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This book deals with the study of sequence spaces, matrix transformations, measures of noncompactness and their various applications. The notion of measure of noncompactness is one of the most useful ones available and has many applications. The book discusses some of the existence results for various types of differential and integral equations with the help of measures of noncompactness; in particular, the Hausdorff measure of noncompactness has been applied to obtain necessary and sufficient conditions for matrix operators between BK spaces to be compact operators.
The book consists of eight self-contained chapters. Chapter 1 discusses the theory of FK spaces and Chapter 2 various duals of sequence spaces, which are used to characterize the matrix classes between these sequence spaces (FK and BK spaces) in Chapters 3 and 4. Chapter 5 studies the notion of a measure of noncompactness and its properties. The techniques associated with measures of noncompactness are applied to characterize the compact matrix operators in Chapters 6. In Chapters 7 and 8, some of the existence results are discussed for various types of differential and integral equations, which are obtained with the help of argumentations based on compactness conditions.
"This book deals with some aspects of the theory of sequence spaces and operators acting between them. . The book consists of eight chapters. Each chapter contains exercises and references. . the book will be useful for all specialists in functional analysis since it contains a lot of new and interesting results." (P. P. Zabreiko, Mathematical Reviews, August, 2015)
"This is one of few books out there in which various types of measures of noncompactness are covered so extensively with the inclusion of the applications to compact matrix operators, infinite systems of differential equations and integral equations. . The presentation is clear and self-contained throughout the entire treatment. Proofs of the main results are detailed and elegant. The book is suitable for graduate students and researchers with interest in generalized compactness methods for solving differential and integral equations." (Dhruba Adhikari, zbMATH 1323.47001, 2015)
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