The authors compare oscillation properties of solutions to Sturm-Liouville equations and , where is of the type
and its perturbation is of the form , , .
Under certain conditions on , if is nonoscillatory near (resp. for some (with denoting a positive solution to , the authors establish two theorems for that be nonoscillatory or be oscillatory near (resp. .
The special case , in the first theorem stands for the original oscillation criterion by A. Kneser [Math. Ann. Qd 42, 409-435 (1893; JFM 25.0533.01)] and in the second one represents a generalization of Kneser’s result due to H. Weber (1912).
Finally, making use of certain types of factorizations of general Sturm-Liouville differential expressions on , the authors prove a natural generalization of Hardy’s inequality.