## Abstract and Applied Analysis

### New Fixed Point Results with PPF Dependence in Banach Spaces Endowed with a Graph

#### Abstract

We introduce the concept of an ${\alpha }_{c}$-admissible non-self-mappings with respect to ${\eta }_{c}$ and establish the existence of PPF dependent fixed and coincidence point theorems for ${\alpha }_{c}{\eta }_{c}$-$\psi$-contractive non-self-mappings in the Razumikhin class. As applications of our PPF dependent fixed point and coincidence point theorems, we derive some new fixed and coincidence point results for $\psi$-contractions whenever the range space is endowed with a graph or with a partial order. The obtained results generalize, extend, and modify some PPF dependent fixed point results in the literature. Several interesting consequences of our theorems are also provided.

#### Article information

Source
Abstr. Appl. Anal., Volume 2013, Special Issue (2013), Article ID 827205, 9 pages.

Dates
First available in Project Euclid: 26 February 2014

https://projecteuclid.org/euclid.aaa/1393443689

Digital Object Identifier
doi:10.1155/2013/827205

Mathematical Reviews number (MathSciNet)
MR3143545

Zentralblatt MATH identifier
07095403

#### Citation

Hussain, N.; Khaleghizadeh, S.; Salimi, P.; Akbar, F. New Fixed Point Results with PPF Dependence in Banach Spaces Endowed with a Graph. Abstr. Appl. Anal. 2013, Special Issue (2013), Article ID 827205, 9 pages. doi:10.1155/2013/827205. https://projecteuclid.org/euclid.aaa/1393443689

#### References

• 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. 1–8, 2008.
• R. P. Agarwal, N. Hussain, and M. A. Taoudi, “Fixed point theorems in ordered Banach spaces and applications to nonlinear integral equations,” Abstract and Applied Analysis, vol. 2012, Article ID 245872, 15 pages, 2012.
• M. A. Abbas and T. Nazir, “Common fixed point of a power graphic contraction pair in partial metric spaces endowed with graph,” Fixed Point Theory and Applications, vol. 2013, p. 20, 2013.
• A. Gh. B. Ahmad, Z. Fadail, H. K. Nashine, Z. Kadelburg, and S. Radenovic, “Some new common fixed point results through generalized altering distances on partial metric spaces,” Fixed Point Theory and Applications, vol. 2012, p. 120, 2012.
• S. R. Bernfeld, V. Lakshmikatham, and Y. M. Reddy, “Fixed point theorems of operators with PPF dependence in Banach spaces,” Applicable Analysis, vol. 6, no. 4, pp. 271–280, 1977.
• F. Bojor, “Fixed point theorems for Reich type contractions on metric spaces with a graph,” Nonlinear Analysis, Theory, Methods and Applications, vol. 75, no. 9, pp. 3895–3901, 2012.
• L. Ćirić, M. Abbas, R. Saadati, and N. Hussain, “Common fixed points of almost generalized contractive mappings in ordered metric spaces,” Applied Mathematics and Computation, vol. 217, no. 12, pp. 5784–5789, 2011.
• L. B. Ćirić, S. M. A. Alsulami, P. Salimi, and P. Vetro, “PPF dependent fixed point results for triangular ${\propto }_{c}$-admissible mapping,” The Scientific World Journal. In press.
• B. C. Dhage, “Some basic random fixed point theorems with PPF dependence and functional random differential equations,” Differential Equations & Applications, vol. 4, pp. 181–195, 2012.
• M. Geraghty, “On contractive mappings,” Proceedings of the American Mathematical Society, vol. 40, pp. 604–608, 1973.
• N. Hussain, S. Al-Mezel, and P. Salimi, “Fixed points for $\psi$-graphic contractions with application to integral equations,” Abstract and Applied Analysis, vol. 2013, Article ID 575869, 11 pages, 2013.
• N. Hussain, A. R. Khan, and R. P. Agarwal, “Krasnosel'skii and Ky Fan type fixed point theorems in ordered Banach spaces,” Journal of Nonlinear and Convex Analysis, vol. 11, no. 3, pp. 475–489, 2010.
• M. A. Kutbi, N. Hussain, and P. Salimi, “Best proximity point results for modified $\alpha$-$\psi$-proximal rational contractions,” Abstract and Applied Analysis, vol. 2013, Article ID 927457, 14 pages, 2013.
• N. Hussain, P. Salimi, and A. Latif, “Fixed point results for single and set-valued $\alpha$-$\eta$-$\psi$-contractive mappings,” Fixed Point Theory and Applications, vol. 2013, p. 212, 2013.
• J. Jachymski, “The contraction principle for mappings on a metric space with a graph,” Proceedings of the American Mathematical Society, vol. 136, no. 4, pp. 1359–1373, 2008.
• R. Johnsonbaugh, Discrete Mathematics, Prentice-Hall, New Jersey, NJ, USA, 1997.
• A. Kaewcharoen, “PPF dependent common fixed point theorems for mappings in Bnach spaces,” Journal of Inequalities and Applications, vol. 2013, p. 287, 2013.
• E. Karapinar, P. Kumam, and P. Salimi, “On $\alpha$-$\psi$-Meir-Keeler contractive mappings,” Fixed Point Theory and Applications, vol. 2013, p. 94, 2013.
• E. Karapinar and B. Samet, “Generalized ($\alpha$-$\psi$) contractive type mappings and related fixed point theorems with applications,” Abstract and Applied Analysis, vol. 2012, Article ID 793486, 17 pages, 2012.
• F. Khojasteh, E. Karapinar, and S. Radenovic, “$\theta$-metric space: a generalization,” Mathematical Problems in Engineering, vol. 2013, Article ID 504609, 7 pages, 2013.
• M. Mursaleen, S. Mohiuddine, and R. P. Agarwal, “Coupled fixed point theorems for $\alpha$-$\psi$-contractive type mappings in partially ordered metric spaces,” Fixed Point Theory and Applications, vol. 2012, p. 228, 2012.
• B. Samet, C. Vetro, and P. Vetro, “Fixed point theorems for $\alpha$-$\psi$-contractive type mappings,” Nonlinear Analysis, Theory, Methods and Applications, vol. 75, no. 4, pp. 2154–2165, 2012.
• M. Samreen and T. Kamran, “Fixed point theorems for integral \emphG-contraction,” Fixed Point Theory and Applications, vol. 2013, p. 149, 2013.
• P. Salimi, A. Latif, and N. Hussain, “Modified $\alpha$-$\psi$-contractive mappings with applications,” Fixed Point Theory and Applications, vol. 2013, p. 151, 2013.
• P. Salimi, C. Vetro, and P. Vetro, “Some new fixed point results in non-Archimedean fuzzy metric spaces,” Nonlinear Analysis: Modelling and Control, vol. 18, no. 3, pp. 344–358, 2013.
• P. Salimi, C. Vetro, and P. Vetro, “Fixed point theorems for twisted ($\alpha$,$\beta$)-$\psi$-contractive type mappings and applications,” Filomat, vol. 27, no. 4, pp. 605–615, 2013.
• 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.
• N. Hussain, M. H. Shah, and M. A. Kutbi, “Coupled coincidence point theorems for nonlinear contractions in partially ordered quasi-metric spaces with a Q-function,” Fixed Point Theory and Applications, vol. 2011, Article ID 703938, 21 pages, 2011.
• Y. J. Cho, M. H. Shah, and N. Hussain, “Coupled fixed points of weakly F-contractive mappings in topological spaces,” Applied Mathematics Letters, vol. 24, no. 7, pp. 1185–1190, 2011.
• M. A. Kutbi, J. Ahmad, and A. Azam, “On fixed points of $\alpha \text{-}\psi$-contractive multi-valued mappings in cone metric spaces,” Abstract and Applied Analysis, vol. 2013, Article ID 313782, 6 pages, 2013.
• R. P. Agarwal, P. Kumam, and W. Sintunavarat, “PPF dependent fixed point theorems for an ${\alpha }_{c}$-admissible non-self mapping in the Razumikhin class,” Fixed Point Theory and Applications, vol. 2013, article 280, 2013. \endinput