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

TL;DR

This paper investigates the existence, positivity, and asymptotic behavior of semi-bounded global solutions to a class of functional differential equations with Volterra-type operators, with applications to natural science models.

Contribution

It establishes new sufficient conditions for the existence of semi-bounded and positive solutions to these equations, extending previous results to more general operators.

Findings

Existence of semi-bounded solutions under Volterra-type operators.

Conditions for solutions to be positive on the entire real line.

Analysis of asymptotic behavior near negative infinity.

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.