Generalization of Banach contraction principle

[ozkan12]

Let $\left(X,d\right)$ be a complete metric space and suppose that $f:X\to X$ satisfies

$d\left(fx,fy\right)\le\beta\left(d\left(x,y\right)\right)d\left(x,y\right)$ for all x,y $\in$ X where $\beta$ is a decreasing function on ${R}^{+}$ to $[0,1)$. Then $f$ has a unique fixed point.

The mapping $d\left(fx,fy\right)\le\beta\left(d\left(x,y\right)\right)d\left(x,y\right)$ more general than banach contraction...How this happens ? In my opinion, If we take $\beta\left(t\right)=c$, $c\in [0,1)$ we get banach principle...İs this true ? Can you help me ? Thank you for your attention...Best wishes...

 

[Ackbach]

Let $\left(X,d\right)$ be a complete metric space and suppose that $f:X\to X$ satisfies

$d\left(fx,fy\right)\le\beta\left(d\left(x,y\right)\right)d\left(x,y\right)$ for all x,y $\in$ X where $\beta$ is a decreasing function on ${R}^{+}$ to $[0,1)$. Then $f$ has a unique fixed point.

The mapping $d\left(fx,fy\right)\le\beta\left(d\left(x,y\right)\right)d\left(x,y\right)$ more general than banach contraction...How this happens?

A mathematician somewhere wanted to know if he could generalize the Banach Fixed Point Theorem - still get guaranteed fixed points while relaxing the hypotheses of the theorem.

In my opinion, If we take $\beta\left(t\right)=c$, $c\in [0,1)$ we get banach principle...İs this true?

Yes - so $d\left(fx,fy\right)\le\beta\left(d\left(x,y\right)\right)d\left(x,y\right)$ is more general than a contraction, because $\beta(t)$ wouldn't need to be constant, necessarily.

 

[math771]

Hi there,

Yes, you are correct. This mapping is more general than the Banach contraction principle. The Banach contraction principle states that if we have a mapping $f: X \to X$ on a complete metric space $X$ such that $d\left(fx,fy\right)\le k d\left(x,y\right)$ for some constant $k<1$, then $f$ has a unique fixed point. This is a special case of the mapping given in the post, where $\beta\left(t\right)=k$.

However, in the general case, $\beta$ can take on any value in the interval $[0,1)$, not just a constant. This allows for a wider range of mappings that still satisfy the condition, and therefore have a unique fixed point. So, in a sense, this mapping is a more general version of the Banach contraction principle.

I hope this helps clarify things for you. Let me know if you have any further questions. Best wishes to you as well!