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

2015/07/30 by Aguerrea, Maitere, Hakl, Robert
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1507.08573

Abstract

Consider the equation u'(t)=ℓ₀(u)(t)-ℓ₁(u)(t)+f(u)(t)\qquadfor~a.~e.~ t∈ℝ where ℓ_i:C_loc(ℝ;ℝ)→ L_loc(ℝ;ℝ) (i=0,1) are linear positive continuous operators and f:C_loc(ℝ;ℝ)→ L_loc(ℝ;ℝ) is a continuous operator satisfying the local Carathéodory conditions. The efficient conditions guaranteeing the existence of a global solution, which is bounded and non-negative in the neighbourhood of -∞, to the equation considered are established provided ℓ₀, ℓ₁, 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 -∞. The results are applied to certain models appearing in natural sciences.

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