Abstract and Applied Analysis

New Existence Results and Generalizations for Coincidence Points and Fixed Points without Global Completeness

Wei-Shih Du

Abstract

Some new existence theorems concerning approximate coincidence point property and approximate fixed point property for nonlinear maps in metric spaces without global completeness are established in this paper. By exploiting these results, we prove some new coincidence point and fixed point theorems which generalize and improve Berinde-Berinde's fixed point theorem, Mizoguchi-Takahashi's fixed point theorem, Kikkawa-Suzuki's fixed point theorem, and some well known results in the literature. Moreover, some applications of our results to the existence of coupled coincidence point and coupled fixed point are also presented.

Article information

Source
Abstr. Appl. Anal., Volume 2013 (2013), Article ID 214230, 12 pages.

Dates
First available in Project Euclid: 18 April 2013

https://projecteuclid.org/euclid.aaa/1366306767

Digital Object Identifier
doi:10.1155/2013/214230

Mathematical Reviews number (MathSciNet)
MR3035194

Zentralblatt MATH identifier
1267.54042

Citation

Du, Wei-Shih. New Existence Results and Generalizations for Coincidence Points and Fixed Points without Global Completeness. Abstr. Appl. Anal. 2013 (2013), Article ID 214230, 12 pages. doi:10.1155/2013/214230. https://projecteuclid.org/euclid.aaa/1366306767

References

• 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 Mathe-matical 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.
• T. Suzuki, “A generalized Banach contraction principle that characterizes metric completeness,” Proceedings of the American Mathematical Society, vol. 136, no. 5, pp. 1861–1869, 2008.
• W. Takahashi, Nonlinear Functional Analysis: Fixed Point Theory and Its Applications, Yokohama Publishers, Yokohama, Japan, 2000.
• M. Kikkawa and T. Suzuki, “Three fixed point theorems for generalized contractions with constants in complete metric spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 69, no. 9, pp. 2942–2949, 2008.
• S. B. Nadler Jr., “Multi-valued contraction mappings,” Pacific Journal of Mathematics, vol. 30, pp. 475–488, 1969.
• W.-S. Du, “Some new results and generalizations in metric fixedpoint 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, 9 pages, 2010.
• W.-S. Du, “Nonlinear contractive conditions for coupled cone fixed point theorems,” Fixed Point Theory and Applications, vol. 2010, Article ID 190606, 16 pages, 2010.
• W.-S. Du, “New cone fixed point theorems for nonlinear multi-valued maps with their applications,” Applied Mathematics Let-ters, 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 and S.-X. Zheng, “New nonlinear conditions and inequalities for the existence of coincidence points and fixed points,” Journal of Applied Mathematics, Article ID 196759, 12 pages, 2012.
• Z. He, W.-S. Du, and I.-J. Lin, “The existence of fixed points for new nonlinear multivalued maps and their applications,” Fixed Point Theory and Applications, vol. 2011, 84, 2011.
• 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.
• 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.
• 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.
• T. Suzuki, “Mizoguchi-Takahashi's fixed point theorem is a real generalization of Nadler's,” Journal of Mathematical Analysis and Applications, vol. 340, no. 1, pp. 752–755, 2008.
• 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.
• 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.
• O. Kada, T. Suzuki, and W. Takahashi, “Nonconvex minimization theorems and fixed point theorems in complete metric spaces,” Mathematica Japonica, vol. 44, no. 2, pp. 381–391, 1996.
• L.-J. Lin and W.-S. Du, “Some equivalent formulations of the generalized Ekeland's variational principle and their applications,” Nonlinear Analysis: Theory, Methods & Applications, vol. 67, no. 1, pp. 187–199, 2007.
• W.-S. Du, “On Latif's fixed point theorems,” Taiwanese Journal of Mathematics, vol. 15, no. 4, pp. 1477–1485, 2011.
• L.-J. Lin and W.-S. Du, “Ekeland's variational principle, minimax theorems and existence of nonconvex equilibria in complete metric spaces,” Journal of Mathematical Analysis and Applications, vol. 323, no. 1, pp. 360–370, 2006.
• L.-J. Lin and W.-S. Du, “On maximal element theorems, variants of Ekeland's variational principle and their applications,” Nonlinear Analysis: Theory, Methods & Applications, vol. 68, no. 5, pp. 1246–1262, 2008.
• W.-S. Du, “Critical point theorems for nonlinear dynamical systems and their applications,” Fixed Point Theory and Applications, vol. 2010, Article ID 246382, 16 pages, 2010.
• V. Lakshmikantham and L. Ćirić, “Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 70, no. 12, pp. 4341–4349, 2009.
• F. Sabetghadam, H. P. Masiha, and A. H. Sanatpour, “Some coupled fixed point theorems in cone metric spaces,” Fixed Point Theory and Applications, vol. 2009, Article ID 125426, 8 pages, 2009.
• E. Karap\inar, “Couple fixed point theorems for nonlinear con-tractions in cone metric spaces,” Computers & Mathematics with Applications, vol. 59, no. 12, pp. 3656–3668, 2010.
• W. Shatanawi, “Partially ordered cone metric spaces and coupled fixed point results,” Computers & Mathematics with Appli-cations, vol. 60, no. 8, pp. 2508–2515, 2010.
• B. Samet, “Coupled fixed point theorems for a generalized Meir-Keeler contraction in partially ordered metric spaces,” Non-linear Analysis: Theory, Methods & Applications, vol. 72, no. 12, pp. 4508–4517, 2010.
• N. V. Luong and N. X. Thuan, “Coupled fixed points in partially ordered metric spaces and application,” Nonlinear Analysis: Theory, Methods & Applications, vol. 74, no. 3, pp. 983–992, 2011.
• B. S. Choudhury and A. Kundu, “A coupled coincidence pointresult in partially ordered metric spaces for compatible mappings,” Nonlinear Analysis: Theory, Methods & Applications, vol. 73, no. 8, pp. 2524–2531, 2010.
• V. Berinde, “Generalized coupled fixed point theorems for mixed monotone mappings in partially ordered metric spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 74, no. 18, pp. 7347–7355, 2011.
• Y. J. Cho, R. Saadati, and S. Wang, “Common fixed point theorems on generalized distance in ordered cone metric spaces,” Computers & Mathematics with Applications, vol. 61, no. 4, pp. 1254–1260, 2011.
• E. Graily, S. M. Vaezpour, R. Saadati, and Y. J. Cho, “Generalization of fixed point theorems in ordered metric spaces concerning generalized distance,” Fixed Point Theory and Applications, vol. 2011, article 30, 2011.
• W. Sintunavarat, Y. J. Cho, and P. Kumam, “Common fixed point theorems for c-distance in ordered cone metric spaces,” Computers & Mathematics with Applications, vol. 62, no. 4, pp. 1969–1978, 2011.
• 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.
• V. Berinde and F. Vetro, “Common fixed points of mappings satisfying implicit contractive conditions,” Fixed Point Theory and Applications, vol. 2012, article 105, 2012.
• B. Damjanović, B. Samet, and C. Vetro, “Common fixed point theorems for multi-valued maps,” Acta Mathematica Scientia B, vol. 32, no. 2, pp. 818–824, 2012.
• D. Paesano and P. Vetro, “Suzuki's type characterizations of completeness for partial metric spaces and fixed points for par-tially ordered metric spaces,” Topology and Its Applications, vol. 159, no. 3, pp. 911–920, 2012.
• B. Samet and C. Vetro, “Coupled fixed point theorems for multi-valued nonlinear contraction mappings in partially ordered metric spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 74, no. 12, pp. 4260–4268, 2011.
• F. Vetro and S. Radenović, “Nonlinear $\psi$-quasi-contractions of Ćirić-type in partial metric spaces,” Applied Mathematics and Computation, vol. 219, no. 4, pp. 1594–1600, 2012.
• F. Vetro, “On approximating curves associated with nonexpansive mappings,” Carpathian Journal of Mathematics, vol. 27, no. 1, pp. 142–147, 2011.