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 

$$\underset{k\to \mathrm{\infty }}{\lim }f{x}_k=\underset{k\to \mathrm{\infty }}{\lim }g{x}_k=gx,$$ 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 $X=[0,\mathrm{\infty }).$ 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

$$\underset{k\to \mathrm{\infty }}{\lim }f{x}_k=\underset{k\to \mathrm{\infty }}{\lim }g{x}_k=g0=0,$$

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 $X=[0,\mathrm{\infty }).$ 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

$$\underset{k\to \mathrm{\infty }}{\lim }f{x}_k=\underset{k\to \mathrm{\infty }}{\lim }g{x}_k=g1=2,$$

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