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Fixed point theorems for weakly contractive mappings in relational metric spaces with an application

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Abstract

In this paper, we prove fixed point theorem for weakly contractive mappings using locally T-transitivity of binary relation and presenting a variant of Harjani and Sadarangani theorem involving more general relation theoretic metrical notions. Our results under universal relation reduces to Harjani and Sadarangani (Nonlinear Anal 71:3403–3410, 2009) fixed point theorems. We also generalize some recent fixed point theorems for weak contraction such as Alam and Imdad (J Fixed Point Theory Appl 17(4):693–702, 2015) and Kutbi et al. (Fixed Point Theory Appl 2015:198, 2015) besides some others.

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References

  1. Agarwal, R.P., M.A. El-Gebeily, and D. O’Regan. 2008. Generalized contractions in partially ordered metric spaces. Applicable Analysis 87 (1): 109–116.

    Article  MathSciNet  MATH  Google Scholar 

  2. Alam, A., and M. Imdad. 2015. Relation-theoretic contraction principle. Journal of Fixed Point Theory and Applications 17 (4): 693–702.

    Article  MathSciNet  MATH  Google Scholar 

  3. Alam, A., and M. Imdad. 2018. Nonlinear contractions in metric spaces under locally T-transitive binary relations. Fixed Point Theory 19 (1): 13–24.

    Article  MathSciNet  MATH  Google Scholar 

  4. Alam, A., A.R. Khan, and M. Imdad. 2014. Some coincidence theorems for genralised nonlinear contractions in ordered metric spaces with applications. Fixed Point Theory and Applications 2014: 216.

    Article  MATH  Google Scholar 

  5. Alber, Ya.I., and S. Guerre-Delabriere. 1997. Principles of weakly contractive maps in Hilbert spaces, new results in operator theory. In Advances and Appl., eds. I. Gohberg, Yu. Lyubich, 98, 7–22. Birkhauser, Basel (1997).

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

    Article  MATH  Google Scholar 

  7. Boyd, D.W., and J.S.W. Wong. 1969. On nonlinear contractions. Proceedings of the American Mathematical Society 20: 458–464.

    Article  MathSciNet  MATH  Google Scholar 

  8. Ciric, L., N. Cakic, M. Rajovic, and J.S. Ume. 2008. Monotone generalized nonlinear contractions in partially ordered metric spaces. Fixed Point Theory and Applications 2008: 131294.

    Article  MathSciNet  MATH  Google Scholar 

  9. Dutta, P.N., and B.S. Choudhury. 2008. A Generalisation of Contraction Principle in Metric Spaces. Fixed Point Theory and Applications. https://doi.org/10.1155/2008/406368.

  10. Harjani, J., and K. Sadarangani. 2009. Fixed point theorems for weakly contractive mappings in partially ordered sets. Nonlinear Analysis 71: 3403–3410.

    Article  MathSciNet  MATH  Google Scholar 

  11. Karapinar, E., and A.F. Roldan-Lopez-de-Hierro. 2014. A note on (G, F)-Closed set and tripled point of coincidence theorems for generalized compatibility in partially metric spaces. Journal of Inequalities and Applications 2014: 522.

    Article  MathSciNet  MATH  Google Scholar 

  12. Karapinar, E., A. Roldan, N. Shahzad, and W. Sintunavarat. 2014. Discussion of coupled and tripled coincidence point theorems for ’\(\varphi \)-contractive mappings without the mixed \(g\)-monotone property. Fixed Point Theory and Applications 2014: 92.

    Article  MathSciNet  MATH  Google Scholar 

  13. Kolman, B., R.C. Busby, and S. Ross. 2000. Discrete mathematical structures, 3rd ed. New Delhi: PHI Pvt. Ltd.

    Google Scholar 

  14. Kutbi, M.A., A. Alam, and M. Imdad. 2015. Sharpening some core theorems of Nieto and Rodriguez-Lopez withapplication to boundary value problem. Fixed Point Theory and Applications 2015: 198.

    Article  MATH  Google Scholar 

  15. Lakshmikantham, V., and L. Ciric. 2009. Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces. Nonlinear Analysis 70: 4341–4349.

    Article  MathSciNet  MATH  Google Scholar 

  16. Lipschutz, S. 1964. Schaums outlines of theory and problems of set theory and related topics. New York: McGraw-Hill.

    MATH  Google Scholar 

  17. Nieto, J.J., and R. Rodriguez-Lopez. 2005. Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations. Order 22(3): 223–239. https://doi.org/10.1007/s11083-005-9018-5.

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

    Article  MathSciNet  MATH  Google Scholar 

  19. Ran, A.C.M., and M.C.B. Reurings. 2004. A fixed point theorem in partially ordered sets and some applications to matrix equations. Proceedings of the American Mathematical Society 132 (5): 1435–1443.

    Article  MathSciNet  MATH  Google Scholar 

  20. Samet, B., and M. Turinici. 2012. Fixed point theorems on a metric space endowed with an arbitrary binary relation and applications. Communications in Mathematical Analysis 13 (2): 82–97.

    MathSciNet  MATH  Google Scholar 

  21. Turinici, M. 1986. Abstract comparison principles and multivariable Gronwall–Bellman inequalities. Journal of Mathematical Analysis and Applications 117 (1): 100–127.

    Article  MathSciNet  MATH  Google Scholar 

  22. Turinici, M. 2014. Contractive maps in locally transitive relational metric spaces. The Scientific World Journal 2014: 169358.

    Article  Google Scholar 

  23. Turinici, M. 2014. Contractive operators in relational metric spaces. In Handbook of Functional Equations, Springer Optimization and its Applications, ed. T. Rassias, 95, 419–458. New York: Springer.

    Google Scholar 

  24. Turinici, M. 1986. Fixed points for monotone iteratively local contractions. Demonstratio Mathematica 19 (1): 171–180.

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors thank the referees for their careful reading of the manuscript and insightful comments.

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Authors and Affiliations

  1. Department of Mathematics, H.N.B. Garhwal University, Srinagar, Garhwal, 246174, India

    Gopi Prasad & Ramesh Chandra Dimri

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  1. Gopi Prasad
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  2. Ramesh Chandra Dimri
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All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

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Correspondence to Gopi Prasad.

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Prasad, G., Dimri, R.C. Fixed point theorems for weakly contractive mappings in relational metric spaces with an application. J Anal 26, 151–162 (2018). https://doi.org/10.1007/s41478-018-0076-7

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