{"id":{"repo_id":"baylor","oai_identifier":"oai:baylor-ir.tdl.org:2104/8235"},"canonical_url":"https://search.dev.ndltd.org/etd/baylor/oai:baylor-ir.tdl.org:2104/8235","repository":{"repo_id":"baylor","name":"Baylor University","base_url":"https://baylor-ir.tdl.org/server/oai/request"},"display":{"title":"Comparison of smallest eigenvalues and extremal points for third and fourth order three point boundary value problems.","abstract":"The theory of u₀-positive operators with respect to a cone in a Banach space is applied to the linear differential equations u⁽⁴⁾ + λ₁p(x)u = 0 and u⁽⁴⁾ + λ₂q(x)u = 0, 0 ≤ x ≤ 1, with each satisfying the boundary conditions u(0) = u′(r) = u″(r) = u‴(1) = 0, 0 < r < 1. The existence of smallest positive eigenvalues is established, and a comparison theorem for smallest positive eigenvalues is obtained. These results are then extended to the nth order problem using two different methods. One method involves finding the Green's function for –u⁽ⁿ⁾ = 0 satisfying the higher order boundary conditions, and the other involves making a substitution that allows us to work with a variation of the fourth order problem. Extremal points via Krein-Rutman theory are then found. Analogous results are then obtained for the eigenvalue problems u‴ + λ₁p(x)u = 0 and u‴ + λ₂q(x)u = 0, with each satisfying u(0) = u′(r) = u″(1) = 0, 0 < 1/2 < r < 1.","abstract_html":"The theory of u₀-positive operators with respect to a cone in a Banach space is applied to the linear differential equations u⁽⁴⁾ + λ₁p(x)u = 0 and u⁽⁴⁾ + λ₂q(x)u = 0, 0 ≤ x ≤ 1, with each satisfying the boundary conditions u(0) = u′(r) = u″(r) = u‴(1) = 0, 0 < r < 1. The existence of smallest positive eigenvalues is established, and a comparison theorem for smallest positive eigenvalues is obtained. These results are then extended to the nth order problem using two different methods. One method involves finding the Green's function for –u⁽ⁿ⁾ = 0 satisfying the higher order boundary conditions, and the other involves making a substitution that allows us to work with a variation of the fourth order problem. Extremal points via Krein-Rutman theory are then found. Analogous results are then obtained for the eigenvalue problems u‴ + λ₁p(x)u = 0 and u‴ + λ₂q(x)u = 0, with each satisfying u(0) = u′(r) = u″(1) = 0, 0 < 1/2 < r < 1.","abstract_has_math":false,"creators":["Neugebauer, Jeffrey T."],"institution":"Baylor University.","degree_name":"Ph.D.","degree_level":"Doctoral","degree_discipline":null,"degree_department":null,"school":null,"contributors":[],"advisors":["Henderson, Johnny."],"committee_chairs":[],"committee_members":[],"year":2011,"date_issued":"2011-09","date_published":"2011-09","updated_at":"2026-07-24T01:07:56Z","subjects":["Differential equations.","Eigenvalue problems.","Extremal points."],"languages":["en"],"rights":["Baylor University works are protected by copyright. They may be viewed from this source for any purpose, but reproduction or distribution in any format is prohibited without written permission. Contact libraryquestions@baylor.edu for inquiries about permission."],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://hdl.handle.net/2104/8235","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Henderson, Johnny."]},{"key":"dc:creator","label":"Author","values":["Neugebauer, Jeffrey T."]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2011-09-14T13:01:51Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2011-09-14T13:01:51Z"]},{"key":"dc:date.issued","label":"Date","values":["2011-09"]},{"key":"dc:type","label":"Dc Type","values":["Thesis"]},{"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":["Baylor University."]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Differential equations.","Eigenvalue problems.","Extremal points."]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language.iso","label":"Language (ISO)","values":["en"]},{"key":"dc:rights","label":"Dc Rights","values":["Baylor University works are protected by copyright. They may be viewed from this source for any purpose, but reproduction or distribution in any format is prohibited without written permission. Contact libraryquestions@baylor.edu for inquiries about permission."]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.uri","label":"Identifier URI","values":["https://hdl.handle.net/2104/8235"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["The theory of u₀-positive operators with respect to a cone in a Banach space is applied to the linear differential equations u⁽⁴⁾ + λ₁p(x)u = 0 and u⁽⁴⁾ + λ₂q(x)u = 0, 0 ≤ x ≤ 1, with each satisfying the boundary conditions u(0) = u′(r) = u″(r) = u‴(1) = 0, 0 < r < 1. The existence of smallest positive eigenvalues is established, and a comparison theorem for smallest positive eigenvalues is obtained. These results are then extended to the nth order problem using two different methods. One method involves finding the Green's function for –u⁽ⁿ⁾ = 0 satisfying the higher order boundary conditions, and the other involves making a substitution that allows us to work with a variation of the fourth order problem. Extremal points via Krein-Rutman theory are then found. Analogous results are then obtained for the eigenvalue problems u‴ + λ₁p(x)u = 0 and u‴ + λ₂q(x)u = 0, with each satisfying u(0) = u′(r) = u″(1) = 0, 0 < 1/2 < r < 1."]},{"key":"dc:format.mimetype","label":"Dc Format Mimetype","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["Comparison of smallest eigenvalues and extremal points for third and fourth order three point boundary value problems."]}]}],"canonical_facts":{"dc:contributor.advisor":["Henderson, Johnny."],"dc:creator":["Neugebauer, Jeffrey T."],"dc:date.accessioned":["2011-09-14T13:01:51Z"],"dc:date.available":["2011-09-14T13:01:51Z"],"dc:date.issued":["2011-09"],"dc:description.abstract":["The theory of u₀-positive operators with respect to a cone in a Banach space is applied to the linear differential equations u⁽⁴⁾ + λ₁p(x)u = 0 and u⁽⁴⁾ + λ₂q(x)u = 0, 0 ≤ x ≤ 1, with each satisfying the boundary conditions u(0) = u′(r) = u″(r) = u‴(1) = 0, 0 < r < 1. The existence of smallest positive eigenvalues is established, and a comparison theorem for smallest positive eigenvalues is obtained. These results are then extended to the nth order problem using two different methods. One method involves finding the Green's function for –u⁽ⁿ⁾ = 0 satisfying the higher order boundary conditions, and the other involves making a substitution that allows us to work with a variation of the fourth order problem. Extremal points via Krein-Rutman theory are then found. Analogous results are then obtained for the eigenvalue problems u‴ + λ₁p(x)u = 0 and u‴ + λ₂q(x)u = 0, with each satisfying u(0) = u′(r) = u″(1) = 0, 0 < 1/2 < r < 1."],"dc:format.mimetype":["application/pdf"],"dc:identifier.uri":["https://hdl.handle.net/2104/8235"],"dc:language.iso":["en"],"dc:rights":["Baylor University works are protected by copyright. They may be viewed from this source for any purpose, but reproduction or distribution in any format is prohibited without written permission. Contact libraryquestions@baylor.edu for inquiries about permission."],"dc:subject":["Differential equations.","Eigenvalue problems.","Extremal points."],"dc:title":["Comparison of smallest eigenvalues and extremal points for third and fourth order three point boundary value problems."],"dc:type":["Thesis"],"thesis:degree_level":["Doctoral"],"thesis:degree_name":["Ph.D."],"thesis:institution_name":["Baylor University."]},"updated_at":"2026-07-24T01:07:56Z"}