In this paper we present some existence results and topological structure of the solution set for a class of Caputo implicit fractional q-difference inclusions in Banach spaces. Firstly, using the set-valued analysis, we study some global existence results and we present a new version of Filippov's Theorem. Further, we obtain results in the cases where the nonlinearity is upper as well as lower semi-continuous with respect to the second argument by using Mönch's and Schauder-Tikhonov fixed point theorems and the concept of measure of noncompactness. In the last section, we illustrate our results by an example.

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In this paper, we study the following fractional differential equation involving the Atangana-Baleanu-Caputo fractional derivative: { A B C a D τ θ [ x ( ϑ ) − F ( ϑ , x ( ϑ ) ) ] = G ( ϑ , x ( ϑ ) ) ,         ϑ ∈ J : = [ a , b ] , x ( a ) = φ a ∈ ℝ . $$\left\{ {\matrix{ {AB{C_a}D_\tau ^\theta [x(\vartheta ) - F(\vartheta ,x(\vartheta ))] = G(\vartheta ,x(\vartheta )),\;\;\;{\kern 1pt} \vartheta \in J: = [a,b],} \hfill \cr {x(a) = {\varphi _a} \in .} \hfill \cr } } \right.$$ The result is based on a Dhage fixed point theorem. Further, an example is provided for the justification of our main result.