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The main concentration of the present research is to explore several theoretical criteria for proving the existence results for the suggested boundary problem. In fact, for the first time, we formulate a new hybrid fractional differential inclusion in the  φ -Caputo settings depending on an increasing function  φ subject to separated mixed  φ -hybrid-integro-derivative boundary conditions. In addition to this, we discuss a special case of the proposed  φ -inclusion problem in the non- φ -hybrid structure with the help of the endpoint notion. To confirm the consistency of our findings, two specific numerical examples are provided which simulate both  φ -hybrid and non- φ -hybrid cases.

Fixed Point and Endpoint Theories for Two Hybrid Fractional Differential Inclusions with Operators Depending on an Increasing Function