Abstract
This paper is concerned with the abstract evolution equation with delay. Firstly, we establish some sufficient conditions to ensure the existence results for the S -asymptotically periodic solutions by means of the compact semigroup. Secondly, we consider the global asymptotic behavior of the delayed evolution equation by using the Gronwall-Bellman integral inequality involving delay. These results improve and generalize the recent conclusions on this topic. Finally, we give an example to exhibit the practicability of our abstract results.
Highlights
Let X be a Banach space with norm ∥·∥ and r > 0 be a constant
Delayed partial differential equations play a major role in evolution equations
The existence of S-asymptotically ω-periodic mild solutions of delayed evolution equation (DEE) (1) under the nonlinear function F satisfying some growth conditions is explored by applying the semigroup theory of operators and fixed point theorem
Summary
Let X be a Banach space with norm ∥·∥ and r > 0 be a constant. Let B ≔ Cð1⁄2−r, 0, XÞ be the Banach space of continuous functions from 1⁄2−r, 0 into X provided with the uniform norm ∥φ∥B = sup ∥φðsÞ∥. Some scholars have discussed the existence results about S-asymptotically ω-periodic solutions for differential equations (one can see [3,4,5,6,7,8,9,10,11,12,13,14,15]). Some scholars study the global exponential stability of differential equations by constructing Lyapunov functions or applying matrix theory (one can see [16,17,18,19,20,21] and the references therein). It is hard to establish Lyapunov functions or apply the matrix theory to study the global exponential stability for delayed partial differential equations. The existence of S-asymptotically ω-periodic mild solutions of DEE (1) under the nonlinear function F satisfying some growth conditions is explored by applying the semigroup theory of operators and fixed point theorem. Some notions, definitions, and preliminary facts that we need are provided
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
