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Contents
Chapter 1 Introduction and Background 1
1.1 Nonlocal problem of abstract evolution equations 1
1.2 Monotone iterative method based on lower and upper solutions 5
1.3 Impulsive di.erential equations 7
1.4 Fractional di.erential equations 8
1.5 Non-autonomous evulution equations 10
1.6 Nonlocal evolution equations with delay 12
Reference 15
Chapter 2 Basic Definitions and Theorems 24
2.1 Theory of operator semigroups 24
2.2 Measure of noncompactness 27
2.3 Fixed point theorems 28
2.4 Cone and partial order 30
2.5 Basic properties of fractional integral and derivative 31
Reference 33
Chapter 3 Strong Solutions for Nonlocal Evolution Equations 36
3.1 Existence and uniqueness of strong solutions for semilinear evolution equations with nonlocal initial conditions 36
3.2 Regularity for evolution equations with nonlocal initial conditions on infinite interval 51
3.3 Asymptotic stability of strong solutions for evolution equations with nonlocal initial conditions 64
3.4 Notes 72
Reference 73
Chapter 4 Monotone Iterative Method Based on Lower and Upper Solutions 75
4.1 Monotone iterative technique for semilinear evolution equations with nonlocal initial conditions 76
4.2 Perturbation method for nonlocal evolution equations with instantaneous impulses 89
4.2.1 Initial value problem of linear evolution equations 91
4.2.2 g is compact in PC(J;E) 92
4.2.3 g is continuous in PC(J;E) 96
4.2.4 The case of lower and upper solutions do not exist 97
4.2.5 Applications 99
4.3 Iterative method for a new class of evolution equations with non-
instantaneous impulses 102
4.3.1 Linear evolution equation with non-instantaneous impulses 104
4.3.2 T(·) and gk are compact 105
4.3.3 T(·) is not compact, gk is compact 113
4.3.4 T(·) and gk are not compact 115
4.3.5 Applications 120
4.4 Notes 124
Reference 125
Chapter 5 Nonlocal Evolution Equation of Fractional Order 128
5.1 Fractional evolution equations with nonlocal conditions and noncompact semigroup 129
5.2 Nonlocal problem for fractional evolution equations of mixed type 141
5.2.1 Existence of mild solutions 142
5.2.2 Existence of positive mild solutions 151
5.3 Fractional evolution equations with mixed monotone nonlocal conditions 155
5.3.1 T(t) is compact for t > 0 157
5.3.2 T(t)(t > 0) is a C0-semigroup 162
5.3.3 Existence of mild solution 165
5.3.4 Applications 169
5.4 Notes 172
Reference 172
Chapter 6 Fractional Non-autonomous Evolution Equations 175
6.1 Fractional non-autonomous evolution equation with nonlocal conditions 176
6.2 Application 194
6.3 Notes and comments 196
Reference 198
Chapter 7 Nonlocal Evolution Equation with Delay 200
7.1 Existence of solutions for delay evolution equations with nonlocal
conditions 201
7.1.1 Existence results under the situation that g is Lipschitz continuous 203
7.1.2 Existence results under the situation that g is compact 210
7.1.3 An example 213
7.2 Neutral delay evolution equations with mixed nonlocal plus local initial conditions 214
7.2.1 Existence of mild solutions 216
7.2.2 The regularity of solutions 224
7.2.3 An example 229
7.3 Fractional retarded evolution equations subjected to mixed nonlocal plus local initial conditions 233
7.3.1 Existence of mild solutions 234
7.3.2 Existence of positive mild solutions 243
7.3.3 An example 245
7.4 Notes 247
Reference 248
Index 251
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