator in L2(Ω) with the kernel
p(x, s, α). Hence, by Lemma 2, the PIE f − κ0Wf = g0 has a solution f0 ∈ H of the
form
f0 = g0(x, y) + κ0Sf0(x, y)− κ0
m
∑
j=1
〈f0, fj〉 · gj(x, y).
Therefore, we obtain that
〈f0, gk〉 = 〈g0, gk〉+ 〈κ0Sf0, gk〉 −
m
∑
j=1
〈f0, fj〉 · 〈κ0gj , gk〉
= 〈f0, κ0S
∗gk〉 − κ0〈f0, fk〉 = 〈f0, gk〉 − κ0〈f0, fk〉,
i.e., 〈f0, fk〉 = 0, since κ0 �= 0. Thus, the solution f0 of the equation f −κ0Wf = g0 has
the form f0 = g0 + κ0Sf0 and, hence, the function f0 is also a solution of the PIE (1) at
κ = κ0. �
If there exists a number C such that
(II) |b(t)| ≤ C for almost all t ∈ Ω,
then the PIO S is a bounded operator on the space L2(Ω
2), i.e., Sf ∈ L2(Ω
2), ∀f ∈
L2(Ω
2) ⊂ L2,0(Ω
2) and ‖Sf‖L2(Ω2) ≤ C0‖f‖L2(Ω2) for all f ∈ L2(Ω
2), where C0 is a
positive number,
b(t) =
∫ ∫
|q(x, s, t)|2dxds.
Let k(x, s, y) ∈ C(Ω3). Then the subspace L2(Ω
2) is invariant for the PIO T1 :
(T1f)(x, y) =
∫
k(x, s, y)f(s, y)ds. Therefore it is possible to study solvability for the
PIE
(7) f − κT1f = g0
in the space L2(Ω
2) where f is an unknown function from L2(Ω
2), go ∈ L2(Ω
2) is a given
(known) function, κ ∈ C is a parameter of the equation.
Let χT1
be a set of characteristic numbers for the PIE (7) (see [9]). the definition of
a characteristic number [9] and the obtained Theorem imply the following.
Theorem 4. Let κ0 ∈ χT1
. Then
a) the homogeneous equation f − κT1f = θ has a non-trivial solution, moreover,
the set of all solutions of the homogeneous equation is an infinite dimensional
subspace of L2(Ω
2);
b) PIE (7) is solvable if and only if the given function g0 satisfies the condition
(III)
∫
g0(s, t)g(s, t) ds = 0 for almost all t ∈ Ω,
where g ∈ L2(Ω
2) is an arbitrary solution of the adjoint homogeneous equation
f − κ0T
∗
1 f = θ.
THE NECESSARY AND SUFFICIENT CONDITION FOR SOLVABILITY . . . 73
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14. K. K. Kudaybergenov, ∇-Fredholm operators in Banach-Kantorovich spaces, Methods Funct.
Anal. Topology 12 (2006), no. 3, 234–242.
National University of Uzbekistan, Tashkent, Uzbekistan
E-mail address: yusup62@rambler.ru
Received 07/03/2007; Revised 12/12/2008
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| id | nasplib_isofts_kiev_ua-123456789-5698 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1029-3531 |
| language | English |
| last_indexed | 2025-12-07T15:10:33Z |
| publishDate | 2009 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Eshkabilov, Yu.Kh. 2010-02-02T12:57:23Z 2010-02-02T12:57:23Z 2009 Necessary and sufficient condition for solvability of a partial integral equation / Yu.Kh. Eshkabilov // Methods of Functional Analysis and Topology. — 2009. — Т. 15, № 1. — С. 67-73. — Бібліогр.: 14 назв. — англ. 1029-3531 https://nasplib.isofts.kiev.ua/handle/123456789/5698 en Інститут математики НАН України Necessary and Sufficient Condition for Solvability of a Partial Integral Equation Article published earlier |
| spellingShingle | Necessary and Sufficient Condition for Solvability of a Partial Integral Equation Eshkabilov, Yu.Kh. |
| title | Necessary and Sufficient Condition for Solvability of a Partial Integral Equation |
| title_full | Necessary and Sufficient Condition for Solvability of a Partial Integral Equation |
| title_fullStr | Necessary and Sufficient Condition for Solvability of a Partial Integral Equation |
| title_full_unstemmed | Necessary and Sufficient Condition for Solvability of a Partial Integral Equation |
| title_short | Necessary and Sufficient Condition for Solvability of a Partial Integral Equation |
| title_sort | necessary and sufficient condition for solvability of a partial integral equation |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/5698 |
| work_keys_str_mv | AT eshkabilovyukh necessaryandsufficientconditionforsolvabilityofapartialintegralequation |