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Title:Degree Theory and Nonlinear Boundary Value Problems at Resonance
Author(s):Lefton, Lew Edward
Department / Program:Mathematics
Degree Granting Institution:University of Illinois at Urbana-Champaign
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.
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.
Issue Date:1987
Description:54 p.
Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1987.
Other Identifier(s):(UMI)AAI8721690
Date Available in IDEALS:2014-12-16
Date Deposited:1987

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