1. Preliminaries
The Cauchy problem for functional differential equations in the non-Volterra case [
1] (§ 2.2.3, p. 50) has been studied quite intensively in recent years [
1,
2,
3,
4,
5,
6,
7,
8,
9,
10]. We consider the Cauchy problem for the linear second order functional differential equation
where the operators
,
are linear and positive,
,
,
,
and
are the spaces of all continuous and integrable functions equipped with the standard norms respectively. An operator from
into
is called positive if it maps each non-negative function into an almost everywhere non-negative one. Equalities and inequalities with integrable functions will be considered as equalities and inequalities that are valid almost everywhere on the corresponding interval. Let
be the Banach space of all functions
such that
x and the derivative
are absolutely continuous on the interval
with the norm
. We will say that a function
is a solution to problem (
1) and (
2) if
,
x satisfies Equation (
1) almost everywhere on
, and
x satisfies initial conditions (
2).
In many results on solvability conditions for the Cauchy problem and other boundary value problems for functional differential equations, some smallness conditions are imposed on the positive part
and the negative part
of the functional operator
(see [
1,
3,
11,
12,
13,
14,
15]). Generally, the results are close to those obtained using the Banach contraction principle.
Using one-sided a priori estimates, A. Lomtatidze and S. Mukhigulashvili [
16,
17,
18] managed to weaken the conditions on one of the operators
,
, guaranteeing the unique solvability of some boundary value problems. Similar relaxations of the solvability conditions were obtained by A. Lomtatidze, R. Hakl, B. Půža [
19,
20], A. Lomtatidze, R. Hakl, E. Bravyi [
21] for the Cauchy problem, J. Šemr and R. Hakl [
10,
22,
23] for the Cauchy problem for systems of functional differential equations, and by R. Hakl, A. Lomtatidze, S. Mukhigulashvili, B. Půža, for some other boundary value problems [
16,
24,
25].
In these early works, integral restrictions were imposed on both functional operators in the form of integral inequalities
(see works on solvability conditions for the Cauchy problem [
21] by A. Lomtatidze, R. Hakl, E. Bravyi and [
10,
19,
20], where the solvability conditions for the operator
were weakened and optimal solvability conditions for the Cauchy problem were obtained). Later, pointwise restrictions
for some given constants
,
were imposed on both functional operators [
26] and the similar weaker solvability conditions on the operator
were obtained.
However, apparently, for arbitrary pointwise constraints, necessary and sufficient solvability conditions for all equations in the family have not been obtained for a long time. But it is pointwise restrictions that give the narrowest families of equations and, therefore, the necessary and sufficient conditions for the solvability of the Cauchy problem for all equations from these families give the strongest results. Only in the work [
27] were various types of pointwise constraints used to form families of equations.
Here we take a more general approach, using both types of constraints together (point and integral), so we get a new class of solvability conditions. And the obtained necessary and sufficient conditions for the solvability of the Cauchy problem for all equations from these families will exceed the known results.
We define new families of functional operators using two types of restrictions, integral for one from operators , and pointwise restrictions for another operator. Then we find the necessary and sufficient conditions for a unique solvability of the Cauchy problem for all equations with operators from the chosen family. The obtained sufficient solvability conditions are unimprovable in the following sense. If these conditions are violated, then there exists an equation in the given family for which the Cauchy problem is not uniquely solvable.
All operators and considered here will belong to some families of the operators defined by pointwise and integral restrictions we impose on the functions and , where is the unit function.
Let non-negative functions
,
and non-negative numbers
,
be given. Let us introduce the following kinds of restrictions on the functional operators
and
:
Note that only conditions (
3) [
27] (it corresponds to pointwise restrictions) and conditions (
6) [
20,
21,
22,
23,
25,
28,
29] (corresponds to integral restrictions) were studied in earlier works (primarily in the case of the first order equations). The author is almost unaware of any works where mixed constraints (
4) or (
5) were used to obtain conditions for the solvability of the Cauchy problem (the only exception is the work [
30], published during the preparation of this article).
Definition 1. We will say that Cauchy problem (1) and (2) possesses the property , , if problem (1) and (2) is uniquely solvable for all positive linear operators , satisfying conditions (3), (4), (5), (6) respectively. In the study of boundary value problems for functional differential equations, the Fredholm property is often useful (see, for example, [
1,
3,
31]). For the convenience of readers, we will give a definition of the Fredholm property and show that the Cauchy problem (
1) and (
2) possesses this property.
Below we present some information from [
32]. Let
,
be Banach spaces, and
a linear operator. The set of all solutions to the equation
is called the null-space of the operator
F. An operator
F is called normal if the equation
is solvable for those and only those
for which
for all solutions
g of the homogeneous adjoint equation
, where
is the adjoint operator. For the operator
F to be normal, it is necessary and sufficient that the range of values of the operator
F be closed.
A normal operator is called Noetherian if it and its adjoint operator have null-spaces of finite dimension. The difference between those dimensions is called the operator index.
A Noetherian operator of zero index is called a Fredholm operator.
For a Fredholm operator
F, the Fredholm alternative [
31,
32] is valid. In particular, the equation
is uniquely solvable for all
if and only if the homogeneous equation
has only the trivial solution.
For a bounded operator F to be Fredholm, it is necessary and sufficient that the operator F be representable in the form , where the linear bounded operator is invertible, and the operator is completely continuous or finite-dimensional (we will call an operator with a finite-dimensional domain of values finite-dimensional). Thus, a finite-dimensional or completely continuous perturbation of the operator does not affect the Fredholm property.
Cauchy problem (
1) and (
2) can be rewritten in the form of one equation [
5] (p. 14).
where
the linear operator
acts from the space
into
.
Let us represent the operator
of the Cauchy problem (
1) and (
2) as
where
. Obviously, the operator
is invertible. Indeed, the Cauchy problem for the ordinary differential equation
has a unique solution
,
.
Here we consider differences of linear positive operators
,
. Each such operator is bounded. Indeed, the norm of the linear operator
is not greater than
where
,
, is the unit function. Note that the norm of a positive operator
is equal to
.
Further, the space
is compactly embedded into the space
. This can be proved by direct application of the Arzela–Ascoli theorem [
31] (p. 27). Consequently, the bounded operator
is compact as an operator acting from the space
into the space
. Thus, the operator
is compact. So, the operator
of the Cauchy problem (
1) and (
2) has the Fredholm property and the following assertion is valid.
Lemma 1 (The Fredholm alternative).
Cauchy problem (1) and (2) is uniquely solvable if and only if the homogeneous problemhas only the trivial solution. The class of differences of linear positive operators from
to
includes operators with “deviated argument”:
where
,
are measurable functions,
. These operators can be taken as illustrative examples for all statements of the work.
Note, every linear positive operator
has the representation [
33] (pp. 303–304) in the form of the Riemann–Stieltjes integral:
where for each
the function
does not decrease, for each
the function
is integrable on
,
.
Remark 1. It is easy to see that all equalities in the definitions of properties , , can be replaced by non-strict inequalities less than or equal to “⩽”.
Indeed, from the Fredholm property of the Cauchy problem (Lemma 1) it follows that it is sufficient to consider the homogeneous Cauchy problems. Then the unique solvability is equivalent to the absence of nontrivial solutions. If the problem does not possess some property in Definition 1, then it does not have this property for all greater or equal parameters. This follows from the fact that any additives in the form of a positive operator , , where , , , preserve a nontrivial solution to the homogeneous problem.
Our aim is to obtain necessary and sufficient conditions for the unique solvability of the Cauchy problem for all equations of the family to be uniquely solvable, that is, we search criteria for properties , .
It should be emphasized that we consider generally speaking non-Volterra operators
,
, so the solvability of the Cauchy problem under natural assumptions is not guaranteed, unlike the Cauchy problem for ordinary differential equations. Note, the results can be used in the study of applied, in particular, computational problems such as in, for example [
34,
35]. The statements obtained in Theorems 1–4 improve all results known to the author (see [
10,
19,
20,
21,
22,
23,
28,
29,
30]).
The work is organized as follows.
Section 2 presents the main results.
Section 3 contains a proof of Theorem 1.
Section 4 contains proofs of Theorems 2 and 3 and Corollaries 1 and 2. Theorem 4 and Corollary 3 is proved in
Section 5.
Section 6 provides an example illustrating applications of Theorem 1.
Section 7 discusses the results obtained.
4. Proof of Theorems 2, 3 and Corollaries 1, 2
We use Theorem 1 and find out when its conditions are satisfied for given non-negative
and all non-negative
from the family of functions defined by condition (
4), as well for some non-negative
and all non-negative
from the family of functions defined by condition (
5).
To do this, we consider expression (
10) for the quantity
, which for unique solvability of (
1) and (
2) must be positive for all
,
.
For fixed sets
,
, and for all points
,
,
, the value of
depends on each of the restrictions
,
,
,
linearly and continuously. Thus, its greatest lower bound over all sets of admissible
,
with given integrals on these sets is
accepts if each of the functions
and
is “concentrated” at two points:
in
and
;
to
and
.
In particular, from representation (
10), we obtain
where
,
, for some constants
,
,
that do not depend on
. Therefore,
where the point
is the minimum point:
. The function
r is linear on
, hence
can only be at the ends of the segment
, that is
Find
. From representation (
10) we get
where
The function
q is linear on
and on
, therefore,
. If
, then
. But we can get this value taking
, therefore we may not consider this point. Further, since (
9), we have
for
. Therefore, we need consider only the case
. Similar arguments show that
,
. In all these cases, the infimum of
is not achieved on integrable functions
,
.
Therefore, we obtain the following statements.
Lemma 9. Problem (1) and (2) has the property if and only if inequality (9) is fulfilled and the inequalityholds for all , , , , , , . Lemma 10. Problem (1) and (2) has the property if and only if and the inequalityholds for all , , , , , , . When we minimize
in (
26), it is easy to show that it suffices to consider only the case
. Note that
, defined by (
8), is positive for
if inequality (
9) holds. If
, then
. For
, minimizing
with respect to the quadratic variable
gives us the minimum of
as a quadratic function of
. From the condition
we obtain the assertion of Theorem 2.
Now we prove Corollary 1. Let the conditions of Theorem 2 be fulfilled and
be constant. Inequality (
9) means that
. The second condition of Theorem 2 gives the following inequality
for all
,
.
It is obvious that for
the statement of the corollary is true. Let
. We have
The function
has zeros at the points
The point
is the minimum point of
on the interval
. After substituting
into the function
we obtain Corollary 1 (which was also obtained in [
30] in another way).
Now let us finish the proof of Theorem 3. When we minimize
in (
27) with respect to
and
, we reduce the problem to minimization of the quadratic function
with respect to
for some constants
,
. Therefore, the minimum is taken at
or
. So, we have to consider the following cases: (i)
,
,
; (ii)
,
,
; (iii)
,
,
does not depend on
. It is easy to show that if the minimum of
is negative in the case (i), then the minimum of
is negative in the case (ii). So, it suffices to consider the cases (ii) and (iii). Here the dependence on
is linear. This gives us Theorem 3.
Let us prove Corollary 2. Let the conditions of Theorem 3 be fulfilled and
be constant. Then the minimum of
is taken for the case (ii). In this case Theorem 3 gives the following solvability conditions
for all
,
.
If , then for the function takes its minimum at . Then .
Let now
. Then the function
takes its minimum at
. In this case, we have
Minimizing this expression with respect to
, we conclude that if
, then
. If
, then
takes its minimum at
. We get
Moreover,
if and only if
. This implies Corollary 2.
6. Example
We present an example that illustrates the application of Theorems 1 and 2 and shows that the solvability conditions obtained using Theorem 1 significantly improve the conditions obtained using the Banach contraction principle.
Let constants
,
be given. Define the non-negative functions
,
:
Then
,
. Therefore, if Cauchy problem (
1) and (
2) possesses the property
, then the Cauchy problem
is uniquely solvable for all measurable functions
. If problem (
1) and (
2) does not possesses the property
, then there exists a measurable function
such that problem (
31) is not uniquely solvable.
It should be noted that the application of the Banach contraction principle to this problem gives the following result: Cauchy problem (
31) is uniquely solvable if
that is
This solvability condition will be significantly improved by the following condition (
33). Condition (
32) coincides with condition (
33) only in two cases (when the coefficient
is non-negative, that is, for
,
and for
,
0 (see
Figure 1)). In other cases, condition (
33) is much weaker, than (
32), moreover, the constants in (
33) are unimprovable.
Direct verification of the conditions of Theorem 1 makes it possible to obtain necessary and sufficient conditions for the constants
and
a under which inequality (
10) is satisfied for all
. We find that for such
,
by Theorem 1 problem (
1) and (
2) enjoys the property
if and only if
where the functions
,
are defined by equalities (see
Figure 1)
is a unique real solution of the equation
(
),
is a unique solution of the equation
(
),
If the parameter
a satisfies the inequality (
33), then Cauchy problem (
31) is uniquely solvable for all measurable deviations of the argument
. If the condition (
33) is not satisfied, then there is a measurable function
such that Cauchy problem (
31) is not uniquely solvable.
It follows that, for
, problem (
1) and (
2) possesses property
if and only if
In particular, if
, then the inequality
is necessary and sufficient for problem (
1) and (
2) to enjoy property
. Thus, for
the Cauchy problem
is uniquely solvable for all measurable
if
or
and the constants 48 and
cannot be increased.
Let us apply Theorem 2 to the Cauchy problem
where
for
,
for
,
for all
. So, here we have not changed the operator
from Cauchy problem (
34), but consider an arbitrary operator
. Application of Theorem 2 gives the following solvability condition: the Cauchy problem (
35) is uniquely solvable for all measurable functions
h,
if
,
for
, and
For
, we have the solvability condition
, which is expected to be significantly less than
from the solvability conditions of problem (
34). This is explained by the fact that when considering problem (
35) we imposed not pointwise restrictions on the operator
, but weaker integral restrictions. All constants in these solvability conditions cannot be increased.