Unified Theory for Fractional and Entire Differential Operators: An Approach via Differential Quadruplets and Boundary Restriction Operators (Frontiers in Mathematics)

Contents
1 Introduction
	1.1 Motivations
	1.2 Some Issues in Fractional Calculus
	1.3 Unified Theories
		1.3.1 How to Construct a Unified Theory?
		1.3.2 Outline of Our Unified Theory
	1.4 About This Book
		1.4.1 Preliminary Remarks
		1.4.2 Outline of the Book
		1.4.3 How to Read This Book?
	1.5 Notation and Conventions
		1.5.1 Sets of Numbers
		1.5.2 Miscellaneous Notation
		1.5.3 Matrices
		1.5.4 Spaces
2 Background on Functional Analysis
	2.1 Vector Spaces
		2.1.1 Basic Definitions
		2.1.2 Quotient Spaces
		2.1.3 Direct Sums
	2.2 Normed Vector Spaces
		2.2.1 Generalities on Normed Spaces
		2.2.2 Two Hahn–Banach Theorems and Consequences
		2.2.3 The Second Anti-Dual Space
		2.2.4 Annihilators
		2.2.5 Operators on Normed Spaces
		2.2.6 Adjoint of Operators
			2.2.6.1 Introduction and Definition
			2.2.6.2 Properties
		2.2.7 Identifications
			2.2.7.1 Introduction
			2.2.7.2 Identification by a Surjective Isometry
			2.2.7.3 Identification Through a Linear Embedding
			2.2.7.4 Basic Examples of Identification
		2.2.8 Operators on Cartesian Powers of a Space
			2.2.8.1 Induced Operators on Xn
			2.2.8.2 Tensor Products in Xn
	2.3 Banach Spaces
		2.3.1 Generalities on Banach Spaces
		2.3.2 Projections
		2.3.3 Reflexive Banach Spaces
	2.4 Operators on Banach Spaces
		2.4.1 Operators on Reflexive Spaces
		2.4.2 Operators with Closed Range
		2.4.3 Product of Operators
		2.4.4 Linear Embeddings
		2.4.5 An Extension of Adjoint Operators
			2.4.5.1 Extension of T
			2.4.5.2 Extension of T on Reflexive Spaces
		2.4.6 Neumann Series
	2.5 Hilbert Spaces
		2.5.1 Definitions and First Properties
		2.5.2 Identification and Consequences
		2.5.3 Gelfand Triplets
		2.5.4 Self-Adjoint Operators
		2.5.5 On the Brézis Paradox
3 Background on Fractional Calculus
	3.1 An Overview of the Bochner Integral
		3.1.1 Measurability
			3.1.1.1 Definition and First Properties
			3.1.1.2 Caratheodory Functions
		3.1.2 The Quotient Spaces L0(Ω, X)
			3.1.2.1 Definitions and First Properties
			3.1.2.2 Measurability of g(·-·)f
			3.1.2.3 Essential Limits
		3.1.3 Integration
	3.2 Functional Spaces
		3.2.1 Lebesgue and Sobolev Spaces
			3.2.1.1 Definitions
			3.2.1.2 Properties
			3.2.1.3 Direct Sums in Lebesgue Spaces
		3.2.2 Hölder Spaces
			3.2.2.1 Definitions
			3.2.2.2 Properties
		3.2.3 The Space Cln([0, b], X)
	3.3 Convolution
		3.3.1 Introduction
		3.3.2 Definitions and Basic Properties
	3.4 Special Functions of Fractional Calculus
		3.4.1 The Euler Gamma Function
		3.4.2 The Riemann–Liouville Kernels
		3.4.3 Mittag–Leffler Functions
	3.5 Marchaud Fractional Derivatives
	3.6 Fractional Riemann–Liouville Primitives
4 Differential Triplets on Hilbert Spaces
	4.1 The Analysis of the Operator ddx Revisited
		4.1.1 The Operator D1L2 and the Fundamental Theorem of Calculus
		4.1.2 Revisited Proof of Basic Results
	4.2 Differential Triplets
		4.2.1 Definitions
		4.2.2 A One-Parameter Family of Differential Triplets
	4.3 Boundary Restriction Operators
		4.3.1 Definition and First Results
		4.3.2 Large and Small Restrictions of A
		4.3.3 Minimal Operators
		4.3.4 The Domain Structure Theorem
	4.4 Maximal Operators with Finite-Dimensional Kernel
		4.4.1 Abstract Endogenous Boundary Coordinates
			4.4.1.1 Endogenous Boundary Conditions of A*
			4.4.1.2 Dimension of Es
			4.4.1.3 Endogenous Boundary Conditions of A
			4.4.1.4 Limit Cases and Endogenous Boundary Coordinates
		4.4.2 Boundary Restriction Operators of D1L2
5 Differential Quadruplets on Banach Spaces
	5.1 Differential Quadruplets
		5.1.1 Definitions
		5.1.2 Differential Triplets and Quadruplets
	5.2 Boundary Restriction Operators
		5.2.1 Preliminaries
		5.2.2 Large and Small Restrictions of A
		5.2.3 Boundary Restriction Operators of Realizations of Quadruplets
	5.3 The Domain Structure Theorem
		5.3.1 Regular Differential Quadruplets
		5.3.2 The Domain Structure Theorem
		5.3.3 Boundary Restriction Operators of Regular Quadruplets
	5.4 On the Adjoint of Boundary Restriction Operators
		5.4.1 A Variational Method
		5.4.2 A Direct Method
	5.5 Differential Quadruplets with Finite Dimensional Kernels
		5.5.1 Abstract Endogenous Boundary Conditions
			5.5.1.1 Endogenous Boundary Conditions of A*
			5.5.1.2 Dimension of Es
			5.5.1.3 Endogenous Boundary Conditions of A
		5.5.2 Properties of Boundary Restriction Operators
	5.6 Algebraic Calculus of Differential Triplets and Quadruplets
		5.6.1 Involution-Induced Transforms
			5.6.1.1 Transforms of Quadruplets
			5.6.1.2 Transforms of Triplets
		5.6.2 Conjugation by an Isomorphism
			5.6.2.1 General Setting
			5.6.2.2 Conjugation by an Involution
		5.6.3 Adjoint and Reverse Transforms
		5.6.4 Caputo Extensions
			5.6.4.1 Definitions
			5.6.4.2 Boundary Restriction Operators
			5.6.4.3 Differential Triplets
			5.6.4.4 Finite Dimensional Kernels
		5.6.5 Products of Differential Quadruplets and Triplets
			5.6.5.1 Definition and First Results for Quadruplets
			5.6.5.2 Maximal, Minimal, and Pivot Operators
			5.6.5.3 Compatibility and Differential Triplets
		5.6.6 Caputo Extensions of Products of Operators
	5.7 Reflexive Lebesgue Spaces of Vector Valued Functions
		5.7.1 Endogenous Boundary Values in Lebesgue Spaces
			5.7.1.1 Definitions
			5.7.1.2 The Boundary Values Trick
			5.7.1.3 The Quadruplet Qp, Y, 2 and Its Adjoint
			5.7.1.4 Change of Complementary Subspace of kerA
			5.7.1.5 Integration by Parts
		5.7.2 Boundary Restriction Operators and Their Adjoint
		5.7.3 The Quadruplet Qp, Y, n
			5.7.3.1 Endogenous Boundary Values
	5.8 Quadruplets on Cartesian Powers of a Space
		5.8.1 Definition of
		5.8.2 Properties of Induced Quadruplets
6 Fractional Differential Triplets and Quadruplets on Lebesgue Spaces
	6.1 Introduction
	6.2 Two Toy Models
		6.2.1 The Triplet RLTα, K
			6.2.1.1 Definition
			6.2.1.2 Endogenous Boundary Values
			6.2.1.3 The Adjoint Triplet of RLTα
			6.2.1.4 Boundary Restriction Operators and Their Adjoint
		6.2.2 The Triplet Tα, K
			6.2.2.1 Definition
			6.2.2.2 Endogenous Boundary Values
			6.2.2.3 The Adjoint Triplet of Tα
			6.2.2.4 Boundary Restriction Operators and Their Adjoint
	6.3 Sonine Kernels
	6.4 Entire and Fractional Differential Operators on Lebesgue Spaces
	6.5 The Quadruplet RLQp, Y,
		6.5.1 Introduction
		6.5.2 Endogenous Boundary Values
			6.5.2.1 Case Where  Lies in Lp(0, b)Lq(0, b)
			6.5.2.2 Case Where  Lies in Lq(0, b)Lp(0, b)
			6.5.2.3 Case Where  Lies in Lp(0, b)Lq(0, b)
			6.5.2.4 Case Where  Lies in L1(0, b)(Lq(0, b)Lp(0, b))
		6.5.3 The Adjoint Quadruplet of RLQp, Y,
			6.5.3.1 Case Where  Lies in Lp(0, b)Lq(0, b)
			6.5.3.2 Case Where  Lies in Lq(0, b)Lp(0, b)
			6.5.3.3 Case Where  Lies in Lp(0, b)Lq(0, b)
			6.5.3.4 Case Where  Lies in L1(0, b)(Lq(0, b)Lp(0, b))
		6.5.4 Boundary Restriction Operators of RLQp,Y,  and Their Adjoint
			6.5.4.1 Intrinsic Representations
			6.5.4.2 Case Where  Lies in Lp(0, b)Lq(0, b)
			6.5.4.3 Case Where  Lies in Lq(0, b)Lp(0, b)
			6.5.4.4 Case Where  Lies in Lp(0, b)Lq(0, b)
			6.5.4.5 Case Where  Lies in L1(0, b)(Lq(0, b)Lp(0, b))
	6.6 The Quadruplet RLQp, Y, n+
		6.6.1 Introduction
		6.6.2 Endogenous Boundary Values
		6.6.3 The Adjoint Quadruplet of RLQp, Y, n+
	6.7 The Quadruplet Qp, Y,
		6.7.1 Introduction
		6.7.2 Endogenous Boundary Values
			6.7.2.1 The Case Where  Lies in Lp(0, b)Lq(0, b)
			6.7.2.2 Case Where  Lies in Lq(0, b)Lp(0, b)
			6.7.2.3 Case Where  Lies in Lp(0, b)Lq(0, b)
			6.7.2.4 Case Where  Lies in L1(0, b)(Lq(0, b)Lp(0, b))
			6.7.2.5 Regularity of Qp, Y,
		6.7.3 The Adjoint Quadruplet of Qp, Y,
			6.7.3.1 Case Where  Lies in Lp(0, b)Lq(0, b)
			6.7.3.2 Case Where  Lies in Lq(0, b)Lp(0, b)
			6.7.3.3 Case Where  Lies in Lp(0, b)Lq(0, b)
			6.7.3.4 Case Where  Lies in L1(0, b)(Lq(0, b)Lp(0, b))
		6.7.4 Boundary Restriction Operators of Qp,Y,  and Their Adjoint
			6.7.4.1 Two Special Boundary Restriction Operators
			6.7.4.2 Case Where  Lies in Lp(0, b)Lq(0, b)
			6.7.4.3 Case Where  Lies in Lq(0, b)Lp(0, b)
	6.8 The Quadruplet Qp, Y, n+
		6.8.1 Introduction
		6.8.2 Endogenous Boundary Values
	6.9 The Riemann-Liouville Quadruplet and Its Extensions
		6.9.1 The Riemann-Liouville Quadruplet RLQp,K, α and Its Adjoint
			6.9.1.1 Endogenous Boundary Values
			6.9.1.2 Case Where gα Lies in L1(0, b)(Lq(0, b)Lp(0, b))
		6.9.2 Caputo Extensions of RLQp,K, α
			6.9.2.1 The Quadruplet Qp,K, αγ, δ
			6.9.2.2 Endogenous Boundary Values
		6.9.3 The Quadruplet Qp,K, n+α
			6.9.3.1 Introduction
			6.9.3.2 Two Special Boundary Restriction Operators
7 Endogenous Boundary Value Problems
	7.1 Introduction
	7.2 Abstract Linear Problems
		7.2.1 The Resolvent Map  R LB
			7.2.1.1 The Resolvent Map  R LB on Banach Spaces
			7.2.1.2 The Map Resolvent  R LBf, X on Lebesgue Spaces
		7.2.2 The Generalized Exponential Functions on Lebesgue Spaces
			7.2.2.1 Position of the Problem
			7.2.2.2 Definition
			7.2.2.3 Delsarte Representation of Generalized Exponential Functions
			7.2.2.4 Properties of Generalized Exponential Functions
		7.2.3 Well-Posed Problems on Banach Spaces
			7.2.3.1 The Projection PkerA
			7.2.3.2 Formulation of the Problem
			7.2.3.3 Solvability
		7.2.4 Problems on Cartesian Powers of a Banach Space
			7.2.4.1 Introduction
			7.2.4.2 Preliminaries
			7.2.4.3 A Well-Posed Inhomogeneous Problem
			7.2.4.4 Representation of the Solutions to the Homogeneous Equation
		7.2.5 Higher-Order Problems
			7.2.5.1 Formulation of the Problem
			7.2.5.2 The Order Reducing Method
			7.2.5.3 Well Posedness of (7.2.30)
			7.2.5.4 Structure of the Solution Set of Homogeneous Equation
		7.2.6 Problems Relying on Regular Quadruplets
			7.2.6.1 Formulation of the Problem
			7.2.6.2 Solvability of (7.2.48)
			7.2.6.3 Unconditionally Solvability and Well Posedness
			7.2.6.4 Maximal Operators with Finite-Dimensional Kernel
	7.3 Initial Value Problems I: Linear Equations
		7.3.1 The Framework
		7.3.2 Problems with A:=D1X
			7.3.2.1 Formulation of the Problem
			7.3.2.2 Well Posedness
		7.3.3 Problems with A:=RLDX
			7.3.3.1 Case Where  Lies in Lp(0, b)
			7.3.3.2 Case Where  Lies in L1(0, b)Lp(0, b)
		7.3.4 Problems with A:=DX
			7.3.4.1 Case Where  Lies in Lp(0, b)Lq(0, b)
			7.3.4.2 Case Where  Lies in Lq(0, b)Lp(0, b)
			7.3.4.3 Case Where  Lies in Lp(0, b)Lq(0, b)
			7.3.4.4 Case Where  Lies in L1(0, b)(Lq(0, b)Lp(0, b))
	7.4 Initial Value Problems II: Systems and Higher-Order Equations
		7.4.1 The Framework
		7.4.2 Systems with A:=D1X
			7.4.2.1 A Well-Posed Problem
			7.4.2.2 Representation of the Solutions to the Homogeneous Equation
		7.4.3 Higher Order Equations with A:=D1X
		7.4.4 Systems with A:=DX
			7.4.4.1 A Well-Posed Problem
			7.4.4.2 Representation of the Solutions to the Homogeneous Equation
		7.4.5 Higher-Order Equations with A:=DX
	7.5 Boundary Value Problems with Finite-Dimensional Phase Spaces
		7.5.1 First-Order Problems
			7.5.1.1 Formulation of the Problem
			7.5.1.2 Solvability of (7.5.1)
		7.5.2 Problems with AM:=RLDV
			7.5.2.1 Formulation of the Problem
			7.5.2.2 Solvability of (7.5.7)
		7.5.3 Problems with AM:=DV
			7.5.3.1 Case Where  Lies in Lp(0, b)Lq(0, b)
			7.5.3.2 Case Where  Lies in Lq(0, b)Lp(0, b)
	7.6 Abstract Sublinear Problems
		7.6.1 Formulation of the Problem
		7.6.2 Solvability Under a Strong Assumption on F
		7.6.3 Invariance with Respect to Conjugation and Well Posedness
	7.7 Sublinear Initial Value Problems
		7.7.1 Bielecki\'s Norms on Lp(Y)
		7.7.2 The Nonlinear Term
		7.7.3 First-Order Problems
			7.7.3.1 Formulation of the Problem
			7.7.3.2 Well Posedness of (7.7.6)
			7.7.3.3 Nonautonomous Linear Problems
		7.7.4 Problems with A:=DZ
			7.7.4.1 Case Where  Lies in Lp(0, b)Lq(0, b)
			7.7.4.2 Case Where  Lies in Lq(0, b)Lp(0, b)
			7.7.4.3 Case Where  Lies in Lp(0, b)Lq(0, b)
			7.7.4.4 Case Where  Lies in L1(0, b)(Lq(0, b)Lp(0, b))
8 Abstract and Fractional Laplace Operators
	8.1 Introduction
	8.2 Weak Products on V
		8.2.1 Definition
		8.2.2 Properties
	8.3 Abstract Laplace Operators
		8.3.1 Definitions
		8.3.2 Properties
		8.3.3 Endogenous Boundary Conditions
			8.3.3.1 Underlying Quadruplet (SQ)2
			8.3.3.2 Underlying Quadruplet SQ
			8.3.3.3 Endogenous Boundary Conditions of -SQ, D
	8.4 Abstract Dirichlet Problems
		8.4.1 Formulation of the Problem
		8.4.2 Well Posedness Under kerAM-Regularity
			8.4.2.1 Solvability with Maximal Operators Having Finite-Dimensional Kernel
		8.4.3 On the Homogeneous Dirichlet Problem
	8.5 Fractional Dirichlet Problems
		8.5.1 Problems Involving Qp, Yf, 1
			8.5.1.1 Laplace Operators on Lp(Yf)
			8.5.1.2 Extensions of Laplace Operators
			8.5.1.3  Dirichlet Problems on Lp(Yf)
			8.5.1.4 Representation of the Solution
		8.5.2 Problems Involving Qp, Yf,
			8.5.2.1 The Case Where  Lies in Lp(0, b)Lq(0, b)
			8.5.2.2 Case Where  Lies in Lq(0, b)Lp(0, b)
	8.6 Regularity Issues
		8.6.1 Formulation of the Problem
		8.6.2 Preliminaries
		8.6.3 Regularity Analysis
			8.6.3.1 Case Where hb.c, 2= 0
			8.6.3.2 Case Where hb.c, 2= 0
		8.6.4 A Regularity Result with Respect to the Data
Nomenclature
Bibliography
Index