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Generalised Rational αs-Meir-Keeler Contraction Mapping in S-metric Spaces

Shanu Poddar, Yumnam Rohen
American Journal of Applied Mathematics and Statistics. 2021, 9(2), 48-52. DOI: 10.12691/ajams-9-2-2
Received March 04, 2021; Revised April 07, 2021; Accepted April 16, 2021

Abstract

In this paper, we introduce the concept of generalised rational αs-Meir-Keeler contraction mapping on S-metric spaces. The existence of fixed points is also discussed.

1. Introduction and Preliminaries

Banach contraction principle is one of the most interested topics for many researchers because of its applications in various fields, simplicity and easiness. They attempted to generalise Banach contraction principle in different dimensions. Samet et. al. 1 made an attempt by introducing the idea of α-admissible mappings and further introducing the concept of α-ψ-contractive type mappings in metric spaces. The results of Samet et. al. 1, show that Banach fixed point theorem and a large number of results in the literature are consequences of their results. On the other hand, as one the result of generalisation of metric space, Sedghi et. al. 2 introduced the definition of S-metric space. There are various works on generalisation of Banach contraction principle and generalisation of metric space in the literature. Some of these works can be found in the research papers through 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29 and references mentioned in these papers.

Definition 1.1. 2 In a non-empty set X let S: X × X × X → [0, ∞) be a mapping satisfying

(1) S (a, b, c) ≥ 0,

(2) S (a, b, c) = 0 if and if a = b = c,

(3) S (a, b, c) ≤ S (a, a, e) + S (b, b, e) + S (c, c, e)

for all a, b, c, e in X then S is known as S- metric and pair (X, S) is known as S-metric space.

Definition 1.2. 2 We have for an S-metric space

Definition 1.3. 2 Let (X, S) be an S-metric space.

i) A sequence in X converges to a if and only if S( a) → 0 as n→ ∞. That is there exists ∈ N for each such that S( a) for all and denoted as

ii) A sequence in X is said to be a Cauchy sequence if there exists ∈ N for each such that for each

iii) If every Cauchy sequence is convergent then S-metric space (X, S) is said to be complete.

Meir-Keeler 25 introduced a generalisation of Banach contraction principle. According to them, a self mapping U in a metric space (X, d) is said to be Meir-Keeler contraction if for an ε > 0 there exists δ > 0 such that ε ≤ d(a, b) < ε + δ(ε) implies d(Ua, Ub) < ε for all a, b in X, They also state and prove that in a metric space, if a self mapping U is a Meir-Keeler contraction, then U has a fixed point in X.

In this paper we introduced the concept of -Meir-Keeler contraction on S-metric space and proved a fixed point theorem.

Now, we recall the definition of α-admissible mappings and its generalisations in metric space, G-metric space, S-metric space and -metric space.

Definition 1.4. 1 In a metric space (X, d), let U be a self mapping and let α : X × X → [0, +∞) be a function. U is said to be an α-admissible if a, b ∈ X, α(a, b) ≥1 becomes α(Ua, Ub) ≥ 1.

Example 1. Let X = [0, +∞), and U: X → X be a self mapping defined by Ua = 5a for all a, b ∈ X. Let α : X × X → [0, +∞) be defined by

Then, U is an α-admissible mapping.

Let X be a metric space for further consideration if otherwise stated.

Definition 1.5. 3 Suppose U, V: X → X and α: X × X → [0, +∞) then (U, V) pair is said to be α-admissible if α(a, b) ≥ 1 for a, b ∈ X, then α(Ua, Vb) ≥ 1 and α(Va, Ub) ≥ 1.

Definition 1.6. 4 Suppose U: X → X and α : X × X → (−∞, +∞) then mapping U is said to be a triangular α-admissible if

(i) α(a, b) ≥ 1, implies α(Ua, Ub) ≥ 1, a, b ∈ X,

(ii) α(a, c) ≥ 1, α(c, b) ≥ 1, implies α(a, b) ≥ 1, a, b, c ∈ X.

Definition 1.7. 3 Suppose U, V: X → X and α : X × X → [0, +∞) then (U, V) pair is said to be a triangular α-admissible mapping if

(i) α(a, b) ≥ 1, implies α(Ua, Vb) ≥ 1 and α(Va, Ub) ≥ 1, a, b ∈ X,

(ii) α(a, c) ≥ 1, α(c, b) ≥ 1, implies α(a, b) ≥ 1, a, b, c ∈ X.

Definition 1.8. 5 Suppose U : X → X and let α, η : X ×X → [0, +∞) be two functions then U is said to be α-admissible mapping with respect to η if α(a, b) ≥ η(a, b) implies α(Ua, Ub) ≥ η(Ua, Ub) for all a, b ∈ X.

If we consider η(a, b) = 1, in definition 1.8 then we get definition 1.4. Also, U is said to be a η-sub admissible mapping if α(a, b) = 1.

Lemma 1.1. 6 Let (X, d) be a metric space and U: X → X be a triangular α-admissible mapping. Suppose ∈ X such that α(, U) ≥ 1. Let us define a sequence as Then α ≥1for all m, n ∈ N ∪ {0} with n < m.

Lemma 1.2. 7 Let (X, d) be a metric space and U, V : X → X be a triangular α-admissible mapping. Suppose ∈ X such that α(, U) ≥ 1. Let us define sequences = U and = V, where i = 0, 1, 2, . . . . Then α ≥ 1 for all m, n ∈ N ∪ {0} with n < m.

The notion of α-admissible mappings was generalised by Alghamdi and Karapinar 8 to make it suitable for G-metric space and termed it as β-admissible. The definition itself is as follows.

Definition 1.9. 8 Let (X, G) be a G-metric space, U: X → X and then U is said to be β-admissible if for all implies

They had given suitable example for β-admissible mappings. Further, they also generalised the α-ψ contractive mappings by introducing generalised G-β-ψ contractive mappings of type I and II.

Further, Hussain et. al. 9 generalised the concept of α-admissible mappings in G-metric space. They introduced the concept of rectangular G-α-admissible and extended this concept for two mappings.

By introducing G-η-sub admissible mapping and α-dominating map Ansari et. al. 10 also studied α-admissible mappings in G-metric space. Another type called η-sub dominating map is also introduced by them. They also introduced α-regular in the context of G-metric space, partially weakly G-α-admissible and partially weakly G-η-sub admissible mappings, etc.

Zhou et. al. 11 also extended the concept of α-admissible mappings in S-metric space under the name γ-admissible. It is defined as follows:

Definition 1.10. 11 Let U : X → X and γ : → [0, +∞) then U is said to be γ- admissible if for all a, b, c ∈ X, γ (a, b, c) ≥ 1 implies γ(Ua, Ub, Uc) ≥ 1.

The notion of generalised S-β-ψ contractive type mappings was introduced by Bulbul et. al. 12 on the similar way as that of generalised G-β-γ contractive type mappings, but in S-metric space. The notion of α-admissible mappings in -metric space was also introduced by Nabil et. al. 13.

Definition 1.11. Let (X, S) be an S-metric space, U: X → X and : X × X × X → [0, +∞), then U is called -admissible if a, b, c ∈ X, (a, b, c) ≥ 1 implies (Ua, Ub, Uc) ≥ 1.

Example 2. Let X = [0, +∞) and define U: X → X and : X × X × X → [0, +∞) by Ua = 4a, for all a, b, c ∈ X and

Then U is -admissible.

Definition 1.12. Let (X, S) be an S-metric space, U, V: X → X and : X×X×X→ [0, +∞). We say that the pair (U, V) is -admissible if a, b, c ∈ X such that (a, b, c) ≥ 1, then we have (Ua, Ub, Vc) ≥ 1 and (Va, Vb, Uc) ≥ 1.

Definition 1.13. Let (X, S) be an S-metric space, U : X → X and : X×X×X → [0, +∞). We say that U is triangular -admissible mapping if

(i) (a, b, c) ≥ 1 implies (Ua, Ub, Uc) ≥ 1, a, b, c ∈ X.

(ii) (a, c, e) ≥ 1, (b, b, e) ≥ 1 and (c, c, e) ≥ 1 implies (a, b, c) ≥ 1, a, b, c, e ∈ X.

Definition 1.14. Let (X, S) be an S-metric space, U : X → X and let , : X × X × X → [0, +∞) be two functions. We say that U is -admissible mapping with respect to if a, b, c ∈ X, (a, b, c) ≥ (a, b, c) implies (Ua, Ub, Uc)≥ (Ua, Ub, Uc).

Note that if we take (a, b, c) = 1, then this definition reduces to definition 1.11. Also, if we take (a, b, c) = 1, then we say that U is an -subadmissible mapping.

Now we state the following two lemmas in the line of Lemma 1.1 and Lemma 1.2.

Lemma 1.3. Let (X, S) be an S-metric space, U: X→X be a triangular -admissible mapping. Assume that there exists ∈ X such that (, , U) ≥ 1. Define a sequence by Then we have ≥ 1, for all m, n ∈ N ∪ {0}.

Lemma 1.4. Let (X, S) be an S-metric space, U, V: X → X be a triangular -admissible mapping. Assume that there exists ∈ U such that (, , U) ≥ 1. Define sequences = U and = V, where i = 0, 1, 2, . . . . Then we have ≥1 for all m, n∈ N∪{0} with n < m.

2. Main Result

We present the following results.

Definition 2.1. In an S-metric space (X, S) let U: X → X be a triangular -admissible mapping. Suppose that for every ε >0 there exists δ > 0 such that

implies

(1)

where

(2)

for all a, b, c ∈X. Then U is called a generalised rational αs-Meir Keeler contraction of type-I.

Definition 2.2. In an S-metric space (X, S) let U: X → X be a triangular -admissible mapping. Suppose that for every ε > 0 there exists δ > 0 such that

implies

(3)

where

(4)

for all a, b ∈ X. Then U is called a generalised rational αs-Meir Keeler contraction of type-II.

Theorem 2.1. In a complete S-metric space (X, S), U: X → X be a mapping satisfying

(i) U is generalised rational -Meir-Keeler contraction of type-I.

(ii) U is triangular -admissible.

(iii) there exists ∈ X such that (, , U) ≥ 1.

(iv) U is continuous.

Then U has a fixed point in X.

Proof. Let ∈ X be such that = U Then, we construct a sequence of points in X such that,

(5)

where i = 0, 1, 2, 3, . . . .

By assumption (, , ) ≥ 1 and U is -admissible, by Lemma 1.3, we have

(6)

Then,

(7)

for all i ∈ N ∪ {0}. Now,

If

then

Therefore, we deduce that

which is a contradiction. Hence

(8)

This implies that

(9)

for all n ∈ N ∪ {0}.

So, sequence {S} is nonnegative and nonincreasing. Now, we have to show that S→ 0. It is clear that{S} is a decreasing sequence. Therefore, = r for some positive number r.

Let r > 0. Then we must have

(10)

for all n ∈ N ∪ {0}.

Since the condition (7) holds for every ε > 0, we may choose ε = r then there exists δ(ε) > 0 satisfying (3). In other words,

However, this implies

(11)

A contradiction and hence r = 0, that is

(12)

Next, we have to show that sequence is Cauchy. If possible let is not a Cauchy sequence. Then there exists sequences and with ε> 0 such that,

(13)

and

(14)

where

Using the triangle inequality and (14),

Applying k → ∞ we obtain

(15)

Also, from the triangular inequality, we have

and

Applying k → ∞ we obtain

and

(16)

By Lemma 1.3, α ≥ 1, we have

a contradiction. This shows that is a Cauchy sequence. By completeness of X, there exists a ∈ X such that → a and hence → a. By continuity of U we get U → Ua. Thus a = Ua showing that ‘a’ is a fixed point of U.

3. Conclusion

The concept of generalised rational -Meir-Keeler contraction mapping on S-metric spaces is introduced by giving two contractive definitions. The existence of fixed points for the new contractive type mappings is discussed. Further study for -Meir-Keeler contraction mapping in S-metric, Sb-metric G-metric and Gb-metric spaces can be carried out.

Acknowledgements

We would like to express our thanks to the Editor and Reviewers for valuable advices in helping to improve the manuscript.

Conflict of Interest

There is no conflict of interest.

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In article      View Article
 
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[16]  Qasim Mahmood, Aqeel Shahzad, Abdullah Shoaib, Arslan Hojat Ansari, Stojan Radenovi´c, Common fixed point results for α-ψ -contractive mappings via (F; h) mappings via pair of upper class functions, J. Math. Anal. Vol. 10, 4 (2019), 1-10.
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[22]  A. Gholidahneh, S. Sedghi, T. Doˇsenovi´c and S. Radenovi´c, Ordered S-metric spaces and coupled common fixed point theorems of integral type contraction, Mathematical Interdisciplinary Research, 2 (2017), 71-84.
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In article      View Article
 
[27]  Kifayat Ullah, Bakht Ayaz Khan, Ö. Özer and Zubair Nisar, 2019, Some Convergence Results Using K* Iteration Process In Busemann Spaces, Malaysian Journal of Mathematical Sciences 13(2): 231-249.
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Published with license by Science and Education Publishing, Copyright © 2021 Shanu Poddar and Yumnam Rohen

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Normal Style
Shanu Poddar, Yumnam Rohen. Generalised Rational αs-Meir-Keeler Contraction Mapping in S-metric Spaces. American Journal of Applied Mathematics and Statistics. Vol. 9, No. 2, 2021, pp 48-52. http://pubs.sciepub.com/ajams/9/2/2
MLA Style
Poddar, Shanu, and Yumnam Rohen. "Generalised Rational αs-Meir-Keeler Contraction Mapping in S-metric Spaces." American Journal of Applied Mathematics and Statistics 9.2 (2021): 48-52.
APA Style
Poddar, S. , & Rohen, Y. (2021). Generalised Rational αs-Meir-Keeler Contraction Mapping in S-metric Spaces. American Journal of Applied Mathematics and Statistics, 9(2), 48-52.
Chicago Style
Poddar, Shanu, and Yumnam Rohen. "Generalised Rational αs-Meir-Keeler Contraction Mapping in S-metric Spaces." American Journal of Applied Mathematics and Statistics 9, no. 2 (2021): 48-52.
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[1]  B. Samet, C. Vetro, P.Vetro, Fixed point theorems for α−ψ-contractive type mappings, Nonlinear Anal. 75(2012), 2154-2165.
In article      View Article
 
[2]  Shaban Sedghi, Nabi Shobe and Abdelkrim Aliouche, A generalization of fixed point theorems in S-metric spaces, Matematiki Vesnik, 64, 3 (2012), 258-266.
In article      
 
[3]  T. Abdeljwad, Meir-Keeler α-contractive fixed and common fixed point theorems, Fixed Point Theory and Applications, 2013, 2013:19.
In article      View Article
 
[4]  E. Karapinar, Poom Kumam and Peyman Salimi, On α-ψ-Meir-Keeler contractive mappings, Fixed Point Theory and Applications, 2013, 2013:94.
In article      View Article
 
[5]  P. Salimi, A. Latif and N. Hussain, Modified α-ψ-contractive mappings with applications, Fixed Point Theory and Applications, 2013, 2013:151.
In article      View Article
 
[6]  S. Cho, J. Bae and E. Karapinar, Fixed point theorems of α-Geraghaty contraction type in metric space, Fixed Point Theory and Applications, 2013, 2013:329.
In article      View Article
 
[7]  M. Arshad, Aftab Hussain and Akbar Azam, Fixed point of α-Geraghaty contraction with application, UPB Sci. Bull. Series A, 78(2), 2016, 67-78.
In article      
 
[8]  M. A. Alghamdi and E. Karapnar, G − β − ψ contractive-type mappings and related fixed point theorems, Journal of Inequalities and Applications, 2013, 2013: 70.
In article      View Article
 
[9]  N. Hussain, V. Parvaneh and F. Golkarmanesh, Coupled and tripled coincidence point results under (F, g)-invariant sets in Gb-metric spaces and G−α-admissible mappings, Math Sci. (2015) 9:11-26.
In article      View Article
 
[10]  A. H. Ansari, S. Chandok, N. Hussain, Z. Mustafa and M.M.M. Jaradat, Some common fixed point theorems for weakly α-admissible pairs in G-metric spaces with auxiliary functions, Journal of Mathematical Analysis, 8(3) (2017), 80-107.
In article      
 
[11]  Mi Zhou, Xiao-lan Liu, Stojan Radenovi´c, S-γ-φ-ϕ-contractive type mappings in Smetric spaces, J. Nonlinear Sci. Appl., 10(2017), 1613-1639. 16.
In article      View Article
 
[12]  Bulbul Khomdram, Yumnam Rohen, Yumnam Mahendra Singh, Mohammad Saeed Khan, Fixed point theorems of generalised S-β-ψ contractive type mappings, Mathematica Moravica, 22(1), (2018), 81-92.
In article      
 
[13]  N. Mlaiki, A. Mukheimer, Y. Rohen, N. Souayah and T. Abdeljawad, Fixed point theorems for α-ψ-contractive mapping in Sb-metric spaces, Journal of Mathematical Analysis, 8(5)(2017), 40-46.
In article      
 
[14]  S.Phiangsungnoen, W. Sintunavarat and P. Kumam, Fuzzy fixed point theorems for fuzzy mappings via β-admissible with applications, Fixed Point Theory and Applications, 2014, 2014:190.
In article      View Article
 
[15]  P. Debnath, M. Neog, S. Radenovi´c, Set valued Reich type G-contractions in a complete metric space with graph, Rendiconti del Circolo Matematico di Palermo Series, 2 (2019).
In article      View Article
 
[16]  Qasim Mahmood, Aqeel Shahzad, Abdullah Shoaib, Arslan Hojat Ansari, Stojan Radenovi´c, Common fixed point results for α-ψ -contractive mappings via (F; h) mappings via pair of upper class functions, J. Math. Anal. Vol. 10, 4 (2019), 1-10.
In article      
 
[17]  A S. Babu, T. Doˇsenovi´c, MD. Mustaq Ali, S. Radenovi´c, K.P.R. Rao, Some Preˇsi´c type results in b-dislocated metric spaces, Constructive Mathematical Analysis, 2 (2019), No. 1, pp. 40-48.
In article      View Article
 
[18]  T. Doˇsenovi´c, S. Radenovi´c, S. Sedghi, Generalized metric spaces: Survey, TWMS. J. Pure Appl. Math., Vol.9, 1 (2018), pp. 3-17.
In article      
 
[19]  S. Sedghi, A. Gholidahneh, T. Doˇsenovi´c, J. Esfahani, S. Radenovi´c, Common fixed point of four maps in Sb-metric spaces, Journal of Linear and Topol. Algebra, Vol. 05, No. 02(2016), 93-104.
In article      
 
[20]  A. H. Ansari, D. D. Djeki´c, Feng Gu, B. Z. Popovi´c and S. Radenovi´c, C-class functions and remarks on fixed points of weakly compatible mappings in G-metric spaces satisfying common limit range property, Mathematical Interdisciplinary Research, 1 (2016), 279-290.
In article      View Article
 
[21]  Z. Aleksi´c, Z.D.Mitrovi´c and S. Radenovi´c, Picard sequences in b-metric spaces, Fixed Point Theory, 21 (2020), No. 1, 35-46.
In article      
 
[22]  A. Gholidahneh, S. Sedghi, T. Doˇsenovi´c and S. Radenovi´c, Ordered S-metric spaces and coupled common fixed point theorems of integral type contraction, Mathematical Interdisciplinary Research, 2 (2017), 71-84.
In article      
 
[23]  D. Dhamodharan, R. Krishnakumar and S. Radenovi´c, Coupled fixed point theorems of integral type contraction in Sb- metric spaces, Results in Fixed Point Theory and Applications, Volume 2019, Article ID 2018032, 28 pages.
In article      View Article
 
[24]  Ravi P. Agarwal, Erdal Karapinar, Donal O’Regan, Antonio Francisco Roldan-Lopez-de-Hiero, Fixed Point Theory in Metric Type Spaces, Springer International Publishing Switzerland 2015.
In article      View Article
 
[25]  A. Meir and E. Keeler, A Theorm on Contraction Mappings Journal of Mathematical Analysis and Applications, 28, (1969) 326-329.
In article      View Article
 
[26]  Özen Özer and Saleh OMRAN, 2016. “Common Fixed Point Theorems in C*- Algebra Valued bMetric Spaces” AIP Conference Proceedings 1773, 050005 (2016).
In article      View Article
 
[27]  Kifayat Ullah, Bakht Ayaz Khan, Ö. Özer and Zubair Nisar, 2019, Some Convergence Results Using K* Iteration Process In Busemann Spaces, Malaysian Journal of Mathematical Sciences 13(2): 231-249.
In article      
 
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