- Research Article
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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.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F141959/MediaObjects/13660_2009_Article_1894_Equ1_HTML.gif)
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:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F141959/MediaObjects/13660_2009_Article_1894_Equ2_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F141959/MediaObjects/13660_2009_Article_1894_Equ3_HTML.gif)
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]
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F141959/MediaObjects/13660_2009_Article_1894_Equ4_HTML.gif)
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:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F141959/MediaObjects/13660_2009_Article_1894_Equ5_HTML.gif)
or
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F141959/MediaObjects/13660_2009_Article_1894_Equ6_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F141959/MediaObjects/13660_2009_Article_1894_Equ7_HTML.gif)
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 operatorand the Cauchy function
of equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F141959/MediaObjects/13660_2009_Article_1894_Equ8_HTML.gif)
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 operatorand 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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F141959/MediaObjects/13660_2009_Article_1894_Equ9_HTML.gif)
be less than one, then Green's functionof 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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F141959/MediaObjects/13660_2009_Article_1894_Equ10_HTML.gif)
and its general solution can be written in the form
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F141959/MediaObjects/13660_2009_Article_1894_Equ11_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F141959/MediaObjects/13660_2009_Article_1894_Equ12_HTML.gif)
2. About Maximum Boundaries Principles in the Case of Difference of Two Positive Volterra Operators
In this paragraph we consider the equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F141959/MediaObjects/13660_2009_Article_1894_Equ13_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F141959/MediaObjects/13660_2009_Article_1894_Equ14_HTML.gif)
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:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F141959/MediaObjects/13660_2009_Article_1894_Equ15_HTML.gif)
We can study properties of solution of (2.3) on each finite interval since every solution
of (2.1) satisfies also the equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F141959/MediaObjects/13660_2009_Article_1894_Equ16_HTML.gif)
Consider also the homogeneous equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F141959/MediaObjects/13660_2009_Article_1894_Equ17_HTML.gif)
and the following auxiliary equations (which are analogs of the so-called s-trancated equations defined first in [5])
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F141959/MediaObjects/13660_2009_Article_1894_Equ18_HTML.gif)
where the operators and
are defined by the formulas
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F141959/MediaObjects/13660_2009_Article_1894_Equ19_HTML.gif)
and the operator is defined by the equality
where
for
and
for
We have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F141959/MediaObjects/13660_2009_Article_1894_Equ20_HTML.gif)
for the operator described by formula (1.5), and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F141959/MediaObjects/13660_2009_Article_1894_Equ21_HTML.gif)
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]
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F141959/MediaObjects/13660_2009_Article_1894_Equ22_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F141959/MediaObjects/13660_2009_Article_1894_Equ23_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F141959/MediaObjects/13660_2009_Article_1894_Equ24_HTML.gif)
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:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F141959/MediaObjects/13660_2009_Article_1894_Equ25_HTML.gif)
then the Cauchy function of (2.12) and its derivative satisfy the inequalities
for
Consider now the equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F141959/MediaObjects/13660_2009_Article_1894_Equ26_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F141959/MediaObjects/13660_2009_Article_1894_Equ27_HTML.gif)
be fulfilled, then the Cauchy function of (2.14) and its derivative
satisfy the inequalities
and
for
Consider the delay equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F141959/MediaObjects/13660_2009_Article_1894_Equ28_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F141959/MediaObjects/13660_2009_Article_1894_Equ29_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F141959/MediaObjects/13660_2009_Article_1894_Equ30_HTML.gif)
demonstrates. This is a particular case of (2.16), where is the zero operator,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F141959/MediaObjects/13660_2009_Article_1894_Equ31_HTML.gif)
The function
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F141959/MediaObjects/13660_2009_Article_1894_Equ32_HTML.gif)
is a nontrivial solution of (2.18) and its Cauchy function satisfies the equality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F141959/MediaObjects/13660_2009_Article_1894_Equ33_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F141959/MediaObjects/13660_2009_Article_1894_Equ34_HTML.gif)
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:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F141959/MediaObjects/13660_2009_Article_1894_Equ35_HTML.gif)
then the Cauchy function of (2.22) and its derivative satisfy the inequalities
and
for
Consider the equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F141959/MediaObjects/13660_2009_Article_1894_Equ36_HTML.gif)
Let us define
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F141959/MediaObjects/13660_2009_Article_1894_Equ37_HTML.gif)
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:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F141959/MediaObjects/13660_2009_Article_1894_Equ38_HTML.gif)
then the Cauchy function of (2.24) and its derivative satisfy the inequalities
and
for
Consider the equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F141959/MediaObjects/13660_2009_Article_1894_Equ39_HTML.gif)
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:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F141959/MediaObjects/13660_2009_Article_1894_Equ40_HTML.gif)
then the Cauchy function of (2.27) and its derivative satisfy the inequalities
and
for
Consider now the equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F141959/MediaObjects/13660_2009_Article_1894_Equ41_HTML.gif)
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:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F141959/MediaObjects/13660_2009_Article_1894_Equ42_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F141959/MediaObjects/13660_2009_Article_1894_Equ43_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F141959/MediaObjects/13660_2009_Article_1894_Equ44_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F141959/MediaObjects/13660_2009_Article_1894_Equ45_HTML.gif)
is equivalent to (2.11) with the condition
The spectral radius of the operator defined by the equality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F141959/MediaObjects/13660_2009_Article_1894_Equ46_HTML.gif)
is zero for every positive number [4]. Let us build the sequence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F141959/MediaObjects/13660_2009_Article_1894_Equ47_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F141959/MediaObjects/13660_2009_Article_1894_Equ48_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F141959/MediaObjects/13660_2009_Article_1894_Equ49_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F141959/MediaObjects/13660_2009_Article_1894_Equ50_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F141959/MediaObjects/13660_2009_Article_1894_Equ51_HTML.gif)
If , the solutions
are decreasing.
3. About Nondecreasing Solutions of Neutral Equations
Let us consider the equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F141959/MediaObjects/13660_2009_Article_1894_Equ52_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F141959/MediaObjects/13660_2009_Article_1894_Equ53_HTML.gif)
Let us start with the equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F141959/MediaObjects/13660_2009_Article_1894_Equ54_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F141959/MediaObjects/13660_2009_Article_1894_Equ55_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F141959/MediaObjects/13660_2009_Article_1894_Equ56_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F141959/MediaObjects/13660_2009_Article_1894_Equ57_HTML.gif)
for then the solution
of the equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F141959/MediaObjects/13660_2009_Article_1894_Equ58_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F141959/MediaObjects/13660_2009_Article_1894_Equ59_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F141959/MediaObjects/13660_2009_Article_1894_Equ60_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F141959/MediaObjects/13660_2009_Article_1894_IEq417_HTML.gif)
then the solution of the equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F141959/MediaObjects/13660_2009_Article_1894_Equ61_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F141959/MediaObjects/13660_2009_Article_1894_Equ62_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F141959/MediaObjects/13660_2009_Article_1894_Equ63_HTML.gif)
then the solution of the equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F141959/MediaObjects/13660_2009_Article_1894_Equ64_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F141959/MediaObjects/13660_2009_Article_1894_Equ65_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F141959/MediaObjects/13660_2009_Article_1894_Equ66_HTML.gif)
be fulfilled, then the solution of the equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F141959/MediaObjects/13660_2009_Article_1894_Equ67_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F141959/MediaObjects/13660_2009_Article_1894_Equ68_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F141959/MediaObjects/13660_2009_Article_1894_Equ69_HTML.gif)
for then the solution
of the equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F141959/MediaObjects/13660_2009_Article_1894_Equ70_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F141959/MediaObjects/13660_2009_Article_1894_Equ71_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F141959/MediaObjects/13660_2009_Article_1894_Equ72_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F141959/MediaObjects/13660_2009_Article_1894_Equ73_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F141959/MediaObjects/13660_2009_Article_1894_Equ74_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F141959/MediaObjects/13660_2009_Article_1894_Equ75_HTML.gif)
and the condition
The following integral equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F141959/MediaObjects/13660_2009_Article_1894_Equ76_HTML.gif)
is equivalent to (3.24) with the condition
The spectral radius of the operator defined by the equality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F141959/MediaObjects/13660_2009_Article_1894_Equ77_HTML.gif)
is zero for every positive number [4]. Let us build the sequence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F141959/MediaObjects/13660_2009_Article_1894_Equ78_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F141959/MediaObjects/13660_2009_Article_1894_Equ79_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F141959/MediaObjects/13660_2009_Article_1894_Equ80_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F141959/MediaObjects/13660_2009_Article_1894_Equ81_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F141959/MediaObjects/13660_2009_Article_1894_Equ82_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F141959/MediaObjects/13660_2009_Article_1894_Equ83_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F141959/MediaObjects/13660_2009_Article_1894_Equ84_HTML.gif)
the inequality for
does not avoid the changes of signs of
and its derivative:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F141959/MediaObjects/13660_2009_Article_1894_Equ85_HTML.gif)
and for
and
for
This example demonstrates that we cannot set very natural inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F141959/MediaObjects/13660_2009_Article_1894_Equ86_HTML.gif)
instead of
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F141959/MediaObjects/13660_2009_Article_1894_Equ87_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F141959/MediaObjects/13660_2009_Article_1894_Equ88_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F141959/MediaObjects/13660_2009_Article_1894_Equ89_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F141959/MediaObjects/13660_2009_Article_1894_Equ90_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F141959/MediaObjects/13660_2009_Article_1894_Equ91_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F141959/MediaObjects/13660_2009_Article_1894_Equ92_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F141959/MediaObjects/13660_2009_Article_1894_Equ93_HTML.gif)
with , and there exists an index
such that
is uniquely solvable for each
(4)The boundary value problem (4.1), (4.7), where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F141959/MediaObjects/13660_2009_Article_1894_Equ94_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F141959/MediaObjects/13660_2009_Article_1894_Equ95_HTML.gif)
cannot be set instead of condition (4.4) as the following example demonstrates.
Example 4.5.
Consider the boundary value problem
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F141959/MediaObjects/13660_2009_Article_1894_Equ96_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F141959/MediaObjects/13660_2009_Article_1894_Equ97_HTML.gif)
According to Lemma 4.1, problem (4.9) cannot be uniquely solvable for each
Remark 4.6.
The periodic problem
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F141959/MediaObjects/13660_2009_Article_1894_Equ98_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F141959/MediaObjects/13660_2009_Article_1894_Equ99_HTML.gif)
and reference to Lemma 4.1 completes this example.
Remark 4.7.
According to Lemma 4.1, the problem
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F141959/MediaObjects/13660_2009_Article_1894_Equ100_HTML.gif)
demonstrates that the condition about existence of such that
in the assertion 3 is essential, and the problem
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F141959/MediaObjects/13660_2009_Article_1894_Equ101_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F141959/MediaObjects/13660_2009_Article_1894_Equ102_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F141959/MediaObjects/13660_2009_Article_1894_Equ103_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F141959/MediaObjects/13660_2009_Article_1894_Equ104_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F141959/MediaObjects/13660_2009_Article_1894_Equ105_HTML.gif)
Condition (4.18) is essential as the following example demonstrates.
Example 4.11.
The homogeneous boundary value problem
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F141959/MediaObjects/13660_2009_Article_1894_Equ106_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F141959/MediaObjects/13660_2009_Article_1894_Equ107_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F141959/MediaObjects/13660_2009_Article_1894_Equ108_HTML.gif)
has a nontrivial solution
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F141959/MediaObjects/13660_2009_Article_1894_Equ109_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F141959/MediaObjects/13660_2009_Article_1894_Equ110_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F141959/MediaObjects/13660_2009_Article_1894_Equ111_HTML.gif)
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