Abstract and Applied Analysis

Theorems for Boyd-Wong-Type Contractions in Ordered Metric Spaces

Hassen Aydi, Wasfi Shatanawi, Mihai Postolache, Zead Mustafa, and Nedal Tahat

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Abstract

We give some fixed point results using an ICS mapping and involving Boyd-Wong-type contractions in partially ordered metric spaces. Our results generalize, extend, and unify several well-known comparable theorems in the literature. Also, we present some examples to support our results.

Article information

Source
Abstr. Appl. Anal., Volume 2012 (2012), Article ID 359054, 14 pages.

Dates
First available in Project Euclid: 28 March 2013

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1364475848

Digital Object Identifier
doi:10.1155/2012/359054

Mathematical Reviews number (MathSciNet)
MR2975319

Zentralblatt MATH identifier
1253.54036

Citation

Aydi, Hassen; Shatanawi, Wasfi; Postolache, Mihai; Mustafa, Zead; Tahat, Nedal. Theorems for Boyd-Wong-Type Contractions in Ordered Metric Spaces. Abstr. Appl. Anal. 2012 (2012), Article ID 359054, 14 pages. doi:10.1155/2012/359054. https://projecteuclid.org/euclid.aaa/1364475848


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References

  • S. Banach, “Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales,” Fundamenta Mathematicae, vol. 3, pp. 133–181, 1922.
  • D. S. Jaggi, “Some unique fixed point theorems,” Indian Journal of Pure and Applied Mathematics, vol. 8, no. 2, pp. 223–230, 1977.
  • 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.
  • J. J. Nieto and R. Rodríguez-López, “Existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equations,” Acta Mathematica Sinica, vol. 23, no. 12, pp. 2205–2212, 2007.
  • 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.
  • 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.
  • 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.
  • H. Aydi, “Coincidence and common fixed point results for contraction type maps in partially ordered metric spaces,” International Journal of Mathematical Analysis, vol. 5, no. 13, pp. 631–642, 2011.
  • H. Aydi, “Some fixed point results in ordered partial metric spaces,” Journal of Nonlinear Science and its Applications, vol. 4, no. 3, pp. 210–217, 2011.
  • H. Aydi, “Fixed point results for weakly contractive mappings in ordered partial metric spaces,” Journal of Advanced Mathematical Studies, vol. 4, no. 2, pp. 1–12, 2011.
  • H. Aydi, “Fixed point theorems for generalized weakly contractive condition in ordered partial metric spaces,” Journal of Nonlinear Analysis and Optimization: Theory and Applications, vol. 2, no. 2, pp. 33–48, 2011.
  • H. Aydi, “Common fixed point results for mappings satisfying ($\Psi $, $\phi $)-weak contractions in ordered partial metric spaces,” International JMathematics and Statistics, vol. 12, no. 2, pp. 53–64, 2012.
  • H. Aydi, H. K. Nashine, B. Samet, and H. Yazidi, “Coincidence and common fixed point results in partially ordered cone metric spaces and applications to integral equations,” Nonlinear Analysis. Theory, Methods & Applications, vol. 74, no. 17, pp. 6814–6825, 2011.
  • L. Gholizadeh, R. Saadati, W. Shatanawi, and S. M. Vaezpour, “Contractive mapping in generalized, ordered metric spaces with application in integral equations,” Mathematical Problems in Engineering, vol. 2011, Article ID 380784, 14 pages, 2011.
  • 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.
  • J. Harjani and K. Sadarangani, “Fixed point theorems for weakly contractive mappings in partially ordered sets,” Nonlinear Analysis. Theory, Methods & Applications, vol. 71, no. 7-8, pp. 3403–3410, 2009.
  • J. Harjani, B. López, and K. Sadarangani, “A fixed point theorem for mappings satisfying a contractive condition of rational type on a partially ordered metric space,” Abstract and Applied Analysis, vol. 2010, Article ID 190701, 8 pages, 2010.
  • E. Karapinar, “Weak $\varphi $-contraction on partial metric spaces and existence of fixed points in partially ordered sets,” Mathematica Æ terna, vol. 1, no. 3-4, pp. 237–244, 2011.
  • M. S. Khan, M. Swaleh, and S. Sessa, “Fixed point theorems by altering distances between the points,” Bulletin of the Australian Mathematical Society, vol. 30, no. 1, pp. 1–9, 1984.
  • N. V. Luong, N. X. Thuan, and T. T. THai, “Coupled fixed point theorems in partially ordered metric spaces,” Bulletin of Mathematical Analysis and Applications, vol. 3, no. 3, pp. 129–140, 2011.
  • N. V. Luong and N. X. Thuan, “Fixed point theorem for generalized weak contractions satisfying rational expressions in ordered metric spaces,” Fixed Point Theory and Applications, vol. 2011, article 46, 2011.
  • S. Moradi and M. Omid, “A fixed-point theorem for integral type inequality depending on another function,” International Journal of Mathematical Analysis, vol. 4, no. 29-32, pp. 1491–1499, 2010.
  • J. J. Nieto and R. Rodríguez-López, “Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations,” A Journal on the Theory of Ordered Sets and its Applications, vol. 22, no. 3, p. 223–239, 2005.
  • 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.
  • A. Petruşel and I. A. Rus, “Fixed point theorems in ordered $L$-spaces,” Proceedings of the American Mathematical Society, vol. 134, no. 2, pp. 411–418, 2006.
  • W. Shatanawi, Z. Mustafa, and N. Tahat, “Some coincidence point theorems for nonlinear contraction in ordered metric spaces,” Fixed Point Theory and Applications, vol. 2011, article 68, 2011.
  • W. Shatanawi and B. Samet, “On $(\psi ,\varphi )$-weakly contractive condition in partially ordered metric spaces,” Computers & Mathematics with Applications, vol. 62, no. 8, pp. 3204–3214, 2011.
  • W. Shatanawi and B. Samet, “Coupled fixed point theorems for mixed monotone mappings in čommentComment on ref. [33?]: Please update the information of this reference, if possible.ordered ordered partial metric spaces,” Mathematical and Computer Modelling. In press.
  • W. Shatanawi, “Fixed point theorems for nonlinear weakly \emphC-contractive mappings in metric spaces,” Mathematical and Computer Modelling, vol. 54, no. 11-12, pp. 2816–2826, 2011.
  • K. P. Chi, “On a fixed point theorem for certain class of maps satisfying a contractive condition depended on an another function,” Lobachevskii Journal of Mathematics, vol. 30, no. 4, pp. 289–291, 2009.
  • D. W. Boyd and J. S. W. Wong, “On nonlinear contractions,” Proceedings of the American Mathematical Society, vol. 20, pp. 458–464, 1969.
  • J. Jachymski, “Equivalent conditions for generalized contractions on (ordered) metric spaces,” Nonlinear Analysis. Theory, Methods & Applications, vol. 74, no. 3, pp. 768–774, 2011.