Abstract and Applied Analysis

New Results and Generalizations for Approximate Fixed Point Property and Their Applications

Wei-Shih Du and Farshid Khojasteh

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Abstract

We first introduce the concept of manageable functions and then prove some new existence theorems related to approximate fixed point property for manageable functions and α -admissible multivalued maps. As applications of our results, some new fixed point theorems which generalize and improve Du's fixed point theorem, Berinde-Berinde's fixed point theorem, Mizoguchi-Takahashi's fixed point theorem, and Nadler's fixed point theorem and some well-known results in the literature are given.

Article information

Source
Abstr. Appl. Anal., Volume 2014, Special Issue (2013), Article ID 581267, 9 pages.

Dates
First available in Project Euclid: 6 October 2014

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1412606981

Digital Object Identifier
doi:10.1155/2014/581267

Mathematical Reviews number (MathSciNet)
MR3178876

Zentralblatt MATH identifier
07022651

Citation

Du, Wei-Shih; Khojasteh, Farshid. New Results and Generalizations for Approximate Fixed Point Property and Their Applications. Abstr. Appl. Anal. 2014, Special Issue (2013), Article ID 581267, 9 pages. doi:10.1155/2014/581267. https://projecteuclid.org/euclid.aaa/1412606981


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References

  • S. Banach, “Sur les opérations dans les ensembles abstraits et leurs applications aux équations integrales,” Fundamenta Mathematicae, vol. 3, pp. 133–181, 1922.
  • S. B. Nadler Jr., “Multi-valued contraction mappings,” Pacific Journal of Mathematics, vol. 30, pp. 475–488, 1969.
  • N. Mizoguchi and W. Takahashi, “Fixed point theorems for multivalued mappings on complete metric spaces,” Journal of Mathematical Analysis and Applications, vol. 141, no. 1, pp. 177–188, 1989.
  • S. Reich, “Some problems and results in fixed point theory,” Contemporary Mathematics, vol. 21, pp. 179–187, 1983.
  • M. Berinde and V. Berinde, “On a general class of multi-valued weakly Picard mappings,” Journal of Mathematical Analysis and Applications, vol. 326, no. 2, pp. 772–782, 2007.
  • W.-S. Du, “On coincidence point and fixed point theorems for nonlinear multivalued maps,” Topology and its Applications, vol. 159, no. 1, pp. 49–56, 2012.
  • W. Takahashi, Nonlinear Functional Analysis, Yokohama Publishers, Yokohama, Japan, 2000.
  • A. Uderzo, “Fixed points for directional multi-valued $k(\cdot\,\!)$-contractions,” Journal of Global Optimization, vol. 31, no. 3, pp. 455–469, 2005.
  • A. Petruşel and A. Sîntămărian, “Single-valued and multi-valued Caristi type operators,” Publicationes Mathematicae Debrecen, vol. 60, no. 1-2, pp. 167–177, 2002.
  • Y. Feng and S. Liu, “Fixed point theorems for multi-valued contractive mappings and multi-valued Caristi type mappings,” Journal of Mathematical Analysis and Applications, vol. 317, no. 1, pp. 103–112, 2006.
  • L.-G. Huang and X. Zhang, “Cone metric spaces and fixed point theorems of contractive mappings,” Journal of Mathematical Analysis and Applications, vol. 332, no. 2, pp. 1468–1476, 2007.
  • W.-S. Du, “A note on cone metric fixed point theory and its equivalence,” Nonlinear Analysis: Theory, Methods & Applications, vol. 72, no. 5, pp. 2259–2261, 2010.
  • W.-S. Du, “Some new results and generalizations in metric fixed point theory,” Nonlinear Analysis: Theory, Methods & Applications, vol. 73, no. 5, pp. 1439–1446, 2010.
  • W.-S. Du, “Coupled fixed point theorems for nonlinear contractions satisfied Mizoguchi-Takahashi's condition in quasiordered metric spaces,” Fixed Point Theory and Applications, vol. 2010, Article ID 876372, 2010.
  • W.-S. Du, “Nonlinear contractive conditions for coupled cone fixed point theorems,” Fixed Point Theory and Applications, vol. 2010, Article ID 190606, 2010.
  • W.-S. Du, “New cone fixed point theorems for nonlinear multivalued maps with their applications,” Applied Mathematics Letters, vol. 24, no. 2, pp. 172–178, 2011.
  • W.-S. Du and S.-X. Zheng, “Nonlinear conditions for coincidence point and fixed point theorems,” Taiwanese Journal of Mathematics, vol. 16, no. 3, pp. 857–868, 2012.
  • W.-S. Du, “On Caristi type maps and generalized distances with applications,” Abstract and Applied Analysis, vol. 2013, Article ID 407219, 8 pages, 2013.
  • W.-S. Du, E. Karap\inar, and N. Shahzad, “The study of fixed point theory for various multivalued non-self-maps,” Abstract and Applied Analysis, vol. 2013, Article ID 938724, 9 pages, 2013.
  • I. J. Lin and T. H. Chen, “New existence theorems of coincidence points approach to generalizations of Mizoguchi-Takahashi's fixed point theorem,” Fixed Point Theory and Applications, vol. 2012, article 156, 2012.
  • F. Khojasteh, S. Shukla, and S. Radenovic, “A new approach to the study of fixed point theory for simulation function,” Filomat. In press.
  • A. A. Eldred and P. Veeramani, “Existence and convergence of best proximity points,” Journal of Mathematical Analysis and Applications, vol. 323, no. 2, pp. 1001–1006, 2006.
  • E. Karapinar, “Generalizations of Caristi Kirk's theorem on partial metric spaces,” Fixed Point Theory and Applications, vol. 2011, article 4, 2011.
  • M. Jleli, E. Karapinar, and B. Samet, “On best proximity points under the $P$-property on partially ordered metric spaces,” Abstract and Applied Analysis, vol. 2013, Article ID 150970, 6 pages, 2013.
  • E. Karap\inar, “On best proximity point of $\psi $-Geraghty contractions,” Fixed Point Theory and Applications, vol. 2013, article 200, 2013.
  • M. Jleli, E. Karap\inar, and B. Samet, “Best proximity points for generalized $\alpha $-$\psi $-proximal contractive type mappings,” Journal of Applied Mathematics, vol. 2013, Article ID 534127, 10 pages, 2013.
  • W.-S. Du and H. Lakzian, “Nonlinear conditions for the existence of best proximity points,” Journal of Inequalities and Applications, vol. 2012, article 206, 2012.
  • W.-S. Du and E. Karapinar, “A note on Caristi-type cyclic maps: related results and applications,” Fixed Point Theory and Applications, vol. 2013, article 344, 2013.
  • N. Hussain, A. Amini-Harandi, and Y. J. Cho, “Approximate endpoints for set-valued contractions in metric spaces,” Fixed Point Theory and Applications, vol. 2010, Article ID 614867, 13 pages, 2010.
  • M. A. Khamsi, “On asymptotically nonexpansive mappings in hyperconvex metric spaces,” Proceedings of the American Mathematical Society, vol. 132, no. 2, pp. 365–373, 2004.
  • W.-S. Du, “On approximate coincidence point properties and their applications to fixed point theory,” Journal of Applied Mathematics, vol. 2012, Article ID 302830, 17 pages, 2012.
  • W.-S. Du, Z. He, and Y. L. Chen, “New existence theorems for approximate coincidence point property and approximate fixed point property with applications to metric fixed point theory,” Journal of Nonlinear and Convex Analysis, vol. 13, no. 3, pp. 459–474, 2012.
  • W.-S. Du, “On generalized weakly directional contractions and approximate fixed point property with applications,” Fixed Point Theory and Applications, vol. 2012, article 6, 2012.
  • W.-S. Du, “New existence results and generalizations for coincidence points and fixed points without global completeness,” Abstract and Applied Analysis, vol. 2013, Article ID 214230, 12 pages, 2013.
  • B. Samet, C. Vetro, and P. Vetro, “Fixed point theorems for $\alpha $-$\psi $-contractive type mappings,” Nonlinear Analysis: Theory, Methods & Applications, vol. 75, no. 4, pp. 2154–2165, 2012.
  • J. H. Asl, S. Rezapour, and N. Shahzad, “On fixed points of $\alpha $-$\psi $-contractive multifunctions,” Fixed Point Theory and Applications, vol. 2012, article 212, 2012.
  • B. Mohammadi, S. Rezapour, and N. Shahzad, “Some results on fixed points of $\alpha $-$\psi $-Ciric generalized multifunctions,” Fixed Point Theory and Applications, vol. 2013, article 24, 2013.
  • H. Alikhani, Sh. Rezapour, and N. Shahzad, “Fixed points of a new type of contractive mappings and multifunctions,” Filomat, vol. 27, no. 7, pp. 1319–1315, 2013.
  • H. Alikhani, V. Rakocevic, Sh. Rezapour, and N. Shahzad, “Fixed points of proximinal valued $\beta $-$\psi $-contractive multifunctions,” Journal of Nonlinear and Convex Analysis. In press.
  • H.-K. Xu, “Metric fixed point theory for multivalued mappings,” Dissertationes Mathematicae, vol. 389, 39 pages, 2000. \endinput