This work introduces readers to the topic of maximal regularity for difference equations. The authors systematically present the method of maximal regularity, outlining basic linear difference equations along with relevant results. They address recent advances in the field, as well as basic semigroup and cosine operator theories in the discrete setting. The authors also identify some open problems that readers may wish to take up for further research. This book is intended for graduate students and researchers in the area of difference equations, particularly those with advance knowledge of and interest in functional analysis.

Presents the basic discrete semigroup theory and introduces the discrete cosine and sine operators Addresses applications of the theory of discrete maximal regularity to stability of concrete dynamics Introduces recent advances on theory of difference equations on Banach spaces