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.