Filippov-type existence theorems for $q$-fractional differential inclusions with nonlinear $q$-integral conditions in Banach spaces

This study focused on boundary value problems involving fractional $q$-difference inclusions subject to nonlinear $q$-integral conditions:

$$\begin{array}{c}{}^c{D}_{q,\gamma }\left(\nu \left(\iota \right)-h\left(\iota ,\nu \left(\iota \right)\right)\right)\in F\left(\iota ,\nu \left(\iota \right)\right),\iota \in \left[0,\ell \right],1<\gamma \le 2,\\ \nu \left(0\right)-{\nu }^{\mathrm{\prime }}\left(0\right)=a\left(\iota \right){\int }_0^{\ell }{\mathcal{G}}_1\left(\tau ,\nu \left(\tau \right)\right)d_q\tau ,\\ \nu \left(\ell \right)-{\nu }^{\mathrm{\prime }}\left(\ell \right)=b\left(\iota \right){\int }_0^{\ell }{\mathcal{G}}_2\left(\tau ,\nu \left(\tau \right)\right)d_q\tau .\end{array}$$

By applying appropriate fixed point theorems, we established the existence of solutions to the problem considered. In addition, a Filippov-type result was provided. This work continued the study recently presented in [N. Allouch, J. R. Graef, S. Hamani, Boundary value problem for fractional $q$-difference equations with integral conditions in Banach spaces, Fractal Fract., 6 (2022), 237].