Mild Differentiability Conditions for Newton’s Method in Banach Spaces

Preface......Page 7
Contents......Page 11
1 The Newton-Kantorovich Theorem......Page 14
1.1 Brief History of Newton\'s Method......Page 15
1.2 Kantorovich\'s Theory......Page 22
1.2.1 The Newton-Kantorovich Theorem......Page 23
1.2.2 Application to a Hammerstein Integral Equation......Page 24
1.3 Accessibility of Newton\'s Method......Page 28
1.3.1 Basin of Attraction......Page 29
1.3.3 Domain of Parameters......Page 30
1.4 Mild Alternatives to Kantorovich\'s Conditions......Page 32
1.5 A Technique Based on Recurrence Relations......Page 34
2 Operators with Lipschitz Continuous First Derivative......Page 36
2.1.1 Recurrence Relations......Page 37
2.1.2 Analysis of the Scalar Sequence......Page 39
2.1.3 Proof of the Semilocal Convergence......Page 40
2.2 Application to a Chandrasekhar Integral Equation......Page 44
3 Operators with Hölder Continuous First Derivative......Page 48
3.1.1 Recurrence Relations......Page 49
3.1.2 Analysis of the Scalar Sequence......Page 50
3.1.3 A Semilocal Convergence Result......Page 53
3.2.2 Case p=1......Page 57
3.2.3 Scalar Sequence......Page 58
3.2.4.1 Region of Accessibility......Page 59
3.2.4.3 Comparative Study......Page 60
3.3 Application to a Hammerstein Integral Equation......Page 62
3.3.1 Existence and Uniqueness of a Solution......Page 64
3.3.2 Localization of Solutions......Page 65
3.3.3 Approximation of Solutions......Page 68
4 Operators with Hölder-Type Continuous First Derivative......Page 70
4.1.1 Recurrence Relations......Page 72
4.1.2 Analysis of the Scalar Sequence......Page 74
4.1.3 Semilocal Convergence Result......Page 76
4.2.1 Case p=0......Page 79
4.3.1 Existence and Uniqueness of a Solution......Page 80
4.3.2 Localization of Solutions......Page 85
4.3.3 Approximation of Solutions......Page 86
5 Operators with ω-Lipschitz Continuous First Derivative......Page 90
5.1.1 Recurrence Relations......Page 92
5.1.2 Analysis of the Scalar Sequences......Page 94
5.1.3 Semilocal Convergence Result......Page 95
5.2.1 On the R-order of Convergence......Page 99
5.3 Application to a Hammerstein Integral Equation......Page 103
5.3.1 Existence and Uniqueness of a Solution......Page 104
5.3.2 Localization of Solutions......Page 105
5.3.3 Approximation of Solution......Page 106
6 Improving the Domain of Starting Points Based on Center Conditions for the First Derivative......Page 108
6.1.1.1 Recurrence Relations......Page 110
6.1.1.2 Analysis of the Scalar Sequence......Page 112
6.1.1.3 Semilocal Convergence Result......Page 113
6.1.2 Domain of Parameters......Page 116
6.1.3 Application to a Conservative Problem......Page 120
6.1.3.1 Setting up a Finite Difference Scheme......Page 121
6.2 Operators with ω-Lipschitz Continuous First Derivative......Page 124
6.2.1.1 Recurrence Relations......Page 125
6.2.1.2 Analysis of the Scalar Sequence......Page 127
6.2.1.3 Semilocal Convergence Result......Page 128
6.2.2.1 Domain of Parameters......Page 130
6.2.2.2 Application to a Conservative Problem......Page 136
6.2.2.3 Particular Cases......Page 138
7 Operators with Center ω-Lipschitz Continuous First Derivative......Page 151
7.1 Convergence Analysis......Page 155
7.2 Application to an Elliptic Problem......Page 160
7.3 Particular Case......Page 162
8 Using Center ω-Lipschitz Conditions for the First Derivative at Auxiliary Points......Page 167
8.1 Convergence Analysis......Page 170
8.2 Particular Cases and Applications to Elliptic Problems......Page 174
8.2.2 Domain of Parameters......Page 175
8.2.3 Region of Accessibility......Page 177
8.2.4 Applications......Page 179
8.2.5 The Hölder Case......Page 182
8.2.6 General Case......Page 183
Bibliography......Page 186