# A General Iterative Scheme for Fixed Points of Nonlinear Operators

**A General Iterative Scheme for Fixed Points of Nonlinear Operators**.

**ABSTRACT**

This thesis examines a general iterative scheme for fixed points of nonlinear operators using the special cases of the Krasnoselskii-Mann (KM) iterative procedure for particular choices of the nonexpansive operator N.

We prove many theorems and lemmas that help to see the relationship between some iterative algorithms. The iterative scheme *x*^{k+1} =Tx^{k} =

(1- α)x^{k}+ αNx^{k} has been shown to be one of the schemes that can be generally used to represent some iterative schemes for finding fixed points of nonlinear operators.

**TABLE OF CONTENTS**

Title page..…. i

Declaration page…ii

Certification… iii

Dedication… iv

Acknowledgement…. v

Abstract… vi

Content… vii

**1 .0 INTRODUCTION AND PRELIMINARIES**

1.1 Introduction . .…. 1

1.2 Research questions . .…1

1.3 Aim and Objectives of the Study . . .. . 2

1.4 Statement of the Research Problem .. . 2

1.5 Research Methodology . . . .. 3

1.6 Convergence in the normed space . 3

1.7 Nonexpansive operator . . . … 3

1.8 Hilbert space . 3

1.9 Fixed Point . . .…….3

1.10 Operators on Hilbert Space . . 4

1.11 Firmly nonexpansive operator . .…… 5

1.12 Monotone Operator . …. 5

1.13 Strict contraction . . 5

1.14 Average operator . . . .5

1.15 V-ism operator . . . .… 6

1.16 Sequences in a Metric Space . . . 6

1.17 Subsequences in a Metric Space . . .. 6

1.18 Convergence of a Sequence in a Metric Spac…. 7

1.19 Bounded Sequences . . . . …….7

1.20 Attracting mapping . …..8

1.21 Outline of the thesis . . …….9

**2.0 Literature Review**

2.1 A Superior Implementation of the Algebraic Reconstruction Technique (ART) Algorithm . .10

2.2 The Multiple-sets Split Feasibility Problem and its Applications for Inverse Problems . 11

2.3 Projection Methods and their advantages . . . 12

2.4 The Split Feasibility Problem . . . …. 13

2.5 Cimmino’s Method and the Algebraic Reconstruction Technique . 14

2.6 Bandlimited Extrapolation Methods . . … . 15

2.7 The Landweber algorithms . . . . . . 19

2.8 The Geometric Properties of Banach Spaces and Nonlinear Iterations . . . .21

2.9 Interior point optimization algorithms . . .. . . . 24

**3.0 SOME NONLINEAR OPERATORS AND THEIR RELATIONS**

3.1 Introduction. . . 25

3.2 Firmly nonexpansive operator . . . . 31

3.3 Prototype of a strongly attracting mapping . . .. . . 35

3.4 A system of generalized equilibrium problems . . 36

**4.0 SOME ITERATIVE ALGORITHMS FOR FIXED POINT OF NONLINEAR OPERATORS**

4.1 Introduction . . . .40

4.2 Constrained Optimization Algorithms . . . . 40

4.3 Orthogonal Projection onto Sets C and Q . . . 41

4.4 The convex feasibility problem . . .…. 43

4.5 The split feasibility problem . . . .. 44

**5.0 THE MAIN RESULT**

5.1 Introduction . . . 48

5.2 The Main result..48

**6.0 SUMMARY, CONCLUSION AND RECOMMENDATIONS**

6.1 Summary . . . . . ..51

6.2 Conclusion . … 52

6.3 Recommendations…..52

REFERENCES……..53

** INTRODUCTION**

Many problems in Mathematics and related field can be solved by finding fixed point of a particular operator, and algorithms for finding such points play a prominent role in a number of applications.

The article by Bauschke (1996) is fundamental to this work. This section deals with definitions of some basic terms used in the subsequent discussions while some examples are also given to make these definitions clearer.

**Statement of the Research Problem**

In the algorithms of interest here, the operator T is selected so that the set Fix(T) contains those vectors *z *that possess the properties we desire that is ‘find a general iterative scheme for some iterative algorithms for nonlinear operators’ finding a fixed point of the some iterative schemes leads to a solution of the problem.

Some applications involve constrained optimization, in which we seek a vector *x *in a given convex set C that minimizes a certain function *f*. For suitable y > 0 the fixed points of the operator T = P_{C}(I – y∇*f *) will solve the problem under conditions to be discussed below.

REFERENCES

Anderson, A. and Kak, A. (1984) Simultaneous algebraic reconstruction technique a superior implementation of the ART algorithm Ultrason. Orlando, FL: Aca Imaging 6, 81-94

Aubin, J-P. (1993) Optima and Equilibria: an Introduction to Nonlinear Analysis (Mathematical Society Books in Mathematics (Berlin: Springer)

Bauschke, H. and Borwein, J. (1996) On projection algorithms for solving convex feasibility problems SIAM Rev. 38, 367-426

Bertero, M. and Boccacci, P. (1998) Introduction to Inverse Problems in Imaging (Bristol: Institute of Physics Publishing)

Bertsekas, D. P. (1997) A new class of incremental gradient methods for least squares problems SIAM J. Optim. 7, 913- 926

Borwein, J. and Lewis, A.(2000) Convex Analysis and Nonlinear Optimization (Canadian Mathematical Society Books in Mathematics) (New York: Springer)

Bregman, L. M. (1967) The relaxation method of finding the common point of convex sets and its application to the solution of problems in convex programming, USSR Compt. Math. Phys.7,200- 217

Byrne, C. L. (1995) Erratum and addendum to ‘Iterative image reconstruction algorithms based on cross-entropy minimization’ IEEE Trans. Image Process. 4, 225-6

Byrne, C. L. (1997) Convergent block-iterative algorithms for image reconstruction from inconsistent data IEEE Trans. Image Process. 6, 1296- 1304

Byrne, C. L .(1998) Accelerating the EMML algorithm and related iterative algorithms by rescaled block-iterative (RBI) methods IEEE Trans. Image Process. 7, 100-9