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

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

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
3.4 Creating Materials with a Desired Refraction Coefficient 71
3.5 Scattering by Small Particles Embedded in an Inhomogeneous Medium 72
3.6 Conclusions 72

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 E;;Epigraph 84
4.4 Generalized s-Convex Functions 85
4.5 Applications to Special Means 96

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
5.2 The Catalan–Qi Function 111
5.3 The Fuss–Catalan Numbers 119
5.4 The Fuss–Catalan–Qi Function 121
5.5 Some Properties for Ratios of Two Gamma Functions 124
5.6 Some New Results on the Catalan Numbers 126
5.7 Open Problems 126

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
6.2 Jensen's Type Trace Inequalities 141
6.3 Reverses of Jensen's Trace Inequality 157
6.4 Slater's Type Trace Inequalities 177

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

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
8.4 Stability Results in Felbin's Type Spaces 267
8.5 Intuitionistic Fuzzy Normed Space: Stability Results 270

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
9.3 A Note on the Split Equality Fixed-Point Problems in Hilbert Spaces 315
9.4 Numerical Example 322
9.5 The Split Feasibility and Fixed Point Problems for Quasi-Nonexpansive Mappings in Hilbert Spaces 328
9.6 Ishikawa-Type Extra-Gradient Iterative Methods for Quasi-Nonexpansive Mappings in Hilbert Spaces 329
9.7 Conclusion 336

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
10.2 Ulam Stability Problem for Functional Equation 344
10.3 Various Forms of Functional Equations 348
10.4 Preliminaries 353
10.5 Rational Functional Equations 355
10.6 Euler-Lagrange–Jensen (a, b; k = a + b)-Sextic Functional Equations 384

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

12 Regular and Rapid Variations and Some Applications 429
Ljubiša D.R. Koèinac, Dragan Djurèiæ, and Jelena V. Manojloviæ

12.1 Introduction and Historical Background 429
12.2 Regular Variation 431
12.3 Rapid Variation 437
12.4 Applications to Selection Principles 453
12.5 Applications to Differential Equations 463

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
13.3 Orthogonality in n-Normed Spaces 500
13.4 Angles Between Two Subspaces 505

14 Proximal Fiber Bundles on Nerve Complexes 517
James F. Peters

14.1 Brief Introduction 517
14.2 Preliminaries 518
14.3 Sewing Regions Together 527
14.4 Some Results for Fiber Bundles 530
14.5 Concluding Remarks 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

16 Well-Posed Minimization Problems via the Theory of Measures of Noncompactness 553
Józef Banaœ and Tomasz Zaj¹c

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

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
17.4 Common Fixed Points Theorems 608
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

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

19 Coupled Fixed Points and Coupled Coincidence Points via Fixed Point Theory 661
Adrian Petruºel and Gabriela Petruºel

19.1 Introduction and Preliminaries 661
19.2 Fixed Point Results 665
19.3 Coupled Fixed Point Results 680
19.4 Coincidence Point Results 689
19.5 Coupled Coincidence Results 699

20 The Corona Problem, Carleson Measures, and Applications 709
Alberto Saracco

20.1 The Corona Problem 709
20.2 Carleson's Proof and Carleson Measures 711
20.3 The Corona Problem in Higher Henerality 712
20.4 Results on Carleson Measures 718

References 728
Index 731