{"id":{"repo_id":"vt","oai_identifier":"oai:vtechworks.lib.vt.edu:10919/49966"},"canonical_url":"https://search.dev.ndltd.org/etd/vt/oai:vtechworks.lib.vt.edu:10919/49966","repository":{"repo_id":"vt","name":"Virginia Tech","base_url":"https://vtechworks.lib.vt.edu/oai/request"},"display":{"title":"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∫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) 0τ→∞ 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).","abstract_html":"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).","abstract_has_math":false,"creators":["Noren, Richard Dennis"],"institution":"Virginia Polytechnic Institute and State University","degree_name":"Ph. D.","degree_level":"doctoral","degree_discipline":"Mathematics","degree_department":"Mathematics","school":null,"contributors":[],"advisors":[],"committee_chairs":["Hannsgen, Kenneth B."],"committee_members":["Wheeler, Robert","Prather, Carl","Herdman, Terry L.","McCoy, Robert A."],"year":1985,"date_issued":"1985","date_published":"1985","updated_at":"2026-07-22T22:19:54Z","subjects":[],"languages":[],"rights":["In Copyright"],"rights_urls":["http://rightsstatements.org/vocab/InC/1.0/"],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/10919/49966","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.committeechair","label":"Committee Chair","values":["Hannsgen, Kenneth B."]},{"key":"dc:contributor.committeemember","label":"Committee Member","values":["Wheeler, Robert","Prather, Carl","Herdman, Terry L.","McCoy, Robert A."]},{"key":"dc:contributor.department","label":"Department","values":["Mathematics"]},{"key":"dc:creator","label":"Author","values":["Noren, Richard Dennis"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2014-08-13T14:38:54Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2014-08-13T14:38:54Z"]},{"key":"dc:date.issued","label":"Date","values":["1985"]},{"key":"dc:publisher","label":"Institution","values":["Virginia Polytechnic Institute and State University"]},{"key":"dc:type","label":"Dc Type","values":["Dissertation"]},{"key":"dc:type.dcmitype","label":"Dc Type Dcmitype","values":["Text"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["doctoral"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph. D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["Virginia Polytechnic Institute and State University"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:rights","label":"Dc Rights","values":["In Copyright"]},{"key":"dc:rights.uri","label":"Rights URI","values":["http://rightsstatements.org/vocab/InC/1.0/"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.uri","label":"Identifier URI","values":["http://hdl.handle.net/10919/49966"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["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) 0τ→∞ 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)."]},{"key":"dc:description.degree","label":"Dc Description Degree","values":["Ph. D."]},{"key":"dc:format.mimetype","label":"Dc Format Mimetype","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["Uniform L¹ behavior for the solution of a volterra equation with a parameter"]}]}],"canonical_facts":{"dc:contributor.committeechair":["Hannsgen, Kenneth B."],"dc:contributor.committeemember":["Wheeler, Robert","Prather, Carl","Herdman, Terry L.","McCoy, Robert A."],"dc:contributor.department":["Mathematics"],"dc:creator":["Noren, Richard Dennis"],"dc:date.accessioned":["2014-08-13T14:38:54Z"],"dc:date.available":["2014-08-13T14:38:54Z"],"dc:date.issued":["1985"],"dc:description.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) 0τ→∞ 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)."],"dc:description.degree":["Ph. D."],"dc:format.mimetype":["application/pdf"],"dc:identifier.uri":["http://hdl.handle.net/10919/49966"],"dc:publisher":["Virginia Polytechnic Institute and State University"],"dc:rights":["In Copyright"],"dc:rights.uri":["http://rightsstatements.org/vocab/InC/1.0/"],"dc:title":["Uniform L¹ behavior for the solution of a volterra equation with a parameter"],"dc:type":["Dissertation"],"dc:type.dcmitype":["Text"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["doctoral"],"thesis:degree_name":["Ph. D."],"thesis:institution_name":["Virginia Polytechnic Institute and State University"]},"updated_at":"2026-07-22T22:19:54Z"}