Properties of the solution map for a first order linear problem

Abstract

We are interested in establishing properties of the general mathematical model $$frac{dvec{u}}{dt}=T(t,vec{u})+vec{b}+vec{g}(t),quad vec{u}(t_0)=vec{u}_0 $$ for the dynamical system defined by the (possibly nonlinear) operator $T(t,cdot):Vo V$ 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).

Keywords

• First order linear ordinary differential equation.