Associated Spaces Defined by Ordinary Differential Equations

Abstract

The paper deals with initial value problems for desired functions $u(t,x)$ depending on the time $t$ and one spacelike variable $x$. In case the initial function $\phi (x)$ satisfies an associated (ordinary) differential equation, the solution $u(t,x)$ satisfies the associated differential equation for each $t$. If the general solution of the associated differential equation is known, one gets a system of ordinary differential equations for the desired coefficients depending on $t$. In any case, the solution can be obtained as fixed point of a related integro-differential operator.