transactions of the american mathematical society Volume 279, Number 2, October 1983 ON THE LOCATIONOF ZEROS OF OSCILLATORY SOLUTIONS BY H. GINGOLD Abstract. The location of zeros of solutions of second order singular differential equations is provided bv a new asymptotic decomposition formula. The approximate location of zeros is provided with high accuracy error estimates in the neighbour- hood of the point at infinity. The same asymptotic formula suggested is applicable to the neighbourhood of most types of singularities as well as to the neighbourhoods of regular points. 1. Introduction. In the oscillation theory of second order differential equations one may distinguish three types of problems. Given the differential equation (Li) y" = q*y on an interval (a, oo), let y(t) be a real nontrivial solution of (1.1). Then the following three problems are raised. (1) Isy(?) oscillatory on (a, oo)? Namely does y(t) possess an infinite number of zeros? (2) Find an estimation of the number of zeros of y(t) on (a, T). (3) Find the location of the zeros of y(t) on a given interval (a,T). Each of the above three problems is intimately connected with the other two. They ascend in difficulty from (1) to (3), problem (3) being most delicate and its solution most desired. An answer to problem (3) provides an answer to (2) and (1). An answer to (2) provides an answer to (1). Therefore, the conditions to guarantee answers to the three problems differ respectively. The more smoothness assumed on q4, the more accurate is the location of the zeros of y(t) by a single given formula. Problems (l)-(3) are difficult because we have to deal with a singular differential equation. The singularity of the differential equation is manifested in the fact that we have to describe the behaviour of solutions y(t) of an equation, which may have an unbounded coefficient on a noncompact domain, (a, oo). Classical asymptotic techniques are a major tool in the investigation of singular differential equations. It is surprising to notice the small amount of classical asymptotic techniques applied specifically to oscillation problems compared with the other techniques Received by the editors April 5, 1982. 1980 Mathematics Subject Classification.Primary 34C10; Secondary 34E05. Key words and phrases. Oscillation, zeros, second order differential equations, oscillatory solution. ©1983 American Mathematical Society 0002-9947/83 $1.00 + $.25 per page 471 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use