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Generalized Bloch type periodicity and applications to semi-linear differential equations in banach spaces

Part of: Spaces and manifolds of mappings Qualitative theory Entire and meromorphic functions, and related topics

Published online by Cambridge University Press:  08 March 2022

Yanyan Wei  and
Yong-Kui Chang
Yanyan Wei
Affiliation:
School of Mathematics and Statistics, Xidian University, Xi'an 710071, Shaanxi, P. R. China (yywei@stu.xidian.edu.cn; lzchangyk@163.com)
Yong-Kui Chang
Affiliation:
School of Mathematics and Statistics, Xidian University, Xi'an 710071, Shaanxi, P. R. China (yywei@stu.xidian.edu.cn; lzchangyk@163.com)
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Abstract

In this paper, we mainly introduce some new notions of generalized Bloch type periodic functions namely pseudo Bloch type periodic functions and weighted pseudo Bloch type periodic functions. A Bloch type periodic function may not be Bloch type periodic under certain small perturbations while it can be quasi Bloch type periodic in sense of generalized Bloch type periodic functions. We firstly show the completeness of spaces of generalized Bloch type periodic functions and establish some further properties such as composition and convolution theorems of such functions. We then apply these results to investigate existence results for generalized Bloch type periodic mild solutions to some semi-linear differential equations in Banach spaces. The obtained results show that for each generalized Bloch type periodic input forcing disturbance, the output mild solutions to reference evolution equations remain generalized Bloch type periodic.

Keywords

weighted pseudo Bloch type periodic functionscompletenesscomposition and convolution theoremevolution equations

MSC classification

Primary: 34C25: Periodic solutions
Secondary: 30D45: Bloch functions, normal functions, normal families 58D25: Equations in function spaces; evolution equations
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 65 , Issue 2 , May 2022 , pp. 326 - 355
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Copyright
Copyright © The Author(s), 2022. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society

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