On multi-step methods for singular fractional <i>q</i>-integro-differential equations

Abstract The objective of this paper is to investigate, by applying the standard Caputo fractional q -derivative of order <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>α</m:mi> </m:math> \alpha , the existence of solutions for the singular fractional q -integro-differential equation <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msubsup> <m:mrow> <m:mi class="MJX-tex-caligraphic" mathvariant="script">D</m:mi> </m:mrow> <m:mrow> <m:mi>q</m:mi> </m:mrow> <m:mrow> <m:mi>α</m:mi> </m:mrow> </m:msubsup> <m:mrow> <m:mo>[</m:mo> <m:mrow> <m:mi>k</m:mi> </m:mrow> <m:mo>]</m:mo> </m:mrow> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>t</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>=</m:mo> <m:mi mathvariant="normal">Ω</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>t</m:mi> <m:mo>,</m:mo> <m:msub> <m:mrow> <m:mi>k</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:mo>,</m:mo> <m:msub> <m:mrow> <m:mi>k</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> <m:mo>,</m:mo> <m:msub> <m:mrow> <m:mi>k</m:mi> </m:mrow> <m:mrow> <m:mn>3</m:mn> </m:mrow> </m:msub> <m:mo>,</m:mo> <m:msub> <m:mrow> <m:mi>k</m:mi> </m:mrow> <m:mrow> <m:mn>4</m:mn> </m:mrow> </m:msub> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> {{\mathcal{D}}}_{q}^{\alpha }\left[k]\left(t)=\Omega \left(t,{k}_{1},{k}_{2},{k}_{3},{k}_{4}) , under some boundary conditions where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="normal">Ω</m:mi> </m:math> \Omega is singular at some point <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mn>0</m:mn> <m:mo>≤</m:mo> <m:mi>t</m:mi> <m:mo>≤</m:mo> <m:mn>1</m:mn> </m:math> 0\le t\le 1 , on a time scale <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mi mathvariant="double-struck">T</m:mi> </m:mrow> <m:mrow> <m:msub> <m:mrow> <m:mi>t</m:mi> </m:mrow> <m:mrow> <m:mn>0</m:mn> </m:mrow> </m:msub> </m:mrow> </m:msub> <m:mo>=</m:mo> <m:mrow> <m:mo>{</m:mo> <m:mrow> <m:mi>t</m:mi> <m:mo>:</m:mo> <m:mi>t</m:mi> <m:mo>=</m:mo> <m:msub> <m:mrow> <m:mi>t</m:mi> </m:mrow> <m:mrow> <m:mn>0</m:mn> </m:mrow> </m:msub> <m:msup> <m:mrow> <m:mi>q</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msup> </m:mrow> <m:mo>}</m:mo> </m:mrow> <m:mo>∪</m:mo> <m:mrow> <m:mo>{</m:mo> <m:mrow> <m:mn>0</m:mn> </m:mrow> <m:mo>}</m:mo> </m:mrow> </m:math> {{\mathbb{T}}}_{{t}_{0}}=\left\{t:t={t}_{0}{q}^{n}\right\}\cup \left\{0\right\} , for <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>n</m:mi> <m:mo>∈</m:mo> <m:mi mathvariant="double-struck">N</m:mi> </m:math> n\in {\mathbb{N}} where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mi>t</m:mi> </m:mrow> <m:mrow> <m:mn>0</m:mn> </m:mrow> </m:msub> <m:mo>∈</m:mo> <m:mi mathvariant="double-struck">R</m:mi> </m:math> <ja