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

Abstract

Let I ⊆ R be an interval and β : I → I a strictly increasing and continuous function with a unique fixed point s 0 ∈ I which satisfies (s 0 − t)(β(t) − t) ≥ 0 for all t ∈ I, where the equality holds only when t = s 0 .The general quantum operator defined in 2015 by Hamza et al.,Jackson q-operator Dq and also the Hahn (q, ω)-operator, Dq,ω.Regarding a β−Sturm Liouville eigenvalue problem associated with the above operator D β , we construct the β−Lagrange's identity, show that it is self-adjoint in L 2 β ([a, b]), and exhibit some properties for the corresponding eigenvalues and eigenfunctions.