Sequence Spaces and Measures of Noncompactness with Applications to Differential and Integral Equations

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.

• Compact operators
• Differential and integral equations
• Matrix transformations
• Measures of noncompactness
• Sequence spaces
• partial differential equations
• ordinary differential equations

“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. Zabreĭko, 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)

• Department of Mathematics, Rzeszów University of Technology, Rzeszow, Poland

  Józef Banaś

• Department of Mathematics, Aligarh Muslim University, Aligarh, India

  Mohammad Mursaleen

JÓZEF BANAŚ is professor of mathematics and chair at the Department of Mathematics in Rzeszow University of Technology, Poland. He is on the editorial committee of many journals of international repute: Commentationes Mathematicae (Polish Mathematical Society), Abstract and Applied Analysis (Hindawi Corporation), Journal of Inequalities and Applications (SpringerOpen), World Scientific Journal (Hindawi Corporation), Mathematica Applicanda (section: Mathematical Economics) and many others. Prof. Banaś is also editor-in-chief of the Journal of Mathematics and Applications, Rzeszow. He has over 140 published research papers to his credit in journals such as Journal of Integral Equations and Applications, Rocky Mountain Journal of Mathematics, Proceedings of AMS, Journal of Mathematical Analysis and Applications, Applied Mathematics and Computation, Bulletin of the London Mathematical Society, Nonlinear Analysis, Abstract and Applied Analysis, among others. Prof. Banaś is also coauthor of two books: Measures of Noncompactness in Banach Spaces, Lecture Notes in Pure and Applied Mathematics 60, Marcel Dekker, New York and Basel, 1980 (with K. Goebel) and Bounded Variation and Around, De Gruyter Series in Nonlinear Analysis and Applications 17, Walter de Gruyter, Berlin/Boston 2014 (with J. Appell and N. Merentes). His fields of interest include geometry of Banac

M. MURSALEEN is professor of mathematics at Aligarh Muslim University, Aligarh. Earlier, he worked as professor of mathematics at King Abdulaziz University, Jeddah, Saudi Arabia, during 2004–2006. An active researcher, Prof. Mursaleen has authored two books and five book chapters, in addition to his contributions to 190 research papers to various international journals such as Proc. Amer. Math. Soc., Quarterly J. Math. (Oxford), Studia Math., Information Sciences, and Chaos, Solitons & Fractals to name a few. Prof. Mursaleen is reviewer for Mathematical Reviews (USA) since 1982, as well as referee of about 100 scientific journals (most of them are SCI/SCI expended journals). He has also guided 12 PhD students so far. He has visited about 16 countries including USA & UK and delivered about 32 talks, He has also participated in joint research work with faculty members of the host institutions. Prof. Mursaleen is member of the editorial board of various scientific journals and has served as member of various international scientific bodies and organizing committees that include International Council of Scientists “Global World Communicator Education and Science”. He has worked on a number of joint projects of international collaboration<. His main research interests are Sequence Spaces, Summability Theory, Approximation Theory, Functional Equations, Measures of Non-compactness and Fixed-Point Theory.

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