MHB Generalization of Banach contraction principle

ozkan12
Messages
145
Reaction score
0
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...
 
Physics news on Phys.org
ozkan12 said:
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.
 
A sphere as topological manifold can be defined by gluing together the boundary of two disk. Basically one starts assigning each disk the subspace topology from ##\mathbb R^2## and then taking the quotient topology obtained by gluing their boundaries. Starting from the above definition of 2-sphere as topological manifold, shows that it is homeomorphic to the "embedded" sphere understood as subset of ##\mathbb R^3## in the subspace topology.
Back
Top