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 .
Suppose is a non-empty set, and is a distance metric; thus, is a metric space. Now suppose we have self mappings These mappings are said to fulfil CLRg property ifÃÂ
for some ÃÂ
Next, we give some examples that satisfy the CLRg property.
Source:
Suppose is a usual metric space, with Now, if the mappings are defined respectively as follows:
for all Now, if the following sequence is considered. We can see that
thus, the mappings and fulfilled the CLRg property.
Another example that shades more light to this CLRg property is given below
Let is a usual metric space, with Now, if the mappings are defined respectively as follows:
for all Now, if the following sequence is considered. We can easily see that
hence, the mappings and fulfilled the CLRg property.