16th Conference on Applied Mathematics, Univ. of Central Oklahoma,
Electron. J. Diff. Eqns., Conf. 07, 2001, pp. 89-97.

Properties of the solution map for a first order linear problem

James L. Moseley

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).

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James L. Moseley
West Virginia University
Morgantown, West Virginia 26506-6310 USA
e-mail: moseley@math.wvu.edu
Telephone: 304-293-2011

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