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In this paper, we prove fixed point theorems of a generalization which is related to the concept of Meir-Keeler function in a complete b
_{2}
-metric space. And we know it extends and generalizes some known results in metric space to b
_{2}
-metric space.

Many mathematicians have studied fixed point theory over the last several decades since Banach contraction principle [_{2}-metric space, and this space was generalized from both 2-metric space [

Throughout this paper N will denote the set of all positive integers and R will denote the set of all real numbers.

Before stating our main results, some necessary definitions might be introduced as follows.

Definition 2.1 [

1) A i , i = 1 , 2 , ⋯ , m are empty subsets of X,

2) f ( A i ) ⊂ A 2 , f ( A 2 ) ⊂ A 3 , ⋯ , f ( A m − 1 ) ⊂ A m , f ( A m ) ⊂ A 1 .

Definition 2.2 [

Definition 2.3 [

Definition 2.4 [

1) For every pair of distinct points x , y ∈ X , there exists a point z ∈ X such that d ( x , y , z ) ≠ 0 .

2) If at least two of three points x , y , z are the same, then d ( x , y , z ) = 0 ,

3) The symmetry:

d ( x , y , z ) = d ( x , z , y ) = d ( y , x , z ) = d ( y , z , x ) = d ( z , x , y ) = d ( z , x , y ) for all x , y , z ∈ X .

4)The rectangle inequality:

d ( x , y , z ) ≤ d ( x , y , a ) + d ( y , z , a ) + d ( z , x , a ) for all x , y , z , a ∈ X .

Then d is called a 2 metric on X and ( X , d ) is called a 2 metric space.

Definition 2.5 [

function d : X × X → R + is a b metric on X if for all x , y , z ∈ X , the following conditions hold:

1) d ( x , y ) = 0 if and only if x = y .

2) d ( x , y ) = d ( y , x ) .

3) d ( x , y ) ≤ s [ d ( x , y ) + d ( y , z ) ] .

In this case, the pair ( X , d ) is called a b metric space.

Definition 2.6 [

1) For every pair of distinct points x , y ∈ X , there exists a point z ∈ X such that d ( x , y , z ) ≠ 0 .

2) If at least two of three points x , y , z are the same, then d ( x , y , z ) = 0 ,

3) The symmetry:

d ( x , y , z ) = d ( x , z , y ) = d ( y , x , z ) = d ( y , z , x ) = d ( z , x , y ) = d ( z , x , y ) for all x , y , z ∈ X .

4) The rectangle inequality:

d ( x , y , z ) ≤ s [ d ( x , y , a ) + d ( y , z , a ) + d ( z , x , a ) ] , for all x , y , z , a ∈ X .

Then d is called a b_{2} metric on X and ( X , d ) is called a b_{2} metric space with parameter s. Obviously, for s = 1 , b_{2} metric reduces to 2-metric.

Definition 2.7 [_{2} metric space ( X , d ) .

1) A sequence { x n } is said to be b_{2}-convergent to x ∈ X , written as lim n → ∞ x n = x , if all a ∈ X lim n → ∞ d ( x n , x , a ) = 0 .

2) { x n } is Cauchy sequence if and only if d ( x n , x m , a ) → 0 , when n , m → ∞ . for all a ∈ X .

3) ( X , d ) is said to be complete if every b_{2}-Cauchy sequence is a b_{2}-convergent sequence.

Definition 2.8 [_{2}-metric spaces and let f : X → X ′ be a mapping. Then f is said to be b_{2}-continuous,at a point z ∈ X if for a given ε > 0 , there exists δ > 0 such that x ∈ X and d ( z , x , a ) < δ for all a ∈ X imply that d ′ ( f z , f x , a ) < ε . The mapping f is b_{2}-continuous on X if it is b_{2}-continuous at all z ∈ X .

Definition 2.9 [_{2}-metric spaces. Then a mapping f : X → X ′ is b_{2}-continuous at a point x ∈ X ′ if and only if it is b_{2}-sequentially continuous at x; that is, whenever { x n } is b_{2}-convergent to x, { f x n } is b_{2}-convergent to f ( x ) .

In this section, we give and prove a generalization of the Meir-Keeler fixed point theorem [

Theorem 3.1. Let ( X , d ) be a complete b_{2}-metric space and let f be a mapping on X, for each ε > 0 , there exists δ ∈ ( s ε , ( 2 s − 1 ) ε ) such that

(a) 1 2 s d ( x , f x , a ) < d ( x , y , a ) and d ( x , y , a ) < ε + δ imply d ( f x , f y , a ) ≤ ε

(b) 1 2 s d ( x , f x , a ) < d ( x , y , a ) implies d ( f x , f y , a ) < d ( x , y , a ) for all x , y ∈ X . Then there exists a unique fixed point z of f. Moreover lim n → ∞ f n x = z for all x ∈ X .

Proof If f x ≠ x , then we can easily get that d ( x , f x , a ) < 2 s d ( x , f x , a ) . So, by hypothesis, d ( f x , f 2 x , a ) < d ( x , f x , a ) holds for all x ∈ X with f x ≠ x . We also get

d ( f x , f 2 x , a ) ≤ d ( x , f x , a ) for all x ∈ X (3.1)

Fix point x 0 in X and define a sequence { x n } in X by x n + 1 = f x n = f n x 0 for n ∈ N . From the above (3.1) we get d ( x n , x n + 1 , a ) ≤ d ( x n − 1 , x n , a ) , so we know that { d ( x n , x n + 1 , a ) } is a decreasing sequence, and the sequence { d ( x n , x n + 1 , a ) } converges to some β ≥ 0 . We assume that β > 0 , then we know that d ( x n , x n + 1 , a ) > β for every n ∈ N , then there exists δ such that (a) is true with ε = β , for the definition of β , there exists i ∈ N such that d ( x i , x i + 1 , a ) < β + δ , so we have d ( x i + 1 , x i + 2 , a ) ≤ β , which is a contraction. Therefore β = 0 , and that is:

lim n → ∞ d ( x n , x n + 1 , a ) = 0 .

Now we show that d ( x i , x j , x k ) = 0 .

From part 2 of Definition 2.6, the equation d ( x m , x m , x m − 1 ) = 0 is obtained. Since { d ( x n , x n + 1 , a ) } is decreasing, if d ( x n − 1 , x n , a ) = 0 , then d ( x n , x n + 1 , a ) = 0 , then it is easy to get

d ( x n , x n + 1 , x m ) = 0 , for all n + 1 ≥ m . (3.2)

For 0 ≤ n + 1 < m , we get m − 1 ≥ n + 1 and that is m − 2 ≥ n , from (3.2)

d ( x m − 1 , x m , x n + 1 ) = d ( x m − 1 , x m , x n ) = 0 , (3.3)

From (3.2) and triangular inequality,

And since

Now for all

From (3.5) and triangular inequality, therefore

In conclusion, the result below is true

Now we fix

Now we will show that

By induction, when

In one case

From (3.6) and (3.7) we have

In other case, where

We get

So for (3.9) and (3.10), (3.8) is true for every

Since X is complete, there exists a point

Case one: There exists

Case two:

In the first case, we know that

In the second case, we know that

for some

Then we have

This is a contraction. So we get either

This is a contraction. Hence z is a unique fixed point of f. £

In this section, we prove a fixed point theory for the cyclic weaker Meir-Keeler function in b_{2}-metric space. Now we give some comments as follows:

(

(

(

We now introduce the following definition of cyclic weaker _{2}-metric space:

Definition 3.2 Let _{2}-metric space, _{2}-metric space if satisfying the following condition:

1)

2)

Theorem 3.3 Let _{2}-metric space, _{2}-metric space, then f has a unique fixed point in

Proof Let

Since sequence

First we assume that

Now we prove that

Suppose to the contrary, that is,

From the part 4 of Definition 3.6 and (3.6), we get

Taking

Now by using the condition that f is a cyclic weaker

Letting

From (3.13) and (3.14)

Since X is a complete set, there exists a point

From the above inequality, letting

Now we prove the fixed point is unique for f. Suppose there exists another fixed point y, since f gets the cyclic character, we have

then we get

The authors declare no conflicts of interest regarding the publication of this paper.

Tian, Z.Y., Cui, J.X. and Zhong, L.N. (2019) Fixed Point Theorem for Meir-Keeler Type Function in b_{2}-Metric Spaces. Open Access Library Journal, 6: e5973. https://doi.org/10.4236/oalib.1105973