On the resolvent of singular q-Sturm-Liouville operators

In this paper, we investigate the resolvent operator of singular q-Sturm-Liouville problem defined as − ( 1 / q ) D ⁻ ¹ [D y x )] + [r - λ ]y )=0 −(1/q)Dq⁻¹Dqy(x)+r(x)y(x)=λy(x) , with boundary condition 0 c o s β i n = y(0,λ)cosβ+Dq⁻¹y(0,λ)sinβ=0 where ∈ C λ∈C $r$ is a real function on $[0,∞)$, continuous at zero and r L l ∞ r∈Lq,loc¹(0,∞) . We give an integral representation for some properties operator. Furthermore, obtain formula Titchmarsh-Weyl $q$-Sturm-Liouville problem.