# Some fixed point theorems for G-rational Geraghty contractive mappings in ordered generalized b-metric spaces

Volume 8, Issue 6, pp 1212--1227
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### Authors

Abdul Latif - Department of Mathematics, King Abdulaziz University, P. O. Box 80203, Jeddah 21589, Saudi Arabia. Zoran Kadelburg - Faculty of Mathematics, University of Belgrade, Studentski trg 16, 11000 Beograd, Serbia. Vahid Parvaneh - Department of Mathematics, Gilan-E-Gharb Branch, Islamic Azad University, Gilan-E-Gharb, Iran. Jamal Rezaei Roshan - Department of Mathematics, Qaemshahr Branch, Islamic Azad University, Qaemshahr, Iran.

### Abstract

In this paper, we introduce the notion of G-rational Geraghty contractive mappings in the setup of ordered generalized b-metric spaces and investigate the existence of fixed points for such mappings. We also provide an example to illustrate the presented results and show that they are more general then some existing ones.

### Keywords

• Geraghty-type condition
• rational contractive condition
• ordered generalized b-metric space
• fixed point.

•  47H10
•  54H25

### References

• [1] A. Aghajani, M. Abbas, J. R. Roshan, Common fixed point of generalized weak contractive mappings in partially ordered $G_b$-metric spaces, Filomat, 28 (2014), 1087-1101.

• [2] M. A. Alghamdi, N. Hussain, P. Salimi , Fixed point and coupled fixed point theorems on b-metric like spaces, J. Inequal. Appl., 2013 (2013), 25 pages.

• [3] A. Amini-Harandi, H. Emami , A fixed point theorem for contraction type maps in partially ordered metric spaces and application to ordinary differential equations, Nonlinear Anal., 72 (2010), 2238-2242.

• [4] T. Van An, N. Van Dung, Z. Kadelburg, S. Radenović, Various generalizations of metric spaces and fixed point theorems, Rev. Real Acad. Cienc. Exac. Fis. Nat. Ser. A, Mat., 109 (2015), 175-198.

• [5] M. Arshad, E. Karapínar, J. Ahmad, Some unique fixed point theorems for rational contractions in partially ordered metric spaces, J. Inequal. Appl., 2013 (2013), 16 pages.

• [6] M. Asadi, E. Karapinar, P. Salimi, A new approach to G-metric and related fixed point theorems, J. Inequal. Appl., 2013 (2013), 14 pages.

• [7] S. Czerwik, Contraction mappings in b-metric spaces, Acta Math. Inf. Univ. Ostrav., 1 (1993), 5-11.

• [8] D. Dukić, Z. Kadelburg, S. Radenović, Fixed points of Geraghty-type mappings in various generalized metric spaces, Abstract Appl. Anal., 2011 (2011), 13 pages.

• [9] M. Geraghty , On contractive mappings, Proc. Amer. Math. Soc., 40 (1973), 604-608.

• [10] N. Hussain, V. Parvaneh, J. R. Roshan, Z. Kadelburg , Fixed points of cyclic weakly ($\psi,\varphi,L;,A,B$)-contractive mappings in ordered b-metric spaces with applications, Fixed Point Theory Appl., 2013 (2013), 18 pages.

• [11] N. Hussain, J. R. Roshan, V. Parvaneh, A. Latif, A unification of G-metric, partial metric, and b-metric spaces, Abstract Appl. Anal., 2014 (2014), 14 pages.

• [12] D. S. Jaggi , Some unique fixed point theorems, Indian J. Pure Appl. Math., 8 (1977), 223-230.

• [13] M. A. Kutbi, N. Hussain, J. R. Roshan, V. Parvaneh , Coupled and tripled coincidence point results with application to Fredholm integral equations, Abstract Appl. Anal., 2014 (2014), 18 pages.

• [14] Z. Mustafa, B. Sims, A new approach to generalized metric spaces, J. Nonlinear Convex Anal., 7 (2006), 289-297.

• [15] Z. Mustafa, J. R. Roshan, V. Parvaneh, Coupled coincidence point results for ($\psi,\varphi$)-weakly contractive mappings in partially ordered $G_b$-metric spaces, Fixed Point Theory Appl., 2013 (2013), 21 pages.

• [16] J. J. Nieto, R. Rodríguez-López , Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations, Order, 22 (2005), 223-239.

• [17] J. J. Nieto, R. Rodríguez-López, Existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equations, Acta Math. Sin., 23 (2007), 2205-2212.

• [18] A. C. M. Ran, M. C. B. Reurings, A fixed point theorem in partially ordered sets and some applications to matrix equations, Proc. Amer. Math. Soc., 132 (2004), 1435-1443.

• [19] J. R. Roshan, N. Shobkolaei, Sh. Sedghi, V. Parvaneh, S. Radenović, Common fixed point theorems for three maps in discontinuous $G_b$-metric spaces, Acta Math. Sci., 34 (2014), 1643-1654.

• [20] P. Salimi, P. Vetro, A result of Suzuki type in partial G-metric spaces, Acta Math. Sci., 34 (2014), 274-284.

• [21] F. Zabihi, A. Razani, Fixed point theorems for hybrid rational Geraghty contractive mappings in orderd b-metric spaces, J. Appl. Math., 2014 (2014), 9 pages.