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In this paper, we prove common fixed point results for quadruple self-mappings satisfying an implicit function which is general enough to cover a multitude of known as well as unknown contractions. Our results modify, unify, extend and generalize many relevant results existing in literature. Also, we define the concept of compatible maps and its variants in the setting of digital metric space and establish some common fixed point results for these maps. Also, an application of the proposed results is quoted in this note.

Digital topology is an emerging area based on general topology and functional analysis and focuses on studying digital topological properties of n-dimensional digital spaces, where as Euclidean topology deals with topological properties of subspaces of the n-dimensional real space, which has contributed to the study of some areas of computer sciences such as computer graphics, image processing, approximation theory, mathematical morphology, optimization theory etc. Rosenfield [

Fixed point theory leads to lots of applications in mathematics, computer science, engineering, game theory, fuzzy theory, image processing and so forth. In metric spaces, this theory begins with the Banach fixed-point theorem which provides a constructive method of finding fixed points and an essential tool for solution of some problems in mathematics and engineering and consequently has been generalized in many ways. Up to now, several developments have occurred in this area. A major shift in the arena of fixed point theory came in 1976 when Jungck [

In 2014, He et al. [

Theorem 1. Let S , T , A and B be mappings of a complete multiplicative metric space ( X , d ) into itself satisfying the following conditions:

1) S ( X ) ⊆ B ( X ) , T ( X ) ⊆ A ( X ) ,

2) d ( S x , T y ) [ max = { d ( A x , B y ) , d ( S x , T y ) , d ( S x , A x ) ,

d ( B y , T y ) , d ( S x , B y ) , d ( A x , T y ) } ] λ ,

for all x , y ∈ X , λ ∈ ( 0, 1 2 ) .

3) one of the mappings S , T , A and B is continuous. Assume that the pairs ( A , S ) and ( B , T ) are weakly commuting.Then S , T , A and B have a unique common fixed point.

The objective of this paper is to give digital version of above theorem using an implicit function which is general enough to cover several linear as well as some nonlinear contractions. Our results generalize and extend the many results existing in literature.

This paper is organized as follows. In the first part, we give the required background about the digital topology and fixed point theory. In the next section, we state and prove main results for compatible mappings and compatible mappings of types (A) and (P) in digital metric spaces. Our results improve and generalize many other results existing in literature. Finally, we give an important application of fixed point theorem to digital images. Lastly, we make some conclusions.

Let X be subset of Z n for a positive integer n where Z n is the set of lattice points in the n-dimensional Euclidean space and ρ represent an adjacency relation for the members of X. A digital image consists of ( X , ρ ) .

Definition 2. [

| a i − b i | = 1

and for all other indices j such that

| a j − b j | ≠ 1 , a j = b j .

A r-neighbour of a ∈ Z n is a point of Z n that is r-adjacent to a where ρ = { 2 , 4 , 6 , 8 , 18 , 26 } and n ∈ { 1,2,3 } . The set N p ( a ) = { b / b is ρ -adjacent to a } is called the r-neighbourhood of a. A digital interval is defined by

[ p , q ] z = { z ∈ Z / p ≤ z ≤ q } , where p , q ∈ Z and p < q .

A digital image X ⊂ Z n is r-connected if and only if for every pair of different points u , v ∈ X , there is a set { u 0 , u 1 , ⋯ , u r } of points of digital image X such that u = u 0 , v = u r and u i and u i + 1 are r-neighbour where i = 0 , 1 , 2 , ⋯ , r − 1 .

Definition 3. [

1) T is said to be ( ρ 0 , ρ 1 ) -continuous, if for all ρ 0 -connected subset E of X, f ( E ) is a ρ 1 -connected subset of Y.

2) For all ρ 0 -adjacent points { u 0 , u 1 } of X, either T ( u 0 ) = T ( u 1 ) or T ( u 0 ) and T ( u 1 ) are a ρ 1 -adjacent in Y if and only if T is ( ρ 0 , ρ 1 ) -continuous.

3) If T is ( ρ 0 , ρ 1 ) -continuous, bijective and T − 1 is ( ρ 0 , ρ 1 ) -continuous, then T is called ( ρ 0 , ρ 1 ) -isomorphism and denoted by X ≅ ( ρ 0 , ρ 1 ) Y .

Definition 4. [

Definition 5. [

Definition 6. [

Definition 7. [

Proposition 8. [

Theorem 9. [

Definition 10. [

S ∘ T = T ∘ S , ∀ x ∈ X .

Definition 11. [

d ( S ( T ( x ) ) , T ( S ( x ) ) ) ≤ ( S ( x ) , T ( x ) ) for all x ∈ X .

Remark 1. [

Definition 12. [

l i m n → ∞ S ( x n ) = l i m n → ∞ T ( X n ) = t for some t in X .

Then

1) S and T and are said to be compatible if l i m n → ∞ d ( S T x n , T S x n ) = 0

2) S and T and are said to be compatible of type (A) if lim n → ∞ d ( S T x n , T T x n ) = 0 and lim n → ∞ d ( T S x n , S S x n ) = 0

3) S and T and are said to be compatible of type (P) if lim n → ∞ d ( S S x n , T T x n ) = 0

4) O ¯ -compatible if for any sequence { x n } ⊂ X with S x n ↑ t and T x n ↑ t (for some t ∈ X ) implies lim n → ∞ d ( S T x n , T S x n ) = 0 ,

5) O _ -compatible if for any sequence { x n } ⊂ X with S x n ↓ t and T x n ↓ t (for some t ∈ X ) implies lim n → ∞ d ( S T x n , T S x n ) = 0 ,

6) O-compatible if for any sequence { x n } ⊂ X with S x n ↓ t and T x n ↑ t (for some t ∈ X ) implies lim n → ∞ d ( S T x n , T S x n ) = 0 .

7) weakly compatible if S ( T x ) = T ( S x ) , ∀ x ∈ X .

Remark 2. [

Definition 13. [

Remark 3. In a digital metric space, completeness Þ O-completeness Þ O ¯ -completeness (as well as O _ -completeness).

Remark 4. Every pair of weakly commuting maps is compatible but the converse is not true.

Definition 14. [

Notice that, on setting S = I X , Definition 14 reduces to the usual definition of continuity.

Definition 15. [

Notice that, on setting S = I X , Definition 15 reduces to the usual definition of O-continuity (resp. O ¯ -continuity, O _ -continuity) of a self-mapping (T on X).

Remark 5. In digital metric space, S-continuity Þ ( S , O ) -continuity Þ ( S , O ¯ ) -continuity (as well as ( S , O _ ) -continuity).

Proposition 16. Let S and T be compatible maps of type (A) on a digital metric space ( X , d , ρ ) . If one of S and T is continuous, then S and T are compatible.

Proof. Since S and T are compatible maps of type (A), so lim n → ∞ d ( S T x n , T T x n ) = 0 and lim n → ∞ d ( T S x n , S S x n ) = 0 , whenever { x n } is a sequence in X such that

lim n → ∞ S x n = lim n → ∞ T x n = t for some t in X .

Let S is continuous, then

l i m n → ∞ S T x n → S t and l i m n → ∞ S S x n → S t

and hence

d ( S T x n , T S x n ) ≤ d ( S T x n , S S x n ) + d ( S S x n , T S x n ) .

Letting n → ∞ , we have lim n → ∞ d ( S T x n , T S x n ) = 0 implies S and T are compatible.

Proposition 17. Let S and T be compatible maps on a digital metric space ( X , d , ρ ) into itself. Suppose that

l i m n → ∞ S x n = l i m n → ∞ T x n = t for some t ∈ X .

Then

1) lim n → ∞ S T x n = T t if T is continuous at t.

2) lim n → ∞ T S x n = S t if S is continuous at t.

Proof.

1) Suppose that T is continuous at t. Since lim n → ∞ S x n = lim n → ∞ T x n = t for some t ∈ X , we have

lim n → ∞ T S x n = T t .

Since S and T are compatible maps, we have

d ( S T x n , T t ) ≤ d ( S T x n , T S x n ) + d ( T S x n , T t ) → 0

as n → ∞ and hence the proof.

2) The proof of lim n → ∞ T S x n = S t follows by similar arguments as in (a).

Proposition 18. Let S and T be compatible maps on a digital metric space ( X , d , ρ ) into itself. If S t = T t for some t ∈ X , then

S T t = T S t = S S t = T T t .

In recent years, Popa [

In the literature, there are several types of implicit contraction mappings where many nice consequences of fixed point theorems could be derived. First, denote Φ the set of functions ϕ : [ 0, ∞ ) → [ 0, ∞ ) satisfying:

1) ϕ is nondecreasing,

2) ∑ n = 0 ∞ ϕ n < ∞ for each t ∈ [ 0, ∞ ) , where ϕ n is the nth iterate of ϕ .

Remark 6. It is easy to see that if ϕ ∈ Φ , then ϕ ( 0 ) = 0 and ϕ ( t ) < t , . ∀ t > 0

Definition 19. [

1) For each u , v ≥ 0 , T ( u , v , u , v , u + v ,0 ) ≤ 0 ⇒ u ≤ ϕ ( v ) .

2) T ( u , 0 , 0 , u , u , 0 ) > 0 or T ( u , v , 0 , 0 , v , u ) > 0 .

Example 20. T ( t 1 , t 2 , t 3 , t 4 , t 5 , t 6 ) = t 1 − α max { t 2 , t 3 , t 4 } − ( 1 − α ) [ a t 5 + b t 6 ] where 0 ≤ α ≤ 1 , 0 ≤ a < 1 2 , 0 ≤ b < 1 2 .

Example 21. T ( t 1 , t 2 , t 3 , t 4 , t 5 , t 6 ) = t 1 − k max { t 2 , t 3 , t 4 , 1 2 [ a t 5 + b t 6 ] } , where k ∈ ( 0,1 ) .

Example 22. T ( t 1 , t 2 , t 3 , t 4 , t 5 , t 6 ) = t 1 − ϕ m a x { t 2 , t 3 , t 4 , 1 2 [ a t 5 + b t 6 ] } , where ϕ : R + → R + is right continuous and ϕ ( 0 ) = 0 and ϕ ( t ) < t , ∀ t > 0 .

Theorem 23. Let A , B , S and T be four self-mappings of a complete digital metric space ( X , d , ρ ) satisfying the following conditions:

a) S ( X ) ⊂ B ( X ) and T ( X ) ⊂ A ( X ) ;

b) the pairs ( A , S ) and ( B , T ) are compatible;

c) one of S , T , A and B is continuous;

d) F ( d ( A x , B y ) , d ( S x , T y ) , d ( A x , S x ) , d ( B y , T y ) , d ( A x , T y ) , d ( B y , S x ) ) , ∀ x , y ∈ X .

Then A , B , S and T have a unique common fixed point in X.

Proof. Since S ( X ) ⊂ B ( X ) , we can consider a point x 0 ∈ X , there exists x 1 ∈ X such that

S x 0 = B x 1 = y 0 .

Also, for this point x 1 ,there exists x 2 ∈ X such that

T x 1 = A x 2 = y 1 .

Continuing in this way, we can construct a sequence { y n } in X such that

y 2 n = S x 2 n = B x 2 n + 1 , y 2 n + 1 = T x 2 n + 1 = A x 2 n + 2 for each n ≥ 0.

Now, we have to show that { y n } is Cauchy sequence in ( X , d , ρ ) . Using condition (d), we have

F ( d ( A x 2 n , B x 2 n + 1 ) , d ( S x 2 n , T x 2 n + 1 ) , d ( A x 2 n , S x 2 n ) , d ( B x 2 n + 1 , T x 2 n + 1 ) , d ( A x 2 n , T x 2 n + 1 ) , d ( B x 2 n + 1 , S x 2 n ) ) ≤ 0

i .e F ( d ( y 2 n − 1 , y 2 n ) , d ( y 2 n , y 2 n + 1 ) , d ( y 2 n − 1 , y 2 n ) , d ( y 2 n , y 2 n + 1 ) , d ( x 2 n − 1 , y 2 n + 1 ) , d ( y 2 n , y 2 n ) ) ≤ 0,

which is,

F ( d ( y 2 n − 1 , y 2 n ) , d ( y 2 n , y 2 n + 1 ) , d ( y 2 n − 1 , y 2 n ) , d ( y 2 n , y 2 n + 1 ) , d ( y 2 n − 1 , y 2 n ) , d ( y 2 n , y 2 n + 1 ) ,0 ) ≤ 0,

implies that

d ( y 2 n , y 2 n + 1 ) ≤ ϕ ( d ( y 2 n − 1 , y 2 n ) ) < d ( y 2 n − 1 , y 2 n ) .

Therefore, the sequence { d ( y 2 n , y 2 n + 1 ) } is strictly decreasing. Then there exists r ≥ 0 such that

l i m n → ∞ d ( y 2 n , y 2 n + 1 ) = r .

Suppose that r > 0 , then, letting n → ∞ in above equation, r ≤ ϕ ( r ) < r , which is imposible. Hence r = 0 , that is,

l i m n → ∞ d ( y 2 n , y 2 n + 1 ) = 0.

By the completeness of X, there exists z ∈ X such that y n → ∞ as n → ∞ . Consequently, the subsequences S x 2 n , A x 2 n , T x 2 n + 1 and B x 2 n + 1 of { y n } also converge to a point z ∈ X . Now, suppose that A is continuous.Then { A A x 2 n } and { A S x 2 n } converge to A z as n → ∞ . Since the mappings A and S are compatible on X, it follows from Proposition 17 that { S A x 2 n } converge to A z as n → ∞ .

Now, we claim that A z = z . Consider,

F ( d ( A A x 2 n , B x 2 n + 1 ) , d ( S A x 2 n , T x 2 n + 1 ) , d ( A A x 2 n , S A x 2 n ) , d ( B x 2 n + 1 , T x 2 n + 1 ) , d ( A A x 2 n , T x 2 n + 1 ) , d ( B x 2 n + 1 , S A x 2 n ) ) ≤ 0.

Letting n → ∞ we have

F ( d ( A z , z ) , d ( A z , z ) , d ( A z , A z ) , d ( z , z ) , d ( A z , z ) , d ( z , A z ) ) ≤ 0 ⇒ F ( d ( A z , z ) , d ( A z , z ) ,0,0, d ( A z , z ) , d ( z , A z ) ) ≤ 0,

yields A z = z . Again, from (d), we obtain

F ( d ( A z , B x 2 n + 1 ) , d ( S z , T x 2 n + 1 ) , d ( A z , S z ) , d ( B x 2 n + 1 , T x 2 n + 1 ) , d ( A z , T x 2 n + 1 ) , d ( B x 2 n + 1 , S z ) ) ≤ 0.

Letting n → ∞ we get

F ( d ( z , z ) , d ( S z , z ) , d ( z , S z ) , d ( z , z ) , d ( z , z ) , d ( z , S z ) ) ≤ 0 = F ( 0, d ( S z , z ) , d ( z , S z ) ,0,0, d ( z , S z ) ) ≤ 0 ⇒ S z = z .

Since S ( X ) ⊂ B ( X ) , there exists u ∈ X such that

z = A z = S z = B u .

By using (d), we can obtain

F ( d ( A z , B u ) , d ( S z , T u ) , d ( A z , S z ) , d ( B u , T u ) , d ( A z , T u ) , d ( B u , S z ) ) ≤ 0 = F ( d ( z , z ) ) , d ( z , T u ) , d ( z , z ) , d ( z , T u ) , d ( z , T u ) , d ( z , z ) ) ≤ 0. = F ( 0, d ( z , T u ) ,0, d ( z , T u ) , d ( z , T u ) ,0 ) ≤ 0 ⇒ z = T u .

As B and T are compatible on X and z = T u = B u , by Proposition 18, we have T B u = B T u and hence B z = T B u = B T u = T z . Also, we have

F ( d ( A z , B z ) , d ( S z , T z ) , d ( A z , S z ) , d ( B z , T z ) , d ( A z , T z ) , d ( B z , S z ) ) ≤ 0. ⇒ F ( 0 , d ( z , T z ) , 0 , 0 , d ( z , T z ) , d ( T z , z ) ) ≤ 0 ⇒ z = T z .

Thus z = T z = B z and so z is a common fixed point of A , B , S and T. Similarly, we can use above assertion in case of continuity of B or S or T and uniqueness of the common fixed point follows directly from the condition (d) and hence the proof follows.

Theorem 24. Let A , B , S and T be four self-mappings of a complete digital metric space ( X , d , ρ ) satisfying the conditions (a), (c) and (d). If the pairs ( A , S ) and ( B , T ) are compatible of type (A) or of type (P) then A , B , S and T have a unique common fixed point in X.

Proof. Theorem is direct consequence of Proposition 16 and Theorem 23. ☐

Example 25. Let X = [ 1 , ∞ ) with usual metric and A , B , S , T : X → X be self mappings such that

T ( x ) = { 2 x − 1 , x ≤ 2 3 , x > 2 , A ( x ) = { 3 x − 2 , x ≤ 2 x , x > 2 (1)

B ( x ) = { x , x ≤ 2 x + 1 , x > 2 , S ( x ) = 1. (2)

Then we see that S ( X ) ⊂ B ( X ) and T ( X ) ⊂ A ( X ) and if we consider a sequence 〈 x n 〉 = 1 + 1 n , then

l i m n → ∞ d ( A S x n , S A x n ) = 0 and l i m n → ∞ d ( B T x n , T B x n ) = 0

implies that the pairs ( A , S ) and ( B , T ) are compatible and S is continuous. Also the condition (d) holds. Hence all the conditions of Theorem 23 are satisfied and x = 1 is common fixed point of the mappings.

In this section, we give an application of digital contractions to solve the problem related to image compression. The aim of image compression is to reduce redundant image information in the digital image. When we store an image we may come across certain type of problems like either memory data is usually too large or stored image has not more information than original image. Also, the quality of compressed image can be poor. For this reason, we must pay attention to compress a digital image. Fixed point theorem can be used for image compression of a digital image.

The purpose of this paper is four fold which can be described as follows.

1) We slightly modify the implicit relation of Popa so that contraction conditions obtained involving functional inequalities.

2) Proved some common fixed point theorems by using modified implicit relation, compatibility and its variants in the setting of digital-metric spaces.

3) Generalized the Theorem 1 and many others too existing in literature, derived related results and furnished illustrative examples.

4) Our results generate scope for other researchers to prove the results by utilizing other contractions, weak compatibility, control function and admissible maps in a digital framework.

Dalal, S., Masmali, I.A. and Alhamzi, G.Y. (2018) Common Fixed Point Results for compatible Map in Digital Metric Space. Advances in Pure Mathematics, 8, 362-371. https://doi.org/10.4236/apm.2018.83019