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

Abstract

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 $(\omega ,k)$-Bloch periodic and pseudo S-asymptotically $(\omega ,k)$-Bloch periodic mild solutions to the referenced equation on $\mathbb{R}$ by 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) $(\omega ,k)$-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 $\mathbb{R}$.

Keywords:

• Pseudo $(\omega ,k)$-Bloch periodicity
• pseudo S-asymptotically $(\omega ,k)$-Bloch periodicity
• fractional evolution equations

Mathematics Subject Classifications (2020)::

• 34K13
• 58D25
• 34K37

Acknowledgments

Authors would like to thank the anonymous referee for carefully reading this manuscript and giving valuable comments to improve this paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was partially supported by NSF of Shaanxi Province (2020JM-183).