Abstract
The equation (1) (ry')'= qy, where r, q are locally integrable and r(t) 50 is of limit point type on (0, oo) i.e. it has a solution which is not in L2 (0, oo ) if (i) q &L2(O, 00). This result, for the case r1, follows from a theorem of P. Hartman [1, ??1 and 2]. A simple proof of this special case, which can readily be adapted to yield the result mentioned above, is given in [2, Satz 7, p. 305]. In this note we show that the equation (2) Ly=qy where L has the form (P) Ly = r,, (r.-I ... (r, (roy) /) /** with n even, q, ri locally integrable and ri having no zeros in (0, oo) for i=0, * *, n has a solution which is not in L2(0, oo ) if (i)' q/r.&L2(0, oo), 1/ro($L2(0, oo) and (ii) ri = rn_i for i = 1, . . . , n. PROOF. Define Diy=ri(Di-ly)' for i= 1, * , n with Doy=roy. Note that if y is an L2(0, oo) solution of (2) then Dn_ly is bounded since (D.-jy)t = constant +f'qy/r,. Let yi, * * , y. be a set of linearly independent solutions of (2) and consider their generalized Wronskian
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