CLRg property

In mathematics, the notion of “common limit in the range” property denoted by CLRg property^[1][2][3] 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 ${\displaystyle X}$.

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

$${\displaystyle \lim _{k\to \infty }fx_{k}=\lim _{k\to \infty }gx_{k}=gx,}$$ for some ${\displaystyle x\in X.}$ 

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

Examples

Source:^[1]

Example 1

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

• ${\displaystyle fx={\frac {x}{4}}}$
• ${\displaystyle gx={\frac {3x}{4}}}$

for all ${\displaystyle x\in X.}$ Now, if the following sequence ${\displaystyle \{x_{k}\}=\{1/k\}}$ is considered. We can see that

$${\displaystyle \lim _{k\to \infty }fx_{k}=\lim _{k\to \infty }gx_{k}=g0=0,}$$

thus, the mappings ${\displaystyle f}$ and ${\displaystyle g}$ fulfilled the CLRg property.

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

Example 2

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

• ${\displaystyle fx=x+1}$
• ${\displaystyle gx=2x}$

for all ${\displaystyle x\in X.}$ Now, if the following sequence ${\displaystyle \{x_{k}\}=\{1+1/k\}}$ is considered. We can easily see that

$${\displaystyle \lim _{k\to \infty }fx_{k}=\lim _{k\to \infty }gx_{k}=g1=2,}$$

hence, the mappings ${\displaystyle f}$ and ${\displaystyle g}$ fulfilled the CLRg property.