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The current work is devoted to investigate evolution problem of nonlocal Cauchy type under “Mittag–Leffler” type fractional derivative which works as nonsingular and nonlocal kernel. For the considered problem, we establish some appropriate results devoted to the qualitative theory of existence and stability analysis. Upon using Banach and Krasnoselskii’s fixed point theorems, we establish the required analysis. Further stability theory of Ulam’s type is constructed by sing nonlinear analysis. Some interesting examples are provided to support our established results.