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Abstract

In this paper, we introduce the notion of cyclic coupled proximal type fuzzy contraction (CCPTF-contraction) mappings in fuzzy metric spaces (FM-spaces). The condition of $C$ -class functions is used to establish some strong coupled proximal point (SCPP) results under the CCPTF-contraction mappings. The concept of closeness and the triangular condition of fuzzy metric is considered to prove our main results. We establish an illustrative example to support our work. In addition, we present an application of nonlinear integral equations (NIEs) to unify our work. This new concept of best proximal coupled fixed point results on FM-spaces with application will play a very best role for researchers in the theory of fixed point.

Keywords

best proximity point contraction conditions strong coupled proximal point integral equations

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References

Abkar

Gabeleh

(2013). Some best proximity results in partially ordered metric spaces. UPB Scientific Bulletin, Series A, 75, 91–100.

Ansari

A. H.

(2014). Note on

ϕ

ψ

-contractive type mappings and related fixed point. In The 2nd regional conference on mathematics and applications (pp. 377–380). Payame Noor University.

Ansari

A. H.

Jacob

G. K.

Marudai

Kumam

(2016). On the

C

-class functions of fixed point and best proximity point results for generalized cyclic coupled mappings. Cogent Mathematics, 3, 1–11.

Aydi

Felhi

(2016). Best proximity points for cyclic Kannan-Chatterjea-Ciric type contractions on metric-like spaces. The Journal of Nonlinear Science and Applications, 9, 2458–2466.

Balraja

Piramananthamb

Marudaic

(2019). Cyclic coupled best proximity points for generalized rational proximal contractions. In AIP conference proceedings (vol. 2177, no. 010001, p. 10). https://doi.org/10.1063/1.5135175

Bari

C. D.

Vetro

(2005). Fixed points, attractors and weak fuzzy contractive mappings in a fuzzy metric space. Journal of Fuzzy Mathematics, 13, 973–982.

Chen

G-X.

Jabeen

Rehman

S. U.

Khalil

A. M.

Abbas

Kanwal

Ullah

(2020). Coupled fixed point analysis in fuzzy cone metric spaces with an application to nonlinear integral equations. Advances in Difference Equations, 2020, 671. https://doi.org/10.1186/s13662-020-03132-8

Choudhury

B. S.

Das

Bhandari

S. K.

(2011). A generalized cyclic C-contraction principal in Menger spaces using a control function. International Journal of Applied Mathematics, 24, 663–673.

Choudhury

B. S.

Maity

(2014). Cyclic coupled fixed point result using Kannan type contractions. Journal of Operators, 2014, 876749.

10.

Eldred

A. A.

Veeramani

(2006). Existence and convergence of best proximity points. Journal of Mathematical Analysis and Applications, 323, 1001–1006.

11.

George

Veeramani

(1994). On some results in fuzzy metric spaces. Fuzzy Sets Systems, 64, 395–399.

12.

Grabiec

(1988). Fixed point in fuzzy metric spaces. Fuzzy Sets Systems, 27, 385–389.

13.

Gupta

Rajput

S. S.

Kaurav

P. S.

(2014). Coupled best proximity theorem in metric spaces. International Journal of Analysis and Applications, 4, 201–215.

14.

Hadzic

Pap

(2002). Fixed point theorem for multivalued mappings in probabilistic metric spaces and an applications in fuzzy metric spaces. Fuzzy Sets Systems, 127, 333–344.

15.

Ivanova

Zlatanov

(2019). Implementation of fuzzy functions aimed at fairer grading of students’ tests. Education Sciences, 9(3), 214.

16.

Karapinar

(2011). Fixed point theory for cyclic weak

ϕ

-contraction. Applied Mathematics Letters, 24, 822–825.

17.

Kiany

Amini-Harandi

(2011). Fixed point and endpoint theorems for set-valued fuzzy contraction maps in fuzzy metric spaces. Fixed Point Theory and Applications, 94, 1–9.

18.

Kirk

W. A.

Srinivasan

P. S.

Veermani

(2003). Fixed points for mappings satisfying cyclic contractive conditions. Fixed Point Theory, 4, 79–89.

19.

Kramosil

Michalek

(1975). Fuzzy metric and statistical metric spaces. Kybernetika, 11, 336–344.

20.

Lakshikantham

Ciric

(2009). Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces. Nonlinear Analysis: Theory, Methods & Applications, 12, 4341–4349.

21.

Latif

Abbas

Hussain

(2016). Coincidence best proximity point of

F g

-weak contractive mappings in partially ordered metric spaces. The Journal of Nonlinear Science and Applications, 9, 2448–2457.

22.

Liu

Ansari

A. H.

Chandok

Park

(2016). Some new fixed point results in partial ordered metric spaces via admissible mappings and two new functions. The Journal of Nonlinear Science and Applications, 9, 1564–1580.

23.

Moussaoui

Hussain

Melliani

(2021). Global optimal solutions for proximal fuzzy contractions involving control functions. Journal of Mathematics, 2021(1), 6269304.

24.

Moussaoui

Hussain

Melliani

Hayel

Imdad

(2022b). Fixed point results via extended FZ-simulation functions in fuzzy metric spaces. Journal of Inequalities and Applications, 2022, 69.

25.

Moussaoui

Saleem

Melliani

Zhou

(2022a). Fixed point results for new types of fuzzy contractions via admissible functions and FZ-simulation functions. Axioms, 11, 87.

26.

Mursaleen

Srivastava

H. M.

Sharma

S. K.

(2016). Generalized statistically convergent sequences of fuzzy numbers. Journal of Intelligent and Fuzzy Systems, 30, 1511–1518.

27.

Pǎcurar

Rus

I. A.

(2012). Fixed point theory for cyclic

φ

-contractions. Nonlinear Analysis: Theory, Methods & Applications, 75, 3218–3228.

28.

Razani

(2005). A contraction theorem in fuzzy metric spaces. Fixed Point Theory and Applications, 3, 257–265.

29.

Razani

Shirdaryazdi

(2006). Some results on fixed points in the fuzzy metric space. Journal of Applied Mathematics and Computing, 20, 401–408.

30.

Rehman

S. U.

Jabeen

Khan

S. U.

Jaradat

M. M. M.

(2021). Some

α - ϕ

-fuzzy cone contraction results with integral type application. Journal of Mathematics, 2021, 1566348.

31.

Rusmiari

Putra

Sasmita

(2013). Fuzzy logic method for evaluation of diffculty level of exam and student graduation. International Journal of Computer Science, 10(2), 223–229.

32.

Sadeghi

Vaezpour

S. M.

Park

Saadati

Vetro

(2014). Set-valued mappings in partially ordered fuzzy metric spaces. Journal of Inequalities and Applications, 157, 1–17.

33.

Schweizer

Sklar

(1960). Statical metric spaces. Pacific Journal of Mathematics, 10, 314–334.

34.

Shamas

Rehman

S. U.

Abdeljawad

Sattar

Khan

S. U.

Mlaiki

(2022). Generalized contraction theorems approach to fuzzy differential equations in fuzzy metric spaces. AIMS Mathematics, 7(6), 11243–11275. https://doi.org/10.3934/math.2022628

35.

Shamas

Rehman

S. U.

Aydi

Mahmood

Ameer

(2021). Unique fixed-point results in fuzzy metric spaces with an application to Fredholm integral equations. Journal of Function Spaces, 2021, 4429173. https://doi.org/10.1155/2021/4429173

36.

Sobrino

(2013). Fuzzy logic and education: Teaching the basics of fuzzy logic through an example (by way of cycling). Education Sciences, 3, 75–97.

37.

Udo-Utun

(2014). Existence of strong coupled fixed points for cyclic coupled Ciric-type mappings. Journal of Operators, 2014(1), 381685.

38.

Waheed

M. T.

Rehman

S. U.

Jan

Gumaei

Al-Rakhami

(2021a). Some new coupled fixed-point findings depending on another function in fuzzy cone metric spaces with application. Mathematical Problems in Engineering, 2021(1), 4144966. https://doi.org/10.1155/2021/4144966

39.

Waheed

M. T.

Rehman

S. U.

Jan

Gumaei

Al-Rakhami

(2021b). An approach of Lebesgue integral in fuzzy cone metric spaces via unique coupled fixed point theorems. Journal of Function Spaces, 2021(1), 8766367. https://doi.org/10.1155/2021/8766367

40.

Zadeh

L. A.

(1965). Fuzzy sets. Information and Control, 8, 338–353.

Best Proximal Coupled Fixed Point Results on Fuzzy Metric Spaces Using C -Class Functions with an Application

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