Existence and regularity of periodic solutions for a class of neutral evolution equation with delay

Shengbin Yang; Yongxiang Li; Shengbin Yang; Yongxiang Li

doi:10.3934/math.2025689

AIMS Mathematics

2025, Volume 10, Issue 7: 15370-15389. doi: 10.3934/math.2025689

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Research article Topical Sections

Existence and regularity of periodic solutions for a class of neutral evolution equation with delay

Shengbin Yang ,
Yongxiang Li ^,

College of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, China

Received: 15 May 2025 Revised: 23 June 2025 Accepted: 25 June 2025 Published: 03 July 2025
MSC : 34K30, 35K55, 35K90, 47D03

The purpose of this paper is to investigate the existence and $ C^{1} $-regularity of $ \omega $-periodic mild solutions for a class of neutral evolution equation with two-constant delays in Banach space $ X $
$ \frac{d}{dt}(z(t)-cz(t-\delta))+A(z(t)-cz(t-\delta)) = f(t, \ z(t), \ z(t-\tau)), \quad t\in\mathbb{R}, $
where $ \mid c\mid < 1 $, the constants $ \tau, \ \delta > 0 $ are defined as time lags, $ A:\mathcal{D}(A)\subset X\rightarrow X $ is a sectorial operator and has compact resolvent, that is, $ -A $ generates exponentially stable, compact analytic operator semigroup $ T(t)(t\geqslant0) $, and $ f:\mathbb{R}\times X\times X\rightarrow X $ is nonlinear mapping which is $ \omega $-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 $ \omega $-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.
- neutral evolution equation with delay,
- periodic mild solution,
- analytic operator semigroup,
- existence and $ C^{1} $-regularity,
- fixed point theorem
Citation: Shengbin Yang, Yongxiang Li. Existence and regularity of periodic solutions for a class of neutral evolution equation with delay[J]. AIMS Mathematics, 2025, 10(7): 15370-15389. doi: 10.3934/math.2025689

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Abstract

The purpose of this paper is to investigate the existence and $ C^{1} $-regularity of $ \omega $-periodic mild solutions for a class of neutral evolution equation with two-constant delays in Banach space $ X $

$ \frac{d}{dt}(z(t)-cz(t-\delta))+A(z(t)-cz(t-\delta)) = f(t, \ z(t), \ z(t-\tau)), \quad t\in\mathbb{R}, $

where $ \mid c\mid < 1 $, the constants $ \tau, \ \delta > 0 $ are defined as time lags, $ A:\mathcal{D}(A)\subset X\rightarrow X $ is a sectorial operator and has compact resolvent, that is, $ -A $ generates exponentially stable, compact analytic operator semigroup $ T(t)(t\geqslant0) $, and $ f:\mathbb{R}\times X\times X\rightarrow X $ is nonlinear mapping which is $ \omega $-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 $ \omega $-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.

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