What question did this study set out to answer?

The study aims to explore the existence and asymptotic stability of periodic solutions within a specific class of neutral delayed evolution equations.

May 11, 2026Open Access

Existence and asymptotic stability of periodic solutions for a class of neutral delayed evolution equation

Key Points

The study aims to explore the existence and asymptotic stability of periodic solutions within a specific class of neutral delayed evolution equations.
Utilized the theory of operator semigroup and fixed point theorems to derive existence results.
Analyzed both cases of compact and non-compact operators to study periodic behavior.
Focused on ω-periodic mild solutions in the context of Banach space.
Established existence of ω-periodic mild solutions under the assumption of compact operators.
Provided conditions for the asymptotic stability of periodic solutions when non-compact operators are considered.

Abstract

Abstract The purpose of this work is to study the existence and asymptotic stability of ω -periodic mild solutions to a class of neutral delayed evolution equation 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, ddt (z (t) -cz (t-) ) +A (z (t) -cz (t-) ) =f (t, z (t), z (t-) ), t R, where ∣ c ∣ 0 are defined as time lags, A: D (A) ⊂ X → X A: D (A) X X is a closed linear operator, − A generates a strongly continuous semigroup T (t) (t ⩾ 0), and f: R × X 2 → X f: RX^2 X is continuous function which is ω -periodic in t. Firstly, by using theory of operator semigroup and fixed point theorems, the existence results of ω -periodic mild solutions to the above equation are obtained under the assumption that the T (t) (t ⩾ 0) is compact. Next, in the case that T (t) (t ⩾ 0) is non-compact, we investigate the asymptotic stability of periodic solution for the equation based on the contract

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