On multi-step methods for singular fractional q-integro-differential equations

The objective of this paper is to investigate, by applying the standard Caputo fractional q-derivative of order α α , the existence of solutions for the singular fractional q-integro-differential equation D q α [ k ] ( t ) = Ω ( t , k 1 , k 2 , k 3 , k 4 ) {{D}}qα[k](t)=Ω (t,{k}₁,{k}₂,{k}₃,{k}₄) , under some boundary conditions where Ω Ω is singular at some point 0 ≤ t ≤ 1 0≤ t≤ 1 , on a time scale T t 0 = { t : t = t 0 q n } ∪ { 0 } {{T}}_{{t}₀}=\{t:t={t}₀{q}ⁿ\}∪ \{0\} , for n ∈ N n∈ {N} where t 0 ∈ R