CLRg property explained

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 

$$\backslash lim\_\; f\; x\_\; =\; \backslash lim\_\; g\; x\_\; =\; gx,$$for some

x \in X.  

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

Examples

Example 1.

Suppose

(X,d) is a usual metric space, with X = [ 0 , inf ty). Now, if the mappings f,g: X \to X are defined respectively as follows:fx = (x)/(4) gx = (3 x)/(4)

for all

x\in X. Now, if the following sequence \{x _{k\}=\{1/k\}} is considered. We can see that

$$\backslash lim\_fx\_\; =\; \backslash lim\_gx\_\; =\; g0\; =\; 0,$$

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 , inf ty). Now, if the mappings f,g: X \to X are defined respectively as follows:fx = x +1 gx = 2 x

for all

x\in X. Now, if the following sequence \{x _{k\}=\{1+1/k} \} is considered. We can easily see that

$$\backslash lim\_fx\_\; =\; \backslash lim\_gx\_\; =\; g1\; =\; 2,$$

f and g fulfilled the CLRg property.

Notes and References

1. Common Fixed Point Theorems for a Pair of Weakly Compatible Mappings in Fuzzy Metric Spaces. Wutiphol. Sintunavarat. Poom. Kumam. August 14, 2011. Journal of Applied Mathematics. 2011. e637958. 10.1155/2011/637958. free .
2. FIXED POINT THEOREMS IN MENGER SPACES USING THE $(CLR\_$\$) $ PROPERTY AND APPLICATIONS. MDAD . MOHAMMAD. Pant . BD. CHAUHAN . SUNNY. 2012. Journal of Nonlinear Analysis and Optimization: Theory \& Applications. 3. 225–237. 10.1186/1687-1812-2012-55. free .
3. A comparison of various noncommuting conditions in metric fixed point theory and their applications. Ravi. P Agarwal. Ravindra . K Bisht. Naseer . Shahzad. February 13, 2014. Fixed Point Theory and Applications. 2014. 1–33. 10.1186/1687-1812-2014-38. free .