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Mathematical Analysis and Applications: Selected Topics

Mathematical Analysis and Applications: Selected Topics

Michael Ruzhansky (Editor), Hemen Dutta (Editor), Ravi P. Agarwal (Editor)

ISBN: 978-1-119-41434-6

May 2018

768 pages

In Stock

$140.00

Description

An authoritative text that presents the current problems, theories, and applications of mathematical analysis research

Mathematical Analysis and Applications: Selected Topics offers the theories, methods, and applications of a variety of targeted topics including: operator theory, approximation theory, fixed point theory, stability theory, minimization problems, many-body wave scattering problems, Basel problem, Corona problem, inequalities, generalized normed spaces, variations of functions and sequences, analytic generalizations of the Catalan, Fuss, and Fuss–Catalan Numbers, asymptotically developable functions, convex functions, Gaussian processes, image analysis, and spectral analysis and spectral synthesis. The authors—a noted team of international researchers in the field— highlight the basic developments for each topic presented and explore the most recent advances made in their area of study. The text is presented in such a way that enables the reader to follow subsequent studies in a burgeoning field of research.

This important text:

  • Presents a wide-range of important topics having current research importance and interdisciplinary applications such as game theory, image processing, creation of materials with a desired refraction coefficient, etc.
  • Contains chapters written by a group of esteemed researchers in mathematical analysis Includes problems and research questions in order to enhance understanding of the information provided
  • Offers references that help readers advance to further study

Written for researchers, graduate students, educators, and practitioners with an interest in mathematical analysis, Mathematical Analysis and Applications: Selected Topics includes the most recent research from a range of mathematical fields.

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 fD → 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