Part I Split feasibility problem
Chapter 1 Introduction to split feasibility problem
The theory of nonlinear operators is theoretical basis and basic tools of nonlinear science， and it has already been an important branch of modern mathematics and plays an important role in the other branches. The fixed point theory of nonlinear operators is a constituent important part of onlinear functional analysis， especially， the problem of approximating to solutions of nonlinear operator equations (systems) becomes the active topic that people study in recent years.
The fixed points of nonlinear operators are closely related to the equilibrium problems， variational inequalities， feasibility problems， zero points of nonlinear operators， which can be also converted to one another. Based on the transformation of the relationship between them， some new nonlinear operators and related iteration algorithms are constructed， through approximating fixed points of the new nonlinear operators， the equilibrium problems or variational inequalities and some other related problems are solved， and the strong (weak) convergence theorems are obtained. The nonlinear operators theory is studied mainly by generalizing the space， improving the iterative algorithm and reducing the restrictions of coefficient or the constraint of operators， then the more meaningful and the more generalized results are also obtained.
Especially in recent years， with the rapid development of modern science and technology and the continuous improvement in computer performance， nonlinear operator theory has been widely used in many fields. For example， on the background of practical application for image reconstructions and the intensity modulated radiation therapy， the split feasibility problems become a research hot spot of nonlinear function analysis. Therefore， the study of nonlinear operators becomes very important and significant.
This chapter contains the basic definition and initial results needed for a study of Banach space and Hilbert space. In order to explain certain notations， terminologies and elementary results used throughout this book， first we need to introduce the definition of norms.
1.1 Abstract space and their property
Metric space: Let X be a nonempty set， a metric on X is a real function d of ordered pairs of elements of X which satisfies the following three conditions:
(1) d(x， y)≥ 0 and d(x， y)=0x= y;
(2) d(x， y) = d(y， x);
(3) d(x， y)≤d(x， z) + d(z， y).
Then， d(x， y) is called the distance between x and y.
The function d assigns to each pair (x， y) of elements of X is a nonnegative real number d(x， y)， which does not depend on the order of the elements.
A metric space consists of two objects: A nonempty set X and a metric d on X. Whenever it can be done without causing confusion， we denote the metric space (X， d) by the symbol X which is used for the underlying set of points.
Let X be a metric space with metric d， if x0 is a point of X and r is a positive real number.
The open ball Sr(x0) with center x0 and radius r is the subset of X defined by
Sr(x0) = {x ∈ X : d(x0， x) < r}.
The closed ball Sr(x0) is defined by
Sr(x0) = {x ∈ X : d(x0， x)≤r}，
where r is a nonnegative real number. A subset G of the metric space X is called an open set， if given any point x in C， there exists a positive real number r such that Sr(x)G.
A subset F of X is called a closed set if the complement FC of F is open.
Let X and Y be metric spaces with metrics d1 and d2， and let f be a mapping of X into Y .
Definition 1.1 f is said to be continuous at a point x0 in X， if for each ε > 0， there exists δ > 0 such that
A mapping of X into Y is said to be continuous if it is continuous at each point in its domain X.
Definition 1.2 A mapping f of X into Y is called nonexpansive， if
Definition 1.3 A mapping f of X into Y is called contractive， if there exists a nonnegative number r < 1 such that
