Existence and uniqueness of solutions for multi-term fractional q-integro-differential equations via quantum calculus

Abstract

AbstractIn this investigation, by applying the definition of the fractional q-derivative of the Caputo type and the fractional q-integral of the Riemann–Liouville type, we study the existence and uniqueness of solutions for a multi-term nonlinear fractional q-integro-differential equations under some boundary conditions ${}^{c}D_{q}^{\alpha} x(t) = w ( t, x(t), (\varphi_{1} x)(t), (\varphi_{2} x)(t), {}^{c}D_{q} ^{ \beta_{1}} x(t), {}^{c}D_{q}^{\beta_{2}} x(t), \ldots, {}^{c}D _{q}^{ \beta_{n}}x(t) )$Dqαcx(t)=w(t,x(t),(φ1x)(t),(φ2x)(t),cDqβ1x(t),cDqβ2x(t),…,cDqβnx(t)). Our results are based on some classical fixed point techniques, as Schauder’s fixed point theorem and Banach contraction mapping principle. Besides, some instances are exhibited to illustrate our results and we report all algorithms required along with the numerical result obtained.