how to know about different types of contraction mapping to prove fixed point theorem
like
ciric, rus, f-contraction, suzuki contraction etc
Appana Kiran Kumar
To prove the fixed-point theorem using different types of contraction mappings, it's essential to understand what a contraction mapping is and how it relates to the fixed-point theorem. A contraction mapping is a function on a metric space that shrinks the distance between two points. Specifically, a function f: X → X, where X is a metric space with distance metric d, is called a contraction mapping if there exists a constant k (0 < k < 1) such that:
d(f(x), f(y)) ≤ k * d(x, y) for all x, y ∈ X.
The fixed-point theorem states that every contraction mapping on a complete metric space has a unique fixed point, i.e., a point x ∈ X such that f(x) = x.
Different types of contraction mappings can be used to prove the fixed-point theorem, and here are a few examples:
Ciric (Ćirić) Contraction: A function f: X → X is a Ciric contraction if there exists a constant k (0 < k < 1) such that:
d(f(x), f(y)) ≤ k * [d(f(x), x) + d(f(y), y)] for all x, y ∈ X.
Rus-Contractions: A function f: X → X is a Rus contraction if there exists a constant k (0 < k < 1) such that:
d(f(x), f(y)) ≤ k * max[d(f(x), x), d(f(y), y)] for all x, y ∈ X.
F-Contraction (F-Map): A function f: X → X is an F-contraction if there exists a function g: [0, ∞) → [0, ∞) such that g(0) = 0 and:
d(f(x), f(y)) ≤ g(d(x, y)) for all x, y ∈ X.
Suzuki Contraction: A function f: X → X is a Suzuki contraction if there exists a constant k (0 < k < 1) such that:
d(f(x), f(y)) ≤ k * [d(x, f(x)) + d(y, f(y)) + d(x, y)] for all x, y ∈ X.
These are just a few examples of different types of contraction mappings. There are other variations and generalizations that researchers have explored.
When you want to prove the fixed-point theorem using any of these contraction mappings, you'll typically show that the mapping satisfies the required contraction condition, and then you can invoke the fixed-point theorem to conclude the existence of a unique fixed point.
Remember that the fixed-point theorem and its applications have significance in various fields, including functional analysis, optimization, computer science, and various areas of mathematics.
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