Convergence in Norm of Modified Krasnoselskii-Mann Iteration for Fixed Points of Asymptotically Demi Contractive Mappings

Abstract 

This project report deals with the class of asymptotically demicontractive mappings in Hilbert spaces. We noted some historical aspects concerning the concept of asymptotically demicontractivity and studied a regularized variant of the Krasnoselskii-Mann iteration scheme, which ensured the strong convergence of the generated sequence towards the least norm element of the set of fixed points of asymptotically demicontractive mapping.

Contents

Certification ii

Dedication iii

Acknowledgement iv

Abstract viii

1 Introduction 1

1.1 General Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Aim and Scope of Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Some Useful Identities and Inequalities in Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.4 Weak and Strong Convergence . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.4.1 Norm (strong) convergence . . . . . . . . . . . . . . . . . . . . . . . 9

1.4.2 Weak convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.4.3 Some Fundamental Results on Weak and Strong Convergence . . . 10

1.5 Nearest Point Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.6 Reflexive Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

1.6.1 Canonical Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2 SOME IMPORTANT CLASSES OF OPERATORS AND ITERATIVE PROCESSES 22

2.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.2 OPERATORS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.2.1 Lipschitz operators: . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.2.2 Accretive Operators . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.2.3 Strictly Pseudocontractive Mappings . . . . . . . . . . . . . . . . . 24

2.2.4 Pseudocontractive Mappings . . . . . . . . . . . . . . . . . . . . . . 25

2.2.5 Demicontractive Mappings . . . . . . . . . . . . . . . . . . . . . . . 26

2.2.6 Asymptotically Nonexpansive operators: . . . . . . . . . . . . . . . 28

2.2.7 Uniformly L-Lipschitzian Mappings: . . . . . . . . . . . . . . . . . . 28

2.2.8 k-strictly Asymptotically Pseudocontractive operators: . . . . . . . 28

2.2.9 Asymptotically demicontractive operators: . . . . . . . . . . . . . . 28

2.3 ITERATIVE PROCESSES . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.3.1 The Picard Iteration Process . . . . . . . . . . . . . . . . . . . . . . 30

2.3.2 The Krasnoselskii Iteration Process . . . . . . . . . . . . . . . . . . 30

2.3.3 The Mann Iteration Process . . . . . . . . . . . . . . . . . . . . . . 31

2.3.4 The Ishikawa Iteration Process . . . . . . . . . . . . . . . . . . . . 31

2.3.5 Mann-type Iteration Process with Errors in the sense of Liu . . . . 32

2.3.6 Ishikawa-type Iteration Process with Errors in the sense of Liu . . . 32

2.3.7 Mann-type and Ishikawa-type Iteration Process with Errors in the sense of Xu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.4 REVIEW OF SOME CONVERGENCE THEOREMS . . . . . . . . . . . . . . . . . . . . . . . . 33

3 MODIFIED KRASNOSELSKII-MANN ALGORITHM FOR A CLASS OF NONLINEAR MAPPING 38

3.1 MAIN RESULTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

References 58