<em>µ</em>-extended fuzzy <em>b</em>-metric spaces and related fixed point results

Badshah-e-Rome; Muhammad Sarwar; Thabet Abdeljawad; Badshah-e-Rome; Muhammad Sarwar; Thabet Abdeljawad

doi:10.3934/math.2020333

AIMS Mathematics

2020, Volume 5, Issue 5: 5184-5192. doi: 10.3934/math.2020333

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Research article

µ-extended fuzzy b-metric spaces and related fixed point results

1.
Department of Mathematics, University of Malakand, Chakdara Dir(L), Pakistan
2.
Department mathematics and General Sciences, Prince Sultan University, P. O. Box 66833, Riyadh 11586, Saudi Arabia
3.
Department of Medical Research, China Medical University, Taichung, Taiwan
4.
Department of Computer Science and Information Engineering, Asia University, Taichung, Taiwan

Received: 01 February 2020 Accepted: 01 June 2020 Published: 15 June 2020
MSC : 47H10, 54H25

This paper introduces the notion of $\mu$-extended fuzzy $b$-metric space for extending the concept of fuzzy $b$-metric space and obtains an analogue of Banach fixed point result. Using functions $\alpha(x, y)$ and $\mu(x, y)$, the corresponding triangle inequality in $\mu$-extended fuzzy $b$-metric space is given as follows $ M( \upsilon,\omega,\alpha(\upsilon,\omega)s+\mu(\upsilon,\omega)t)\geq M(\upsilon,\nu,s)*M(\nu,\omega,t)\ \ \forall \upsilon,\nu,\omega \in X. $ An analogue of Banach fixed point result is established. Besides, an example is given to confirm validity of this theorem.
- fuzzy metric space,
- extended fuzzy b-metric space,
- µ-extended fuzzy b-metric space,
- fixed point
Citation: Badshah-e-Rome, Muhammad Sarwar, Thabet Abdeljawad. µ-extended fuzzy b-metric spaces and related fixed point results[J]. AIMS Mathematics, 2020, 5(5): 5184-5192. doi: 10.3934/math.2020333

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Abstract

This paper introduces the notion of $\mu$-extended fuzzy $b$-metric space for extending the concept of fuzzy $b$-metric space and obtains an analogue of Banach fixed point result. Using functions $\alpha(x, y)$ and $\mu(x, y)$, the corresponding triangle inequality in $\mu$-extended fuzzy $b$-metric space is given as follows $ M( \upsilon,\omega,\alpha(\upsilon,\omega)s+\mu(\upsilon,\omega)t)\geq M(\upsilon,\nu,s)*M(\nu,\omega,t)\ \ \forall \upsilon,\nu,\omega \in X. $ An analogue of Banach fixed point result is established. Besides, an example is given to confirm validity of this theorem.

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