Mathematical Analysis and Applications

Name: Mathematical Analysis and Applications | Selected Topics
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Selected Topics

Michael Ruzhansky Hemen Dutta Ravi P. Agarwal(Herausgeber*in)

Wiley-Scrivener (Verlag)

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Erschienen am 5. April 2018

768 Seiten

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Inhalt

Cover
Title Page
Copyright
Contents
Preface
About the Editors
List of Contributors
Chapter 1 Spaces of Asymptotically Developable Functions and Applications
1.1 Introduction and Some Notations
1.2 Strong Asymptotic Expansions
1.3 Monomial Asymptotic Expansions
1.4 Monomial Summability for Singularly Perturbed Differential Equations
1.5 Pfaffian Systems
References
Chapter 2 Duality for Gaussian Processes from Random Signed Measures
2.1 Introduction
2.2 Reproducing Kernel Hilbert Spaces (RKHSs) in the Measurable Category
2.3 Applications to Gaussian Processes
2.4 Choice of Probability Space
2.5 A Duality
2.A Stochastic Processes
2.B Overview of Applications of RKHSs
Acknowledgments
References
Chapter 3 Many-Body Wave Scattering Problems for Small Scatterers and Creating Materials with a Desired Refraction Coefficient
3.1 Introduction
3.2 Derivation of the Formulas for One-Body Wave Scattering Problems
3.3 Many-Body Scattering Problem
3.3.1 The Case of Acoustically Soft Particles
3.3.2 Wave Scattering by Many Impedance Particles
3.4 Creating Materials with a Desired Refraction Coefficient
3.5 Scattering by Small Particles Embedded in an Inhomogeneous Medium
3.6 Conclusions
References
Chapter 4 Generalized Convex Functions and their Applications
4.1 Brief Introduction
4.2 Generalized E-Convex Functions
4.3 Ea- Epigraph
4.4 Generalized s-Convex Functions
4.5 Applications to Special Means
References
Chapter 5 Some Properties and Generalizations of the Catalan, Fuss, and Fuss-Catalan Numbers
5.1 The Catalan Numbers
5.1.1 A Definition of the Catalan Numbers
5.1.2 The History of the Catalan Numbers
5.1.3 A Generating Function of the Catalan Numbers
5.1.4 Some Expressions of the Catalan Numbers
5.1.5 Integral Representations of the Catalan Numbers
5.1.6 Asymptotic Expansions of the Catalan Function
5.1.7 Complete Monotonicity of the Catalan Numbers
5.1.8 Inequalities of the Catalan Numbers and Function
5.1.9 The Bell Polynomials of the Second Kind and the Bessel Polynomials
5.2 The Catalan-Qi Function
5.2.1 The Fuss Numbers
5.2.2 A Definition of the Catalan-Qi Function
5.2.3 Some Identities of the Catalan-Qi Function
5.2.4 Integral Representations of the Catalan-Qi Function
5.2.5 Asymptotic Expansions of the Catalan-Qi Function
5.2.6 Complete Monotonicity of the Catalan-Qi Function
5.2.7 Schur-Convexity of the Catalan-Qi Function
5.2.8 Generating Functions of the Catalan-Qi Numbers
5.2.9 A Double Inequality of the Catalan-Qi Function
5.2.10 The q-Catalan-Qi Numbers and Properties
5.2.11 The Catalan Numbers and the k-Gamma and k-Beta Functions
5.2.12 Series Identities Involving the Catalan Numbers
5.3 The Fuss-Catalan Numbers
5.3.1 A Definition of the Fuss-Catalan Numbers
5.3.2 A Product-Ratio Expression of the Fuss-Catalan Numbers
5.3.3 Complete Monotonicity of the Fuss-Catalan Numbers
5.3.4 A Double Inequality for the Fuss-Catalan Numbers
5.4 The Fuss-Catalan-Qi Function
5.4.1 A Definition of the Fuss-Catalan-Qi Function
5.4.2 A Product-Ratio Expression of the Fuss-Catalan-Qi Function
5.4.3 Integral Representations of the Fuss-Catalan-Qi Function
5.4.4 Complete Monotonicity of the Fuss-Catalan-Qi Function
5.5 Some Properties for Ratios of Two Gamma Functions
5.5.1 An Integral Representation and Complete Monotonicity
5.5.2 An Exponential Expansion for the Ratio of Two Gamma Functions
5.5.3 A Double Inequality for the Ratio of Two Gamma Functions
5.6 Some New Results on the Catalan Numbers
5.7 Open Problems
Acknowledgments
References
Chapter 6 Trace Inequalities of Jensen Type for Self-adjoint Operators in Hilbert Spaces: A Survey of Recent Results
6.1 Introduction
6.1.1 Jensen's Inequality
6.1.2 Traces for Operators in Hilbert Spaces
6.2 Jensen's Type Trace Inequalities
6.2.1 Some Trace Inequalities for Convex Functions
6.2.2 Some Functional Properties
6.2.3 Some Examples
6.2.4 More Inequalities for Convex Functions
6.3 Reverses of Jensen's Trace Inequality
6.3.1 A Reverse of Jensen's Inequality
6.3.2 Some Examples
6.3.3 Further Reverse Inequalities for Convex Functions
6.3.4 Some Examples
6.3.5 Reverses of Hölder's Inequality
6.4 Slater's Type Trace Inequalities
6.4.1 Slater's Type Inequalities
6.4.2 Further Reverses
References
Chapter 7 Spectral Synthesis and Its Applications
7.1 Introduction
7.2 Basic Concepts and Function Classes
7.3 Discrete Spectral Synthesis
7.4 Nondiscrete Spectral Synthesis
7.5 Spherical Spectral Synthesis
7.6 Spectral Synthesis on Hypergroups
7.7 Applications
Acknowledgments
References
Chapter 8 Various Ulam-Hyers Stabilities of Euler-Lagrange-Jensen General (a,b
k=a+b)-Sextic Functional Equations
8.1 Brief Introduction
8.2 General Solution of Euler-Lagrange-Jensen General (a,b
k=a+b)-Sextic Functional Equation
8.3 Stability Results in Banach Space
8.3.1 Banach Space: Direct Method
8.3.2 Banach Space: Fixed Point Method
8.4 Stability Results in Felbin's Type Spaces
8.4.1 Felbin's Type Spaces: Direct Method
8.4.2 Felbin's Type Spaces: Fixed Point Method
8.5 Intuitionistic Fuzzy Normed Space: Stability Results
8.5.1 IFNS: Direct Method
8.5.2 IFNS: Fixed Point Method
References
Chapter 9 A Note on the Split Common Fixed Point Problem and its Variant Forms
9.1 Introduction
9.2 Basic Concepts and Definitions
9.2.1 Introduction
9.2.2 Vector Space
9.2.3 Hilbert Space and its Properties
9.2.4 Bounded Linear Map and its Properties
9.2.5 Some Nonlinear Operators
9.2.6 Problem Formulation
9.2.7 Preliminary Results
9.2.8 Strong Convergence for the Split Common Fixed-Point Problems for Total Quasi-Asymptotically Nonexpansive Mappings
9.2.9 Strong Convergence for the Split Common Fixed-Point Problems for Demicontractive Mappings
9.2.10 Application to Variational Inequality Problems
9.2.11 On Synchronal Algorithms for Fixed and Variational Inequality Problems in Hilbert Spaces
9.2.12 Preliminaries
9.3 A Note on the Split Equality Fixed-Point Problems in Hilbert Spaces
9.3.1 Problem Formulation
9.3.2 Preliminaries
9.3.3 The Split Feasibility and Fixed-Point Equality Problems for Quasi-Nonexpansive Mappings in Hilbert Spaces
9.3.4 The Split Common Fixed-Point Equality Problems for Quasi-Nonexpansive Mappings in Hilbert Spaces
9.4 Numerical Example
9.5 The Split Feasibility and Fixed Point Problems for Quasi-Nonexpansive Mappings in Hilbert Spaces
9.5.1 Problem Formulation
9.5.2 Preliminary Results
9.6 Ishikawa-Type Extra-Gradient Iterative Methods for Quasi-Nonexpansive Mappings in Hilbert Spaces
9.6.1 Application to Split Feasibility Problems
9.7 Conclusion
References
Chapter 10 Stabilities and Instabilities of Rational Functional Equations and Euler-Lagrange-Jensen (a,b)-Sextic Functional Equations
10.1 Introduction
10.1.1 Growth of Functional Equations
10.1.2 Importance of Functional Equations
10.1.3 Functional Equations Relevant to Other Fields
10.1.4 Definition of Functional Equation with Examples
10.2 Ulam Stability Problem for Functional Equation
10.2.1 ?-Stability of Functional Equation
10.2.2 Stability Involving Sum of Powers of Norms
10.2.3 Stability Involving Product of Powers of Norms
10.2.4 Stability Involving a General Control Function
10.2.5 Stability Involving Mixed Product-Sum of Powers of Norms
10.2.6 Application of Ulam Stability Theory
10.3 Various Forms of Functional Equations
10.4 Preliminaries
10.5 Rational Functional Equations
10.5.1 Reciprocal Type Functional Equation
10.5.2 Solution of Reciprocal Type Functional Equation
10.5.3 Generalized Hyers-Ulam Stability of Reciprocal Type Functional Equation
10.5.4 Counter-Example
10.5.5 Geometrical Interpretation of Reciprocal Type Functional Equation
10.5.6 An Application of Equation (10.41) to Electric Circuits
10.5.7 Reciprocal-Quadratic Functional Equation
10.5.8 General Solution of Reciprocal-Quadratic Functional Equation
10.5.9 Generalized Hyers-Ulam Stability of Reciprocal-Quadratic Functional Equations
10.5.10 Counter-Examples
10.5.11 Reciprocal-Cubic and Reciprocal-Quartic Functional Equations
10.5.12 Hyers-Ulam Stability of Reciprocal-Cubic and Reciprocal-Quartic Functional Equations
10.5.13 Counter-Examples
10.6 Euler-Lagrange-Jensen (a,b
k=a+b)-Sextic Functional Equations
10.6.1 Generalized Ulam-Hyers Stability of Euler-Lagrange-Jensen Sextic Functional Equation Using Fixed Point Method
10.6.2 Counter-Example
10.6.3 Generalized Ulam-Hyers Stability of Euler-Lagrange-Jensen Sextic Functional Equation Using Direct Method
References
Chapter 11 Attractor of the Generalized Contractive Iterated Function System
11.1 Iterated Function System
11.2 Generalized F-contractive Iterated Function System
11.3 Iterated Function System in b-Metric Space
11.4 Generalized F-Contractive Iterated Function System in b-Metric Space
References
Chapter 12 Regular and Rapid Variations and Some Applications
12.1 Introduction and Historical Background
12.2 Regular Variation
12.2.1 The Class Tr(RVs)
12.2.2 Classes of Sequences Related to Tr(RVs)
12.2.3 The Class ORVs and Seneta Sequences
12.3 Rapid Variation
12.3.1 Some Properties of Rapidly Varying Functions
12.3.2 The Class ARVs
12.3.3 The Class KRs,8
12.3.4 The Class Tr(Rs,8)
12.3.5 Subclasses of Tr(Rs,8)
12.3.6 The Class s
12.4 Applications to Selection Principles
12.4.1 First Results
12.4.2 Improvements
12.4.3 When ONE has a Winning Strategy?
12.5 Applications to Differential Equations
12.5.1 The Existence of all Solutions of (A)
12.5.2 Superlinear Thomas-Fermi Equation (A)
12.5.3 Sublinear Thomas-Fermi Equation (A)
12.5.4 A Generalization
References
Chapter 13 n-Inner Products, n-Norms, and Angles Between Two Subspaces
13.1 Introduction
13.2 n-Inner Product Spaces and n-Normed Spaces
13.2.1 Topology in n-Normed Spaces
13.3 Orthogonality in n-Normed Spaces
13.3.1 G-, P-, I-, and BJ- Orthogonality
13.3.2 Remarks on the n-Dimensional Case
13.4 Angles Between Two Subspaces
13.4.1 An Explicit Formula
13.4.2 A More General Formula
References
Chapter 14 Proximal Fiber Bundles on Nerve Complexes
14.1 Brief Introduction
14.2 Preliminaries
14.2.1 Nerve Complexes and Nerve Spokes
14.2.2 Descriptions and Proximities
14.2.3 Descriptive Proximities
14.3 Sewing Regions Together
14.3.1 Sewing Nerves Together with Spokes to Construct a Nervous System Complex
14.4 Some Results for Fiber Bundles
14.5 Concluding Remarks
References
Chapter 15 Approximation by Generalizations of Hybrid Baskakov Type Operators Preserving Exponential Functions
15.1 Introduction
15.2 Baskakov-Szász Operators
15.3 Genuine Baskakov-Szász Operators
15.4 Preservation of eAx
15.5 Conclusion
References
Chapter 16 Well-Posed Minimization Problems via the Theory of Measures of Noncompactness
16.1 Introduction
16.2 Minimization Problems and Their Well-Posedness in the Classical Sense
16.3 Measures of Noncompactness
16.4 Well-Posed Minimization Problems with Respect to Measures of Noncompactness
16.5 Minimization Problems for Functionals Defined in Banach Sequence Spaces
16.6 Minimization Problems for Functionals Defined in the Classical Space C([a,b])
16.7 Minimization Problems for Functionals Defined in the Space of Functions Continuous and Bounded on the Real Half-Axis
References
Chapter 17 Some Recent Developments on Fixed Point Theory in Generalized Metric Spaces
17.1 Brief Introduction
17.2 Some Basic Notions and Notations
17.3 Fixed Points Theorems
17.3.1 Fixed Points Theorems for Monotonic and Nonmonotonic Mappings
17.3.2 PPF-Dependent Fixed-Point Theorems
17.3.3 Fixed Points Results in b-Metric Spaces
17.3.4 The generalized Ulam-Hyers Stability in b-Metric Spaces
17.3.5 Well-Posedness of a Function with Respect to a-Admissibility in b-Metric Spaces
17.3.6 Fixed Points for F-Contraction
17.4 Common Fixed Points Theorems
17.4.1 Common Fixed-Point Theorems for Pair of Weakly Compatible Mappings in Fuzzy Metric Spaces
17.5 Best Proximity Points
17.6 Common Best Proximity Points
17.7 Tripled Best Proximity Points
17.8 Future Works
References
Chapter 18 The Basel Problem with an Extension
18.1 The Basel Problem
18.2 An Euler Type Sum
18.3 The Main Theorem
18.4 Conclusion
References
Chapter 19 Coupled Fixed Points and Coupled Coincidence Points via Fixed Point Theory
19.1 Introduction and Preliminaries
19.2 Fixed Point Results
19.2.1 The Single-Valued Case
19.2.2 The Multi-Valued Case
19.3 Coupled Fixed Point Results
19.3.1 The Single-Valued Case
19.3.2 The Multi-Valued Case
19.4 Coincidence Point Results
19.5 Coupled Coincidence Results
References
Chapter 20 The Corona Problem, Carleson Measures, and Applications
20.1 The Corona Problem
20.1.1 Banach Algebras: Spectrum
20.1.2 Banach Algebras: Maximal Spectrum
20.1.3 The Algebra of Bounded Holomorphic Functions and the Corona Problem
20.2 Carleson's Proof and Carleson Measures
20.2.1 Wolff's Proof
20.3 The Corona Problem in Higher Henerality
20.3.1 The Corona Problem in C
20.3.2 The Corona Problem in Riemann Surfaces: A Positive and a Negative Result
20.3.3 The Corona Problem in Domains of Cn
20.3.4 The Corona Problem for Quaternionic Slice-Regular Functions
20.3.4.1 Slice-Regular Functions f:DH
20.3.4.2 The Corona Theorem in the Quaternions
20.4 Results on Carleson Measures
20.4.1 Carleson Measures of Hardy Spaces of the Disk
20.4.2 Carleson Measures of Bergman Spaces of the Disk
20.4.3 Carleson Measures in the Unit Ball of Cn
20.4.4 Carleson Measures in Strongly Pseudoconvex Bounded Domains of Cn
20.4.5 Generalizations of Carleson Measures and Applications to Toeplitz Operators
20.4.6 Explicit Examples of Carleson Measures of Bergman Spaces
20.4.7 Carleson Measures in the Quaternionic Setting
20.4.7.1 Carleson Measures on Hardy Spaces of B?H
20.4.7.2 Carleson Measures on Bergman Spaces of B?H
References
Index
EULA

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