Uniform L¹ behavior for the solution of a volterra equation with a parameter
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Abstract
The solution u=u(t)=u(t,λ) of
(E) u′(t)+λ∫0tu(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 ≥ 1sup|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∫01/τsa(s)ds / ∫01/τ-sa′(s)ds ≤ lim sup τ→∞ τ∫01/τsa(s)ds / ∫01/τ-sa′(s)ds < ∞,
(iii) lim τ→∞ τ∫01/τsa(s)ds / ∫01/τa(s)ds = 0,
(iv) lim τ→∞ ∫01/τ-sa′(s)ds / ∫01/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→∞ ∫01/τ-sa′(s)ds / ∫01/τa(s)ds = 0 and τ(∫01/τ a(s)ds)³ / ∫01/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).