The fixed point theory has numerous applications within mathematics and also in the diverse fields as biology, chemistry, economics, engineering, game theory, management, social sciences, etc. The Banach contraction principle is one of the most widely applicable fixed point theorems in all of analysis.

In this chapter, we focus on applications of the Banach contraction principle and its variants to the following problems:

• • System of linear equations

• • Differential equations and second-order two-point boundary-value problems

• • Linear and nonlinear Volterra integral equations, Fredholm integral equations, and mixed Volterra–Fredholm integral equations

Application to System of Linear Equations

In this section, we present an application of the Banach contraction principle to the system of linear equations (2.1).

As we have seen in Example 2.3, the system of linear equations (2.1) can be reformulated as follows:

and I is the identity matrix. If Tn → n is a matrix transformation defined by

then finding a solution of the system (6.1) is equivalent to find a fixed point of T. So, if T, defined by (6.2), can be proved to be a contraction mapping, then one can use the Banach contraction principle to obtain a unique fixed point of T, and hence, a unique solution of the system (6.1).

The conditions under which T is a contraction mapping depend on the choice of the metric on X = n. Here we discuss only one case and have left two others for exercise.

then the linear system (6.1) of n linear equations in n unknowns has a unique solution.

Proof Since X = n with respect to the metric d∞ is complete, it is sufficient to prove that the mapping T defined by (6.2) is a contraction. Indeed,

Thus, T is a contraction mapping. By Banach contraction principle (Theorem 2.1), the linear systems (6.1) has a unique solution.

Example 6.1 Consider the following system of linear equations:

the system of linear equations (6.4) can be written as Note that Here, Thus, the matrix transformation Tdefined by

is a contraction under the metric d∞. Therefore, the system of linear equations given by (6.4) has a unique solution.