Abstract

Samet et al. (Nonlinear Anal. 75, 2012, 2154-2165) introduced a new, simple and unified approach by using the concepts of α-ψ-contractive type mappings and α-admissible mappings in metric spaces and presented some nice fixed point results. Recently, Sridevi et.al (International Journal of Mathematics Trends and Technology, Volume 48, Number 3 August 2017) proposed the concept of α-ψ-φ contraction and generalized α-ψ-φ for self map in digital metric spaces. The purpose of this paper is to present a new class of contractive pair of mappings called α-β-ψ-φ contraction and generalized α-β-ψ-φ contractive pair of mappings and study various fixed point theorems for such mappings in digital metric spaces. For this, we introduce a new notion of α-β-admissible w.r.t T mapping which in turn generalizes the concept of g-monotone mapping recently given by “Ciric et al. (Fixed Point Theory Appl. 2008 (2008), Article ID 131294, 11 pages)”. Also, we give some fixed point theorems for cyclic contractive mapping in such spaces. The presented theorems hold without using completeness of the space and without the assumption of continuity of the given mappings. Our results extend, generalize and subsumes digital version of various known comparable results [[1-4,8,13,16,18-22], worth to mention here]. Some illustrative examples are quoted to demonstrate the main results.

1. Introduction

In metric fixed point theory the contractive conditions on underlying functions play an important role for finding solution of fixed point problems. Banach contraction principle ⁸ is a fundamental result in metric fixed point theory. Due to its importance and simplicity, several authors have obtained many interesting extensions and generalizations of the Banach contraction principle.

Recently, Rosenfeld ¹⁷ proposed an impressive generalizations of the notion of a metric as digital metric. Digital topology is the study of the topological properties of images arrays. The results provide a sound mathematical basis for image processing operations such as image thinning, border following, contour filling and object counting. Digital topology is important for computer vision, image processing and computer graphics. Kong ¹⁵, then introduced the digital fundamental group of a discrete object. The digital versions of the topological concepts were given by Boxer ⁴, who later studied digital continuous functions ⁵. Later, he established results of digital homology groups of 2D digital images in ⁶. Ege and Karaca ^{9, 10, 11, 12} give relative and reduced Lefschetz fixed point theorem for digital images.

Samet et al. ¹⁹ extended and generalized the Banach contraction principle by introducing a new category of contractive type mappings known as contractive type mapping. Further, Karapinar and Samet ¹⁶, Gulyaz ¹³, Bilgili ³, Bota ⁴ generalized the -contractive type mappings and obtained various fixed point theorems for this generalized class of contractive mappings. Recently, Sridevi et. al ²², introduced digital -contraction and proved fixed point results for self maps in digital metric spaces.

In this paper we generalize the concept of -contractive mappings in the setting of digital metric space, as digital--contractive mappings. We introduce a new notion of -admissible w.r.t mapping and establish some coincidence and common fixed point theorems for the generalized digital--contractive pair of mappings. Our results unify and generalize the results derived by Sridevi et.al ²², Karapinar and Samet ¹⁶, Gulyaz ¹³, Bilgili ³, Bota ⁴ and various other related results in the literature. Also, we furnish some non-trivial examples to elicit the usability of the obtained theorems.

2. Preliminaries

This document unfolds with preliminaries section, where we review some definitions, examples and notable results that are involved in the sequel. Before referring to the works, first of all, we need to recall the following:

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

Definition 2.1 ⁷: Let be positive integers, and two distinct points

and are - adjacent if there are at most indices such that and for all other indices such that .

A -neighbour ⁷ of is a point of that is - adjacent to where and . The set

is called the - neighbourhood of . A digital interval ¹⁴ is defined by where and .

A digital image is - connected ³ if and only if for every pair of different points , there is a set of points of digital image such that and and and are - neighbours where

Definition 2.2: ⁷ Let be digital images and be a function, then

i) is said to be - continuous ⁷, if for all - connected subset of , is a - connected subset of .

ii) For all - adjacent points of , either or and are a - adjacent in if and only if is -continuous .

iii) If is - continuous, bijective and is - continuous, then is called - isomorphism ⁶ and denoted by

Definition 2.3 ²² Let be the Euclidean metric on is a metric space. Suppose is a digital image with - adjacency then is called a digital metric space.

Definition 2.4 ²²: A sequence of points of a digital metric space is a Cauchy sequence if for all , there exists such that for all , then .

Definition 2.5 ²²: A sequence of points of a digital metric space converges to a limit if for all , there exists such that for all , then .

Definition 2.6 ²²: A digital metric space is a complete digital metric space if any Cauchy sequence of points of converges to a point of .

Definition 2.7 ²²: Let be any digital metric space and be a self digital map. If there exists such that for all then is called a digital contraction map.

Proposition 2.8 ⁷: Every digital contraction map is digitally continuous.

Theorem 2.9 ⁷: (Banach Contraction principle) Let be a complete digital metric space which has a usual Euclidean metric in . Let be a digital contraction map. Then has a unique fixed point, i.e. there exists a unique such that

Definition 2.10 ²² Let be a self mapping on and let be a function. We say is an -admissible mapping if

Let be such that is increasing, and iff

Definition 2.11 ²² Let be a digital metric space and and Suppose for all Then is called a digital contraction.

Definition 2.12 ²² Let be a digital metric space and be a digital -contractive type mapping if there exist functions and such that

Sridevi et.al ²² proved the following results:

Theorem 2.1 ²² Let be a digital metric space and be a digital -contractive type mapping. Suppose satisfies the following conditions

i) is -admissible ;

ii) There exist such that ;

Then

i) has a fixed point.

ii) If further are fixed points of with , then

Theorem 2.2 ²² Let be a digital metric space and be a digital -contractive type mapping. Suppose satisfies the following conditions

i) is -admissible ;

ii) There exist such that ;

iii) is digital continuous.

iv)

where

For all Then has a fixed point. If further are fixed points of with then . It is in this case has unique fixed point.

3. Main Results

We start the main section by introducing the new concepts of -admissible mapping, -admissible w.r.t mapping, - contractive and generalized - contractive pair of mappings.

Definition 3.1 Let be a self mapping on and let be a function. We say is an -admissible mapping if and implies and for all

Example Let Define the mapping and by

for all

Now we prove that is -admissible. Let and implies that

and

Definition 3.2 Let be a digital metric space and be a digital -contractive type mapping if there exist functions and such that

Definition 3.3 Let be complete digital metric space and and be the mappings. is -regular if is a sequence in such that for all , there exists a subsequence of such that for all and

Definition 3.4 Let be complete digital metric space and and be the mappings. is regular if is a sequence in such that for all , there exists a subsequence of such that for all and

Theorem 3.1 Let be complete digital metric space and and be the mappings satisfying the following conditions:

i) is -admissible ;

ii) There exist such that and

iii) Either is continuous or is regular

iv)

and Then has a fixed point. Further if are fixed points of with and then

Proof: Let be such that and Now we can construct a sequence in by

(1)

Moreover, we assume that if for some , then is a fixed point of . Consequently, we suppose that for all . Since is -admissible, so

and by induction we have for all . Similarly, we have for all

From (iii),

(2)

As is non decreasing, we get is a non-increasing sequence. Hence there is such that Letting in (2), we have and hence .

Step 2 We will prove that is a Cauchy sequence. Suppose to contrary; that is, is not a Cauchy sequence. Then there exists for which we can find two subsequences of positive integers and such that such that

(3)

This means that

(4)

From (3) and (4) and triangular inequality, we get

(5)

On letting in above inequality and using (3), we have

(6)

Also,

and

Therefore,

(7)

Using (4), (6) and (7) and , we get

(8)

On putting and in (iii), we have

(9)

Letting , in (9) and using continuity of , we obtain a contradiction. Thus is a Cauchy sequence in Since is complete, so there exists such that .

First, we suppose that is continuous, then

Now, suppose that is regular. Therefore, there exists a subsequence of such that and for all and and Using (iii) for , we get

Letting we get Further, suppose and be two fixed points of such that and

Again from (iii), we have

Hence has unique fixed point.

Definition 3.5 Let and . We say that is -admissible w.r.t if for all we have and implies that and

Example: Let Define the mappings and by

for all

Now we prove that is -admissible w.r.t Let and implies that

and and

Remark: Clearly, every -admissible mapping is -admissible w.r.t mapping when The following example shows that a mapping which is -admissible w.r.t may not be -admissible.

Example: Let . Define the mappings and by

for all

Suppose then we get , which shows is not -admissible. Now we prove that is -admissible w.r.t . Let

Therefore, is -admissible w.r.t .

Definition 3.5 Let be a digital metric space and and be the mappings. Then we say that the pair satisfy -contractive type mapping if

(10)

and

Theorem 3.2 Let be a complete digital metric space and and be mappings satisfying the following:

i) ;

ii) is -admissible w.r.t ;

iii) There exist such that and ;

iv)

v) If is a sequence in such that and for all and as , then there exists a subsequence of such that and for all

Also suppose is closed. Thenand have a coincide point.

Proof In view of condition, (iii) let be such that and Since , we can choose a point such that . Continuing this process having chosen we choose in such that

(11)

We complete the proof in following steps.

Step 1 First we prove that

(12)

Since is -admissible w.r.t we have

Using mathematical induction, we get

(13)

Similarly, we can prove that,

If for some , then by (11),

that is and have a coincide point at and hence the proof. For this, we suppose that for all n. Applying (10)and (13),

(14)

As is non decreasing, we get

is a non-increasing sequence. Hence there is such that Letting in (14), we have and hence .

Step 2: We will prove that is a Cauchy sequence. From (12), it is sufficient to show that is a Cauchy sequence. Suppose to contrary; that is, is not a Cauchy sequence. Then there exists for which we can find two sub sequences of positive integers and such that is smallest index for which

(15)

This means that

(16)

From (14)and (16) and triangular inequality, we get

On letting in above inequality and using (12), we have

(17)

Also,

Using (3),(8) and , we get

(18)

On the other hand, we have

Letting , and using continuity of , we have

(19)

From (10)

(20)

From (18),

Again from (9), we have to prove that

Using triangular inequality, we get

Letting in the above inequality and using (12) and (18), we obtain

From (11), we have

a contradiction. Thus is a Cauchy sequence in , which gives that is a Cauchy sequence in .

Step 3 Since by (11), we have and is closed, there existsuch that

(21)

Now, we show that is a coincidence point of and . On contrary, assume that Since by condition (iii) and (21), we have and for all then by use of triangle inequality and (10)we have

Letting , we get , a contradiction. Hence our supposition is wrong and , that is . This shows that and have a coincidence point.

Definition: Let be a digital metric space. Then the self maps are called generalized -contractive type 1 mappings if there exist three functions and such that

(22)

where

Let be two mappings. We denote by the set of coincidence points of and ; that is,

Theorem 3.3: Let be a complete digital metric space and and be mappings satisfying the following conditions :

i) ;

ii) is -admissible w.r.t ;

iii) There exist such that and

iv)

and where

v) If is a sequence in such that and for all and as , then there exists a subsequence of such that and for all

Also suppose is closed. Thenand have a coincide point.

Proof: Proceeding as in theorem 3.2, we can have

(23)

If for some then by (11), , that is and have a coincide point at and hence the proof. For this, we suppose that for all n. Applying (22)and (23),

(24)

where

If for some we have

then

a contradiction and hence

As is non decreasing, we get

is a non-increasing sequence. Hence there is such that Letting in (14), we have

and hence

Step 2: We will prove that is a Cauchy sequence. From (3), it is sufficient to show that is a Cauchy sequence. Suppose to contrary; that is, is not a Cauchy sequence. Then there exists for which we can find two subsequences of positive integers and such that is smallest index for which

(25)

This means that

(26)

From (25)and (26)and triangular inequality, we get

On letting in above inequality and using (12), we have

(27)

Also,

Using (12), (27)and , we get

(28)

On the other hand, we have

Letting , and using continuity of , we have

(29)

From (13)

(30)

where,

From (19),

Also, using triangular inequality, we get

Letting in the above inequality and using (12) and (28), we obtain

From (21), we have

a contradiction. Thus is a Cauchy sequence in , which gives that is a Cauchy sequence in .

Step 3 Since by (11), we have and is closed, there exist such that

Now, we show that is a coincidence point of and On contrary, assume that Since by condition (v) and (21), we have and for all , then by use of triangle inequality and (1) we have

(31)

Also,

Letting in (22), we get , a contradiction. Hence our supposition is wrong and , that is . This shows that and have a coincidence point.

Theorem 3.4: Let be a complete digital metric space and and be mappings satisfying the following conditions :

i) ;

ii) is -admissible w.r.t ;

iii) There exist such that ;

iv)

and

v) If is a sequence in such that for all and as , then there exists a subsequence of such that for all

Also suppose is closed. Then and have a coincide point.

Proof: Follows directly from theorem 3.2.

The next theorem shows that under additional hypotheses we can deduce the existence and uniqueness of a common fixed point.

Theorem 3.5: In addition to the hypotheses of theorem 3.2, suppose that for all , there exists such that and and commute at coincidence points. Then and have a unique common fixed point.

Proof: We will complete the result in three steps :

Step1 Uniqueness of coincidence point:

Following as in theorem 3.2, we can have , a Cauchy sequence in such that

for all and. As is closed, there exists such that then existence of coincidence point is guaranteed. Now, we prove that and have a unique coincidence point i.e if then

As and ,

(32)

Since is -admissible w.r.t we have from (32)

(33)

for all . Let if possible . Applying (1) and (24), we have

(34)

Letting in above inequality, we get

a contradiction and hence Similarly we can show that that yields

Step 2: Existence of common fixed point:

Let i.e Owing to the commutativity of and at their coincidence points, we get

(35)

Let us denote , then from (26), . Thus is a coincidence point of and Now, from Step 1, we have and hence is a common fixed point of and

Step 3: Uniqueness: Assume that is another common fixed point of and Then . By Step 1 we have . This completes the proof.

Example: Let and be a digital metric space in with 1-adjacency. Define the self maps by

Also, define the mappings as and

Clearly, the pair is contractive with and for all . In fact for all we have

Moreover, there exist such that . In fact for we have

Now, we show that is -admissible w.r.t . For this, let , such that i.e and therefore and

i.e and therefore

Also and is closed. Lastly, let be a sequence in such that

for all and as .Since , for all , by definition of and we have for all and . Then and hence all the conditions of theorem 3.5 are satisfied and consequently, is common fixed point of and .

Remarks

Letting in theorem 3.1, we obtain Theorem 3.10 in ²²

Letting in theorem 3.2, we obtain Theorem 3.12 in ²²

Fixed Point Theorems for Cyclic Contractive Mappings. As a generalization of the Banach contraction mapping principle, Kirk et al. ²² in 2003 introduced cyclic representations and cyclic contractions. A mapping is called cyclic if and where are nonempty subsets of a metric space Moreover, is called a cyclic contraction if there exists such that for all and Notice that although a contraction is continuous, cyclic contractions need not be. This is one of the important gains of this theorem. In the last decade, several authors have used the cyclic representations and cyclic contractions to obtain various fixed point results. see for example.

Theorem 4.1 Let be complete digital metric space, and be two nonempty closed subsets of Suppose that and be the mappings such that and , when , satisfying the following conditions:

i) There exist such that

ii) Either is continuous or is -regular

and Then has a fixed point in Further if are fixed points of with and then

Proof: Let and defined as then be complete digital metric space. Now if such that then also and hence all the hypotheses of theorem 3.1 are satisfied with , consequently, has a fixed point in , say If implies and implies hence . Further if are fixed points of then Then and , thus we can say We deduce that all the conditions of theorem 3.1 are satisfied with and hence has a fixed point.

Theorem 4.2: Let be complete digital metric space, and be two nonempty closed subsets of and be the mappings, where , satisfying the following conditions:

i) and are closed ;

ii) and ;

iii) is one one ;

iv) There exist functions such that

and

Then and have a coincidence point in Further, if commute at their coincidence point , then and have a unique common fixed point in

Proof: Since and are closed subsets of so be complete digital metric space. Define the mappings defined as

Now if then we need to prove that and If then and and thus , which proves and

As is one one, condition (iv) is equivalent to

By using (ii), we can show that

From (ii), we have Moreover, is closed. Now, we proceed to show that is -admissible w.r.t . Let

Since is one one, we have

thus

which gives that . Similarly, we can show that implies

Now, let be a sequence in such that for all and as . From the definition of and , we have

Since is closed set, thus

therefore, for all . Thus all the hypothesis of theorem 3.2 are satisfied. Hence, we deduce that and have a coincidence point that is . If then On the other hand Then we get using one one property of we have Similarly, we have

Notice that if is a coincidence point of and then Finally, let that is and From the above observation, we have implies that due to the fact that is one one, we get and Then our claims holds

Now, all the hypotheses of Theorem 3.3 are satisfied. So we deduce that is the unique common fixed point of and This completes the proof.

The following results are immediate consequences of above theorem.

Corollary 4.3: Let be complete digital metric space, and be two nonempty closed subsets of and be the mappings, where , satisfying the following conditions:

i) and are closed ;

ii) and ;

ii) is one one;

iv) There exist functions such that

and

Then and have a coincidence point in Further, if commute at their coincidence point, then and have a unique common fixed point in

Corollary 4.4: Let be complete digital metric space, and be two nonempty closed subsets of and be the mappings, where , satisfying the following conditions:

i) and are closed;

ii) and ;

iii) is one one ;

iv) There exist functions such that

Then and have a coincidence point in Further, if commute at their coincidence point, then and have a unique common fixed point in

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