1. Introduction
A kind of Volterra integral equations with weakly singular kernels arisen in 1975 [1] from some heat condition problems with mixed-type boundary conditions is transformed by Watson transforms [2] and the convolution theorem [3]. In [4], the author generalizes such kind of equations into cordial Volterra integral equations (CVIEs) with the form
(1)
where
, the core
and the cordial Volterra integral operator is defined by
![](//html.scirp.org/file/74149x9.png)
CVIEs appear in a lot of application models, such as Diogo core
, linear Lighthill’s equation (
), and so on.
It is shown that the cordial Volterra integral operator
in the Banach space
is noncompact and its spectrum is a non-countable set, i.e.,
![](//html.scirp.org/file/74149x14.png)
where
![](//html.scirp.org/file/74149x15.png)
In [5], the author describes the eigenvalues and eigenfucntions of the operator
on the space
when
with some
:
1) the point spectrum of
is exactly the set
;
2) the dimension of the null space
is the sum of the multiplicities of the roots of
in the complex plane
;
3) the linearly independent eigenfunctions are given by
![]()
where
is the multiplicity of the root
of
.
The pure Volterra integral equations with vanishing delay (VIEwND) are initially studied in [6] and a special form of VIEwND, proportional delay differential equations, is widely used in practical applications, for example, electrodynamics [7] [8], nonlinear dynamical systems [9] [10], and also the survey papers [11] [12]. In this paper, we consider the CVIEs with a vanishing delay,
(2)
where
is a continuous delay function such that
and
for all
and the operator with delay is similarly defined by
(3)
Besides the existence and uniqueness of solutions to (2), it is more interesting how the eigenvalues and eigenfunctions of the operators are influenced by vanishing delays. In Section 2, we show that the proportional delay
,
, is the only one that replicates all eigenfunctions
,
or
. For such a delay, we describe the spectrum, eigenvalues and eigenfunctions of the operator
. In Section 3, we present a necessary and sufficient condition for the compactness of the operator
with a vanishing delay. Based on these discussions, we present the existence, uniqueness and the construction of solutions to (2).
2. Propositional Delays
For a vanishing delay
satisfying that
(D1)
,
(D2)
for all
,
(D3)
is a continuous function in the interval
and
exists,
the operator (3) is rewritten as the following form
(4)
where the function
is a well-defined continuous function in the whole interval
. Obviously
![]()
and
for all
.
The cordial Volterra integral operator with a vanishing delay (3) is also written as a cordial Volterra integral operator with a variable kernel, i.e.,
![]()
where the discontinuous kernel
is defined by
![]()
The properties of the operator
with continuous kernels are investigated in [13] such as it is compact if and only if
. From the above definition, the discontinuous function
always satisfies
, but the compactness of the operator
is influenced not only by the core but also by the value of
(see in Corollary 2.3 and Theorem 3.1).
Theorem 2.1. Assume that the function
.
1) The operator
is a bounded operator from
to
.
2) If all power-functions
or
, are eigenfunctions of
, then
![]()
where for
, the integration is defined by
![]()
Proof. (i) For
,
and
, there exists a
such that
![]()
and for all
with ![]()
,
since
is uniformly continuous on the closed interval. The uniform continuity of
implies that there exists a
such that
for all
with
.
We, without loss of generality, assume that
in the following estimation. Then
![]()
Hence
maps
to
and its boundedness comes from
![]()
2) Without loss of generality, suppose that
and
for some
. Then similarly to the approach in [4], there exists a polynomial
such that
![]()
Since
, is an eigenfunction of
,
![]()
is also independent of
for
. Thus
![]()
![]()
and hence
![]()
This contradiction implies the proof is complete. ![]()
Remark 2.2. In [4], the author shows that an operator
mapping
to
has the two properties:
1)
is a bounded operator;
2) all power-functions
,
or
, are eigenfunctions of
;
if and only if
is a cordial Volterra integral operator. While including vanishing delays, the two properties only hold for a proportional delay
,
.
For a core
, we define an integration function of the core by
![]()
If
for
with some
(or
), then CVIEs naturally reduce to a proportional delay form
(5)
where the corresponding operator has the form
with
and
![]()
Corollary 2.3. Assume that
and
is a strictly increasing function for
. Then a cordial Volterra integral operator with vanishing delays opposites the two properties in Remark 2.2 if and only if the delay
is a proportional delay. Of course it is a noncompact operator.
Proof. By Theorem 2.1, one obtains that
is a constant. Thus the proof is completed by
. ![]()
Based on
, some more detailed properties on cordial Volterra integral operators with a proportional delay are presented in the following theorem.
Theorem 2.4. Assume that a core
with some
,
,
and
for
. Then
1) The spectrum of
is given by
![]()
where
.
2) The point spectrum of
is exactly the set
.
3) The dimension of the null space
is the sum of the multiplicities of the roots of
in the complex plane
.
4) The linearly independent eigenfunctions are given by
![]()
where
is the multiplicity of the root
of
.
5) The range of the operator
is the whole space
if and only if
.
Both the existence and uniqueness of solutions to (5) are valid when the parameter
does not lie in the spectrum of the corresponding operators. On the other hand, for
lying in the spectrum, by the same technique in [5], we are also able to construct solutions to (5). For convenience, we review some notations in [5]:
1)
with different parameters ![]()
2)
with the norm
![]()
Theorem 2.5. Assume that
with some p > 1 and that
,
, and
for
. Let
. Then there exist
distinct points in
such that the following statements are true.
1) For
, there exists a unique solution
to (5) that continuously depends on
, and all solutions have the form
![]()
where
is a linear combination of functions fo functions
,
, and
is a root of
with multiplicity
.
2) For
, there exists at most one solution to (5), and there exists exactly one solution to (5) when
for any
.
3) For
, there exists at most one solution
belonging to
, and there exists a unique solution in
for any
and
. All solutions have the form
![]()
where
is linearly combined by such functions 1 (if
) and
,
, and
is a root of
with multiplicity
.
3. General Vanishing Delays
For a more general vanishing delay, the compactness of the cordial Volterra integral operators is influenced by the value of
.
Theorem 3.1. Assume that
and that the delay function
satisfies the assumptions (D1), (D2), (D3). Then the operator
is compact in
if and only if
.
Proof. From the definition of the function
, it is known that
. In Lemma 3.6, one obtains from
that
for all
. Hence by Ascoli-Arzela theorem, the compactness of the cordial Volterra integral operator
with such a vanishing delay term is shown in Lemma 3.7. The proof will be completed, when the non-compactness of the operator is proved in Lemma 3.8. ![]()
The simplest compact condition according to Theorem 3.1 is
.
Corollary 3.2. Assume that
and that the delay function
satisfies the assumptions (D1), (D2), (D3). Then the operator
is compact in
for any core
provided that
.
Remark 3.3. Consider the constant core
. Then
1)
,
, are non-compact in
.
2) For
,
is compact in
.
The existence and uniqueness of solutions to (2) is similar to the classical second kind of VIEs when the corresponding operator is compact.
Theorem 3.4. Assume that
and that the delay function
sa- tisfies the assumptions (D1), (D2), (D3) and that
. Then for all
and all
, there exists a unique solution to (2).
Proof. In Lemma 3.9, it is shown that the null space of the operator
in
is
, which together with the compactness of
implies that the operator
has a bounded inverse in
(see in [14]). Hence the proof is complete. ![]()
Example 3.5. Consider the following CVIEs with a vanishing delay
1)
and
for
;
2)
(the linear form of Lighthill’s equations) and
for
;
3)
and
for
.
Then the corresponding operators are compact and there exists a unique solution to (2) for
and
.
Theorems 3.1 and 3.4 are proved by the following lemmas.
Lemma 3.6 Assume that
and that
is a continuous function in
. Then one obtains that
for all
if
.
Proof. In view of
![]()
the condition in this lemma yields that for all
,
![]()
The proof is complete. ![]()
Lemma 3.7 Assume that
,
is a continuous function in I and that
. Then
is a compact operator in
.
Proof. By Ascoli-Arzela theorem, the compactness will be proved by the equiv-continuity of
.
Since
is a continuous function of
and
is a continuous function of
, for any given
there exists an
such that
.
Therefore, for
with
,
by Lemma 3.6 and for
,
.
In the following, we let
and we choose
such that for all
implies
![]()
Therefore,
![]()
The proof is complete. ![]()
Lemma 3.8. Assume that
,
and that
. Then
is a noncompact operator in
.
Proof. Without loss of generality, we assume that
(or
) for all
and suppose that the operator
is compact. Then the operator
![]()
or
![]()
is compact by Lemma 3.7. This contradicts to Corollary 2.3 and the proof is complete. ![]()
Lemma 3.9 Assume that
,
is a continuous function in I and that
. Then the null space of
is trivial in
for all
.
Proof. We suppose that
and there exists a
such that
(6)
Then
by
![]()
Thus, (6) reduces to
![]()
For all
and
, it holds
![]()
Hence (6) yields for sufficiently small
and sufficiently large
,
![]()
This implies that
and the proof is complete. ![]()
4. Concluding Remarks
In this paper, we consider CVIEs with a vanishing delay:
1) a proportional delay,
2) a nonlinear vanishing delay
.
The first case reduces to a classical CVIE with a core limited to a subinterval. Hence these results are trivial from [4] [5]. For case 2), we present the compactness of the operators, i.e.,
. In subsequent work, we will investigate the spectrum, eigenvalues and eigenfunctions when
and also numerical methods for CVIEs with vanishing delays.
NOTES
![]()
*This work is supported by the Natural Science Foundation of Heilongjiang Province, China (Grant No. QC2013C019).