# $$\sigma$$-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. $$t\in [a,b]$$.

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1. 1.

By a solution to (3.2), we understand a function $$u\in AC\big ([a,b];\mathbb {R}^n\big )$$ that satisfies the differential equality in (3.2) almost everywhere in [ab] and it satisfies the boundary condition $$\varphi (u)=c$$.

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Correspondence to Robert Hakl.