Uniform L¹ behavior for the solution of a volterra equation with a parameter

Abstract

The solution u=u(t)=u(t,λ) of

(E) u′(t)+λ∫₀^tu(t-τ)(d+a(τ))dτ=0, u(0)=1, t ≥ 0, λ ≥ 1

where d ≥ 0, a is nonnegative, nonincreasing, convex and ∞ ≥ a(0+) > a(∞) = 0 is studied. In particular the question asked is: When is

(F) ∫₀^∞_{λ ≥ 1}^sup|u′′(t, λ)/λ|dt < ∞?

We obtain two necessary conditions for (F). For (F) to hold, it is necessary that (-lnt)a(τ)∈L¹(0,1) and lim sup _τ→∞ (τθ(τ))²/φ(τ) <∞ where â(τ)=∫₀^∞e^-iτta(t)dt=φ(τ)-iτθ(τ) (φ,θ both real).

We obtain sufficient conditions for (F) to hold which involve φ and θ (See Theorem 7). Then we look for direct conditions on a which imply (F). with the addition assumption -a′ is convex, we prove that (F) holds provided any one of the following hold:

(i) a(0+)<∞,

(ii) 0<lim inf _τ→∞ τ∫₀^1/τsa(s)ds / ∫₀^1/τ-sa′(s)ds ≤ lim sup _τ→∞ τ∫₀^1/τsa(s)ds / ∫₀^1/τ-sa′(s)ds < ∞,

(iii) lim _τ→∞ τ∫₀^1/τsa(s)ds / ∫₀^1/τa(s)ds = 0,

(iv) lim _τ→∞ ∫₀^1/τ-sa′(s)ds / ∫₀^1/τa(s)ds = 0, a²(t)/-a′(t) is increasing for small t and a²(t) / -ta′(t)∈L¹(0,∈) for some ∈>0,

(v) lim _τ→∞ ∫₀^1/τ-sa′(s)ds / ∫₀^1/τa(s)ds = 0 and τ(∫₀^1/τ a(s)ds)³ / ∫₀^1/τ-sa′(s)ds ≤ M < ∞ for δ ≤ τ < ∞ (some δ > 0).

Thus (F) holds for wide classes of examples. In particular, (F) holds when d+a(t) = t^-p, 0 < p < 1; a(t)+d = -lnt (small t); a(t)+d = t⁻¹(-lnt)^-q, q > 2 (small t).