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Journal of Function Spaces
Volume 2016, Article ID 6275367, 11 pages
http://dx.doi.org/10.1155/2016/6275367
Research Article

Discussion on Some Recent Order-Theoretic Metrical Coincidence Theorems Involving Nonlinear Contractions

Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India

Received 22 December 2015; Accepted 16 February 2016

Academic Editor: Tomonari Suzuki

Copyright © 2016 Aftab Alam et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. K. Goebel, “A coincidence theorem,” Bulletin de l'Académie Polonaise des Sciences. Série des Sciences Mathématiques, Astronomiques et Physiques, vol. 16, pp. 733–735, 1968. View at Google Scholar · View at MathSciNet
  2. G. Jungck, “Commuting mappings and fixed points,” The American Mathematical Monthly, vol. 83, no. 4, pp. 261–263, 1976. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. M. Turinici, “Abstract comparison principles and multivariable Gronwall-Bellman inequalities,” Journal of Mathematical Analysis and Applications, vol. 117, no. 1, pp. 100–127, 1986. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  4. M. Turinici, “Fixed points for monotone iteratively local contractions,” Demonstratio Mathematica, vol. 19, no. 1, pp. 171–180, 1986. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. A. C. M. Ran and M. C. B. Reurings, “A fixed point theorem in partially ordered sets and some applications to matrix equations,” Proceedings of the American Mathematical Society, vol. 132, no. 5, pp. 1435–1443, 2004. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  6. J. J. Nieto and R. Rodríguez-López, “Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations,” Order, vol. 22, no. 3, pp. 223–239, 2005. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  7. R. P. Agarwal, M. A. El-Gebeily, and D. O'Regan, “Generalized contractions in partially ordered metric spaces,” Applicable Analysis, vol. 87, no. 1, pp. 109–116, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  8. D. O'Regan and A. Petruşel, “Fixed point theorems for generalized contractions in ordered metric spaces,” Journal of Mathematical Analysis and Applications, vol. 341, no. 2, pp. 1241–1252, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  9. L. Ćirić, N. Cakić, M. Rajović, and J. S. Ume, “Monotone generalized nonlinear contractions in partially ordered metric spaces,” Fixed Point Theory and Applications, vol. 2008, Article ID 131294, 11 pages, 2008. View at Publisher · View at Google Scholar
  10. A. Amini-Harandi and H. Emami, “A fixed point theorem for contraction type maps in partially ordered metric spaces and application to ordinary differential equations,” Nonlinear Analysis: Theory, Methods & Applications, vol. 72, no. 5, pp. 2238–2242, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  11. J. Harjani and K. Sadarangani, “Generalized contractions in partially ordered metric spaces and applications to ordinary differential equations,” Nonlinear Analysis, vol. 72, no. 3-4, pp. 1188–1197, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  12. I. Altun and H. Simsek, “Some fixed point theorems on ordered metric spaces and application,” Fixed Point Theory and Applications, vol. 2010, Article ID 621469, 17 pages, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  13. J. Caballero, J. Harjani, and K. Sadarangani, “Contractive-like mapping principles in ordered metric spaces and application to ordinary differential equations,” Fixed Point Theory and Applications, vol. 2010, Article ID 916064, 14 pages, 2010. View at Google Scholar · View at MathSciNet
  14. M. Jleli, V. C. Rajic, B. Samet, and C. Vetro, “Fixed point theorems on ordered metric spaces and applications to nonlinear elastic beam equations,” Journal of Fixed Point Theory and Applications, vol. 12, no. 1-2, pp. 175–192, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  15. S. Hadj Amor, E. Karapinar, and P. Kumam, “A new class of generalized contraction using P-functions in ordered metric spaces,” Analele Stiintifice ale Universitatii Ovidius Constanta, vol. 23, no. 2, pp. 93–106, 2015. View at Google Scholar · View at MathSciNet
  16. E. Karapınar and K. Sadarangani, “Berinde mappings in ordered metric spaces,” Revista de la Real Academia de Ciencias Exactas, Fsicas y Naturales. Serie A. Matemticas, vol. 109, no. 2, pp. 353–366, 2015. View at Publisher · View at Google Scholar
  17. E. Karapinar, I. M. Erhan, and U. Aksoy, “Weak ψ-contractions on partially ordered metric spaces and applications to boundary value problems,” Boundary Value Problems, vol. 2014, article 149, 2014. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  18. A. Alam, A. R. Khan, and M. Imdad, “Some coincidence theorems for generalized nonlinear contractions in ordered metric spaces with applications,” Fixed Point Theory and Applications, vol. 2014, article 216, 30 pages, 2014. View at Publisher · View at Google Scholar · View at MathSciNet
  19. A. Alam, Q. H. Khan, and M. Imdad, “Enriching some recent coincidence theorems for nonlinear contractions in ordered metric spaces,” Fixed Point Theory and Applications, vol. 2015, article 141, 14 pages, 2015. View at Publisher · View at Google Scholar · View at MathSciNet
  20. M. A. Kutbi, A. Alam, and M. Imdad, “Sharpening some core theorems of Nieto and Rodríguez López with application,” Fixed Point Theory and Applications, vol. 2015, p. 198, 2015. View at Google Scholar · View at MathSciNet
  21. D. W. Boyd and J. S. W. Wong, “On nonlinear contractions,” Proceedings of the American Mathematical Society, vol. 20, pp. 458–464, 1969. View at Publisher · View at Google Scholar · View at MathSciNet
  22. S. Lipschutz, Schaum's Outlines of Theory and Problems of Set Theory and Related Topics, McGraw-Hill, New York, NY, USA, 1964.
  23. M. Turinici, “Ran-Reurings fixed point results in ordered metric spaces,” Libertas Mathematica, vol. 31, pp. 49–55, 2011. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  24. G. Jungck, “Common fixed points for noncontinuous nonself maps on non-metric spaces,” Far East Journal of Mathematical Sciences, vol. 4, no. 2, pp. 199–215, 1996. View at Google Scholar · View at MathSciNet
  25. S. Sessa, “On a weak commutativity condition of mappings in fixed point considerations,” Publications of the Research Institute for Mathematical Sciences, vol. 32, pp. 149–153, 1982. View at Google Scholar
  26. G. Jungck, “Compatible mappings and common fixed points,” International Journal of Mathematics and Mathematical Sciences, vol. 9, no. 4, pp. 771–779, 1986. View at Publisher · View at Google Scholar · View at MathSciNet
  27. K. P. R. Sastry and I. S. Krishna Murthy, “Common fixed points of two partially commuting tangential selfmaps on a metric space,” Journal of Mathematical Analysis and Applications, vol. 250, no. 2, pp. 731–734, 2000. View at Publisher · View at Google Scholar · View at MathSciNet
  28. N. V. Luong and N. X. Thuan, “Coupled points in ordered generalized metric spaces and application to integro-dierential equations,” Analele Universitatii “Ovidius” Constanta Seria Matematica, vol. 21, no. 3, pp. 155–180, 2013. View at Publisher · View at Google Scholar
  29. T. Gnana Bhaskar and V. Lakshmikantham, “Fixed point theorems in partially ordered metric spaces and applications,” Nonlinear Analysis: Theory, Methods & Applications, vol. 65, no. 7, pp. 1379–1393, 2006. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  30. V. Lakshmikantham and L. Ćirić, “Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces,” Nonlinear Analysis, vol. 70, no. 12, pp. 4341–4349, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  31. N. Jotic, “Some fixed point theorems in metric spaces,” Indian Journal of Pure and Applied Mathematics, vol. 26, no. 10, pp. 947–952, 1995. View at Google Scholar · View at MathSciNet
  32. S. Radenović, Z. Kadelburg, D. Jandrlić, and A. Jandrlić, “Some results on weakly contractive maps,” Bulletin of the Iranian Mathematical Society, vol. 38, no. 3, pp. 625–645, 2012. View at Google Scholar · View at MathSciNet
  33. M. Berzig, E. Karapinar, and A. Roldán, “Discussion on generalized-(αψ,βφ)-contractive mappings via generalized altering distance function and related fixed point theorems,” Abstract and Applied Analysis, vol. 2014, Article ID 259768, 12 pages, 2014. View at Publisher · View at Google Scholar
  34. R. H. Haghi, S. Rezapour, and N. Shahzad, “Some fixed point generalizations are not real generalizations,” Nonlinear Analysis, vol. 74, no. 5, pp. 1799–1803, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus