Basic Monotonicity Methods with Some Applications

Description

This textbook introduces some basic tools from the theory of monotone operators together with some of their applications. Examples that work for ordinary differential equations are provided. The illustrating material is kept relatively simple, while at the same time offering inspiring applications to the reader. The material will appeal to graduate students in mathematics who want to learn some basics in the theory of monotone operators. Furthermore, it offers a smooth transition to studying more advanced topics pertaining to more refined applications by shifting to pseudomonotone operators, and next, to multivalued monotone operators. Preface Contents 1 Introduction to the Topic of the Course 1.1 Some Outline of the Problem Under Consideration 1.2 The Finite Dimensional Monotonicity Methods 1.3 Applications to Discrete Equations 2 Some Excerpts from Functional Analysis 2.1 On the Weak Convergence 2.2 On the Function Spaces 2.3 On the du Bois-Reymond Lemma and the Regularity of Solutions 2.4 Nemytskii Operator and the Krasnosel'skii Type Theorem 2.5 Differentiation in Banach Spaces 2.6 A Detour on a Direct Method in the Calculus of Variation 3 Monotone Operators 3.1 Monotonicity 3.2 On Some Properties of Monotone Operators 3.3 Different Types of Continuity 3.4 Coercivity 3.5 An Example of a Monotone Mapping 3.6 Condition (S) and Some Other Related Notions 3.7 The Minty Lemma and the Fundamental Lemma for Monotone Operators 4 On the Fenchel-Young Conjugate 4.1 Some Background from Convex Analysis 4.2 On the Conjugate and Its Properties 5 Potential Operators 5.1 Basic Concepts and Properties 5.2 Invertible Potential Operators 5.3 Criteria for Checking the Potentiality 6 Existence of Solutions to Abstract Equations 6.1 Preliminary Result 6.2 The Browder–Minty Theorem 6.3 Some Useful Corollaries 6.4 The Strongly Monotone Principle 6.5 Pseudomonotone Operators 6.6 The Leray–Lions Theorem 7 Normalized Duality Mapping 7.1 Introductory Notions and Properties 7.2 Examples of a Duality Mapping 7.2.1 A Duality Mapping for H01( 0,1) 7.2.2 On a Duality Mapping for Lp( 0,1) 7.2.3 On a Duality Mapping for W01,p( 0,1) 7.3 On the Strongly Monotone Principle in Banach Spaces 7.4 On a Duality Mapping Relative to a Normalization Function 8 On the Galerkin Method 8.1 Basic Notions and Results 8.2 On the Galerkin and the Ritz Method for Potential Equations 9 Some Selected Applications 9.1 On Nonlinear Lax-Milgram Theorem and the Nonlinear Orthogonality 9.2 On a Certain Converse of the Lax-Milgram Theorem 9.3 Applications to the Differentiability of the Fenchel-Young Conjugate 9.4 Applications to Minimization Problems 9.5 Applications to the Semilinear Dirichlet Problem 9.5.1 Examples and Special Cases 9.6 Applications to Problems with the Generalized p-Laplacian 9.7 Applications of the Leray–Lions Theorem 9.8 On Some Application of a Direct Method References Index