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 152 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 453 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 653.4 Creating Materials with a Desired Refraction Coefficient 713.5 Scattering by Small Particles Embedded in an Inhomogeneous Medium 723.6 Conclusions 724 Generalized Convex Functions and their Applications 77Adem Kiliçman and Wedad Saleh4.1 Brief Introduction 774.2 Generalized E-Convex Functions 784.3 E;;Epigraph 844.4 Generalized s-Convex Functions 854.5 Applications to Special Means 965 Some Properties and Generalizations of the Catalan, Fuss, and Fuss–Catalan Numbers 101Feng Qi and Bai-Ni Guo5.1 The Catalan Numbers 1015.2 The Catalan–Qi Function 1115.3 The Fuss–Catalan Numbers 1195.4 The Fuss–Catalan–Qi Function 1215.5 Some Properties for Ratios of Two Gamma Functions 1245.6 Some New Results on the Catalan Numbers 1265.7 Open Problems 1266 Trace Inequalities of Jensen Type for Self-adjoint Operators in Hilbert Spaces: A Survey of Recent Results 135Silvestru Sever Dragomir6.1 Introduction 1356.2 Jensen's Type Trace Inequalities 1416.3 Reverses of Jensen's Trace Inequality 1576.4 Slater's Type Trace Inequalities 1777 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 2488 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 2588.4 Stability Results in Felbin's Type Spaces 2678.5 Intuitionistic Fuzzy Normed Space: Stability Results 2709 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 2849.3 A Note on the Split Equality Fixed-Point Problems in Hilbert Spaces 3159.4 Numerical Example 3229.5 The Split Feasibility and Fixed Point Problems for Quasi-Nonexpansive Mappings in Hilbert Spaces 3289.6 Ishikawa-Type Extra-Gradient Iterative Methods for Quasi-Nonexpansive Mappings in Hilbert Spaces 3299.7 Conclusion 33610 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 34110.2 Ulam Stability Problem for Functional Equation 34410.3 Various Forms of Functional Equations 34810.4 Preliminaries 35310.5 Rational Functional Equations 35510.6 Euler-Lagrange–Jensen (a, b; k = a + b)-Sextic Functional Equations 38411 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 42012 Regular and Rapid Variations and Some Applications 429Ljubiša D.R. Koèinac, Dragan Djurèiæ, and Jelena V. Manojloviæ12.1 Introduction and Historical Background 42912.2 Regular Variation 43112.3 Rapid Variation 43712.4 Applications to Selection Principles 45312.5 Applications to Differential Equations 46313 n-Inner Products, n-Norms, and Angles Between Two Subspaces 493Hendra Gunawan13.1 Introduction 49313.2 n-Inner Product Spaces and n-Normed Spaces 49513.3 Orthogonality in n-Normed Spaces 50013.4 Angles Between Two Subspaces 50514 Proximal Fiber Bundles on Nerve Complexes 517James F. Peters14.1 Brief Introduction 51714.2 Preliminaries 51814.3 Sewing Regions Together 52714.4 Some Results for Fiber Bundles 53014.5 Concluding Remarks 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 54916 Well-Posed Minimization Problems via the Theory of Measures of Noncompactness 553Józef Banaœ and Tomasz Zaj¹c16.1 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 58017 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 59617.4 Common Fixed Points Theorems 60817.5 Best Proximity Points 61117.6 Common Best Proximity Points 61417.7 Tripled Best Proximity Points 61717.8 Future Works 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 65219 Coupled Fixed Points and Coupled Coincidence Points via Fixed Point Theory 661Adrian Petruºel and Gabriela Petruºel19.1 Introduction and Preliminaries 66119.2 Fixed Point Results 66519.3 Coupled Fixed Point Results 68019.4 Coincidence Point Results 68919.5 Coupled Coincidence Results 69920 The Corona Problem, Carleson Measures, and Applications 709Alberto Saracco20.1 The Corona Problem 70920.2 Carleson's Proof and Carleson Measures 71120.3 The Corona Problem in Higher Henerality 71220.4 Results on Carleson Measures 718References 728Index 731

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 152 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 453 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 653.4 Creating Materials with a Desired Refraction Coefficient 713.5 Scattering by Small Particles Embedded in an Inhomogeneous Medium 723.6 Conclusions 724 Generalized Convex Functions and their Applications 77Adem Kiliçman and Wedad Saleh4.1 Brief Introduction 774.2 Generalized E-Convex Functions 784.3 E;;Epigraph 844.4 Generalized s-Convex Functions 854.5 Applications to Special Means 965 Some Properties and Generalizations of the Catalan, Fuss, and Fuss–Catalan Numbers 101Feng Qi and Bai-Ni Guo5.1 The Catalan Numbers 1015.2 The Catalan–Qi Function 1115.3 The Fuss–Catalan Numbers 1195.4 The Fuss–Catalan–Qi Function 1215.5 Some Properties for Ratios of Two Gamma Functions 1245.6 Some New Results on the Catalan Numbers 1265.7 Open Problems 1266 Trace Inequalities of Jensen Type for Self-adjoint Operators in Hilbert Spaces: A Survey of Recent Results 135Silvestru Sever Dragomir6.1 Introduction 1356.2 Jensen's Type Trace Inequalities 1416.3 Reverses of Jensen's Trace Inequality 1576.4 Slater's Type Trace Inequalities 1777 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 2488 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 2588.4 Stability Results in Felbin's Type Spaces 2678.5 Intuitionistic Fuzzy Normed Space: Stability Results 2709 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 2849.3 A Note on the Split Equality Fixed-Point Problems in Hilbert Spaces 3159.4 Numerical Example 3229.5 The Split Feasibility and Fixed Point Problems for Quasi-Nonexpansive Mappings in Hilbert Spaces 3289.6 Ishikawa-Type Extra-Gradient Iterative Methods for Quasi-Nonexpansive Mappings in Hilbert Spaces 3299.7 Conclusion 33610 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 34110.2 Ulam Stability Problem for Functional Equation 34410.3 Various Forms of Functional Equations 34810.4 Preliminaries 35310.5 Rational Functional Equations 35510.6 Euler-Lagrange–Jensen (a, b; k = a + b)-Sextic Functional Equations 38411 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 42012 Regular and Rapid Variations and Some Applications 429Ljubiša D.R. Koèinac, Dragan Djurèiæ, and Jelena V. Manojloviæ12.1 Introduction and Historical Background 42912.2 Regular Variation 43112.3 Rapid Variation 43712.4 Applications to Selection Principles 45312.5 Applications to Differential Equations 46313 n-Inner Products, n-Norms, and Angles Between Two Subspaces 493Hendra Gunawan13.1 Introduction 49313.2 n-Inner Product Spaces and n-Normed Spaces 49513.3 Orthogonality in n-Normed Spaces 50013.4 Angles Between Two Subspaces 50514 Proximal Fiber Bundles on Nerve Complexes 517James F. Peters14.1 Brief Introduction 51714.2 Preliminaries 51814.3 Sewing Regions Together 52714.4 Some Results for Fiber Bundles 53014.5 Concluding Remarks 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 54916 Well-Posed Minimization Problems via the Theory of Measures of Noncompactness 553Józef Banaœ and Tomasz Zaj¹c16.1 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 58017 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 59617.4 Common Fixed Points Theorems 60817.5 Best Proximity Points 61117.6 Common Best Proximity Points 61417.7 Tripled Best Proximity Points 61717.8 Future Works 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 65219 Coupled Fixed Points and Coupled Coincidence Points via Fixed Point Theory 661Adrian Petruºel and Gabriela Petruºel19.1 Introduction and Preliminaries 66119.2 Fixed Point Results 66519.3 Coupled Fixed Point Results 68019.4 Coincidence Point Results 68919.5 Coupled Coincidence Results 69920 The Corona Problem, Carleson Measures, and Applications 709Alberto Saracco20.1 The Corona Problem 70920.2 Carleson's Proof and Carleson Measures 71120.3 The Corona Problem in Higher Henerality 71220.4 Results on Carleson Measures 718References 728Index 731