Fixed Point Results for Weakly C-Contraction Mapping in Modular Metric Spaces

DOI: 10.4236/oalib.1104061    357 Downloads   891 Views   Citations

ABSTRACT

In this paper, we introduce the concept of weakly C-contraction mapping in modular metric spaces. And we established some fixed point results in w-complete spaces. Our results encompass various generalizations of Banach contraction.

Cite this paper

Zhao, J. , Zhao, Q. , Jin, B. and Zhong, L. (2018) Fixed Point Results for Weakly C-Contraction Mapping in Modular Metric Spaces. Open Access Library Journal, 5, 1-9. doi: 10.4236/oalib.1104061.

1. Introduction

Fixed point theory has absorbed many mathematicians since 1922 with the celebrated Banach contraction principle (see [1] ). It is one of the most useful results in nonlinear analysis, functional analysis and topology. Due to its application in mathematics, the Banach contraction principle has been generalized in many directions (see [2] [3] [4] ).

Chatteriea in [5] introduced the notion of C-contraction which is a generalization of the Banach contraction.

Definition 1.1. [5] A mapping T : X X where ( X , d ) is a metric space

is said to be a C-contraction if there exists α [ 0, 1 2 ) such that for all

x , y X the following inequality holds:

d ( T x , T y ) α ( d ( x , T y ) , d ( y , T x ) ) (1)

Chatteriea in [5] proved that if X is complete, then every C-contraction mapping have a unique fixed point.

The notion of C-contraction was generalized to a weak C-contraction by Choudhury in [6] .

Definition 1.2. [6] Let ( X , d ) be a metric space and T : X X be a map. Then T is called a weakly C-contraction (or a weak C-contraction) if there exists φ : [ 0 ) 2 [ 0 ) which is continuous, and φ ( x , y ) = 0 if and only if x = y = 0 such that

d ( T x , T y ) 1 2 [ d ( x , T y ) + d ( y , T x ) ] φ ( d ( x , T y ) , d ( y , T x ) ) , (2)

for all x , y X .

In [6] the author proved that if X is a complete metric space, then every weakly C-contraction has a unique fixed point. This fixed point theory was generalized to a complete, partially ordered metric space in [7] and a ordered 2-metric space in [8] .

In 2006, Chistyakov introduced the notion of modular metric space in [9] . Recently, there have been many interesting results in the field of existence and uniqueness of fixed point in complete modular metric (see [10] [11] ). In this paper, we will establish fixed point theorems for weakly C-contraction in modular metric space. The presented results extend some recent results in the literature.

2. Preliminaries

Throughout this paper will denote the set of natural numbers.

The notion of modular metric space was introduced by Chistyakov in [9] [12] [13] , who proved some fixed point results in such kind of spaces.

Let X be a nonempty set. Throughout this paper, for a function w : ( 0, ) × X × X [ 0, ) , we write

w λ ( x , y ) = w ( λ , x , y ) , (3)

for all λ > 0 and x , y X .

Definition 2.1. [9] Let X be a nonempty set. A function w : ( 0, ) × X × X [ 0, ) is said to be a metric modular on X if it satisfies, for all x , y , z X , the following condition:

1) w λ ( x , y ) = 0 for all λ > 0 if and only if x = y ;

2) w λ ( x , y ) = w λ ( y , x ) for all λ > 0 ;

3) w λ + μ ( x , y ) w λ ( x , z ) + w μ ( z , y ) for all λ , μ > 0 .

If instead of (i) we have only the condition (i')

w λ ( x , x ) = 0 forall λ > 0 , x X ,

then w is said to be a pseudomodular (metric) on X.

An important property of the (metric) pseudomodular on set X is that the mapping λ w λ ( x , y ) is non increasing for all x , y X .

Definition 2.2. [9] Let w is a pseudomodular on X. Fixed x 0 X . The set

X w = X w ( x 0 ) = { x X : w λ ( x , x 0 ) 0 as λ }

is said to be a modular metric space (around x 0 ).

Definition 2.3. [14] Let X w be a modular metric space.

1) The sequence { x n } n in X w is said to be w-convergent to x X w if and only if w λ ( x n , x ) 0 , as n for some λ > 0 ;

2) The sequence { x n } n in X w is said to be w-Cauchy if w λ ( x m , x n ) 0 as m , n for some λ > 0 ;

3) A subset C of X w is said to be w-complete if any w-Cauchy sequence in C is a convergent sequence and its limit is in C.

Definition 2.4. [15] Let w be a metric modular on X and X w be a modular metric space induced by w. If X w is a w-complete modular metric space and T : X w X w be an arbitrary mapping T is called a contraction if for each x , y X w and for all λ > 0 there exists 0 k < 1 such that

w λ ( T x , T y ) k w λ ( x , y ) . (4)

In [15] Chirasak proved that if X w is a w-complete modular metric space, then contraction mapping T has a unique fixed point. At the same time, the author proved the following theorem.

Theorem 2.5. [15] Let w be a metric modular on X, X w be a w-complete modular metric space induced by w and T : X w X w . If

w λ ( T x , T y ) k ( w 2 λ ( T x , x ) + w 2 λ ( T y , y ) ) , (5)

for all x , y X w and for all λ > 0 , where k [ 0, 1 2 ) , then T has a unique

fixed point in X w . Moreover, for any x X w , iterative sequence { T n x } converges to the fixed point.

3. Main Results

Theorem 3.1. Let w be a metric modular on X, X w be a w-complete modular metric space induced by w and T : X w X w . If

w λ ( T x , T y ) k ( w 2 λ ( x , T y ) + w 2 λ ( y , T x ) ) , (6)

for all x , y X w and for all λ > 0 , where k [ 0, 1 2 ) , then T has a unique

fixed point in X w .

Proof. Let x 0 be an arbitrary point in X w and we write x 1 = T x 0 ,

x 2 = T x 1 = T 2 x 0 , and in general, x n = T x n 1 = T 2 x 0 for all n . If T x n 0 1 = T x n 0 for some n 0 , then T x n 0 = x n 0 . Thus x n 0 is a fixed point of T. Suppose that

T x n 1 T x n for all n . For k [ 0, 1 2 ) , we have

w λ ( x n + 1 , x n ) = w λ ( T x n , T x n 1 ) k ( w 2 λ ( x n , T x n 1 ) + w 2 λ ( x n 1 , T x n ) ) = k w 2 λ ( x n 1 , x n + 1 ) k ( w λ ( x n 1 , x n ) + w λ ( x n , x n + 1 ) ) , (7)

for all λ > 0 and all n . Hence,

w λ ( x n + 1 , x n ) k 1 k w λ ( x n , x n 1 ) , (8)

for all λ > 0 and all n . Put β : = k 1 k , since k [ 0, 1 2 ) , we get β [ 0,1 )

and hence

w λ ( x n + 1 , x n ) β w λ ( x n , x n 1 ) β 2 w λ ( x n 1 , x n 2 ) β n w λ ( x 1 , x 0 ) , (9)

for all λ > 0 and each n . Therefore, lim n w λ ( x n + 1 , x n ) = 0 for all λ > 0 . So for each λ > 0 , we have for all ε > 0 there exists n 0 such that w λ ( x n + 1 , x n ) < ε for all n with n n 0 . Without loss of generality, suppose

m , n a 1 2 ) ) = 3 a 2 2 λ + a 1 4 λ , (27)

w λ ( T x , T y ) 2 7 ( w 2 λ ( x , T y ) + w 2 λ ( y , T x ) ) . (28)

Case 2. let x = ( 0 , b 1 ) , y = ( 0 , b 2 ) , we have

w λ ( T x , T y ) = w λ ( T ( 0 , b 1 ) , T ( 0 , b 2 ) ) = w λ ( ( b 1 24 , 0 ) , ( b 2 24 , 0 ) ) = | b 1 b 2 | 8 λ , (29)

w 2 λ ( x , T y ) = w 2 λ ( ( 0 , b 1 ) , T ( 0 , b 2 ) ) = w 2 λ ( ( 0 , b 1 ) , ( b 2 24 , 0 ) ) = b 2 16 λ + b 1 2 λ , (30)

w λ ( T x , T y ) 2 9 ( w 2 λ ( x , T y ) + w 2 λ ( y , T x ) ) . (31)

Case 3. Let x = ( a , 0 ) , y = ( 0 , b ) , then

w λ ( T x , T y ) = w λ ( T ( a , 0 ) , T ( 0 , b ) ) = w λ ( ( 0 , a 2 ) , ( b 24 , 0 ) ) = b 8 λ + a 2 λ , (32)

w 2 λ ( x , T y ) = w 2 λ ( ( a , 0 ) , T ( 0 , b ) ) = w 2 λ ( ( a , 0 ) , ( b 24 , 0 ) ) = | b 16 λ 3 a 2 λ | , (33)

w 2 λ ( y , T x ) = w 2 λ ( ( 0 , b ) , T ( a , 0 ) ) = w 2 λ ( ( 0 , b ) , ( 0 a 2 ) ) = | b 2 λ a 4 λ | , (34)

w λ ( T x , T y ) 2 5 ( w 2 λ ( x , T y ) + w 2 λ ( y , T x ) ) . (35)

φ ( w λ ( x , T y ) , w λ ( y , T x ) ) = 1 20 ( w λ ( ( x , T y ) + w λ ( y , T x ) ) = 1 20 [ 2 ( w 2 λ ( x , T y ) + w 2 λ ( y , T x ) ) ] = 1 10 ( w 2 λ ( x , T y ) + w 2 λ ( y , T x ) ) . (36)

Hence we have

w λ ( T x , T y ) 2 5 ( w 2 λ ( x , T y ) + w 2 λ ( y , T x ) ) , (37)

for all λ > 0 and x , y X . And

1 2 ( w 2 λ ( x , T y ) + w 2 λ ( y , T x ) ) φ ( w λ ( x , T y ) , w λ ( y , T x ) ) = 1 2 ( w 2 λ ( x , T y ) + w 2 λ ( y , T x ) ) 1 10 ( w 2 λ ( x , T y ) + w 2 λ ( y , T x ) ) = 2 5 ( w 2 λ ( x , T y ) + w 2 λ ( y , T x ) ) , (38)

for all λ > 0 and x , y X . We can get

w λ ( T x , T y ) 1 2 ( w 2 λ ( x , T y ) + w 2 λ ( y , T x ) ) φ ( w λ ( x , T y ) , w λ ( y , T x ) ) , (39)

for all x , y X and all λ > 0 . Thus T is a weakly C-contractive mapping. Therefore, T has a unique fixed point that is ( 0,0 ) X w .

On the Euclidean metric d on X w , we see that

d ( T ( 1 , 0 ) , T ( 0 , 1 2 ) ) > 1 2 ( d ( T ( 1 , 0 ) , T ( 0 , 1 2 ) ) + d ( ( 0 , 1 2 ) , T ( 1 , 0 ) ) ) φ ( d ( ( 1 , 0 ) , T ( 0 , 1 2 ) ) , d ( ( 0 , 1 2 ) , T ( 1 , 0 ) ) ) . (40)

Thus, T is not a weak C-contraction on standard metric space.

4. Conclution

In this paper, we extend the fixed point results for the weakly C-contraction in modular metric space. Moreover, as example, we give a unique fixed point theorem for a mapping satisfying a weak C-contractive condition in modular metric space rather than in standard metric space. The main results of this article generalize and unify some recent results given by some authors.

NOTES

*Co-first authors.

#Corresponding authors.

Conflicts of Interest

The authors declare no conflicts of interest.

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