Fractional hybrid initial value problem featuring q-derivatives

Abstract

We have perused about the existence of a solution toward Hybrid initial value problem (HIVP) featuring fractional q-derivative\begin{equation*} \left\{\begin{array}{l} \mathfrak{D}^{\delta}_{q}\Big[\frac{\nu\left(t\right)}{h\big(t,\nu\left(t\right),\ \max \limits_{0\leq\tau\leq t}\left|\nu\left(\tau\right)\right| \big)}\Big]= \rho\left(t,\ \nu\left(t\right)\right), \ t\in(0,1), \ 0<\delta\leq 1, \\\nu(0)=0, \end{array} \right.\end{equation*} in which \mathfrak{D}^{\delta}_{q} denotes the Riemann-Liouville fractional q-derivative in the order of $\delta$. In Banach algebra by making use of a fixed point theorem based Dhage along with mixed Lipschitz and Caratheodory condition, there exists a way of solving toward the above fractional Hybrid initial value problem (FHIVP) featuring q-derivatives is verified.