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In this p aper, 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.

Fixed point theory has absorbed many mathematicians since 1922 with the celebrated Banach contraction principle (see [

Chatteriea in [

Definition 1.1. [

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 [

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

Definition 1.2. [

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 [

In 2006, Chistyakov introduced the notion of modular metric space in [

Throughout this paper ℕ will denote the set of natural numbers.

The notion of modular metric space was introduced by Chistyakov in [

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. [

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. [

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. [

1) The sequence { x n } 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 ∈ 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. [

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

In [

Theorem 2.5. [

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.

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 ∈ ℕ . If T x n 0 − 1 = T x n 0 for some n 0 ∈ ℕ , 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 ∈ 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 ∈ 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 ∈ 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 ∈ 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 ∈ ℕ such that w λ ( x n + 1 , x n ) < ε for all n ∈ ℕ with n ≥ n 0 . Without loss of generality, suppose

m , n ∈ ℕ and m > n . Observe that, for λ m − n > 0 and for above-mentioned

ε , there exists n λ / ( m − n ) ∈ ℕ such that

w λ m − n ( x n + 1 , x n ) < ε m − n , (10)

for all n ≥ n λ / ( m − n ) . Now we have

w λ ( x n , x m ) ≤ w λ m − n ( x n , x n + 1 ) + w λ m − n ( x n + 1 , x n + 2 ) + ⋯ + w λ m − n ( x m − 1 , x m ) < ε m − n + ε m − n + ⋯ + ε m − n = ε , (11)

for all m , n ≥ n λ / ( m − n ) ∈ ℕ . This implies { x n } n ∈ ℕ is a Cauchy sequence. By the completeness of X w , there exists point x ∈ X w , such that x n → x as n → ∞ .

By the notion of metric modular w and the contraction of T, we get

w λ ( T x , x ) ≤ w λ 2 ( T x , T x n ) + w λ 2 ( T x n , x ) ≤ k ( w λ ( x , T x n ) + w λ ( x n , T x ) ) + w λ 2 ( T x n , x ) = k ( w λ ( x , x n + 1 ) + w λ ( x n , T x ) ) + w λ 2 ( x n + 1 , x ) , (12)

for all λ > 0 and for all n ∈ ℕ . Taking n → ∞ in inequality (12), we obtained that

w λ ( T x , x ) ≤ k w λ ( T x , x ) . (13)

Since k ∈ [ 0, 1 2 ) , we have T x = x . Thus, x is a fixed point of T. Next, we

prove that x is a unique fixed point. Suppose that z be another fixed point of T. We note that

w λ ( x , z ) = w λ ( T x , T z ) ≤ k ( w 2 λ ( x , T z ) + w 2 λ ( z , T x ) ) ≤ k ( w λ ( x , z ) + w λ ( z , T z ) + w λ ( z , x ) + w λ ( x , T x ) ) = 2 k w λ ( x , z ) , (14)

for all λ > 0 . Therefore we have

( 1 − 2 k ) w λ ( x , z ) ≤ 0.

Since 1 − 2 k > 0 , we can imply that x = z . Therefore, x is a unique fixed point of T. □

Next, we will introduce the notion of weakly C-contraction in modular metric space.

Definition 3.2. Let w be a metric modular on X, X w be a modular metric space induced by w. A mapping T : X w → X w is said to be a weak C-contraction in X w if for all x , y ∈ X w and for all λ > 0 , the following inequality holds:

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 ) ) , (15)

where φ [ 0, ∞ ) 2 → [ 0, ∞ ) is a continuous mapping such that φ ( x , y ) = 0 if and only if x = y .

Theorem 3.3. Let w be a metric modular on X, X w be a w-complete modular metric space induced by w. Let T : X w → X w be a weak C-contraction in X w such that T is continuous and non-decreasing. Then T has a unique fixed point.

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 ∈ ℕ . If T x n 0 − 1 = T x n 0 for some n 0 ∈ ℕ , 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 ∈ ℕ , we have

w λ ( x n + 1 , x n ) = w λ ( T x n , T x n − 1 ) ≤ 1 2 ( w 2 λ ( x n , T x n − 1 ) + w 2 λ ( x n − 1 , T x n ) ) − φ ( w λ ( x n , T x n − 1 ) , w λ ( x n − 1 , T x n ) ) = 1 2 ( w 2 λ ( x n , x n ) + w 2 λ ( x n − 1 , x n + 1 ) ) − φ ( w λ ( x n , x n ) , w λ ( x n − 1 , x n + 1 ) ) = 1 2 w 2 λ ( x n − 1 , x n + 1 ) − φ ( 0 , w λ ( x n − 1 , x n + 1 ) ) ≤ 1 2 w 2 λ ( x n − 1 , x n + 1 ) ≤ 1 2 ( w λ ( x n − 1 , x n ) + w λ ( x n , x n + 1 ) ) , (16)

for all λ > 0 . The last inequality gives us

w λ ( x n , x n + 1 ) ≤ w λ ( x n − 1 , x n ) ,

for all λ > 0 and for all n ∈ ℕ . Thus { w λ ( x n , x n + 1 ) } is a decreasing sequence of nonnegative real numbers and hence it is convergent.

For each λ > 0 , let

lim n → ∞ w λ ( x n , x n + 1 ) = r . (17)

Letting n → ∞ in (16) we have

r ≤ lim n → ∞ 1 2 w λ ( x n − 1 , x n + 1 ) ≤ 1 2 ( r + r ) = r . (18)

or, equivalently,

lim n → ∞ w λ ( x n − 1 , x n + 1 ) = 2 r . (19)

Again, making n → ∞ in (17), (19) and the continuity of φ we have

r ≤ 1 2 2 r − φ ( 0 , 2 r ) = r − φ ( 0 , 2 r ) ≤ r . (20)

And, consequently, φ ( 0 , 2 r ) = 0 . This gives us that r = 0 by our assumption about φ .

Thus, for all λ > 0 , we have

lim n → ∞ w λ ( x n , x n + 1 ) = 0. (21)

From the proof of theorem 3.1, we can prove that { x n } is a w-Cauchy sequence. By the completeness of X w , there exists a point x ∈ X w , such that x n → x as n → ∞ .

By the notion of metric modular w and the contraction of T, we get

w λ ( T x , x ) ≤ w λ 2 ( T x , T x n ) + w λ 2 ( T x n , x ) ≤ 1 2 ( w λ ( x , T x n ) + w λ ( x n , T x ) ) − φ ( w λ ( x , T x n ) , w λ ( x n , T x ) ) + w λ 2 ( T x n , x ) = 1 2 ( w λ ( x , x n + 1 ) + w λ ( x n , T x ) ) − φ ( w λ ( x , x n + 1 ) , w λ ( x n , T x ) ) + w λ 2 ( x n + 1 , x ) , (22)

for all λ > 0 and for all n ∈ ℕ . Taking n → ∞ by (22), we obtained that

w λ ( T x , x ) ≤ 1 2 w λ ( T x , x ) − φ ( 0, w λ ( T x , x ) ) . (23)

This prove that x = T x . Thus x is a fixed point of T. Next, we prove that x is a unique fixed point. Suppose that z and x are different fixed points of T, then from (15), we have

w λ ( z , x ) = w λ ( T z , T x ) ≤ 1 2 ( w 2 λ ( z , T x ) + w 2 λ ( x , T z ) ) − φ ( w λ ( z , T x ) , w λ ( x , T z ) ) ≤ w 2 λ ( x , z ) − φ ( w λ ( z , x ) , w λ ( x , z ) ) , (24)

for all λ > 0 By the property of the φ , we have x = z . Hence x is a unique fixed point of T. □

Example 3.4 Let X = { ( a , 0 ) ∈ R 2 | a ≥ 0 } ∪ { ( 0 , b ) ∈ R 2 | b ≥ 0 } . Defined the mapping w : ( 0, ∞ ) × X × X → [ 0, ∞ ) by

w λ ( ( a 1 ,0 ) , ( a 2 ,0 ) ) = 3 | a 1 − a 2 | λ ,

w λ ( ( 0, b 1 ) , ( 0, b 2 ) ) = | b 1 − b 2 | λ ,

and

w λ ( ( a , 0 ) , ( 0 , b ) ) = 3 a λ + b λ = w λ ( ( 0 , b ) , ( a , 0 ) ) .

We note that if we take λ → ∞ , then we see that X = X w and also T and φ is define by

T ( ( a , 0 ) ) = ( 0 , a 2 ) ,

T ( ( 0, b ) ) = ( b 24 ,0 ) .

and

φ ( x , y ) = 1 20 ( x + y ) .

We can imply that

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 ) ) for all x , y ∈ X and all λ > 0 .

Indeed, case1. let x = ( a 1 , 0 ) , y = ( a 2 , 0 ) , then

w λ ( T x , T y ) = w λ ( T ( a 1 , 0 ) , T ( a 2 , 0 ) ) = w λ ( ( 0 , a 1 2 ) , ( 0 , a 2 2 ) ) = | a 1 − a 2 | 2 λ , (25)

w 2 λ ( x , T y ) = w 2 λ ( ( a 1 , 0 ) , T ( a 2 , 0 ) ) = w 2 λ ( ( a 1 , 0 ) , ( 0 , a 2 2 ) ) = 3 a 1 2 λ + a 2 4 λ , (26)

w 2 λ ( y , T x ) = w 2 λ ( ( a 2 , 0 ) , T ( a 1 , 0 ) ) = w 2 λ ( ( a 2 , 0 ) , ( 0 , 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.

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.

Zhao, J.W., Zhao, Q.Q., Jin, B. and Zhong, L.N. (2018) Fixed Point Results for Weakly C-Contraction Mapping in Modular Metric Spaces. Open Access Library Journal, 5: e4061. https://doi.org/10.4236/oalib.1104061