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Abstract
The classical Wazewski theorem claims that the condition p ij ≤ 0, j ≠ i, i, j =1,...,n, is necessary and sufficient for non-negativity of all the components of solution vector to a system of the inequalities , x i (0) ≥ 0, i =1, ..., n. Although this result was extent on various boundary value problems and on delay differential systems, analogs of these heavy restrictions on non-diagonal coefficients p ij preserve in all assertions of this sort. It is clear from formulas of the integral representation of the general solution that these theorems claim actually the positivity of all elements of Green's matrix. The method to compare only one component of the solution vector, which does not require such heavy restrictions, is proposed in this article. Note that comparison of only one component of the solution vector means the positivity of elements in a corresponding row of Green's matrix. Necessary and sufficient conditions of this fact are obtained in the form of theorems about differential inequalities. It is demonstrated that the sufficient conditions of positivity of the elements in the nth row of Green's matrix, proposed in this article, cannot be improved in corresponding cases. The main idea of our approach is to construct a first order functional differential equation for the n th component of the solution vector and then to use assertions, obtained recently for first order scalar functional differential equations. 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. Note that in some cases the sufficient conditions, obtained in the article, does not require any smallness of the interval [0, ω], where the system is considered. Mathematics Subject Classification 2000: 34K06; 34K10.
Highlights
Domoshnitsky et al Journal of Inequalities and Applications 2012, 2012:112 http://www.journalofinequalitiesandapplications.com/content/2012/1/112 subjected to the periodic conditions xi(0) = xi(ω) + ci, i = 1, . . . , n, (1:2)
The technique of analysis of the first order scalar functional differential equations, developed, for example, in the works [13,14], is used. On this basis we prove assertions of Section 3
Are fulfilled, where the numbers A+ and A- are given by the relations (3.19) and (3.20), respectively, periodic problem (1.1), (1.4) is uniquely solvable for each right hand side f = col(f1, ..., fn) and the elements Gnj(t, s), j = 1, ..., n, of its Green’s matrix G (t, s) satisfy the inequalities (3.10), (iii) if Bni, i = 1, ..., n - 1, are positive operators and the inequalities (3.18) are fulfilled, where n−1 n−1
Summary
Let the following conditions be fulfilled: (1) Green’s functions gi(t, s), i =1, ..., n − 1, of n − 1 scalar periodic problems (2.1), (2.2) exist, are non-negative, and satisfy inequality (2.4) for each positive measurable essentially bounded function , (2) all non-diagonal operators Bij, i, j =1, ..., n − 1, i ≠ j, are negative, (3) there exists a vector z = col(z1, ..., zn-1) with all positive components such that n−1 (Bijzj)(t) ≥ 1, t ∈ [0, ω], i = 1, .
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