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CLRg property

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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 {\displaystyle X} .

Suppose X {\displaystyle X} is a non-empty set, and d {\displaystyle d} is a distance metric; thus, ( X , d ) {\displaystyle (X,d)} is a metric space. Now suppose we have self mappings f , g : X → X . {\displaystyle f,g:X\to X.} These mappings are said to fulfil CLRg property if 

lim k → ∞ f x k = lim k → ∞ g x k = g x , {\displaystyle \lim _{k\to \infty }fx_{k}=\lim _{k\to \infty }gx_{k}=gx,} for some x ∈ X . {\displaystyle x\in X.}  

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

Examples

Source:

Example 1

Suppose ( X , d ) {\displaystyle (X,d)} is a usual metric space, with X = [ 0 , ∞ ) . {\displaystyle X=[0,\infty ).} Now, if the mappings f , g : X → X {\displaystyle f,g:X\to X} are defined respectively as follows:

  • f x = x 4 {\displaystyle fx={\frac {x}{4}}}
  • g x = 3 x 4 {\displaystyle gx={\frac {3x}{4}}}

for all x ∈ X . {\displaystyle x\in X.} Now, if the following sequence { x k } = { 1 / k } {\displaystyle \{x_{k}\}=\{1/k\}} is considered. We can see that

lim k → ∞ f x k = lim k → ∞ g x k = g 0 = 0 , {\displaystyle \lim _{k\to \infty }fx_{k}=\lim _{k\to \infty }gx_{k}=g0=0,}

thus, the mappings f {\displaystyle f} and g {\displaystyle g} fulfilled the CLRg property.

Another example that shades more light to this CLRg property is given below

Example 2

Let ( X , d ) {\displaystyle (X,d)} is a usual metric space, with X = [ 0 , ∞ ) . {\displaystyle X=[0,\infty ).} Now, if the mappings f , g : X → X {\displaystyle f,g:X\to X} are defined respectively as follows:

  • f x = x + 1 {\displaystyle fx=x+1}
  • g x = 2 x {\displaystyle gx=2x}

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

lim k → ∞ f x k = lim k → ∞ g x k = g 1 = 2 , {\displaystyle \lim _{k\to \infty }fx_{k}=\lim _{k\to \infty }gx_{k}=g1=2,}

hence, the mappings f {\displaystyle f} and g {\displaystyle g} fulfilled the CLRg property.

References

  1. ^ Sintunavarat, Wutiphol; Kumam, Poom (August 14, 2011). "Common Fixed Point Theorems for a Pair of Weakly Compatible Mappings in Fuzzy Metric Spaces". Journal of Applied Mathematics. 2011: e637958. doi:10.1155/2011/637958.
  2. MOHAMMAD, MDAD; BD, Pant; SUNNY, CHAUHAN (2012). "FIXED POINT THEOREMS IN MENGER SPACES USING THE $(CLR\_$\{$ST$\}$) $ PROPERTY AND APPLICATIONS". Journal of Nonlinear Analysis and Optimization: Theory \& Applications. 3: 225–237. doi:10.1186/1687-1812-2012-55. hdl:10397/6571.
  3. P Agarwal, Ravi; K Bisht, Ravindra; Shahzad, Naseer (February 13, 2014). "A comparison of various noncommuting conditions in metric fixed point theory and their applications". Fixed Point Theory and Applications. 2014: 1–33. doi:10.1186/1687-1812-2014-38.
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