CLRg property

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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 

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

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

Examples

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:

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

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

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

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 $\{x_{k}\}=\{1+1/k\}$ is considered. We can easily see that

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

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