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Fuzzy generalized weak contraction and its application to Fredholm non-linear integral equation in fuzzy metric space

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Abstract

The aim of this paper is to introduce fuzzy generalised weak contractive condition for a pair of self maps in a fuzzy metric space,which is in accordance with the theory of metric spaces given by Rhoades (Nonlinear Analysis 47:2683–2693, 2001). Our results generalize existing fuzzy contraction (by Gregori and Sapena (Fuzzy Sets System 125:245–252, 2002), which is for only one self map. Using this, we establish a unique common fixed point theorem for two self-maps through weak compatibility. The article includes an example, in support of our results. Also an application we established, for the existence and uniqueness of a solution of Fredholm non-linear integral equation in the setting of fuzzy metric space.

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References

  1. Abbas, M., M. Imdad, and D. Gopal. 2011. \(\psi\)-weak contractions in fuzzy metric spaces. Iranian Journal of Fuzzy Systems 8 (5): 141–148.

    MathSciNet  MATH  Google Scholar 

  2. Alber, Ya I., and S. Guerre-Delabriere. 1997. Principles of weakly contractive maps in Hilbert spaces. New Results in Operator Theory and Its Applications 98: 7–22.

    Article  MathSciNet  Google Scholar 

  3. Banach, S. 1922. Sur les operations dans les ensembles abstraits et leur application aux equations integrales. Fundamenta Mathematicae 3: 133–181.

    Article  MathSciNet  Google Scholar 

  4. Berinde, V. (1997)Contract ii Generalizate si Aplicat. ii, 22, Editura Cub Press, Baia Mare, Romania.

  5. Di Bari, C., and C. Vetro. 2003. A fixed point theorem for a family of mappings in a fuzzy metric space. Rendiconti del Circolo Matematico di Palermo 52: 315321.

    MathSciNet  MATH  Google Scholar 

  6. George, A., and P. Veeramani. 1994. On some results in fuzzy metric spaces. Fuzzy Sets and Systems 64: 395–399.

    Article  MathSciNet  Google Scholar 

  7. Gopal, D., M. Imdad, C. Vetro, and M. Hasan. 2012. Fixed Point Theory for cyclic weak f-contraction in fuzzy metric spaces. Journal of Nonlinear Analysis and Application 2012: 11.

    Article  Google Scholar 

  8. Gregori, V., Sapena, A. 2002. On fixed-point theorems in fuzzy metric spaces. Fuzzy Sets System 125: 245–252.

  9. Gupta, V., W. Shatanawi, and N. Mani. 2017. Fixed point theorems for \((\psi, \beta )\)-Geraghty contraction type maps in ordered metric spaces and some applications to integral and ordinary differential equations. Journal of Fixed Point Theory and Applications 19: 12511267.

    Article  MathSciNet  Google Scholar 

  10. Gupta, V., A. Kaushikb, and M. Verma. 2019. Some new fixed point results on \(V-\psi\)-fuzzy contractions endowed with graph. Journal of Intelligent and Fuzzy Systems 36: 65496554.

    Google Scholar 

  11. Gupta, V., M. Verma, and J. Kaur. 2020. A new contraction and existence theorems on fuzzy metric space with a graph. Italian Journal of Pure and Applied Mathematics 43: 717–729.

    Google Scholar 

  12. Jain, S., S. Jain, and L.B. Jain. 2009. Compatible mappings of type \((\beta )\)and weak compatibility in fuzzy metric space. Mathematica Bohemica 134: 151–164.

    Article  MathSciNet  Google Scholar 

  13. Jain, S., and S. Jain. 2019. \(Z_s\)-Contractive Mappings and WeakCompatibility in Fuzzy Metric Space. Mathematica Moravica 23 (2): 5968.

    Article  Google Scholar 

  14. Rhoades, B.E. 2001. Some theorems on weakly contractive maps. Nonlinear Analysis 47: 2683–2693.

    Article  MathSciNet  Google Scholar 

  15. Saha, P., B.S. Choudhury, and P. Das. 2016. Weak coupled coincidence point results having a partially ordering in fuzzy metric spaces. Fuzzy Information and Engineering 8: 199–216.

    Article  MathSciNet  Google Scholar 

  16. Schweizer, B., and A. Sklar. 1983. Probabilistic Metric Spaces. New York: Elsevier.

    MATH  Google Scholar 

  17. Shukla, S., D. Gopal, A. Francisco, and R. Lpez-de-Hierro. 2016. Some fixed point theorems in \(1\)-M-complete fuzzy metric-like spaces. International Journal of General Systems 45: 815–829.

    Article  MathSciNet  Google Scholar 

  18. Shukla, S., D. Gopal, and W. Sintunavarat. 2018. A new class of fuzzy contractive mappings and fixed point theorems. Fuzzy Sets and Systems 350: 85–94.

    Article  MathSciNet  Google Scholar 

  19. Vetro, D. 2010. Gopal, common fixed point theorems for \((\phi,\psi )\)-weak contractions in fuzzy metric spaces. Indian Journal of Mathematics 52 (3): 573–590.

    MathSciNet  Google Scholar 

  20. Gregori, V., Minnana, J.J. 2014. Some remarks on fuzzy contractive mappings. Fuzzy Sets System 251:101–103. https://doi.org/10.1016/j .fss.2014.01.002.

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Acknowledgements

The authors are thankful to Dr. Lal Bahadur, Retired Principal, Govt. Arts and Commerce college , Indore, the editor and the referees, for their precise remarks to improve the presentation of the paper.

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  1. Shri Vaishnav Vidyapeeth Vishwavidyalaya, Indore, Madhya Pradesh, India

    Shobha Jain & Shishir Jain

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  1. Shobha Jain
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  2. Shishir Jain
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Correspondence to Shobha Jain.

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Jain, S., Jain, S. Fuzzy generalized weak contraction and its application to Fredholm non-linear integral equation in fuzzy metric space. J Anal 29, 619–632 (2021). https://doi.org/10.1007/s41478-020-00270-w

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  • DOI: https://doi.org/10.1007/s41478-020-00270-w

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