Electronic Journal of Differential Equations,
Conference 07 (2001), pp. 89-97.
Title: Properties of the solution map for a first order linear problem.
Author: James L. Moseley (West Virginia Univ. Morgantown, WV, USA)
Abstract:
We are interested in establishing properties of the general mathematical
model
$$\frac{d\vec{u}}{dt}=T(t,\vec{u})+\vec{b}+\vec{g}(t),\quad
\vec{u}(t_0)=\vec{u}_0
$$
for the dynamical system defined by the (possibly nonlinear) operator
$T(t,\cdot):V\to V$ with state space $V$. For one state
variable where $V=\mathbb{R}$ this may be written as $dy/dx=f(x,y)$,
$y(x_0)=y_0$. This paper establishes some mapping properties
for the operator $L[y]=dy/dx+p(x)y$ with $y(x_0)=y_0$ where
$f(x,y)=-p(x)y+g(x)$ and $T(x,y)=-p(x)y$ is linear. The conditions
for the one-to-one property of the solution map as a function of
$p(x)$ appear to be new or at least undocumented. This property is
needed in the development of a solution technique for a nonlinear model
for the agglomeration of point particles in a confined space (reactor).
Published July 20, 2001.
Math Subject Classifications: 34A99.
Key Words: First order linear ordinary differential equation.