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Maximum Principles and Boundary Value Problems for First-Order Neutral Functional Differential Equations
Journal of Inequalities and Applications volume 2009, Article number: 141959 (2009)
Abstract
We obtain the maximum principles for the first-order neutral functional differential equation 
where 
, and 
are linear continuous operators, 
and 
are positive operators, 
is the space of continuous functions, and 
is the space of essentially bounded functions defined on 
. New tests on positivity of the Cauchy function and its derivative are proposed. Results on existence and uniqueness of solutions for various boundary value problems are obtained on the basis of the maximum principles.
1. Preliminary
This paper is devoted to the maximum principles and their applications for first order neutral functional differential equation.
where 
are linear continuous Volterra operators, the spectral radius 
of the operator 
is less than one,
is the space of continuous functions, 
is the space of essentially bounded functions defined on 
We consider (1.1) with the following boundary condition:
where
is a linear bounded functional defined on the space of absolutely continuous functions 
. By solutions of (1.1) we mean functions 
from the space 
which satisfy this equation almost everywhere in 
and such that 
.
We mean the Volterra operators according to the classical Tikhonov's definition.
Definition 1.1.
An operator 
is called Volterra if any two functions 
and 
coinciding on an interval 
have the equal images on 
that is, 
for 
and for each 
Maximum principles present one of the classical parts of the qualitative theory of ordinary and partial differential equations [1]. Although in many cases, speaking about maximum principles, authors mean quite different definitions of maximum principles (such as e.g., corresponding inequalities, boundedness of solutions and maximum boundaries principles), there exists a deep internal connection between these definitions. This connection was discussed, for example, in the recent paper [2]. Main results of our paper are based on the maximum boundaries principle, that is, on the fact that the maximal and minimal values of the solution can be achieved only at the points 
or 
The boundaries maximum principle in the case of the zero operator 
was considered in the recent papers [2, 3]. In this paper we develop the maximum boundaries principle for neutral functional differential equation (1.1) and on this basis we obtain results on existence and uniqueness of solutions of various boundary value problems.
Although several assertions were presented as the maximum principles for delay differential equations, they can be only interpreted in a corresponding sense as analogs of classical ones for ordinary differential equations and do not imply important corollaries, reached on the basis of the finite-dimensional fundamental systems. For example, results, associated with the maximum principles in contrast with the cases of ordinary and even partial differential equations, do not add so much in problems of existence and uniqueness for boundary value problems with delay differential equations. The Azbelev's definition of the homogeneous delay differential equation [4, 5] allowed his followers to consider questions of existence, uniqueness and positivity of solutions on this basis. The first results about the maximum principles for functional differential equations, which were based on the idea of the finite-dimensional fundamental system, were presented in the paper [2].
Neutral functional differential equations have their own history. Equations in the form
were considered in the known books [6–8] (see also the bibliography therein), where existence and uniqueness of solutions and especially stability and oscillation results for these equations were obtained. There exist problems in applications whose models can be written in the form [9]
This equation is a particular case of (1.1).
Let us note here that the operator 
in (1.1) can be, for example, of the following forms:
or
where 
are essentially bounded measurable functions, 
are measurable functions for 
and 
are summable with respect to 
and measurable essentially bounded with respect to 
for 
. All linear combinations of operators (1.5) and (1.6) and their superpositions are also allowed.
The study of the neutral functional differential equations is essentially based on the questions of the action and estimates of the spectral radii of the operators in the spaces of discontinuous functions, for example, in the spaces of summable or essentially bounded functions. Operator (1.5), which is a linear combination of the internal superposition operators, is a key object in this topic. Properties of this operator were studied by Drakhlin [10, 11]. In order to achieve the action of operator (1.5) in the space of essentially bounded functions 
, we have for each 
to assume that 
for every constant 
Let us suppose everywhere below that this condition is fulfilled. It is known that the spectral radius of the integral operator (1.6)
considered on every finite interval 
is equal to zero (see, e.g., [4]). Concerning the operator (1.5), we can note the sufficient conditions of the fact that its spectral radius 
is less than one
Define the set 
and 
If there exists such 
that 
(
then on every finite interval 
the spectral radius of the operator 
defined by the formula (1.5) for 
is zero. In the case 
(
the spectral radius of the operator 
defined by (1.5) on the finite interval 
is less than one if 
The inequality 
implies that the spectral radius 
of the operator 
considered on the semiaxis 
and defined by (1.5)
satisfies the inequality 
Usually we will also assume that 
are nondecreasing functions for 
.
Various results on existence and uniqueness of boundary value problems for this equation and its stability were obtained in [4], where also the basic results about the representation of solutions were presented. Note also in this connection the papers in [12–15], where results on nonoscillation and positivity of Green's functions for neutral functional differential equations were obtained.
It is known [4] that the general solution of (1.1) has the representation
where the kernel 
is called the Cauchy function, and 
is the solution of the homogeneous equation 
satisfying the condition 
On the basis of representation (1.7), the results about differential inequalities (under corresponding conditions, solutions of inequalities are greater or less than solution of the equation) can be formulated in the following form of positivity of the Cauchy function 
and the solution 
. Results about comparison of solutions for delay differential equations solved with respect to the derivative (i.e., in the case when 
is the zero operator) were obtained in [2, 15, 16], where assertions on existence and uniqueness of solutions of various boundary value problems for first order functional differential equations were obtained.
All results presented in the paper [15] and in the book [16] for equation with the difference of two positive operators are based on corresponding analogs of the following assertion [15]: Let the operator
and the Cauchy function
of equation
be positive for
, then the Cauchy function
of
( 1.1 ) is also positive for
This result was extent on various boundary value problems in [16] in the form: Let the operator
and Green's function
of problem ( 1.8 ), ( 1.2 ) be positive in the square
and the spectral radius of the operator
defined by the equality
be less than one, then Green's function
of problem ( 1.1 ), ( 1.2 ) is positive in the square
The scheme of the proof was based on the reduction of problem (1.8), (1.2) with 
to the equivalent integral equation 
where 
It is clear that the operator 
is positive if the operator 
and the Green's function 
are positive. If the spectral radius 
or, more roughly, the norm 
of the operator 
are less than one, then there exists the inverse bounded operator 
which is of course positive. This implies the positivity of the Green's function 
of problem (1.1), (1.2). In order to get the inequality 
, the classical theorems about estimates of the spectral radius of the operator 
[17] can be used. All these theorems are based on a corresponding "smallness" of the operator 
which is actually close to the condition 
In order to get positivity of 
and 
a corresponding smallness of 
was assumed.
Below we present another approach to this problem starting with the following question: how can one conclude about positivity of Green's function 
in the cases when the spectral radius satisfies the opposite inequality 
or Green's function 
changes its sign? Note, that in the case, when the operator 
is positive and its spectral radius is less than one, the positivity of the Cauchy function 
of (1.8) follows from the nonoscillation of the homogeneous equation 
, and in the case of the zero operator 
the positivity of 
is even equivalent to nonoscillation [15]. This allows us to formulate our question also in the form: how can we make the conclusions about nonoscillation of the equation 
or about positivity of the Cauchy function 
of (1.1) without assumption about nonoscillation of the equation 
In this paper we obtain assertions allowing to make such conclusions. Our assertions are based on the assumption that the operator 
is a dominant among two operators 
and 
.
We assume that the spectral radius of the operator 
is less than one. In this case we can rewrite (1.1) in the equivalent form
and its general solution can be written in the form
where 
is the Cauchy function of (1.10) [4]. Note that this approach in the study of neutral equations was first used in the paper [14]. Below in the paper we use the fact that the Cauchy function 
coincides with the fundamental function of (1.10). It is also clear that
2. About Maximum Boundaries Principles in the Case of Difference of Two Positive Volterra Operators
In this paragraph we consider the equation
where 
and 
are positive linear continuous Volterra operators and the spectral radius 
of the operator 
is less than one.
These operators 
and 
are 
-bounded operators and according to [18], they can be written in the form of the Stieltjes integrals
respectively, where the functions 
and 
are measurable for 
and 
has the bounded variation for almost all 
and 
are essentially bounded.
Consider for convenience (2.1) in the following form:
We can study properties of solution of (2.3) on each finite interval 
since every solution 
of (2.1) satisfies also the equation
Consider also the homogeneous equation
and the following auxiliary equations (which are analogs of the so-called s-trancated equations defined first in [5])
where the operators 
and 
are defined by the formulas
and the operator 
is defined by the equality 
where 
for 
and 
for 
We have
for the operator described by formula (1.5), and
for the operator described by formula (1.6). It is clear that 
for every 
if 
Functions from the space 
of absolutely continuous functions 
satisfy (2.6) almost everywhere in 
, we call solutions of this equation.
It was noted above that the general solution of (2.1) has the representation [4]
where the function 
is the Cauchy function of (2.1).We use also formula (1.12) connecting 
and the Cauchy function 
of (1.10). Note that 
is a solution of (2.6) as a function of the first argument 
for every fixed 
and satisfies also the equation
Let us formulate our results about positivity of the Cauchy function 
and the maximum boundaries principle in the case when the condition 
is not assumed.
Consider the equation
Theorem 2.1.
Let 
be a positive Volterra operator, 
, 
let the functions 
and 
be nondecreasing functions with respect to 
for almost every 
and let the following inequality be fulfilled:
then the Cauchy function 
of (2.12) and its derivative satisfy the inequalities 
for 
Consider now the equation
Theorem 2.2.
Let 
be a positive Volterra operator, 
let the functions 
and 
be nondecreasing functions with respect to 
for
almost every 
and let the following inequalities
be fulfilled, then the Cauchy function 
of (2.14) and its derivative 
satisfy the inequalities 
and 
for 
Consider the delay equation
where
with 
and measurable functions 
and 
This equation is a particular case of (2.14).
Theorem 2.3.
Let 
be a positive Volterra operator, 
and 
for 
then the Cauchy function 
of (2.16) and its derivative 
satisfy the inequalities 
and 
for 
Example 2.4.
The inequality on deviating argument 
is essential as the following equation
demonstrates. This is a particular case of (2.16), where 
is the zero operator, 
The function
is a nontrivial solution of (2.18) and its Cauchy function 
satisfies the equality
that is, 
for 
for 
for 
for 
We see that each interval 
where 
is a nonoscillation one for this equation, but 
changes its sign for 
Consider the integrodifferential equation
as a particular case of (2.14).
Let us define the functions 
and 
where 
Theorem 2.5.
Let 
be a positive Volterra operator, 
for 
and the following inequalities be fulfilled:
then the Cauchy function of (2.22) and its derivative 
satisfy the inequalities 
and 
for 
Consider the equation
Let us define
In the following assertion the integral term is dominant.
Theorem 2.6.
Let 
be a positive Volterra operator, 
for 
and the following inequalities be fulfilled:
then the Cauchy function of (2.24) and its derivative 
satisfy the inequalities 
and 
for 
Consider the equation
In the following assertion the term 
is a dominant one.
Theorem 2.7.
Let 
be a positive Volterra operator, 
for 
and the following inequalities be fulfilled:
then the Cauchy function of (2.27) and its derivative 
satisfy the inequalities 
and 
for 
Consider now the equation
In the following assertion we do not assume inequalities 
or 
Here the sum 
is a dominant term.
Theorem 2.8.
Let 
be a positive Volterra operator, 
for 
and let the following inequalities be fulfilled:
for 
then the Cauchy function 
of (2.29) and its derivative 
satisfy the inequalities 
and 
for 
The proofs of Theorems 2.1–2.8 are based on the following auxiliary lemmas.
Lemma 2.9 ([2]).
Let 
be the zero operator. Then the following two assertions are equivalent:
(1)for every positive 
there exists a positive function 
such that 
for 
(2)the Cauchy function 
of (2.1) is positive for 
Lemma 2.10.
Let 
be a positive Volterra operator, 
and for every positive 
there exist a positive function 
such that 
for 
then the Cauchy function 
of (1.10) is positive for 
Proof.
Using the condition 
we can write (1.1) in the form (1.10). The positivity of the operator 
implies that the inequality 
follows from the inequality 
It is clear that the inequality 
for
implies the inequality 
for
Now according to Lemma 2.9, we obtain that 
for 
Lemma 2.11.
Let 
and 
be positive Volterra operators, 
and for every 
the inequality
be fulfilled. Then 
for 
In order to prove Lemma 2.11 we set 
for every 
in the assertion 1 of Lemma 2.10.
Remark 2.12.
The condition
cannot be set instead of condition (2.31) in Lemma 2.11 as Example 2.4 demonstrates. It is clear that this inequality is fulfilled for (2.18), where 
and 
the functions 
and 
are defined by formula (2.19) respectively. The operator 
is the zero one for every 
and consequently 
for 
for 
and condition (2.31) is not fulfilled for 
.
Lemma 2.13.
Let 
and 
be positive Volterra operators, 
and inequality (2.31) be fulfilled for every 
Then 
and (
for 
Proof.
According to Lemma 2.11, we have 
for 
The positivity of the operator 
and formula (1.12) implies now that 
for 
Now let us prove that (
for 
We use the fact that the function 
, as a function of 
the argument 
for each fixed 
satisfies (2.11) and the condition 
.
The following integral equation
is equivalent to (2.11) with the condition 
The spectral radius of the operator 
defined by the equality
is zero for every positive number 
[4]. Let us build the sequence
where the iterations start with the constant 
for 
.
The sequence of functions 
converges in the space 
to the unique solution 
of (2.33) on the interval 
. It is clear that this solution is absolutely continuous. It follows from the fact that all operators are Volterra ones, that the solution 
of (2.11) with the initial condition 
and the solution 
of (2.33) coincide for 
Positivity of the operator 
, the inequalities 
and (2.31) imply nonnegativity of the derivatives
Let us prove now that the sequence 
of nondecreasing functions converges to the nondecreasing function 
Assume in the contrary that there exist two points 
such that 
Let us choose 
There exists a number 
such that 
for 
, and there exists 
such that 
for 
It is clear that 
for 
This contradicts to the fact that 
nondecreases.
We have proven that for every positive 
the solution 
of (2.33) is nondecreasing for 
. It means that the solution 
of (2.11) is nondecreasing for every 
and consequently 
for 
Positivity of the operator 
the inequality 
and formula (1.12) imply now the inequality 
for 
To prove Theorems 2.1–2.8 it is sufficient to note that the conditions of each theorem imply inequality (2.31).
Remark 2.14.
The space of solutions of the homogeneous equation
in the case 
is one dimensional. All nontrivial solutions of (2.37) are proportional to 
One of the assertions of Lemma 2.11 claims that 
for 
that is, all nontrivial positive solutions do not decrease. This allows us to consider Lemma 2.13 as the maximum boundaries principle for (2.1). Theorems 2.1–2.8 present the sufficient conditions of this maximum principle for the equations (2.12), (2.14), (2.16), (2.22), (2.24), (2.27) and (2.29) respectively.
Remark 2.15.
The condition 
about the spectral radius of the operator 
is essential as the following example demonstrates.
Example 2.16.
Consider the equation
The spectral radius of the operator 
, defined by the formula 
is equal to one. All other conditions of Theorems 2.1–2.8 for the zero operators 
and 
are fulfilled. The space of solutions of this neutral homogeneous equation is infinitely dimensional. Every linear functions 
satisfy the homogeneous equation
If 
, the solutions 
are decreasing.
3. About Nondecreasing Solutions of Neutral Equations
Let us consider the equation
where 
and 
are positive linear continuous Volterra operators, and the spectral radius 
of the operator 
is less than one. If the operator 
is positive, then 
is not generally speaking a positive operator. This is the main difficulty in the study of positivity of the solution 
and its derivative 
. All previous results about the positivity of solutions for this equation assumed the negativity of the operator 
(see, e.g., [12, 13, 15]). In this paragraph we propose results about positivity of solutions in the case of the positive operator 
defined by the equality
Let us start with the equation
Theorem 3.1.
Assume that the spectral radius 
of the operator 
defined by equality (3.2) is less than one, 
and 
are nondecreasing functions, and the coefficients satisfy the inequalities 
and 
for 
then the solution 
of the equation
such that 
satisfies the inequalities 
for 
and in the case, when there exists 
such that 
the solution 
of (3.3) is nonnegative and nondecreasing for every positive nondecreasing function 
Consider the equation
Theorem 3.2.
Let the spectral radius 
of the operator 
defined by equality (3.2) be less than one, 
be a nondecreasing function and the functions 
and 
be nondecreasing functions with respect to 
, and the following inequalities be fulfilled
for 
then the solution 
of the equation
such that 
satisfies the inequalities 
for 
and in the case, when there exists 
such that 
the solution 
of (3.5) is nonnegative and nondecreasing for every positive nondecreasing function 
Consider the integrodifferential equation
Theorem 3.3.
Let the spectral radius 
of the operator 
defined by equality (3.2) be less than one, 
be a nondecreasing function and 
, 
for 
and the following inequalities be fulfilled
then the solution 
of the equation
such that 
satisfies the inequalities 
for 
and in the case, when there exists 
such that 
the solution 
of (3.8) is nonnegative and nondecreasing for every positive nondecreasing function 
Consider the equation
In the following assertion the integral term is dominant.
Theorem 3.4.
Let the spectral radius 
of the operator 
defined by equality (3.2) be less than one, 
be a nondecreasing function, 
for 
and the following inequality be fulfilled
then the solution 
of the equation
such that 
satisfies the inequalities 
for 
and in the case, when there exists 
such that 
the solution 
of (3.11) is nonnegative and nondecreasing for every positive nondecreasing function 
Consider the equation
In the following assertion the term 
is dominant.
Theorem 3.5.
Let the spectral radius 
of the operator 
defined by equality (3.2) be less than one, 
be a nondecreasing function, 
for 
and the following inequality
be fulfilled, then the solution 
of the equation
such that 
satisfies the inequalities 
for 
and in the case, when there exists 
such that 
the solution 
of (3.14) is nonnegative and nondecreasing for every positive nondecreasing 
Consider now the equation
In the following assertion we do not assume inequalities 
or 
Here the sum 
is a dominant term.
Theorem 3.6.
Let the spectral radius 
of the operator 
defined by equality (3.2) be less than one, 
be a nondecreasing function, 
for 
and the following inequalities be fulfilled
for 
then the solution 
of the equation
such that 
satisfies the inequalities 
for 
and in the case, when there exists 
such that 
the solution 
of (3.17) is nonnegative and nondecreasing for every positive nondecreasing 
Let us write (3.1) in the form
The spectral radius 
of the operator 
is less than one, then there exists the bounded operator 
and we can write (3.21) in the form
Denote by 
the Cauchy function of the equation 
which is also the fundamental function of (3.1).
Proofs of Theorems 3.1–3.6 are based on the following auxiliary assertions.
Lemma 3.7.
Let 
and 
be positive Volterra operators, the spectral radius 
of the operator 
be less than one and
for every nonnegative 
Then 
and 
for 
Proof.
Lemma 2.9 is true for (3.22). Let us set 
in the assertion 1 of Lemma 2.9. Condition (3.23) implies, according to Lemma 2.9, that 
for 
Now let us prove that 
for 
We use the fact that the function 
as a function of argument 
for each fixed positive 
satisfies the equation
and the condition 
The following integral equation
is equivalent to (3.24) with the condition 
The spectral radius of the operator 
defined by the equality
is zero for every positive number 
[4]. Let us build the sequence
where the iterations start with the constant 
for 
.
The sequence of functions 
converges in the space 
to the unique solution 
of (3.25) on the interval 
. It is clear that this solution is absolutely continuous. It follows from the fact that all operators are Volterra ones, that the solution 
of (3.24) with the initial condition 
and the solution 
of (3.25) coincide for 
Positivity of the operator 
, the inequalities 
and (3.23) imply nonnegativity of the derivatives
Repeating the argumentation used in the proof of Lemma 2.13, we obtain that this sequence of nondecreasing functions 
converges to the nondecreasing solution 
, that is, 
for 
Concerning nonhomogeneous (3.1) we propose the following assertion.
Lemma 3.8.
Let 
and 
be positive Volterra operators, the spectral radius 
of the operator 
be less than one and inequality (3.23) be fulfilled for every nonnegative 
Then the solution 
of the homogeneous equation
such that 
satisfies inequalities 
for 
If in addition the nonnegative function 
satisfies the inequality 
for 
then the solution 
of (3.1) is nonnegative and nondecreasing for every positive nondecreasing 
Remark 3.9.
The inequality 
for
is fulfilled if a nonnegative function 
is nondecreasing and the norm of the operator 
is less than one.
Proof of Lemma 3.8.
Assertions about nonnegativity of solution 
of the homogeneous equation 
and its derivative follows from the equalities 
and 
and Lemma 3.7. From the representation of solutions of (3.22) we can write
It is clear now that the inequality
for 
positivity of 
and nonnegativity of 
for 
imply the inequalities 
for 
The proofs of Theorems 3.1–3.6 follows from the fact that conditions of every theorem imply the conditions of Lemma 3.8 for corresponding equations.
Remark 3.10.
The condition
cannot be set instead of condition (3.23) as Example 2.4 demonstrates. In this example, the operator 
is the zero one, 
and 
for 
and 
for 
condition (3.31) fulfilled, the Cauchy function 
of (2.18) and its derivative change their signs. We noted that the inequality on delays avoids this situation.
Remark 3.11.
In the case of the neutral equation
where
the inequality 
for 
does not avoid the changes of signs of 
and its derivative:
and 
for 
and 
for 
This example demonstrates that we cannot set very natural inequality
instead of
in Theorem 3.1 even in the case when 
for 
.
4. Maximum Boundaries Principles in Existence and Uniqueness of Boundary Value Problems
Consider the boundary value problems of the following type
where 
is a linear bounded functional and 
It was explained in Remark 2.14 that Lemma 2.13 can be considered as the maximum boundaries principle for (4.1), and Theorems 2.1–2.8 present sufficient conditions of the maximum boundaries principles for equations (2.12), (2.14), (2.16), (2.22), (2.24), (2.27) and (2.29) respectively (i.e., under these conditions the modulus of nontrivial solutions of the corresponding homogeneous equations does not decrease). This allows us to obtain various results about existence and uniqueness of solutions of boundary value problems for these equations without the standard assumption about smallness of the norms of the operators 
and 
.
The assertions about existence and uniqueness are based on the known Fredholm alternative for functional differential equations.
Lemma 4.1 ([4]).
Let the spectral radius of the operator 
be less than one, then boundary value problem (4.1), (4.2) is uniquely solvable for each 
if and only if the homogeneous problem
has only the trivial solution.
Theorem 4.2.
Let 
be a positive Volterra operator and its spectral radius satisfy the inequality 
and for each 
the inequality
be fulfilled. Then the following assertions are true:
(1)If 
is a linear nonzero positive functional, then boundary value problem (4.1), (4.2) is uniquely solvable for each 
(2)The boundary value problem (4.1), (4.5), where
and the norm of the linear functional 
is less than one is uniquely solvable for each 
(3)The boundary value problem (4.1), (4.6), where
with 
, and there exists an index 
such that 
is uniquely solvable for each 
(4)The boundary value problem (4.1), (4.7), where
in the case when 
for 
for 
, 
, and there exists 
such that 
is uniquely solvable for each 
Proof.
If we suppose in the contrary that the assertions 1–4 are not true, then according to Lemma 4.1, the nontrivial solution 
of homogeneous problem (4.3) exists. According to Lemma 2.13 (see also Remark 2.14), the maximum boundaries principle is true, moreover, solutions of the homogeneous equation 
does not decrease on 
Conditions of each of the assertions 1–4 lead us to the inequality 
which contradicts to the existence of the nontrivial solution 
of the homogeneous problem 
.
Theorem 4.3.
Conditions of each of Theorems 2.1–2.8 imply assertions 1–4 of Theorem 4.2 for equations (2.12), (2.14), (2.16), (2.22), (2.24), (2.27) and (2.29) respectively.
In order to prove Theorem 4.3 we have only to note that conditions of Theorem 4.2 for corresponding equations follow from the conditions of each of Theorems 2.1–2.8.
Remark 4.4.
The condition
cannot be set instead of condition (4.4) as the following example demonstrates.
Example 4.5.
Consider the boundary value problem
where 
and 
are defined by formula (2.19) respectively. Here the operator 
is defined as (
and inequality (4.8) is fulfilled, but the operator 
is a zero operator for 
and inequality (4.4) is not true. Formula (2.20) defines the nontrivial solution of the homogeneous problem
According to Lemma 4.1, problem (4.9) cannot be uniquely solvable for each 
Remark 4.6.
The periodic problem
where 
demonstrates that the condition 
in the assertion 2 of Theorem 4.2 is essential: the function 
for 
is a nontrvial solution of the homogeneous boundary value problem
and reference to Lemma 4.1 completes this example.
Remark 4.7.
According to Lemma 4.1, the problem
demonstrates that the condition about existence of such 
that 
in the assertion 3 is essential, and the problem
demonstrates that the condition about existence of such 
that 
in the assertion 4 of Theorem 4.2 is essential. If we suppose that such 
does not exist, then the function 
for 
is a nontrvial solution of each of the homogeneous boundary value problems
Remark 4.8.
The condition 
for 
in the assertion 4 of Theorem 4.2 cannot be omitted that follows from the example of one of the reviewers: the function 
is a nontrivial solution of the boundary value problem
where 
for 
for 
for 
In this case 
and consequently all other conditions of the assertion 4 of Theorem 4.2 are fulfilled.
Remark 4.9.
Let us define the set
and the following condition:
(a)there exists a set 
of nonzero measure (i.e., 
Instead of the condition 
in the assertion 2 of Theorem 4.2 we can assume that the inequality 
and the condition (a) are fulfilled.
Remark 4.10.
Instead of the condition about existence of such 
that 
in the assertion 3 of Theorem 4.2 we can assume that
Condition (4.18) is essential as the following example demonstrates.
Example 4.11.
The homogeneous boundary value problem
where
has a nontrivial solution
In this case we have 
Remark 4.12.
Let us denote as 
the following set: 
Instead of the condition about existence of such 
that 
in the assertion 4, we can assume the following: 
Example 4.13.
Consider (4.19), where the coefficient 
is defined by (4.21), with the boundary condition
Homogeneous problem (4.19), (4.23) has a nontrivial solution defined by (4.22). In this case we have 
Consider now (3.1) with the opposite sign near the neutral term (
Theorem 4.14.
Let the conditions of Lemma 3.8 be fulfilled for (3.1), then assertions 1–4 of Theorem 4.2 are true for (3.1).
Proof follows from the fact that according to Lemma 3.8, solution 
of the homogeneous equation
does not decrease. Conditions of each of the assertions 1–4 of Theorem 4.2 lead us to the maximum boundaries principle and consequently to the inequality 
which contradicts to the existence of the nontrivial solution 
of the homogeneous problem 
.
Theorem 4.15.
Conditions of each of Theorems 3.1–3.6 imply assertions 1–4 of Theorem 4.2 for equations (3.3), (3.5), (3.8), (3.11), (3.14) and (3.17) respectively.
Proof follows from the fact that conditions of each of Theorems 3.1–3.6 imply that the conditions of Lemma 3.8 are fulfilled.
Note that Remarks 4.4–4.12 are relevant also for Theorems 4.14–4.15.
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Acknowlegments
This research was supported by The Israel Science Foundation (Grant no. 828/07). The authors thank the reviewers for their valuable suggestions.
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Domoshnitsky, A., Maghakyan, A. & Shklyar, R. Maximum Principles and Boundary Value Problems for First-Order Neutral Functional Differential Equations. J Inequal Appl 2009, 141959 (2009). https://doi.org/10.1155/2009/141959
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DOI: https://doi.org/10.1155/2009/141959