Mathematical Analysis and Applications: Selected Topics

An authoritative text that presents the current problems, theories, and applications of mathematical analysis research Mathematical Analysis and Applications: Selected Topics offers the theories, methods, and applications of a variety of targeted topics including: operator theory, approximation theory, fixed point theory, stability theory, minimization problems, many–body wave scattering problems, Basel problem, Corona problem, inequalities, generalized normed spaces, variations of functions and sequences, analytic generalizations of the Catalan, Fuss, and Fuss Catalan Numbers, asymptotically developable functions, convex functions, Gaussian processes, image analysis, and spectral analysis and spectral synthesis. The authors a noted team of international researchers in the field highlight the basic developments for each topic presented and explore the most recent advances made in their area of study. The text is presented in such a way that enables the reader to follow subsequent studies in a burgeoning field of research. This important text: Presents a wide–range of important topics having current research importance and interdisciplinary applications such as game theory, image processing, creation of materials with a desired refraction coefficient, etc. Contains chapters written by a group of esteemed researchers in mathematical analysis Includes problems and research questions in order to enhance understanding of the information provided Offers references that help readers advance to further study Written for researchers, graduate students, educators, and practitioners with an interest in mathematical analysis, Mathematical Analysis and Applications: Selected Topics includes the most recent research from a range of mathematical fields. Michael Ruzhansky, Ph.D., is Professor in the Department of Mathematics at Imperial College London, UK. Dr. Ruzhansky was awarded the Ferran Sunyer I Balaguer Prize in 2014. Hemen Dutta, Ph.D., is Senior Assistant Professor of Mathematics at Gauhati University, India. Ravi P. Agarwal, Ph.D., is Professor and Chair of the Department of Mathematics at Texas A&M University–Kingsville, Kingsville, USA. INDICE: Preface xv .About the Editors xxi .List of Contributors xxiii .1 Spaces of Asymptotically Developable Functions and Applications 1Sergio Alejandro Carrillo Torres and Jorge Mozo Fernández .1.1 Introduction and Some Notations 1 .1.2 Strong Asymptotic Expansions 2 .1.3 Monomial Asymptotic Expansions 7 .1.4 Monomial Summability for Singularly Perturbed Differential Equations 13 .1.5 Pfaffian Systems 15 .References 19 .2 Duality for Gaussian Processes from Random Signed Measures 23Palle E.T. Jorgensen and Feng Tian .2.1 Introduction 23 .2.2 Reproducing Kernel Hilbert Spaces (RKHSs) in the Measurable .Category 24 .2.3 Applications to Gaussian Processes 30 .2.4 Choice of Probability Space 34 .2.5 A Duality 37 .2.A Stochastic Processes 40 .2.B Overview of Applications of RKHSs 45 .Acknowledgments 50 .References 51 .3 Many–BodyWave Scattering Problems for Small Scatterers and CreatingMaterials with a Desired Refraction Coefficient 57Alexander G. Ramm .3.1 Introduction 57 .3.2 Derivation of the Formulas for One–BodyWave Scattering Problems 62 .3.3 Many–Body Scattering Problem 65 .3.3.1 The Case of Acoustically Soft Particles 68 .3.3.2 Wave Scattering by Many Impedance Particles 70 .3.4 Creating Materials with a Desired Refraction Coefficient 71 .3.5 Scattering by Small Particles Embedded in an Inhomogeneous Medium 72 .3.6 Conclusions 72 .References 73 .4 Generalized Convex Functions and their Applications 77Adem Kiliçman andWedad Saleh .4.1 Brief Introduction 77 .4.2 Generalized E–Convex Functions 78 .4.3 E –Epigraph 84 .4.4 Generalized s–Convex Functions 85 .4.5 Applications to Special Means 96 .References 98 .5 Some Properties and Generalizations of the Catalan, Fuss, and Fuss Catalan Numbers 101Feng Qi and Bai–Ni Guo .5.1 The Catalan Numbers 101 .5.1.1 A Definition of the Catalan Numbers 101 .5.1.2 The History of the Catalan Numbers 101 .5.1.3 A Generating Function of the Catalan Numbers 102 .5.1.4 Some Expressions of the Catalan Numbers 102 .5.1.5 Integral Representations of the Catalan Numbers 103 .5.1.6 Asymptotic Expansions of the Catalan Function 104 .5.1.7 Complete Monotonicity of the Catalan Numbers 105 .5.1.8 Inequalities of the Catalan Numbers and Function 106 .5.1.9 The Bell Polynomials of the Second Kind and the Bessel Polynomials 109 .5.2 The Catalan Qi Function 111 .5.2.1 The Fuss Numbers 111 .5.2.2 A Definition of the Catalan Qi Function 111 .5.2.3 Some Identities of the Catalan Qi Function 112 .5.2.4 Integral Representations of the Catalan Qi Function 114 .5.2.5 Asymptotic Expansions of the Catalan Qi Function 115 .5.2.6 Complete Monotonicity of the Catalan Qi Function 116 .5.2.7 Schur–Convexity of the Catalan Qi Function 118 .5.2.8 Generating Functions of the Catalan Qi Numbers 118 .5.2.9 A Double Inequality of the Catalan Qi Function 118 .5.2.10 The q–Catalan Qi Numbers and Properties 119 .5.2.11 The Catalan Numbers and the k–Gamma and k–Beta Functions 119 .5.2.12 Series Identities Involving the Catalan Numbers 119 .5.3 The Fuss Catalan Numbers 119 .5.3.1 A Definition of the Fuss Catalan Numbers 119 .5.3.2 A Product–Ratio Expression of the Fuss Catalan Numbers 120 .5.3.3 Complete Monotonicity of the Fuss Catalan Numbers 120 .5.3.4 A Double Inequality for the Fuss Catalan Numbers 121 .5.4 The Fuss Catalan Qi Function 121 .5.4.1 A Definition of the Fuss Catalan Qi Function 121 .5.4.2 A Product–Ratio Expression of the Fuss Catalan Qi Function 122 .5.4.3 Integral Representations of the Fuss Catalan Qi Function 123 .5.4.4 Complete Monotonicity of the Fuss Catalan Qi Function 124 .5.5 Some Properties for Ratios of Two Gamma Functions 124 .5.5.1 An Integral Representation and Complete Monotonicity 125 .5.5.2 An Exponential Expansion for the Ratio of Two Gamma Functions 125 .5.5.3 A Double Inequality for the Ratio of Two Gamma Functions 125 .5.6 Some NewResults on the Catalan Numbers 126 .5.7 Open Problems 126 .Acknowledgments 127 .References 127 .6 Trace Inequalities of Jensen Type for Self–adjoint Operators in Hilbert Spaces: A Survey of Recent Results 135Silvestru Sever Dragomir .6.1 Introduction 135 .6.1.1 Jensen s Inequality 135 .6.1.2 Traces for Operators in Hilbert Spaces 138 .6.2 Jensen s Type Trace Inequalities 141 .6.2.1 Some Trace Inequalities for Convex Functions 141 .6.2.2 Some Functional Properties 145 .6.2.3 Some Examples 151 .6.2.4 More Inequalities for Convex Functions 154 .6.3 Reverses of Jensen s Trace Inequality 157 .6.3.1 A Reverse of Jensen s Inequality 157 .6.3.2 Some Examples 163 .6.3.3 Further Reverse Inequalities for Convex Functions 165 .6.3.4 Some Examples 169 .6.3.5 Reverses of Hölder s Inequality 174 .6.4 Slater s Type Trace Inequalities 177 .6.4.1 Slater s Type Inequalities 177 .6.4.2 Further Reverses 180 .References 188 .7 Spectral Synthesis and Its Applications 193László Székelyhidi .7.1 Introduction 193 .7.2 Basic Concepts and Function Classes 195 .7.3 Discrete Spectral Synthesis 203 .7.4 Nondiscrete Spectral Synthesis 217 .7.5 Spherical Spectral Synthesis 219 .7.6 Spectral Synthesis on Hypergroups 238 .7.7 Applications 248 .Acknowledgments 252 .References 252 .8 Various Ulam Hyers Stabilities of Euler Lagrange Jensen General (a, b; k = a + b)–Sextic Functional Equations 255JohnMichael Rassias and Narasimman Pasupathi .8.1 Brief Introduction 255 .8.2 General Solution of Euler Lagrange Jensen General (a, b; k = a + b)–Sextic Functional Equation 257 .8.3 Stability Results in Banach Space 258 .8.3.1 Banach Space: Direct Method 258 .8.3.2 Banach Space: Fixed Point Method 261 .8.4 Stability Results in Felbin s Type Spaces 267 .8.4.1 Felbin s Type Spaces: Direct Method 268 .8.4.2 Felbin s Type Spaces: Fixed Point Method 269 .8.5 Intuitionistic Fuzzy Normed Space: Stability Results 270 .8.5.1 IFNS: Direct Method 272 .8.5.2 IFNS: Fixed Point Method 279 .References 281 .9 A Note on the Split Common Fixed Point Problem and its Variant Forms 283A. K l çman and L.B. Mohammed .9.1 Introduction 283 .9.2 Basic Concepts and Definitions 284 .9.2.1 Introduction 284 .9.2.2 Vector Spaces 284 .9.2.3 Hilbert Space and Its Properties 286 .9.2.4 Bounded Linear Map and Its Properties 288 .9.2.5 Some Nonlinear Operators 289 .9.2.6 Problem Formulation 294 .9.2.7 Preliminary Results 294 .9.2.8 Strong Convergence for the Split Common Fixed–Point Problems for Total Quasi–Asymptotically Nonexpansive Mappings 296 .9.2.9 Strong Convergence for the Split Common Fixed–Point Problems for Demicontractive Mappings 302 .9.2.10 Application to Variational Inequality Problems 306 .9.2.11 On Synchronal Algorithms for Fixed and Variational Inequality Problems in Hilbert Spaces 307 .9.2.12 Preliminaries 307 .9.3 A Note on the Split Equality Fixed–Point Problems in Hilbert Spaces 315 .9.3.1 Problem Formulation 315 .9.3.2 Preliminaries 316 .9.3.3 The Split Feasibility and Fixed–Point Equality Problems for Quasi–Nonexpansive Mappings in Hilbert Spaces 316 .9.3.4 The Split Common Fixed–Point Equality Problems for Quasi–Nonexpansive Mappings in Hilbert Spaces 320 .9.4 Numerical Example 322 .9.5 The Split Feasibility and Fixed Point Problems for Quasi–Nonexpansive Mappings in Hilbert Spaces 328 .9.5.1 Problem Formulation 328 .9.5.2 Preliminary Results 328 .9.6 Ishikawa–Type Extra–Gradient IterativeMethods for Quasi–Nonexpansive Mappings in Hilbert Spaces 329 .9.6.1 Application to Split Feasibility Problems 334 .9.7 Conclusion 336 .References 337 .10 Stabilities and Instabilities of Rational Functional Equations and Euler Lagrange Jensen (a, b)–Sextic Functional Equations 341John M. Rassias, Krishnan Ravi and Beri V. Senthil Kumar .10.1 Introduction 341 .10.1.1 Growth of Functional Equations 342 .10.1.2 Importance of Functional Equations 342 .10.1.3 Functional Equations Relevant to Other Fields 343 .10.1.4 Definition of Functional Equation with Examples 343 .10.2 Ulam Stability Problem for Functional Equation 344 .10.2.1 –Stability of Functional Equation 344 .10.2.2 Stability Bounded by Sum of Powers of Norms 345 .10.2.3 Stability Bounded by Product of Powers of Norms 346 .10.2.4 Stability Bounded by a General Control Function 347 .10.2.5 Stability Bounded by Mixed Product Sum of Powers of Norms 347 .10.2.6 Application of Ulam Stability Theory 348 .10.3 Various Forms of Functional Equations 348 .10.4 Preliminaries 353 .10.5 Rational Functional Equations 355 .10.5.1 Reciprocal Type Functional Equation 355 .10.5.2 Solution of Reciprocal Type Functional Equation 356 .10.5.3 Generalized Hyers Ulam Stability of Equation 357 .10.5.4 Counter–Example 360 .10.5.5 Geometrical Interpretation of Equation 362 .10.5.6 An Application of Equation to Electric Circuits 364 .10.5.7 Reciprocal–Quadratic Functional Equation 364 .10.5.8 General Solution of Reciprocal–Quadratic Functional Equation 366 .10.5.9 Generalized Hyers Ulam Stability of Reciprocal–Quadratic Functional Equation 368 .10.5.10 Counter–Examples 373 .10.5.11 Reciprocal–Cubic and Reciprocal–Quartic Functional Equations 375 .10.5.12 Hyers Ulam Stability of Equation 375 .10.5.13 Counter–Examples 380 .10.6 Euler–Lagrange Jensen (a, b; k = a + b)–Sextic Functional Equations 384 .10.6.1 Generalized Ulam Hyers Stability of Equation Using Fixed Point Method 384 .10.6.2 Counter–Example 387 .10.6.3 Generalized Ulam Hyers Stability of Equation Using Direct Method 389 .References 395 .11 Attractor of the Generalized Contractive Iterated Function System 401Mujahid Abbas and Talat Nazir .11.1 Iterated Function System 401 .11.2 Generalized F–contractive Iterated Function System 407 .11.3 Iterated Function System in b–Metric Space 414 .11.4 Generalized F–Contractive Iterated Function System in b–Metric Space 420 .References 426 .12 Regular and Rapid Variations and Some Applications 429Ljubisa D.R. Kocinac, Dragan Djurcic, Jelena V. Manojlovic .12.1 Introduction and Historical Background 429 .12.2 Regular Variation 431 .12.2.1 The Class Tr(RVs) 432 .12.2.2 Classes of Sequences Related to Tr(RVs) 434 .12.2.3 The Class ORVs and Seneta Sequences 436 .12.3 Rapid Variation 437 .12.3.1 Some Properties of Rapidly Varying Functions 438 .12.3.2 The Class ARVs 440 .12.3.3 The Class KRs, 442 .12.3.4 The Class Tr(Rs, ) 447 .12.3.5 Subclasses of Tr(Rs, ) 448 .12.3.6 The Class s 451 .12.4 Applications to Selection Principles 453 .12.4.1 First Results 455 .12.4.2 Improvements 455 .12.4.3 When ONE has aWinning Strategy? 460 .12.5 Applications to Differential Equations 463 .12.5.1 The Existence of all Solutions of (A) 464 .12.5.2 Superlinear Thomas Fermi Equation (A) 466 .12.5.3 Sublinear Thomas Fermi Equation (A) 470 .12.5.4 A Generalization 480 .References 486 .13 n–Inner Products, n–Norms, and Angles Between Two Subspaces 493Hendra Gunawan .13.1 Introduction 493 .13.2 n–Inner Product Spaces and n–Normed Spaces 495 .13.2.1 Topology in n–Normed Spaces 499 .13.3 Orthogonality in n–Normed Spaces 500 .13.3.1 G–, P–, I–, and BJ– Orthogonality 503 .13.3.2 Remarks on the n–Dimensional Case 505 .13.4 Angles Between Two Subspaces 505 .13.4.1 An Explicit Formula 509 .13.4.2 A More General Formula 511 .References 513 .14 Proximal Fiber Bundles on Nerve Complexes 517James F. Peters .14.1 Brief Introduction 517 .14.2 Preliminaries 518 .14.2.1 Nerve Complexes and Nerve Spokes 518 .14.2.2 Descriptions and Proximities 521 .14.2.3 Descriptive Proximities 523 .14.3 Sewing Regions Together 527 .14.3.1 Sewing Nerves Together with Spokes to Construct a Nervous System .14.4 Some Results for Fiber Bundles 530 .14.5 Concluding Remarks 534 .References 534 .15 Approximation by Generalizations of Hybrid Baskakov Type Operators Preserving Exponential Functions 537Vijay Gupta .15.1 Introduction 537 .15.2 Baskakov Szász Operators 539 .15.3 Genuine Baskakov Szász Operators 542 .15.4 Preservation of eAx 545 .15.5 Conclusion 549 .References 550 .16 Well–PosedMinimization Problems via the Theory of Measures of Noncompactness 553 .Józef Bana´s and Tomasz Zaja c .16.1 Introduction 553 .16.2 Minimization Problems and TheirWell–Posedness in the Classical Sense 554 .16.3 Measures of Noncompactness 556 .16.4 Well–Posed Minimization Problems with Respect to Measures of Noncompactness 565 .16.5 Minimization Problems for Functionals Defined in Banach Sequence Spaces 568 .16.6 Minimization Problems for Functionals Defined in the Classical Space C([a, b])) 576 .16.7 Minimization Problems for Functionals Defined in the Space of Functions Continuous and Bounded on the Real Half–Axis 580 .References 584 .17 Some Recent Developments on Fixed Point Theory in GeneralizedMetric Spaces 587Poom Kumam and Somayya Komal .17.1 Brief Introduction 587 .17.2 Some Basic Notions and Notations 593 .17.3 Fixed Points Theorems 596 .17.3.1 Fixed Points Theorems for Monotonic and Nonmonotonic Mappings 597 .17.3.2 PPF–Dependent Fixed–Point Theorems 600 .17.3.3 Fixed Points Results in b–Metric Spaces 602 .17.3.4 The generalized Ulam Hyers Stability in b–Metric Spaces 604 .17.3.5 Well–Posedness of a Function with Respect to –Admissibility in b–Metric Spaces 605 .17.3.6 Fixed Points for F–Contraction 606 .17.4 Common Fixed Points Theorems 608 .17.4.1 Common Fixed–Point Theorems for Pair ofWeakly Compatible Mappings in Fuzzy Metric Spaces 609 .17.5 Best Proximity Points 611 .17.6 Common Best Proximity Points 614 .17.7 Tripled Best Proximity Points 617 .17.8 FutureWorks 624 .References 624 .18 The Basel Problem with an Extension 631Anthony Sofo .18.1 The Basel Problem 631 .18.2 An Euler Type Sum 640 .18.3 The Main Theorem 645 .18.4 Conclusion 652 .References 652 .19 Coupled Fixed Points and Coupled Coincidence Points via Fixed Point Theory 661Adrian Petru¸sel and Gabriela Petru¸sel .19.1 Introduction and Preliminaries 661 .19.2 Fixed Point Results 665 .19.2.1 The Single–Valued Case 665 .19.2.2 The Multi–Valued Case 673 .19.3 Coupled Fixed Point Results 680 .19.3.1 The Single–Valued Case 680 .19.3.2 The Multi–Valued Case 686 .19.4 Coincidence Point Results 689 .19.5 Coupled Coincidence Results 699 .References 704 .20 The Corona Problem, Carleson Measures, and Applications 709Alberto Saracco .20.1 The Corona Problem 709 .20.1.1 Banach Algebras: Spectrum 709 .20.1.2 Banach Algebras: Maximal Spectrum 710 .20.1.3 The Algebra of Bounded Holomorphic Functions and the Corona Problem 710 .20.2 Carleson s Proof and Carleson Measures 711 .20.2.1 Wolff s Proof 712 .20.3 The Corona Problem in Higher Henerality 712 .20.3.1 The Corona Problem in 712 .20.3.2 The Corona Problem in Riemann Surfaces: A Positive and a Negative Result 713 .20.3.3 The Corona Problem in Domains of n 714 .20.3.4 The Corona Problem for Quaternionic Slice–Regular Functions 715 .20.3.4.1 Slice–Regular Functions f D 715 .20.3.4.2 The CoronaTheorem in the Quaternions 717 .20.4 Results on Carleson Measures 718 .20.4.1 Carleson Measures of Hardy Spaces of the Disk 718 .20.4.2 Carleson Measures of Bergman Spaces of the Disk 719 .20.4.3 Carleson Measures in the Unit Ball of n 720 .20.4.4 Carleson Measures in Strongly Pseudoconvex Bounded Domains of n 722 .20.4.5 Generalizations of Carleson Measures and Applications to Toeplitz Operators 723 .20.4.6 Explicit Examples of Carleson Measures of Bergman Spaces 724 .20.4.7 Carleson Measures in the Quaternionic Setting 725 .20.4.7.1 Carleson Measures on Hardy Spaces of 725 .20.4.7.2 Carleson Measures on Bergman Spaces of 726 .References 728 .Index 731

• ISBN: 978-1-119-41434-6
• Editorial: Wiley–Blackwell
• Encuadernacion: Cartoné
• Páginas: 768
• Fecha Publicación: 25/05/2018
• Nº Volúmenes: 1
• Idioma: Inglés