Taiwanese Journal of Mathematics

APPROXIMATE FIXED POINT THEOREMS FOR PARTIAL GENERALIZED CONVEX CONTRACTION MAPPINGS IN $\alpha$-COMPLETE METRIC SPACES

Abdul Latif, Wutiphol Sintunavarat, and Aphinat Ninsri

Full-text: Open access

Abstract

In this paper, we introduce the new concept called partial generalized convex contractions and partial generalized convex contractions of order 2. Also, we establish some approximate fixed point theorems for such mappings in $\alpha$-complete metric spaces. Our results extend and unify the results of Miandaragh et al. [M. A. Miandaragh, M. Postolache, S. Rezapour, Approximate fixed points of generalized convex contractions, Fixed Point Theory and Applications 2013, 2013 :255] and several well-known results in literature. We give some examples of a nonlinear contraction mapping, which is not applied to the existence of approximate fixed point and fixed point by using the results of Miandaragh et al. We also consider approximate fixed point results in metric space endowed with an arbitrary binary relation and approximate fixed point results in metric space endowed with graph.

Article information

Source
Taiwanese J. Math., Volume 19, Number 1 (2015), 315-333.

Dates
First available in Project Euclid: 4 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1499133631

Digital Object Identifier
doi:10.11650/tjm.19.2015.4746

Mathematical Reviews number (MathSciNet)
MR3313418

Zentralblatt MATH identifier
1357.54034

Subjects
Primary: 47H09: Contraction-type mappings, nonexpansive mappings, A-proper mappings, etc. 47H10: Fixed-point theorems [See also 37C25, 54H25, 55M20, 58C30]

Keywords
approximate fixed point property binary relations $\varepsilon$-Fixed points partial generalized convex contraction mappings

Citation

Latif, Abdul; Sintunavarat, Wutiphol; Ninsri, Aphinat. APPROXIMATE FIXED POINT THEOREMS FOR PARTIAL GENERALIZED CONVEX CONTRACTION MAPPINGS IN $\alpha$-COMPLETE METRIC SPACES. Taiwanese J. Math. 19 (2015), no. 1, 315--333. doi:10.11650/tjm.19.2015.4746. https://projecteuclid.org/euclid.twjm/1499133631


Export citation

References

  • S. Banach, Sur les opérations dans les ensembles abstraits et leurs applications aux équations intégrales, Fund. Math., 3 (1922), 133-181.
  • M. Berinde, Approximate fixed point theorems, Stud. Univ. Babes Bolyai, Math., 51(1) (2006), 11-25.
  • F. E. Browder and W. V. Petrysyn, The solution by iteration of nonlinear functional equation in Banach spaces, Bull. Amer. Math. Soc., 72 (1966), 571-576.
  • D. Dey and M. Saha, Approximate fixed point of Reich operator, Acta Mathematica Universitatis Comenianae, 82(1) (2013), 119-123.
  • D. Dey, A. K. Laha and M. Saha, Approximate coincidence point of two nonlinear mappings, Journal of Mathematics, Vol. 2013, Article ID 962058, 4 pages.
  • N. Hussain, M. A. Kutbi and P. Salimi, Fixed Point Theory in $\alpha$-complete Metric Spaces with Applications, Abstract and Applied Analysis, Vol. 2014, Article ID 280817, 11 pages.
  • V. I. Istratescu, Some fixed point theorems for convex contraction mappings and mappings with convex diminishing diameters, I. Ann. Mat. Pura Appl., 130(4) (1982), 89- 104.
  • R. Kannan, Some results on fixed points II, Amer. Math. Monthly, 76 (1969), 405-408.
  • U. Kohlenbach and L. Leustean, The approximate fixed point property in product spaces, Nonlinear Anal., 66 (2007), 806-818.
  • M. A. Miandaragh, M. Postolache and S. Rezapour, Approximate fixed points of generalized convex contractions, Fixed Point Theory and Applications, 2013, 2013:255.
  • S. Reich, Kannan's Fixed Point Theorem, 4 (1971), 1-11.
  • S. Reich, Approximate selections, best approximations, fixed points, and invariant sets, J. Math. Anal. Appl., 62 (1978), 104-113.
  • B. Samet, C. Vetro and P. Vetro, Fixed-point theorems for $\alpha$-$\psi$-contractive type mappings, Nonlinear Anal., 75 (2012), 2154-2165.
  • S. Tijs, A. Torre and R. Branzei, Approximate fixed point theorems, Libertas Mathematica, 23 (2003), 35-39.