# Differential Inequalities for One Component of Solution Vector for Systems of Linear Functional Differential Equations

- Alexander Domoshnitsky
^{1}Email author

**2010**:478020

https://doi.org/10.1155/2010/478020

© Alexander Domoshnitsky. 2010

**Received: **24 December 2009

**Accepted: **26 April 2010

**Published: **30 May 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.

## Keywords

## 1. Introduction

where are linear continuous operators, and are the spaces of continuous and summable functions , respectively.

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.

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].

where are summable functions, and are measurable functions such that for

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.

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.

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

where are linear bounded operators for and , are linear boundary functionals

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.

satisfying condition (2.4).

Proof.

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

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.

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

and it is clear that It is known from Theorem of the paper [14] that for This implies that 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

Theorem 4.1.

- (1)
- (2)
- (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.

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

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 .

From Theorem 4.1 as a particular case for , we obtain the following assertion.

Theorem 4.2.

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.

Remark 4.4.

## 5. Sufficient Conditions of Nonpositivity of the Elements in the th Row of Green's Matrices for Systems with Delay

We introduce the denotations: , , , and

Theorem 5.1.

- (1)
- (2)
- (3)
- (4)

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.

and the boundary conditions are nonnegative.

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.

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.

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.

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.

- (1)
- (2)
- (3)
- (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.

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.

## Declarations

### Acknowledgments

The author thanks the referees for their available remarks. This research was supported by The Israel Science Foundation (Grant no. 828/07).

## Authors’ Affiliations

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