CLRg Property

In mathematics, the notion of ''“common limit in the range”'' property denoted by CLRg property is a theorem that unifies, generalizes, and extends the contractive mappings in fuzzy metric spaces, where the range of the mappings does not necessarily need to be a closed subspace of a non-empty set $X$.

Suppose $X$ is a non-empty set, and $d$ is a distance metric; thus, $(X, d)$ is a metric space. Now suppose we have self mappings $f,g : X \to X.$ These mappings are said to fulfil CLRg property if 

$\lim_ f x_ = \lim_ g x_ = gx,$
for some $x \in X.$ 

Next, we give some examples that satisfy the CLRg property.

Examples

Source:

Example 1

Suppose $(X,d)$ is a usual metric space, with $X=[0,\infty).$ Now, if the mappings $f,g: X \to X$ are defined respectively as follows:

*$fx = \frac$
*$gx = \frac$

for all $x\in X.$ Now, if the following sequence $\=\$ is considered. We can see that

$\lim_fx_ = \lim_gx_ = g0 = 0,$

thus, the mappings $f$ and $g$ fulfilled the CLRg property.

Another example that shades more light to this CLRg property is given below

Example 2

Let $(X,d)$ is a usual metric space, with $X=[0,\infty).$ Now, if the mappings $f,g: X \to X$ are defined respectively as follows:

*$fx = x+1$
*$gx = 2x$

for all $x\in X.$ Now, if the following sequence $\=\$ is considered. We can easily see that

$\lim_fx_ = \lim_gx_ = g1 = 2,$

hence, the mappings $f$ and $g$ fulfilled the CLRg property.

References

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Fixed points (mathematics)