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Lefton, Lew Edward 2014-12-16T06:18:20Z 2014-12-16T06:18:20Z 10000-01-01 1987 1987 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. Made available in DSpace on 2014-12-16T06:18:20Z (GMT). No. of bitstreams: 1 8721690.pdf: 1857492 bytes, checksum: 9ed6452d4bee0b6b91099e402c8ab713 (MD5) Previous issue date: 1987 Embargo set by: Seth Robbins for item 71424 Lift date: Forever Reason: Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs U of I Only 54 p. Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1987. http://hdl.handle.net/2142/71258 (UMI)AAI8721690 Mathematics Degree Theory and Nonlinear Boundary Value Problems at Resonance text Mathematics Mathematics University of Illinois at Urbana-Champaign Dissertation Ph.D.