Existence and Properties of Semi-Bounded Global Solutions to the Functional Differential Equation with Volterra's Type Operators on the Real Line

Abstract

Consider the equation $$ u'(t)=\ell_0(u)(t)-\ell_1(u)(t)+f(u)(t)\qquad\mbox{for~a.~e.~}\,t\in\mathbb{R} $$ where $\ell_i:C_{loc}\big(\mathbb{R};\mathbb{R}\big)\to L_{loc}\big(\mathbb{R};\mathbb{R}\big)$ $(i=0,1)$ are linear positive continuous operators and $f:C_{loc}\big(\mathbb{R};\mathbb{R}\big)\to L_{loc}\big(\mathbb{R};\mathbb{R}\big)$ is a continuous operator satisfying the local Carath\'eodory conditions. The efficient conditions guaranteeing the existence of a global solution, which is bounded and non-negative in the neighbourhood of $-\infty$, to the equation considered are established provided $\ell_0$, $\ell_1$, and $f$ are Volterra's type operators. The existence of a solution which is positive on the whole real line is discussed, as well. Furthermore, the asymptotic properties of such solutions are studied in the neighbourhood of $-\infty$. The results are applied to certain models appearing in natural sciences.