Differential Inequalities for One Component of Solution Vector for Systems of Linear Functional Differential Equations
© Alexander Domoshnitsky. 2010
Received: 24 December 2009
Accepted: 26 April 2010
Published: 30 May 2010
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.
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 , were devoted to the various aspects of Tchaplygin's approximate method. The well-known monograph by Lakshmikantham and Leela  was one of the most important in this area. The known book by Krasnosel'skii et al.  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 .
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 , 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.
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.
satisfying condition (2.4).
and it is clear that It is known from Theorem of the paper  that for This implies that 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
imply that the conditions and of Theorem of the paper  are fulfilled. Assertion (a) of Theorem  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 , 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 , we can conclude that for .
5. Sufficient Conditions of Nonpositivity of the Elements in the th Row of Green's Matrices for Systems with Delay
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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