### Fixed Point Results for $$(\alpha$$-$$\beta_k,\phi$$-$$\psi)$$ Integral Type Contraction Mappings in Fuzzy Metrics

#### Abstract

In this paper, we introduce the notion of a modified $$(\alpha$$-$$\beta_k,\phi$$-$$\psi)$$ integral type contraction mappings in fuzzy metric spaces. We study and prove the existence and uniqueness of fixed points theorems in generalized fuzzy contractive mappings of integral type in fuzzy metric spaces. Our main result generalizes the fuzzy Banach contraction theorem and we validate our results by some suitable examples which reveal that our results are proper generalization and modification of some researchers' integral contraction works.

#### Keywords

Fixed point; Metric spaces; Fuzzy metric spaces; Integral type contraction; Modified $$(\alpha$$-$$\beta_k,\phi$$-$$\psi)$$ Contraction

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#### References

S. Banach, Sur les operations dans les ensembles abstraits et leur application aux equations integrals, Fundamenta Mathematicae 3 (1922), 133 – 181, URL: http://matwbn.icm.edu.pl/ksiazki/or/or2/or215.pdf.

A. Branciari, A fixed point theorem for mappings satisfying a general contractive condition of integral type, International Journal of Mathematics and Mathematical Sciences 29(9) (2002), 531 – 536, DOI: 10.1155/S0161171202007524.

T. Dosenovic, D. Rakic and M. Brdara, Fixed point theorem in fuzzy metric spaces using altering distance, Filomat 28(7) (2014), 1517 – 1524, DOI: 10.2298/FIL1407517D.

A. George and P. Veeramani, On some results in fuzzy metric spaces, Fuzzy Sets and Systems 64 (1994), 395 – 399, DOI: 10.1016/0165-0114(94)90162-7.

M. Grabiec, Fixed points in fuzzy metric spaces, Fuzzy Sets and Systems 27 (1988), 385 – 389, DOI: 10.1016/0165-0114(88)90064-4.

V. Gregori and A. Sapena, On fixed-point theorems in fuzzy metric spaces, Fuzzy Sets and Systems 125 (2002), 245 – 252, DOI: 10.1016/S0165-0114(00)00088-9.

S. Hussain and M. Samreen, A fixed point theorem satisfying integral type contraction in fuzzy metric space, Results in Fixed Point and Applications, (2018), DOI: 10.30697/rfpta-2018-013.

M. S. Khan, M. Swaleh and S. Sessa, Fixed point theorems by altering distance between the points, Bulletin of the Australian Mathematical Society 30 (1984), 1 – 9, DOI: 10.1017/S0004972700001659.

I. Kramosil and J. Michalek, Fuzzy metric and statistical metric spaces, Kybernetika 11 (1975), 326 – 334, URL: http://dml.cz/dmlcz/125556.

D. Mihet, A Banach contraction theorem in fuzzy metric spaces, Fuzzy Sets and Systems 144(3) (2004), 431 – 439, DOI: 10.1016/S0165-0114(03)00305-1.

S. Phiangsungnoen, P. Thounthong and P. Kumam, Fixed point results in fuzzy metric spaces via ® and ¯k-admissible mappings with application to integral types Journal of Intelligent and Fuzzy Systems 34 (2018), 467 – 475, DOI: 10.3233/JIFS-17350.

P. Salimi, C. Vetro and P. Vetro, Some new fixed point results in non-Archimedean fuzzy metric spaces, Nonlinear Analysis Modelling and Control 18(3) (2013), 344 – 358, DOI: 10.15388/NA.18.3.14014.

B. Samet, C. Vetro and P. Vetro, Fixed point theorems for ®-Ã-contractive type mappings, Nonlinear Analysis 75 (2012), 2154 – 2165, DOI: 10.1016/j.na.2011.10.014.

Y. Shen, D. Qiu and W. Chen, Fixed point theorems in fuzzy metric spaces, Applied Mathematics Letters 25 (2012), 138 – 141, DOI: 10.1016/j.aml.2011.08.002.

L. A. Zadeh, Fuzzy sets, Inform and Control 8 (1965), 338 – 353, DOI: 10.1016/S0019-9958(65)90241-X.

DOI: http://dx.doi.org/10.26713%2Fcma.v12i1.1476

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