Draft: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

$$\lim_{k \to \infty} f x_{k} = \lim_{k \to \infty} g x_{k} = gx,$$ for some $$x \in X.$$

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

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.