Abstract and Applied Analysis

Fixed Points of Multivalued Contractive Mappings in Partial Metric Spaces

Abdul Rahim Khan, Mujahid Abbas, Talat Nazir, and Cristiana Ionescu

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Abstract

The aim of this paper is to present fixed point results of multivalued mappings in the framework of partial metric spaces. Some examples are presented to support the results proved herein. Our results generalize and extend various results in the existing literature. As an application of our main result, the existence and uniqueness of bounded solution of functional equations arising in dynamic programming are established.

Article information

Source
Abstr. Appl. Anal., Volume 2014, Special Issue (2013), Article ID 230708, 9 pages.

Dates
First available in Project Euclid: 6 October 2014

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1412606985

Digital Object Identifier
doi:10.1155/2014/230708

Mathematical Reviews number (MathSciNet)
MR3173272

Zentralblatt MATH identifier
07021957

Citation

Khan, Abdul Rahim; Abbas, Mujahid; Nazir, Talat; Ionescu, Cristiana. Fixed Points of Multivalued Contractive Mappings in Partial Metric Spaces. Abstr. Appl. Anal. 2014, Special Issue (2013), Article ID 230708, 9 pages. doi:10.1155/2014/230708. https://projecteuclid.org/euclid.aaa/1412606985


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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.
  • T. L. Hicks, “Fixed point theorems for quasimetric spaces,” Mathematica Japonica, vol. 33, no. 2, pp. 231–236, 1988.
  • B. S. Choudhury and N. Metiya, “Coincidence point and fixed point theorems in ordered cone metric spaces,” Journal of Advanced Mathematical Studies, vol. 5, no. 2, pp. 20–31, 2012.
  • M. O. Olatinwo and M. Postolache, “Stability results of Jungck-type iterative processes in convex metric spaces,” Applied Mathematics and Computation, vol. 218, no. 12, pp. 6727–6732, 2012.
  • H. Aydi, W. Shatanawi, M. Postolache, Z. Mustafa, and N. Tahat, “Theorems for Boyd-Wong-type contractions in ordered metric spaces,” Abstract and Applied Analysis, vol. 2012, Article ID 359054, 14 pages, 2012.
  • H. Aydi, E. Karapinar, and M. Postolache, “Tripled coincidence point theorems for weak $\varphi $-contractions in partially ordered metric spaces,” Fixed Point Theory and Applications, vol. 2012, article 44, 12 pages, 2012.
  • S. Chandok and M. Postolache, “Fixed point theorem for weakly Chatterjea-type cyclic contractions,” Fixed Point Theory and Applications, vol. 2013, article 28, 9 pages, 2013.
  • B. S. Choudhury, N. Metiya, and M. Postolache, “A generalized weak contraction principle with applications to coupled coincidence point problems,” Fixed Point Theory and Applications, vol. 2013, article 152, 21 pages, 2013.
  • W. Shatanawi and M. Postolache, “Common fixed point results for mappings under nonlinear contraction of cyclic form in ordered metric spaces,” Fixed Point Theory and Applications, vol. 2013, article 60, 13 pages, 2013.
  • W. Shatanawi and M. Postolache, “Common fixed point theorems for dominating and weakannihilator mappings in ordered metric spaces,” Fixed Point Theory and Applications, vol. 2013, article 271, 2013.
  • H. Aydi, M. Postolache, and W. Shatanawi, “Coupled fixed point results for $(\psi ,\phi )$-weakly contractive mappings in ordered $G$-metric spaces,” Computers & Mathematics with Applications, vol. 63, no. 1, pp. 298–309, 2012.
  • S. Chandok, Z. Mustafa, and M. Postolache, “Coupled common fixed point theorems for mixed g-monotone mappings in partially ordered G-metric spaces,” University Politehnica of Bucharest Scientific Bulletin Series A, vol. 75, no. 4, pp. 11–24, 2013.
  • W. Shatanawi and M. Postolache, “Some fixed-point results for a $G$-weak contraction in $G$-metric spaces,” Abstract and Applied Analysis, vol. 2012, Article ID 815870, 19 pages, 2012.
  • W. Shatanawi and A. Pitea, “Fixed and coupled fixed point theorems of omega-distance for nonlinear contraction,” Fixed Point Theory and Applications, vol. 2013, article 275, 16 pages, 2013.
  • W. Shatanawi and A. Pitea, “Omega-distance and coupled fixed point in G-metric spaces,” Fixed Point Theory and Applications, vol. 2013, article 208, 15 pages, 2013.
  • 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.
  • W. Shatanawi and M. Postolache, “Coincidence and fixed point results for generalized weak contractions in the sense of Berinde on partial metric spaces,” Fixed Point Theory and Applications, vol. 2013, article 54, 17 pages, 2013.
  • W. Shatanawi and A. Pitea, “Some coupled fixed point theorems in quasi-partial metric spaces,” Fixed Point Theory and Applications, vol. 2013, article 153, 15 pages, 2013.
  • M. Grabiec, “Fixed points in fuzzy metric spaces,” Fuzzy Sets and Systems, vol. 27, no. 3, pp. 385–389, 1988.
  • K. Menger, “Statistical metrics,” Proceedings of the National Academy of Sciences of the United States of America, vol. 28, pp. 535–537, 1942.
  • R. H. Haghi, M. Postolache, and Sh. Rezapour, “On T-stability of the Picard iteration for generalized $\varphi $-contraction mappings,” Abstract and Applied Analysis, vol. 2012, Article ID 658971, 7 pages, 2012.
  • M. A. Miandaragh, M. Postolache, and Sh. Rezapour, “Some approximate fixed point results for generalized $\alpha $-contractive mappings,” Scientific Bulletin A, vol. 75, no. 2, pp. 3–10, 2013.
  • M. A. Miandaragh, M. Postolache, and Sh. Rezapour, “Approximate fixed points of generalized convex contractions,” Fixed Point Theory and Applications, vol. 2013, article 255, 8 pages, 2013.
  • S. G. Matthews, “Partial metric topology,” in Proceedings of the 8th Summer Conference on General Topology and Applications, vol. 728 of Annals of the New York Academy of Sciences, pp. 183–197, 1994.
  • C. D. Bari and P. Vetro, “Fixed points for weak $\varphi $-contractions on partial metric spaces,” International Journal of Engineering, Contemporary Mathematics and Sciences, vol. 1, pp. 5–13, 2011.
  • D. Paesano and P. Vetro, “Suzuki's type characterizations of completeness for partial metric spaces and fixed points for partially ordered metric spaces,” Topology and its Applications, vol. 159, no. 3, pp. 911–920, 2012.
  • L. J. Ćirić, B. Samet, H. Aydi, and C. Vetro, “Common fixed points of generalized contractions on partial metric spaces and an application,” Applied Mathematics and Computation, vol. 218, no. 6, pp. 2398–2406, 2011.
  • R. Heckmann, “Approximation of metric spaces by partial metric spaces,” Applied Categorical Structures, vol. 7, no. 1-2, pp. 71–83, 1999.
  • S. Romaguera, “A Kirk type characterization of completeness for partial metric spaces,” Fixed Point Theory and Applications, vol. 2010, Article ID 493298, 6 pages, 2010.
  • S. Romaguera and O. Valero, “A quantitative computational model for complete partial metric spaces via formal balls,” Mathematical Structures in Computer Science, vol. 19, no. 3, pp. 541–563, 2009.
  • M. P. Schellekens, “The correspondence between partial metrics and semivaluations,” Theoretical Computer Science, vol. 315, no. 1, pp. 135–149, 2004.
  • R. H. Haghi, Sh. Rezapour, and N. Shahzad, “Be careful on partial metric fixed point results,” Topology and its Applications, vol. 160, no. 3, pp. 450–454, 2013.
  • M. Jleli, E. Karap\inar, and B. Samet, “Further remarks on fixed-point theorems in the context of partial metric spaces,” Abstract and Applied Analysis, vol. 2013, Article ID 715456, 6 pages, 2013.
  • B. Samet, C. Vetro, and F. Vetro, “From metric spaces to partial metric spaces,” Fixed Point Theory and Applications, vol. 2013, article 5, 11 pages, 2013.
  • H. K. Nashine and Z. Kadelburg, “Cyclic contractions and fixed point results via control functions on partial metric spaces,” International Journal of Analysis, vol. 2013, Article ID 726387, 9 pages, 2013.
  • H. K. Nashine, Z. Kadelburg, and S. Radenović, “Fixed point theorems via various cyclic contractive conditions in partial metric spaces,” De L'Institut Mathematique, vol. 93, no. 107, pp. 69–93, 2013.
  • H. Aydi, M. Abbas, and C. Vetro, “Partial Hausdorff metric and Nadler's fixed point theorem on partial metric spaces,” Topology and its Applications, vol. 159, no. 14, pp. 3234–3242, 2012.
  • 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.
  • D. Klim and D. Wardowski, “Fixed point theorems for set-valued contractions in complete metric spaces,” Journal of Mathematical Analysis and Applications, vol. 334, no. 1, pp. 132–139, 2007.
  • A. Latif and A. A. N. Abdou, “Fixed point results for generalized contractive multimaps in metric spaces,” Fixed Point Theory and Applications, vol. 2009, Article ID 432130, 16 pages, 2009.
  • A. Nicolae, “Fixed point theorems for multi-valued mappings of Feng-Liu type,” Fixed Point Theory, vol. 12, no. 1, pp. 145–154, 2011.
  • I. Altun and H. Simsek, “Some fixed point theorems on dualistic partial metric spaces,” Journal of Advanced Mathematical Studies, vol. 1, no. 1-2, pp. 1–8, 2008.
  • M. Abbas and T. Nazir, “Common fixed point of a power graphic contraction pair in partial metric spaces endowed with a graph,” Fixed Point Theory and Applications, vol. 2013, article 20, 8 pages, 2013.
  • M. Abbas and T. Nazir, “Fixed point of generalized weakly contractive mappings in ordered partial metric spaces,” Fixed Point Theory and Applications, vol. 2012, article 1, 19 pages, 2012.
  • M. A. Bukatin and S. Y. Shorina, “Partial metrics and co-continuous valuations,” in Foundations of Software Science and Computation Structure, M. Nivat et al., Ed., vol. 1378 of Lecture Notes in Computer Science, pp. 125–139, Springer, Berlin, Germany, 1998.
  • 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.
  • Y. J. Cho, S. Hirunworakit, and N. Petrot, “Set-valued fixed-point theorems for generalized contractive mappings without the Hausdorff metric,” Applied Mathematics Letters, vol. 24, no. 11, pp. 1959–1967, 2011.
  • L. Ćirić, “Fixed point theorems for multi-valued contractions in complete metric spaces,” Journal of Mathematical Analysis and Applications, vol. 348, no. 1, pp. 499–507, 2008.
  • S. B. Nadler, Jr., “Multi-valued contraction mappings,” Pacific Journal of Mathematics, vol. 30, pp. 475–488, 1969.
  • R. Baskaran and P. V. Subrahmanyam, “A note on the solution of a class of functional equations,” Applicable Analysis, vol. 22, no. 3-4, pp. 235–241, 1986.
  • R. Bellman and E. S. Lee, “Functional equations in dynamic programming,” Aequationes Mathematicae, vol. 17, no. 1, pp. 1–18, 1978.
  • P. C. Bhakta and S. Mitra, “Some existence theorems for functional equations arising in dynamic programming,” Journal of Mathematical Analysis and Applications, vol. 98, no. 2, pp. 348–362, 1984.
  • M. Abbas and B. Ali, “Fixed point of Suzuki-Zamfirescu hybrid contractions in partial metric spaces via partial Hausdorff metric,” Fixed Point Theory and Applications, vol. 2013, article 21, 16 pages, 2013. \endinput