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

Abstract

The objective of this paper is to investigate, by applying the standard Caputo fractional q-derivative of order α\alpha , the existence of solutions for the singular fractional q-integro-differential equation Dqα[k](t)=Ω(t,k1,k2,k3,k4){{\mathcal{D}}}_{q}^{\alpha }\left[k]\left(t)=\Omega \left(t,{k}_{1},{k}_{2},{k}_{3},{k}_{4}), under some boundary conditions where Ω\Omega is singular at some point 0≤t≤10\le t\le 1, on a time scale Tt0={t:t=t0qn}∪{0}{{\mathbb{T}}}_{{t}_{0}}=\left\{t:t={t}_{0}{q}^{n}\right\}\cup \left\{0\right\}, for n∈Nn\in {\mathbb{N}} where t0∈R{t}_{0}\in {\mathbb{R}} and q∈(0,1)q\in \left(0,1). We consider the compact map and avail the Lebesgue dominated theorem for finding solutions of the addressed problem. Besides, we prove the main results in context of completely continuous functions. Our attention is concentrated on fractional multi-step methods of both implicit and explicit type, for which sufficient existence conditions are investigated. Finally, we present some examples involving graphs, tables and algorithms to illustrate the validity of our theoretical findings.

Keywords

• singularity
• multi-step methods
• q-integro-differential equation
• 34a08
• 34b16
• 39a13