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Albrecht, Felix Pollack, David Howard 2011-05-07T12:31:58Z 2011-05-07T12:31:58Z 10000-01-01 1989 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 AAI8924922 (UMI)AAI8924922 http://hdl.handle.net/2142/20198 eng Copyright 1989 Pollack, David Howard Mathematics Nonlinear, conditionally stable, singularly perturbed boundary-relation problems text Mathematics Mathematics University of Illinois at Urbana-Champaign Dissertation Ph.D.