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

Abstract

The classical Ważewski 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 inequalitiesAlthough 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 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, ω], where the system is considered.

Positive periodic solutions for multiparameter nonlinear differential systems with delays

“…1 For other research work on periodic solutions of functional differential equations and systems, we refer the readers to [15][16][17] and references therein.…”

New existence results for nonlinear delayed differential systems at resonance

“…Recently, Domoshnitsky et al [ 14 ] studied the following system of periodic functional differential equations: where , are linear bounded operator for , and for . Note that if the corresponding homogeneous problem has only trivial solution, then, for any and , system ( 1.5 ), ( 1.6 ) admits a unique solution x defined as [ 15 ] Here is a matrix and is called Green’s matrix of ( 1.5 ), the matrix is the fundamental matrix of ( 1.7 ) satisfying , where E denotes the unit matrix.…”

Infinitely Many Periodic Solutions to Delay Differential Equations via Critical Point Theory

“…Delay differential equations have widely been applied to describe the dynamics phenomena in both natural and manmade processes such as chemistry, physics, engineering, and economics. The existence of the periodic solutions for delay differential equations has been extensively investigated by using various methods, including fixed point theorems [1][2][3][4][5], Hopf bifurcation theorems [6][7][8], variational methods [9][10][11][12][13][14], the methods of differential inequalities [15][16][17][18][19][20][21], and other effective approaches (e.g., see [22][23][24]). In [25][26][27][28][29][30][31], the minimal periods of the periodic solutions to Lipschitzian differential equations are estimated through the Lipschitz constants (see Remark 4).…”

“…From this perspective, Theorem 1 complements the information in the case of non-Lipschitzian differential equations. For the unique solvability of the periodic problems on functional differential equations, we refer the reader to [1,[15][16][17][18][19][20][21].…”