Coupled system of sequential partial $\sigma \left( .,.\right)-$Hilfer fractional differential equationswith weighted double phase operator: Existence, Hyers-Ulam stability and controllability

In this paper, we are concerned by a sequential partial Hilfer fractional di erential system with weighted double phase operator. First, we introduce the concept of Hyers-Ulam stability with respect to an operator L for an abstract equation of the form u = LFu in Banach lattice by using the xed point ar- guments and spectral theory. Then, we prove the controllability and apply the previous results obtained for abstract equation to prove existence and Hyers- Ulam stability of a coupled system of sequential fractional partial di erential equations involving a weighted double phase operator. Finally, example illustrat- ing the main results is constructed. This work contains several new ideas, and gives a uni ed approach applicable to many types of di erential equations.