## Abstract and Applied Analysis

### Discussions on Recent Results for $\alpha$-$\psi$-Contractive Mappings

#### Abstract

We establish certain fixed point results for $\alpha \text{-}\eta$-generalized convex contractions, $\alpha \text{-}\eta$-weakly Zamfirescu mappings, and $\alpha \text{-}\eta$-Ćirić strong almost contractions. As an application, we derive some Suzuki type fixed point theorems and certain new fixed point theorems in metric spaces endowed with a graph and a partial order. Moreover, we discuss some illustrative examples to highlight the realized improvements.

#### Article information

Source
Abstr. Appl. Anal., Volume 2014, Special Issue (2014), Article ID 456482, 13 pages.

Dates
First available in Project Euclid: 6 October 2014

https://projecteuclid.org/euclid.aaa/1412606045

Digital Object Identifier
doi:10.1155/2014/456482

Mathematical Reviews number (MathSciNet)
MR3178870

Zentralblatt MATH identifier
07022409

#### Citation

Hussain, N.; Kutbi, M. A.; Khaleghizadeh, S.; Salimi, P. Discussions on Recent Results for $\alpha$ - $\psi$ -Contractive Mappings. Abstr. Appl. Anal. 2014, Special Issue (2014), Article ID 456482, 13 pages. doi:10.1155/2014/456482. https://projecteuclid.org/euclid.aaa/1412606045

#### References

• R. Espínola and W. A. Kirk, “Fixed point theorems in $R$-trees with applications to graph theory,” Topology and its Applications, vol. 153, no. 7, pp. 1046–1055, 2006.
• 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. Abbas, I. Altun, and S. Romaguera, “Common fixed points of Ćirić-type contractions on partial metric spaces,” Publicationes Mathematicae Debrecen, vol. 82, no. 2, pp. 425–438, 2013.
• L. B. Ćirić, “A generalization of Banach's contraction principle,” Proceedings of the American Mathematical Society, vol. 45, pp. 267–273, 1974.
• L. Ćirić, N. Hussain, and N. Cakić, “Common fixed points for Ćirić type $f$-weak contraction with applications,” Publicationes Mathematicae Debrecen, vol. 76, no. 1-2, pp. 31–49, 2010.
• 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.
• M. A. Geraghty, “On contractive mappings,” Proceedings of the American Mathematical Society, vol. 40, pp. 604–608, 1973.
• M. Gopal, M. Imdad, C. Vetro, and M. Hasan, “Fixed point theory for cyclic weak$\phi$-contraction in fuzzy metric spaces,” Journal Nonlinear Analysis and Application, Article ID jnaa-00110, 11 pages, 2012.
• N. Hussain, E. Karap\inar, P. Salimi, and F. Akbar, “$\alpha$-admissible mappings and related fixed point theorems,” Journal of Inequalities and Applications, vol. 2013: article 114, 2013.
• N. Hussain and M. A. Khamsi, “On asymptotic pointwise contractions in metric spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 10, pp. 4423–4429, 2009.
• J.-C. Yao and L.-C. Zeng, “Fixed point theorem for asymptotically regular semigroups in metric spaces with uniform normal structure,” Journal of Nonlinear and Convex Analysis, vol. 8, no. 1, pp. 153–163, 2007.
• V. I. Istrăţescu, “Some fixed point theorems for convex contraction mappings and mappings with convex diminishing diameters. I,” Annali di Matematica Pura ed Applicata, vol. 130, no. 4, pp. 89–104, 1982.
• M. A. Miandaragh, M. Postolache, and Sh. Rezapour, “Approximate fixed points of generalized convex contractions,” Fixed Point Theory and Applications, vol. 2013, article 255, 2013.
• P. Salimi, A. Latif, and N. Hussain, “Modified alpha-psi-con-tractive mappings with applications,” Fixed Point Theory and Applications, vol. 2013151, 2013.
• B. Samet, C. Vetro, and P. Vetro, “Fixed point theorems for $\alpha$-$\psi$-contractive type mappings,” Nonlinear Analysis: Theory, Methods & Applications, vol. 75, no. 4, pp. 2154–2165, 2012.
• E. Karap\inar 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.
• P. Salimi and E. Karap\inar, “Suzuki-Edelstein type contractions via auxiliary functions,” Mathematical Problems in Engineering, vol. 2013, Article ID 648528, 8 pages, 2013.
• M. Berinde, “Approximate fixed point theorems,” Studia Universitatis Babeş-Bolyai, vol. 51, no. 1, pp. 11–25, 2006.
• 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.
• D. Ariza-Ruiz, A. Jiménez-Melado, and G. López-Acedo, “A fixed point theorem for weakly Zamfirescu mappings,” Nonlinear Analysis: Theory, Methods & Applications, vol. 74, no. 5, pp. 1628–1640, 2011.
• V. Berinde, “Some remarks on a fixed point theorem for Ćirić-type almost contractions,” Carpathian Journal of Mathematics, vol. 25, no. 2, pp. 157–162, 2009.
• 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.
• N. Hussain, M. A. Kutbi, and P. Salimi, “Fixed point theory in alpha-psi-complete metric spaces with applications,” Abstract and Applied Analysis. In press.
• 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.
• 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, M. A. Kutbi, and P. Salimi, “Best proximity pointresults for modified $\alpha$-$\psi$-proximal rational contractions,” Abstract and Applied Analysis, Article ID 927457, 14 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.
• N. Hussain and M. A. Taoudi, “Krasnosel'skii-type fixed point theorems with applications to Volterra integralequations,” Fixed Point Theory and Applications, vol. 2013, article 196, 2013.
• Q. H. Ansari, A. Idzik, and J.-C. Yao, “Coincidence and fixedpoint theorems with applications,” Topological Methods in Nonlinear Analysis, vol. 15, no. 1, pp. 191–202, 2000.
• J. J. Nieto and R. Rodríguez-López, “Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations,” Order, vol. 22, no. 3, pp. 223–239, 2005. \endinput