and Applied Analysis 3 The final solution to problems in 1.3 can yield optimal lower and upper bounds of eigenvalues. Actually, from 1.3 , we have In ∥q ∥ ∥ 1 ) ≤ λn ( q ) ≤ Mn ∥q ∥ ∥ 1 ) ∀q ∈ L1, 1.4 which are optimal. Several extremal problems on Dirichlet, Neumann, and periodic eigenvalues of the Sturm-Liouville operator and the p-Laplace operator have been studied recently, where the potent...

In this paper, inverse Laplace transform method is applied to analytical solution of the fractional Sturm-Liouville problems. The method introduces a powerful tool for solving the eigenvalues of the fractional Sturm-Liouville problems. The results  how that the simplicity and efficiency of this method.

In this paper, linear second-order differential equations of Sturm-Liouville type having a finite number of singularities and turning points in a finite interval are investigated. First, we obtain the dual equations associated with the Sturm-Liouville equation. Then, we prove the uniqueness theorem for the solutions of dual initial value problems.

In this paper, we give the spectral theory for eigenvalues and eigenfunctions of a boundary value problem consisting of the linear fractional Bessel operator. Moreover, we show that this operator is self-adjoint, the eigenvalues of the problem are real, and the corresponding eigenfunctions are orthogonal. In this paper, we give the spectral theory for eigenvalues and eigenfunctions...

We prove a natural generalization of Kneser's oscillation criterion and Hardy's inequality for Sturm{Liouville diierential expressions. In particular, assuming ? d dx p 0 (x) d dx + q 0 (x), x 2 (a; b), ?1 a < b 1 to be nonoscillatory near a (or b), we determine conditions on q(x) such that ? d dx p 0 (x) d dx + q 0 (x) + q(x) is nonoscillatory, respectively, oscillatory near a (or b).

We give a comprehensive treatment of Sturm–Liouville operators whose coefficients are measures including a full discussion of self-adjoint extensions and boundary conditions, resolvents, and Weyl–Titchmarsh–Kodaira theory. We avoid previous technical restrictions and, at the same time, extend all results to a larger class of operators. Our operators include classical Sturm– Liouville operators,...

We consider the spectral function ρ(µ) (µ ≥ 0) for the Sturm-Liouville equation y ′′ + (λ − q)y = 0 on [0, ∞) with the boundary condition y(0) = 0 and where q has slow decay O(x −α) (a > 0) as x → ∞. We develop our previous methods of locating spectral concentration for q with rapid exponential decay (this Journal 81 (1997) 333-348) to deal with the new theoretical and computational complexitie...