An approximation theory for oscillations of differential equations

Abstract:

In an earlier paper we gave an approximation theory of focal points and focal intervals. The fundamental purpose of this paper is to show that the ideas and methods of that paper can be used to give an approximation theory for oscillation for linear selfadjoint differential (and integral-differential) equations. These methods follow from the theory of quadratic forms given by Hestenes. In §1 we give the preliminaries needed for this paper. In §2 we define oscillation for quadratic control problems and discuss their connection with differential equations. In §3 we give our main approximation results relating the oscillation points ${\lambda _n}({\sigma _0})(n = i,2,3, \cdots )$ for a given problem to the oscillation points ${\lambda _n}(\sigma )(n = 1,2,3, \cdots )$ of an approximating problem. The element $\sigma$ belongs to a metric space $(\Sigma ,d)$. The main result is to show that the $n$th oscillation point ${\lambda _n}(\sigma )$ is a continuous function of $\sigma (n = i,2,3, \cdots )$. For completeness in §4 we present an example for fourth order differential equations where the approximation is by discrete problems. Thus oscillation points can be “easily” computed by numerical algorithms.