Understanding Banach Spaces

Daniel González Sánchez (Editor)
Escuela de Ciencias Físicas y Matemáticas, Universidad de Las Américas, Quito, Ecuador

Series: Mathematics Research Developments
BISAC: MAT003000

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Edited by I Leslie Rubin, Robert J Geller, Abby Mutic, Benjamin A Gitterman, Nathan Mutic, Wayne Garfinkel, Claire D Coles, Kurt Martinuzzi, and Joav Merrick

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This book focuses on the study of several properties of Banach spaces applied to diverse problems in functional and numerical analysis. Many problems in science, engineering and other disciplines can be expressed in the form of equations, inequalities or systems of equations using mathematical modelling. In particular, a large number of these problems can be solved using these spaces. A great multitude of examples showing the theoretical application developed appears throughout the work. Researchers and practitioners will find this book very useful as a source book, utilize its methods and also use it as a classroom text for a senior undergraduate or graduate course. It is certainly an excellent must read book. (Imprint: Nova)

Preface

Chapter 1. Right Complex Caputo Fractional Inequalities
(George A. Anastassiou, Department of Mathematical Sciences, University of Memphis, Memphis, TN, US)

Chapter 2. Mixed Complex Fractional Inequalities
(George A. Anastassiou, Department of Mathematical Sciences, University of Memphis, Memphis, TN, US)

Chapter 3. Advanced Complex Fractional Ostrowski Inequalities
(George A. Anastassiou, Department of Mathematical Sciences, University of Memphis, Memphis, TN, US)

Chapter 4. Improved Qualitative Analysis for Newton–like Methods with R–Order of Convergence at Least Three in Banach Spaces
(Ioannis K. Argyros and Santhosh George, Department of Mathematical Sciences, Cameron University, Lawton, OK, US, and others)

Chapter 5. Developments on the Convergence Region of Newton–like Methods with Generalized Inverses in Banach Spaces
(Ioannis K. Argyros and Santhosh George, Department of Mathematical Sciences, Cameron University, Lawton, OK, US, and others)

Chapter 6. Modified Newton–Type Compositions for Solving Equations in Banach Spaces
(Ioannis K. Argyros and Santhosh George, Department of Mathematical Sciences, Cameron University, Lawton, OK, US, and others)

Chapter 7. Ball Convergence for Optimal Derivative Free Methods
(Ioannis K. Argyros and Ramandeep Behl, Department of Mathematical Sciences, Cameron University, Lawton, OK, US, and others)

Chapter 8. Ball Convergence for a Derivative Free Method with Memory
(Ioannis K. Argyros and Ramandeep Behl, Department of Mathematical Sciences, Cameron University
Lawton, OK, US, and others)

Chapter 9. Weaker Convergence Conditions of an Iterative Method for Nonlinear Ill–Posed Equations
(Ioannis K. Argyros and Santhosh George, Department of Mathematical Sciences, Cameron University
Lawton, OK, US, and others)

Chapter 10. Ball Convergence Theorem for a Fifth Order Method in Banach Spaces
(Ioannis K. Argyros and Santhosh George, Department of Mathematical Sciences, Cameron University, Lawton, OK, US, and others)

Chapter 11. Extended Convergence of King–Werner–like Methods without Derivatives
(Ioannis K. Argyros and Santhosh George, Department of Mathematical Sciences, Cameron University
Lawton, OK, US, and others)

Chapter 12. On an Eighth Order Steffensen–Type Solver Free of Derivatives
(Ioannis K. Argyros and Santhosh George, Department of Mathematical Sciences, Cameron University, Lawton, OK, US, and others)

Chapter 13. Local Convergence of Osada’s Method for Finding Zeros with Multiplicity
(Ioannis K. Argyros and Santhosh George, Department of Mathematical Sciences, Cameron University
Lawton, OK, US, and others)

Chapter 14. Expanding the Applicability of an Eighth–Order Method in Banach Space under Weak Conditions
(Ioannis K. Argyros, Ramandeep Behl, Daniel González and S. S. Motsa, Department of Mathematical Sciences, Cameron University, Lawton, OK, US, and others)

Chapter 15. Local Convergence for Three Step Eighth Order Method under Weak Conditions
(Ioannis K. Argyros, Ramandeep Behl and Daniel González, Department of Mathematical Sciences
Cameron University, Lawton, OK, US, and others)

Chapter 16. Ball Convergence for a Three–Step One Parameter Efficient Method in Banach Space under Generalized Conditions
(Ioannis K. Argyros, Ramandeep Behl and Daniel González, Department of Mathematical Sciences

Chapter 17. On the Convergence of Newton–Moser Method from Data at One Point
(José M. Gutiérrez and Miguel A. Hernández-Verón, Department of Mathematics and Computer Sciences, University of La Rioja, Logrono, Spain)

Chapter 18. Approximating Inverse Operators by a Fourth–Order Iterative Method
(J. A. Ezquerro, Miguel A. Hernández-Verón and J. L. Varona, Department of Mathematics and Computer Sciences, University of La Rioja, Logrono, Spain, and others)

Chapter 19. Initial Value Problems in Clifford Analysis Using Associated Spaces
(David Armendáriz, Johan Ceballos and Antonio Di Teodoro, Departamento de Matemáticas, Colegio de Ciencias e Ingenierías, Universidad San Francisco de Quito, Quito, Ecuador, and others)

Chapter 20. Computational Approach of Initial Value Problems in Clifford Analysis Using Associated Spaces
(David Armendáriz, Johan Ceballos and Antonio Di Teodoro, Departamento de Matemáticas, Colegio de Ciencias e Ingenierías, Universidad San Francisco de Quito, Quito, Ecuador, and others)

Chapter 21. Asymmetric Convexity and Smoothness: Quantitative Lyapunov Theorems, Lozanovskii Factorisation, Walsh Chaos and Convex Duality
(Sergey S. Ajiev, University of Technology, Sydney, Australia)

Chapter 22. Copies of Sequence Spaces and Basic Properties of Anisotropic Function Spaces
(Sergey S. Ajiev, University of Technology, Sydney, Australia)

Chapter 23. Interpolation and Bounded Extension of Hӧlder–Lipschitz Mappings between Function, Non–Commutative and Other Banach Spaces
(Sergey S. Ajiev, University of Technology, Sydney, Australia)

Chapter 24. Differential Equations with a Small Parameter in a Banach Space
(Vasiliy Kachalov, National Research University “MPEI”, Moscow, Russia)

Chapter 25. Role of Hanson–Antczak–Type V –Invex Functions in Sufficient Efficiency Conditions for Semiinfinite Multiobjective Fractional Programming
(Ram U. Verma, International Publications USA, Mathematical Sciences Division, Denton, TX, US)

Index

"Banach spaces as well as their specializations such as Hilbert or Euclidean spaces to mention a few are of great interest in Pure as well as Applied Mathematics. In particular Dr. González is connecting Banach spaces to applications using iterative methods. Researchers and practitioners consider Banach spaces general enough for studying operator equations, since problems from diverse disciplines can be reduced to locating a zero of a Banach space valued operator. These zeros are needed in closed form but this is achieved in rare situations. That is why he has presented a plethora of alternative iterate methods generating a sequence approximating a zero under certain initial conditions about a zero (local convergence) or about in an initial point (semilocal convergence). These methods converge with linear, quadratic or higher order of convergence and under weak conditions. Researchers and practitioners will find this book very useful as a source book, utilize its methods and also use it as a classroom text for a senior undergraduate or graduate course. It is certainly an excellent must read book." - Ioannis K. Argyros, Cameron University, Department of Mathematics, Lawton, Oklahoma, USA

Researchers and practitioners will find this book very useful as a source book, utilize its methods and also use it as a classroom text for a senior undergraduate or graduate course.

• Banach spaces
• Iterative method
• Local and semilocal convergence
• Order of convergence
• Clifford analysis

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