 ## Fixed Point Theory and Banach Contraction Principle

The source of the fixed point theory, which dates to the later piece of the nineteenth-century, vigorously lays on the utilization of successive approximations to set up the existence and uniqueness of solutions, especially differential conditions. This method is associated with many big names which include Cauchy, Liouville, Lipschitz, Peano, Fredholm, and above all Picard. Indeed, the antecedents of a fixed point theoric approach are express in crafted by Picard. Be that as it may, the Polish mathematician Stefan Banach is credited for setting the fundamental thoughts into a theoretical edge reasonable for wide applications well beyond the scope of elementary differential and integral equations.

As expected, a self-mapping f on a nonempty set X is said to have a fixed point x if x remains invariant under f (i.e., fx=x). In order to illustrate the fact, let us consider the simple quadratic equation x^2-6x+5=0: Clearly, x= 1 and x= 5 are the roots of this equation. Also, we can rewrite this equation in the following form:

x=(x^2+ 5)/6

Define a real-valued mapping f by fx=(x^2+5)/6, then the above equation reduces to fx=x. We notice that x=1 and x=5 are the two fixed points of f.

Thus, from the above observations, we conclude that the problem of finding the solution of a functional equation fx-x=0 is the same as finding a fixed point of the mapping f. A mapping f on a non-empty set X can have no fixed point, unique fixed point, finite fixed point, and infinite fixed point as given below:

Example: Let X=R be a non-empty set and f:X→X a mapping defined as:

1. fx=x+a, for a≠0;
2. fx=x/2
3. fx=x^2
4. fx=x

Notice that, in conditions

1.  f has no fixed point,
2.  f has unique fixed point,
3.  f has two (finite) fixed points and
4.  f has infinite fixed points.

Fixed point theory has acquired a driving force, because of its wide scope of relevance, to determine assorted issues exuding from nonlinear differential conditions, a theory of nonlinear indispensable conditions, game theory, control theory, mathematical economics, etc. For instance, in theoretical economics, such as general equilibrium theory, a circumstance emerges where one has to know whether the answer for an arrangement of conditions fundamentally exists. All the more explicitly, under what conditions will an answer fundamentally exist. The numerical investigation of this inquiry generally depends on fixed point theory. Thus discovering fundamental and adequate conditions for the existence of fixed points is an interesting aspect.

The Banach contraction mapping principle is one of the pivotal results of the analysis. It is generally considered as the good spirit of metric fixed point theory. Additionally, its importance lies in its huge relevance in various parts of Mathematics.
Theorem: Let (X, d)be complete metric space and f:X→X a mapping satisfy:

d(fx, fy)≤αd(x,y) ∀x, y ∈X,

where, α∈[0,1). Then f has a unique fixed point in X.

The above theorem has been generalized and improved by numerous researchers predominantly by supplanting contraction mappings with moderately more broad contractive mappings, by enlarging the class of metric and this practice is still going on.

Moreover, the classical Banach contraction principle guarantees that Picard sequence of f based at any point converges to the fixed point, i.e., starting at any point x_0∈X, the repeated iterations of the mapping at x_0 yields a sequence that converges to the unique fixed point of f. The advantage of this principle is that its hypotheses are very simple and always give a unique fixed point that can be found using a straightforward method. The only disadvantage attached to this principle is that assuming the mapping to be contraction forces the mapping f to be continuous at each point of the space. However, this principle is widely considered as the source of metric fixed point theory and one of the most fundamental and powerful tools of nonlinear analysis.