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Perturbative oscillation criteria and Hardy-type inequalities. (English) Zbl 0903.34030

The authors compare oscillation properties of solutions to Sturm-Liouville equations τ 0 ψ 0 =λψ 0 and τψ=λψ, where τ 0 is of the type

τ 0 =-d dxp 0 (x)d dx+q 0 (x)

and its perturbation τ is of the form τ=τ 0 +q(x), x(a,b), -a<b.

Under certain conditions on q(x), if (τ 0 -λ 0 ) is nonoscillatory near b (resp. a) for some λ 0 (with ψ 0 (λ 0 ,x) denoting a positive solution to τ 0 ψ=λ 0 ψ), the authors establish two theorems for that (τ-λ 0 ) be nonoscillatory or be oscillatory near b (resp. a).

The special case p 0 =ψ 0 =1, q 0 =λ 0 =0 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 (a,b), the authors prove a natural generalization of Hardy’s inequality.

MSC:
34C10Qualitative theory of oscillations of ODE: zeros, disconjugacy and comparison theory
34B24Sturm-Liouville theory