This section introduces fixed-point theorems, particularly focusing on the Brouwer and Kakutani fixed-point theorems. It begins by highlighting the significance of these theorems in game theory, where they were used by John von Neumann and J. F. Nash to prove fundamental theorems. The concept of a fixed-point theorem is explained through the equation \( x = f(x) \), where a solution \( x \) is a point that remains unchanged by the function \( f \). The section also discusses the variety of fixed-point theorems, including those that provide conditions for uniqueness or multiplicity of solutions. The Brouwer fixed-point theorem is noted for its non-constructive nature, meaning it only guarantees the existence of a solution without providing a method to find it. However, the contraction mapping principle is an exception, being constructive and allowing for the construction of a solution through successive approximations. The elementary form of the contraction mapping theorem states that if \( M \) is a closed set of real numbers and \( f \) is a contraction mapping on \( M \), then \( f \) has a unique fixed point in \( M \). The uniqueness is straightforward, and the existence of a fixed point is proven using an iterative scheme, where the sequence of approximations \( x_n \) converges to the fixed point.This section introduces fixed-point theorems, particularly focusing on the Brouwer and Kakutani fixed-point theorems. It begins by highlighting the significance of these theorems in game theory, where they were used by John von Neumann and J. F. Nash to prove fundamental theorems. The concept of a fixed-point theorem is explained through the equation \( x = f(x) \), where a solution \( x \) is a point that remains unchanged by the function \( f \). The section also discusses the variety of fixed-point theorems, including those that provide conditions for uniqueness or multiplicity of solutions. The Brouwer fixed-point theorem is noted for its non-constructive nature, meaning it only guarantees the existence of a solution without providing a method to find it. However, the contraction mapping principle is an exception, being constructive and allowing for the construction of a solution through successive approximations. The elementary form of the contraction mapping theorem states that if \( M \) is a closed set of real numbers and \( f \) is a contraction mapping on \( M \), then \( f \) has a unique fixed point in \( M \). The uniqueness is straightforward, and the existence of a fixed point is proven using an iterative scheme, where the sequence of approximations \( x_n \) converges to the fixed point.