{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/20198"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/20198","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Nonlinear, conditionally stable, singularly perturbed boundary-relation problems","abstract":"Consider the nonlinear, singularly perturbed, vector boundary relation problem x$\\sp\\prime$ = f(t,x,y,$\\epsilon$), $\\epsilon$y$\\sp\\prime$ = g(t,x,y,$\\epsilon$), L(x(0),y(0),$\\epsilon$) = $\\alpha\\sb0$, R(x(1),y(1),$\\epsilon$) = $\\alpha\\sb1$. Suppose that there exist smooth maps $\\phi$,P,A$\\sb1$,A$\\sb2$ such that for all appropriate (t,x): (1) g(t,x,$\\phi$(t,x),0) = 0, (2) P(t,x) $\\cdot$ D$\\sb3$g(t,x,$\\phi$(t,x),0)P(t,x)$\\sp{-1}$ = Diag(A$\\sb1$(t,x),A$\\sb2$(t,x)), and (3) the spectrum of A$\\sb1$(t,x) is bounded away and to the left of the imaginary axis and the spectrum of A$\\sb2$(t,x) is bounded away and to the right of the imaginary axis. Suppose also that p$\\sb0$(t) is a solution of the reduced differential equation p$\\sb0\\sp\\prime$(t) = f(t,p$\\sb0$(t),$\\phi$(t,p$\\sb0$(t)),0) and that L(p$\\sb0$(0),$\\phi$(0,p$\\sb0$(0),0) = 0 and R(p$\\sb0$(1),$\\phi$(1,p$\\sb0$(1)),0) = 0. If L and R are of a special type (projections onto complementary sets of variables) Hadlock has shown (J. Diff. Eq. 14, 498-517) that the full problem has a bounded family of solutions (x(t,$\\epsilon$),y(t,$\\epsilon$)) defined for all $\\epsilon$ sufficiently small and $\\alpha$ in some neighborhood of 0. It is also clear in this special case what reduced set of boundary conditions (cancellation law) determine p$\\sb\\alpha$(t) = x(t,0+). A corollary of the main result of this paper extends Hadlock's result to allow for an arbitrary set of nonlinear boundary relations L and R subject to the invertibility of a certain linear operator. The proof makes use of the local invariant manifolds of the boundary layer equation along solutions of the reduced differential equation. The vectors x and y may belong to arbitrary Banach spaces.","abstract_html":"Consider the nonlinear, singularly perturbed, vector boundary relation problem x$\\sp\\prime$ = f(t,x,y,<span class=\"etd-inline-math\">&epsilon;</span>), <span class=\"etd-inline-math\">&epsilon;</span>y$\\sp\\prime$ = g(t,x,y,<span class=\"etd-inline-math\">&epsilon;</span>), L(x(0),y(0),<span class=\"etd-inline-math\">&epsilon;</span>) = <span class=\"etd-inline-math\">&alpha;\\sb0</span>, R(x(1),y(1),<span class=\"etd-inline-math\">&epsilon;</span>) = <span class=\"etd-inline-math\">&alpha;\\sb1</span>. Suppose that there exist smooth maps $\\phi$,P,A$\\sb1$,A$\\sb2$ such that for all appropriate (t,x): (1) g(t,x,$\\phi$(t,x),0) = 0, (2) P(t,x) $\\cdot$ D$\\sb3$g(t,x,$\\phi$(t,x),0)P(t,x)$\\sp{-1}$ = Diag(A$\\sb1$(t,x),A$\\sb2$(t,x)), and (3) the spectrum of A$\\sb1$(t,x) is bounded away and to the left of the imaginary axis and the spectrum of A$\\sb2$(t,x) is bounded away and to the right of the imaginary axis. Suppose also that p$\\sb0$(t) is a solution of the reduced differential equation p$\\sb0\\sp\\prime$(t) = f(t,p$\\sb0$(t),$\\phi$(t,p$\\sb0$(t)),0) and that L(p$\\sb0$(0),$\\phi$(0,p$\\sb0$(0),0) = 0 and R(p$\\sb0$(1),$\\phi$(1,p$\\sb0$(1)),0) = 0. If L and R are of a special type (projections onto complementary sets of variables) Hadlock has shown (J. Diff. Eq. 14, 498-517) that the full problem has a bounded family of solutions (x(t,<span class=\"etd-inline-math\">&epsilon;</span>),y(t,<span class=\"etd-inline-math\">&epsilon;</span>)) defined for all <span class=\"etd-inline-math\">&epsilon;</span> sufficiently small and <span class=\"etd-inline-math\">&alpha;</span> in some neighborhood of 0. It is also clear in this special case what reduced set of boundary conditions (cancellation law) determine p<span class=\"etd-inline-math\">\\sb&alpha;</span>(t) = x(t,0+). A corollary of the main result of this paper extends Hadlock&#x27;s result to allow for an arbitrary set of nonlinear boundary relations L and R subject to the invertibility of a certain linear operator. The proof makes use of the local invariant manifolds of the boundary layer equation along solutions of the reduced differential equation. The vectors x and y may belong to arbitrary Banach spaces.","abstract_has_math":true,"creators":["Pollack, David Howard"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Albrecht, Felix"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2011,"date_issued":"2011-05-07T12:31:58Z","date_published":"2011-05-07T12:31:58Z","updated_at":"2026-07-22T22:25:15Z","subjects":["Mathematics"],"languages":["eng"],"rights":["Copyright 1989 Pollack, David Howard"],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["AAI8924922","(UMI)AAI8924922"],"render_values":[{"text":"AAI8924922","href":null,"code":true},{"text":"(UMI)AAI8924922","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/2142/20198","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Albrecht, Felix"]},{"key":"dc:creator","label":"Author","values":["Pollack, David Howard"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2011-05-07T12:31:58Z","10000-01-01","1989"]},{"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":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["eng"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 1989 Pollack, David Howard"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["AAI8924922","(UMI)AAI8924922","http://hdl.handle.net/2142/20198"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["Consider the nonlinear, singularly perturbed, vector boundary relation problem x$\\sp\\prime$ = f(t,x,y,$\\epsilon$), $\\epsilon$y$\\sp\\prime$ = g(t,x,y,$\\epsilon$), L(x(0),y(0),$\\epsilon$) = $\\alpha\\sb0$, R(x(1),y(1),$\\epsilon$) = $\\alpha\\sb1$. Suppose that there exist smooth maps $\\phi$,P,A$\\sb1$,A$\\sb2$ such that for all appropriate (t,x): (1) g(t,x,$\\phi$(t,x),0) = 0, (2) P(t,x) $\\cdot$ D$\\sb3$g(t,x,$\\phi$(t,x),0)P(t,x)$\\sp{-1}$ = Diag(A$\\sb1$(t,x),A$\\sb2$(t,x)), and (3) the spectrum of A$\\sb1$(t,x) is bounded away and to the left of the imaginary axis and the spectrum of A$\\sb2$(t,x) is bounded away and to the right of the imaginary axis. Suppose also that p$\\sb0$(t) is a solution of the reduced differential equation p$\\sb0\\sp\\prime$(t) = f(t,p$\\sb0$(t),$\\phi$(t,p$\\sb0$(t)),0) and that L(p$\\sb0$(0),$\\phi$(0,p$\\sb0$(0),0) = 0 and R(p$\\sb0$(1),$\\phi$(1,p$\\sb0$(1)),0) = 0. If L and R are of a special type (projections onto complementary sets of variables) Hadlock has shown (J. Diff. Eq. 14, 498-517) that the full problem has a bounded family of solutions (x(t,$\\epsilon$),y(t,$\\epsilon$)) defined for all $\\epsilon$ sufficiently small and $\\alpha$ in some neighborhood of 0. It is also clear in this special case what reduced set of boundary conditions (cancellation law) determine p$\\sb\\alpha$(t) = x(t,0+). A corollary of the main result of this paper extends Hadlock's result to allow for an arbitrary set of nonlinear boundary relations L and R subject to the invertibility of a certain linear operator. The proof makes use of the local invariant manifolds of the boundary layer equation along solutions of the reduced differential equation. The vectors x and y may belong to arbitrary Banach spaces.","Made available in DSpace on 2011-05-07T12:31:58Z (GMT). No. of bitstreams: 2 license.txt: 4922 bytes, checksum: 910b249b4beec47e7ab768910c8f966f (MD5) 8924922.pdf: 7485874 bytes, checksum: fe46760a4573e6e4ddf42b12e74d0b99 (MD5) Previous issue date: 1989","Item marked as restricted to the 'UIUC Users [automated]' Group (id=2) by Howard Ding (hding2@illinois.edu) on 2011-05-07T14:42:17Z Item is restricted indefinitely.","Restriction data tranferred 2014-07-01T11:18:22-05:00 Original Data Group with Access UIUC Users [automated] Release Date: none Reason: ETDs are only available to UIUC Users without author permission","ETDs are only available to UIUC Users without author permission","U of I Only"]},{"key":"dc:title","label":"Title","values":["Nonlinear, conditionally stable, singularly perturbed boundary-relation problems"]}]}],"canonical_facts":{"dc:contributor":["Albrecht, Felix"],"dc:creator":["Pollack, David Howard"],"dc:date":["2011-05-07T12:31:58Z","10000-01-01","1989"],"dc:description":["Consider the nonlinear, singularly perturbed, vector boundary relation problem x$\\sp\\prime$ = f(t,x,y,$\\epsilon$), $\\epsilon$y$\\sp\\prime$ = g(t,x,y,$\\epsilon$), L(x(0),y(0),$\\epsilon$) = $\\alpha\\sb0$, R(x(1),y(1),$\\epsilon$) = $\\alpha\\sb1$. Suppose that there exist smooth maps $\\phi$,P,A$\\sb1$,A$\\sb2$ such that for all appropriate (t,x): (1) g(t,x,$\\phi$(t,x),0) = 0, (2) P(t,x) $\\cdot$ D$\\sb3$g(t,x,$\\phi$(t,x),0)P(t,x)$\\sp{-1}$ = Diag(A$\\sb1$(t,x),A$\\sb2$(t,x)), and (3) the spectrum of A$\\sb1$(t,x) is bounded away and to the left of the imaginary axis and the spectrum of A$\\sb2$(t,x) is bounded away and to the right of the imaginary axis. Suppose also that p$\\sb0$(t) is a solution of the reduced differential equation p$\\sb0\\sp\\prime$(t) = f(t,p$\\sb0$(t),$\\phi$(t,p$\\sb0$(t)),0) and that L(p$\\sb0$(0),$\\phi$(0,p$\\sb0$(0),0) = 0 and R(p$\\sb0$(1),$\\phi$(1,p$\\sb0$(1)),0) = 0. If L and R are of a special type (projections onto complementary sets of variables) Hadlock has shown (J. Diff. Eq. 14, 498-517) that the full problem has a bounded family of solutions (x(t,$\\epsilon$),y(t,$\\epsilon$)) defined for all $\\epsilon$ sufficiently small and $\\alpha$ in some neighborhood of 0. It is also clear in this special case what reduced set of boundary conditions (cancellation law) determine p$\\sb\\alpha$(t) = x(t,0+). A corollary of the main result of this paper extends Hadlock's result to allow for an arbitrary set of nonlinear boundary relations L and R subject to the invertibility of a certain linear operator. The proof makes use of the local invariant manifolds of the boundary layer equation along solutions of the reduced differential equation. The vectors x and y may belong to arbitrary Banach spaces.","Made available in DSpace on 2011-05-07T12:31:58Z (GMT). No. of bitstreams: 2 license.txt: 4922 bytes, checksum: 910b249b4beec47e7ab768910c8f966f (MD5) 8924922.pdf: 7485874 bytes, checksum: fe46760a4573e6e4ddf42b12e74d0b99 (MD5) Previous issue date: 1989","Item marked as restricted to the 'UIUC Users [automated]' Group (id=2) by Howard Ding (hding2@illinois.edu) on 2011-05-07T14:42:17Z Item is restricted indefinitely.","Restriction data tranferred 2014-07-01T11:18:22-05:00 Original Data Group with Access UIUC Users [automated] Release Date: none Reason: ETDs are only available to UIUC Users without author permission","ETDs are only available to UIUC Users without author permission","U of I Only"],"dc:identifier":["AAI8924922","(UMI)AAI8924922","http://hdl.handle.net/2142/20198"],"dc:language":["eng"],"dc:rights":["Copyright 1989 Pollack, David Howard"],"dc:subject":["Mathematics"],"dc:title":["Nonlinear, conditionally stable, singularly perturbed boundary-relation problems"],"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":"2026-07-22T22:25:15Z"}