@article{Yang2025, 
author = {Shengbin Yang and Yongxiang Li},
title = {Existence and regularity of periodic solutions for a class of neutral evolution equation with delay},
year = {2025},
journal = {AIMS Mathematics},
volume = {10},
number = {7},
pages = {15370-15389},
keywords = {neutral evolution equation with delay, periodic mild solution, analytic operator semigroup, existence and C1-regularity, fixed point theorem},
url = {https://www.sciopen.com/article/10.3934/math.2025689},
doi = {10.3934/math.2025689},
abstract = {The purpose of this paper is to investigate the existence and        C          1      -regularity of    ω-periodic mild solutions for a class of neutral evolution equation with two-constant delays in Banach space    X         d          d      t        (  z  (  t  )  −  c  z  (  t  −  δ  )  )  +  A  (  z  (  t  )  −  c  z  (  t  −  δ  )  )  =  f  (  t  ,    z  (  t  )  ,    z  (  t  −  τ  )  )  ,    t  ∈      R    ,where    ∣  c  ∣<  1, the constants    τ  ,    δ  >  0 are defined as time lags,    A  :      D    (  A  )  ⊂  X  →  X is a sectorial operator and has compact resolvent, that is,    −  A generates exponentially stable, compact analytic operator semigroup    T  (  t  )  (  t  ⩾  0  ), and    f  :      R    ×  X  ×  X  →  X is nonlinear mapping which is    ω-periodic in    t. By using the theory of analytic operator semigroups, fixed point theorems, and the fractional powers of the sectorial operator, we establish the existence and        C          1      -regularity results of    ω-periodic mild solutions for the equation for the first time when    f satisfies the appropriate growth conditions. In the end, we present an example to demonstrate the applications of our main results.}
}