$\sigma$-Increasing Positive Solutions for Systems of Linear Functional Differential Inequalities of Non-Metzler Type

Abstract

Consider the system of functional differential inequalities:

$$\begin{aligned} \mathcal {D}\big (\sigma \big )\big [u'(t)-\ell (u)(t)\big ]\ge 0\qquad \text{ for } \text{ a. } \text{ e. } \,t\in [a,b],\quad \varphi (u)\ge 0, \end{aligned}$$

where $\ell :C\big ([a,b];\mathbb {R}^n\big )\rightarrow L\big ([a,b];\mathbb {R}^n\big )$ is a linear bounded operator, $\varphi :C\big ([a,b];\mathbb {R}^n\big )\rightarrow \mathbb {R}^n$ is a linear bounded functional, $\sigma =(\sigma _i)_{i=1}^n$, where $\sigma _i\in \{-1,1\}$, and $\mathcal {D}\big (\sigma \big )={\text {diag}}(\sigma _1,\dots ,\sigma _n)$. In the present paper, we establish conditions guaranteeing that every absolutely continuous vector-valued function u satisfying the above-mentioned inequalities admits also the inequalities $u(t)\ge 0$ for $t\in [a,b]$ and $\mathcal {D}\big (\sigma \big )u'(t)\ge 0$ for a. e. $t\in [a,b]$.