The solution u=u(t)=u(t,λ) of
(E) u′(t)+λ∫<sub>0</sub><sup>t</sup>u(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) ∫<sub>0</sub><sup>∞</sup><sub>λ ≥ 1</sub><sup>sup</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 <sub>τ→∞</sub> (τθ(τ))²/φ(τ) <∞ where â(τ)=∫<sub>0</sub><sup>∞</sup>e<sup>-iτt</sup>a(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 <sub>τ→∞</sub> τ∫<sub>0</sub><sup>1/τ</sup>sa(s)ds / ∫<sub>0</sub><sup>1/τ</sup>-sa′(s)ds ≤ lim sup <sub>τ→∞</sub> τ∫<sub>0</sub><sup>1/τ</sup>sa(s)ds / ∫<sub>0</sub><sup>1/τ</sup>-sa′(s)ds < ∞,
(iii) lim <sub>τ→∞</sub> τ∫<sub>0</sub><sup>1/τ</sup>sa(s)ds / ∫<sub>0</sub><sup>1/τ</sup>a(s)ds = 0,
(iv) lim <sub>τ→∞</sub> ∫<sub>0</sub><sup>1/τ</sup>-sa′(s)ds / ∫<sub>0</sub><sup>1/τ</sup>a(s)ds = 0, a²(t)/-a′(t) is increasing for small t and a²(t) / -ta′(t)∈L¹(0,∈) for some ∈>0,
(v) lim <sub>τ→∞</sub> ∫<sub>0</sub><sup>1/τ</sup>-sa′(s)ds / ∫<sub>0</sub><sup>1/τ</sup>a(s)ds = 0 and τ(∫<sub>0</sub><sup>1/τ</sup> a(s)ds)³ / ∫<sub>0</sub><sup>1/τ</sup>-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<sup>-p</sup>, 0 < p < 1; a(t)+d = -lnt (small t); a(t)+d = t⁻¹(-lnt)<sup>-q</sup>, q > 2 (small t). / Ph. D. / incomplete_metadata
| Identifer | oai:union.ndltd.org:VTETD/oai:vtechworks.lib.vt.edu:10919/49966 |
| Date | January 1985 |
| Creators | Noren, Richard Dennis |
| Contributors | Mathematics, Hannsgen, Kenneth B., Wheeler, Robert, Prather, Carl, Herdman, Terry L., McCoy, Robert A. |
| Publisher | Virginia Polytechnic Institute and State University |
| Source Sets | Virginia Tech Theses and Dissertation |
| Detected Language | English |
| Type | Dissertation, Text |
| Format | iii, 69 leaves ;, application/pdf, application/pdf |
| Rights | In Copyright, http://rightsstatements.org/vocab/InC/1.0/ |
| Relation | OCLC# 13131395 |
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