Solutions of two fractional q-integro-differential equations under sum and integral boundary value conditions on a time scale

Abstract

AbstractIn this manuscript, by using the Caputo and Riemann–Liouville type fractional q-derivatives, we consider two fractional q-integro-differential equations of the forms ${}^{c}\mathcal{D}_{q}^{\alpha }[x](t) + w_{1} (t, x(t), \varphi (x(t)) )=0$Dqαc[x](t)+w1(t,x(t),φ(x(t)))=0 and $$ {}^{c}\mathcal{D}_{q}^{\alpha }[x](t) = w_{2} \biggl( t, x(t), \int _{0}^{t} x(r) \,\mathrm{d}r, {}^{c} \mathcal{D}_{q}^{\alpha }[x](t) \biggr) $$Dqαc[x](t)=w2(t,x(t),∫0tx(r)dr,cDqα[x](t)) for $t \in [0,l]$t∈[0,l] under sum and integral boundary value conditions on a time scale $\mathbb{T}_{t_{0}}= \{ t: t =t_{0}q^{n}\}\cup \{0\}$Tt0={t:t=t0qn}∪{0} for $n\in \mathbb{N}$n∈N where $t_{0} \in \mathbb{R}$t0∈R and q in $(0,1)$(0,1). By employing the Banach contraction principle, sufficient conditions are established to ensure the existence of solutions for the addressed equations. Examples involving algorithms and illustrated graphs are presented to demonstrate the validity of our theoretical findings.