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Component-wise positivity of solutions to periodic boundary problem for linear functional differential system. (English) Zbl 1278.34070

Summary: The classical Ważewski theorem claims that the condition \(p_{ij}\leq0\), \(j\neq i\), \(i,j=1,\dots,n\), is necessary and sufficient for non-negativity of all the components of the solution vector to a system of the inequalities \[ x'(t)+\sum_{j=1}^{n}p_{ij}(t)x(t)\geq 0,\,x_{i}(0)\geq 0,\, i=1,\dots, n. \] Although this result has been extent to various boundary value problems and to delay differential systems, analogs of these heavy restrictions on the non-diagonal coefficients are \(p_{ij}\) preserved in all assertions of this sort. It is clear from the 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 the 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 \(n\)th 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, do not require any smallness of the interval \([0, \omega]\), where the system is considered.

MSC:

34K10 Boundary value problems for functional-differential equations
34K06 Linear functional-differential equations
34K12 Growth, boundedness, comparison of solutions to functional-differential equations
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References:

[1] doi:10.1023/A:1022829931363 · Zbl 0930.34047 · doi:10.1023/A:1022829931363
[2] doi:10.1017/S0017089509990218 · Zbl 1200.34073 · doi:10.1017/S0017089509990218
[3] doi:10.1023/A:1022304626385 · Zbl 1098.34560 · doi:10.1023/A:1022304626385
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