Positive periodic solutions for systems of linear functional differential inequalities

Abstract

Consider the system of functional differential inequalities$D\left(\sigma \right)\left[u^{\prime }\right(t)-\ell (u\left)\right(t\left)\right]\ge 0\text{for a. e. }t\in R,$ where $\ell :C_{\omega }\left(R^n\right)\to L_{\omega }\left(R^n\right)$ is a linear bounded operator, $\sigma ={\left({\sigma }_i\right)}_{i=1}^n$ where ${\sigma }_i\in \{-1,1\}$, and $D\left(\sigma \right)=\mathrm{diag}({\sigma }_1,\dots ,{\sigma }_n)$. In the present paper, we establish conditions guaranteeing that there exists $c\in ]0,1[$ such that every absolutely continuous ω-periodic vector-valued function $u={\left(u_i\right)}_{i=1}^n$ satisfying the above-mentioned differential inequality belongs to a cone$K_c^n=\{u\in C_{\omega }(R^n):u_i(s)\ge c{u}_i(t)\text{ for }s,t\in R(i=1,\dots ,n\left)\right\}.$ We denote the set of periodic linear operators ℓ with the above property by $U_c^n\left(\sigma \right)$. We further show that the set $U_c^n\left(\sigma \right)$ is bounded from above, i.e., for every operator $\ell \in U_c^n\left(\sigma \right)$ and every σ-positive periodic operator $\tilde{\ell }$ there exists a positive threshold value ${\lambda }^{⁎}$ such that $\ell +\lambda \tilde{\ell }$ belongs to $U_c^n\left(\sigma \right)$ or does not, depending on whether the parameter λ is below or above ${\lambda }^{⁎}$. Finally, we propose a numerical method how to estimate the threshold ${\lambda }^{⁎}$ in particular cases. This method is described and illustrated by several examples. Possible applications to population models are discussed in the end of the paper.