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1 Introduction to the Topic of the Course |
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1 | (14) |
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1.1 Some Outline of the Problem Under Consideration |
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1 | (3) |
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1.2 The Finite Dimensional Monotonicity Methods |
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4 | (7) |
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1.3 Applications to Discrete Equations |
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11 | (4) |
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2 Some Excerpts from Functional Analysis |
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15 | (40) |
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2.1 On the Weak Convergence |
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15 | (8) |
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2.2 On the Function Spaces |
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23 | (12) |
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2.3 On the du Bois-Reymond Lemma and the Regularity of Solutions |
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35 | (1) |
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2.4 Nemytskii Operator and the Krasnosel'skii Type Theorem |
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36 | (3) |
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2.5 Differentiation in Banach Spaces |
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39 | (7) |
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2.6 A Detour on a Direct Method in the Calculus of Variation |
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46 | (9) |
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55 | (26) |
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55 | (4) |
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3.2 On Some Properties of Monotone Operators |
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59 | (6) |
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3.3 Different Types of Continuity |
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65 | (4) |
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69 | (2) |
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3.5 An Example of a Monotone Mapping |
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71 | (4) |
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3.6 Condition (S) and Some Other Related Notions |
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75 | (2) |
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3.7 The Minty Lemma and the Fundamental Lemma for Monotone Operators |
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77 | (4) |
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4 On the Fenchel-Young Conjugate |
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81 | (10) |
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4.1 Some Background from Convex Analysis |
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81 | (4) |
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4.2 On the Conjugate and Its Properties |
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85 | (6) |
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91 | (16) |
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5.1 Basic Concepts and Properties |
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91 | (7) |
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5.2 Invertible Potential Operators |
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98 | (3) |
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5.3 Criteria for Checking the Potentiality |
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101 | (6) |
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6 Existence of Solutions to Abstract Equations |
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107 | (18) |
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107 | (2) |
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6.2 The Browder--Minty Theorem |
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109 | (7) |
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6.3 Some Useful Corollaries |
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116 | (1) |
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6.4 The Strongly Monotone Principle |
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117 | (1) |
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6.5 Pseudomonotone Operators |
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118 | (3) |
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6.6 The Leray--Lions Theorem |
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121 | (4) |
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7 Normalized Duality Mapping |
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125 | (12) |
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7.1 Introductory Notions and Properties |
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125 | (5) |
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7.2 Examples of a Duality Mapping |
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130 | (2) |
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7.2.1 A Duality Mapping for H01 (0, 1) |
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130 | (1) |
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7.2.2 On a Duality Mapping for LP (0, 1) |
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131 | (1) |
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7.2.3 On a Duality Mapping for W1,p0 (0, 1) |
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132 | (1) |
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7.3 On the Strongly Monotone Principle in Banach Spaces |
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132 | (2) |
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7.4 On a Duality Mapping Relative to a Normalization Function |
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134 | (3) |
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137 | (6) |
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8.1 Basic Notions and Results |
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137 | (3) |
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8.2 On the Galerkin and the Ritz Method for Potential Equations |
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140 | (3) |
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9 Some Selected Applications |
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143 | (32) |
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9.1 On Nonlinear Lax-Milgram Theorem and the Nonlinear Orthogonality |
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143 | (4) |
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9.2 On a Certain Converse of the Lax-Milgram Theorem |
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147 | (1) |
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9.3 Applications to the Differentiability of the Fenchel-Young Conjugate |
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148 | (2) |
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9.4 Applications to Minimization Problems |
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150 | (4) |
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9.5 Applications to the Semilinear Dirichlet Problem |
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154 | (6) |
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9.5.1 Examples and Special Cases |
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159 | (1) |
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9.6 Applications to Problems with the Generalized p---Laplacian |
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160 | (5) |
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9.7 Applications of the Leray---Lions Theorem |
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165 | (4) |
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9.8 On Some Application of a Direct Method |
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169 | (6) |
| References |
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175 | (4) |
| Index |
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179 | |
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