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 Title: Nonlinear, conditionally stable, singularly perturbed boundary-relation problems Author(s): Pollack, David Howard Doctoral Committee Chair(s): Albrecht, Felix Department / Program: Mathematics Discipline: Mathematics Degree Granting Institution: University of Illinois at Urbana-Champaign Degree: Ph.D. Genre: Dissertation Subject(s): Mathematics 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. Issue Date: 1989 Type: Text Language: English URI: http://hdl.handle.net/2142/20198 Rights Information: Copyright 1989 Pollack, David Howard Date Available in IDEALS: 2011-05-07 Identifier in Online Catalog: AAI8924922 OCLC Identifier: (UMI)AAI8924922
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