# On some common coupled fixed point results in rectangular $b$-metric spaces

Volume 10, Issue 8, pp 4085--4098 Publication Date: August 07, 2017       Article History
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### Authors

Feng Gu - Institute of Applied Mathematics and Department of Mathematics, Hangzhou Normal University, Hangzhou, Zhejiang 310036, China.

### Abstract

In this paper, by using the $w$-compatible conditions of mapping pair, we discuss the existence and uniqueness problem of the common coupled fixed point for mappings defined on a set equipped with two rectangular $b$-metrics. Some new common coupled fixed point theorems are obtained. We also provide illustrative examples in support of our new results. As application, we provide an existence and uniqueness theorem of common solution for a class of nonlinear integral equations by using the obtained new result. The results presented in this paper generalize the well-known comparable results in the literature.

### Keywords

• Rectangular b-metric space
• coupled coincidence point
• common coupled fixed point
• w-compatible mapping pairs.

•  47H10
•  54H25

### References

• [1] M. Abbas, M. A. Khan, S. Radenović, Common coupled fixed point theorems in cone metric spaces for w-compatible mappings, Appl. Math. Comput., 217 (2010), 195–202.

• [2] T. Abdeljawad, D. Türkoğlu , Locally convex valued rectangular metric spaces and the Kannan’s fixed point theorem, J. Comput. Anal. Appl., 14 (2012), 484–494.

• [3] M. Arshad, J. Ahmad, E. Karapınar , Some common fixed point results in rectangular metric spaces, Int. J. Anal., 2013 (2013), 7 pages.

• [4] H. Aydi, A. Felhi, S. Sahmim, Common fixed points in rectangular b-metric spaces using (E.A) property, J. Adv. Math. Stud., 8 (2015), 159–169.

• [5] H. Aydi, E. Karapınar, H. Lakzian, Fixed point results on a class of generalized metric spaces, Math. Sci. (Springer), 2012 (2012), 6 pages.

• [6] A. Branciari, A fixed point theorem of Banach-Caccioppoli type on a class of generalized metric spaces, Publ. Math. Debrecen, 57 (2000), 31–37.

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

• [8] C. Di Bari, P. Vetro , Common fixed points in generalized metric spaces , Appl. Math. Comput., 218 (2012), 7322–7325.

• [9] H.-S. Ding, M. Imdad, S. Radenović, J. Vujaković , On some fixed point results in b-metric, rectangular and b-rectangular metric spaces, Arab J. Math. Sci., 22 (2016), 151–164.

• [10] ˙I. M. Erhan, E. Karapınar, T. Sekulić, Fixed points of ($\psi,\phi$) contractions on rectangular metric spaces , Fixed Point Theory Appl., 2012 (2012), 12 pages.

• [11] R. George, S. Radenović, K. P. Reshma, S. Shukla, Rectangular b-metric space and contraction principles, J. Nonlinear Sci. Appl., 8 (2015), 1005–1013.

• [12] R. George, R. Rajagopalan, Common fixed point results for $\psi-\phi$ contractions in rectangular metric spaces, Bull. Math. Anal. Appl., 5 (2013), 44–52.

• [13] T. Gnana Bhaskar, V. Lakshmikantham, Fixed point theorems in partially ordered metric spaces and applications, Nonlinear Anal., 65 (2006), 1379–1393.

• [14] H. Isık, D. Türkoğlu, Common fixed points for ( $\psi,\alpha,\beta$)-weakly contractive mappings in generalized metric spaces, Fixed Point Theory Appl., 2013 (2013), 6 pages.

• [15] W. A. Kirk, N. Shahzad, Generalized metrics and Caristi’s theorem, Fixed Point Theory Appl., 2013 (2013), 9 pages.

• [16] B. K. Lahiri, P. Das , Fixed point of a Ljubomir Ćirić’s quasi-contraction mapping in a generalized metric space, Publ. Math. Debrecen, 61 (2002), 589–594.

• [17] V. Lakshmikantham, L. Ćirić , Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces, Nonlinear Anal., 70 (2009), 4341–4349.

• [18] H. Lakzian, B. Samet, Fixed points for ($\psi,\phi$)-weakly contractive mappings in generalized metric spaces, Appl. Math. Lett., 25 (2012), 902–906.

• [19] V. La Rosa, P. Vetro, Common fixed points for $\alpha-\psi-\phi$-contractions in generalized metric spaces, Nonlinear Anal. Model. Control, 19 (2014), 43–54.

• [20] J. R. Roshan, V. Parvaneh, Z. Kadelburg, H. Zoran , New fixed point results in b-rectangular metric spaces, Nonlinear Anal. Model. Control, 21 (2016), 614–634.

• [21] B. Samet, A fixed point theorem in a generalized metric space for mappings satisfying a contractive condition of integral type, Int. J. Math. Anal. (Ruse), 3 (2009), 1265–1271.