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

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

(1.1)

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

(1.2)

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

(1.3)

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

(1.4)

it follows that

(1.5)

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:

(1.6)

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

The classical Wazewskii's theorem claims [9] that the condition

(1.7)

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

(1.8)

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

(1.9)

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

(1.10)

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

(1.11)

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

(2.1)

(2.2)

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:

(2.3)

of the order and the boundary conditions

(2.4)

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:

(2.5)

where the operator and the function are defined by the equalities

(2.6)

(2.7)

where is the solution of the system

(2.8)

satisfying condition (2.4).

Proof.

Using Green's matrix of problem (2.3), (2.4), we obtain

(2.9)

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

(3.1)

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

(3.2)

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

(3.3)

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

(3.4)

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:

(3.5)

where ( is a solution to the problem

(3.6)

It is clear that the functions ( satisfy the homogeneous system

(3.7)

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 theth Row of Green's Matrices for System of Ordinary Differential Equations

In this paragraph, we consider the system of the ordinary differential equations

(4.1)

with the boundary conditions

(4.2)

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

(4.4)

are nonnegative. The conditions , , and the inequality

(4.5)

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:

(4.6)

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;

(4.7)

with the conditions

(4.8)

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:

(4.10)

Remark 4.4.

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

(4.11)

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

(4.12)

with constant coefficients is as follows:

(4.13)

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

(4.14)

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

(5.1)

(5.2)

with the boundary conditions

(5.3)

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

(5.4)

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.

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

(5.5)

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:

(5.6)

(5.7)

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

(5.8)

where for with the boundary conditions

(5.9)

and the corresponding differential system of the second order

(5.10)

where for with the boundary conditions

(5.11)

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

(5.12)

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

(5.13)

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:

(5.14)

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

(5.15)

(5.16)

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.

(5.17)

In the left-hand side, we have the inequality

(5.18)

which is fulfilled when

(5.19)

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:

(5.20)

(5.21)

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

(5.22)

It implies that the homogeneous problem

(5.23)

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

(5.24)

where is any positive number instead of inequality (5.15).