Variational methods for fractional q-Sturm-Liouville problems

Abstract

In this paper, we formulate a regular q-fractional Sturm-Liouville problem (qFSLP) which includes the left-sided Riemann-Liouville and the right-sided Caputo q-fractional derivatives of the same order α, α ∈ (0, 1). We introduce the essential q-fractional variational analysis needed in proving the existence of a countable set of real eigenvalues and associated orthogonal eigenfunctions for the regular qFSLP when α > 1/2 associated with the boundary condition y(0) = y(a) = 0. A criterion for the first eigenvalue is proved. Examples are included. These results are a generalization of the integer regular q-Sturm-Liouville problem introduced by

On fractional q-Sturm–Liouville problems

“…This work generalizes the study of integer Sturm-Liouville problem introduced by Annaby and Mansour in [1]. It is worth mentioning that a q-fractional variational calculus is developed and used in [21] to prove that the qFSLp (3.1) with the boundary condition y(0) = y(a) = 0 has a countable set of real eigenvalues and associated orthogonal eigenfunctions when α > 1/2 and 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.…”

Fractional q-Calculus Operators Pertaining to the q-Analogue of M-Function and Its Application to Fractional q-Kinetic Equations

“…From [35], for ε > 0 and ⌈ε⌉ = m, the right-and left-sided R-L fractional q-derivatives of order ε are given as…”

A $β$-Sturm Liouville problem associated with the general quantum operator

“…Those difference operators together with its inverse operators are very important in mathematics investigation and in applications, with a large number of publications and a variety of topics including, but not limited to, the quantum calculus [16], the quantum variational calculus [41,25,8,42,40], q−difference equations properties [6, 7, ?, 14], Sturm-Liouville problems [29,15,10,13,40,14], Paley-Wiener results [27,4,26], Sampling theory [9,37,2,26,13,11,32], q−exponential, trigonometric, hyperbolic and other important families of functions associated with Fourier expansions and corresponding properties connected and derived from its orthogonality feature [39,17,5,44,3,18,38,19,23,20,21,1,22]. These and many other subjects has attracted many researchers.…”