g(x) - g(y) ≤ k |x - y|

This condition implies that the function g is a contraction mapping in a neighborhood around the fixed point p. If this condition is met, the Banach Fixed-Point Theorem ensures that the sequence {x_n} converges to p.

Additional Considerations

1. Continuity: The function g(x) should be continuous in the neighborhood of the fixed point.

2. Initial Guess: The choice of the initial guess x_0 must be sufficiently close to the fixed point p to ensure the iterative sequence converges.

Example: Newton's Method

For methods like Newton's method, a sufficient condition for convergence is that the function f is continuously differentiable in a neighborhood of the root and that f'(p) ≠ 0, where p is the root. Additionally, if the derivative f' is bounded away from zero near p, convergence can be ensured.

Special Cases: Gauss-Jacobi and Gauss-Siedel Iterative Methods

Speaking of specific iterative methods, such as the Gauss-Jacobi and Gauss-Siedel methods, there is an additional sufficient condition for their convergence:

The coefficient matrix of the system must be diagonally dominant.

This condition is not necessary but can be proven using matrix analysis and linear algebra. For a more in-depth understanding, I prepared a set of notes using Mathcha - Online Math Editor. You can access them here.

In conclusion, while specific conditions for convergence can vary, understanding the concept of a contraction mapping provides a robust and widely applicable sufficient condition for the convergence of iterative methods.