Mathematical Analysis and Applications

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.

• Preface xvAbout the Editors xxiList of Contributors xxiii1 Spaces of Asymptotically Developable Functions and Applications 1Sergio Alejandro Carrillo Torres and Jorge Mozo Fernández1.1 Introduction and Some Notations 11.2 Strong Asymptotic Expansions 21.3 Monomial Asymptotic Expansions 71.4 Monomial Summability for Singularly Perturbed Differential Equations 131.5 Pfaffian Systems 15References 192 Duality for Gaussian Processes from Random Signed Measures 23Palle E.T. Jorgensen and Feng Tian2.1 Introduction 232.2 Reproducing Kernel Hilbert Spaces (RKHSs) in the Measurable Category 242.3 Applications to Gaussian Processes 302.4 Choice of Probability Space 342.5 A Duality 372.A Stochastic Processes 402.B Overview of Applications of RKHSs 45Acknowledgments 50References 513 Many-Body Wave Scattering Problems for Small Scatterers and Creating Materials with a Desired Refraction Coefficient 57Alexander G. Ramm3.1 Introduction 573.2 Derivation of the Formulas for One-Body Wave Scattering Problems 623.3 Many-Body Scattering Problem 653.3.1 The Case of Acoustically Soft Particles 683.3.2 Wave Scattering by Many Impedance Particles 703.4 Creating Materials with a Desired Refraction Coefficient 713.5 Scattering by Small Particles Embedded in an Inhomogeneous Medium 723.6 Conclusions 72References 734 Generalized Convex Functions and their Applications 77Adem Kiliçman and Wedad Saleh4.1 Brief Introduction 774.2 Generalized E-Convex Functions 784.3 E𝛼-Epigraph 844.4 Generalized s-Convex Functions 854.5 Applications to Special Means 96References 985 Some Properties and Generalizations of the Catalan, Fuss, and Fuss–Catalan Numbers 101Feng Qi and Bai-Ni Guo5.1 The Catalan Numbers 1015.1.1 A Definition of the Catalan Numbers 1015.1.2 The History of the Catalan Numbers 1015.1.3 A Generating Function of the Catalan Numbers 1025.1.4 Some Expressions of the Catalan Numbers 1025.1.5 Integral Representations of the Catalan Numbers 1035.1.6 Asymptotic Expansions of the Catalan Function 1045.1.7 Complete Monotonicity of the Catalan Numbers 1055.1.8 Inequalities of the Catalan Numbers and Function 1065.1.9 The Bell Polynomials of the Second Kind and the Bessel Polynomials 1095.2 The Catalan–Qi Function 1115.2.1 The Fuss Numbers 1115.2.2 A Definition of the Catalan–Qi Function 1115.2.3 Some Identities of the Catalan–Qi Function 1125.2.4 Integral Representations of the Catalan–Qi Function 1145.2.5 Asymptotic Expansions of the Catalan–Qi Function 1155.2.6 Complete Monotonicity of the Catalan–Qi Function 1165.2.7 Schur-Convexity of the Catalan–Qi Function 1185.2.8 Generating Functions of the Catalan–Qi Numbers 1185.2.9 A Double Inequality of the Catalan–Qi Function 1185.2.10 The q-Catalan–Qi Numbers and Properties 1195.2.11 The Catalan Numbers and the k-Gamma and k-Beta Functions 1195.2.12 Series Identities Involving the Catalan Numbers 1195.3 The Fuss–Catalan Numbers 1195.3.1 A Definition of the Fuss–Catalan Numbers 1195.3.2 A Product-Ratio Expression of the Fuss–Catalan Numbers 1205.3.3 Complete Monotonicity of the Fuss–Catalan Numbers 1205.3.4 A Double Inequality for the Fuss–Catalan Numbers 1215.4 The Fuss–Catalan–Qi Function 1215.4.1 A Definition of the Fuss–Catalan–Qi Function 1215.4.2 A Product-Ratio Expression of the Fuss–Catalan–Qi Function 1225.4.3 Integral Representations of the Fuss–Catalan–Qi Function 1235.4.4 Complete Monotonicity of the Fuss–Catalan–Qi Function 1245.5 Some Properties for Ratios of Two Gamma Functions 1245.5.1 An Integral Representation and Complete Monotonicity 1255.5.2 An Exponential Expansion for the Ratio of Two Gamma Functions 1255.5.3 A Double Inequality for the Ratio of Two Gamma Functions 1255.6 Some New Results on the Catalan Numbers 1265.7 Open Problems 126Acknowledgments 127References 1276 Trace Inequalities of Jensen Type for Self-adjoint Operators in Hilbert Spaces: A Survey of Recent Results 135Silvestru Sever Dragomir6.1 Introduction 1356.1.1 Jensen’s Inequality 1356.1.2 Traces for Operators in Hilbert Spaces 1386.2 Jensen’s Type Trace Inequalities 1416.2.1 Some Trace Inequalities for Convex Functions 1416.2.2 Some Functional Properties 1456.2.3 Some Examples 1516.2.4 More Inequalities for Convex Functions 1546.3 Reverses of Jensen’s Trace Inequality 1576.3.1 A Reverse of Jensen’s Inequality 1576.3.2 Some Examples 1636.3.3 Further Reverse Inequalities for Convex Functions 1656.3.4 Some Examples 1696.3.5 Reverses of Hölder’s Inequality 1746.4 Slater’s Type Trace Inequalities 1776.4.1 Slater’s Type Inequalities 1776.4.2 Further Reverses 180References 1887 Spectral Synthesis and Its Applications 193László Székelyhidi7.1 Introduction 1937.2 Basic Concepts and Function Classes 1957.3 Discrete Spectral Synthesis 2037.4 Nondiscrete Spectral Synthesis 2177.5 Spherical Spectral Synthesis 2197.6 Spectral Synthesis on Hypergroups 2387.7 Applications 248Acknowledgments 252References 2528 Various Ulam–Hyers Stabilities of Euler–Lagrange–Jensen General (a, b; k = a + b)-Sextic Functional Equations 255John Michael Rassias and Narasimman Pasupathi8.1 Brief Introduction 2558.2 General Solution of Euler–Lagrange–Jensen General(a, b; k = a + b)-Sextic Functional Equation 2578.3 Stability Results in Banach Space 2588.3.1 Banach Space: Direct Method 2588.3.2 Banach Space: Fixed Point Method 2618.4 Stability Results in Felbin’s Type Spaces 2678.4.1 Felbin’s Type Spaces: Direct Method 2688.4.2 Felbin’s Type Spaces: Fixed Point Method 2698.5 Intuitionistic Fuzzy Normed Space: Stability Results 2708.5.1 IFNS: Direct Method 2728.5.2 IFNS: Fixed Point Method 279References 2819 A Note on the Split Common Fixed Point Problem and its Variant Forms 283Adem Kiliçman and L.B. Mohammed9.1 Introduction 2839.2 Basic Concepts and Definitions 2849.2.1 Introduction 2849.2.2 Vector Space 2849.2.3 Hilbert Space and its Properties 2869.2.4 Bounded Linear Map and its Properties 2889.2.5 Some Nonlinear Operators 2899.2.6 Problem Formulation 2949.2.7 Preliminary Results 2949.2.8 Strong Convergence for the Split Common Fixed-Point Problems for Total Quasi-Asymptotically Nonexpansive Mappings 2969.2.9 Strong Convergence for the Split Common Fixed-Point Problems for Demicontractive Mappings 3029.2.10 Application to Variational Inequality Problems 3069.2.11 On Synchronal Algorithms for Fixed and Variational Inequality Problems in Hilbert Spaces 3079.2.12 Preliminaries 3079.3 A Note on the Split Equality Fixed-Point Problems in Hilbert Spaces 3159.3.1 Problem Formulation 3159.3.2 Preliminaries 3169.3.3 The Split Feasibility and Fixed-Point Equality Problems for Quasi-Nonexpansive Mappings in Hilbert Spaces 3169.3.4 The Split Common Fixed-Point Equality Problems for Quasi-Nonexpansive Mappings in Hilbert Spaces 3209.4 Numerical Example 3229.5 The Split Feasibility and Fixed Point Problems for Quasi-Nonexpansive Mappings in Hilbert Spaces 3289.5.1 Problem Formulation 3289.5.2 Preliminary Results 3289.6 Ishikawa-Type Extra-Gradient Iterative Methods for Quasi-Nonexpansive Mappings in Hilbert Spaces 3299.6.1 Application to Split Feasibility Problems 3349.7 Conclusion 336References 33710 Stabilities and Instabilities of Rational Functional Equations and Euler–Lagrange–Jensen (a, b)-Sextic Functional Equations 341John Michael Rassias, Krishnan Ravi, and Beri V. Senthil Kumar10.1 Introduction 34110.1.1 Growth of Functional Equations 34210.1.2 Importance of Functional Equations 34210.1.3 Functional Equations Relevant to Other Fields 34310.1.4 Definition of Functional Equation with Examples 34310.2 Ulam Stability Problem for Functional Equation 34410.2.1 𝜖-Stability of Functional Equation 34410.2.2 Stability Involving Sum of Powers of Norms 34510.2.3 Stability Involving Product of Powers of Norms 34610.2.4 Stability Involving a General Control Function 34710.2.5 Stability Involving Mixed Product–Sum of Powers of Norms 34710.2.6 Application of Ulam Stability Theory 34810.3 Various Forms of Functional Equations 34810.4 Preliminaries 35310.5 Rational Functional Equations 35510.5.1 Reciprocal Type Functional Equation 35510.5.2 Solution of Reciprocal Type Functional Equation 35610.5.3 Generalized Hyers–Ulam Stability of Reciprocal Type Functional Equation 35710.5.4 Counter-Example 36010.5.5 Geometrical Interpretation of Reciprocal Type Functional Equation 36210.5.6 An Application of Equation (10.41) to Electric Circuits 36410.5.7 Reciprocal-Quadratic Functional Equation 36410.5.8 General Solution of Reciprocal-Quadratic Functional Equation 36610.5.9 Generalized Hyers–Ulam Stability of Reciprocal-Quadratic Functional Equations 36810.5.10 Counter-Examples 37310.5.11 Reciprocal-Cubic and Reciprocal-Quartic Functional Equations 37510.5.12 Hyers–Ulam Stability of Reciprocal-Cubic and Reciprocal-Quartic Functional Equations 37510.5.13 Counter-Examples 38010.6 Euler-Lagrange–Jensen (a, b; k = a + b)-Sextic Functional Equations 38410.6.1 Generalized Ulam–Hyers Stability of Euler-Lagrange-Jensen Sextic Functional Equation Using Fixed Point Method 38410.6.2 Counter-Example 38710.6.3 Generalized Ulam–Hyers Stability of Euler-Lagrange-Jensen Sextic Functional Equation Using Direct Method 389References 39511 Attractor of the Generalized Contractive Iterated Function System 401Mujahid Abbas and Talat Nazir11.1 Iterated Function System 40111.2 Generalized F-contractive Iterated Function System 40711.3 Iterated Function System in b-Metric Space 41411.4 Generalized F-Contractive Iterated Function System in b-Metric Space 420References 42612 Regular and Rapid Variations and Some Applications 429Ljubiša D.R. Kočinac, Dragan Djurčić, and Jelena V. Manojlović12.1 Introduction and Historical Background 42912.2 Regular Variation 43112.2.1 The Class Tr(RVs) 43212.2.2 Classes of Sequences Related to Tr(RVs) 43412.2.3 The Class ORVs and Seneta Sequences 43612.3 Rapid Variation 43712.3.1 Some Properties of Rapidly Varying Functions 43812.3.2 The Class ARVs 44012.3.3 The Class KRs,∞ 44212.3.4 The Class Tr(Rs,∞) 44712.3.5 Subclasses of Tr(Rs,∞) 44812.3.6 The Class Γs 45112.4 Applications to Selection Principles 45312.4.1 First Results 45512.4.2 Improvements 45512.4.3 When ONE has a Winning Strategy? 46012.5 Applications to Differential Equations 46312.5.1 The Existence of all Solutions of (A) 46412.5.2 Superlinear Thomas–Fermi Equation (A) 46612.5.3 Sublinear Thomas–Fermi Equation (A) 47012.5.4 A Generalization 480References 48613 n-Inner Products, n-Norms, and Angles Between Two Subspaces 493Hendra Gunawan13.1 Introduction 49313.2 n-Inner Product Spaces and n-Normed Spaces 49513.2.1 Topology in n-Normed Spaces 49913.3 Orthogonality in n-Normed Spaces 50013.3.1 G-, P-, I-, and BJ- Orthogonality 50313.3.2 Remarks on the n-Dimensional Case 50513.4 Angles Between Two Subspaces 50513.4.1 An Explicit Formula 50913.4.2 A More General Formula 511References 51314 Proximal Fiber Bundles on Nerve Complexes 517James F. Peters14.1 Brief Introduction 51714.2 Preliminaries 51814.2.1 Nerve Complexes and Nerve Spokes 51814.2.2 Descriptions and Proximities 52114.2.3 Descriptive Proximities 52314.3 Sewing Regions Together 52714.3.1 Sewing Nerves Together with Spokes to Construct a Nervous System Complex 52914.4 Some Results for Fiber Bundles 53014.5 Concluding Remarks 534References 53415 Approximation by Generalizations of Hybrid Baskakov Type Operators Preserving Exponential Functions 537Vijay Gupta15.1 Introduction 53715.2 Baskakov–Szász Operators 53915.3 Genuine Baskakov–Szász Operators 54215.4 Preservation of eAx 54515.5 Conclusion 549References 55016 Well-Posed Minimization Problems via the Theory of Measures of Noncompactness 553Józef Banaś and Tomasz Zając16.1 Introduction 55316.2 Minimization Problems and Their Well-Posedness in the Classical Sense 55416.3 Measures of Noncompactness 55616.4 Well-Posed Minimization Problems with Respect to Measures of Noncompactness 56516.5 Minimization Problems for Functionals Defined in Banach Sequence Spaces 56816.6 Minimization Problems for Functionals Defined in the Classical Space C([a, b])) 57616.7 Minimization Problems for Functionals Defined in the Space of Functions Continuous and Bounded on the Real Half-Axis 580References 58417 Some Recent Developments on Fixed Point Theory in Generalized Metric Spaces 587Poom Kumam and Somayya Komal17.1 Brief Introduction 58717.2 Some Basic Notions and Notations 59317.3 Fixed Points Theorems 59617.3.1 Fixed Points Theorems for Monotonic and Nonmonotonic Mappings 59717.3.2 PPF-Dependent Fixed-Point Theorems 60017.3.3 Fixed Points Results in b-Metric Spaces 60217.3.4 The generalized Ulam–Hyers Stability in b-Metric Spaces 60417.3.5 Well-Posedness of a Function with Respect to 𝛼-Admissibility in b-Metric Spaces 60517.3.6 Fixed Points for F-Contraction 60617.4 Common Fixed Points Theorems 60817.4.1 Common Fixed-Point Theorems for Pair of Weakly Compatible Mappings in Fuzzy Metric Spaces 60917.5 Best Proximity Points 61117.6 Common Best Proximity Points 61417.7 Tripled Best Proximity Points 61717.8 Future Works 624References 62418 The Basel Problem with an Extension 631Anthony Sofo18.1 The Basel Problem 63118.2 An Euler Type Sum 64018.3 The Main Theorem 64518.4 Conclusion 652References 65219 Coupled Fixed Points and Coupled Coincidence Points via Fixed Point Theory 661Adrian Petruşel and Gabriela Petruşel19.1 Introduction and Preliminaries 66119.2 Fixed Point Results 66519.2.1 The Single-Valued Case 66519.2.2 The Multi-Valued Case 67319.3 Coupled Fixed Point Results 68019.3.1 The Single-Valued Case 68019.3.2 The Multi-Valued Case 68619.4 Coincidence Point Results 68919.5 Coupled Coincidence Results 699References 70420 The Corona Problem, Carleson Measures, and Applications 709Alberto Saracco20.1 The Corona Problem 70920.1.1 Banach Algebras: Spectrum 70920.1.2 Banach Algebras: Maximal Spectrum 71020.1.3 The Algebra of Bounded Holomorphic Functions and the Corona Problem 71020.2 Carleson’s Proof and Carleson Measures 71120.2.1 Wolff’s Proof 71220.3 The Corona Problem in Higher Henerality 71220.3.1 The Corona Problem in ℂ 71220.3.2 The Corona Problem in Riemann Surfaces: A Positive and a Negative Result 71320.3.3 The Corona Problem in Domains of ℂn 71420.3.4 The Corona Problem for Quaternionic Slice-Regular Functions 71520.3.4.1 Slice-Regular Functions f ∶ D → ℍ 71520.3.4.2 The Corona Theorem in the Quaternions 71720.4 Results on Carleson Measures 71820.4.1 Carleson Measures of Hardy Spaces of the Disk 71820.4.2 Carleson Measures of Bergman Spaces of the Disk 71920.4.3 Carleson Measures in the Unit Ball of ℂn 72020.4.4 Carleson Measures in Strongly Pseudoconvex Bounded Domains of ℂn 72220.4.5 Generalizations of Carleson Measures and Applications to Toeplitz Operators 72320.4.6 Explicit Examples of Carleson Measures of Bergman Spaces 72420.4.7 Carleson Measures in the Quaternionic Setting 72520.4.7.1 Carleson Measures on Hardy Spaces of 𝔹 ⊂ ℍ 72520.4.7.2 Carleson Measures on Bergman Spaces of 𝔹 ⊂ ℍ 726References 728Index 731