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 Fern¡ndez
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 Kili§man and Wedad Saleh
4.1 Brief Introduction 77
4.2 Generalized E-Convex Functions 78
4.3 E°¼-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 H¶lder¢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
L¡szl³ Sz©kelyhidi
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 Kili§man 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 °--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
Ljubi¡a D.R. Koinac, Dragan Djuri, and Jelena V. Manojlovi
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 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¢"Sz¡sz Operators 539
15.3 Genuine Baskakov¢"Sz¡sz 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
J³zef Bana and Tomasz Zajc
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 °¼-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 Petru?el and Gabriela Petru?el
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 ¢ 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 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 ¢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 °"¹ ¢ ¢ 725
20.4.7.2 Carleson Measures on Bergman Spaces of °"¹ ¢ ¢ 726
References 728
Index 731
