Korean J. Math. Vol. 26 No. 2 (2018) pp.307-326
DOI: https://doi.org/10.11568/kjm.2018.26.2.307

Fixed point theorems in fuzzy metric spaces for mappings with some contractive type conditions

Article Sidebar

PDF
Published: Jun 30, 2018
Keywords:
fuzzy metric space, altering distance, t-norm, fixed point
Mathematics Subject Classification:
54H25, 47H10.

Main Article Content

Bijoy Patir
Department of Mathematics Gauhati University Guwahati-781014, Assam, India
Nilakshi Goswami
Department of Mathematics Gauhati University Guwahati-781014, Assam, India
Lakshmi Narayan Mishra
Department of Mathematics School of Advanced Sciences Vellore Institute of Technology (VIT) University Vellore 632 014, Tamil Nadu, India

Abstract

In this paper, we derive some fixed point theorems in fuzzy metric spaces for self mappings satisfying different contractive type conditions. Some of these theorems generalize some results of Wairojjana et al. \textit{(Fixed Point Theory and Applications (2015) 2015:69)}. Several examples in support of the theorems are also presented here.


Article Details

References

[1] M. Abbas, T. Nazir, B.E. Rhoades,Fixed points of multivalued mapping satisfying ciric type contractive conditions in G-metric spaces, Hacettepe Journal of Mathematics and Statistic, Volume 42 (1) (2013), 21–29. Google Scholar

[2] A. Aghajani, M. Abbas, J. R. Roshan,Common fixed point of generalized weak contractive mapping in partially ordered Gb-metric spaces, Filomat 28:6 (2014), 1087–1101. Google Scholar

[3] S.Banach, Sur les oprations dans les ensembles abstraits et leur application aux quations intgrales, Fund. Math. 3 (1922), 133–181. Google Scholar

[4] R. Chen, X. Wang,Fixed point of nonlinear contractions in modular spaces, Journal of Inequalities and Applications (2013), 2013:399. Google Scholar

[5] D. Das and N. Goswami, Fixed points of different contractive type mappings on tensor product spaces, IJIRSET 3 (2009), 14512–14519. Google Scholar

[6] D. Das and N. Goswami, Fixed points of mappings satisfying a weakly contractive type condition, Journal of Mathematical Research with Applications 36 (1) (2016), 70–78. Google Scholar

[7] D. Das and N. Goswami, Some fixed point theorems on the sum and product of operators in tensor product spaces, International Journal of Pure and Applied Mathematics 109 (3) (2016), 651–663. Google Scholar

[8] D. Das, N. Goswami, V.N. Mishra, Some Results on Fixed Point Theorems in Banach Algebras, Int. J. Anal. Appl. 13 (1) (2017), 32–40. Google Scholar

[9] N.R. Das and M.L. Saha,On fixed points in fuzzy metric spaces, Annals of Fuzzy Mathematics and Informatics 7 (2) (Feb 2014), 313–318. Google Scholar

[10] Deepmala,A Study on Fixed Point Theorems for Nonlinear Contractions and its Applications, Ph.D. Thesis (2014), Pt. Ravishankar Shukla University, Raipur 492 010, Chhatisgarh, India. Google Scholar

[11] D. Dey, M. Saha,An extension of Banach fixed point theorem in fuzzy metric space, Bol. Soc. Paran. Mat. 32 (1) (2014), 299–304. Google Scholar

[12] M. Dinarvand,Some fixed point results for admissible Geraghty contraction type mappings in fuzzy metric spaces, Iranian journal of fuzzy systems 14 (3) (2017), 161–177. Google Scholar

[13] T. DoVsenovi c, D. Raki c, M. Brdar, Fixed point theorem in fuzzy metric space using altering distance, Filomat 28:7 (2014), 1517-1524. Google Scholar

[14] M. Edelstein,On fixed and periodic points under contractive mappings, J. Lond. Math. Soc. 37 (1962), 74–79. Google Scholar

[15] L. Gajic, M. Stojakovic,Sehgal-Thomas type fixed point theorems in generalized metric spaces, Filomat 31:11 (2017), 3347–3356. Google Scholar

[16] A. George, P. Veeramani, On some results in fuzzy metric spaces, Fuzzy Set. Syst. 64 (1994), 395–399. Google Scholar

[17] A. George, P. Veeramani,On some results of analysis for fuzzy metric spaces, Fuzzy set. Syst. 90 (1997), 365–368. Google Scholar

[18] M. Grabiec,Fixed points in fuzzy metric spaces, Fuzzy Set. Syst. 27 (1988), 385– 389. Google Scholar

[19] J. Harjani, B. L opez and K. Sadarangani, Fixed point theorems for cyclic ph- contractions in ordered metric space, Fixed Point Theory 14 (2) (2013), 359- 368. Google Scholar

[20] N. Hussain, V. Parvaneh, B. Samet and C. Vetro, Some fixed point theorems for generalized contractive mappings in complete metric spaces, Fixed Point Theory and Application(2015), 2015:185. Google Scholar

[21] E. Karapinar, Edelstein type fixed point theorems, Fixed Point Theory and Application(2012), 2012:107. Google Scholar

[22] M. S. Khan, M. Swaleh, S. Sessa,Fixed point theorems by altering distances between the points, Bull. Aust. Math. Soc. 30 (1984), 1–9. Google Scholar

[23] I. Kramosil and J. Michalek,Fuzzy metrics and statistical metric spaces, Kybernetika(Prague) 11 (5)(1975), 336–344. Google Scholar

[24] F. Lael and Z. Heidarpour, Fixed point theorems for a class of generalized nonexpansive mappings, Fixed Point Theory and Applications (2016), 2016:82. Google Scholar

[25] X. Li, M. Guo, Y. Su,On the intuitionistic fuzzy metric spaces and the intuitionistic fuzzy normed spaces, J. Nonlinear Sci. Appl. www.tjnsa.com. Google Scholar

[26] V.N. Mishra, Some Problems on Approximations of Functions in Banach Spaces, Ph.D. Thesis (2007), Indian Institute of Technology, Roorkee 247 667, Uttarakhand, India. Google Scholar

[27] L.N. Mishra,On existence and behavior of solutions to some nonlinear integral equations with Applications, Ph.D. Thesis (2017), National Institute of Technology, Silchar 788 010, Assam, India. Google Scholar

[28] V.N. Mishra, L.N. Mishra,Trigonometric Approximation of Signals (Functions) in Lp (p≥1)-norm, International Journal of Contemporary Mathematical Sciences 7 (19) (2012), 909–918. Google Scholar

[29] L.N. Mishra, K. Jyoti, A. Rani, Vandana,Fixed point theorems with digital contractions image processing, Nonlinear Sci. Lett. A 9 (2) (2018), 104–115. Google Scholar

[30] L.N. Mishra, H.M. Srivastava, M. Sen, On existence results for some nonlinear functional-integral equations in Banach algebra with applications, Int. J. Anal. Appl. 11 (1) (2016), 1–10. Google Scholar

[31] L.N. Mishra, S.K. Tiwari, V.N. Mishra, Fixed point theorems for generalized weakly S-contractive mappings in partial metric spaces, Journal of Applied Analysis and Computation 5 (4) (2015), 600–612. doi:10.11948/2015047. Google Scholar

[32] L.N. Mishra, S.K. Tiwari, V.N. Mishra, I.A. Khan,Unique Fixed Point Theorems for Generalized Contractive Mappings in Partial Metric Spaces, Journal of Function Spaces, Volume 2015 (2015), Article ID 960827, 8 pages. Google Scholar

[33] J. R. Morales, E. M. Rojas,Some generalizations of Jungck’s fixed point theorem, IJMMS (2012), Vol:2012, Article ID 213876. Google Scholar

[34] R. M. Patel, R. Bhardwaj,Some common fixed point theorems for compatible mapping in fuzzy metric spaces for Integral type mapping, Mathematical Theory and modeling 3 (6) (2013). Google Scholar

[35] H.K. Pathak and Deepmala,Common fixed point theorems for PD-operator pairs under Relaxed conditions with applications, Journal of Computational and Applied Mathematics,239 (2013), 103-113. Google Scholar

[36] A. Sadiku,Schauder bases and locally complemented subspaces of Banach spaces, Master’s thesis, University of Agder, (2014). Google Scholar

[37] B. Schweizer, A. Sklar,Statistical metric spaces, Pacific J. Math. 10 (1960), 314–334. Google Scholar

[38] Y. Shen, D. Qiu, W. Chen,Fixed point theorems in fuzzy metric spaces, Appl. Math. Lett. 25 (2012), 138–141. Google Scholar

[39] S. Shukla and M. Abbas,Fixed point results in fuzzy metric-like spaces, Iranian Journal of Fuzzy Systems 11 (5) (2014), 81–92. Google Scholar

[40] H. Vosoughi, S. J. Hosseini Ghoncheh,A remark on fuzzy contractions, ICM (2012), 11-14 March, Faculty.uaeu.ac.ae. Google Scholar

[41] N. Wairojjana, T. DoVsenovi c, D. Raki c, D. Gopal and P. Kumam,An altering distance function in fuzzy metric space theorems, Fixed Point Theory and Ap- plications (2015), 2015:69. Google Scholar