Existence theorems and periodic solutions of neutral integral equations

摘要

In this paper we study the equations x(t) = a(t) + ∫ from x=0 to x=t of D(t,s,x(s))ds + ∫ from x=t to x=∞ E(t,s,x(s))ds, t ∈ R~+ (1A) and x(t) = a(t) + ∫ from x=-∞ to x=t D(t,s,x(s))ds + ∫ from x=t to x=∞ E(t,s,x(s))ds, t ∈ R, (1B) where R~+:=[0,∞), R:=(-∞,∞), a:R → R~n, D:Δ~- * R~n → R~n, and E:Δ~+ * R~n → R~n are continuous, and where Δ~-:={(t,s):s ≤ t} and Δ~+:={(t,s):s ≥ t}.