A Singular Functional

in which the functions r, p and q are continuous on (0, °o) while r is positive there. The point x = 0 is singular for the functional in the sense that the conditions on r, p, and q may not hold for an interval of the form [O, b]. Finally, all integrals which appear are Lebesgue integrals. We denote by: F[0, b], the class of all functions y such that y is absolutely continuous and y'ÇzL2 on every closed subinterval of (0, b] while yib) =0; F'[0, b], the class of all functions y such that yGF[0, b] and y is bounded on [O, b]; F0[0, b], the class of all y£F[0, b] for which the point x = 0 is a limit point of zeros of y; A [O, b], the class of all y£F[0, b] for which y is continuous on [O, b] and for which y(0) =0; -4o[0, b), the class of all yÇEA [O, b] for which the point x = 0 is a limit point of zeros of y. For a function y in any one of the above classes Jiy) \ \ exists as a Lebesgue integral and is finite for 0 <x—^b. We extend its definition by the relation