On fractional q-Sturm–Liouville problems

Abstract

In this paper, we formulate a regular q-fractional Sturm-Liouville problem (qF-SLP) which includes the left-sided Riemann-Liouville and the right-sided Caputo q-fractional derivatives of the same order α, α ∈ (0, 1). The properties of the eigenvalues and the eigenfunctions are investigated. A q-fractional version of the Wronskian is defined and its relation to the simplicity of the eigenfunctions is verified. We use the fixed point theorem to introduce a sufficient condition on eigenvalues for the existence and uniqueness of the associated eigenfunctions when α > 1/2. These results are a generalization of the integer regular q-SturmLiouville problem introduced by Annaby and Mansour in [1]. An example for a qFSLP whose eigenfunctions are little q-Jacobi polynomials is introduced.Keywords: Left and right sided Riemann-Liouville and Caputo q-derivatives, eigenvalues and eigenfunctions, existence and uniqueness theorem, q, α Wronskian. 2000 MSC: 39A13, 26A33, 34L10.

Variational methods for fractional q-Sturm-Liouville problems

“…We use these results in recasting the qFSLP under consideration as a q-isoperimetric problem, and then we solve it by a technique similar to the one used in solving regular Sturm-Liouville problems in [10] and fractional Sturm-Liouville problems in [4]. This complete the work started by the author in [28], and generalizes the study of integer Sturm-Liouville problem introduced by Annaby and Mansour in [1]. A similar study for the fractional Sturm-Liouville problem c D q,a − p(x)D α q,0 + y(x) + r(x)y(x) = λw α (x)y(x), is in progress.…”

“…(2.2) see [28]. The left sided Riemann-Liouville q-fractional operator satisfies the semigroup property I α q,a + I β q,a + f (x) = I α+β q,a + f (x).…”

“…Step 5 In the following, we show that (y (1) m ) m∈N itself converges to y (1) . First, from Theorem [28,Theorem 3.12], for a given λ the solution of is unique except for a sign. Let us assume that y (1) solves (5.20) and the corresponding eigenvalue is λ = λ (1) .…”

Two classes of conformable fractional Sturm‐Liouville problems: Theory and applications

“…We point out that our results are extensions to those in previous studies. 89,90 Definition 3.6. The multiplicity of an eigenvalue is defined to be the number of linearly independent eigenfunctions associated with it.…”

“…Thus, solution u is of the form (89) and it also obeys boundary conditions (88) as each eigenfunction n fulfills…”