Existence of Global Mild Solutions for Nonautonomous Abstract Evolution Equations

Zhou, Mian; Liang, Yong; Zhou, Yong

doi:10.3390/math13111722

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Open AccessArticle

Existence of Global Mild Solutions for Nonautonomous Abstract Evolution Equations

by

Mian Zhou

,

Yong Liang

and

Yong Zhou

^*

Faculty of Innovation Engineering, Macau University of Science and Technology, Macau 999078, China

^*

Author to whom correspondence should be addressed.

Mathematics 2025, 13(11), 1722; https://doi.org/10.3390/math13111722

Submission received: 30 April 2025 / Revised: 20 May 2025 / Accepted: 21 May 2025 / Published: 23 May 2025

(This article belongs to the Section C1: Difference and Differential Equations)

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Abstract

In this paper, we investigate the Cauchy problem for nonautonomous abstract evolution equations of the form

y^{'} (t) = A (t) y (t) + f (t, y (t)), t \geq 0

,

y (0) = y_{0}

. We obtain new existence theorems for global mild solutions under both compact and noncompact evolution families

U (t, s)

. Our key method relies on the generalized Ascoli–Arzela theorem we previously obtained. Finally, an example is provided to illustrate the applicability of our results.

Keywords: nonautonomous evolution equation; global mild solutions; existence; fixed-point theorem; measure of noncompactness

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MDPI and ACS Style

Zhou, M.; Liang, Y.; Zhou, Y. Existence of Global Mild Solutions for Nonautonomous Abstract Evolution Equations. Mathematics 2025, 13, 1722. https://doi.org/10.3390/math13111722

AMA Style

Zhou M, Liang Y, Zhou Y. Existence of Global Mild Solutions for Nonautonomous Abstract Evolution Equations. Mathematics. 2025; 13(11):1722. https://doi.org/10.3390/math13111722

Chicago/Turabian Style

Zhou, Mian, Yong Liang, and Yong Zhou. 2025. "Existence of Global Mild Solutions for Nonautonomous Abstract Evolution Equations" Mathematics 13, no. 11: 1722. https://doi.org/10.3390/math13111722

APA Style

Zhou, M., Liang, Y., & Zhou, Y. (2025). Existence of Global Mild Solutions for Nonautonomous Abstract Evolution Equations. Mathematics, 13(11), 1722. https://doi.org/10.3390/math13111722

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Existence of Global Mild Solutions for Nonautonomous Abstract Evolution Equations

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