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Applications of cone theory to boundary value problems
Abstract
We are concerned with the existence and comparison of eigenvalues for the eigenvalue problem ($-$1)$\sp{n-1}$Lu = $\lambda P(t)u, Tu$ = 0, where $Tu$ = 0 are appropriate boundary conditions at points in the interval ($a,b$). Here $u(t)$ is an $m$-column vector function, $P(t)$ is a continuous $m \times m$ matrix function on ($a,b$) and $Lu = u\sp{(n)}$ + $p\sb1(t)u\sp{(n-1)}$ +$\cdots$+ $p\sb{n}(t)$. We will assume that the corresponding scalar equation $Ly$ = 0 is right disfocal on $\lbrack a,b\rbrack$. We get our existence and comparison results by using several abstract theorems from cone theory in a Banach space. We first consider the boundary value problem $u\sp{(n)} + r(t)u = 0, u\sp{(i)}(a) = 0, i = 0,1,\..., k-1$ and $u\sp{(i\sb j)}(b)$ = 0, j = 1,2, ...,$n-k$. Using comparison theorems for Green's functions due to Peterson and Ridenhour we are able to apply cone theory to get the existence and uniqueness of an eigenvector in a cone. Further, we can give comparison results between the smallest positive eigenvalues of different eigenvalue problems We also examine the $n$-point right focal eigenvalue problem $(-1)\sp{n-1}Lu$ = $\lambda P(t)u, u\sp{(i-1)}(t\sb{i})$ = 0, for $i = 1,2,\..., n$. Assuming that $Ly = 0$ is right disfocal we give an explicit form for the Green's function. Under certain sign conditions on the Green's function and conditions on $P(t)$, we can show the existence of a smallest positive eigenvalue. And with further conditions on $P(t)$, that its corresponding eigenvector is essentially unique with respect to a 'cone'. We also have comparison results for the eigenvalue problem above the and the problem $Lu = \Lambda Q(t)u, u\sp{(i-1)}(t\sb{i}) = 0$ for $i = 1,2,\..., n$. We close this chapter by giving examples where the Green's function has the desired sign conditions. We also give results for the difference equation analog on this problem.
Subject Area
Mathematics
Recommended Citation
Diaz, Gerald, "Applications of cone theory to boundary value problems" (1989). ETD collection for University of Nebraska-Lincoln. AAI9013602.
https://digitalcommons.unl.edu/dissertations/AAI9013602