## Abstract and Applied Analysis

### Coupled and Tripled Coincidence Point Results with Application to Fredholm Integral Equations

#### Abstract

The aim of this paper is to define weak $\alpha$-$\psi$-$\phi$-contractive mappings and to establish coupled and tripled coincidence point theorems for such mappings defined on ${G}_{b}$-metric spaces using the concept of rectangular $G$-$\alpha$-admissibility. As an application, we derive new coupled and tripled coincidence point results for weak $\psi$-$\phi$-contractive mappings in partially ordered ${G}_{b}$-metric spaces. Our results are generalizations and extensions of some recent results in the literature. We also present an example as well as an application to nonlinear Fredholm integral equations in order to illustrate the effectiveness of our results.

#### Article information

Source
Abstr. Appl. Anal., Volume 2014, Special Issue (2013), Article ID 568718, 18 pages.

Dates
First available in Project Euclid: 6 October 2014

https://projecteuclid.org/euclid.aaa/1412607380

Digital Object Identifier
doi:10.1155/2014/568718

Mathematical Reviews number (MathSciNet)
MR3214438

Zentralblatt MATH identifier
07022625

#### Citation

Kutbi, Marwan Amin; Hussain, Nawab; Rezaei Roshan, Jamal; Parvaneh, Vahid. Coupled and Tripled Coincidence Point Results with Application to Fredholm Integral Equations. Abstr. Appl. Anal. 2014, Special Issue (2013), Article ID 568718, 18 pages. doi:10.1155/2014/568718. https://projecteuclid.org/euclid.aaa/1412607380

#### References

• Z. Mustafa and B. Sims, “A new approach to generalized metric spaces,” Journal of Nonlinear and Convex Analysis, vol. 7, no. 2, pp. 289–297, 2006.
• A. Aghajani, M. Abbas, and J. R. Roshan, “Common fixed pointčommentComment on ref. [1?]: Please update the information of this reference, if possible. of generalized weak contractive mappings in partially ordered ${G}_{b}$-metric spaces,” Filomat. In press.
• S. Czerwik, “Contraction mappings in $b$-metric spaces,” Acta Mathematica et Informatica Universitatis Ostraviensis, vol. 1, pp. 5–11, 1993.
• Z. Mustafa, J. R. Roshan, and V. Parvaneh, “Coupled coincidence point results for $(\psi ,\phi )$-weakly contractive mappings in partially ordered ${G}_{b}$-metric spaces,” Fixed Point Theory and Applications, vol. 2013, article 206, 2013.
• N. Hussain, D. Dorić, Z. Kadelburg, and S. Radenović, “Suzuki-type fixed point results in metric type spaces,” Fixed Point Theory and Applications, vol. 2012, article 126, 2012.
• H. Aydi, M. Postolache, and W. Shatanawi, “Coupled fixed point results for $(\psi ,\varphi )$-weakly contractive mappings in ordered $G$-metric spaces,” Computers & Mathematics with Applications, vol. 63, no. 1, pp. 298–309, 2012.
• T. G. 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.
• 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.
• B. S. Choudhury and A. Kundu, “A coupled coincidence point result in partially ordered metric spaces for compatible mappings,” Nonlinear Analysis: Theory, Methods & Applications, vol. 73, no. 8, pp. 2524–2531, 2010.
• B. S. Choudhury and P. Maity, “Coupled fixed point results in generalized metric spaces,” Mathematical and Computer Modelling, vol. 54, no. 1-2, pp. 73–79, 2011.
• L. Ćirić, B. Damjanović, M. Jleli, and B. Samet, “Coupled fixed point theorems for generalized Mizoguchi-Takahashi contractions with applications,” Fixed Point Theory and Applications, vol. 2012, article 51, 2012.
• H.-S. Ding, L. Li, and S. Radenović, “Coupled coincidence point theorems for generalized nonlinear contraction in partially ordered metric spaces,” Fixed Point Theory and Applications, vol. 2012, article 96, 2012.
• D. J. Guo and V. Lakshmikantham, “Coupled fixed points of nonlinear operators with applications,” Nonlinear Analysis: Theory, Methods & Applications, vol. 11, no. 5, pp. 623–632, 1987.
• N. Hussain, M. Abbas, A. Azam, and J. Ahmad, “Coupled coincidence point results for a generalized compatible pair with applications,” Fixed Point Theory and Applications, vol. 2014, article 62, 2014.
• N. Hussain, P. Salimi, and S. al-Mezel, “Coupled fixed point results on quasi-Banach spaces with application to a system of integral equations,” Fixed Point Theory and Applications, vol. 2013, article 261, 2013.
• N. Hussain, A. Latif, and M. H. Shah, “Coupled and tripled coincidence point results without compatibility,” Fixed Point Theory and Applications, vol. 2012, article 77, 2012.
• 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.
• 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.
• J. J. Nieto and R. Rodríguez-López, “Existence andčommentComment on ref. [29?]: This reference is a repetition of [27]. Please check. 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.
• 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. R. Roshan, V. Parvaneh, and I. Altun, “Some coincidence point results in ordered $b$-metric spaces and applications in a system of integral equations,” Applied Mathematics and Computation, vol. 226, pp. 725–737, 2014.
• J. R. Roshan, V. Parvaneh, S. Sedghi, N. Shobkolaei, and W. Shatanawi, “Common fixed points of almost generalized ${(\psi ,\varphi )}_{s}$-contractive mappings in ordered $b$-metric spaces,” Fixed Point Theory and Applications, vol. 2013, article 159, 2013.
• M. Mursaleen, S. A. 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, article 228, 2012.
• V. Parvaneh, J. R. Roshan, and S. Radenović, “Existence of tripled coincidence points in ordered $b$-metric spaces and an application to a system of integral equations,” Fixed Point Theory and Applications, vol. 2013, article 130, 2013.
• V. Berinde and M. Borcut, “Tripled fixed point theorems for contractive type mappings in partially ordered metric spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 74, no. 15, pp. 4889–4897, 2011.
• M. Borcut, “Tripled coincidence theorems for contractive type mappings in partially ordered metric spaces,” Applied Mathematics and Computation, vol. 218, no. 14, pp. 7339–7346, 2012.
• M. Borcut and V. Berinde, “Tripled coincidence theorems for contractive type mappings in partially ordered metric spaces,” Applied Mathematics and Computation, vol. 218, no. 10, pp. 5929–5936, 2012.
• B. S. Choudhury, E. Karap\inar, and A. Kundu, “Tripled coincidence point theorems for nonlinear contractions in partially ordered metric spaces,” International Journal of Mathematics and Mathematical Sciences, vol. 2012, Article ID 329298, 14 pages, 2012.
• Y. J. Cho, B. E. Rhoades, R. Saadati, B. Samet, and W. Shatanawi, “Nonlinear coupled fixed point theorems in ordered generalized metric spaces with integral type,” Fixed Point Theory and Applications, vol. 2012, article 8, 2012.
• H. Aydi, E. Karap\inar, and W. Shatanawi, “Tripled coincidence point results for generalized contractions in ordered generalized metric spaces,” Fixed Point Theory and Applications, vol. 2012, article 101, 2012.
• 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.
• M. A. Alghamdi and E. Karap\inar, “$G$-$\beta$-$\psi$ contractive-type mappings and related fixed point theorems,” Journal of Inequalities and Applications, vol. 2013, article 70, 2013.
• 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.
• E. Karap\inar, P. Kumam, and P. Salimi, “On $\alpha$-$\psi$-Meir-Keeler contractive mappings,” Fixed Point Theory and Applications, vol. 2013, article 94, 2013.
• N. Hussain, E. Karap\inar, P. Salimi, and P. Vetro, “Fixed point results for ${G}^{m}$-Meir-Keeler contractive and $G$-$(\alpha ,\psi )$-Meir-Keeler contractive mappings,” Fixed Point Theory and Applications, vol. 2013, article 34, 2013.
• Z. Mustafa, J. R. Roshan, and V. Parvaneh, “Existence of tripled coincidence point in ordered ${G}_{b}$-metric spaces and applications to a system of integral equations,” Journal of Inequalities and Applications, vol. 2013, article 453, 2013.
• V. Berinde, “Coupled fixed point theorems for $\phi$-contractive mixed monotone mappings in partially ordered metric spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 75, no. 6, pp. 3218–3228, 2012.
• N. V. Luong and N. X. Thuan, “Coupled fixed point theorems in partially ordered $G$-metric spaces,” Mathematical and Computer Modelling, vol. 55, no. 3-4, pp. 1601–1609, 2012. \endinput