Existence of solutions for a class of the boundary value problem of fractional \(q\)-difference inclusions (Q1677144)

This paper studies the existence of solutions for a class of boundary value problems (BVP) of fractional \(q\)-difference inclusions \[ \left(^cD_q^\alpha x\right)(t)\in F\left (t, ^cD_q^\sigma x(t)\right), \;\;0\leq t\leq 1, \] subject to the boundary conditions \[ x(0)=g(x),\;\;D_q^2 x(0)=0,\;\;\gamma \left(D_q x\right)(1)=\beta D_q^2 x(1), \] with \(2<\alpha\leq 3\), \(0<q<1\), \(0<\sigma<1\), \(\beta\gamma\geq 0\), \(g\) is a continuous function, and \(F:[0,1]\times \mathbb{R}\rightarrow \mathcal{P}(\mathbb{R})\) is a multivalued map. The first section is devoted to literature reviews. In the second section, the authors provide some introductions and a few basic properties and lemmas. The main result is given in the third section. The authors obtain sufficient conditions such that the main BVP has at least one solution on \([0, 1]\). Moreover, they provide an example in the last section to demonstrate the application. This paper will be of interest to anyone who is studying fractional \(q\)-difference equations.