Mathematical Analysis and Applications

Preface xv About the Editors xxi List of Contributors xxiii 1 Spaces of Asymptotically Developable Functions and Applications 1 Sergio 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 23 Palle 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-Body Wave Scattering Problems for Small Scatterers and Creating Materials with a Desired Refraction Coefficient 57 Alexander G. Ramm 3.1 Introduction 57 3.2 Derivation of the Formulas for One-Body Wave 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 77 Adem Kiliçman and Wedad Saleh 4.1 Brief Introduction 77 4.2 Generalized E-Convex Functions 78 4.3 E𝛼-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 101 Feng 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 New Results 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 135 Silvestru 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 193 László 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 255 John Michael 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 283 Adem Kiliçman and L.B. Mohammed 9.1 Introduction 283 9.2 Basic Concepts and Definitions 284 9.2.1 Introduction 284 9.2.2 Vector Space 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 Iterative Methods 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 341 John Michael 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 𝜖-Stability of Functional Equation 344 10.2.2 Stability Involving Sum of Powers of Norms 345 10.2.3 Stability Involving Product of Powers of Norms 346 10.2.4 Stability Involving a General Control Function 347 10.2.5 Stability Involving 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 Reciprocal Type Functional Equation 357 10.5.4 Counter-Example 360 10.5.5 Geometrical Interpretation of Reciprocal Type Functional Equation 362 10.5.6 An Application of Equation (10.41) 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 Equations 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 Reciprocal-Cubic and Reciprocal-Quartic Functional Equations 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 Euler-Lagrange-Jensen Sextic Functional Equation Using Fixed Point Method 384 10.6.2 Counter-Example 387 10.6.3 Generalized Ulam–Hyers Stability of Euler-Lagrange-Jensen Sextic Functional Equation Using Direct Method 389 References 395 11 Attractor of the Generalized Contractive Iterated Function System 401 Mujahid 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 429 Ljubiša D.R. Kočinac, Dragan Djurčić, and Jelena V. Manojlović 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 Γs 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 a Winning 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 493 Hendra 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 517 James 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 Complex 529 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 537 Vijay 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-Posed Minimization Problems via the Theory of Measures of Noncompactness 553 Józef Banaś and Tomasz Zając 16.1 Introduction 553 16.2 Minimization Problems and Their Well-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 Generalized Metric Spaces 587 Poom 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 𝛼-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 of Weakly 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 Future Works 624 References 624 18 The Basel Problem with an Extension 631 Anthony 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 661 Adrian Petruşel and Gabriela Petruşel 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 709 Alberto 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 ℂ 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 ℂn 714 20.3.4 The Corona Problem for Quaternionic Slice-Regular Functions 715 20.3.4.1 Slice-Regular Functions f ∶ D → ℍ 715 20.3.4.2 The Corona Theorem 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 ℂn 720 20.4.4 Carleson Measures in Strongly Pseudoconvex Bounded Domains of ℂn 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 𝔹 ⊂ ℍ 725 20.4.7.2 Carleson Measures on Bergman Spaces of 𝔹 ⊂ ℍ 726 References 728 Index 731