On solutions of certain self-adjoint differential equations of fourth order

Abstract

In this paper the author considers the differential equation [ r( x) y″]″ − p( x) y = 0, where r( x) and p( x) are continuous, r( x) > 0, and p( x) ≠ 0 on an interval [ϵ, ∞), ϵ > 0. The cases for which p( x) is positive and p( x) is negative on [ϵ, ∞) are treated separately. In the first part of the paper it is assumed that p( x) is positive on [ϵ, ∞). A study is made of the general properties of solutions of the equation, particularly, oscillatory solutions. Special emphasis is given the case when all oscillatory solutions are bounded. A rather simple representation for all bounded oscillatory solutions is given. The function p( x) is assumed to be negative in the second part of the paper. Some of Marko Svec's results on the behavior of oscillatory solutions are extended to this more general equation. Also two theorems analogus to the Bôcher-Osgood theorem for second-order equations are given. Finally, the author gives some necessary conditions that certain types of nonoscillatory solutions exist.