Existence of Urysohn and Atangana–Baleanu fractional integral inclusion systems solutions via common fixed point of multi-valued operators

This paper aims to prove that the solution of the integral equation systems exists via the concepts of fixed point and multi-valued operators. For this purpose, at first, some new theoretical results are presented and proved for the existence of a common fixed point for a pair of multi-valued operators. The multi-valued operators are αℂ-admissible and hold a new generalized contractive condition in complex-valued double-controlled metric spaces. In applying the considered generalized contractive condition on multi-valued operators, an important principle of complex numbers has complied, which is inadvertently neglected in some research This unneglectable principle is that the maximum of two complex numbers is not necessarily one of them; Rather, it can be greater than both. Thus, in our applied contractive condition, satisfying the contraction condition is considered with any member of the assumed set instead of the set maximum. Consequently, the presented results in this work improve and generalize some results mentioned in the literature. In the Applications section, two existence theorems for the solution of Urysohn integral equations system and Atangana–Baleanu fractional integral inclusions system are provided and proved based on our obtained theoretical results. Finally, analytical and numerical examples are provided to confirm the applicability of the obtained theoretical results.