Abstract
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 D q α [ k ] ( t ) = Ω ( t , k 1 , k 2 , k 3 , k 4 ) {{\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 ≤ 1 0\le t\le 1 , on a time scale T t 0 = { t : t = t 0 q n } ∪ { 0 } {{\mathbb{T}}}_{{t}_{0}}=\left\{t:t={t}_{0}{q}^{n}\right\}\cup \left\{0\right\} , for n ∈ N n\in {\mathbb{N}} where t 0 ∈ 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.
Highlights
The field of fractional calculus plays a fundamental role in mathematical analysis
Ahmad et al investigated the existence of solutions for a q-antiperiodic boundary value problem of fractional q-difference inclusions given by c qα[k](t) ∈ F(t, k(t), q[k](t), q2[k](t)), for t ∈ [0, 1], q ∈ (0, 1), 2 < α ≤ 3, 0 < β ≤ 3 with conditions k(0) + k(1) = 0, q[k](0) + q[k](1) = 0, q2k(0) + q2[k](1) = 0, where c α q denotes
1380 Sayyedeh Narges Hajiseyedazizi et al In this paper and motivated by the aforementioned achievements, we investigate the singular fractional q-integro-differential equation of the form t σq[k](t) = Ω⎜⎜t, k(t), k′(t), ζq[k](t), f (r)k(r)dr⎟⎟, (1)
Summary
The field of fractional calculus plays a fundamental role in mathematical analysis. It provides efficient techniques to solve fractional differential equations and inclusions [1,2,3,4,5,6,7,8,9,10]. Ahmad et al investigated the existence of solutions for a q-antiperiodic boundary value problem of fractional q-difference inclusions given by c qα[k](t) ∈ F(t, k(t), q[k](t), q2[k](t)), for t ∈ [0, 1], q ∈ (0, 1), 2 < α ≤ 3, 0 < β ≤ 3 with conditions k(0) + k(1) = 0, q[k](0) + q[k](1) = 0, q2k(0) + q2[k](1) = 0, where c α q denotes q-derivative of order α and. In 2019, Ntouyas et al in [20], by applying definition of the fractional q-derivative of the Caputo-type and the fractional q-integral of the Riemann-Liouville-type, studied the existence and uniqueness of solutions for a multi-term nonlinear fractional q-integro-differential equations under some boundary conditions c σq[k](r) = Ω(r, k(r), (φ1k)(r), (φ2k)(r), c qβ1[k](r), c qβ2[k](r), ...,c qβn[k](t)).
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