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In this paper, we mainly investigate some new asymptotic properties on mild solutions to a fractional evolution equation in Banach spaces. Under local, global and mixed Lipschitz type conditions on the second variable for neutral and forced functions respectively, we establish some existence results for pseudo-Bloch periodic and pseudo  S -asymptotically-Bloch periodic mild solutions to the referenced equation onby suitable superposition theorems. The results show that the strict contraction of the neutral function for its second variable takes a dominated part in the existence and uniqueness of such solutions compared with the forced function. As subordinate results, we derive existence results of pseudo ( S -asymptotically)-Bloch periodic mild solutions for the sublinear growth of forced function with its second variable. As special cases, we also deduce some existence results for pseudo  ω -antiperiodic and pseudo  S -asymptotically  ω -antiperiodic mild solutions to the considered equation on .

Some new asymptotic properties on solutions to fractional evolution equations in Banach spaces