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

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).

Name thesis:degree_name

Ph. D.

Level thesis:degree_level

doctoral

Discipline thesis:degree_discipline

Mathematics

Department dc:contributor.department

Mathematics

Grantor dc:publisher

Virginia Polytechnic Institute and State University

Year dc:date.issued

1985

Author dc:creator

• Noren, Richard Dennis

Chair dc:contributor.committeechair

• Hannsgen, Kenneth B.

Committee members dc:contributor.committeemember

• Wheeler, Robert
• Prather, Carl
• Herdman, Terry L.
• McCoy, Robert A.

Statement dc:rights

• In Copyright

Licence dc:rights.uri

• http://rightsstatements.org/vocab/InC/1.0/

Handle dc:identifier.uri

http://hdl.handle.net/10919/49966

OAI identifier oai:identifier

oai:vtechworks.lib.vt.edu:10919/49966