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# Monotone Iterative Technique for the Periodic Solutions of High-Order Delayed Differential Equations in Abstract Spaces

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College of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, Gansu, China
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Academic Editor: Miguel Ángel López Guerrero
Symmetry 2021, 13(3), 449; https://doi.org/10.3390/sym13030449
Received: 27 February 2021 / Revised: 5 March 2021 / Accepted: 5 March 2021 / Published: 10 March 2021
This paper deals with the existence of $\omega$-periodic solutions for nth-order ordinary differential equation involving fixed delay in Banach space E. ${L}_{n}u\left(t\right)=f\left(t,u\left(t\right),u\left(t-\tau \right)\right),\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}t\in \mathbb{R},$ where ${L}_{n}u\left(t\right):={u}^{\left(n\right)}\left(t\right)+\sum _{i=0}^{n-1}{a}_{i}{u}^{\left(i\right)}\left(t\right)$, ${a}_{i}\in \mathbb{R}$, $i=0,1,\cdots ,n-1$, are constants, $f\left(t,x,y\right):\mathbb{R}×E×E⟶E$ is continuous and $\omega$-periodic with respect to t, $\tau >0$. By applying the approach of upper and lower solutions and the monotone iterative technique, some existence and uniqueness theorems are proved under essential conditions. View Full-Text
MDPI and ACS Style

Yang, H.; Li, Y. Monotone Iterative Technique for the Periodic Solutions of High-Order Delayed Differential Equations in Abstract Spaces. Symmetry 2021, 13, 449. https://doi.org/10.3390/sym13030449

AMA Style

Yang H, Li Y. Monotone Iterative Technique for the Periodic Solutions of High-Order Delayed Differential Equations in Abstract Spaces. Symmetry. 2021; 13(3):449. https://doi.org/10.3390/sym13030449

Chicago/Turabian Style

Yang, He, and Yongxiang Li. 2021. "Monotone Iterative Technique for the Periodic Solutions of High-Order Delayed Differential Equations in Abstract Spaces" Symmetry 13, no. 3: 449. https://doi.org/10.3390/sym13030449

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