Advanced Topics on Semilinear Evolution Equations

Differential evolution equations serve as mathematical representations that capture the progression or transformation of functions or systems as time passes. Currently, differential equations continue to be an active and thriving area of study, with continuous advancements in mathematical methodologies and their practical applications spanning diverse fields such as physics, engineering, and economics. In the late 20th century, the notion of 'Differential Evolution Equations' emerged as a distinct field applied to optimization and machine learning challenges. Evolution equations hold immense importance in numerous realms of applied mathematics and have experienced notable prominence in recent times.

This book delves into the study of several classes of equations, aiming to investigate the existence of mild and periodic mild solutions and their properties such as approximate controllability, complete controllability and attractivity, under various conditions. By examining diverse problems involving second-order semilinear evolution equations, differential and integro-differential equations with state-dependent delay, random effects, and functional differential equations with delay and random effects, we hope to contribute to the advancement of mathematical knowledge and provide researchers, academicians, and students with a solid foundation for further exploration in this field. Throughout this book, we explore different mathematical frameworks, employing Frechet spaces and Banach spaces to provide a comprehensive analysis. Our investigation extends beyond traditional solutions, encompassing the study of asymptotically almost automorphic mild solutions, periodic mild solutions, and impulsive integro-differential equations. These topics shed light on the behavior of equations in both bounded and unbounded domains, offering valuable insights into the dynamics of functional evolution equations.

Contents:

• Preface
• Introduction
• Preliminary Background
• Semilinear Evolution Equations
• Semilinear Evolution Equations with State-Dependent Delay
• Semilinear Differential Evolution Equations with Random Effects
• Semilinear Differential Evolution Equations with Random Effects and Delay
• S-Asymptotically ?-Periodic Mild Solutions for Differential Evolution Equations
• Impulsive Differential Evolution Equations
• Periodic Mild Solutions of Evolution Equations with Impulses and Delay
• Periodic Mild Solutions of Evolution Inclusions with Impulses and Delay
• Qualitative and Quantitative Analysis of Dynamical Models
• Bibliography
• Index

Readership: Researchers and graduate students that are interested in the study of differential equations, particularly, evolution differential equations and inclusions, and their relevant fields will find this monograph extremely useful for their future research. This book is appropriate for use in advanced graduate courses, seminars and research projects in numerous applied sciences.