Sequence Space Theory with Applications

The book features original chapters on sequence spaces involving the idea of ideal convergence, modulus function, multiplier sequences, Riesz mean, Fibonacci difference matrix etc., and illustrate their involvement in various applications. The preliminaries have been presented in the beginning of each chapter and then the advanced discussion takes place, so it is useful for both expert and nonexpert on aforesaid topics. The book consists of original thirteen research chapters contributed by the well-recognized researchers in the field of sequence spaces with associated applications. Features Discusses the Fibonacci and vector valued difference sequence spaces Presents the solution of Volterra integral equation in Banach algebra Discusses some sequence spaces involving invariant mean and related to the domain of Jordan totient matrix Presents the Tauberian theorems of double sequences Discusses the paranormed Riesz difference sequence space of fractional order Includes a technique for studying the existence of solutions of infinite system of functional integro-differential equations in Banach sequence spaces The subject of book is an active area of research of present time internationally and would serve as a good source for researcher and educators involved with the topic of sequence spaces. This book contains advance and modern techniques to define sequence spaces and obtain their applications.This book is aimed primarily at graduates and researchers studying sequence spaces. Students in mathematics and engineering would also find this book useful. Cover 1 Half Title 2 Title Page 4 Copyright Page 5 Contents 6 Preface 10 Editors 12 Contributors 14 1. Hahn-Banach and Duality Type Theorems for Vector Lattice-Valued Operators and Applications to Subdifferential Calculus and Optimization 16 1.1. Introduction 16 1.2. Basic Notions and Results 18 1.2.1. Relative interior points and convexity 18 1.2.2. Dual spaces of vector lattices and representation as spaces of continuous functions 19 1.2.3. The p-integral in vector lattice setting 22 1.2.4. A Chojnacki-type integral for vector lattice-valued functions 24 1.2.5. Basic assumptions and properties 24 1.3. The Main Results 27 1.4. Applications to Set Functions 50 Bibliography 53 2. Application of Measure of Noncompactness on Infinite System of Functional Integro-differential Equations with Integral Initial Conditions 60 2.1. Introduction 60 2.1.1. Preliminaries 61 2.1.2. Kuratowski measure of noncompactness 62 2.1.3. Axiomatic approach to the concept of a measure of noncompactness 62 2.1.4. Hausdor measure of noncompactness 63 2.1.5. Condensing operators, compact operators and related results 65 2.2. Existence of Solution C(I, c0) 68 2.3. Existence of Solution C(I, l1) 71 2.4. Illustrative Example 74 2.5. Conclusion 76 Bibliography 76 3. -Statistical Convergence of Interval Numbers of Order a 78 3.1. Introduction 78 3.2. Main Results 79 Bibliography 87 4. Necessary and Sufficient Tauberian Conditions under which Convergence follows from (Ar,s,p,q; 1,1), (Ar,*,p,*; 1,0) and (A*,s,*,q; 0,1) Summability Methods of Double Sequences 90 4.1. Introduction 90 4.2. Auxiliary Results 92 4.3. Tauberian Theorems for the (Ar,s,p,q; 1,1) Summability Method 94 4.3.1. Proofs 97 4.4. Tauberian Theorems for the (Ar,*,p,*; 1,0) Summability Method 101 4.4.1. Proofs 104 4.5. Tauberian Theorems for the (A*,s,*,q; 0,1) Summability Method 107 Bibliography 108 5. On New Sequence Spaces Related to Domain of the Jordan Totient Matrix 110 5.1. Introduction and Background 110 5.2. The Domains of the Jordan Totient Matrix in the Spaces c0, c,l 114 5.3. The a-, b- and y-Duals 116 5.4. Certain Matrix Transformations 119 Bibliography 126 6. A Study of Fibonacci Difference I-Convergent Sequence Spaces 129 6.1. Introduction and Preliminaries 129 6.1.1. Fibonacci sequence 130 6.2. Fibonacci Difference Sequence Spaces 136 6.3. Orlicz Fibonacci Difference Sequence Spaces 144 6.4. Paranormed Fibonacci Difference Sequence Spaces 147 Bibliography 150 7. Theory of Approximation for Operators in Intuitionistic Fuzzy Normed Linear Spaces 154 7.1. Introduction 154 7.1.1. Background 155 7.1.2. Main goal 156 7.2. Basic Definitions 156 7.3. Definitions and Main Results 158 7.3.1. Essential definitions 158 7.3.2. Main results 159 7.3.3. Modified version of de nitions of AP and BAP 160 7.3.4. Certain related results and examples 161 7.4. Conclusion 165 Bibliography 165 8. Solution of Volterra Integral Equations in Banach Algebras using Measure of Noncompactness 169 8.1. Introduction and Preliminaries 169 8.2. Fixed Point Results 171 8.3. Solvability of Volterra integral equation in Banach algebra 177 Bibliography 181 9. Solution of a pair of Nonlinear Matrix Equation using Fixed Point Theory 184 9.1. Introduction and Preliminaries 184 9.2. Result 1 185 9.3. Result 2 190 9.3.1. Consequences 193 9.4. Application 194 9.5. Numerical Experiment 197 Bibliography 205 10. Sequence Spaces and Matrix Transformations 206 10.1. Introduction 206 10.2. On Strong o-Convergence 208 10.3. o-Regular Dual Summability Methods 213 10.3.1. Dual summability methods 214 10.3.2. o-Regular summability methods 214 10.4. Some New Sequence Spaces 217 10.5. Some New Sequence Spaces Defined by Modulus 224 10.6. Matrix Transformations 231 Bibliography 235 11. Caratheodory Theory of Dynamic Equations on Time Scales 239 11.1. Introduction and Preliminaries 239 11.2. Caratheodory Solutions 243 11.3. Generalized Dynamic Equations 251 11.3.1. Henstock-Kurzweil
-integral 251 11.3.2. Existence and uniqueness of solutions 253 11.4. Dependency and Convergence of Solutions 259 Bibliography 265 12. Vector Valued Ideal Convergent Generalized Difference Sequence Spaces Associated with Multiplier Sequences 267 12.1. Introduction 267 12.2. Definitions and Preliminaries 268 12.2.1. Difference sequence spaces 269 12.2.2. Matrix transformation between sequence spaces 270 12.2.3. Vector valued sequence spaces 270 12.3. Ideal Convergence of Sequences 271 12.3.1. Statistically convergent sequence space 272 12.4. Sequence Spaces Associated with the Multiplier Sequences 273 12.4.1. Relation with real-life problems 274 12.4.2. Advantages 274 12.4.3. Vector valued generalized difference ideal convergent sequence spaces associated with the multiplier sequences 274 12.5. Main Results 275 12.6. Conclusion 278 Bibliography 279 13. Domain of Generalized Riesz Difference Operator of Fractional Order in Maddox's Space l(p) and Certain Geometric Properties 283 13.1. Introduction 283 13.2. Paranormed Riesz Di erence Sequence Space rt(p,
Bq) of Fractional Order 286 13.3. The a-, b- and y-Duals 290 13.4. Matrix Transformations 293 13.5. Certain Geometric Properties 294 Bibliography 298 Index 304 Numerical;,Functional;,Analysis;,Optimization;,Mathematical,Methods;,Inequalities Numerical,Functional,Analysis,Optimization,Mathematical Methods,Inequalities "The book features original chapters on sequence spaces involving the idea of ideal convergence, modulus function, multiplier sequences, Riesz mean, Fibonacci difference matrix etc., and illustrate their involvement in various applications. The preliminaries have been presented in the beginning of each chapter and then the advanced discussion takes place, so it is useful for both expert and nonexpert on aforesaid topics. The book consists of original thirteen research chapters contributed by the well-recognized researchers in the field of sequence spaces with associated applications"-- Provided by publisher