In Section 2, we give first an existence result for problems where the second, the functional argument u of g, is replaced in (1.1) by fixed functions v ∈ C(J), and study the dependence of solution sets of these problems on v. The so obtained results and a fixed point result for multifunctions proved recently in [7] are then used in Section 3 to derive existence results for minimal and maximal ...

In this paper, we present a comparative study of Sinc-Galerkin method and differential transform method to solve Sturm-Liouville eigenvalue problem. As an application, a comparison between the two methods for various celebrated Sturm-Liouville problems are analyzed for their eigenvalues and solutions. The study outlines the significant features of the two methods. The results show that these me...

The eigenvalues of Sturm-Liouville (SL) problems depend not only continuously but smoothly on boundary points. The derivative of the nth eigenvalue as a function of an endpoint satisfies a first order differential equation. This for arbitrary (separated or coupled) self-adjoint regular boundary conditions. In addition, as the length of the interval shrinks to zero all higher eigenvalues march o...

A new algorithm is proposed for solving the inverse Sturm–Liouville problem of reconstructing a symmetric potential from eigenvalues. It uses Numerov’s method instead of the second order method of the related algorithm of Fabiano, Knobel and Lowe. An extension by Andrew and Paine of the asymptotic correction technique of Paine, de Hoog and Anderssen is the key to the success of the new algorith...

This article concerns the existence and multiplicity of positive solutions for the singular Sturm-Liouville boundary value problem (p(t)u′(t))′ + h(t)f(t, u(t)) = 0, 0 < t <∞, au(0)− b lim t→0+ p(t)u′(t) = 0, c lim t→∞ u(t) + d lim t→∞ p(t)u′(t) = 0. We use fixed point index theory to establish our main results based on a priori estimates derived by utilizing spectral properties of associated l...

Consider the minimal Sturm-Liouville operator A = Amin generated by the differential expression

We establish the results on the existence of three positive solutions to Sturm-Liouville boundary value problems of the singular fractional differential equation with the nonlinearity depending on D 0+u ⎪⎨ ⎪⎩ D0+u(t)+ f (t,u(t),D μ 0+ u(t)) = 0, t ∈ (0,1),1 < α < 2, a limt→0 I2−α 0+ u(t)−b limt→0 [ I2−α 0+ u(t) ]′ = ∫ 1 0 g(s,u(s),D μ 0+u(s))ds, c D 0+u(1)+du(1) = ∫ 1 0 h(s,u(s),D μ 0+u(s))ds. ...

wherem≥ 3 and 1≤ q≤m – 2. We note that the nonlinear term F involves derivatives. This makes the problem challenging, and such cases are seldom investigated in the literature. In this paper we develop a new technique to obtain existence criteria for one or multiple positive solutions of the boundary value problem. Several examples with known positive solutions are presented to dwell upon the us...

We examine properties of solutions to a 2n-dimensional Stieltjes Sturm-Liouville initial-value problem. Existence and uniqueness of a solution has been previously proven, but we present a proof in order to establish properties of boundedness, bounded variation, and continuity. These properties are then used to prove that the solutions depend continuously on the coefficients and on the initial c...