Abstract and Applied Analysis

Implicit-Relation-Type Cyclic Contractive Mappings and Applications to Integral Equations

Hemant Kumar Nashine, Zoran Kadelburg, and Poom Kumam

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Abstract

We introduce an implicit-relation-type cyclic contractive condition for a map in a metric space and derive existence and uniqueness results of fixed points for such mappings. Examples are given to support the usability of our results. At the end of the paper, an application to the study of existence and uniqueness of solutions for a class of nonlinear integral equations is presented.

Article information

Source
Abstr. Appl. Anal., Volume 2012 (2012), Article ID 386253, 15 pages.

Dates
First available in Project Euclid: 28 March 2013

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1364475857

Digital Object Identifier
doi:10.1155/2012/386253

Mathematical Reviews number (MathSciNet)
MR2984018

Zentralblatt MATH identifier
06116383

Citation

Nashine, Hemant Kumar; Kadelburg, Zoran; Kumam, Poom. Implicit-Relation-Type Cyclic Contractive Mappings and Applications to Integral Equations. Abstr. Appl. Anal. 2012 (2012), Article ID 386253, 15 pages. doi:10.1155/2012/386253. https://projecteuclid.org/euclid.aaa/1364475857


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References

  • S. Banach, “Sur les operations dans les ensembles abstraits et leur application aux equations integrales,” Fundamenta Mathematicae, vol. 3, pp. 133–181, 1922.
  • H. K. Nashine, “New fixed point theorems for mappings satisfying a generalized weakly contractive condition with weaker control functions,” Annales Polonici Mathematici, vol. 104, no. 2, pp. 109–119, 2012.
  • W. A. Kirk, P. S. Srinivasan, and P. Veeramani, “Fixed points for mappings satisfying cyclical contractive conditions,” Fixed Point Theory, vol. 4, no. 1, pp. 79–89, 2003.
  • M. Păcurar and I. A. Rus, “Fixed point theory for cyclic $\varphi $-contractions,” Nonlinear Analysis: Theory, Methods & Applications, vol. 72, no. 3-4, pp. 1181–1187, 2010.
  • R. P. Agarwal, M. A. Alghamdi, and N. Shahzad, “Fixed point theory for cyclic generalized contractions in partial metric spaces,” Fixed Point Theory and Applications, vol. 2012, article 40, 2012.
  • E. Karap\inar, “Fixed point theory for cyclic weak $\phi $-contraction,” Applied Mathematics Letters, vol. 24, no. 6, pp. 822–825, 2011.
  • E. Karap\inar and K. Sadaranagni, “Fixed point theory for cyclic ($\phi $-$\varphi $)-contractions,” Fixed Point Theory and Applications, vol. 2011, article 69, 2011.
  • M. A. Petric, “Some results concerning cyclical contractive mappings,” General Mathematics, vol. 18, no. 4, pp. 213–226, 2010.
  • I. A. Rus, “Cyclic representations and fixed points,” Annals of the Tiberiu Popoviciu Seminar of Functional Equations, Approximation and Convexity, vol. 3, pp. 171–178, 2005.
  • C. Mongkolkeha and P. Kumam, “Best proximity point theorems for generalized cyclic contractions in ordered metric spaces,” Journal of Optimization Theory and Applications. In press.
  • W. Sintunavarat and P. Kumam, “Common fixed point theorem for hybrid generalized multi-valued contraction mappings,” Applied Mathematics Letters, vol. 25, no. 1, pp. 52–57, 2012.
  • H. Aydi, C. Vetro, W. Sintunavarat, and P. Kumam, “Coincidence and fixed points for contractions and cyclical contractions in partial metric spaces,” Fixed Point Theory and Applications, vol. 2012, article 124, 2012.
  • V. Popa, “A fixed point theorem for mapping in d-complete topological spaces,” Mathematica Moravica, vol. 3, pp. 43–48, 1999.
  • I. Altun and D. Turkoglu, “Some fixed point theorems for weakly compatible multivalued mappings satisfying an implicit relation,” Filomat, vol. 22, no. 1, pp. 13–21, 2008.
  • I. Altun and D. Turkoglu, “Some fixed point theorems for weakly compatible mappings satisfying an implicit relation,” Taiwanese Journal of Mathematics, vol. 13, no. 4, pp. 1291–1304, 2009.
  • V. Popa, “A general coincidence theorem for compatible multivalued mappings satisfying an implicit relation,” Demonstratio Mathematica, vol. 33, no. 1, pp. 159–164, 2000.
  • V. Popa and M. Mocanu, “Altering distance and common fixed points under implicit relations,” Hacettepe Journal of Mathematics and Statistics, vol. 38, no. 3, pp. 329–337, 2009.
  • M. Imdad, S. Kumar, and M. S. Khan, “Remarks on some fixed point theorems satisfying implicit relations,” Radovi Matematički, vol. 11, no. 1, pp. 135–143, 2002.
  • S. Sharma and B. Deshpande, “On compatible mappings satisfying an implicit relation in common fixed point consideration,” Tamkang Journal of Mathematics, vol. 33, no. 3, pp. 245–252, 2002.
  • 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.
  • J. Matkowski, “Fixed point theorems for mappings with a contractive iterate at a point,” Proceedings of the American Mathematical Society, vol. 62, no. 2, pp. 344–348, 1977.
  • C.-M. Chen, “Fixed point theory for the cyclic weaker Meir-Keeler function in complete metric spaces,” Fixed Point Theory and Applications, vol. 2012, article 17, 2012.