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 Title: Degree Theory and Nonlinear Boundary Value Problems at Resonance Author(s): Lefton, Lew Edward 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 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 Type: Text Description: 54 p.Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1987. URI: http://hdl.handle.net/2142/71258 Other Identifier(s): (UMI)AAI8721690 Date Available in IDEALS: 2014-12-16 Date Deposited: 1987
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