TY - JOUR

T1 - Component-wise positivity of solutions to periodic boundary problem for linear functional differential system

AU - Domoshnitsky, Alexander

AU - Hakl, Robert

AU - Šremr, Jiří

PY - 2012

Y1 - 2012

N2 - The classical Ważewski theorem claims that the condition pij ≤ 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'(t)+ Σnj=1 pij(t)x(t) ≥ 0, xi (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 pij preserve in all assertions of this sort. It is clear from formulas of the integra 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 nth 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, w], where the system is considered.

AB - The classical Ważewski theorem claims that the condition pij ≤ 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'(t)+ Σnj=1 pij(t)x(t) ≥ 0, xi (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 pij preserve in all assertions of this sort. It is clear from formulas of the integra 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 nth 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, w], where the system is considered.

KW - Green's matrix

KW - Linear functional differential system

KW - Negative solution

KW - Periodic problem

KW - Positive solution

UR - http://www.scopus.com/inward/record.url?scp=84870470563&partnerID=8YFLogxK

U2 - 10.1186/1029-242X-2012-112

DO - 10.1186/1029-242X-2012-112

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AN - SCOPUS:84870470563

SN - 1025-5834

VL - 2012

JO - Journal of Inequalities and Applications

JF - Journal of Inequalities and Applications

M1 - 112

ER -