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Abstract In this article, more general types of fractional proportional integrals and derivatives are proposed. Some properties of these operators are discussed.

This article deals with existence results of Caputo fractional neutral inclusions without compactness in Banach space using weak topology. In fact, for weakly sequentially closed maps we apply fixed point theorems to obtain the existence of the solution. Furthermore, the results are manifested for fractional neutral system held by nonlocal conditions. To justify the application of the reported results an illustration is presented.

In this article, we discuss the existence and uniqueness of solution of a delay Caputo q-fractional difference system. Based on the q-fractional Gronwall inequality, we analyze the Ulam-Hyers stability and the Ulam-Hyers-Rassias stability. An example is provided to support the theoretical results.

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