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Differential Inequalities for One Component of Solution Vector for Systems of Linear Functional Differential Equations
Advances in Difference Equations volume 2010, Article number: 478020 (2010)
Abstract
The method to compare only one component of the solution vector of linear functional differential systems, which does not require heavy sign restrictions on their coefficients, is proposed in this paper. Necessary and sufficient conditions of the positivity of elements in a corresponding row of Green's matrix are obtained in the form of theorems about differential inequalities. The main idea of our approach is to construct a first order functional differential equation for the th component of the solution vector and then to use assertions about positivity of its Green's functions. This demonstrates the importance to study scalar equations written in a general operator form, where only properties of the operators and not their forms are assumed. It should be also noted that the sufficient conditions, obtained in this paper, cannot be improved in a corresponding sense and does not require any smallness of the interval
, where the system is considered.
1. Introduction
Consider the following system of functional differential equations

where   
are linear continuous operators,
and
are the spaces of continuous and summable functions
, respectively.
Let be a linear bounded functional. If the homogeneous boundary value problem
has only the trivial solution, then the boundary value problem

has for each where
and
a unique solution, which has the following representation [1]:

where the matrix
is called Green's matrix of problem (1.2), and
is the
fundamental matrix of the system
such that
(
is the unit
-matrix). It is clear from the solution representation (1.3) that the matrices
and
determine all properties of solutions.
The following property is the basis of the approximate integration method by Tchaplygin [2]: from the conditions

it follows that

Series of papers, started with the known paper by Luzin [3], were devoted to the various aspects of Tchaplygin's approximate method. The well-known monograph by Lakshmikantham and Leela [4] was one of the most important in this area. The known book by Krasnosel'skii et al. [5] was devoted to approximate methods for operator equations. These ideas have been developing in scores of books on the monotone technique for approximate solution of boundary value problems for systems of differential equations. Note in this connection the important works by Kiguradze and Puza [6, 7] and Kiguradze [8].
As a particular case of system (1.1), let us consider the following delay system:

where are summable functions, and
are measurable functions such that
for
The classical Wazewskii's theorem claims [9] that the condition

is necessary and sufficient for the property (1.4)(1.5) for the Cauchy problem for system of ordinary differential equations

From formula of solution representation (1.3), it is clear that property (1.4)(1.5) is true if all elements of the matrices
and
are nonnegative.
We focus our attention upon the problem of comparison for only one of the components of solution vector. Let be either 1 or 2. In this paper we consider the following property: from the conditions

it does follow that for a corresponding fixed component of the solution vector the inequality

is satisfied. This property is a weakening of the property (1.4)(1.5) and, as we will obtain below, leads to essentially less hard limitations on the given system. From formula of solution's representation (1.3), it follows that this property is reduced to sign-constancy of all elements standing only in the
th row of Green's matrix.
The main idea of our approach is to construct a corresponding scalar functional differential equation of the first order

for th component of a solution vector, where
is a linear continuous operator,
This equation is built in Section 2. Then the technique of analysis of the first-order scalar functional differential equations, developed, for example, in the works [10–12], is used. On this basis in Section 3 we obtain necessary and sufficient conditions of nonpositivity/nonnegativity of elements in
th row of Green's matrices in the form of theorems about differential inequalities. Simple coefficient tests of the sign constancy of the elements in the
th row of Green's matrices are proposed in Section 4 for systems of ordinary differential equations and in Section 5 for systems of delayed differential equations. It should be stressed that in our results a smallness of the interval
is not assumed.
Note that results of this sort for the Cauchy problem (i.e., ) and Volterra operators
were proposed in the recent paper [13], where the obtained operator
became a Volterra operator. In this paper we consider other boundary conditions that imply that the operator
is not a Volterra one even in the case when all
are Volterra operators.
2. Construction of Equation for
th Component of Solution Vector
In this paragraph, we consider the boundary value problem


where are linear bounded operators for
and
, are linear boundary functionals
Together with problem (2.1), (2.2) let us consider the following auxiliary problem consisting of the system:

of the order and the boundary conditions

Let us assume that problem (2.3), (2.4) is uniquely solvable; denote by its Green's matrix and by
Green's matrix of the problem (2.1), (2.2).
Let us start with the following assertion, explaining how the scalar functional differential equation for one of the components of the solution vector can be constructed.
Lemma 2.1.
The component of the solution vector of system (2.1) satisfies the following scalar functional differential equation:

where the operator and the function
are defined by the equalities


where is the solution of the system

satisfying condition (2.4).
Proof.
Using Green's matrix of problem (2.3), (2.4), we obtain

for every Substitution of these representations in the
th equation of the system (2.1) leads to (2.5), where the operator
and the function
are described by formulas (2.6) and (2.7), respectively.
3. Positivity of the Elements in the Fixed
th Row of Green's Matrices
Consider the boundary value problem

where are linear continuous operators for
Theorem 3.1.
Let problem (2.3), (2.4) be uniquely solvable, all elements of its ( Green's matrix
nonnegative, and the operators
and
positive operators for
Then the following 2 assertions are equivalent:
-
(1)
there exists an absolutely continuous vector function
such that
   for
and the solution of the homogeneous equation (
   for
satisfying the conditions
   is nonpositive;
-
(2)
the boundary value problem (3.1) is uniquely solvable for every summable
and
and elements of the nth row of its Green's matrix satisfy the inequalities:
for
while
for
Proof.
Let us start with the implication By virtue of Lemma 2.1, the component
of the solution vector of problem (3.1) satisfies (2.5). Condition
by virtue of Theorem
of the paper [14] implies that Green's function G
of the boundary value problem

exists and satisfies the inequalities for
while
for
. Lemma 2.1, the representations of solutions of boundary value problem (3.1) and the scalar one-point problem (3.2) imply the equality

If is a negative operator for every
and
for
then
The nonpositivity of
implies that
is nonnegative and consequently
for
and
If we set for
and
for
then

and it is clear that It is known from Theorem
of the paper [14] that
for
This implies that
for
In order to prove , let us define
(
by the following way:

where (
is a solution to the problem

It is clear that the functions (
satisfy the homogeneous system

and for
Theorem 3.2.
Let problem (2.3), (2.4) be uniquely solvable, all elements of its ( Green's matrix
nonpositive, and
and
positive operators for
Then the following 2 assertions are equivalent:
-
(1*)
there exists an absolutely continuous vector function
such that
for
 and the solution of the homogeneous equation (
   for
  
  satisfying the conditions
  is nonnegative;
-
(2*)
the boundary value problem (3.1) is uniquely solvable for every summable
and
and elements of the nth row of its Green's matrix satisfies the inequalities:
for
  
for
while
for
The proof of this theorem is analogous to the proof of Theorem 3.1.
4. Sufficient Conditions of Nonpositivity of the Elements in the
th Row of Green's Matrices for System of Ordinary Differential Equations
In this paragraph, we consider the system of the ordinary differential equations

with the boundary conditions

Theorem 4.1.
Let the following conditions be fulfilled:
-
(1)
for
;
-
(2)
for
;
-
(3)
there exists a positive number
such that
(4.3)
Then problem (4.1), (4.2) is uniquely solvable for every summable and
  and the elements of the
th row of Green's matrix of boundary value problem (4.1), (4.2) satisfy the inequalities:
for
  for
  for
Proof.
Let us prove that all elements of Green's matrix of the auxiliary boundary value problem

are nonnegative. The conditions ,
, and the inequality

imply that the conditions and
of Theorem
of the paper [13] are fulfilled. Assertion (a) of Theorem
[13] is fulfilled. To prove it, we set
for
in this assertion. Now according to equivalence of assertions (a) and (b) in Theorem
of the paper [13], we get the nonnegativity of all elements of its Green's matrix
Let us set for
and
in the condition
of Theorem 3.1. We obtain that this condition is satisfied if
satisfies the following system of the inequalities:

Now by virtue of Theorem 3.1, all elements of the th row of Green's matrix satisfy the inequalities
for
and, using [14], we can conclude that
for
.
Consider now the following ordinary differential system of the second order;

with the conditions

From Theorem 4.1 as a particular case for , we obtain the following assertion.
Theorem 4.2.
Let the following two conditions be fulfilled:
-
(1)
-
(2)
there exists a positive
such that
(4.9)
Then problem (4.7), (4.8) is uniquely solvable for every summable and
and the elements of the second row of Green's matrix of problem (4.7), (4.8) satisfy the inequalities:
for
for
Remark 4.3.
If coefficients are constants, the second condition in Theorem 4.2 is as follows:

Remark 4.4.
Let us demonstrate that inequality (4.10) is best possible in a corresponding case and the condition

cannot be set instead of (4.10). The characteristic equation of the system

with constant coefficients is as follows:

If we set ,
then the roots are
,
and the problem

has nontrivial solution for
5. Sufficient Conditions of Nonpositivity of the Elements in the
th Row of Green's Matrices for Systems with Delay
Let us consider the system of the delay differential equations


with the boundary conditions

We introduce the denotations: ,
,  
, and
Theorem 5.1.
Let the following conditions be fulfilled:
-
(1)
for
-
(2)
for
-
(3)
for
-
(4)
there exists a positive number
such that

Then problem (5.1), (5.3) is uniquely solvable for every summable and
and the elements of the
th row of Green's matrix of problem (5.1), (5.3) satisfy the inequalities:
for
for
Proof.
Repeating the explanations in the beginning of the proof of Theorem 4.1, we can obtain on the basis of Theorem of the paper [13] that all the elements of
Green's matrix
of the auxiliary problem, consisting of the system

and the boundary conditions are nonnegative.
Let us set for
and
in the condition (
) of Theorem 3.1. We obtain that the condition (
) of Theorem 3.1 is satisfied if
satisfies the following system of the inequalities:


Now by virtue of Theorem 3.1, all elements of the th row of Green's matrix of problem (5.1), (5.3) satisfy the inequalities
for
while
for
Remark 5.2.
It was explained in the previous paragraph that in the case of ordinary system ( with constant coefficients
, inequality (5.4) is best possible in a corresponding case.
Let us consider the second-order scalar differential equation

where for
with the boundary conditions

and the corresponding differential system of the second order

where for
with the boundary conditions

It should be noted that the element of Green's matrix of system (5.10), (5.11) coincides with Green's function
of the problem (5.8), (5.9) for scalar second-order equation.
Theorem 5.3.
Assume that and there exists a positive number
such that

Then problem (5.10), (5.11) is uniquely solvable for every summable and
and the elements of the second row of Green's matrix of this problem satisfy the inequalities:
  
 while
  for
In order to prove Theorem 5.3, we set in the assertion (
) of Theorem 3.1.
Remark 5.4.
Inequality (5.12) is best possible in the following sense. Let us add in its right hand side. We get that the inequality

and the assertion of Theorem 5.3 is not true. Let us set that coefficients are constants: and
It is clear that the inequality (5.13) is fulfilled if we set
small enough. Consider the following homogeneous boundary value problem:

The components of the solution vector are periodic and for
the boundary value problem (5.14) has a nontrivial solution.
Let us prove the following assertions, giving an efficient test of nonpositivity of the elements in the th row of Green's matrix in the case when the coefficients
are small enough for
Theorem 5.5.
Let the following conditions be fulfilled:
-
(1)
for
-
(2)
  for
-
(3)
and other delays
are zeros;
-
(4)
the inequalities


are fulfilled.
Then problem (5.1), (5.3) is uniquely solvable for every summable and
, and the elements of the
th row of its Green's matrix satisfy the inequalities:
for
while
for
Proof.
Let us set for
and
in the condition
of Theorem 3.1.

In the left-hand side, we have the inequality

which is fulfilled when

The right-hand side in inequality (5.18) gets its maximum for Substituting this
into (5.19) and the right part of (5.17), we obtain inequalities (5.15) and (5.16).
Remark 5.6.
It can be stressed that we do not require a smallness of the interval in Theorems 5.1–5.5.
Remark 5.7.
It can be noted that inequality (5.15) is best possible in the following sense. If for
then system (5.1) and inequality (5.15) become of the following forms:


respectively. The opposite to (5.21) inequality implies oscillation of all solutions [15] of the equation

It implies that the homogeneous problem

has nontrivial solutions for corresponding Now it is clear that we cannot substitute

where is any positive number instead of inequality (5.15).
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Acknowledgments
The author thanks the referees for their available remarks. This research was supported by The Israel Science Foundation (Grant no. 828/07).
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Domoshnitsky, A. Differential Inequalities for One Component of Solution Vector for Systems of Linear Functional Differential Equations. Adv Differ Equ 2010, 478020 (2010). https://doi.org/10.1155/2010/478020
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DOI: https://doi.org/10.1155/2010/478020