The existence of smallest positive eigenvalues is established for the linear differential equations $u^{(4)}+\\lambda_{1} q(t)u=0$ and $u^{(4)}+\\lambda_{2} r(t)u=0$, $0\\leq t \\leq 1$, with each satisfying the boundary conditions $u(0)=u'(p)=u''(1)=u'''(1)=0$ where $1-\\frac{\\sqrt{3}}{3}\\le p < 1$. A comparison theorem for smallest positive eigenvalues is then obtained. Using the same theorems, we will extend the problem to the fifth order via the Green's Function and again via Substitution. Applying the comparison theorems and the properties of $u_0$-positive operators to determine the existence of smallest eigenvalues. The existence of these smallest eigenvalues is then applied to characterize extremal points of the differential equation $u^{(4)} + q(t)u = 0$ satisfying boundary conditions $u(0) = u'(p) = u''(b) = u'''(b)= 0$ where $1-\\frac{
","abstract_html":"<p>The existence of smallest positive eigenvalues is established for the linear differential equations u(4)+\\lambda1 q(t)u=0 and u(4)+\\lambda2 r(t)u=0, $0\\leq t \\leq 1$, with each satisfying the boundary conditions $u(0)=u'(p)=u''(1)=u'''(1)=0$ where $1-\\frac{\\sqrt{3}}{3}\\le p < 1$. A comparison theorem for smallest positive eigenvalues is then obtained. Using the same theorems, we will extend the problem to the fifth order via the Green's Function and again via Substitution. Applying the comparison theorems and the properties of u0-positive operators to determine the existence of smallest eigenvalues. The existence of these smallest eigenvalues is then applied to characterize extremal points of the differential equation u(4) + q(t)u = 0 satisfying boundary conditions $u(0) = u'(p) = u''(b) = u'''(b)= 0$ where $1-\\frac{ </p>","abstract_has_math":true,"creators":["King`, Sarah Schulz"],"institution":"Eastern Kentucky University","degree_name":"Master of Science (MS)","degree_level":"Master's","degree_discipline":"Mathematics and Statistics","degree_department":null,"school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2013,"date_issued":"2013-01-01T08:00:00Z","date_published":"2013-01-01T08:00:00Z","updated_at":"2026-07-24T02:15:19Z","subjects":["boundary value","differential","eigenvalue","extremal points","fifth order","fourth order","Mathematics"],"languages":[],"rights":["Copyright 2013 Sarah Schulz King`"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://encompass.eku.edu/etd/185","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:creator","label":"Author","values":["King`, Sarah Schulz"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:publisher","label":"Institution","values":["Encompass Digital Archive, Eastern Kentucky University"]},{"key":"dc:type","label":"Dc Type","values":["Master Thesis"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics and Statistics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Master's"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Master of Science (MS)"]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["Eastern Kentucky University"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["boundary value","differential","eigenvalue","extremal points","fifth order","fourth order","Mathematics"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:rights","label":"Dc Rights","values":["Copyright 2013 Sarah Schulz King`"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://encompass.eku.edu/etd/185"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["The existence of smallest positive eigenvalues is established for the linear differential equations $u^{(4)}+\\lambda_{1} q(t)u=0$ and $u^{(4)}+\\lambda_{2} r(t)u=0$, $0\\leq t \\leq 1$, with each satisfying the boundary conditions $u(0)=u'(p)=u''(1)=u'''(1)=0$ where $1-\\frac{\\sqrt{3}}{3}\\le p < 1$. A comparison theorem for smallest positive eigenvalues is then obtained. Using the same theorems, we will extend the problem to the fifth order via the Green's Function and again via Substitution. Applying the comparison theorems and the properties of $u_0$-positive operators to determine the existence of smallest eigenvalues. The existence of these smallest eigenvalues is then applied to characterize extremal points of the differential equation $u^{(4)} + q(t)u = 0$ satisfying boundary conditions $u(0) = u'(p) = u''(b) = u'''(b)= 0$ where $1-\\frac{
"]},{"key":"dc:format","label":"Dc Format","values":["application/pdf"]},{"key":"dc:source","label":"Dc Source","values":["Encompass Digital Archive: Online Theses and Dissertations"]},{"key":"dc:title","label":"Title","values":["Positive Solutions, Existence Of Smallest Eigenvalues, And Comparison Of Smallest Eigenvalues Of A Fourth Order Three Point Boundary Value Problem"]}]}],"canonical_facts":{"dc:creator":["King`, Sarah Schulz"],"dc:description.abstract":["The existence of smallest positive eigenvalues is established for the linear differential equations $u^{(4)}+\\lambda_{1} q(t)u=0$ and $u^{(4)}+\\lambda_{2} r(t)u=0$, $0\\leq t \\leq 1$, with each satisfying the boundary conditions $u(0)=u'(p)=u''(1)=u'''(1)=0$ where $1-\\frac{\\sqrt{3}}{3}\\le p < 1$. A comparison theorem for smallest positive eigenvalues is then obtained. Using the same theorems, we will extend the problem to the fifth order via the Green's Function and again via Substitution. Applying the comparison theorems and the properties of $u_0$-positive operators to determine the existence of smallest eigenvalues. The existence of these smallest eigenvalues is then applied to characterize extremal points of the differential equation $u^{(4)} + q(t)u = 0$ satisfying boundary conditions $u(0) = u'(p) = u''(b) = u'''(b)= 0$ where $1-\\frac{
"],"dc:format":["application/pdf"],"dc:identifier":["https://encompass.eku.edu/etd/185"],"dc:publisher":["Encompass Digital Archive, Eastern Kentucky University"],"dc:rights":["Copyright 2013 Sarah Schulz King`"],"dc:source":["Encompass Digital Archive: Online Theses and Dissertations"],"dc:subject":["boundary value","differential","eigenvalue","extremal points","fifth order","fourth order","Mathematics"],"dc:title":["Positive Solutions, Existence Of Smallest Eigenvalues, And Comparison Of Smallest Eigenvalues Of A Fourth Order Three Point Boundary Value Problem"],"dc:type":["Master Thesis"],"thesis:degree_discipline":["Mathematics and Statistics"],"thesis:degree_level":["Master's"],"thesis:degree_name":["Master of Science (MS)"],"thesis:institution_name":["Eastern Kentucky University"]},"updated_at":"2026-07-24T02:15:19Z"}