{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/71258"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/71258","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Degree Theory and Nonlinear Boundary Value Problems at Resonance","abstract":"Consider the second order non-linear differential operator ${\\cal L}y$ = Ly + $\\eta y\\sp3$, where $\\eta$ = $\\pm$1 and L, the linear part of ${\\cal L}$, is of the form Ly = $y\\sp{\\prime\\prime}$ + $p(x)y\\sp\\prime$ + q(x)y. Assume p(x) and q(x) are integrable on (a,b). We study the existence and uniqueness of solutions of ${\\cal L}y$ = f satisfying linear boundary conditions on (a,b). The function f is an element of $L\\sp1$ (a,b) Define BC = $\\{y \\in L\\sp\\infty$ (a,b): $y\\sp\\prime$ is absolutely continuous on (a,b), and y satisfies the boundary conditions$\\}$. Assume the null space of L:BC $\\to$ $L\\sp1$ (a,b) is one-dimensional and spanned by $\\varphi$. This is what is called the problem at resonance.","abstract_html":"Consider the second order non-linear differential operator ${\\cal L}y$ = Ly + $\\eta y\\sp3$, where $\\eta$ = $\\pm$1 and L, the linear part of ${\\cal L}$, is of the form Ly = $y\\sp{\\prime\\prime}$ + $p(x)y\\sp\\prime$ + q(x)y. Assume p(x) and q(x) are integrable on (a,b). We study the existence and uniqueness of solutions of ${\\cal L}y$ = f satisfying linear boundary conditions on (a,b). The function f is an element of $L\\sp1$ (a,b) Define BC = $\\{y \\in L\\sp\\infty$ (a,b): $y\\sp\\prime$ is absolutely continuous on (a,b), and y satisfies the boundary conditions$\\}$. Assume the null space of L:BC $\\to$ $L\\sp1$ (a,b) is one-dimensional and spanned by $\\varphi$. This is what is called the problem at resonance.","abstract_has_math":true,"creators":["Lefton, Lew Edward"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2014,"date_issued":"2014-12-16T06:18:20Z","date_published":"2014-12-16T06:18:20Z","updated_at":"2026-07-22T22:26:04Z","subjects":["Mathematics"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["(UMI)AAI8721690"],"render_values":[{"text":"(UMI)AAI8721690","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/2142/71258","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:creator","label":"Author","values":["Lefton, Lew Edward"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2014-12-16T06:18:20Z","10000-01-01","1987"]},{"key":"dc:type","label":"Dc Type","values":["text"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Illinois at Urbana-Champaign"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Mathematics"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/71258","(UMI)AAI8721690"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["Consider the second order non-linear differential operator ${\\cal L}y$ = Ly + $\\eta y\\sp3$, where $\\eta$ = $\\pm$1 and L, the linear part of ${\\cal L}$, is of the form Ly = $y\\sp{\\prime\\prime}$ + $p(x)y\\sp\\prime$ + q(x)y. Assume p(x) and q(x) are integrable on (a,b). We study the existence and uniqueness of solutions of ${\\cal L}y$ = f satisfying linear boundary conditions on (a,b). The function f is an element of $L\\sp1$ (a,b) Define BC = $\\{y \\in L\\sp\\infty$ (a,b): $y\\sp\\prime$ is absolutely continuous on (a,b), and y satisfies the boundary conditions$\\}$. Assume the null space of L:BC $\\to$ $L\\sp1$ (a,b) is one-dimensional and spanned by $\\varphi$. This is what is called the problem at resonance.","We show that ${\\cal L}y$ = f has at least one "small" solution in BC provided that $\\Vert f\\Vert\\sb1$ is small enough and that $\\varphi\\sp3 \\notin$ R(L) (the range of L). This last hypothesis can be weakened slightly, however, some restriction on R(L) will be necessary, in general. If the operator $L$:$BC\\rightarrow L\\sp1\\lbrack a,b\\rbrack$ is self-adjoint, then ${\\cal L}y = f$ actually has a unique small solution in BC for small $\\Vert f\\Vert\\sp1.$ Examples are given to demonstrate that existence and uniqueness do not always hold. In the final chapter, a generalization of the $y\\sp3$ nonlinearity is given.","The main technique used is topological degree theory, specifically Leray-Schauder degree defined for compact perturbations of the identity.","Made available in DSpace on 2014-12-16T06:18:20Z (GMT). No. of bitstreams: 1 8721690.pdf: 1857492 bytes, checksum: 9ed6452d4bee0b6b91099e402c8ab713 (MD5) Previous issue date: 1987","Embargo set by: Seth Robbins for item 71424 Lift date: Forever Reason: Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","U of I Only","54 p.","Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1987."]},{"key":"dc:title","label":"Title","values":["Degree Theory and Nonlinear Boundary Value Problems at Resonance"]}]}],"canonical_facts":{"dc:creator":["Lefton, Lew Edward"],"dc:date":["2014-12-16T06:18:20Z","10000-01-01","1987"],"dc:description":["Consider the second order non-linear differential operator ${\\cal L}y$ = Ly + $\\eta y\\sp3$, where $\\eta$ = $\\pm$1 and L, the linear part of ${\\cal L}$, is of the form Ly = $y\\sp{\\prime\\prime}$ + $p(x)y\\sp\\prime$ + q(x)y. Assume p(x) and q(x) are integrable on (a,b). We study the existence and uniqueness of solutions of ${\\cal L}y$ = f satisfying linear boundary conditions on (a,b). The function f is an element of $L\\sp1$ (a,b) Define BC = $\\{y \\in L\\sp\\infty$ (a,b): $y\\sp\\prime$ is absolutely continuous on (a,b), and y satisfies the boundary conditions$\\}$. Assume the null space of L:BC $\\to$ $L\\sp1$ (a,b) is one-dimensional and spanned by $\\varphi$. This is what is called the problem at resonance.","We show that ${\\cal L}y$ = f has at least one "small" solution in BC provided that $\\Vert f\\Vert\\sb1$ is small enough and that $\\varphi\\sp3 \\notin$ R(L) (the range of L). This last hypothesis can be weakened slightly, however, some restriction on R(L) will be necessary, in general. If the operator $L$:$BC\\rightarrow L\\sp1\\lbrack a,b\\rbrack$ is self-adjoint, then ${\\cal L}y = f$ actually has a unique small solution in BC for small $\\Vert f\\Vert\\sp1.$ Examples are given to demonstrate that existence and uniqueness do not always hold. In the final chapter, a generalization of the $y\\sp3$ nonlinearity is given.","The main technique used is topological degree theory, specifically Leray-Schauder degree defined for compact perturbations of the identity.","Made available in DSpace on 2014-12-16T06:18:20Z (GMT). No. of bitstreams: 1 8721690.pdf: 1857492 bytes, checksum: 9ed6452d4bee0b6b91099e402c8ab713 (MD5) Previous issue date: 1987","Embargo set by: Seth Robbins for item 71424 Lift date: Forever Reason: Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","U of I Only","54 p.","Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1987."],"dc:identifier":["http://hdl.handle.net/2142/71258","(UMI)AAI8721690"],"dc:subject":["Mathematics"],"dc:title":["Degree Theory and Nonlinear Boundary Value Problems at Resonance"],"dc:type":["text"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Ph.D."],"thesis:institution_name":["University of Illinois at Urbana-Champaign"]},"updated_at":"2026-07-22T22:26:04Z"}