University of Illinois Urbana-Champaign

Degree Theory and Nonlinear Boundary Value Problems at Resonance

Lefton, Lew Edward

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https://hdl.handle.net/2142/71258

Description

Title
Degree Theory and Nonlinear Boundary Value Problems at Resonance
Author(s)
Lefton, Lew Edward
Issue Date
1987
Department of Study
Mathematics
Discipline
Mathematics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Mathematics
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.
Graduation Semester
1987
Type of Resource
text
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http://hdl.handle.net/2142/71258

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