# Mathematical Analysis and Applications: Selected Topics

# Mathematical Analysis and Applications: Selected Topics

ISBN: 978-1-119-41433-9

Apr 2018

768 pages

$112.99

## 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.

- 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 Topic*s includes the most recent research from a range of mathematical fields.

**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 ***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-BodyWave Scattering Problems for Small Scatterers and CreatingMaterials with a Desired Refraction Coefficient ***57*

*Alexander G. Ramm*

3.1 Introduction *57*

3.2 Derivation of the Formulas for One-BodyWave 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 andWedad 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*

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 NewResults 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*

*JohnMichael 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*

*A. K*

*��*

*l*

*��*

*çman and L.B. Mohammed*

9.1 Introduction *283*

9.2 Basic Concepts and Definitions *284*

9.2.1 Introduction *284*

9.2.2 Vector Spaces *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 IterativeMethods 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 M. 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 Bounded by Sum of Powers of Norms *345*

10.2.3 Stability Bounded by Product of Powers of Norms *346*

10.2.4 Stability Bounded by a General Control Function *347*

10.2.5 Stability Bounded by 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 Equation *357*

10.5.4 Counter-Example *360*

10.5.5 Geometrical Interpretation of Equation *362*

10.5.6 An Application of Equation 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 Equation *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 Equation *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 Equation Using Fixed Point Method *384*

10.6.2 Counter-Example *387*

10.6.3 Generalized Ulam–Hyers Stability of 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*

*Ljubisa D.R. Kocinac, Dragan Djurcic, Jelena V. Manojlovic*

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 Γ*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 aWinning 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

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 *e**Ax **545*

15.5 Conclusion *549*

References *550*

**16 Well-PosedMinimization Problems via the Theory of Measures of Noncompactness ***553*

*Józef Bana´s and Tomasz Zaja˛c*

16.1 Introduction *553*

16.2 Minimization Problems and TheirWell-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 GeneralizedMetric 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 *��*-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 ofWeakly 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 FutureWorks *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¸sel and Gabriela Petru¸sel*

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 ℂ*n **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 CoronaTheorem 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 ℂ*n **720*

20.4.4 Carleson Measures in Strongly Pseudoconvex Bounded Domains of ℂ*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 �� *⊂ *ℍ *725*

20.4.7.2 Carleson Measures on Bergman Spaces of �� *⊂ *ℍ *726*

References *728*

**Index ***731*