CLRg property

In mathematics, the notion of “common limit in the range” property denoted by CLRg property{{Cite journal|title=Common Fixed Point Theorems for a Pair of Weakly Compatible Mappings in Fuzzy Metric Spaces|first1=Wutiphol|last1=Sintunavarat|first2=Poom|last2=Kumam|date=August 14, 2011|journal=Journal of Applied Mathematics|volume=2011|pages=e637958|doi=10.1155/2011/637958|doi-access=free }}{{Cite journal|title=Common fixed points for R-weakly commuting in fuzzy metric spaces

|first1=Wutiphol |last1=Sintunavarat|first2=Poom |last2=Kumam|date=2012|journal=Annali dell'Universita'di Ferrara|volume=58|pages=389-406|doi=10.1007/s11565-012-0150-z|doi-access=free}}{{Cite journal|title=A comparison of various noncommuting conditions in metric fixed point theory and their applications

date= February 13, 2014|journal=Fixed Point Theory and Applications|volume=2014|pages=1–33|doi=10.1186/1687-1812-2014-38|doi-access=free }} 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.

Examples

Source:

=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.

References

Category:Fixed points (mathematics)