Current Trends in Mathematical Analysis and Its Interdisciplinary Applications

Preface
Contents
Contributors
1 Frictional Contact Problems for Steady Flow of Incompressible Fluids in Orlicz Spaces
	1.1 Introduction
	1.2 Preliminaries
		1.2.1 Operators of Monotone Type
		1.2.2 Orlicz and Orlicz–Sobolev Spaces
		1.2.3 Generalized Gradient
	1.3 Subdifferential Operator Inclusions
	1.4 Hemivariational Inequalities
		1.4.1 Tangential Superpotential
		1.4.2 Normal Superpotential
	1.5 Steady Flows of Non-Newtonian Fluids Under Slip Boundary Conditions of Frictional Type
		1.5.1 Existence and Uniqueness
			1.5.1.1 Setting of the Flow Problem
			1.5.1.2 Weak Formulation and Main Result
		1.5.2 Slow Flows
	1.6 Steady Flows of Newtonian Fluids Under Leak Boundary Conditions of Frictional Type
		1.6.1 Existence and Uniqueness
			1.6.1.1 Setting of the Flow Problem
			1.6.1.2 Weak Formulation and Main Result
		1.6.2 Slow Flows
		1.6.3 Optimal Control Problem
			1.6.3.1 Continuous Dependence on External Forces
			1.6.3.2 Optimal Control Problem
	1.7 Concluding Remarks
	References
2 Discrete Fourier Transform and Theta Function Identities
	2.1 Introduction
	2.2 Spectral Theory of Discrete Fourier Transform
		2.2.1 The Discrete Fourier Transform
		2.2.2 Spectral Decomposition of the Matrix or Operator Roots of Unity
		2.2.3 Eigenvectors of (n)
		2.2.4 Eigenvectors of the DFT from Any Absolutely Summable Series
	2.3 DFT (2) and Jacobi Theta Function Identities
		2.3.1 Jacobi Theta Functions
		2.3.2 DFT and Theta Functions
		2.3.3 DFT (2) and Jacobi Theta Function Identities
		2.3.4 The Identity θ4(0,τ)- θ40,12(0,τ) = θ12,04(0,τ)
		2.3.5 Extended Watson Addition Formula
		2.3.6 Riemann\'s Identity
	2.4 DFT (3) and Theta Function Identities
		2.4.1 θa,b(x,τ) with a,b  13Z and (3)
		2.4.2 Extended Watson Addition Formula Corresponding to (3)
		2.4.3 Extended Riemann\'s Identity Corresponding to (3)
	References
3 On Some Combinatorics of Rogers–Ramanujan Type Identities Using Signed Color Partitions
	3.1 Introduction
	3.2 Main Proof
	3.3 Conclusion
	References
4 Piecewise Continuous Stepanov-Like Almost Automorphic Functions with Applications to Impulsive Systems
	4.1 Introduction
	4.2 Preliminaries
	4.3 Composition Theorem
	4.4 Impulsive Delay Differential Equations
	4.5 Examples
	4.6 Discussion
	References
5 On the Convergence of Secant-Like Methods
	5.1 Introduction
	5.2 Preliminaries
	5.3 Convergence of Secant-Like Methods for Fréchet Differentiable Operators
		5.3.1 Divided Differences of First Order Lipschitz Continuous
		5.3.2 Divided Differences of First Order Hölder Continuous
		5.3.3 Application: A Special Case of Conservative Problems
			5.3.3.1 Existence of the Solution
			5.3.3.2 Location of the Solution
			5.3.3.3 Numerical Solution of the Finite-Difference Equations
			5.3.3.4 Final Remark
	5.4 Convergence of Secant-Like Methods for Non-Differentiable Operators
		5.4.1 Numerical Example
	5.5 Convergence of Secant-Like Methods Whatever the Operator
		5.5.1 A Semilocal Convergence Result
		5.5.2 Applications
			5.5.2.1 Example 1
			5.5.2.2  Example 2
	5.6 Convergence for the Secant-Like Methods from Auxiliary Points
		5.6.1 Local Convergence Analysis
		Remarks 1
		5.6.2 Semilocal Convergence Analysis
		Remarks 2
		5.6.3 Numerical Example
	References
6 Spacetimes as Topological Spaces, and the Need to Take Methods of General Topology More Seriously
	6.1 Introduction
		6.1.1 The Manifold Topology vs. Finer or Incomparable Topologies
		6.1.2 On Name-Giving and Notation
	6.2 Topologies Coarser Than or Equal to the Manifold Topology
	6.3 The Class Z of Zeeman-Göbel Topologies
	6.4 Topologies Different Than the Manifold Topology
	6.5 In the Beginning Was the Metric…or the Topology?
	6.6 Ambient Cosmology: A Failure Due to a Topological Misconception
	6.7 Towards an Evolving Topology and a Quantum Theory of Gravity
	6.8 The Need to Take Methods of General Topology More Seriously
	References
7 Analysis of Generalized BBM Equations: Symmetry Groups and Conservation Laws
	7.1 Introduction
	7.2 Lie Point Symmetries
		7.2.1 Lie Point Symmetries of Eq.(7.12)
		7.2.2 Lie Point Symmetries of Eq.(7.13)
	7.3 Optimal System
		7.3.1 Optimal System for Eq.(7.12)
		7.3.2 Optimal System for Eq.(7.13)
	7.4 Reductions and Exact Solutions
		7.4.1 Reductions for Eq.(7.12)
		7.4.2 Reductions for Eq.(7.13)
	7.5 Travelling Waves
	7.6 Conservation Laws
	7.7 Potential Symmetries
	7.8 Conclusions
	References
8 Symmetry Analysis and Conservation Laws for Some Boussinesq Equations with Damping Terms
	8.1 Introduction
	8.2 Lie Classical Symmetries and Reductions
		8.2.1 Lie Symmetries and Reductions for Eq.(8.4)
		8.2.2 Lie Symmetries and Reductions for Eq.(8.5)
	8.3 Multiplier Conservation Laws Method
		8.3.1 Multipliers and Conservation Laws for Eq.(8.4)
		Conservation Laws
		8.3.2 Multipliers and Conservation Laws for Eq.(8.5)
		Conservation Laws
	8.4 Double Reduction Method and Exact Solutions
	8.5 Conclusions
	References
9 On Some Variable Exponent Problems with No-Flux Boundary Condition
	9.1 Introduction
	9.2 Functional Framework
	9.3 Critical Point Tools
	9.4 Problems with (Isotropic) Variable Exponent
	9.5 Problems with Anisotropic Variable Exponent
	9.6 Final Comments
	References
10 On the General Decay for a System of Viscoelastic Wave Equations
	10.1 Introduction
	10.2 Preliminaries
	10.3 Technical Lemmas
	10.4 General Decay Result
	References
11 Mathematical Theory of Incompressible Flows: Local Existence, Uniqueness, and Blow-Up of Solutions in Sobolev–Gevrey Spaces
	11.1 Local Existence and Uniqueness of Solutions
	11.2 Blow-Up Criteria for the Solution
		11.2.1 Limit Superior Related to Hsa,σ(R3)
		11.2.2 Blow-Up of the Integral Related to L1(R3)
		11.2.3 Blow-Up Inequality Involving L1(R3)
		11.2.4 Blow-Up Inequality Involving Hsa,σ(R3)
		11.2.5 Generalization of the Blow-Up Criteria
		11.2.6 Main Blow-Up Criterion Involving Hsa,σ(R3)
	References
12 Mathematical Research for Models Which is Related to Chemotaxis System
	12.1 Introduction
	12.2 The (Quasilinear) Keller–Segel Model
		12.2.1 The Quasilinear Parabolic–Elliptic Keller–Segel System (τ=0)
		12.2.2 The Quasilinear Parabolic–Parabolic Keller–Segel System (τ=1)
	12.3 The (Quasilinear) Chemotaxis System with Consumption of Chemoattractant
		12.3.1 A Priori Estimates
	12.4 The (Quasilinear) Chemotaxis–Haptotaxis Model
	12.5 The (Quasilinear) Keller–Segel–Navier–Stokes System
		12.5.1 Preliminaries and Theorems
		12.5.2 A Priori Estimates
		12.5.3 The Global Solvability of Regularized Problem (12.5.13)
			12.5.3.1 Regularity Properties of Time Derivatives
			12.5.3.2 Passing to the Limit: Proof of Theorem 12.5.1
	12.6 Open Problem
	References
13 Optimal Control of Quasivariational Inequalities with Applications to Contact Mechanics
	13.1 Introduction
	13.2 Quasivariational Inequalities
		13.2.1 Notation and Preliminaries
		13.2.2 Existence and Uniqueness
		13.2.3 A Convergence Result
	13.3 Optimal Control of Quasivariational Inequalities
		13.3.1 Existence of Optimal Pairs
		13.3.2 Convergence of Optimal Pairs
		13.3.3 A Relevant Particular Case
	13.4 A Frictional Contact Problem
		13.4.1 Function Spaces
		13.4.2 The Model
		13.4.3 Weak Solvability
		13.4.4 Optimal Control
		13.4.5 A One-Dimensional Example
	13.5 Conclusion
	References
14 On Generalized Derivative Sampling Series Expansion
	14.1 Introduction and Motivation
	14.2 Master Sampling Theorem for Deterministic Signals
	14.3 Discussion of Certain Special Cases
	14.4 Brief Invitation to Piranashvili Processes
	14.5 Master Sampling Theorem for Stochastic Signals
	14.6 Generalized Sampling Series for Random Fields
	References
15 Voronoi Polygonal Hybrid Finite Elements and Their Applications
	15.1 Introduction
	15.2 Basics of Voronoi Polygons
	15.3 Formulations of Polygonal Hybrid Finite Element
		15.3.1 Governing Equations
		15.3.2 Mesh Discretization
		15.3.3 Displacement Interpolations
		15.3.4 Double-Variable Hybrid Functional
			15.3.4.1 Stationary Condition of the Proposed Variational Functional
			15.3.4.2 Theorem on the Existence of Extremum
		15.3.5 Formation of Resulting Linear Equations
			15.3.5.1 Element Equations
			15.3.5.2 Assembly of Global Equation
			15.3.5.3 Imposition of Displacement Constraints
		15.3.6 Recovery of Rigid-Body Motion
		15.3.7 Algorithm for Implementing the Solution Procedure
	15.4 Applications
		15.4.1 Cook\'s Problem
		15.4.2 Thick Cylinder Under Internal Pressure
		15.4.3 Infinite Plate with a Centered Elliptical Hole Under Tension
	15.5 Conclusions
	References
16 Variational Methods for Schrödinger Type Equations
	16.1 Introduction
	16.2 Background Material
		16.2.1 Recalling Sobolev Spaces
		16.2.2 Basic Notions of Differential Calculus in Hilbert Spaces
		16.2.3 The Ljusternick-Schnirelmann Category
		16.2.4 Schrödinger Type Equations
	16.3 The Case of Given Potential: The Fractional Schrödinger Equation
		16.3.1 The Variational Setting
		16.3.2 Compactness for I and Eμ: Existence of a Ground State Solution
		16.3.3 The Barycenter Map
		16.3.4 Proof of Theorem 16.3.2
	16.4 The Case of Unknown Potential: The Fractional Schrödinger-Poisson System
		16.4.1 The Variational Setting
			16.4.1.1 The Problem at ``Infinity\'\'
		16.4.2 Compactness for I and Eμ: Existence of a Ground State Solution
		16.4.3 The Barycenter Map
		16.4.4 Proof of Theorem 16.4.2
	References
17 Nonlinear Nonhomogeneous Elliptic Problems
	17.1 Introduction
	17.2 Regularity and Auxiliary Results
	17.3 Maximum Principle: Comparison Results
	17.4 Eigenvalue Problems
	17.5 Superlinear Problems
	17.6 Nodal Solutions
	17.7 Dirichlet (p,2)-Equations
	17.8 Remarks
	References
18 Summability of Double Sequences and Double Series Over Non-Archimedean Fields: A Survey
	18.1 Double Sequences and Double Series
	18.2 Silverman–Toeplitz, Schur, and Steinhaus Theorems
	18.3 Characterization of 2-Dimensional Schur Matrices
	18.4 The Nörlund Method for Double Sequences
	18.5 Weighted Mean Method for Double Sequences
	18.6 (M, λm,n) Method (or Natarajan Method) for Double Sequences
	References
19 On Approximate Solutions of Linear and Nonlinear Singular Integral Equations
	19.1 Introduction
	19.2 Newton–Kantorovich Method for Two-Dimensional Nonlinear Singular Integral Equations
		19.2.1 Newton–Kantorovich Method for Eq.(19.1)
	19.3 Some Integral Operators in Holder Space
		19.3.1 Some Properties of the Integral Operator (19.27)
		19.3.2 Existence of the Solutions due to Banach Contraction Principle
	19.4 Fixed Point Theory and Approximate Solutions of Nonlinear Singular Integral Equations
		19.4.1 On the Solution of Nonlinear Singular Integral Equations
	19.5 Nonlinear Singular Integro-Differential Equations
		19.5.1 On the Solution of Eq.(19.65)
	19.6 The Collocation Method for the Solution of Boundary Integral Equations
		19.6.1 On the Existence and Uniqueness of the Solution
		19.6.2 Collocation Method
	19.7 On the Approximate Solution of Singular Integral Equations with Negative Index
		19.7.1 Collocation Method
		19.7.2 Conclusions for Sect.19.7
	References
20 On Difference Double Sequences and Their Applications
	20.1 Introduction
	20.2 Definitions
	20.3 Related Difference Double Sequence Spaces
	20.4 Applications
	References
21 Pointwise Convergence Analysis for Nonlinear Double m-Singular Integral Operators
	21.1 Introduction
	21.2 Preliminaries
	21.3 Pointwise Convergence
	21.4 Fatou Type Convergence
	21.5 Rate of Convergence
	21.6 Concluding Remarks
	References
22 A Survey on p-Adic Integrals
	22.1 Introduction
	22.2 p-Adic Integrals on Z p
		22.2.1 Volkenborn Integral and Its Some Properties
		22.2.2 Fermionic p-Adic Integral and Its Some Properties
	22.3 p-Adic q-Integrals on Z p
		22.3.1 q-Volkenborn Integral and Its Some Properties
		22.3.2 Fermionic p-Adic q-Integral and Its Some Properties
	22.4 Weighted p-Adic Integrals on Z p
		22.4.1 (α,β)-Volkenborn Integral and Its Some Properties
			22.4.1.1 The q-Daehee Polynomials with Weight ( α,β)
		22.4.2 (α,β)-Fermionic Integral and Its Some Properties
			22.4.2.1 The q-Changhee Polynomials with Weight ( α,β)
			22.4.2.2 The q-Boole Polynomials with Weight ( α,β)
	22.5 Conclusion
	References
23 On Statistical Deferred Cesàro Summability
	23.1 Introduction
	23.2 Cesàro and Deferred Cesàro Summability Methods
	23.3 A Korovkin-Type Theorem
	23.4 Rate of Statistical Deferred Cesàro Summability
	23.5 Concluding Remarks and Observations
	References