Some Existence Results of Coupled Hilfer Fractional Differential System and Differential Inclusion on the Circular Graph

Circular network structure is widely used in neural network, image processing, computer vision and bioinformatics. For example, recurrent neural network is a kind of neural network with a circular structure that can be used to process temporal data. It has a wide range of applications in natural language processing, speech recognition, music generation, etc. In this paper, in order to reduce the complexity of the presentation, we study a class of Hilfer-type fractional differential system and differential inclusion with coupled integral boundary value conditions on the simplest circular graph. First, two existence results of Hilfer-type fractional differential system are proved by some known fixed point theorems. Further, the existence results of convex and non-convex multivalued mappings are obtained by using Leray–Schauder nonlinear alternative and Covitz–Nadler fixed point theorem, respectively. At last, two examples are given to verify our theoretical results.